initial commit of ebook 36525HEADmain

151 files changed, 31935 insertions, 0 deletions

diff --git a/.gitattributes b/.gitattributes
new file mode 100644
index 0000000..6833f05
--- /dev/null
+++ b/.gitattributes
@@ -0,0 +1,3 @@
+* text=auto
+*.txt text
+*.md text
diff --git a/36525-pdf.pdf b/36525-pdf.pdf
new file mode 100644
index 0000000..1410a9c
--- /dev/null
+++ b/36525-pdf.pdf
diff --git a/36525-pdf.zip b/36525-pdf.zip
new file mode 100644
index 0000000..f6ef694
--- /dev/null
+++ b/36525-pdf.zip
diff --git a/36525-t.zip b/36525-t.zip
new file mode 100644
index 0000000..076a5b7
--- /dev/null
+++ b/36525-t.zip
diff --git a/36525-t/36525-t.tex b/36525-t/36525-t.tex
new file mode 100644
index 0000000..48d812c
--- /dev/null
+++ b/36525-t/36525-t.tex
@@ -0,0 +1,31919 @@
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% The Project Gutenberg EBook of Notes on Recent Researches in Electricity%
+% and Magnetism, by J. J. Thomson                                         %
+%                                                                         %
+% This eBook is for the use of anyone anywhere at no cost and with        %
+% almost no restrictions whatsoever.  You may copy it, give it away or    %
+% re-use it under the terms of the Project Gutenberg License included     %
+% with this eBook or online at www.gutenberg.net                          %
+%                                                                         %
+%                                                                         %
+% Title: Notes on Recent Researches in Electricity and Magnetism          %
+%        Intended as a Sequel to Professor Clerk-Maxwell\'s Treatise      %
+%        on Electricity and Magnetism                                     %
+%                                                                         %
+% Author: J. J. Thomson                                                   %
+%                                                                         %
+% Release Date: June 27, 2011 [EBook #36525]                              %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: ISO-8859-1                                      %
+%                                                                         %
+% *** START OF THIS PROJECT GUTENBERG EBOOK RECENT RESEARCHES--ELECTRICITY, MAGNETISM ***
+%                                                                         %
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+
+\def\ebook{36525}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%                                                                  %%
+%% Packages and substitutions:                                      %%
+%%                                                                  %%
+%% book:        Document class. Required.                           %%
+%% geometry:    Enhanced page layout package. Required.             %%
+%% inputenc:    Standard DP encoding. Required.                     %%
+%% amsmath:     AMS mathematics enhancements. Required.             %%
+%% amssymb:     AMS symbols e.g. \therefore. Required.              %%
+%% fancyhdr:    Enhanced running headers and footers. Required.     %%
+%% extramarks:  Display article number in header. Required.         %%
+%% verbatim:    For PG license text. Required.                      %%
+%% graphicx:    Allows inclusion of images. Required.               %%
+%% wrapfig:     Wrap text around images. Required.                  %%
+%% rotating:    For sideways images. Required.                      %%
+%% caption:     Provides customised caption format. Required.       %%
+%% multirow:    For table cells spanning several rows. Required.    %%
+%% makeidx:     Allows index creation.  Required.                   %%
+%% multicol:    Automatically balance index columns. Required.      %%
+%% calc:        Used for length calculations. Required.             %%
+%% array:       Enhancement to arrays.  Required.                   %%
+%% mhchem:      For chemical formulae. Required.                    %%
+%% indentfirst: Indent after headings. Required.                    %%
+%% ifthen:      Logical conditionals. Required.                     %%
+%% longtable    Allows multipage tables. Required.                  %%
+%% footmisc:    For better footnote handling. Required.             %%
+%% hyperref:    Hypertext embellishments for pdf output. Required.  %%
+%%                                                                  %%
+%% Things to Check:                                                 %%
+%%                                                                  %%
+%% Spellcheck: OK                                                   %%
+%% lacheck: OK                                                      %%
+%%   False positives:                                               %%
+%%     Complaints from preamble                   (before line 525) %%
+%%     Complaints from lprep config and log file (after line 28650) %%
+%%     Whitespace before punctation mark           (many instances) %%
+%%     possible unwanted space at "{"              (many instances) %%
+%%     punctuation mark should be placed after end of math mode(x2) %%
+%%     \ldots should be \cdots in "+ \ldots -"                 (x2) %%
+%%     Use ` to begin quotation, not ' " 'T"                        %%
+%%                                                                  %%
+%% Lprep: OK, 3 warnings about moving code to preamble              %%
+%% Gutcheck: OK                                                     %%
+%% PDF pages: 617                                                   %%
+%% PDF page size: 5.5" x 8.25"                                      %%
+%% PDF bookmarks: point to preface, contents, chapters, appendix,   %%
+%%                 index, PG licensing                              %%
+%% PDF document info: filled in                                     %%
+%% PDF Reader displays document title in window title bar           %%
+%% ToC page numbers: OK                                             %%
+%% Images: 144 PNGs (fig01-fig144) located in the images subfolder  %%
+%%                                                                  %%
+%% Summary of log file:                                             %%
+%%   No errors or warnings.                                         %%
+%%   No overfull hboxes.                                            %%
+%%   23 underfull hboxes (primarily wrapped text around images)     %%
+%%                                                                  %%
+%% Command block:                                                   %%
+%% pdflatex                                                         %%
+%% makeindex                                                        %%
+%% pdflatex x2                                                      %%
+%%                                                                  %%
+%% Compile History:                                                 %%
+%% June 2011: windymilla (Nigel Blower)                             %%
+%%           MiKTeX 2.9, TeXnicCenter, Windows XP Pro               %%
+%%           Compiled (includes makeindex) three times              %%
+%%                                                                  %%
+%%                                                                  %%
+%% June 2011: pglatex.                                              %%
+%%   Compile this project with:                                     %%
+%%   pdflatex 36525-t.tex                                           %%
+%%   makeindex 36525-t.idx                                          %%
+%%   pdflatex 36525-t.tex ..... TWO times                           %%
+%%                                                                  %%
+%%   pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6)                 %%
+%%                                                                  %%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\listfiles
+
+\makeindex
+
+\makeatletter
+
+\documentclass[12pt,oneside]{book}[2005/09/16]
+
+\usepackage[paperwidth=5.5in,paperheight=8.25in,
+            headsep=0.2in, headheight=0.2in,
+            top=0.525in, bottom=0.25in,
+            left=0.125in, right=0.125in]{geometry}[2008/11/13]
+\usepackage[latin1]{inputenc}[2006/05/05]
+
+\usepackage{amsmath}[2000/07/18]
+\usepackage{amssymb}[2002/01/22]
+\usepackage{fancyhdr}% no date stamp
+\usepackage{extramarks}% no date stamp
+\usepackage{verbatim}[2003/08/22]
+\usepackage{graphicx}[1999/02/16]
+\usepackage{wrapfig}[2003/01/31]
+\usepackage{rotating}[2009/3/28]
+\usepackage[justification=centering, font={footnotesize}]{caption}[2008/08/24]
+\usepackage{multirow}% no date stamp
+\usepackage{makeidx}[2000/03/29]
+\usepackage{multicol}[2006/05/18]
+\usepackage{calc}[2005/08/06]
+\usepackage{array}[2005/08/23]
+\usepackage[version=3]{mhchem}[2007/05/19]
+\usepackage{indentfirst}[1995/11/23]
+\usepackage{ifthen}[2001/05/26]
+\usepackage{longtable}[2004/02/01]
+\usepackage[perpage, symbol]{footmisc}[2009/09/15]
+
+% PDF attributes
+\providecommand{\ebook}{00000}
+\usepackage[pdftex,
+  hyperfootnotes=false,
+  pdftitle={The Project Gutenberg eBook \#\ebook: Notes on Recent Researches in Electricity and Magnetism},
+  pdfsubject={Electricity, Magnetism},
+  pdfauthor={J. J. Thomson},
+  pdfkeywords={Robert Cicconetti, Nigel Blower, Project Gutenberg Online Distributed Proofreading Team},
+  pdfpagelayout=SinglePage,
+  pdfdisplaydoctitle,
+  pdfpagelabels=true,
+  bookmarksopen=true,
+  bookmarksopenlevel=2,
+  colorlinks=false,
+  linkcolor=blue]{hyperref}[2008/11/18]
+
+% redefine hyperref's re-definition
+% so that chapter anchor is above chapter title
+\AtBeginDocument{% in case hyperref clobbers this
+\def\@schapter#1{%
+  \begingroup
+    \let\@mkboth\@gobbletwo
+    \Hy@GlobalStepCount\Hy@linkcounter
+    \xdef\@currentHref{\Hy@chapapp*.\the\Hy@linkcounter}%
+    \Hy@raisedlink{%
+      \hyper@anchorstart{\@currentHref}\hyper@anchorend
+    }%
+  \endgroup
+  \H@old@schapter{#1}%
+}}
+
+% Used to fix typos in the original text
+\newboolean{FixTypos}
+% Comment the following line to leave typos as in original. Uncomment it to fix them
+\setboolean{FixTypos}{true}
+\newcommand{\DPtypo}[2]{\ifthenelse{\boolean{FixTypos}}{#2}{#1}}
+
+\newcommand{\Ditto}{\text{,,}}
+
+% For sensible insertion of boilerplate/licence,
+% Overlong lines will wrap and be indented 0.25in
+\def\@xobeysp{~\hfil\discretionary{}{\kern\z@}{}\hfilneg}
+\renewcommand\verbatim@processline{\leavevmode
+  \null\kern-0.25in\the\verbatim@line\par}
+\addto@hook\every@verbatim{\@totalleftmargin0.25in\scriptsize}
+
+% Headers & footers
+\pagestyle{fancy}
+\renewcommand{\headrulewidth}{0pt}
+\setlength\headheight{14.5pt}
+\fancyhf{}
+\fancyhead[L]{\footnotesize\firstleftmark}
+\fancyhead[R]{\footnotesize\thepage}
+
+% Degree and centered dot symbols
+\DeclareInputText{176}{\ifmmode{{}^\circ}\else\textdegree\fi}
+\DeclareInputText{183}{\ifmmode\cdot\else\textperiodcentered\fi}
+
+% command to provide stretchy vertical space in proportion
+\newcommand\nbvspace[1][1]{\vspace*{\stretch{#1}}}
+
+% command to put first word of chapter in small-caps
+\newcommand\Firstsc[1]{\textsc{#1}}
+
+\newlength\TmpLen % for general use
+
+% For miscellaneous math alignment - usage : \PadTo{Longer Math}{Short Math}
+% Will center Short Math in space large enough for Longer Math
+% (use optional [l] or [r] for left/right alignment)
+\newlength\TmpPadLen % for PadTo
+\newcommand{\PadTo}[3][c]{%
+  \settowidth{\TmpPadLen}{$#2$}%
+  \makebox[\TmpPadLen][#1]{$#3$}%
+}
+
+% \includegraphicsmid[placement]{graphics filename}{caption}
+\newcommand{\includegraphicsmid}[3][!ht]{%
+  \begin{figure}[#1]%
+    \centering
+    \figurelabel{#2}%
+    \includegraphics{./images/#2.png}%
+    \ifthenelse{\equal{#3}{}}{}{\caption*{#3}}%
+  \end{figure}}
+% \includegraphicstwo[placement]{graphics filename 1}{caption 1}{graphics filename 2}{caption 2}
+\newcommand{\includegraphicstwo}[5][!ht]{%
+  \begin{figure}[#1]%
+    \centering%
+    \figurelabel{#2}%
+    \hspace*{\stretch{1}}%
+    \includegraphics{./images/#2.png}%
+    \hspace*{\stretch{2}}
+    \figurelabel{#4}%
+    \includegraphics{./images/#4.png}%
+    \hspace*{\stretch{1}}%
+    \caption*{\hspace*{\stretch{1}}#3\hspace*{\stretch{2}}#5\hspace*{\stretch{1}}}%
+  \end{figure}}
+% \includegraphicsouter[height in lines]{graphics filename}{caption}
+\newcommand{\includegraphicsouter}[3][]{%
+  \begin{wrapfigure}[#1]{o}{0pt}%
+    \figurelabel{#2}%
+    \includegraphics{./images/#2.png}%
+    \ifthenelse{\equal{#3}{}}{}{\caption*{#3}}%
+  \end{wrapfigure}}
+% \includegraphicssideways[placement]{graphics filename}{caption}
+\newcommand{\includegraphicssideways}[3][!ht]{%
+  \begin{sideways}
+  \settowidth{\TmpLen}{\includegraphics{./images/#2.png}}
+  \begin{minipage}{\TmpLen}
+    \figurelabel{#2}%
+    \centering\includegraphics{./images/#2.png}\\[0.5ex]
+    \ifthenelse{\equal{#3}{}}{}{\footnotesize #3}%
+  \end{minipage}
+  \end{sideways}}
+
+% Reduce spacing around wrapfigs
+\setlength\intextsep{4pt plus 4pt minus 2pt}
+
+% Phantom section before label so hyperref links to correct place
+\newcommand\nblabel[1]{\phantomsection\label{#1}}
+
+% Commands to label and hyperref to chapters
+\newcommand{\chaplabel}[1]{\label{chap:#1}}% No phantom section because immediately preceding chapter heading
+\newcommand{\chapref}[2]{\hyperref[chap:#1]{#2}}
+
+% Commands to label and hyperref to figures
+\newcommand{\figurelabel}[1]{\nblabel{fig:#1}}
+\newcommand{\figureref}[2]{\hyperref[fig:#1]{#2}}
+
+% Commands to label and hyperref to articles
+\newcommand{\artlabel}[1]{\nblabel{art:#1}}
+\newcommand{\artref}[2]{\hyperref[art:#1]{#2}}
+
+% Commands to label and hyperref to equations
+\newcommand{\eqnart}{} % needed because equation numbers are not unique
+\newcommand{\eqnlabel}[1]{\nblabel{eqn:#1}}
+\newcommand{\eqnref}[2]{\hyperref[eqn:#1.#2]{#2}}
+\newcommand{\Tag}[2][]{\ifthenelse{\equal{#1}{}} % label equation with "article.equation"
+                         {\eqnlabel{\eqnart.#2}}%
+                         {\eqnlabel{\eqnart.#1}}%
+                       \tag{#2}}
+% Very occasional use to squeeze an equation tag into the page width - labelling is done manually
+\newcommand{\nbtag}[1]{~(#1)}
+
+% Command to simplify placing repeated footnotes in tables
+\newcommand{\tabfootmark}[1][1]{\addtocounter{footnote}{#1}\footnotemark[\value{footnote}]}
+\renewcommand\footnoterule{\vfill}
+
+% For math functions dn, cn, sn
+\DeclareMathOperator{\dn}{dn}
+\DeclareMathOperator{\cn}{cn}
+\DeclareMathOperator{\sn}{sn}
+\DeclareMathOperator{\cosec}{cosec}
+
+% Text size integral and sum operators (\strut ensures surrounding brackets are big enough)
+\newcommand{\tint}{\mathop{\strut\textstyle\int}}
+\newcommand{\tiint}{\mathop{\strut\textstyle\iint}}
+\newcommand{\tiiint}{\mathop{\strut\textstyle\iiint}}
+\newcommand{\tsum}{\mathop{\strut\textstyle\sum}}
+
+% \centerdot is too heavy
+\renewcommand{\centerdot}{\mathbin{.}}
+
+% For small bold or sans-serif variables.
+\newcommand{\smallbold}[1]{\mathbf{#1}}
+\newcommand{\smallsanscap}[1]{\textsf{\scriptsize #1}}
+
+% Chapter headings
+\newcommand{\Heading}{\centering\normalfont\normalsize}
+\newcommand{\Chapter}[2]{\ifthenelse{\equal{#2}{}}%
+                           {\ChapCmd{\MakeUppercase{#1}}{\rule{0.5in}{0.5pt}}{\MakeUppercase{#1}}{#1}{#1}}%
+                           {\ChapCmd{\MakeUppercase{#1}}{\MakeUppercase{#2}}{\MakeUppercase{#2}}{#1 --- #2}{#1}}}
+\newcommand{\ChapCmd}[5]{\chapter*{\chaplabel{#5}\Heading\large #1\\[1ex]\footnotesize #2}%
+                         \addcontentsline{toc}{section}{\textsc{#4}}%
+                         \fancyhead[C]{\footnotesize #3}}
+
+% Small caps section headings
+\newcommand{\Section}[1]{\section*{\Heading\textsc{#1}}}
+% Italic subsection headings
+\newcommand{\Subsection}[1]{\subsection*{\Heading\textit{#1}}}
+% Article numbers
+\newcommand{\Article}[1]{#1\artlabel{#1}.]%
+                         \renewcommand{\eqnart}{#1}% used for equation referencing
+                         \markboth{#1.]}{}}
+\renewcommand\paragraph{\@startsection{paragraph}{4}%
+  {\parindent}{0pt}{-0.75em}{\normalfont\normalsize}}
+
+% Chapter heading
+\def\@makeschapterhead#1{%
+  \vspace*{70\p@}%
+  {\parindent \z@ \raggedright
+    \interlinepenalty\@M
+    #1\par\nobreak
+    \vskip 15\p@
+  }}
+
+% List definitions
+\newenvironment{olist}%
+  {\begin{list}{}{\setlength\topsep{0pt}%
+                 \setlength\itemsep{0pt}%
+                 \setlength\parsep{0pt}%
+                 \setlength\leftmargin{2\parindent}}}%
+  {\end{list}}
+\newenvironment{ilist}%
+  {\begin{list}{}{\setlength\topsep{0pt}%
+                 \setlength\itemsep{0pt}%
+                 \setlength\parsep{0pt}%
+                 \setlength\leftmargin{0pt}}}%
+  {\end{list}}
+
+% Hanging indent used in Additions and Corrections section
+\newlength{\HangPadLen}
+\newlength{\HangLen}
+\settowidth{\HangPadLen}{Page 99.\@ }
+\settowidth{\HangLen}{For}
+\addtolength{\HangLen}{\HangPadLen}
+\newcommand{\hangindentbox}[1]{\noindent\hangindent=\HangLen\makebox[\HangPadLen][r]{#1\@ }}
+
+% Factorial can be changed to modern notation (n!) by toggling commenting out of the following two lines
+%\newcommand{\nbfactorial}[1]{#1!} % Modern factorial notation
+\newcommand{\nbfactorial}[1]{\renewcommand\arraystretch{0.8}\begin{tabular}{|@{\;}c@{\,}}#1\\\hline\end{tabular}} % Original factorial notation
+
+% Use this for small-sized text in tables, and normal size italic captions
+\newcommand{\tabletextsize}{\footnotesize}
+\newcommand{\tablespaceup}{\rule{0pt}{2.5ex}}
+\newcommand{\tablespacedown}{\rule[-1ex]{0pt}{2.5ex}}
+\captionsetup{aboveskip=2pt}
+
+% Wider spaced \dotfills
+\newcommand{\mdotfill}{\leaders\hbox{\kern 0.4em.\kern 0.4em}\hfill\null}
+\newcommand{\wdotfill}{\leaders\hbox{\kern 0.7em.\kern 0.7em}\hfill\null}
+
+% Hanging indent used in tables
+\newcommand{\tabhang}{\hangindent 1em}
+
+% Used to force pagebreaks - if repaginated, remove and re-insert as necessary
+\newcommand{\nbpagebreak}{\pagebreak}
+
+% Space the same width as a digit - for alignments
+\newcommand{\Z}{\phantom{0}}
+
+% Used to raise lintertext - first argument is number of line spacings to raise
+\newcommand{\raiseboxlint}[2][0.5]{\raisebox{#1\baselineskip}{#2}}
+
+% Occasionally lengthen or shorten page to improve pagebreaks
+\newcommand{\longpage}[1][1]{\enlargethispage{#1\baselineskip}}
+\newcommand{\shortpage}[1][1]{\enlargethispage{-#1\baselineskip}}
+
+% Allow some slack to avoid under/overfull boxes
+\newcommand\nbstretchyspace{\spaceskip0.5em plus 0.3em minus 0.25em}
+
+% Stop names being hyphenated
+\hyphenation{Cremona Watson}
+
+% Definitions for Table of Contents
+
+\newcommand{\TocChapter}[2]{{\centering\vspace{3ex}\large #1\\[1ex]\footnotesize#2\\[1ex]}}
+
+\newlength{\ToCLen}
+\newlength{\ToCLenA}
+\newlength{\ToCLenB}
+\newcommand{\PadTxt}[3][c]{%
+  \settowidth{\TmpPadLen}{#2}%
+  \makebox[\TmpPadLen][#1]{#3}%
+}
+\newcommand{\ToCBox}[2][]{%
+  \ifthenelse{\not\equal{#1}{}}{\artref{#1}{#1}--}{}%
+  \settowidth{\ToCLenA}{#2}%
+  \settowidth{\ToCLenB}{99}%
+  \ifthenelse{\equal{#2}{App}}{\PadTxt{999.}{}}{% Special case for Appendix
+  \ifthenelse{\ToCLenA<\ToCLenB}% Put small article numbers into a wider box for better alignment
+    {\PadTxt[r]{99}{\artref{#2}{#2}}}{\artref{#2}{#2}}%
+  .}%
+  \hspace{0.75em}%
+}
+\newcommand{\ToCRange}[2][]{%
+  \ifthenelse{\not\equal{#1}{}}{%
+    \ifthenelse{\not\equal{\pageref{art:#1}}{\pageref{art:#2}}}
+      {\pageref{art:#1}--}{}}{}%
+  \pageref{art:#2}%
+}
+\newcommand{\ToCHead}{{\noindent\scriptsize Art.\hfill Page\smallskip}}
+\newcommand{\ToCHeadPage}{\pagebreak\par\ToCHead}
+
+%\ToCLine[extra]{start number}{end number}{Title}
+\newcommand{\ToCLineInt}[4][]{%
+  \par\ifthenelse{\equal{#1}{}}{}{\smallskip}%
+  \settowidth{\TmpLen}{9999}%
+  % set up ToCLen to give extra space for page range if needed
+  \ifthenelse{\equal{#2}{}}%
+    {\setlength{\ToCLen}{0pt}}%
+    {\ifthenelse{\equal{\pageref{art:#2}}{\pageref{art:#3}}}%
+       {\setlength{\ToCLen}{0pt}}%
+       {\ifthenelse{\pageref{art:#2}<100}{\settowidth{\ToCLen}{99}}{\settowidth{\ToCLen}{--999}}}}%
+%  \ifthenelse{\TmpLen<\ToCLen}{\setlength{\TmpLen}{\ToCLen}}{}
+  % output hanging indent parbox with article number, title, dotfill, and optional extra space for page range
+  \noindent\parbox[b]{\linewidth-\TmpLen}{\ToCBox[#2]{#3}%
+    \hangindent3em #4\mdotfill\rule{\ToCLen}{0pt}}%
+  % output page (range)
+   \makebox[\TmpLen][r]{\ToCRange[#2]{#3}}\par%
+%  \PadTxt[r]{9999}{\ToCRange[#2]{#3}}%
+}
+% Non-empty option argument gives extra spacing before line to align bottom-aligned parbox correctly
+\newcommand{\ToCLine}[3][]{\ToCLineInt[#1]{}{#2}{#3}}
+\newcommand{\ToCLinR}[4][]{\ToCLineInt[#1]{#2}{#3}{#4}}
+
+% Definitions for advert pages at end
+\newcommand{\selectedwork}[2]{\noindent\hangindent 1.75em%
+  \textbf{\ifthenelse{\equal{#1}{}}{---}{#1.}\hspace{1em}#2.\hspace{0.5em plus 0.25em}}}
+\newcommand{\selectedpart}[2]{\hangindent 5em%
+  \settowidth{\TmpLen}{II\@.}%
+  \makebox[\TmpLen][l]{\textbf{#1.}}\hspace{1em}\textbf{#2.\hspace{0.5em plus 0.25em}}}
+\newcommand{\selectedauthor}[2][]{\ifthenelse{\equal{#1}{}}{}{#1~}{\textsc{#2}}}
+\newcommand{\selectedprice}[3][]{%
+  \ifthenelse{\equal{#1}{}}{}{#1\textit{l.}~}%
+  \ifthenelse{\equal{#2}{}}{}{#2\textit{s.}~}%
+  \ifthenelse{\equal{#3}{}}{}{#3\textit{d.}}}
+\newcommand{\selectedvol}[1]{\settowidth{\TmpPadLen}{Vol.~III\@.~}%
+  \indent\indent\hangindent 8em\makebox[\TmpPadLen][l]{#1.}}
+
+% Definitions for Index
+\renewenvironment{theindex}{%
+  \small%
+  \setlength\columnseprule{0.5pt}%
+  \setlength\columnsep{1.5em}%
+  \begin{multicols}{2}[%
+                        \chapter*{\Heading\large INDEX.\\[2ex]\normalsize\itshape The numbers refer to the pages.}%
+                        \addcontentsline{toc}{section}{\textsc{Index.}}]
+  \fancyhead[C]{\footnotesize INDEX.}
+  \setlength\parindent{0pt}%
+  \setlength\parskip{0pt plus 0.1pt}%
+  \nbstretchyspace%
+  \renewcommand{\@idxitem}{\par\hangindent 1em}%
+  \let\item\@idxitem%
+}%
+{\end{multicols}}
+\newcommand{\subdashone}{---\hspace{0.3em}}
+\newcommand{\subdashtwo}{---\hspace{0.3em}---\hspace{0.3em}}
+\newcommand{\indexetseq}[1]{#1\ et seq.}
+
+% Definitions for DPalign*, DPgather* and lintertext
+\providecommand\shortintertext\intertext
+\newcount\DP@lign@no
+\newtoks\DP@lignb@dy
+\newif\ifDP@cr
+\newif\ifbr@ce
+\def\f@@zl@bar{\null}
+\def\addto@DPbody#1{\global\DP@lignb@dy\@xp{\the\DP@lignb@dy#1}}
+\def\parseb@dy#1{\ifx\f@@zl@bar#1\f@@zl@bar
+    \addto@DPbody{{}}\let\@next\parseb@dy
+  \else\ifx\end#1%
+    \let\@next\process@DPb@dy
+\ifDP@cr\else\addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&}\fi
+    \addto@DPbody{\end}%
+  \else\ifx\intertext#1%
+    \def\@next{\eat@command0}%
+  \else\ifx\shortintertext#1%
+    \def\@next{\eat@command1}%
+\else\ifDP@cr\addto@DPbody{&\DP@lint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\DPh@@kl}%
+          \DP@crfalse\fi
+    \ifx\begin#1\def\begin@stack{b}%
+      \let\@next\eat@environment
+  \else\ifx\lintertext#1%
+    \let\@next\linter@text
+  \else\ifx\rintertext#1%
+    \let\@next\rinter@text
+  \else\ifx\\#1%
+\addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\\}\DP@crtrue
+    \global\advance\DP@lign@no\@ne
+    \let\@next\parse@cr
+  \else\check@braces#1!Q!Q!Q!\ifbr@ce\addto@DPbody{{#1}}\else
+    \addto@DPbody{#1}\fi
+    \let\@next\parseb@dy
+  \fi\fi\fi\fi\fi\fi\fi\fi\@next}
+\def\process@DPb@dy{\let\lintertext\@gobble\let\rintertext\@gobble
+  \@xp\start@align\@xp\tw@\@xp\st@rredtrue\@xp\m@ne\the\DP@lignb@dy}
+\def\linter@text#1{\@xp\DPlint\@xp{\the\DP@lign@no}{#1}\parseb@dy}
+\def\rinter@text#1{\@xp\DPrint\@xp{\the\DP@lign@no}{#1}\parseb@dy}
+\def\DPlint#1#2{\@xp\def\csname DP@lint:#1\endcsname{\text{#2}}}
+\def\DPrint#1#2{\@xp\def\csname DP@rint:#1\endcsname{\text{#2}}}
+\def\DP@lint#1{\ifbalancedlrint\@xp\ifx\csname DP@lint:#1\endcsname\relax\phantom
+  {\csname DP@rint:#1\endcsname}\else\csname DP@lint:#1\endcsname\fi
+  \else\csname DP@lint:#1\endcsname\fi}
+\def\DP@rint#1{\ifbalancedlrint\@xp\ifx\csname DP@rint:#1\endcsname\relax\phantom
+  {\csname DP@lint:#1\endcsname}\else\csname DP@rint:#1\endcsname\fi
+  \else\csname DP@rint:#1\endcsname\fi}
+\def\eat@command#1#2{\ifcase#1\addto@DPbody{\intertext{#2}}\or
+  \addto@DPbody{\shortintertext{#2}}\fi\DP@crtrue
+  \global\advance\DP@lign@no\@ne\parseb@dy}
+\def\parse@cr{\new@ifnextchar*{\parse@crst}{\parse@crst{}}}
+\def\parse@crst#1{\addto@DPbody{#1}\new@ifnextchar[{\parse@crb}{\parseb@dy}}
+\def\parse@crb[#1]{\addto@DPbody{[#1]}\parseb@dy}
+\def\check@braces#1#2!Q!Q!Q!{\def\dp@lignt@stm@cro{#2}\ifx
+  \empty\dp@lignt@stm@cro\br@cefalse\else\br@cetrue\fi}
+\def\eat@environment#1{\addto@DPbody{\begin{#1}}\begingroup
+  \def\@currenvir{#1}\let\@next\digest@env\@next}
+\def\digest@env#1\end#2{%
+  \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}%
+  \ifx\@empty\begin@stack
+    \@checkend{#2}
+    \endgroup\let\@next\parseb@dy\fi
+    \addto@DPbody{#1\end{#2}}
+    \@next}
+\def\lintertext{lint}\def\rintertext{rint}
+\newif\ifbalancedlrint
+\let\DPh@@kl\empty\let\DPh@@kr\empty
+\def\DPg@therl{&\omit\hfil$\displaystyle}
+\def\DPg@therr{$\hfil}
+
+\newenvironment{DPalign*}[1][a]{%
+  \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi
+  \global\DP@lign@no\z@\DP@crfalse
+  \DP@lignb@dy{&\DP@lint0&}\parseb@dy
+}{%
+  \endalign
+}
+\newenvironment{DPgather*}[1][a]{%
+  \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi
+  \global\DP@lign@no\z@\DP@crfalse
+  \let\DPh@@kl\DPg@therl
+  \let\DPh@@kr\DPg@therr
+  \DP@lignb@dy{&\DP@lint0&\DPh@@kl}\parseb@dy
+}{%
+  \endalign
+}
+\makeatother
+%%%%%%%%%%%% end of preamble %%%%%%%%%%%%
+
+\begin{document}
+
+% reduce space around displayed maths
+\setlength\abovedisplayskip{6pt plus 6pt minus 3pt}
+\setlength\belowdisplayskip{\abovedisplayskip}
+\setlength\belowdisplayshortskip{\belowdisplayskip}
+
+%set up page headers
+\fancypagestyle{plain}{\fancyhf{}}
+
+\pagestyle{empty}
+\pagenumbering{alph}
+
+\pdfbookmark[0]{Project Gutenberg Boilerplate}{Project Gutenberg Boilerplate}
+
+\begin{verbatim}
+The Project Gutenberg EBook of Notes on Recent Researches in Electricity
+and Magnetism, by J. J. Thomson
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.net
+
+
+Title: Notes on Recent Researches in Electricity and Magnetism
+       Intended as a Sequel to Professor Clerk-Maxwell\'s Treatise
+       on Electricity and Magnetism
+
+Author: J. J. Thomson
+
+Release Date: June 27, 2011 [EBook #36525]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK RECENT RESEARCHES--ELECTRICITY, MAGNETISM ***
+\end{verbatim}
+
+\clearpage
+%% -----File: 001.png-----
+
+\pagestyle{empty}
+\pagenumbering{roman}
+\pdfbookmark[0]{Notes on Recent Researches in Electricity and Magnetism}{Notes on Recent Researches in Electricity and Magnetism}
+
+\begin{center}
+\nbvspace[1]
+{\large NOTES}\\[2.5ex]
+{\scriptsize ON}\\[3ex]
+{\large RECENT RESEARCHES IN}\\[4ex]
+{\Large ELECTRICITY AND MAGNETISM}
+
+\nbvspace[1]
+{\scriptsize INTENDED AS A SEQUEL TO}
+
+\nbvspace[1]
+PROFESSOR CLERK-MAXWELL'S TREATISE\\[1ex]
+ON ELECTRICITY AND MAGNETISM
+
+\nbvspace[2]
+{\scriptsize BY}\\[2ex]
+J. J. THOMSON, M.A., F.R.S.
+
+\textsc{Hon. Sc.D. Dublin}
+
+{\scriptsize FELLOW OF TRINITY COLLEGE}
+
+{\scriptsize PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE}
+
+\nbvspace[3]
+
+$\mathfrak{Oxford}$\\[1ex]
+AT THE CLARENDON PRESS\\[1ex]
+1893
+\nbvspace[1]
+\end{center}
+\clearpage
+%% -----File: 002.png-----
+% Publisher page
+\nbvspace
+\begin{center}
+$\mathfrak{Oxford}$
+
+{\footnotesize PRINTED AT THE CLARENDON PRESS}
+
+{\tiny BY HORACE HART, PRINTER TO THE UNIVERSITY}
+\nbvspace
+\end{center}
+
+\clearpage
+\thispagestyle{empty}
+\begin{center}
+\small
+Produced by Robert Cicconetti, Nigel Blower and the Online
+Distributed Proofreading Team at http://www.pgdp.net (This
+file was produced from images generously made available
+by Cornell University Digital Collections)
+\end{center}
+
+\vfill
+{%
+  \setlength{\parindent}{0pt}
+	\setlength{\parskip}{6pt plus 2pt minus 1pt}
+  \subsection*{{\normalsize\centering\itshape Transcriber's Notes}}
+  \small
+  A small number of minor typographical errors and inconsistencies
+  have been corrected. See the {\ttfamily\footnotesize DPtypo} command
+  in the \LaTeX\ source for more information.
+}
+
+\clearpage
+%% -----File: 003.png---Folio v-------
+\pagestyle{fancy}
+
+\Chapter{Preface}{}
+
+
+\Firstsc{In} the twenty years which have elapsed since the first
+appearance of Maxwell's Treatise on \textit{Electricity and Magnetism}
+great progress has been made in these sciences.  This progress
+has been largely---perhaps it would not be too much to say
+mainly---due to the influence of the views set forth in that
+Treatise, to the value of which it offers convincing testimony.
+
+In the following work I have endeavoured to give an account
+of some recent electrical researches, experimental as well as
+theoretical, in the hope that it may assist students to gain some
+acquaintance with the recent progress of Electricity and yet
+retain Maxwell's Treatise as the source from which they learn
+the great principles of the science.  I have adopted exclusively
+Maxwell's theory, and have not attempted to discuss the consequences
+which would follow from any other view of electrical
+action.  I have assumed throughout the equations of the Electromagnetic
+Field given by Maxwell in the ninth chapter of the
+second volume of his Treatise.
+
+The \chapref{Chapter I.}{first chapter} of this work contains an account of a method
+of regarding the Electric Field, which is geometrical and physical
+rather than analytical. I have been induced to dwell on this
+because I have found that students, especially those who commence
+the subject after a long course of mathematical studies,
+have a great tendency to regard the whole of Maxwell's theory
+as a matter of the solution of certain differential equations, and
+to dispense with any attempt to form for themselves a mental
+picture of the physical processes which accompany the phenomena
+they are investigating.  I think that this state of things
+is to be regretted, since it retards the progress of the science of
+%% -----File: 004.png---Folio vi-------
+Electricity and diminishes the value of the mental training
+afforded by the study of that science.
+
+In the first place, though no instrument of research is more
+powerful than Mathematical Analysis, which indeed is indispensable
+in many departments of Electricity, yet analysis works to
+the best advantage when employed in developing the suggestions
+afforded by other and more physical methods.  One example of
+such a method, and one which is very closely connected with the
+initiation and development of Maxwell's Theory, is that of the
+`tubes of force' used by Faraday.  Faraday interpreted all the
+laws of Electrostatics in terms of his tubes, which served him in
+place of the symbols of the mathematician, while in his hands
+the laws according to which these tubes acted on each other
+served instead of the differential equations satisfied by such
+symbols.  The method of the tubes is distinctly physical, that
+of the symbols and differential equations is analytical.
+
+The physical method has all the advantages in vividness which
+arise from the use of concrete quantities instead of abstract
+symbols to represent the state of the electric field; it is more easily
+wielded, and is thus more suitable for obtaining rapidly the main
+features of any problem; when, however, the problem has to be
+worked out in all its details, the analytical method is necessary.
+
+In a research in any of the various fields of electricity we shall
+be acting in accordance with Bacon's dictum that the best results
+are obtained when a research begins with Physics and ends with
+Mathematics, if we use the physical theory to, so to speak, make
+a general survey of the country, and when this has been done
+use the analytical method to lay down firm roads along the line
+indicated by the survey.
+
+The use of a physical theory will help to correct the tendency---which
+I think all who have had occasion to examine in Mathematical
+Physics will admit is by no means uncommon---to look on
+analytical processes as the modern equivalents of the Philosopher's
+Machine in the Grand Academy of Lagado, and to regard as the
+normal process of investigation in this subject the manipulation
+of a large number of symbols in the hope that every now and
+then some valuable result may happen to drop out.
+%% -----File: 005.png---Folio vii-------
+
+Then, again, I think that supplementing the mathematical
+theory by one of a more physical character makes the study of
+electricity more valuable as a mental training for the student.
+Analysis is undoubtedly the greatest thought-saving machine
+ever invented, but I confess I do not think it necessary or desirable
+to use artificial means to prevent students from thinking too
+much. It frequently happens that more thought is required,
+and a more vivid idea of the essentials of a problem gained, by
+a rough solution by a general method, than by a complete
+solution arrived at by the most recent improvements in the
+higher analysis.
+
+The method of illustrating the properties of the electric field
+which I have given in \chapref{Chapter I.}{Chapter~I} has been devised so as to lead
+directly to the distinctive feature in Maxwell's Theory, that
+changes in the polarization in a dielectric produce magnetic
+effects analogous to those produced by conduction currents.
+Other methods of viewing the processes in the Electric Field,
+which would be in accordance with Maxwell's Theory, could, I
+have no doubt, be devised; the question as to which particular
+method the student should adopt is however for many purposes
+of secondary importance, provided that he does adopt one, and
+acquires the habit of looking at the problems with which he
+is occupied as much as possible from a physical point of view.
+
+It is no doubt true that these physical theories are liable
+to imply more than is justified by the analytical theory they
+are used to illustrate. This however is not important if we
+remember that the object of such theories is suggestion and not
+demonstration. Either Experiment or rigorous Analysis must
+always be the final Court of Appeal; it is the province of these
+physical theories to supply cases to be tried in such a court.
+
+\chapref{Chapter II.}{Chapter~II} is devoted to the consideration of the discharge of
+electricity through gases; \chapref{Chapter III.}{Chapter~III} contains an account of the
+application of Schwarz's method of transformation to the solution
+of two-dimensional problems in Electrostatics. The rest of
+the book is chiefly occupied with the consideration of the properties
+of alternating currents; the experiments of Hertz and the
+development of electric lighting have made the use of these
+%% -----File: 006.png---Folio viii-------
+currents, both for experimental and commercial purposes, much
+more general than when Maxwell's Treatise was written; and
+though the principles which govern the action of these currents
+are clearly laid down by Maxwell, they are not developed to the
+extent which the present importance of the subject demands.
+
+\chapref{Chapter IV.}{Chapter~IV} contains an investigation of the theory of such
+currents when the conductors in which they flow are cylindrical
+or spherical, while in \chapref{Chapter V.}{Chapter~V} an account of Hertz's
+experiments on Electromagnetic Waves is given. This Chapter
+also contains some investigations on the Electromagnetic Theory
+of Light, especially on the scattering of light by small metallic
+particles; on reflection from metals; and on the rotation of the
+plane of polarization by reflection from a magnet. I regret that
+it was only when this volume was passing through the press that
+I became acquainted with a valuable paper by Drude (Wiedemann's
+\textit{Annalen}, 46, p.~353, 1892) on this subject.
+
+\chapref{Chapter VI.}{Chapter~VI} mainly consists of an account of Lord Rayleigh's
+investigations on the laws according to which alternating
+currents distribute themselves among a network of conductors;
+while the \chapref{Chapter VII.}{last Chapter} contains a discussion of the equations
+which hold when a dielectric is moving in a magnetic field,
+and some problems on the distribution of currents in rotating
+conductors.
+
+I have not said anything about recent researches on Magnetic
+Induction, as a complete account of these in an easily accessible
+form is contained in Professor Ewing's `Treatise on Magnetic
+Induction in Iron and other Metals.'
+
+I have again to thank Mr.~Chree, Fellow of King's College,
+Cambridge, for many most valuable suggestions, as well as for
+a very careful revision of the proofs.
+
+\begin{flushright}
+J.~J. THOMSON.\qquad\null
+\end{flushright}
+
+%% -----File: 007.png---Folio ix-------
+
+%% Manually placed "Art./Page" headers using \ToCHead and \ToCHeadPage may need moving if repaginated
+%% Lines marked with %%[**manual] may also need adjusting
+
+\Chapter{Contents}{}
+
+\TocChapter{CHAPTER I.}{ELECTRIC DISPLACEMENT AND FARADAY TUBES OF FORCE.}
+
+\ToCHead
+\ToCLine{1}{Electric displacement}
+\ToCLine{2}{Faraday tubes}
+\ToCLine{3}{Unit Faraday tubes}
+\ToCLine{4}{Analogy with kinetic theory of gases}
+\ToCLine{5}{Reasons for taking tubes of electrostatic induction as the unit}
+\ToCLine{6}{Energy in the electric field}
+\ToCLine{7}{Behaviour of Faraday tubes in a conductor}
+\ToCLine{8}{Connection between electric displacement and Faraday tubes}
+\ToCLine[+]{9}{Rate of change of electric polarization expressed in terms of the velocity of Faraday tubes}
+\ToCLine{10}{Momentum due to Faraday tubes}
+\ToCLine{11}{Electromotive intensity due to induction}
+\ToCLine{12}{Velocity of Faraday tubes}
+\ToCLine{13}{Systems of tubes moving with different velocities}
+\ToCLine{14}{Mechanical forces in the electric field}
+\ToCLine{15}{Magnetic force due to alteration in the dielectric polarization}
+\ToCLine[+]{16}{Application of Faraday tubes to find the magnetic force due to a moving charged sphere}
+\ToCLine{17}{Rotating electrified plates}
+\ToCLine{18}{Motion of tubes in a steady magnetic field}
+\ToCLine{19}{Induction of currents due to changes in the magnetic field}
+\ToCLine{20}{Induction due to the motion of the circuit}
+\ToCHeadPage
+\ToCLine{21}{Effect of soft iron in the field}
+\ToCLine{22}{Permanent magnets}
+\ToCLine{23}{Steady current along a straight wire}
+\ToCLine{24}{Motion of tubes when the currents are rapidly alternating}
+\ToCLine{25}{Discharge of a Leyden jar}
+\ToCLine{26}{Induced currents}
+\ToCLine{27}{Electromagnetic theory of light}
+%% -----File: 008.png---Folio x-------
+\ToCLinR{28}{32}{Behaviour of tubes in conductors}
+\ToCLine{33}{Galvanic cell}
+\ToCLine{34}{Metallic and electrolytic conduction}
+
+\TocChapter{CHAPTER II.}{PASSAGE OF ELECTRICITY THROUGH GASES.}
+
+\ToCLine{35}{Introduction}
+\ToCLine{36}{Can the molecules of a gas be electrified?}
+\ToCLine{37}{Hot gases}
+\ToCLine{38}{Electric properties of flames}
+\ToCLine{39}{Effect of ultra-violet light on the discharge}
+\ToCLine{40}{Electrification by ultra-violet light}
+\ToCLine{41}{Disintegration of the negative electrode}
+\ToCLine{42}{Discharge of electricity from illuminated metals}
+\ToCLine{43}{Discharge of electricity by glowing bodies}
+\ToCLine{44}{Volta-potential}
+\ToCLine{45}{Electrification by sun-light}
+\ToCLine{46}{`Electric Strength' of a gas}
+\ToCLine{47}{Effect of the nature of the electrodes on the spark length}
+\ToCLine{48}{Effect of curvature of the electrodes on the spark length}
+\ToCLine[+]{49}{Baille's experiments on the connection between potential difference and spark length}
+\ToCLine{50}{Liebig's on the same subject}
+\ToCLine{51}{Potential difference expressed in terms of spark length}
+\ToCLinR{52}{53}{Minimum potential difference required to produce a spark}
+\ToCLinR{54}{61}{Discharge when the field is not uniform}
+\ToCLinR[+]{62}{65}{Peace's experiments on the connection between pressure and spark potential}
+\ToCHeadPage
+\ToCLinR{66}{68}{Critical pressure}
+\ToCLinR{69}{71}{Potential difference required to spark through various\\gases} %%[**manual]
+\ToCLinR{72}{76}{Methods of producing electrodeless discharges}
+\ToCLine{77}{Appearance of such discharges}
+\ToCLinR{78}{80}{Critical pressure for such discharges}
+\ToCLine{81}{Difficulty of getting the discharge to pass across from gas to metal}
+\ToCLinR{82}{86}{High conductivity of rarefied gases}
+\ToCLine{87}{Discharge through a mixture of gases}
+\ToCLinR{88}{93}{Action of a magnet on the electrodeless discharges}
+\ToCLine{94}{Appearance of discharge when electrodes are used}
+\ToCLine{95}{Crookes' theory of the dark space}
+%% -----File: 009.png---Folio xi-------
+\ToCLine{96}{Length of dark space}
+\ToCLinR{97}{98}{Negative glow}
+\ToCLinR{99}{103}{Positive column and striations}
+\ToCLinR{104}{107}{Velocity of discharge along positive column}
+\ToCLinR{108}{116}{Negative rays}
+\ToCLine{117}{Mechanical effects produced by negative rays}
+\ToCLinR{118}{123}{Shadows cast by negative rays}
+\ToCLinR{124}{125}{Relative magnitudes of time quantities in the discharge}
+\ToCLinR{126}{128}{Action of a magnet on the discharge}
+\ToCLine{129}{Action of a magnet on the negative glow}
+\ToCLinR{130}{133}{Action of a magnet on the negative rays}
+\ToCLine{134}{Action of a magnet on the positive column}
+\ToCLine{135}{Action of a magnet on the negative rays in very high vacua}
+\ToCLine{136}{Action of a magnet on the course of the discharge}
+\ToCLinR{137}{138}{Action of a magnet on the striations}
+\ToCLinR{139}{147}{Potential gradient along the discharge tube}
+\ToCLinR{148}{151}{Effect of the strength of the current on the cathode fall}
+\ToCLinR[+]{152}{155}{Small potential difference sufficient to maintain current when once started}
+\ToCLinR{156}{162}{Warburg's experiments on the cathode fall}
+\ToCLinR{163}{165}{Potential gradient along positive column}
+\ToCLinR{166}{168}{Discharge between electrodes placed close together}
+\ToCLinR{169}{176}{The arc discharge}
+\ToCHeadPage
+\ToCLinR{177}{178}{Heat produced by the discharge}
+\ToCLinR[+]{179}{182}{Difference between effects at positive and negative electrodes}
+\ToCLinR{183}{186}{Lichtenberg's and Kundt's dust figures}
+\ToCLinR{187}{193}{Mechanical effects due to the discharge}
+\ToCLinR{194}{201}{Chemical action of the discharge}
+\ToCLine{202}{Phosphorescent glow due to the discharge}
+\ToCLinR[+]{203}{206}{Discharge facilitated by rapid changes in the strength of the field}
+\ToCLinR{207}{229}{Theory of the discharge}
+
+\TocChapter{CHAPTER III.}{CONJUGATE FUNCTIONS.}
+
+\ToCLinR{230}{233}{Schwarz and Christoffel's transformation}
+\ToCLine{234}{Method of applying it to electrostatics}
+%% -----File: 010.png---Folio xii-------
+\ToCLine[+]{235}{Distribution of electricity on a plate placed parallel to an infinite plate}
+\ToCLine{236}{Case of a plate between two infinite parallel plates}
+\ToCLine{237}{Correction for thickness of plate}
+\ToCLine{238}{Case of one cube inside another}
+\ToCLinR{239}{240}{Cube over an infinite plate}
+\ToCLine{241}{Case of condenser with guard-ring when the slit is shallow}
+\ToCLine[+]{242}{Correction when guard-ring is not at the same potential as the plate}
+\ToCLine{243}{Case of condenser with guard-ring when the slit is deep}
+\ToCLine[+]{244}{Correction when guard-ring is not at the same potential as the plate}
+\ToCLine{245}{Application of elliptic functions to problems in electrostatics}
+\ToCLine{246}{Capacity of a pile of plates}
+\ToCLine{247}{Capacity of a system of radial plates}
+\ToCLine{248}{Finite plate at right angles to two infinite ones}
+\ToCLine{249}{Two sets of parallel plates}
+\ToCLine{250}{Two sets of radial plates}
+\ToCLine{251}{Finite strip placed parallel to two infinite plates}
+\ToCLine{252}{Two sets of parallel plates}
+\ToCHeadPage
+\ToCLine{253}{Two sets of radial plates}
+\ToCLine{254}{Limitation of problems solved}
+
+\TocChapter{CHAPTER IV.}{ELECTRICAL WAVES AND OSCILLATIONS.}
+
+\ToCLine{255}{Scope of the chapter}
+\ToCLine{256}{General equations}
+\ToCLine{257}{Alternating currents in two dimensions}
+\ToCLine{258}{Case when rate of alternation is very rapid}
+\ToCLinR{259}{260}{Periodic currents along cylindrical conductors}
+\ToCLine[+]{261}{Value of Bessel's functions for very large or very small values of the variable}
+\ToCLine{262}{Propagation of electric waves along wires}
+\ToCLinR{263}{264}{Slowly alternating currents}
+\ToCLine{265}{Expansion of $xJ_0(x) / J_0'(x)$}
+\ToCLine{266}{Moderately rapid alternating currents}
+\ToCLine{267}{Very rapidly alternating currents}
+\ToCLine{268}{Currents confined to a thin skin}
+\ToCLine{269}{Magnetic force in dielectric}
+\ToCLine{270}{Transmission of disturbances along wires}
+%% -----File: 011.png---Folio xiii-------
+\ToCLine{271}{Relation between external electromotive force and current}
+\ToCLine{272}{Impedance and self-induction}
+\ToCLinR{273}{274}{Values of these when alternations are rapid}
+\ToCLinR{275}{276}{Flat conductors}
+\ToCLine{277}{Mechanical force between flat conductors}
+\ToCLine{278}{Propagation of longitudinal magnetic waves along wires}
+\ToCLine{279}{Case when the alternations are very rapid}
+\ToCLine{280}{Poynting's theorem}
+\ToCLine{281}{Expression for rate of heat production in a wire}
+\ToCLine{282}{Heat produced by slowly varying current}
+\ToCLinR{283}{284}{Heat produced by rapidly varying currents}
+\ToCLine[+]{285}{Heat in a transformer due to Foucault currents when the rate of alternation is slow}
+\ToCLine{286}{When the rate of alternation is rapid}
+\ToCLine{287}{Heat produced in a tube}
+\ToCHeadPage
+\ToCLine{288}{Vibrations of electrical systems}
+\ToCLine{289}{Oscillations on two spheres connected by a wire}
+\ToCLine{290}{Condition that electrical system should oscillate}
+\ToCLine{291}{Time of oscillation of a condenser}
+\ToCLine{292}{Experiments on electrical oscillations}
+\ToCLinR[+]{293}{297}{General investigation of time of vibration of a condenser\\\null} %[**manual]
+\ToCLinR{298}{299}{Vibrations along wires in multiple arc}
+\ToCLine{300}{Time of oscillations on a cylindrical cavity}
+\ToCLine{301}{On a metal cylinder surrounded by a dielectric}
+\ToCLine{302}{State of the field round the cylinder}
+\ToCLine{303}{Decay of currents in a metal cylinder}
+\ToCLinR[+]{304}{305}{When the lines of magnetic force are parallel to the axis of the cylinder}
+\ToCLinR{306}{307}{When the lines of force are at right angles to the axis}
+\ToCLine{308}{Electrical oscillations on spheres}
+\ToCLine{309}{Properties of the functions $S$ and $E$}
+\ToCLine{310}{General solution}
+\ToCLine{311}{Equation giving the periods of vibration}
+\ToCLine{312}{Case of the first harmonic distribution}
+\ToCLine{313}{Second and third harmonics}
+\ToCLine{314}{Field round vibrating sphere}
+\ToCLine{315}{Vibration of two concentric spheres}
+\ToCLine{316}{When the radii of the spheres are nearly equal}
+\ToCLine{317}{Decay of currents in spheres}
+\ToCLine{318}{Rate of decay when the currents flow in meridional planes}
+%% -----File: 012.png---Folio xiv-------
+\ToCLinR{319}{320}{Effect of radial currents in the sphere}
+\ToCLine[+]{321}{Currents induced in a sphere by the annihilation of a uniform magnetic field}
+\ToCLine[+]{322}{Magnetic effects of these currents when the sphere is not made of iron}
+\ToCLine{323}{When the sphere is made of iron}
+
+\pagebreak
+\TocChapter{CHAPTER V.}{ELECTROMAGNETIC WAVES.}
+
+\ToCHead
+\ToCLine{324}{Hertz's experiments}
+\ToCLinR{325}{327}{Hertz's vibrator}
+\ToCLine{328}{The resonator}
+\ToCLine{329}{Effect of altering the position of the air gap}
+\ToCLinR{330}{331}{Explanation of these effects}
+\ToCLine{332}{Resonance}
+\ToCLinR{333}{335}{Rate of decay of the vibrations}
+\ToCLinR{336}{339}{Reflection of waves from a metal plate}
+\ToCLinR{340}{342}{Sarasin's and De la Rive's experiments}
+\ToCLine{343}{Parabolic mirrors}
+\ToCLinR{344}{346}{Electric screening}
+\ToCLine{347}{Refraction of electromagnetic waves}
+\ToCLine{348}{Angle of polarization}
+\ToCLinR[+]{349}{350}{Theory of reflection of electromagnetic waves by a dielectric\\\null} %[**manual]
+\ToCLine[+]{351}{Reflection of these waves from and transmission through a thin metal plate}
+\ToCLinR{352}{354}{Reflection of light from metals}
+\ToCLine{355}{Table of refractive indices of metals}
+\ToCLine{356}{Inadequacy of the theory of metallic reflection}
+\ToCLine{357}{Magnetic properties of iron for light waves}
+\ToCLine{358}{Transmission of light through thin films}
+\ToCLinR{359}{360}{Reflection of electromagnetic waves from a grating}
+\ToCLinR{361}{368}{Scattering of these waves by a wire}
+\ToCLine{369}{Scattering of light by metal spheres}
+\ToCLine{370}{Lamb's theorem}
+\ToCLine{371}{Expressions for magnetic force and electric polarization}
+\ToCLine[+]{372}{Polarization in plane wave expressed in terms of spherical harmonics}
+\ToCLinR{373}{376}{Scattering of a plane wave by a sphere of any size}
+\ToCLine{377}{Scattering by a small sphere}
+\ToCLine{378}{Direction in which the scattered light vanishes}
+%% -----File: 013.png---Folio xv-------
+\ToCLinR{379}{384}{Hertz's experiments on waves along wires}
+\ToCHeadPage
+\ToCLine{385}{Sarasin's and De la Rive's experiments on waves along wires}
+\ToCLinR[+]{390}{392}{Comparison of specific inductive capacity with refractive index}
+\ToCLinR[+]{393}{401}{Experiments to determine the velocity of electromagnetic waves through various dielectrics}
+\ToCLine{402}{Effects produced by a magnetic field on light}
+\ToCLine{403}{Kerr's experiments}
+\ToCLine{404}{Oblique reflection from a magnetic pole}
+\ToCLine{405}{Reflection from tangentially magnetized iron}
+\ToCLine{406}{Kundt's experiments on films}
+\ToCLine{407}{Transverse electromotive intensity}
+\ToCLine{408}{Hall effect}
+\ToCLinR[+]{409}{414}{Theory of rotation of plane of polarization by reflection from a magnet}
+\ToCLinR{415}{416}{Passage of light through thin films in a magnetic field}
+
+\TocChapter{CHAPTER VI.}{DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS.}
+
+\ToCLinR[+]{417}{418}{Very rapidly alternating currents distribute themselves so as to make the Kinetic Energy a minimum}
+\ToCLine{419}{Experiments to illustrate this}
+\ToCLine{420}{Distribution of alternating currents between two wires in parallel}
+\ToCLine{421}{Self-induction and impedance of the wires}
+\ToCLine{422}{Case of any number of wires in parallel}
+\ToCLinR{423}{426}{General case of any number of circuits}
+\ToCLinR{427}{428}{Case of co-axial solenoids}
+\ToCLine{429}{Wheatstone's bridge with alternating currents}
+\ToCLinR{430}{432}{Combination of self-induction and capacity}
+\ToCLine{432*}{Effect of two adjacent vibrators on each other's periods}
+
+\TocChapter{CHAPTER VII.}{ELECTROMOTIVE INTENSITY IN MOVING BODIES.}
+
+\ToCLine{433}{Equations of electromotive intensity for moving bodies}
+\ToCLinR{434}{439}{Sphere rotating in a symmetrical magnetic field}
+\ToCLine{440}{Propagation of light through a moving dielectric}
+\ToCHeadPage
+\ToCLine{441}{Currents induced in a sphere rotating in an unsymmetrical field}
+%% -----File: 014.png---Folio xvi-------
+\ToCLine{442}{Special case when the field is uniform}
+\ToCLine{443}{Case when the rotation is very rapid}
+\ToCLine{444}{Magnetic force outside the sphere}
+\ToCLine{445}{Couples and Forces on the Rotating Sphere}
+\ToCLine{446}{The magnetic force is tangential when the rotation is rapid}
+\ToCLine{447}{Force on the sphere}
+\ToCLine[+]{448}{Solution of the previous case gives that of a sphere at rest in an alternating field}
+
+\TocChapter{~}{\textsc{\large Appendix.}}
+
+\ToCLine{App}{The Electrolysis of Steam}
+
+\vfill
+
+\begin{center}
+\rule{0.5\textwidth}{0.5pt}
+\end{center}
+
+\vfill
+
+\Section{ADDITIONS AND CORRECTIONS.}
+
+\hangindentbox{Page \pageref{add:1}.} For further remarks on electrification by incandescent bodies see
+Appendix, p.~\pageref{add:2}.
+
+\hangindentbox{\Ditto\quad\pageref{add:3}.} E.~Wiedemann and Ebert have shown (\textit{Wied.~Ann.}~46, p.~158, 1892) that
+the repulsion between two pencils of negative rays is due to the
+influence which the presence of one cathode exerts on the emission of
+rays from a neighbouring cathode.
+
+\hangindentbox{\Ditto\quad\pageref{add:4}.} Dewar (\textit{Proc.~Roy.~Soc}.~33, p.~262, 1882) has shown that the interior of
+the gaseous envelope of the electric arc always shows a fixed pressure
+amounting to about that due to a millimetre of water above that of
+the surrounding atmosphere.
+
+\hangindentbox{\Ditto\quad\pageref{add:5}.} \textit{For} $90°$\,C. \textit{read} $100°$\,C.
+
+\vfill
+%% -----File: 015.png---Folio 1-------
+\clearpage
+
+\thispagestyle{empty}
+\nbvspace
+\begin{center}\Large
+NOTES ON\\ELECTRICITY AND MAGNETISM.
+\end{center}
+\nbvspace[2]
+
+\mainmatter
+
+\Chapter{Chapter I.}{Electric Displacement and Faraday Tubes of Force.}
+\index{Electric displacement}%
+\index{Displacement@`Displacement', electric}%
+
+\Article{1} \Firstsc{The} influence which the notation and ideas of the fluid
+theory of electricity have ever since their introduction exerted
+over the science of Electricity and Magnetism, is a striking
+illustration of the benefits conferred upon this science by a
+concrete representation or `\emph{construibar vorstellung}' of the symbols,
+which in the Mathematical Theory of Electricity define
+the state of the electric field. Indeed the services which the
+old fluid theory has rendered to Electricity by providing a language
+in which the facts of the science can be clearly and
+briefly expressed can hardly be over-rated. A descriptive theory
+of this kind does more than serve as a vehicle for the clear expression
+of well-known results, it often renders important services
+by suggesting the possibility of the existence of new phenomena.
+
+The descriptive hypothesis, that of displacement in a dielectric,
+used by Maxwell to illustrate his mathematical theory, seems
+to have been found by many readers neither so simple nor so
+easy of comprehension as the old fluid theory; indeed this seems
+to have been one of the chief reasons why his views did not
+sooner meet with the general acceptance they have since received.
+As many students find the conception of `displacement' difficult,
+I venture to give an alternative method of regarding the processes
+occurring in the electric field, which I have often found
+useful and which is, from a mathematical point of view, equivalent
+to Maxwell's Theory.
+%% -----File: 016.png---Folio 2-------
+
+\Article{2} This method is based on the conception, introduced by
+Faraday, of tubes of electric force, or rather of electrostatic
+\index{Tubes of electric force}%
+\index{Closed@`Closed' Faraday tubes}%
+\index{Faraday, lines of force}%
+\index{Faraday, tubes@\subdashone tubes|(}%
+induction. Faraday, as is well known, used these tubes
+as the language in which to express the phenomena of the
+electric field. Thus it was by their tendency to contract, and
+the lateral repulsion which similar tubes exert on each other,
+that he explained the mechanical forces between electrified
+bodies, while the influence of the medium on these tubes was
+on his view indicated by the existence of specific inductive
+capacity in dielectrics. Although the language which Faraday
+used about lines of force leaves the impression that he usually
+regarded them as chains of polarized particles in the dielectric,
+yet there seem to be indications that he occasionally regarded
+them from another aspect; i.e.~as something having an existence
+apart from the molecules of the dielectric, though these were
+polarized by the tubes when they passed through the dielectric.
+Thus, for example, in §~1616 of the \textit{Experimental Researches} he
+seems to regard these tubes as stretching across a vacuum. It is
+this latter view of the tubes of electrostatic induction which we
+shall adopt, we shall regard them as having their seat in the
+ether, the polarization of the particles which accompanies their
+passage through a dielectric being a secondary phenomenon.
+We shall for the sake of brevity call such tubes Faraday
+Tubes.
+
+In addition to the tubes which stretch from positive to negative
+electricity, we suppose that there are, in the ether, multitudes
+of tubes of similar constitution but which form discrete closed
+curves instead of having free ends; we shall call such tubes
+`closed' tubes. The difference between the two kinds of tubes is
+similar to that between a vortex filament with its ends on the
+free surface of a liquid and one forming a closed vortex ring
+inside it. These closed tubes which are supposed to be present
+in the ether whether electric forces exist or not, impart a fibrous
+structure to the ether.
+
+In his theory of electric and magnetic phenomena Faraday
+made use of tubes of magnetic as well as of electrostatic
+induction, we shall find however that if we keep to the conception
+of tubes of electrostatic induction we can explain the
+phenomena of the magnetic field as due to the motion of such
+tubes.
+%% -----File: 017.png---Folio 3-------
+
+\Subsection{The Faraday Tubes.}
+
+\Article{3} As is explained in Art.~82 of Maxwell's \textit{Electricity and
+Magnetism}, these tubes start from places where there is positive
+and end at places where there is negative electricity, the
+quantity of positive electricity at the beginning of the tube
+being equal to that of the negative at the end. If we assume
+that the tubes in the field are all of the same strength,
+the quantity of free positive electricity on any surface will be
+proportional to the number of tubes leaving the surface. In the
+mathematical theory of electricity there is nothing to indicate
+that there is any limit to the extent to which a field of electric
+force can be subdivided up into tubes of continually diminishing
+strength, the case is however different if we regard these tubes
+of force as being no longer merely a form of mathematical expression,
+but as real physical quantities having definite sizes
+and shapes. If we take this view, we naturally regard the tubes
+as being all of the same strength, and we shall see reasons for
+believing that this strength is such that when they terminate on
+a conductor there is at the end of the tube a charge of negative
+electricity equal to that which in the theory of electrolysis we
+associate with an atom of a monovalent element such as chlorine.
+
+This strength of the unit tubes is adopted because the phenomena
+of electrolysis show that it is a natural unit, and that
+fractional parts of this unit do not exist, at any rate in electricity
+that has passed through an electrolyte. We shall assume
+in this chapter that in all electrical processes, and not merely in
+electrolysis, fractional parts of this unit do not exist.
+
+The Faraday tubes either form closed circuits or else begin
+and end on atoms, all tubes that are not closed being tubes that
+stretch in the ether along lines either straight or curved from
+one atom to another. When the length of the tube connecting
+two atoms is comparable with the distance between the
+atoms in a molecule, the atoms are said to be in chemical combination;
+when the tube connecting the atoms is very much
+longer than this, the atoms are said to be `chemically free'.
+
+The property of the Faraday tubes of always forming closed
+circuits or else having their ends on atoms may be illustrated
+by the similar property possessed by tubes of vortex motion in
+a frictionless fluid, these tubes either form closed circuits or
+%% -----File: 018.png---Folio 4-------
+have their ends on the boundary of the liquid in which the
+vortex motion takes place.
+
+The Faraday tubes may be supposed to be scattered throughout
+space, and not merely confined to places where there is a
+finite electromotive intensity, the absence of this intensity being
+due not to the absence of the Faraday tubes, but to the want of
+arrangement among such as are present: the electromotive intensity
+at any place being thus a measure, not of the whole
+number of tubes at that place, but of the excess of the number
+pointing in the direction of the electromotive intensity over the
+number of those pointing in the opposite direction.
+
+\Article{4} In this chapter we shall endeavour to show that the various
+phenomena of the electromagnetic field may all be interpreted
+as due to the motion of the Faraday tubes, or to changes in their
+position or shape. Thus, from our point of view, this method
+of looking at electrical phenomena may be regarded as forming
+a kind of molecular theory of Electricity, the Faraday tubes
+taking the place of the molecules in the Kinetic Theory of
+Gases: the object of the method being to explain the phenomena
+of the electric field as due to the motion of these tubes,
+just as it is the object of the Kinetic Theory of Gases to explain
+the properties of a gas as due to the motion of its molecules.
+
+These tubes also resemble the molecules of a gas in another respect,
+as we regard them as incapable of destruction or creation.
+
+\Article{5} It may be asked at the outset, why we have taken the tubes
+of electrostatic induction as our molecules, so to speak, rather
+than the tubes of magnetic induction? The answer to this question
+is, that the evidence afforded by the phenomena which accompany
+the passage of electricity through liquids and gases shows
+that molecular structure has an exceedingly close connection
+with tubes of electrostatic induction, much closer than we have
+any reason to believe it has with tubes of magnetic induction.
+The choice of the tubes of electrostatic induction as our molecules
+seems thus to be the one which affords us the greatest facilities
+for explaining those electrical phenomena in which matter as
+well as the ether is involved.
+
+\Article{6} Let us consider for a moment on this view the origin of
+the energy in the electrostatic and electromagnetic fields. We
+suppose that associated with the Faraday tubes there is a distribution
+of velocity of the ether both in the tubes themselves
+%% -----File: 019.png---Folio 5-------
+and in the space surrounding them. Thus we may have rotation
+in the ether inside and around the tubes even when the tubes
+themselves have no translatory velocity, the kinetic energy due
+to this motion constituting the potential energy of the electrostatic
+field: while when the tubes themselves are in motion we
+have super-added to this another distribution of velocity whose
+energy constitutes that of the magnetic field.
+
+The energy we have considered so far is in the ether, but when
+a tube falls on an atom it may modify the internal motion
+of the atom and thus affect its energy. Thus, in addition to
+the kinetic energy of the ether arising from the electric
+field, there may also be in the atoms some energy arising
+from the same cause and due to the alteration of the internal
+motion of the atoms produced by the incidence of the Faraday
+tubes. If the change in the energy of an atom produced by
+the incidence of a Faraday tube is different for atoms of different
+substances, if it is not the same, for example, for an atom of
+hydrogen as for one of chlorine, then the energy of a number
+of molecules of hydrochloric acid would depend upon whether
+the Faraday tubes started from the hydrogen and ended on the
+chlorine or vice versâ. Since the energy in the molecules thus
+depends upon the disposition of the tubes in the molecule, there
+will be a tendency to make all the tubes start from the hydrogen
+and end on the chlorine or vice versâ, according as the first or
+second of these arrangements makes the difference between the
+kinetic and potential energies a maximum. In other words,
+there will, in the language of the ordinary theory of electricity,
+be a tendency for all the atoms of hydrogen to be charged with
+electricity of one sign, while all the atoms of chlorine are charged
+with equal amounts of electricity of the opposite sign.
+
+The result of the different effects on the energy of the
+atom produced by the incidence of a Faraday tube will be the
+same as if the atoms of different substances attracted electricity
+with different degrees of intensity: this has been shown
+\index{Helmholtz va@Helmholtz, v.\ H., attraction of electricity by different substances}%
+by v.~Helmholtz to be sufficient to account for contact and frictional
+electricity. It also, as we shall see in \chapref{Chapter II.}{Chapter~II}, accounts
+for some of the effects observed when electricity passes from a
+gas to a metal or vice versâ.
+\index{Faraday, tubes@\subdashone tubes|)}%
+
+\Article{7} The Faraday tubes when they reach a conductor shrink to
+molecular dimensions. We shall consider the processes by which
+%% -----File: 020.png---Folio 6-------
+this is effected at the end of this chapter, and in the meantime
+proceed to discuss the effects produced by these tubes when
+moving through a dielectric.
+
+\Article{8} In order to be able to fix the state of the electric field at any
+point of a dielectric, we shall introduce a quantity which we shall
+call the `polarization' of the dielectric, and which while mathematically
+\index{Polarization}%
+\index{Displacement@`Displacement', electric}%
+\index{Electric displacement}%
+identical with Maxwell's `displacement' has a different
+physical interpretation. The `polarization' is defined as follows:
+Let $A$~and~$B$ be two neighbouring points in the dielectric, let a
+plane whose area is unity be drawn between these points and at
+right angles to the line joining them, then the polarization in
+the direction~$AB$ is the excess of the number of the Faraday
+tubes which pass through the unit area from the side~$A$ to the
+side~$B$ over those which pass through the same area from the
+side~$B$ to the side~$A$. In a dielectric other than air we imagine
+the unit area to be placed in a narrow crevasse cut out of the dielectric,
+the sides of the crevasse being perpendicular to~$AB$.
+The polarization is evidently a vector quantity and may be
+resolved into components in the same way as a force or a velocity;
+we shall denote the components parallel to the axes of
+$x$,~$y$,~$z$ by the letters $f$,~$g$,~$h$; these are mathematically identical
+with the quantities which Maxwell denotes by the same letters,
+their physical interpretation however is different.
+
+\Article{9} We shall now investigate the rate of change of the components
+of the polarization in a dielectric. Since the Faraday tubes
+in such a medium can neither be created nor destroyed, a change
+in the number passing through any fixed area must be due to the
+motion or deformation of the tubes. We shall suppose, in the
+first place, that the tubes at one place are all moving with the
+same velocity. Let $u$,~$v$,~$w$ be the components of the velocities
+of these tubes at any point, then the change in~$f$, the number
+of tubes passing at the point $x$,~$y$,~$z$, through unit area at right
+angles to the axis of~$x$, will be due to three causes. The first of
+these is the motion of the tubes from another part of the field
+up to the area under consideration; the second is the spreading
+out or concentration of the tubes due to their relative motion;
+and the third is the alteration in the direction of the tubes due
+to the same cause.
+
+Let $\delta_{1}f$ be the change in~$f$ due to the first cause, then in
+consequence of the motion of the tubes, the tubes which at the
+%% -----File: 021.png---Folio 7-------
+time $t + \delta t$ pass through the unit area will be those which at
+the time~$t$ were at the point
+\[
+x - u \delta t, \quad y - v \delta t, \quad z - w \delta t,
+\]
+hence $\delta _{1}f$ will be given by the equation
+\[
+\delta_{1} f = -  \left(u \frac{df}{dx} + v \frac{df}{dy} + w \frac{df}{dz} \right) \delta t.
+\]
+
+In consequence of the motion of the tubes relatively to one
+another, those which at the time~$t$ passed through unit area at
+right angles to~$x$ will at the time~$t + \delta t$ be spread over an area
+\[
+1 + \delta t \left\{\frac{dv}{dy} + \frac{dw}{dz} \right\};
+\]
+thus $\delta_{2}f$, the change in~$f$ due to this cause, will be given by
+the equation
+\begin{DPalign*}
+\delta_{2}f & = \frac{f}{1 + \delta t \left\{\dfrac{dv}{dy} + \dfrac{dw}{dz} \right\}} - f, \\
+\lintertext{or} \delta_{2}f & = -\delta t f \left\{\frac{dv}{dy} + \frac{dw}{dz} \right\}.
+\end{DPalign*}
+
+In consequence of the deflection of the tubes due to the relative
+motion of their parts some of those which at the time~$t$ were at
+right angles to the axis of~$x$ will at the time~$t + \delta t$ have a
+component along it. Thus, for example, the tubes which at the
+time~$t$ were parallel to~$y$ will after a time~$\delta t$ has elapsed be
+twisted towards the axis of~$x$ through an angle~$\delta t \dfrac{du}{dy}$, similarly
+those parallel to~$z$ will be twisted through an angle~$\delta t \dfrac{du}{dz}$
+towards the axis of~$x$ in the time $\delta t$; hence $\delta_{3}f$, the change in~$f$
+due to this cause, will be given by the equation
+\[
+\delta_{3}f = \delta t \left\{g \frac{du}{dy} + h \frac{du}{dz} \right\}.
+\]
+
+Hence if $\delta f$ is the total change in~$f$ in the time~$\delta t$, since
+\[
+\delta f = \delta_{1}f + \delta_{2}f + \delta_{3}f,
+\]
+we have
+\[
+\delta f = \left[ - \left(u \frac{df}{dx} + v \frac{df}{dy} + w \frac {df}{dz} \right) - f \left(\frac{dv}{dy} + \frac{dw}{dz} \right) + \left(g \frac{du}{dy} + h \frac{du}{dz} \right) \right]\delta t,
+\]
+which may be written as
+\[
+\frac{df}{dt} = \frac{d}{dy} (ug - vf) - \frac{d}{dz} (wf - uh) - u \left(\frac{df}{dx} + \frac{dg}{dy} + \frac{dh}{dz}\right). \Tag{1}
+\]
+%% -----File: 022.png---Folio 8-------
+
+If $\rho$ is the density of the free electricity, then since by the
+definition of \artref{8}{Art.~8} the surface integral of the normal polarization
+taken over any closed surface must be equal to the quantity of
+electricity inside that surface, it follows that
+\[
+\rho = \frac{df}{dx} + \frac{dg}{dy} + \frac{dh}{dz},
+\]
+hence equation~(\eqnref{9}{1}) may be written
+\begin{DPalign*}
+\lintertext{\raiseboxlint[0.75]{Similarly}}
+\left.
+\begin{aligned}
+\frac{df}{dt} + \PadTo{w}{u}\rho &= \frac{d}{dy} (ug - vf) - \frac{d}{dz} (wf - uh). \\[0.75\baselineskip]
+\frac{dg}{dt} + \PadTo{w}{v}\rho &= \frac{d}{dz} (vh - wg) - \frac{d}{dx} (ug - vf), \\
+\frac{dh}{dt} + w\rho &= \frac{d}{dx} (wf - uh) - \frac{d}{dy} (vh - wg).
+\end{aligned}\right\} \Tag{2}
+\end{DPalign*}
+
+If $p$,~$q$,~$r$ are the components of the current parallel to $x$,~$y$,~$z$
+respectively, $\alpha$,~$\beta$,~$\gamma$ the components of the magnetic force in the
+same directions, then we know
+\[
+\left.
+\begin{aligned}
+4\pi p & = \dfrac{d\gamma}{dy} - \dfrac{d\beta}{dz}, \\
+4\pi q & = \dfrac{d\alpha}{dz} - \dfrac{d\gamma}{dx}, \\
+4\pi r & = \dfrac{d\beta}{dx} - \dfrac{d\alpha}{dy}.
+\end{aligned} \right\} \Tag{3}
+\]
+
+Hence, if we regard the current as made up of the convection
+current whose components are $u\rho$,~$v\rho$,~$w\rho$ respectively, and the
+polarization current whose components are $\dfrac{df}{dt}$,~$\dfrac{dg}{dt}$,~$\dfrac{dh}{dt}$, we
+see by comparing equations (\eqnref{9}{2})~and~(\eqnref{9}{3}) that we may regard
+the moving Faraday tubes as giving rise to a magnetic force
+whose components $\alpha$,~$\beta$,~$\gamma$ are given by the equation
+\[
+\left.\begin{aligned}
+\alpha & = 4\pi (vh - wg), \\
+\beta & = 4\pi (wf - uh), \\
+\gamma & = 4\pi (ug - vf).
+\end{aligned} \right\} \Tag{4}
+\]
+
+Thus a Faraday tube when in motion produces a magnetic
+\index{Magnetic force due to the motion of Faraday tubes}%
+force at right angles both to itself and to its direction of motion,
+whose magnitude is proportional to the component of the velocity
+at right angles to the direction of the tube. The magnetic force
+%% -----File: 023.png---Folio 9-------
+and the rotation from the direction of motion to that of the tube
+at any point are related like translation and rotation in a
+right-handed screw.
+
+\Article{10} The motion of these tubes involves kinetic energy, and this
+kinetic energy is the energy of the magnetic field. Now if $\mu$~is the
+magnetic permeability we know that the energy per unit volume is
+\[
+\frac{\mu}{8\pi} (\alpha^2 + \beta^2 + \gamma^2),
+\]
+or substituting the values of~$\alpha$, $\beta$,~$\gamma$ from equations~(\eqnref{9}{4}),
+\[
+2\pi \mu [(hv-gw)^2 + (fw-hu)^2 + (gu-fv)^2].
+\]
+
+\index{Faradayx tubes, momentum of@\subdashtwo momentum of}%
+\index{Momentum of Faraday tubes}%
+The momentum per unit volume of the dielectric parallel to~$x$
+is the differential coefficient of this expression with regard to~$u$,
+hence if $U$,~$V$,~$W$ are the components of the momentum
+parallel to~$x$, $y$,~$z$, we have
+\begin{align*}
+U &= 4\pi \mu \{g(gu - fv) - h(fw - hu)\} \\
+  &= gc - hb,
+\end{align*}
+if $a$,~$b$,~$c$ are the components of the magnetic induction parallel
+to~$x$, $y$,~$z$.
+\begin{DPgather*}
+\lintertext{\raiseboxlint{\indent Similarly}} \left.\begin{aligned}
+V &= ha - fc, \\
+W &= fb - ga.
+\end{aligned}\right\} \Tag{5}
+\end{DPgather*}
+
+Thus the momentum per unit volume in the dielectric, which
+is due to the motion of the tubes, is at right angles to the
+polarization and to the magnetic induction, the magnitude of
+the momentum being equal to the product of the polarization
+and the component of the magnetic induction at right angles
+to it. We may regard each tube as having a momentum
+proportional to the intensity of the component of the magnetic
+induction at right angles to the direction of the tube. It is
+interesting to notice that the components of the momentum
+in the field as given by equations~(\eqnref{10}{5}) are proportional to the
+amounts of energy transferred in unit time across unit planes
+\index{Energy, transfer of}%
+\index{Poynting, transfer of energy in electric field}%
+at right angles to the axes of $x$,~$y$,~$z$ in Poynting's theory
+of the transfer of energy in the electromagnetic field (\textit{Phil.\
+\index{Transfer of energy}%
+Trans.}\ 1884, Part~II. p.~343); hence the direction in which the
+energy in Poynting's theory is supposed to move is the same
+as the direction of the momentum determined by the preceding
+investigation.
+%% -----File: 024.png---Folio 10-------
+
+\Article{11} The electromotive intensities parallel to $x$,~$y$,~$z$ due to the
+\index{Intensity, electromotive|(}%
+motion of the tubes are the differential coefficients of the kinetic
+energy with regard to $f$,~$g$,~$h$ respectively, hence we obtain
+the following expressions for $X$,~$Y$,~$Z$ the components of the
+electromotive intensity,
+\index{Electromotive intensity}%
+\[
+  \left.
+\begin{aligned}
+  X & = wb - vc, \\
+  Y & = uc - wa, \\
+  Z & = va - ub.
+\end{aligned}\right\} \Tag{6}
+\]
+
+Thus the direction of the electromotive intensity due to the
+motion of the tubes is at right angles both to the magnetic
+induction and to the direction of motion of the tubes.
+
+From equations~(\eqnref{11}{6}) we get
+\begin{multline*}
+\frac{dZ}{dy}-\frac{dY}{dz}=v\frac{da}{dy}+w\frac{da}{dz}-u\left(\frac{db}{dy} + \frac{dc}{dz}\right) \\
++a\left(\frac{dv}{dy}+\frac{dw}{dz}\right)-b\frac{du}{dy}-c\frac{du}{dz}.
+\end{multline*}
+
+But since the equation
+\[
+\frac{da}{dx} + \frac{db}{dy} + \frac{dc}{dz} = 0
+\]
+holds, as we shall subsequently show, on the view we have taken
+of the magnetic force as well as on the ordinary view, we have
+\[
+\frac{dZ}{dy} - \frac{dY}{dz} = u \frac{da}{dx} + v \frac{da}{dy} + w \frac{da}{dz} + a\left(\frac{dv}{dy} + \frac{dw}{dz}\right) - b \frac{du}{dy} - c \frac{du}{dz}.
+\]
+
+The right-hand side of this investigation is by the reasoning
+given in \artref{9}{Art.~9} equal to~$-\dfrac{da}{dt}$, the rate of diminution in the
+number of lines of magnetic induction passing through unit
+area at right angles to the axis of~$x$: hence we have
+\begin{DPgather*}
+\lintertext{Similarly} \left.\begin{aligned}
+  \frac{dZ}{dy} - \frac{dY}{dz} & = -\frac{da}{dt}. \\
+  \frac{dX}{dz} - \frac{dZ}{dx} & = -\frac{db}{dt}, \\
+  \frac{dY}{dx} - \frac{dX}{dy} & = -\frac{dc}{dt}.
+\end{aligned} \right\} \Tag{7}
+\end{DPgather*}
+
+Now by Stokes' theorem
+\index{Stokes, theorem}%
+\[
+\tint (X\, dx + Y\, dy + Z\, dz)
+\]
+%% -----File: 025.png---Folio 11-------
+taken round a closed circuit is equal to
+\[
+\iint \left\{l\left(\frac{dZ}{dy} - \frac{dY}{dz}\right) + m \left(\frac{dX}{dz} - \frac{dZ}{dx}\right) + n\left(\frac{dY}{dx} - \frac{dX}{dy}\right)\right\}\,dS,
+\]
+where $l$,~$m$,~$n$ are the direction-cosines of the normal to a surface~$S$
+which is entirely bounded by the closed circuit. Substituting
+the preceding values for~$dZ / dy - dY / dz$, \&c., we see that the line
+integral of the electromotive intensity round a closed circuit
+is equal to the rate of diminution in the number of lines of
+magnetic induction passing through the circuit. Hence the preceding
+view of the origin of magnetic force leads to Faraday's
+rule for the induction of currents by the alteration of the
+magnetic field.
+
+\Article{12} When the electromotive intensity is entirely due to the
+motion of the tubes in an isotropic medium whose specific inductive
+capacity is~$K$, we have
+\begin{align*}
+f & = \frac{K}{4\pi} X \\
+  & = \frac{K}{4\pi} \{wb - vc\},
+\end{align*}
+and since
+\begin{DPalign*}
+&b = 4\pi \mu \{fw - hu\} , \quad c = 4\pi \mu \{gu - fv\}, \\
+\lintertext{we have} &f = \mu K \{f(u^2 + v^2 + w^2) - u(fu + gv + hw)\}; \\
+\lintertext{similarly} &g = \mu K \{g(u^2 + v^2 + w^2) - v(fu + gv + hw)\}, \\
+&h = \mu K \{h(u^2 + v^2 + w^2) - w(fu + gv + hw)\}, \\
+\lintertext{hence} &fu + gv + hw = 0, \\
+\lintertext{\rlap{and therefore}}
+&\PadTo{h = \mu K \{h(u^2 + v^2 + w^2) - w(fu + gv + hw)\},}{u^2 + v^2 + w^2 = \frac{1}{\mu K}.}
+\end{DPalign*}
+
+\index{Faradayx tubes, velocity of@\subdashtwo velocity of}%
+\index{Velocity of Faraday tube}%
+Hence when the electromotive intensity is entirely due to the
+motion of the tubes, the tubes move at right angles to themselves
+with the velocity~$1/\sqrt{\mu K}$, which is the velocity with
+which light travels through the dielectric. In this case the
+momentum is parallel to the direction of motion, and the electromotive
+intensity is in the direction of the polarization. In this
+case the polarization, the direction of motion and the magnetic
+force, are mutually at right angles; their relative disposition is
+shown in \figureref{fig01}{Fig.~1}.
+
+Collecting the preceding results, we see that when a Faraday
+tube is in motion it is accompanied by (1)~a magnetic force
+%% -----File: 026.png---Folio 12-------
+right angles to the tube and to the direction in which it is
+moving, (2)~a momentum at right angles to the tube and to
+the magnetic induction, (3)~an electromotive intensity at right
+angles to the direction of motion and to the magnetic induction;
+this always tends to make the tube set itself at right angles
+to the direction in which it is moving. Thus in an isotropic
+medium in which there is no free electricity and consequently
+no electromotive intensities except those which arise from the
+motion of the tubes, the tubes set themselves at right angles
+to the direction of motion.
+
+\includegraphicsmid{fig01}{Fig.~1.}
+
+\Article{13} We have hitherto only considered the case when the tubes
+at any one place in a dielectric are moving with a common
+velocity. We can however without difficulty extend these results
+to the case when we have different sets of tubes moving
+with different velocities.
+
+Let us suppose that we have the tubes $f_{1}$,~$g_{1}$,~$h_{1}$, moving with a
+velocity whose components are $u_{1}$,~$v_{1}$,~$w_{1}$, while the tubes $f_{2}$,~$g_{2}$,~$h_{2}$
+move with the velocities $u_{2}$,~$v_{2}$,~$w_{2}$, and so on. Then the rate of
+increase in the number of tubes which pass through unit area at
+right angles to the axis of~$x$ is, by the same reasoning as before,
+\[
+\frac{d}{dy} \tsum (ug - vf) - \frac{d}{dz} \tsum (wf - uh) - \tsum (u\rho).
+\]
+%% -----File: 027.png---Folio 13-------
+
+Hence we see as before that the tubes may be regarded as
+\index{Magnetic force due to the motion of Faraday tubes}%
+producing a magnetic force whose components $\alpha$,~$\beta$,~$\gamma$ are
+given by the equations
+\[
+\left.\begin{aligned}
+\alpha & = 4\pi \tsum (vh - wg), \\
+\beta & = 4\pi \tsum (wf - uh), \\
+\gamma & = 4\pi \tsum (ug - vf).
+\end{aligned} \right\} \Tag{8}
+\]
+
+The Kinetic energy per unit volume,~$T$, due to the motion of
+\index{Kinetic energy, due to motion of Faraday tubes}%
+these tubes is given by the equation
+\[
+T = \frac{\mu}{8\pi} \{\alpha^2 + \beta^2 + \gamma^2\},
+\]
+or
+\[
+T = 2\pi \mu \left[ \left\{\tsum (vh - wg)\right\}^2
+                  + \left\{\tsum (wf - uh)\right\}^2
+                  + \left\{\tsum (ug - vf)\right\}^2\right].
+\]
+
+Thus $dT / du_{1}$, the momentum per unit volume parallel to~$x$ due
+to the tube with suffix~$1$, is equal to
+\begin{gather*}
+4\pi \mu \left\{g_{1} \tsum (ug - vf) - h_{1} \tsum (wf - uh)\right\}, \\
+= g_{1} c - h_{1} b,
+\end{gather*}
+where $a$,~$b$,~$c$ are the components of the magnetic induction.
+
+Thus $U$,~$V$,~$W$, the components of the momentum per unit
+volume parallel to the axes of $x$,~$y$,~$z$ respectively, are given
+by the equations
+\[
+\left. \begin{aligned}
+U &= c \tsum g - b \tsum h, \\
+V &= a \tsum h - c \tsum f, \\
+W &= b \tsum f - a \tsum g.
+\end{aligned} \right\} \Tag{9}
+\]
+
+Thus when we have a number of tubes moving about in the
+electric field the resultant momentum at any point is perpendicular
+both to the resultant magnetic induction and to the
+resultant polarization, and is equal to the product of these
+two quantities into the sine of the angle between them.
+
+\index{Electromotive intensity}%
+The electromotive intensities $X$,~$Y$,~$Z$ parallel to the axes of
+$x$,~$y$,~$z$ respectively are equal to the mean values of~$dT / df$,
+$dT / dg$, $dT / dh$, hence we have
+\[
+\left.\begin{aligned}
+X &= b\bar{w} - c\bar{v}, \\
+Y &= c\bar{u} - a\bar{w}, \\
+Z &= a\bar{v} - b\bar{u};
+\end{aligned}\right\} \Tag{10}
+\]
+where a bar placed over any quantity indicates that the mean
+value of that quantity is to be taken.
+\index{Intensity, electromotive|)}%
+%% -----File: 028.png---Folio 14-------
+
+Thus when a system of Faraday tubes is in motion, the
+electromotive intensity is at right angles both to the resultant
+magnetic induction and to the mean velocity of the tubes, and is
+equal in magnitude to the product of these two quantities into
+the sine of the angle between them.
+
+\index{Magnetic force due to the motion of Faraday tubes}%
+We see from the preceding equations that there may be a
+resultant magnetic force due to the motion of the positive
+tubes in one direction and the negative ones in the opposite,
+without either resultant momentum or electromotive intensity;
+for if there are as many positive as negative tubes passing
+through each unit area so that there is no resultant polarization,
+there will, by equations~(\eqnref{13}{9}), be no resultant momentum,
+while if the number of tubes moving in one direction is the
+same as the number moving in the opposite, equations~(\eqnref{13}{10}) show
+that there will be no resultant electromotive intensity due to
+the motion of the tubes. We thus see that when the magnetic
+field is steady the motion of the Faraday tubes in the
+field will be a kind of shearing of the positive past the
+negative tubes; the positive tubes moving in one direction
+and the negative at an equal rate in the opposite. When,
+however, the field is not in a steady state this ceases to be
+the case, and then the electromotive intensities due to induction
+are developed.
+
+\Subsection{Mechanical Forces in the Field.}
+\index{Current, mechanical force on conductor conveying@\subdashone mechanical force on conductor conveying}%
+\index{Mechanical force on current@\subdashone force on current}%
+
+\Article{14} The momentum parallel to~$x$ per unit volume of the
+medium, due to the motion of the Faraday tubes, is by equation~(\eqnref{13}{9})
+\[
+c\tsum g - b\tsum h;
+\]
+thus the momentum parallel to~$x$ which enters a portion of the
+medium bounded by the closed surface~$S$ in unit time is equal to
+\[
+\tiint \left[c\tsum g(lu + mv + nw) - b \tsum h(lu + mv + nw)\right] dS,
+\]
+where $dS$~is an element of the surface and $l$,~$m$,~$n$ the direction-cosines
+of its inwardly directed normal.
+
+If the surface~$S$ is so small that the external magnetic field
+may be regarded as constant over it, the expression may be
+written as
+\begin{DPgather*}
+c \tiint \tsum g(lu + mv + nw)\,dS - b \tiint \tsum h(lu + mv + nw)\,dS.\\[\abovedisplayskip]
+%% -----File: 029.png---Folio 15-------
+\lintertext{\rlap{\indent Now}} \tiint \tsum g(lu + mv + nw)\,dS,\\
+\lintertext{\rlap{and}}         \tiint \tsum h(lu + mv + nw)\,dS,
+\end{DPgather*}
+are the number of Faraday tubes parallel to $y$~and~$z$ respectively
+which enter the element in unit time, that is, they are the
+volume integrals of the components $q$~and~$r$ of the current
+parallel to $y$~and~$z$ respectively: if the medium surrounded
+by~$S$ is a dielectric this is a polarization current, if it is a conductor
+it is a conduction current. Thus the momentum parallel
+to~$x$ communicated in unit time to unit volume of the medium,
+\index{Force acting on a current}%
+in other words the force parallel to~$x$ acting on unit volume of
+the medium, is equal to
+\[
+cq - br;\hspace{0.9em}
+\]
+similarly the forces parallel to $y$~and~$z$ are respectively
+\begin{DPgather*}
+\left.    \begin{aligned}
+ar &- cp,\\
+bp &- aq. \end{aligned}
+\lintertext{\raiseboxlint[-0.5]{and}}
+\right\} \Tag{11}
+\end{DPgather*}
+
+When the medium is a conductor these are the ordinary
+expressions for the components of the force per unit volume of
+the conductor when it is carrying a current in a magnetic
+field.
+
+When, as in the above investigation, we regard the force on a
+conductor carrying a current as due to the communication to
+the conductor of the momentum of the Faraday tubes which enter
+the conductor, the origin of the force between two currents will
+be very much the same as that of the attraction between two
+\index{Sage, Le, theory of gravitation}%
+bodies on Le~Sage's theory of gravitation. Thus, for example, if
+we have two parallel currents $A$~and~$B$ flowing in the same
+direction, then if $A$~is to the left of~$B$ more tubes will enter~$A$ from
+the left than from the right, because some of those which would
+have come from the right if $B$~had been absent will be absorbed
+by~$B$, thus in unit time the momentum having the direction
+left to right which enters~$A$ will exceed that having the opposite
+direction; thus $A$~will tend to move towards the right, that
+is towards~$B$, while for a similar reason $B$~will tend to move
+towards~$A$.
+
+\Article{15} We have thus seen that the hypothesis of Faraday tubes
+in motion explains the properties and leads to the ordinary
+equations of the electromagnetic field. This hypothesis has the
+advantage of indicating very clearly why polarization and conduction
+%% -----File: 030.png---Folio 16-------
+currents produce similar mechanical and magnetic effects.
+For the mechanical effects and the magnetic forces at any
+point in the field are due to the motion of the Faraday tubes at
+that point, and any alteration in the polarization involves motion
+of these tubes just as much as does an ordinary conduction
+current.
+
+\Article{16} We shall now proceed to illustrate this method of regarding
+\index{Electrified sphere@\subdashone sphere, moving}%
+electrical phenomena by applying it to the consideration
+of some simple cases. We shall begin with the case which
+suggested the method; that of a charged sphere moving uniformly
+through the dielectric. Let us suppose the charge on
+the sphere is~$e$ and that it is moving with velocity~$w$ parallel
+to the axis of~$z$. Faraday tubes start from the sphere and are
+carried along with it as it moves through the dielectric; since
+these tubes are moving they will, as we have seen, produce a
+magnetic field. We shall suppose that the system has settled
+down into a steady state, so that the sphere and its tubes are
+all moving with the same velocity~$w$. Let $f$,~$g$,~$h$ be the components
+of the polarization at any point, $\alpha$,~$\beta$,~$\gamma$ those of the
+magnetic force. The expressions for $X$,~$Y$,~$Z$, the components
+of the electromotive intensity, will consist of two parts, one due
+to the motion of the Faraday tubes and given by equations~(\eqnref{11}{6}),
+the other due to the distribution of these tubes and derivable
+from a potential~$\Psi$; we thus have, if the magnetic permeability
+is unity,
+\[
+\left.\begin{aligned}
+X & = \phantom{-}w\beta - \frac{d\Psi}{dx}, \\
+Y & = -w\alpha - \frac{d\Psi}{dy}, \\
+Z & = -\frac{d\Psi}{dz}.
+\end{aligned} \right\} \Tag{12}
+\]
+By equations~(\eqnref{9}{4})
+\begin{align*}
+\alpha & = -4\pi gw, \\
+\beta & = \phantom{-}4\pi fw, \\
+\gamma & = \phantom{-}0.
+\end{align*}
+
+If $K$ is the specific inductive capacity of the medium, we have
+\[
+X = \frac{4\pi}{K} f, \quad Y = \frac{4\pi}{K} g, \quad Z = \frac{4\pi}{K} h.
+\]
+%% -----File: 031.png---Folio 17-------
+
+Since the magnetic permeability of the dielectric is taken as
+unity, we may put $1/K = V^2$, where $V$~is the velocity of light
+through the dielectric.
+
+{\allowdisplaybreaks
+Making these substitutions for the magnetic force and the
+electromotive intensity, equations~(\eqnref{16}{12}) become
+\begin{DPgather*}
+\begin{aligned}
+&4\pi f(V^2 - w^2) = - \frac{d\Psi}{dx},\\
+&4\pi g(V^2 - w^2) = - \frac{d\Psi}{dy},\\
+&4\pi h\PadTo[l]{(V^2 - w^2)}{V^2} = - \frac{d\Psi}{dz};
+\end{aligned} \\
+\lintertext{and since} \frac{df}{dx} + \frac{dg}{dy} + \frac{dh}{dz} = 0,\\
+\lintertext{we get}
+\frac{d^2\Psi}{dx^2} + \frac{d^2\Psi}{dy^2} + \frac{V^2 - w^2}{V^2}\, \frac{d^2\Psi}{dz^2} = 0, \Tag{13}\\
+\lintertext{or putting} z' = \frac{V}{\left\{V^2 - w^2\right\} ^{\frac{1}{2}}} z,
+\end{DPgather*}
+equation~(\eqnref{16}{13}) becomes
+\[
+\frac{d^2\Psi}{dx^2} + \frac{d^2\Psi}{dy^2} + \frac{d^2\Psi}{dz'^2} = 0,
+\]
+a solution of which is
+\begin{align*}
+\Psi &= \frac{A}{\left\{x^2+y^2+z'^2 \right\}^{\frac{1}{2}}} \\
+&= \frac{A}{\left\{{x^2+y^2+\dfrac{V^2}{V^2-w^2}z^2}\right\}^{\frac{1}{2}}}.\Tag{14}
+\end{align*}
+}
+
+To find $A$ we notice that the normal polarization over any
+sphere concentric with the moving one must equal~$e$, the
+charge on the sphere; hence if $a$~is the radius of the moving
+sphere,
+\[
+\iint \left\{\frac{x}{a} f + \frac{y}{a} g + \frac{z}{a} h \right\} dS = e.
+\]
+%% -----File: 032.png---Folio 18-------
+
+Substituting for $f$,~$g$,~$h$ their values, we find
+\begin{DPalign*}
+\frac{Aa}{4\pi (V^2 - w^2)} \iint \frac{dS}{\left\{x^2 + y^2 + \dfrac{V^2}{V^2 - w^2} z^2 \right\}^{\frac{3}{2}}} & = e, \\
+\lintertext{\rlap{\indent\indent or}} \frac{A}{2(V^2 - w^2)} \int_{0}^{\pi} \frac{\sin \theta\,d\theta}{\left\{\sin^2 \theta + \dfrac{V^2}{V^2 - w^2} \cos^2\theta\right\}^{\frac{3}{2}}} & = e.
+\end{DPalign*}
+
+{\allowdisplaybreaks
+The integral, if $V > w$, is equal to
+\begin{DPgather*}
+\frac{2 \{V^2 - w^2\}^{\frac{1}{2}}} {V}; \\
+\lintertext{\rlap{\indent\indent hence}} A = eV\{V^2 - w^2\}^{\frac{1}{2}}, \\
+\lintertext{\raiseboxlint[3.5]{so that}} \left.\begin{aligned}
+f & = \frac{e}{4\pi}\, \frac{V}{\{V^2 - w^2\}^{\frac{1}{2}}}\, \frac{x}{\left\{x^2 + y^2 + \dfrac{V^2}{V^2 - w^2} z^2\right\}^{\frac{3}{2}}}, \\
+g & = \frac{e}{4\pi}\, \frac{V}{\{V^2 - w^2\}^{\frac{1}{2}}}\, \frac{y}{\left\{x^2 + y^2 + \dfrac{V^2}{V^2 - w^2} z^2\right\}^{\frac{3}{2}}}, \\
+h & = \frac{e}{4\pi}\, \frac{V}{\{V^2 - w^2\}^{\frac{1}{2}}}\, \frac{z}{\left\{x^2 + y^2 + \dfrac{V^2}{V^2 - w^2} z^2\right\}^{\frac{3}{2}}}.
+\end{aligned}\right\}\Tag{15} \\
+\lintertext{\indent Thus} \frac{f}{x} = \frac{g}{y} = \frac{h}{z}.
+\end{DPgather*}
+}
+
+The Faraday tubes are radial and the resultant polarization
+varies inversely as
+\[
+r^2 \left\{1 + \frac{w^2}{V^2 - w^2} \cos^2\theta\right\}^{\frac{3}{2}},
+\]
+where $r$ is the distance of the point from the centre, and $\theta$~the
+angle which $r$~makes with the direction of motion of the sphere.
+We see from this result that the polarization is greatest where~$\theta = \pi / 2$,
+least where~$\theta = 0$; the Faraday tubes thus leave the
+poles of the sphere and tend to congregate at the equator. This
+arises from the tendency of these tubes to set themselves at right
+angles to the direction in which they are moving. The surface
+density of the electricity on the moving sphere varies inversely
+as
+\[
+\left\{1 + \frac{w^2}{V^2 - w^2} \cos^2 \theta\right\}^{\frac{3}{2}},
+\]
+%% -----File: 033.png---Folio 19-------
+it is thus a maximum at the equator and a minimum at the
+poles.
+
+The components $\alpha$,~$\beta$,~$\gamma$ of the magnetic force are given by
+the equations
+\[
+\left.\begin{aligned}
+\alpha &= -4\pi wg = - \frac{eVw}{\{V^2 - w^2\}^{\frac{1}{2}}}\, \frac{y}{\left\{x^2 + y^2 + \dfrac{V^2}{V^2 - w^2} z^2\right\}^{\frac{3}{2}}}, \\
+\beta &= \phantom{-}4\pi wf = \frac{eVw}{\{V^2 - w^2\}^{\frac{1}{2}}}\, \frac{x}{\left\{x^2 + y^2 + \dfrac{V^2}{V^2 - w^2} z^2\right\}^{\frac{3}{2}}}, \\
+\gamma &= 0.
+\end{aligned} \right\} \Tag{16}
+\]
+
+These expressions as well as~(\eqnref{16}{15}) were obtained by Mr.\
+Heaviside by another method in the \textit{Phil.\ Mag.}\ for April,
+1889.
+
+Thus the lines of magnetic force are circles with their centres
+in and their planes at right angles to the axis of~$z$. When $w$~is
+so small that $w^2 / V^2$ may be neglected, the preceding equations
+take the simpler forms
+\begin{gather*}
+f = \frac{e}{4\pi} \frac{x}{r^3}, \quad g = \frac{e}{4\pi} \frac{y}{r^3}, \quad h = \frac{e}{4\pi} \frac{z}{r^3}, \\
+\alpha = -\frac{ewy}{r^3}, \quad \beta = \frac{ewx}{r^3}.
+\end{gather*}
+
+(See J.~J. Thomson `On the Electric and Magnetic Effects
+produced by the Motion of Electrified Bodies', \textit{Phil.\ Mag.}\
+April, 1881.)
+
+\index{Electrified sphere, moving@\subdashtwo moving, magnetic force due to}%
+\index{Heaviside, moving electrified sphere}%
+\index{Magnetic xfield due to a moving charged sphere@\subdashone field due to a moving charged sphere}%
+\index{Sphere, charged moving, magnetic force due to}%
+The moving sphere thus produces the same magnetic field as an
+element of current at the centre of the sphere parallel to~$z$ whose
+moment is equal to~$ew$. When as a limiting case~$V = w$, that is
+when the sphere is moving with the velocity of light, we see
+from equations (\eqnref{16}{15})~and~(\eqnref{16}{16}) that the polarization and magnetic
+force vanish except when $z = 0$ when they are infinite. The
+equatorial plane is thus the seat of infinite magnetic force and
+polarization, while the rest of the field is absolutely devoid of
+either. It ought to be noticed that in this case all the Faraday
+tubes have arranged themselves so as to be at right angles to
+the direction in which they are moving.
+
+We shall now consider the momentum in the dielectric due
+to the motion of the Faraday tubes. Since the dielectric is
+%% -----File: 034.png---Folio 20-------
+\index{Electrified sphere, moving@\subdashtwo moving, momentum of}%
+\index{Momentum of xa moving electrified sphere@\subdashtwo a moving electrified sphere}%
+\index{Sphere, charged moving, momentum of@\subdashone charged moving, momentum of}%
+non-magnetic the components $U$,~$V'$,~$W$ of this are by equations~(\eqnref{13}{9})
+given by the following expressions:
+\[
+\left.\begin{aligned}
+U = \PadTo[l]{-\beta f - \alpha g}{-\beta h}
+  & = - \frac{e^{2}}{4\pi}\, \frac{V^{2}w}{V^{2} - w^{2}}\,  \frac{xz}{\left(x^{2} + y^{2} + \frac{V^{2}}{V^{2} - w^{2}} z^{2}\right)^{3}}, \\
+V' = \PadTo[l]{-\beta f - \alpha g}{\phantom{-}\alpha h}
+  & = - \frac{e^2}{4\pi}\, \frac{V^2w}{V^2 - w^2}\, \frac{yz}{\left(x^2 + y^2 + \frac{V^2}{V^2 - w^2} z^2\right)^3}, \\
+W = \PadTo[l]{-\beta f - \alpha g}{\phantom{-}\beta f - \alpha g}
+  & = \phantom{-}\frac{e^2}{4\pi}\, \frac{V^2w}{V^2 - w^2}\, \frac{(x^2 + y^2)}{\left(x^2 + y^2 + \frac{V^2}{V^2 - w^2} z^2\right)^3}.
+\end{aligned} \right\} \Tag{17}
+\]
+
+The resultant momentum at any point is thus at right angles
+to the radius and to the magnetic force; it is therefore in the
+plane through the radius and the direction of motion and at
+right angles to the former. The magnitude of the resultant
+momentum per unit volume at a point at a distance~$r$ from
+the centre of the sphere, and where the radius makes an angle~$\theta$
+with the direction of motion, is
+\[
+\frac{e^{2}w}{4\pi} · \frac{V^{2}}{V^{2} - w^{2}}\, \frac{1}{r^{4}}\, \frac{\sin \theta}{\left\{1 + \frac{w^{2}}{V^{2} - w^{2}} \cos^{2} \theta \right\}^{3}}.
+\]
+
+Thus the momentum vanishes along the line of motion of
+the sphere, where the Faraday tubes are moving parallel to
+themselves, and continually increases towards the equator as the
+tubes get to point more and more at right angles to their
+direction of motion.
+
+The resultant momentum in the whole of the dielectric is
+evidently parallel to the direction of motion; its magnitude~$I$
+is given by the equation
+\begin{align*}
+I &= \frac{e^{2}w}{4\pi}\, \frac{V^{2}}{V^{2} - w^{2}} \int_{a}^{\infty} \int_{0}^{\pi} \int_{0}^{2\pi} \frac{\sin^{2} \theta r^{2}\, dr \sin \theta\, d\theta\, d\phi}{r^{4}\left\{1 + \dfrac{w^{2}}{V^{2} - w^{2}} \cos^{2} \theta \right\}^{3}} \\
+&= \frac{e^{2}w}{a}\, \frac{V^2}{V^2 - w^2} \int_0^{1} \frac{\sin^2 \theta\, d(\cos \theta)}{\left\{1 + \dfrac{w^2}{V^2 - w^2} \cos^2 \theta \right\}^3},
+\end{align*}
+\begin{DPgather*}
+\lintertext{\rlap{or putting}} \frac{w}{\{V^2 - w^2\}^{\frac{1}{2}}} \cos \theta = \tan \psi,\\
+%% -----File: 035.png---Folio 21-------
+\intertext{we see that}
+I = \frac{e^2 V^2}{a\{V^2 - w^2\}^{\frac{1}{2}}}  \int_0^{\tan^{-1} \frac{w}{\{V^2 - w^2\}^{\frac{1}{2}}}} \cos^2 \psi \left(1 - \frac{V^2}{w^2} \sin^2 \psi\right) d\psi; \\
+\lintertext{\rlap{or if}} \tan^{-1} \frac{w}{\{V^2 - w^2\}^{\frac{1}{2}}} = \vartheta, \\
+I = \frac{e^2}{2a}\, \frac{V^2}{\{V^2 - w^2\}^{\frac{1}{2}}} \left\{\vartheta \left(1 - \tfrac{1}{4}\, \frac{V^2}{w^2}\right) + \tfrac{1}{2} \sin 2 \vartheta \left(1 + \tfrac{1}{4}\, \frac{V^2}{w^2} \cos 2 \vartheta\right)\right\}.
+\end{DPgather*}
+
+Thus the momentum of the sphere and dielectric parallel to~$z$
+is~$mw + I$, where $m$~is the mass of the sphere; so that the effect
+of the charge will be to increase the apparent mass of the sphere
+by~$I/w$ or by
+\[
+\tfrac{1}{2}\, \frac{e^2}{a}\, \frac{V^2}{w\{V^2 - w^2\}^{\frac{1}{2}}}
+\left\{\vartheta \left(1 - \tfrac{1}{4}\, \frac{V^2}{w^2}\right) +
+\tfrac{1}{2} \sin 2 \vartheta \left(1 + \tfrac{1}{4}\, \frac{V^2}{w^2} \cos 2 \vartheta\right) \right\}.
+\]
+
+When the velocity of the sphere is very small compared to
+that of light,
+\[
+\vartheta = \frac{w}{V} \left(1 + \tfrac{1}{6}\, \frac{w^2}{V^2}\right)
+\]
+approximately, and the apparent increase in the mass of the
+sphere is
+\[
+\frac{2}{3}\, \frac{e^2}{a}.
+\]
+
+When in the limit~$w = V$ the increase in mass is infinite, thus
+a charged sphere moving with the velocity of light behaves as
+if its mass were infinite, its velocity therefore will remain constant,
+in other words it is impossible to increase the velocity
+of a charged body moving through the dielectric beyond that
+of light.
+
+\index{Electrified sphere, moving@\subdashtwo moving, kinetic energy}%
+\index{Kinetic energy, due to moving charged sphere@\subdashtwo due to moving charged sphere}%
+\index{Sphere, charged moving, xkinetic energy of@\subdashone charged moving, kinetic energy of}%
+The kinetic energy per unit volume of the dielectric is
+\[
+\frac{1}{8\pi} (\alpha^2 + \beta^2),
+\]
+and hence by equations (\eqnref{16}{16})~and~(\eqnref{16}{17}) it is equal to
+\[
+\frac{w}{2}\, W;
+\]
+thus the total kinetic energy in the dielectric is equal to
+\[
+\tfrac{1}{2} wI,
+\]
+%% -----File: 036.png---Folio 22-------
+that is to
+\[
+\frac{e^2}{4a} w · \frac{V^2}{\{V^2 - w^2\}^{\frac{1}{2}}}
+\left\{\vartheta \left(1 - \tfrac{1}{4} \frac{V^2}{w^2}\right) + \tfrac{1}{2} \sin 2 \vartheta
+\left(1 + \tfrac{1}{4}\, \frac{V^2}{w^2} \cos 2 \vartheta\right)\right\}.
+\]
+
+\index{Electrified sphere, moving@\subdashtwo moving, force on in a magnetic field}%
+\index{Mechanical force on xa moving charged sphere@\subdashtwo a moving charged sphere}%
+\index{Sphere, charged moving, yforce acting on@\subdashone charged moving, force acting on}%
+We shall now proceed to investigate the mechanical forces
+acting on the sphere when it is moving parallel to the axis of~$z$
+in a uniform magnetic field in which the magnetic force is
+everywhere parallel to the axis of~$x$ and equal to~$H$.
+
+If $U$,~$V'$,~$W$ are the components of the momentum,
+\begin{align*}
+U & = gc - hb, \\
+V' & = ha - fc, \\
+W & = fb - ga.
+\end{align*}
+
+In this case
+\[
+c = 0, \qquad b = \beta, \qquad a = \alpha + H,
+\]
+where $\alpha$~and~$\beta$ have the values given in equations~(\eqnref{16}{16}).
+
+The momentum transmitted in unit time across the surface of
+a sphere concentric with the moving one has for components
+\[
+\iint wU \cos \theta\,dS, \qquad \iint wV' \cos \theta\,dS, \qquad \iint wW \cos \theta\,dS,
+\]
+the integration being extended over the surface of the sphere.
+Substituting the values of $U$,~$V'$,~$W$, we see that the first and
+third of these expressions vanish, while the second reduces to
+\begin{DPgather*}
+\frac{e}{4\pi}\, \frac{VHw}{\{V^2 - w^2\}^{\frac{1}{2}}} \iint \frac{\cos^2 \theta\,dS}{r^2\left\{1 + \dfrac{w^2}{V^2 - w^2} \cos^2 \theta\right\}^{\frac{3}{2}}}, \\
+\lintertext{or} \tfrac{1}{2}\, \frac{eVHw}{\{V^2 - w^2\}^{\frac{1}{2}}} \int_0^\pi
+\frac{\cos^2 \theta \sin \theta\,d\theta}{\left\{1 + \dfrac{w^2}{V^2 - w^2} \cos^2 \theta \right\}^{\frac{3}{2}}},
+\end{DPgather*}
+which is equal to
+\begin{DPgather*}
+-\frac{eHwV}{\{V^2 - w^2\}^{\frac{1}{2}}} \left\{\frac{(V^2 - w^2)^{\frac{3}{2}}}{Vw^2} - \left(\dfrac{V^2 - w^2}{w^2}\right)^{\frac{3}{2}}
+\log \left(\frac{V + w}{V - w}\right)^{\frac{1}{2}} \right\}, \\
+\lintertext{\rlap{or to}} -eH \frac{(V^2 - w^2)}{w} \left\{1 - \tfrac{1}{2} \frac{V}{w} \log \left(\frac{V + w}{V - w}\right)\right\}.
+\end{DPgather*}
+
+When $w/V$ is very small this expression reduces to
+\[
+\tfrac{1}{3} eHw.
+\]
+%% -----File: 037.png---Folio 23-------
+This is the rate at which momentum is communicated to
+the sphere, in other words it is the force on the sphere; hence
+the force on the charged sphere coincides in direction with the
+force on an element of current parallel to the axis of~$z$, but the
+magnitude of the force on the moving sphere is only one-third
+that of the force on an element of current along~$z$ whose moment
+is~$ew$. By the moment of an element of current we mean the
+product of the intensity of the current and the length of the
+element. When~$w=V$, that is when the sphere moves through
+the magnetic field with the velocity of light, we see from the
+preceding expression that the force acting upon it vanishes.
+
+We can get a general idea of the origin of the mechanical
+force on the moving sphere if we remember that the uniform
+magnetic field is (\artref{13}{Art.~13}) due to the motion of Faraday tubes, the
+positive tubes moving in one direction, the negative ones in the
+opposite, and that in their motion through the field these tubes
+have to traverse the sphere. The momentum due to these tubes
+when they enter the sphere is proportional to the magnetic
+force at the place where they enter the sphere, while their
+momentum when they leave the sphere is proportional to the
+magnetic force at the place of departure. Now the magnetic
+forces at these places will be different, because on one side of
+the sphere the magnetic force arising from its own motion will
+increase the original magnetic field, while on the other side it
+will diminish it. Thus by their passage across the sphere the
+tubes will have gained or lost a certain amount of momentum;
+this will have been taken from or given to the sphere, which
+will thus be subject to a mechanical force.
+
+\Subsection{Rotating Electrified Plates.}
+\index{Rotating electrified plates}%
+\index{Electrified plates, rotating|(}%
+\index{Himstedt, rotating disc}%
+\index{Hutchinson and Rowland, rotating electrified disc}%
+\index{Magnetic xfield due to rotating electrified plates@\subdashtwo due to rotating electrified plates}%
+\index{Plates, rotating electrified|(}%
+\index{Rontgen@Röntgen, rotating disc}%
+\index{Rowland, rotating disc}%
+\index{Rowlandx@Rowland and Hutchinson, rotating disc|(}%
+
+\sloppy
+\Article{17} The magnetic effects due to electrified bodies in motion
+are more conveniently examined experimentally by means of
+electrified rotating plates than by moving electrified spheres. The
+latter have, as far as I know, not been used in any experiments
+on electro-convection, while most interesting experiments with
+rotating plates have been made by Rowland (\textit{Berichte d.~Berl.\ Acad.}\
+1876, p.~211), Rowland and Hutchinson (\textit{Phil.\ Mag.}~27, p.~445,
+1889), Röntgen (\textit{Wied.\ Ann.}~35, p.~264, 1888; 40,~p.~93, 1890),
+Himstedt (\textit{Wied.\ Ann.}~38, p.~560, 1889). The general plan of
+%% -----File: 038.png---Folio 24-------
+these experiments is as follows: an air condenser with circular
+parallel plates is made to rotate about an axis through the
+centres of the plates and at right angles to their planes. To
+prevent induced currents being produced by the rotation of the
+plates in the earth's magnetic field, radial divisions filled with
+insulating material are made in the plates. When the plates
+are charged and set in rotation a magnetic field is found to
+exist in their neighbourhood similar to that which would be
+produced by electric currents flowing in concentric circular
+paths in the plates of the condenser, the centres of these circles
+being the points where the axis of rotation cuts the plates.
+
+\fussy
+Let us now consider how these magnetic forces are produced.
+Faraday tubes at right angles to the plates pass from one
+plate to the other. We shall suppose when the condenser is
+rotating as a rigid body these tubes move as if they were
+rigidly connected with it. Then, taking the axis of rotation as
+the axis of~$z$, the component velocities of a tube at a point whose
+coordinates are $x$,~$y$ are respectively $-\omega y$~and~$\omega x$, where $\omega$~is
+the angular velocity with which the plates are rotating.
+
+If these were the only Faraday tubes in motion the components
+$\alpha$,~$\beta$,~$\gamma$ of the magnetic force would by equations~(\eqnref{9}{4}) be
+given by the equations
+\begin{align*}
+\alpha &= 4\pi\sigma\omega x,\\
+\beta  &= 4\pi\sigma\omega y, \Tag{18} \\
+\gamma &= 0,
+\end{align*}
+where $\sigma$ ($=h$) is the surface-density of the electricity on either
+plate. These values for the components of the magnetic force
+do not however satisfy the relation
+\[
+\frac{d \alpha}{dx} + \frac{d \beta}{dy} + \frac{d \gamma}{dz} = 0,
+\]
+which must be satisfied since the value of
+\[
+\frac{1}{8\pi} \iiint (\alpha^2+\beta^2+\gamma^2)\,dx\,dy\,dz
+\]
+must, in a medium whose magnetic permeability is unity, be
+stationary for all values of $\alpha$,~$\beta$,~$\gamma$ which give assigned values to
+the currents, that is to
+\[
+\frac{d \beta}{dx} - \frac{d\alpha}{dy}, \qquad \frac{d \alpha}{dz} - \frac{d\gamma}{dx}, \qquad \frac{d \gamma}{dy} - \frac{d\beta}{dz}.
+\]
+%% -----File: 039.png---Folio 25-------
+
+For let $\alpha_0$,~$\beta_0$,~$\gamma_0$ be any particular values of the components of
+the magnetic force which satisfy the assigned conditions, then
+the most general values of these components are expressed by
+the equations
+\begin{align*}
+\alpha & = \alpha_0 + \dfrac{d\phi}{dx}, \\
+\beta & = \beta_0 + \dfrac{d\phi}{dy}, \\
+\gamma & = \gamma_0 + \dfrac{d\phi}{dz},
+\end{align*}
+where $\phi$~is an arbitrary function of $x$,~$y$,~$z$.
+
+Then if
+\[
+\tiiint (\alpha^2 + \beta^2 + \gamma^2)\,dx\,dy\,dz
+\]
+is stationary,
+\[
+\tiiint (\alpha\, \delta \alpha + \beta\, \delta \beta + \gamma\, \delta \gamma)\,dx\, dy\, dz = 0. \Tag{19}
+\]
+Let the variations in $\alpha$,~$\beta$,~$\gamma$ be due to the increment of~$\phi$ by
+an arbitrary function~$\delta \phi$, then
+\[
+\delta \alpha = \frac{d\,\delta \phi}{dx}, \qquad \delta \beta = \frac{d\,\delta \phi}{dy}, \qquad \delta \gamma = \frac{d\,\delta \phi}{dz}.
+\]
+
+Substituting these values for $\delta \alpha$,~$\delta \beta$,~$\delta \gamma$, and integrating by
+parts, equation~(\eqnref{17}{19}) becomes
+\begin{multline*}
+\iint \delta \phi(\alpha\,dy\,dz + \beta\,dz\,dx + \gamma\,dx\,dy) \\
+- \iiint \delta \phi \left\{\frac{d \alpha}{dx} + \frac{d \beta}{dy} + \frac{d \gamma}{dz} \right\} dx\,dy\,dz = 0,
+\end{multline*}
+and therefore since~$\delta \phi$ is arbitrary
+\[
+\frac{d\alpha}{dx} + \frac{d\beta}{dy} + \frac{d\gamma}{dz} = 0.
+\]
+
+The values of $\alpha$,~$\beta$,~$\gamma$ given by equation~(\eqnref{17}{18}) cannot therefore be
+the complete expressions for the magnetic force, and since we
+regard all magnetic force as due to the motion of Faraday tubes,
+it follows that the tubes which connect the positive to the negative
+charges on the plates of the condenser cannot be the only
+tubes in the field which are in motion; the motion of these
+tubes must set in motion the closed tubes which, \artref{2}{Art.~2}, exist
+in their neighbourhood. The motion of the closed tubes will
+produce a magnetic field in which the forces can be derived from
+%% -----File: 040.png---Folio 26-------
+a magnetic potential~$\Omega$. When we include the magnetic field
+due to the motion of these closed tubes, we have
+\begin{align*}
+\alpha &= 4\pi \sigma \omega x - \frac{d\Omega}{dx} = \frac{d\Omega'}{dx}, \\
+\beta  &= 4\pi \sigma \omega y - \frac{d\Omega}{dy} = \frac{d\Omega'}{dy}, \\
+\gamma &= \phantom{4\pi \sigma \omega y} -\frac{d\Omega}{dz} = \frac{d\Omega'}{dz},
+\end{align*}
+if $\Omega' = 2\pi \sigma \omega (x^2 + y^2) - \Omega$;
+\begin{DPalign*}
+\lintertext{and since} \frac{d\alpha}{dx} + \frac{d\beta}{dy} + \frac{d\gamma}{dz} = 0,& \\
+\lintertext{we have} \frac{d^2 \Omega'}{dx^2} + \frac{d^2 \Omega'}{dy^2} + \frac{d^2 \Omega'}{dz^2} = 0.&
+\end{DPalign*}
+
+The question now arises, does the motion of the tubes which
+connect the positive and negative electrifications on the plates
+only set those closed tubes in motion which are between the
+plates of the condenser, or does it affect the tubes outside as
+well? Let us examine the consequences of the first hypothesis.
+In this case, since the Faraday tubes outside the condenser are
+at rest, the magnetic force will vanish except between the
+plates of the condenser; it follows, however, from the properties
+of the magnetic potential that it must vanish inside as well, so that
+no magnetic force at all would be produced by the rotation of the
+plates. As this is contrary to the result of Rowland's experiments,
+the Faraday tubes stretching between the plates must by
+their rotation set in motion tubes extending far away from the
+region between the plates. The motion of these closed tubes
+must however be consistent with the condition that the magnetic
+force parallel to the plates due to the motion of the tubes must
+be continuous. Let us consider for a moment the radial magnetic
+force due to the closed tubes: this may arise either from the rotation
+round the axis of tubes which pass through the plates, or
+from the motion at right angles to the plates of tubes parallel to
+them. In the first case, the velocity tangential to the plates of
+the tubes must be continuous, otherwise the tubes would break,
+and since the tangential velocity is continuous, the radial magnetic
+force due to the motion of these tubes will be continuous also. In
+the second case, the product of the normal velocity of the tubes
+%% -----File: 041.png---Folio 27-------
+and their number per unit volume must be the same on the two
+sides of a plate, otherwise there would be an accumulation of
+these tubes in the plate. The product of the normal velocity
+into the number of the tubes is, however, equal to the tangential
+magnetic force due to the motion of the closed tubes, so that
+this must be continuous.
+
+The open tubes which stretch from the positive electricity on
+one plate to the negative on the other will, however, by their
+motion produce a discontinuity in the radial magnetic force,
+since these tubes stop at the plates, and do not pass through
+them. The radial magnetic force at a point due to these tubes
+is~$4\pi\sigma\omega r$, where $r$~is the distance of the point from the axis of rotation.
+The conditions to determine the magnetic field are thus,
+(1)~that except in the substance of the plates there must be a
+magnetic potential satisfying Laplace's equation, and (2)~that
+at either plate the discontinuity in the radial magnetic force
+must be~$4\pi\sigma\omega r$, where $\sigma$~is the surface-density of the electricity
+on the plates. These conditions are, however, exactly those which
+determine the magnetic force produced by a system of electric
+currents circulating in circles in the plates of the condenser, the
+intensity of the currents at a distance~$r$ from the axis of rotation
+being $\sigma\omega r$ for the positive and $-\sigma\omega r$ for the negative plate.
+Hence the magnetic force due to rotating the plates will be the
+same as that produced by this distribution of electric currents.
+
+This conclusion seems to be confirmed by the results of the
+experiments of Rowland and Hutchinson (\textit{Phil.\ Mag.}~27, p.~445,
+1889), as using this hypothesis they found a tolerably accurate
+value of~`$v$', the ratio of the electromagnetic to the
+electrostatic unit of electricity, by means of experiments on a
+rotating plate.
+
+We can see by similar reasoning that if only one of the plates
+is rotating, the other being at rest, the magnetic effect will be
+the same as that due to a system of electric currents circulating
+in the rotating plate, the intensity of the current at a distance~$r$
+from the axis being~$\sigma\omega r$.
+
+Some interesting experiments have been made by Röntgen
+\index{Hutchinson and Rowland, rotating electrified disc}%
+\index{Rontgen@Röntgen, rotating disc}%
+(\index{Magnetic xfield due to rotating electrified plates@\subdashtwo due to rotating electrified plates}%
+\textit{Wied.\ Ann.}\ 35, p.~264, 1888), in which, while the plates of the
+condenser were at rest, a glass disc parallel to the plates and
+situated between them was set in rapid rotation and was found
+to produce a magnetic field. The rotation of the disc must thus
+\index{Rowlandx@Rowland and Hutchinson, rotating disc|)}%
+%% -----File: 042.png---Folio 28-------
+have set in motion the Faraday tubes passing through it, and
+these in turn have affected closed tubes extending into the region
+beyond the condenser.
+
+Experiments of this kind seem to open up a field of enquiry
+which will throw light upon a question which at present is one
+of the most obscure in electricity: that of the relation between
+the velocities of the dielectric and of the Faraday tubes passing
+through it. This question is one of great importance in the
+Electro-magnetic Theory of Light, as but little progress can be
+\index{Aberration@\textsc{Aberration}}%
+made in the Theory of Aberration until we have got an answer
+to it. Another question which we have not touched upon, but
+which is very important in this connexion, is whether the
+motion of the Faraday tubes through ether devoid of matter
+would produce magnetic force, or whether for this purpose it
+is necessary that the tubes should pass across ordinary matter
+as well as ether. The point may be illustrated by the following
+case. Suppose we have a plate of glass between two parallel
+charged plates rigidly electrified, whether uniformly or otherwise,
+and that the whole system is set in rotation and moves like
+a rigid body, then, is or is not the motion of the system
+accompanied by magnetic force? Here the Faraday tubes move
+through the ether (assuming that the velocity of the ether is not
+the same as that of the glass), but do not move relatively to the
+glass.
+
+The motion of the tubes through the \emph{dielectric} will be requisite
+for the production of magnetic effects if we suppose that
+there are no closed tubes in the electric field, and that all the
+tubes connect portions of ordinary matter. The recognition of
+closed tubes in the ether seems to be desirable in the present
+state of our electrical knowledge, as unless we acknowledge the
+existence of such tubes we have to suppose that light being an
+electro-magnetic phenomenon cannot traverse a region wholly
+devoid of ordinary matter, and further that the existence of
+magnetic force depends upon the presence of such matter in the
+field.
+\index{Electrified plates, rotating|)}%
+\index{Plates, rotating electrified|)}%
+
+\Subsection{A Steady Magnetic Field.}
+\index{Faradayx tubes, disposition in a steady magnetic field@\subdashtwo disposition in a steady magnetic field}%
+\index{Magnetic xfield, steady@\subdashtwo steady}%
+
+\Article{18} Magnetic force on the theory we are now discussing is due
+to the motion of the Faraday tubes. When the magnetic field is
+variable the presence of these tubes is rendered evident by the
+%% -----File: 043.png---Folio 29-------
+existence of electromotive intensities in the field: when however
+the field is steady, we have no direct electrical evidence of the
+presence of these tubes, and their disposition and velocities have
+to be deduced from the equations developed in the preceding
+pages. We shall now proceed to examine this very important
+case more in detail.
+
+In a steady magnetic field in which there is no free electricity
+the Faraday tubes must be closed, exception being made of
+course of the short tubes which connect together the atoms in
+the molecules present in the field. Since in such a field there
+is no electromotive intensity, there must pass through each unit
+area of the field the same number of positive as of negative tubes,
+that is, there must be as many tubes pointing in one direction
+as the opposite. These tubes will (\artref{12}{Art.~12}) place themselves
+so as to be at right angles both to the direction in which they
+are moving and to the magnetic force. The distribution of the
+Faraday tubes and the directions in which they are moving
+cannot be determined solely from the magnetic force; but for
+the purpose of forming a clear conception of the way in
+which the magnetic force may be produced, we shall suppose
+that the positive tubes are moving with the velocity of light in
+one direction, the negative tubes with an equal velocity in
+the opposite, and that at any point the direction of a tube,
+its velocity, and the magnetic force are mutually at right
+angles.
+
+In a steady magnetic field surfaces of equal potential exist
+which cut the lines of magnetic force at right angles, so that
+since both the Faraday tubes and the directions in which they are
+moving are at right angles to the lines of magnetic force we may
+suppose that the Faraday tubes form closed curves on the equipotential
+surfaces, a tube always remaining on one equipotential
+surface and moving along it at right angles to itself.
+
+We shall now consider the motion of these tubes in a very
+simple magnetic field: that surrounding an infinitely long circular
+cylinder whose axis is taken as the axis of~$z$, and which is
+uniformly magnetized at right angles to its axis and parallel to
+the axis of~$x$.
+
+The magnetic potential inside the cylinder is equal to
+\[
+Hx,
+\]
+where $H$ is the magnetic force inside the cylinder.
+%% -----File: 044.png---Folio 30-------
+
+The potential outside the cylinder, if $a$~is the radius of the
+cylinder, is equal to
+\[
+H \frac{a^2\cos\theta}{r},
+\]
+where $r$ is the distance from the axis of the cylinder of the point
+at which the potential is reckoned, and $\theta$~the azimuth of~$r$
+measured from the direction of magnetization. Thus inside the
+cylinder the equipotential surfaces are planes at right angles to the
+direction of magnetization, while outside they are a system of circular
+cylinders which if prolonged would pass through the axis of
+the magnet; the axes of all these cylinders are parallel to the
+axis of~$z$ and lie in the plane of~$xz$. The cross sections of the
+original cylinder and the equipotential surfaces are represented
+in \figureref{fig02}{Fig.~2}.
+
+\includegraphicsmid{fig02}{Fig.~2.}
+
+We shall suppose that the Faraday tubes are parallel to the
+axis of the cylinder; then we may regard the magnetic field
+as produced by such tubes travelling round the equipotential
+%% -----File: 045.png---Folio 31-------
+surfaces with uniform velocity, the positive tubes moving in one
+direction, the negative ones in the opposite. We shall show
+that the number of tubes passing through the area bounded
+by unit length of the cross-section of any equipotential surface,
+the normals and the consecutive equipotential surface will be
+constant. For since the magnetic force is at right angles
+both to the Faraday tubes and the direction in which they
+are moving, the magnetic force due to this distribution of
+Faraday tubes will be at right angles to the equipotential
+surfaces; and if $N$~is the number of tubes of one sign between two
+consecutive equipotential surfaces per unit length of cross section
+of one of them, $ds$~the length of a portion of such a cross section,
+$d\nu$~the normal distance between two consecutive \DPtypo{equi-potential}{equipotential}
+surfaces $\Omega_1$~and~$\Omega_2$, then in the cylinder whose base is~$ds\,d\nu$
+the number of Faraday tubes of one sign will be $N\,ds(\Omega_2-\Omega_1)$; but
+since these tubes are distributed over an area~$ds\,d\nu$, the number
+of tubes per unit area of the base of the cylinder is $N(\Omega_2-\Omega_1)/d\nu$.
+These tubes are however all moving at the same rate, so
+that the magnetic force due to them will be proportional to
+the number per unit area of the base of the cylinder, that is to
+$N(\Omega_2-\Omega_1)/d\nu$, so that since the magnetic force due to these tubes
+is proportional to $(\Omega_2-\Omega_1)/d\nu$, $N$~will be constant. Thus the
+magnetic force due to the tubes moving in the way we have
+described coincides both in magnitude and direction with that
+due to the magnetized cylinder.
+
+We see from \figureref{fig02}{Fig.~2} that the directions of motion of these
+tubes change abruptly as they enter the magnetized cylinder.
+The principles by which the amount of this bending of the
+direction of motion of the tubes may be calculated are as follows.
+If $h_1$~and~$h_2$ are the densities of the tubes just inside and outside
+the cylinder, $R_1$,~$R_2$ the corresponding velocities of these tubes
+along the normal to the cylinder, then since there is no accumulation
+of the tubes at the surface of the cylinder we must have
+\[
+\tsum R_1h_1=\tsum R_2h_2.
+\]
+But since $R$ is the radial velocity, $\tsum 4\pi Rh$ is by~(\eqnref{13}{8}), \artref{13}{Art.~13}, the
+tangential magnetic force: hence the preceding equation expresses
+the continuity
+of the tangential magnetic force as we cross the
+surface of the cylinder.
+\includegraphicsouter{fig03}{Fig.~3.}
+Again, when a Faraday tube crosses the
+surface of the cylinder, the tangential component of its momentum
+will not change; but by equations~(\eqnref{13}{9}) the tangential momentum
+%% -----File: 046.png---Folio 32-------
+of the tube is proportional to the normal magnetic induction, so
+that the continuity of the tangential momentum is equivalent to
+that of the normal component of the magnetic induction. We
+have thus deduced from this view of the magnetic field the
+ordinary boundary conditions (1)~that the tangential component
+of the magnetic force is continuous, and (2)~that the normal
+component of the magnetic induction is continuous.
+
+The paths along which the tubes move coincide with the lines
+of flow produced by moving the cylinder uniformly at right
+angles to the direction of magnetization through an incompressible
+fluid.
+
+\Subsection{Induction of Currents due to Changes in the Magnetic
+Field.}
+\index{Induction balance, of currents due to changes in the magnetic field@\subdashone of currents due to changes in the magnetic field}%
+\index{Magnetic xfield, xinduction of current due to change of@\subdashtwo induction of current due to change of}%
+
+\Article{19} Let \figureref{fig03}{Fig.~3} represent a section of the magnetized cylinder
+and one of its equipotential surfaces, the directions of the magnetic
+force round the cylinder being denoted by the dotted lines.
+We shall call those Faraday tubes which point upwards from
+the plane of the paper positive,
+the negative Faraday tubes of
+course pointing downwards.
+The positive and the negative
+tubes circulate round the equipotential
+surface in the directions
+marked in the figure. Let
+\smallsanscap{A}~and~\smallsanscap{B} represent the cross-sections
+of the wires of a circuit,
+the wires being at right
+angles to the plane of the
+paper. When the magnetic
+field is steady no current will
+be produced in this circuit, because
+there are as many positive
+as negative tubes at any point
+in the field. Let us now suppose
+that the magnetic field
+is suddenly destroyed; we may imagine that this is done by
+placing barriers across the equipotential surfaces in the magnetized
+cylinder so as to stop the circulation of the Faraday
+tubes. The inertia of these tubes will for a short time carry
+%% -----File: 047.png---Folio 33-------
+them on in the direction in which they were moving when
+the barrier was interposed, hence the positive tubes will run
+out on the right-hand side of the equipotential surfaces and
+accumulate on the left-hand side, while the negative tubes
+will leave the left-hand side and accumulate on the right.
+The equality which formerly existed between the positive and
+negative tubes will now be destroyed: there will be an excess
+of positive tubes in the neighbourhood of the conductor~\smallsanscap{A},
+and an excess of negative ones round~\smallsanscap{B}. A current will
+therefore be started in the circuit running from~\smallsanscap{A} to~\smallsanscap{B}
+above the plane of the paper, and from~\smallsanscap{B} to~\smallsanscap{A} below it. We
+see in this way how the inertia of the Faraday tubes accounts
+for induced currents arising from variations in the
+intensity of the magnetic field.
+
+\Subsection{Induction due to Motion of the Circuit.}
+\index{Induction, of currents due to changes in the magnetic field, due to motion of the circuit@\subdashtwo due to motion of the circuit}%
+
+\Article{20} We can explain in a similar way the currents induced
+when a conductor is moved about in a magnetic field. Suppose
+we have a straight conductor moving about in the streams of
+Faraday tubes which constitute such a field, the Faraday tubes
+being parallel to each other and to the conductor: let the
+conductor be moved in the opposite direction to that in which
+the positive tubes are moving. This motion of the conductor
+will tend to stop the positive tubes in it and just in front of it;
+the inertia of the tubes further off will make them continue to
+move towards the conductor, and thus the density of the tubes in
+front (i.e.~those entering the conductor) will increase, while the
+density of the tubes behind (i.e.~those leaving the conductor)
+will diminish; the number of positive tubes in the conductor
+will thus be greater than the number which would have been
+present if the conductor had been at rest. Similar reasoning
+will show that there will be a decrease in the number of negative
+tubes in the conductor. Thus the positive tubes in the conductor
+will now outnumber the negative ones, and there will therefore
+be a positive current. The motion of the conductor in the
+direction opposite to that in which the positive Faraday tubes
+are moving will thus be accompanied by the production of a
+positive current. This current is the ordinary induction current
+due to the motion of a conductor in the magnetic field.
+%% -----File: 048.png---Folio 34-------
+
+\Subsection{Effect of the Introduction of Soft Iron into a Magnetic Field.}
+\index{Faradayx tubes, effect of soft iron on their motion@\subdashtwo effect of soft iron on their motion}%
+\index{Iron, effect of, on motion of Faraday tubes}%
+
+\includegraphicsouter{fig04}{Fig.~4.}
+
+\Article{21} Another simple magnetic system which we shall briefly
+consider is that of an infinite cylinder of soft iron, whose axis is
+taken as that of~$z$, placed in what was before its introduction a
+uniform magnetic field parallel to the axis of~$x$. Before the
+cylinder was introduced into the field, the Faraday tubes, which
+we may suppose to be parallel to the axis of~$z$, would all be
+moving parallel to the axis of~$y$; as soon however as the
+cylinder is placed in the field, the tubes will turn so as to avoid
+as much as possible going through it, for since the tangential
+momentum is not altered the tangential velocity of the tubes
+must be smaller inside the cylinder than it is outside, as
+the effective inertia of a tube in a magnetic medium is greater
+than in a non-magnetic one
+(see \artref{10}{Art.~10}). The lines of flow
+of the Faraday tubes will
+thus be deflected by the cylinder
+in much the same way
+as a current of electricity
+flowing through a conducting
+field would be deflected
+by the introduction into
+the field of a cylinder made
+of a worse conductor than
+itself. The Faraday tubes
+bend away from the cylinder
+in the way shown in
+\figureref{fig04}{Figure~4}. The paths of
+the Faraday tubes coincide
+however with equipotential
+surfaces; these surfaces
+therefore bend away from
+the cylinder, and the lines of magnetic force which are at
+right angles to the equipotential surface turn in consequence
+towards the cylinder as indicated in \figureref{fig04}{Fig.~4}, in which the dotted
+lines represent lines of magnetic force.
+%% -----File: 049.png---Folio 35-------
+
+\includegraphicsmid{fig05}{Fig.~5.}
+
+\Subsection{Permanent Magnets.}
+\index{Magnet, permanent}%
+\index{Permanent magnet}%
+
+\Article{22} In the interior of the magnet as well as in the surrounding
+magnetic field there is a shearing of the positive tubes past the
+negative ones. The magnet as it moves about carries this
+system of moving tubes with it, so that the motion of the
+tubes must in some way be maintained by a mechanism connected
+with the magnet: this mechanism exerts a fan-like action, driving
+the positive tubes in one direction, the negative ones in the
+opposite. This effect would be produced if the molecules of the
+magnet had the constitution described below and were in rapid
+rotation about the lines of magnetic force. Let the molecule~\smallsanscap{ABC}
+of a magnet consist of three atoms \smallsanscap{A},~\smallsanscap{B},~\smallsanscap{C}, \figureref{fig05}{Fig.~5}. Let
+one short tube go from~\smallsanscap{B} and end on~\smallsanscap{A}, another start from~\smallsanscap{B}
+and end on~\smallsanscap{C}, then if the molecule rotates in the direction of the
+arrow, about an axis through~\smallsanscap{B} perpendicular to the plane of the
+paper, since two like parallel Faraday tubes repel each other the
+rotation of the molecule will set the Faraday tubes in the ether
+surrounding the molecule in motion, the tubes going from left
+to right will move upwards in the plane of the paper, while
+those from right to left move downwards. This will produce
+a magnetic field in which, since the magnetic force is at right
+angles both to the moving tubes and the direction of motion, it
+will be at right angles to the plane of the paper and upwards;
+thus the magnetic force is parallel to the axis of rotation of the
+molecule. We notice that the atoms in the molecule are of
+different kinds with respect to the number of tubes incident
+upon them; thus \smallsanscap{B}~is the seat of two tubes, \smallsanscap{A}~and~\smallsanscap{C} of one
+each; in chemical language this would be expressed by saying
+that the valency of the atom~\smallsanscap{B} is twice that of either \smallsanscap{A}~or~\smallsanscap{C}.
+
+
+This illustration is only intended to call attention to the
+necessity for some mechanism to be connected with a permanent
+%% -----File: 050.png---Folio 36-------
+magnet to maintain the motion of the Faraday tubes in the field,
+and to point out that the motion of molecular tubes is able to
+furnish such a mechanism.
+
+\Subsection{Steady Current flowing along a Straight Wire.}
+\index{Current, motion of Faraday tubes in neighbourhood of steady@\subdashone motion of Faraday tubes in neighbourhood of steady}%
+\index{Faradayx tubes, round a wire carrying a steady current@\subdashtwo round a wire carrying a steady current}%
+
+\Article{23} We shall now proceed to express in terms of the Faraday
+tubes the phenomena produced by a steady current flowing along
+an infinitely long straight vertical wire. We shall suppose that
+the circumstances are such that there is no free electricity on
+the surface of the wire, so that the Faraday tubes in its neighbourhood
+are parallel to its length. If we take the direction of
+the current as the positive direction, the positive tubes parallel
+to the wire will be moving in radially to keep up the current,
+and this inward radial flow of positive tubes will be accompanied
+by an outward radial flow of negative tubes, a positive tube
+when entering the wire displacing a negative tube which moves
+outward from the wire. This shearing of the positive and
+negative tubes past each other will give rise to a magnetic force
+which will be at right angles both to the direction of the tubes and
+the direction in which they are moving; thus the magnetic force is
+tangential to a circle whose plane is horizontal and whose centre
+is on the axis of the wire. When the positive tubes enter the
+wire they shrink to molecular dimensions in the manner to be
+described in \artref{31}{Art.~31}. At a distance~$r$ from the axis of the
+wire let $N$~be the number of positive tubes passing through unit
+area of a plane at right angles to the wire, $v$~the velocity of
+these tubes inwards, let $N'$~be the number of negative tubes per
+unit area at the same point, $v'$~their velocity outwards. The
+algebraical sum of the number of tubes which cross the circle
+whose radius is~$r$ and whose centre is on the axis of the wire is
+thus
+\[
+(vN + v'N')\, 2\pi r.
+\]
+
+When the field is steady the value of this expression must be
+the same at all distances from the wire, because as many tubes
+must flow into any region as flow out of it. Hence when the
+field is steady this expression must equal the algebraical sum of
+the number of positive tubes which enter the wire in unit time;
+this number is however equal to~$i$, the current through the wire;
+hence we have
+\[
+(vN + v'N')\, 2\pi r = i.
+\]
+%% -----File: 051.png---Folio 37-------
+\index{Steady current, motion of Faraday tubes in neighbourhood of}%
+But by equations~(\eqnref{9}{4})
+\[
+vN + v'N' = \frac{\gamma }{4\pi},
+\]
+where $\gamma$ is the magnetic force at a distance~$r$ from the axis. Substituting
+this value for~$vN + v'N'$, we get
+\[
+\gamma = \frac{2i}{r},
+\]
+the usual expression for the magnetic force outside the wire
+produced by a straight current.
+
+When the field is steady, there will be as many positive as
+negative tubes in each unit area, and therefore no electromotive
+intensity; if however the intensity of the current changes, this will
+no longer hold. To take an extreme case, let us suppose that the
+circuit is suddenly broken, then the inertia of the positive tubes
+will make them continue to move inwards; and since as the
+circuit is broken they can no longer shrink to molecular dimensions
+when they enter it, the positive tubes will accumulate in
+the region surrounding the wire: the inertia of the negative tubes
+carries them out of this region, so that now there will be a preponderance
+of positive tubes in the field around the wire. If any
+conductor is in this field these positive tubes will give rise to a
+positive current, which is the `direct' induced current which
+occurs on breaking the circuit. When the field was steady
+no current would be produced in this secondary circuit, because
+there were as many positive as negative tubes in its
+neighbourhood.
+
+\index{Current, force between two parallel currents@\subdashone force between two parallel currents}%
+\index{Force between two parallel currents@\subdashone between two parallel currents}%
+The Faraday tubes have momentum which they give up when
+they enter the wire. If we consider a single wire where everything
+is symmetrical, the wire is bombarded by these tubes on all sides,
+so that there is no tendency to make it move off in any definite
+direction. Let us suppose, however, that we have \emph{two} parallel
+wires conveying currents in the same direction, let \smallsanscap{A}~and~\smallsanscap{B}
+denote the cross-sections of these wires, \smallsanscap{B}~being to the right of~\smallsanscap{A}.
+Then some of the tubes which if \smallsanscap{B}~were absent would pass
+into~\smallsanscap{A} from the region on the right, will when \smallsanscap{B}~is present be
+absorbed by it, and so prevented from entering~\smallsanscap{A}. The supply of
+positive tubes to~\smallsanscap{A} will thus no longer be symmetrical; more
+will now come into~\smallsanscap{A} from the region on its left than from that
+on its right; hence since each of the tubes has momentum, more
+%% -----File: 052.png---Folio 38-------
+momentum will come to~\smallsanscap{A} from the left than from the right;
+thus \smallsanscap{A}~will be pushed from left to right or towards~\smallsanscap{B}. There
+will thus be an attraction between the parallel currents.
+
+\Article{24} It will be noticed that the tubes in the preceding case
+\index{Polarization}%
+move radially in towards the wire, so that the energy which is
+converted into heat in the circuit comes from the dielectric sideways
+into the wire and is not transmitted longitudinally along
+it. This was first pointed out by Poynting in his paper on the
+Transfer of Energy in the Electromagnetic Field (\textit{Phil.\ Trans.}\
+1884, Part.~II. p.~343).
+
+\index{Alternating currents, motion of Faraday tubes round a wire carrying@\subdashtwo motion of Faraday tubes round a wire carrying}%
+\index{Faradayx tubes, round a wire carrying an alternating current@\subdashtwo round a wire carrying an alternating current}%
+When however the current instead of being constant is
+alternating very rapidly, the motion of the tubes in the dielectric
+is mainly longitudinal and not transversal. We shall show
+in \chapref{Chapter IV.}{Chapter~IV} that if $p$~is the frequency of the current, $\sigma$~the
+specific resistance of the wire, $a$~its radius, and $\mu$~its magnetic
+permeability, then when $4\pi \mu pa^2 / \sigma$ is a large quantity the electromotive
+intensity outside the wire is normal to the wire and
+therefore radial. Thus in this case the Faraday tubes will be
+radial, and they will move at right angles to themselves parallel
+to the wire. There is thus a great contrast between this case
+and the previous one in which the tubes are longitudinal and
+move radially, while in this the tubes are radial and move
+longitudinally.
+
+
+\Subsection{Discharge of a Leyden Jar.}
+\index{Faradayx tubes, motion of during the discharge of a Leyden jar@\subdashtwo motion of during the discharge of a Leyden jar}%
+\index{Leyden jar, motion of Faraday tubes during discharge of}%
+
+\includegraphicsouter{fig06}{Fig.~6.}
+
+\Article{25} We shall now proceed to consider the distribution and
+motion of the Faraday tubes during the discharge of a Leyden jar.
+We shall take the symmetrical
+case in which
+the outside coatings of
+two Leyden jars \smallsanscap{A}~and~\smallsanscap{B}
+(\figureref{fig06}{Fig.~6}) are connected
+by a wire, while the inside
+coating of~\smallsanscap{A} is connected
+to one terminal
+of an electrical machine,
+the inside coating of~\smallsanscap{B}
+to the other. When the
+electrical machine is in action the difference of potential between
+the inside coatings of the jars increases until a spark
+%% -----File: 053.png---Folio 39-------
+passes between the terminals of the machine and electrical
+oscillations are started in the jars.
+
+\includegraphicstwo[!t]{fig07}{Fig.~7.}{fig08}{Fig.~8.}
+\includegraphicstwo[!b]{fig09}{Fig.~9.}{fig10}{Fig.~10.}
+\includegraphicstwo[!t]{fig11}{Fig.~11.}{fig12}{Fig.~12.}
+
+Just before the passage of the spark the Faraday tubes will
+be arranged somewhat as follows. Some tubes will stretch from
+one terminal of the electrical machine to the other, others will go
+from these terminals to neighbouring conductors, such as the
+table on which the machine
+is placed, the floors
+and walls of the room. The
+great majority of the tubes
+will however be short
+tubes going through the
+glass from one coating to
+the other of the jars \smallsanscap{A}~and~\smallsanscap{B}.
+
+Let us consider the behaviour
+of two of these
+tubes, one from~\smallsanscap{A}, the
+other from~\smallsanscap{B}, when a
+spark passes between the
+terminals of the machine:
+while the spark is passing these terminals may be considered to
+be connected by a conductor.
+The tubes which before the spark passed stretched from one terminal
+of the machine to the other,
+will as soon as the air space
+breaks down shrink to
+molecular dimensions; and
+since the repulsion which
+these tubes exerted on
+those surrounding them
+is obliterated, the latter
+crowd into
+the space between
+the terminals. The short tubes which, before the spark
+passed, went from one coating of a jar to the other will now
+occupy some such positions as those shown in \figureref{fig07}{Fig.~7}. These
+tubes being of opposite kinds tend to run together, they approach
+each other until they meet as in \figureref{fig08}{Fig.~8}, the tubes now break up
+as in \figureref{fig09}{Fig.~9}, the upper portion runs into the spark gap where it
+%% -----File: 054.png---Folio 40-------
+contracts, while the lower portion runs towards the wire connecting
+the outside coatings of
+the jars, \figureref{fig10}{Fig.~10}. If this
+wire is a good conductor the
+tubes at their junction with
+the wire will be at right
+angles to it, and a tube
+will move somewhat as in
+\figureref{fig11}{Fig.~11}. The inertia of the
+tube will carry the two
+sides past each other, until
+the tubes are arranged as
+in \figureref{fig12}{Fig.~12}. The tube with
+its ends on the wire will
+travel backwards and will approach the positive tube which
+was emitted from the air
+gap when the negative
+tube (\figureref{fig09}{Fig.~9}) entered it.
+The tubes then go through
+the processes illustrated
+in Figs.\ \figureref{fig09}{9},~\figureref{fig08}{8},~\figureref{fig07}{7} in the reverse
+order, and the jars
+again get charged, but
+with electricity of opposite
+sign to that with
+which they started. After
+a time all the original
+Faraday tubes will be
+replaced by others of opposite sign, and the charges on the jars
+will be equal and opposite
+to the original charges.
+The new charge will then
+proceed to get reversed by
+similar processes to those
+by which the original
+charge was reversed, and
+thus the charges on the jar
+will oscillate from positive
+to negative and back again.
+
+
+\Article{26} When a conducting circuit is placed near the wire
+%% -----File: 055.png---Folio 41-------
+connecting the outer coatings of the jars, the Faraday tubes will
+strike against the circuit on their way to and from the wire.
+The passage of these tubes across the circuit will, since there is
+an excess of tubes of one name, produce a current in this circuit,
+which is the ordinary
+current in the secondary
+due to the variation of
+the intensity of the current
+in the primary circuit.
+
+\includegraphicsmid[!t]{fig13}{Fig.~13}
+
+Some of the tubes as
+they rush from the jar to
+the wire connecting the
+outside coatings of the jar
+strike against the secondary
+circuit, break up into
+two parts, as shown in \figureref{fig13}{Fig.~13}, the ends of these parts run
+along this circuit until they meet again, when the tube reunites
+and goes off as a single tube. The passage of the tube
+across the secondary circuit is thus equivalent to a current in
+the direction of rotation of the hands of a watch; this is
+opposite to that of the current in the wire connecting the
+outside coatings of the jars. The circuit by breaking up the
+%% -----File: 056.png---Folio 42-------
+tubes falling on it prevents them from moving across its interior,
+in other words, it tends to keep the number of lines of magnetic
+induction which pass through the circuit constant; this tendency
+gives the usual rule for finding the direction of the induced
+\index{Induction, of currents due to changes in the magnetic field, due to alternations in the primary circuit@\subdashtwo due to alternations in the primary circuit}%
+current. The introduction of magnetic force for the purpose of
+finding the currents in one circuit induced by alterations of the
+currents in another circuit seems however somewhat artificial.
+
+\Subsection{Electromagnetic Theory of Light.}
+\index{Light, electromagnetic theory of}%
+\index{Electromagnetic theory of light}%
+
+\Article{27} We can by the aid of the Faraday tubes form a mental
+picture of the processes which on the Electromagnetic Theory
+accompany the propagation of light. Let us consider in the
+first place the uninterrupted propagation of a plane wave emitted
+from a plane source. Let $z$~be the direction of propagation and
+let the wave be one of plane polarized light, the plane of polarization
+being that of~$yz$. Then we may suppose that a bundle of
+Faraday tubes parallel to~$x$ are emitted from the plane source,
+and that either these, or other parallel tubes set in motion
+by them, travel at right angles to themselves and parallel to
+the axis of~$z$ with the velocity of light. By the principles we
+have been considering these tubes produce in the region through
+which they are passing a magnetic force whose direction is at
+right angles both to the direction of the tubes and that in which
+they are moving, the magnetic force is thus parallel to the axis of~$y$.
+The magnitude of the magnetic force is by equations~(\eqnref{9}{4})
+equal to~$4\pi v$ times the polarization, where $v$~is the velocity of
+light, and since the electromotive intensity is~$4\pi / K$, or, if the
+medium is non-magnetic, $4\pi v^2$~times the polarization, we see
+that the electromotive intensity is equal to $v$~times the magnetic
+force. If there is no reflection the electromotive intensity and
+the magnetic force travel with uniform velocity~$v$ outwards
+from the plane of disturbance and always bear a constant ratio
+to each other. By supposing the number of tubes issuing from
+the plane source per unit time to vary harmonically we arrive
+at the conception of a divergent wave as a series of Faraday
+tubes travelling outwards with the velocity of light. In this
+case the places of maximum, zero and minimum electromotive
+intensity will correspond respectively to places of maximum, zero
+and minimum magnetic force.
+%% -----File: 057.png---Folio 43-------
+
+The case is different, however, when light is reflected from a
+metallic surface. We shall suppose this surface plane and at
+right angles to the axis of~$z$. In this case since the tangential
+electromotive intensity at the metallic surface vanishes, when a
+bundle of positive tubes enters the reflecting surface, an equal
+number of negative tubes are emitted from it; these travel backwards
+towards the source of light, moving in the opposite direction
+to the positive tubes. If we have a harmonic emission of tubes
+from the source of light we shall evidently also have a harmonic
+emission of tubes from the reflecting surface. Thus, at the
+various places in the path of the light, we may have positive
+tubes moving backwards or forwards accompanied by negative
+tubes moving in either direction. The magnetic effects of the
+positive tubes moving forwards are the same as those of the
+negative tubes moving backwards. Thus, when we have tubes
+of opposite signs moving in opposite directions, their magnetic
+effects conspire while their electromotive effects conflict; so that
+when, as in the case of reflection, we have streams of tubes
+moving in opposite directions the magnetic force will no longer
+be proportional to the electromotive intensity. In fact the
+places where the magnetic force is greatest will be places
+where the electromotive intensity vanishes, for such a place
+will evidently be one where we have the maximum density of
+positive tubes moving in one direction accompanied by the
+maximum density of negative tubes moving in the opposite,
+and since in this case there are as many positive as negative
+tubes the electromotive intensity will vanish. In a similar
+way we can see that the places where the electromotive intensity
+is a maximum will be places where the magnetic force
+vanishes.
+
+This view of the Electromagnetic Theory of Light has some of
+the characteristics of the Newtonian Emission Theory; it is not,
+however, open to the objections to which that theory was liable,
+as the things emitted are Faraday tubes, having definite positions
+at right angles to the direction of propagation of the light. With
+such a structure the light can be polarized, while this could not
+happen if the things emitted were small symmetrical particles as
+on the Newtonian Theory.
+
+\Article{28} Before proceeding to interpret the production of a current
+by a galvanic cell in terms of Faraday tubes it is necessary to
+%% -----File: 058.png---Folio 44-------
+consider a little more in detail the process by which these tubes
+contract when they enter a conductor.
+
+\Article{29} When a Faraday tube is not closed its ends are places
+where electrification exists, and therefore are always situated on
+matter. Now the laws of Electrolysis show that the number of
+Faraday tubes which can fall on an atom is limited; thus only one
+can fall on an atom of a monad element, two on that of a dyad,
+and so on. The atoms in the molecule of a compound which is
+chemically saturated are already connected by the appropriate
+number of tubes, so that no more tubes can fall on such atoms.
+Thus on this view the ends of a tube of finite length are on free
+atoms as distinct from molecules, the atoms in the molecule being
+connected by short tubes whose lengths are of the order of molecular
+distances. Thus, on this view, the existence of free electricity,
+whether on a metal, an electrolyte, or a gas, always requires the
+existence of free atoms. The production of electrification must
+be accompanied by chemical dissociation, the disappearance of
+electrification by chemical combination; in short, on this view,
+changes in electrification are always accompanied by chemical
+changes. This was long thought to be a peculiarity attaching to the
+passage of electricity through electrolytes, but there is strong evidence
+to show that it is also true when electricity passes through
+gases. Reasons for this conclusion will be given in \chapref{Chapter II.}{Chap.~II}, it
+will be sufficient here to mention one or two of the most striking
+instances, the details of which will be found in that chapter.
+
+\index{Perrot, decomposition of steam}%
+Perrot found that when the electric discharge passed through
+steam, oxygen came off in excess at the positive and hydrogen
+at the negative electrode, and that the excesses of oxygen at the
+positive and of hydrogen at the negative electrode were the
+same as the quantities of these gases set free in a water voltameter
+placed in series with the discharge through the steam.
+\index{Grove, chemical action of the discharge}%
+Grove found that when the discharge passed between a point
+and a silver plate through a mixture of hydrogen and oxygen,
+the plate was oxidised when it was the positive electrode, not
+when it was the negative. If the plate was oxidised to begin
+with, it was reduced by the hydrogen when it was the negative
+electrode, not when it was the positive. These and the other
+results mentioned in Chap.~II seem to point unmistakably to
+the conclusion that the passage of electricity through gases is
+necessarily attended by chemical decomposition.
+%% -----File: 059.png---Folio 45-------
+
+\Article{30} Although the evidence that the same is true when electricity
+passes through metals is not so direct, it must be borne
+in mind that here, from the nature of the case, such evidence is
+much more difficult to obtain; there are, however, reasons for
+believing that the passage of electricity through metals is accomplished
+by much the same means as through gases or electrolytes.
+We shall return in \artref{34}{Art.~34} to these reasons after considering the
+behaviour of the Faraday tubes when electricity is passing
+through an electrolyte, liquid or gaseous.
+
+\includegraphicsmid{fig14}{Fig.~14}
+
+\includegraphicsmid{fig15}{Fig.~15}
+
+\Article{31} To fix our ideas, let us take the case of a condenser discharging
+\index{Faradayx tubes, shortening of in a conductor@\subdashtwo shortening of in a conductor|(}%
+through the gas between its plates. Let us consider a
+Faraday tube which before discharge stretched from an atom~\smallsanscap{O}
+(\figureref{fig14}{Fig.~14}) on the positive plate to another atom~\smallsanscap{P} on the negative
+one. The molecules \smallsanscap{AB},~\smallsanscap{CD},~\smallsanscap{EF} of the intervening gas will be
+polarized by induction, and the Faraday tubes which connect
+the atoms in these molecules will point in the opposite direction
+to the long tube~\smallsanscap{OP}. The tube in the molecule~\smallsanscap{AB} will lengthen
+and bend towards the tube~\smallsanscap{OP} (which is supposed to pass near
+to~\smallsanscap{AB}) since these are of opposite signs, until when the field
+is sufficiently strong the tube in the molecule~\smallsanscap{AB} runs up
+into the long tube~\smallsanscap{OP} as in \figureref{fig15}{Fig.~15}. The long tube then breaks
+up into two tubes \smallsanscap{OA}~and~\smallsanscap{BP} as in \figureref{fig16}{Fig.~16}, and the tube~\smallsanscap{OA}
+shortens to molecular dimensions. The result of these operations
+is that the tube~\smallsanscap{OP} has contracted to the tube~\smallsanscap{BP}, and the
+atoms \smallsanscap{O}~and~\textsf{A} have formed a molecule. The process is then
+continued, until the tube~\smallsanscap{OP} has contracted into a tube of
+molecular dimensions at~\smallsanscap{P}.  The above explanation
+\includegraphicsmid{fig16}{Fig.~16}
+is only
+%% -----File: 060.png---Folio 46-------
+intended to represent the general nature of the processes by
+which the Faraday tubes shorten; we must modify it a little in
+order to explain the very great velocity of the discharge along
+the positive column (see Chap.~II, \artref{108}{Art.~108}). If the tubes
+shortened in the preceding manner, we see that the velocity of the
+ends of the tube would only be comparable with the velocity of
+translation of the molecules of the gas, but the experiments alluded
+to above show that it is enormously greater than this. A very
+slight modification of the above process will, however, while
+keeping the essential features of the discharge the same, give a
+much greater velocity of discharge. Instead of supposing that
+the tube~\smallsanscap{OP} jumps from one molecule to the next, we may suppose
+that, under the induction in the field, several of the molecules,
+say \smallsanscap{AB},~\smallsanscap{CD},~\smallsanscap{EF}, form a chain, and that the tubes in these
+molecules instead of being successively affected by the long tube
+and by each other are simultaneously affected, so that the tube~\smallsanscap{OP}
+instead of merely jumping from one molecule to the next,
+moves as in \figureref{fig17}{Fig.~17} from one end of the chain \smallsanscap{AB},~\smallsanscap{CD},~\smallsanscap{EF} to the
+%% -----File: 061.png---Folio 47-------
+other. In this case the long tube would shorten by the length of
+the chain in the same time as on the previous hypothesis it
+shortened by the distance between two molecules, so that on this
+view the velocity of discharge would be greater than that on the
+previous view in the proportion of the length of a chain to the
+distance between two molecules. We shall see in Chap.~II that
+there is considerable evidence that in the electric field chains
+of molecules are formed having a structure much more complex
+than that of the molecules recognized in the ordinary Kinetic
+Theory of Gases.
+\index{Faradayx tubes, shortening of in a conductor@\subdashtwo shortening of in a conductor|)}%
+
+\includegraphicsmid{fig17}{Fig.~17}
+
+\Article{32} We can easily express the resistance of a conductor in
+\index{Resistance of a conductor}%
+\index{Faradayx tubes, duration of in terms of resistance@\subdashtwo duration of in terms of resistance}%
+terms of the time the Faraday tubes take to disappear (i.e.~to
+contract to molecular dimensions). Let us for the sake of clearness
+take the case of a conducting wire, along which $E$~is the
+electromotive intensity at any point, while $K$~is the specific inductive
+capacity of the material of which the wire is made.
+Then the number of Faraday tubes passing through unit area
+of the cross-section of the wire is equal to
+\[
+\frac{K}{4 \pi}\, E.
+\]
+
+Let $T$ be the average life of a tube in the conductor, then
+the number of tubes which disappear from unit area in unit
+time is~$KE/4 \pi T$; and since the current~$c$ across unit area is
+equal to the number of tubes which disappear from unit area
+in unit time, we have
+\[
+c = \frac{KE}{4 \pi T}.
+\]
+
+If $\sigma$ be the specific resistance of the conductor measured in
+electromagnetic units
+\begin{DPalign*}
+E & = \sigma c, \\
+\lintertext{hence} \sigma & = \dfrac{4 \pi T}{K}, \\
+\lintertext{or} T & = \frac{K\sigma}{4 \pi}.
+\end{DPalign*}
+Hence $T$ has the same value as the quantity denoted by the
+same symbol in Maxwell's \textit{Electricity and Magnetism} (Art.~325).
+\index{Time of `relaxation'}%
+It is often called the time of relaxation of the medium.
+
+If $\{K\}$ be the value of~$K$ in electrostatic units,
+%% -----File: 062.png---Folio 48-------
+\begin{DPgather*}
+\lintertext{then since} \{K\} = \frac{K}{9×10^{20}}, \\
+\lintertext{we have} T = \frac{\{K\}}{4 \pi}\, \frac{\sigma}{9×10^{20}}.
+\end{DPgather*}
+
+The approximate values of $T/\{K\}$ for a few substances are
+given in the following table:---
+\begin{center}
+\begin{tabular}{l@{}l}
+ & \multicolumn{1}{c}{$T/\{K\}$} \\
+Silver \wdotfill & $1.5 × 10^{-19}$ \\
+Lead \wdotfill   & $1.8 × 10^{-18}$ \\
+Mercury \wdotfill & $8.7 × 10^{-18}$ \\
+Water with $8.3$ per cent of $H_2SO_4$\qquad\null & $3.1 × 10^{-13}$ \\
+Glass at $200°$\,C \wdotfill & $2\phantom{.1} × 10^{-6}$
+\end{tabular}
+\end{center}
+
+Since the values of~$\{K\}$ have not been determined for substances
+conducting anything like so well as those in the preceding
+\index{Arons and Cohn, specific inductive capacity of water}%
+\index{Cohn and Arons, specific inductive capacity}%
+list we cannot determine the value of~$T$. Cohn and Arons
+have found however that the specific inductive capacity of
+distilled water is about~$76$. Cohn and Arons (\textit{Wied.\ Ann.}~33,
+p.~13, 1888), and Cohn (\textit{Berl.\ Ber.}\ p.~1037, 1891) found that
+the specific inductive capacity of a weak solution differs very
+little from that of the solvent, though the difference in the
+specific resistance is very great. If we suppose that the~$K$
+for water mixed with sulphuric acid is the same as the~$K$
+for water, we should find~$T$ for this electrolyte about~$2 × 10^{-11}$,
+which is about ten thousand times as long as the time of vibration
+of sodium light; hence this electrolyte when exposed to
+electrical vibration of this period will behave as if $T$~were infinite
+or as if it were an insulator, and so will be transparent
+to electrical vibrations as rapid as those of light. We see too
+that if~$\{K\}$ for the metals were as great as~$\{K\}$ for distilled
+water, the values of~$T$ for these substances would not greatly
+exceed the time of vibrations of the rays in the visible spectrum:
+\index{Metals, opacity of}%
+\index{Opacity of metals}%
+this result explains Maxwell's observation, that the opacity of
+thin metallic films is much less than the value calculated on the
+electromagnetic theory, on the assumption that the conductivity
+of the metals for the very rapidly alternating currents which
+constitute light is as great as that for steady currents.
+
+\includegraphicsmid{fig18}{Fig.~18}
+
+\Subsection{Galvanic Cell.}
+\index{Galvanic cell}%
+
+\Article{33} The production of a current by a cell is the reverse process
+to the decomposition of an electrolyte by a current; in the latter
+%% -----File: 063.png---Folio 49-------
+case the chemical processes make a long Faraday tube shrink to
+molecular dimensions, in the former they produce a long tube from
+short molecular tubes. Let \smallsanscap{A}~and~\smallsanscap{B} (\figureref{fig18}{Fig.~18}) represent two
+metal plates immersed in an acid which combines chemically
+with~\smallsanscap{A}. Let~$a$ be a positive atom in the plate~\smallsanscap{A} connected by a
+Faraday tube with a negative atom~$b$, then if $a$~enters into
+chemical combination with a molecule~$cd$ of the acid, after the
+combination $a$~and~$c$ will be connected by a Faraday tube, as will
+also $b$~and~$d$: it will be seen from the second line in the figure that
+the length of the tube~$bd$ has been increased by the chemical action.
+If now $d$~enters into combination with another molecule~$ef$, the
+result of this will be still further to increase the length of the
+tube, and this length will increase as the chemical combination
+%% -----File: 064.png---Folio 50-------
+progresses through the acid. In this way a long tube is produced,
+starting from the metal at which the chemical change occurs.
+This tube will rush to the wire connecting the plates, there shrink
+to molecular dimensions, and produce a current through the wire.
+
+\Article{34} The connection between electric conduction and chemical
+\index{Conduction of electricity through metals and electrolytes}%
+\index{Electrolytes, conduction of electricity through}%
+\index{Metals, xconduction through@\subdashone conduction through}%
+change is much more evident in the cases of liquid electrolytes
+and gases than it is in that of metals. There does not
+seem, however, to be sufficient difference between the \emph{laws} of
+conduction through metals and electrolytes to make it necessary
+to seek an entirely different explanation for metallic conduction.
+The chief points in which metallic conduction differs from
+electrolytic are:---
+
+1. The much greater ease with which electricity passes through
+metals than through electrolytes.
+
+2. The difference of the effects of changes of temperature on the
+conductivity in the two cases. An increase of temperature
+generally diminishes the conductivity of a metal, while it increases
+that of an electrolyte.
+
+3. The appearance of the products of chemical decomposition
+at the electrodes when electricity passes through an electrolyte,
+and the existence of polarization, while neither of these effects
+has been observed in metallic conduction.
+
+With regard to the first of these differences, we may remark
+that though the conductivities of the best conducting metals are
+enormously greater than those of electrolytes, there does not seem
+to be any abrupt change in the values of the conductivities when
+we pass from cases where the conduction is manifestly electrolytic,
+as in fused lead or sodium chlorides, to cases where it is not
+recognised as being of this nature, as in tellurium or carbon.
+The following table, which contains the relative conductivities
+of a few typical substances, is sufficient to show this:---
+\begin{center}
+\begin{tabular}{l l}
+Silver                & $\phantom{8.}63$. \\
+Mercury               & $\phantom{8.0}1$. \\
+Gas Carbon            & $\phantom{8.}1×10^{-2}$. \\
+Tellurium             & $\phantom{8.}4×10^{-4}$. \\
+Fused Lead Chloride   & $\phantom{8.}2×10^{-4}$. \\
+Fused Sodium Chloride & $8.6×10^{-5}$.
+\end{tabular}
+\end{center}
+
+With regard to the second difference between metallic and
+electrolytic conduction, viz.~the effect of temperature on the
+%% -----File: 065.png---Folio 51-------
+\index{Temperature, effect of, on zconductivity@\subdashone effect of, on conductivity}%
+conductivity, though it is true that in most cases the effect of
+an increase in temperature is to diminish the conductivity in
+one case and increase it in the other, this is a rule which is by no
+means without exceptions. There are cases in which, though the
+conduction is not recognised as being electrolytic, the conductivity
+increases as the temperature increases. Carbon is a
+\index{Feussner, temperature coefficient of electrical resistance for alloys}%
+striking instance of this, and Feussner\footnote
+  {\textit{Zeitschrift f.~Instrumentenkunde},~9, p.~233, 1889.}
+has lately prepared alloys
+of manganese, copper and nickel whose conductivities show the
+\index{Sack, temperature coefficients of electrolytes}%
+same peculiarity. On the other hand, Sack (\textit{Wied.\ Ann.}~43, p.~212,
+1891) has lately shown that above $95°$\,C the conductivity of a
+$.5$~per~cent.\ solution of sulphate of copper decreases as the temperature
+increases, and in this respect resembles the conductivity
+of metals. These exceptions are sufficient to show that increase
+of conductivity with temperature is not a sufficient test to
+separate electrolytic from metallic conduction.
+
+With regard to the third and most important point---the
+appearance of the products of chemical decomposition at the
+electrodes---it is evident that we could not expect to get any
+evidence of this in the case of the elementary metals. The case
+\index{Roberts-Austen, conduction through alloys}%
+of alloys looks more hopeful. Roberts-Austen, however, who examined
+several alloys through which a powerful electric current
+had been passed, could not detect any difference in the composition
+of the alloy round the two electrodes. This result does not however
+seem conclusive against the conduction being electrolytic,
+for some alloys are little more than mixtures, while others behave
+as if they were solutions of one metal in another. In neither
+of these cases could we expect to find any separation of the
+constituents produced by the passage of the current; we could
+only expect to find this effect when the connection between the
+constituents was of such a nature that the whole alloy could be
+regarded as a chemical compound, in the molecule of which one
+metal could be regarded as the positive, the other as the negative
+element. The alloys investigated by Roberts-Austen do not
+seem to have been of this character.
+
+\index{Metals, opacity of}%
+One important respect in which metallic resembles electrolytic
+conduction is the way in which electrolytes and metals behave
+to the electrical vibrations which constitute light: an electrolyte,
+though a conductor for steady currents, behaves like an insulator
+to the rapidly alternating luminous electrical currents, and, as
+%% -----File: 066.png---Folio 52-------
+Maxwell's experiments on the transparency of their metallic
+films show, metals show an analogous effect, for their resistance
+for the light vibrations is enormously greater than their resistance
+to steady currents.
+
+The theory of Faraday tubes which we have been considering
+is, as far as we have taken it, geometrical rather than dynamical;
+we have not attempted any theory of the constitution
+of these tubes, though the analogies which exist between their
+properties and those of tubes of vortex motion irresistibly suggest
+that we should look to a rotatory motion in the ether for
+their explanation.
+
+Taking however these tubes for granted, they afford, I think,
+a convenient means of getting a vivid picture of the processes
+occurring in the electromagnetic field, and are especially suitable
+for expressing the relations which exist between chemical change
+and electrical action.
+%% -----File: 067.png---Folio 53------
+
+\Chapter{Chapter II.}{The Passage of Electricity Through Gases.}
+\index{Gases, passage of electricity through|indexetseq}%
+
+\Article{35} \Firstsc{The} importance which Maxwell attached to the study of
+the phenomena attending the passage of electricity through gases,
+as well as the fact that there is no summary in English text books
+of the very extensive literature on this subject, lead me to think
+that a short account of recent researches on this kind of electric
+discharge may not be out of place in this volume.
+
+
+\Subsection{Can the \emph{\textbf{Molecule}} of a Gas be charged with Electricity?}
+
+\Article{36} The fundamental question as to whether a body if charged
+to a low potential and surrounded by dust-free air at a low temperature
+will lose any of its charge, and the very closely
+connected one as to whether it is possible to communicate a
+charge of electricity to air \emph{in this condition}, have occasioned
+considerable divergence of opinion among physicists.
+
+Coulomb (\textit{Mémoires de~l'Académie des~Sciences}, 1785, p.~612),
+\index{Coulomb, leakage of electricity through air}%
+who investigated the loss of electricity from a charged body
+suspended by insulating strings, thought that after allowing for
+the leakage along the supports there was a balance over, which
+he accounted for by a convective discharge through the air; he
+supposed that the particles of air when they came in contact
+with a charged body received a charge of electricity of the same
+sign as that on the body, and that they were then repelled by it.
+On this view the molecules of air, just like small pieces of metal,
+can be charged with electricity.
+
+This theory of the loss of electricity from charged bodies has
+however not been confirmed by subsequent experiments, as
+\index{Nahrwold, leakage of electricity through air}%
+\index{Warburg, leakage of electricity through air}%
+Warburg (\textit{Pogg.\ Ann.}~145, p.~578, 1872) and Nahrwold (\textit{Wied.\
+Ann.}~31, p.~448, 1887) have shown that the loss can be accounted
+%% -----File: 068.png---Folio 54-------
+for by the presence of dust in the air surrounding the bodies;
+and that it is the particles of dust striking against the bodies
+which carry off their electricity, and not the molecules of air.
+
+\index{Dust figures, given off from electrified metals@\subdashone given off from electrified metals}%
+\index{Lenard and Wolf, dust given off under ultra-violet light}%
+\index{Wolf and Lenard, action of ultra-violet light}%
+This dust may either be present in the air originally, or it may
+consist of particles of metal given off from the charged conductors
+themselves, for, as Lenard and Wolf (\textit{Wied.\ Ann.}~37,
+p.~443, 1889) have shown, metals either free from electrification
+or charged with negative electricity give off metallic dust when
+exposed to ultra-violet light. When the metals are positively
+electrified no dust seems to be given off.
+
+The experiments of the physicists above mentioned point to
+the conclusion that the molecules of a gas at ordinary temperatures
+cannot receive a charge of electricity.
+
+\index{Blake, experiment with mercury vapour}%
+This view receives strong support from the results of Blake's
+experiments (\textit{Wied.\ Ann.}~19, p.~518, 1883), which have been
+\index{Sohncke, electrification by evaporation}%
+confirmed by Sohncke (\textit{Wied.\ Ann.}~34, p.~925, 1888), which show
+that not only is there no electricity produced by the evaporation
+of an unelectrified liquid, but that the vapour arising from an
+electrified liquid is not electrified. If the molecules of a vapour
+were capable of receiving a charge of electricity under any circumstances
+we should expect them to do so in this case. This
+experiment is a striking example of the way in which important
+researches may be overlooked, for, as the following extract from
+\index{Priestley@Priestley's \textit{History of Electricity}}%
+Priestley's \textit{History of Electricity}, p.~204, shows, Blake's experiment
+was made and the same result obtained more than one
+\index{Kinnersley, electrification by evaporation}%
+hundred years ago. `Mr.~Kinnersley of Philadelphia, in a letter
+dated March~1761, informs his friend and correspondent Dr.~Franklin,
+then in England, that he could not electrify anything
+by means of steam from electrified boiling water; from whence
+he concluded, that, contrary to what had been supposed by himself
+and his friend, steam was so far from rising electrified that
+it left its share of common electricity behind.'
+
+There does not seem to be any evidence that an electrified
+body can lose any of its charge by radiation through space
+without convection of electricity by charged particles.
+
+
+\Subsection{Hot Gases.}
+\index{Gases, passage of electricity through hot gases@\subdashone passage of electricity through hot gases|(}%
+\index{Hot gases, passage of electricity through|(}%
+
+\Article{37} It is only at moderate temperatures that a conductor
+charged to a low potential retains its charge when surrounded
+\index{Becquerel, conductivity of hot gases}%
+by a gas, for Becquerel (\textit{Annales de Chimie et de Physique}~[3]~39,
+%% -----File: 069.png---Folio 55-------
+p.~355, 1853) found that air at a white heat would allow
+electricity to pass through it even though the potential difference
+was only a few volts. This result has been confirmed by
+\index{Blondlot, conductivity of hot gases}%
+Blondlot (\textit{Comptes Rendus},~104, p.~283, 1887), who found that
+air at a bright red heat was unable to insulate under potential
+differences as low as $1/1000$ of a volt. He found, too, that the
+conduction through the hot gas did not obey Ohm's law.
+
+From some experiments of my own (\textit{Phil.\ Mag.}\ [5]~29,
+pp.~358,~441, 1890) I have come to the conclusion that hot gases
+conduct electricity with very different degrees of facility. Gases
+such as air, nitrogen, or hydrogen which do not experience
+any chemical change when heated conduct electricity only to
+a very small extent when hot, and in this case the conduction,
+as Blondlot supposed, appears to be convective. Gases, however,
+which dissociate at high temperatures, that is gases such as
+iodine, hydriodic acid gas,~\&c., whose molecules split up into
+atoms, conduct with very much greater facility, and the conduction
+does not exhibit that dependence on the material of
+which the electrodes are made which is found when the electricity
+is transmitted by convection.
+
+A large number of gases were examined, and in every case
+where the hot gas possessed any considerable conductivity, I
+was able to detect by purely chemical means that chemical
+decomposition had been produced by the heat. In this connection
+it is necessary to distinguish between two classes of dissociation.
+The first kind is when the molecule is split up into
+atoms, as in iodine, hydriodic acid gas, hydrochloric acid gas
+(when the chlorine, though not the hydrogen, remains partly
+dissociated), and so on. In all cases when dissociation of this
+kind exists, the gas is a good conductor when hot. The second
+kind of dissociation consists in the splitting up of the molecules
+of the gas into simpler molecules but not into atoms.
+This kind of dissociation occurs when a molecule of ammonia
+splits up into molecules of nitrogen and hydrogen, or when a
+molecule of steam splits up into molecules of hydrogen and
+oxygen. In this case the gases only conduct on the very much
+lower scale of the non-dissociable gases.
+
+The first of the following lists contains those gases which only
+conduct badly when heated, the second those which conduct
+comparatively well: chemical analysis showed that all the gases
+%% -----File: 070.png---Folio 56-------
+in the second list were decomposed when they were hot enough
+to conduct electricity:---
+
+(1) Air, Nitrogen, Carbonic Acid, Steam, Ammonia, Sulphuric
+Acid gas, Nitric Acid gas, Sulphur (in an atmosphere of nitrogen),
+Sulphuretted Hydrogen (in an atmosphere of nitrogen).
+
+(2) Iodine, Bromine, Chlorine, Hydriodic Acid gas, Hydrobromic
+Acid gas, Hydrochloric Acid gas, Potassium Iodide,
+Sal-Ammoniac, Sodium Chloride, Potassium Chloride.
+
+The conductivities of the two classes of gases differ so greatly,
+both in amount and in the laws they obey, that the mechanism
+by which the discharge is effected is probably different in the
+two cases.
+
+These experiments seem to show that when electricity passes
+through a gas otherwise than by convection, free atoms, or
+something chemically equivalent to them, must be present. It
+should be noticed that on this view the molecules even of a hot
+gas do not get charged, it is the \emph{atoms} and not the molecules
+which are instrumental in carrying the discharge.
+
+\index{Metallic vapours, conductivity of}%
+I also examined the conductivities of several metallic vapours,
+including those of Sodium, Potassium, Thallium, Cadmium,
+Bismuth, Lead, Aluminium, Magnesium, Tin, Zinc, Silver, and
+Mercury. Of these the vapours of Tin, Mercury, and Thallium
+hardly seemed to conduct at all, the vapours of the other metals
+conducted well, their conductivities being comparable with
+those of the dissociable gases.
+
+The small amount of conductivity which hot gases, which
+are not decomposed by heat, possess, seems to be due to a convective
+discharge carried perhaps by dust produced by the decomposition
+of the electrodes: in some cases perhaps the electricity
+may be carried by atoms produced by the chemical
+action of the electrodes on the adjacent gas.
+
+The temperature of the electrodes seems to exert great influence
+upon the passage of the electricity through the gas into which
+the electrodes dip. In the experiments described above I found
+it impossible to get electricity to pass through the gas, however
+hot it might be, unless the electrodes were hot enough to glow.
+A current passing through a hot gas was immediately stopped by
+placing a large piece of cold platinum foil between the electrodes---though
+a strong up-current of the hot gas was maintained to
+prevent the gas getting chilled by the cold foil. As soon as the
+%% -----File: 071.png---Folio 57-------
+foil began to glow, the passage of the electricity through the
+gas was re-established.
+
+This is one among the many instances we shall meet with in
+this chapter of the difficulty which electricity has in passing
+from a gas to a cold metal.
+\index{Gases, passage of electricity through hot gases@\subdashone passage of electricity through hot gases|)}%
+\index{Hot gases, passage of electricity through|)}%
+
+\nbpagebreak
+\Subsection{Electric Properties of Flames.}
+\index{Flames, electrical properties of}%
+
+\Article{38} The case in which the passage of electricity through hot
+gases has been most studied is that of flames; here the conditions
+are far from simple, and the results that have been
+obtained are too numerous and intricate for us to do more than
+mention their main features. A full account of the experiments
+\index{Wiedemann@Wiedemann's \textit{Elektricität}}%
+\index{Giese, electrical properties of flames}%
+which have been made on this subject will be found in Wiedemann's
+\textit{Lehre von~der Elektricität}, vol.~4,~B\footnote
+  {See also Giese, \textit{Wied.\ Ann.}~38, p.~403, 1889.}.
+
+A flame such as the oxy-hydrogen flame conducts electricity,
+the hotter parts conducting better than the colder: the conductivity
+of the flame is improved by putting volatile salts
+into it, and the increase in the conductivity is greater when the
+salts are placed near the negative electrode than when they
+\index{Arrhenius, conductivity of flames}%
+are placed near the positive\footnotemark.
+  \footnotetext{For an investigation on the effect of putting volatile salts in flames published
+  subsequently to Wiedemann's work, see Arrhenius (\emph{Wied. Ann.}~42, p.~18, 1891).}
+
+The conduction through the flame exhibits polar properties, for
+if the electrodes are of different sizes the flame conducts better
+when the larger electrode is negative than when it is positive.
+
+If wires made of different metals are connected together
+and dipped into the flame, there will be an electromotive force
+round the circuit formed by the flame and the wire; the
+flame apparently behaving in much the same way as the acid
+in a one-fluid battery; the electromotive force in some cases
+amounts to between three and four volts.
+
+A current can also be obtained through a bent piece of wire
+if the ends of the wire are placed in different parts of the flame.
+
+
+\Subsection{Escape of Electricity from a Conductor at Low Potential
+surrounded by Cold Gas.}
+
+\Article{39} Though it seems to be a well-established fact that a
+conductor at a low potential, surrounded by cold air, may retain
+its charge for an indefinitely long time, recent researches have
+%% -----File: 072.png---Folio 58-------
+shown that when the conductor is exposed to certain influences
+leakage of the electricity may ensue.
+
+One of the most striking of these influences is that of ultra-violet
+light. The effect of ultra-violet light on the electric
+\index{Light, xeffect of ultra@\subdashone effect of ultra-violet on electric discharge}%
+\index{Ultra-violet light, effect of, on electric discharge}%
+\index{Hertz, effect of ultra-violet light on the discharge}%
+discharge seems first to have been noticed by Hertz (\textit{Wied.\
+Ann.}\ 31, p.~983, 1887), who found that the disruptive discharge
+between two conductors is facilitated by exposing the air
+space, across which the discharge takes place, to the influence
+of ultra-violet light.
+
+\index{Ebert and E. Wiedemann, effect of ultra-violet light}%
+\index{Wiedemann, E., and Ebert, effect of ultra-violet light}%
+E.~Wiedemann and Ebert (\textit{Wied.\ Ann.}\ 33, p.~241, 1888) subsequently
+proved that the seat of this action is at the cathode;
+they showed that the light produces no effect when the cathode
+is shielded from its influence, however brightly the rest of the
+line of discharge may be illuminated.
+
+They found that if the cathode is surrounded by air the
+effect of the ultra-violet light is greatest when the pressure is
+about $300$~mm.\ of mercury: when the pressure is so low that
+the negative rays (see \artref{108}{Art.~108}) are visible, the effect of the
+ultra-violet light is not at all well marked.
+
+They found also that the magnitude of the effects depends
+upon the gas surrounding the cathode; they tried the effect
+of immersing the cathode in carbonic acid, hydrogen and air,
+and found that for these three gases the effect is greatest in
+carbonic acid, least in air. In carbonic acid the effect is not
+confined to ultra-violet light, as the luminous rays when they
+fall on a cathode also facilitate the discharge.
+
+Great light was thrown on the nature of this effect by an
+\index{Light, xeffect of ultra@\subdashone effect of ultra-violet on electrified metals}%
+\index{Lenard and Wolf, dust given off under ultra-violet light}%
+\index{Wolf and Lenard, action of ultra-violet light}%
+investigation made by Lenard and Wolf (\textit{Wied.\ Ann.}\ 37, p.~443,
+1889), in which it was proved that when ultra-violet light falls
+on a negatively electrified platinum surface, a steam jet in the
+neighbourhood of the surface shows by its change of colour
+that the steam in it has been condensed. This condensation
+always occurs when the negatively electrified surface on
+which the light falls is metallic, or that of a phosphorescent
+liquid, such as a solution of fuchsin or methyl violet. They
+found also that some, but much smaller, effects are produced
+when the surfaces are not electrified, but no effect at all can be
+detected when they are charged with positive electricity.
+
+They attributed this condensation of the jet to dust emitted
+from the illuminated surface, the dust, in accordance with
+%% -----File: 073.png---Folio 59-------
+\index{Aitken, effect of dust}%
+Aitken's experiments (\textit{Trans.\ Roy.\ Soc.\ Edinburgh}, 30, p.~337,
+1881), producing condensation by forming nuclei round which
+the water drops condense.
+
+The indications of a steam jet are not however free from
+\index{Helmholtz vr@Helmholtz, v.\ R., effect of electrification on a steam jet}%
+ambiguity, as R.~v.~Helmholtz (\textit{Wied.\ Ann.}\ 32, p.~1, 1887) has
+shown that condensation occurs in the jet when chemical reactions
+are going on in its neighbourhood, even though no
+dust is present. There is thus some doubt as to whether the
+condensation observed by Lenard and Wolf is due to disintegration
+of the illuminated surface or to chemical action
+taking place close to it. Taking however the interpretation
+which these observers give to their own experiments, the effects
+observed by Hertz, E.~Wiedemann and Ebert can easily be
+explained as due to the carrying of the discharge by particles
+disintegrated from the metallic surface by the action of the
+ultra-violet light.
+
+\index{Electrification of a metal plate by light}%
+\index{Light, xelectrification of a metal plate@\subdashone electrification of a metal plate}%
+\Article{40} Closely connected with this effect is the discovery, made
+\index{Hallwachs, electrification by light}%
+almost simultaneously by Hallwachs (\textit{Phil.\ Mag.}~[5], 26, p.~78,
+\index{Righi, electrification by light}%
+1888) and Righi (\textit{Phil.\ Mag.}~[5], 25, p.~314, 1888), that a metallic
+surface, especially if the metal is zinc and freshly polished,
+becomes positively electrified when exposed to the action of
+ultra-violet light.
+
+Lenard and Wolf's experiments suggest that this is probably
+due to the disintegration of the surface by the light, the metallic
+dust or vapour carrying off the negative electricity and leaving
+the positive behind.
+
+Stoletow (\textit{Phil.\ Mag.}~[5], 30, p.~436, 1890) showed that a kind
+\index{Stoletow, electrification by light}%
+of voltaic battery might be made by taking two plates of
+different metals in metallic connection and exposing one of
+them to the action of ultra-violet light; the plate so exposed becoming
+the negative electrode of the battery. When ultra-violet
+light acts in this way, Stoletow found that, as we should expect,
+the light is powerfully absorbed by the surface on which it falls.
+
+Probably another example of the same effect is the positive
+\index{Crookes on discharge through gases}%
+electrification observed by Crookes (\textit{Phil.\ Trans.}, Part~II. 1879,
+p.~647) on a plate placed inside an exhausted tube in full view of
+the negative electrode. We shall see, when we consider the discharges
+in such tubes, that something proceeds from the cathode
+which resembles ultra-violet light in its power of producing
+phosphorescence in bodies on which it falls. Crookes' experiment,
+%% -----File: 074.png---Folio 60-------
+which was made at Maxwell's suggestion, shows that the
+resemblance of the cathode discharge to ultra-violet light extends
+to its power of producing a positive charge on a metal
+plate exposed to its influence.
+
+\index{Electrodes, yspluttering of@\subdashone spluttering of}%
+\index{Spluttering@`Spluttering' of electrodes}%
+\Article{41} A striking instance of the facility with which a negatively
+electrified surface disintegrates, whilst a positively electrified
+one remains intact, is afforded by the well-known `spluttering'
+of the negative electrode in a vacuum tube. In such a tube
+the glass round the negative electrode is darkened by the
+deposition of a thin film of metal torn from the adjacent
+cathode; the glass round the positive electrode is, on the other
+hand, quite free from any such deposit. The amount of the
+disintegration of the cathode depends greatly upon the metal of
+which it is made. Crookes (\textit{Proc.\ Roy.\ Soc.}\ 50, p.~88, 1891) has
+given the following table, which expresses the relative loss in
+weight in equal times of cathodes of the same size exposed to
+similar electrical conditions:---
+\begin{center}
+\begin{tabular}{l r}
+Palladium &108.00 \\
+Gold      &100.\Z\Z \\
+Silver    &82.68 \\
+Lead      &75.04 \\
+Tin       &56.96 \\
+Brass     &51.58 \\
+Platinum  &44.00 \\
+Copper    &40.24 \\
+Cadmium   &31.99 \\
+Nickel    &10.99 \\
+Indium    &10.49 \\
+Iron      &5.50
+\end{tabular}
+\end{center}
+
+The loss in weight of magnesium and aluminium electrodes
+was too small to be detected. In the same paper Crookes also
+describes an experiment which seems to show that the `spluttering'
+at the negative electrode exists in water even when surrounded
+by air at atmospheric pressure.
+
+\Article{42} Since a metal surface when exposed to the action of
+sunlight emits negative electricity and retains positive, we
+should expect positively electrified bodies when exposed to
+light to behave differently from negatively electrified ones. This
+has been found to be the case. The first observations on this
+%% -----File: 075.png---Folio 61-------
+subject seem to have been made by Hoor (\textit{Repertorium d.\
+Physik.}~25, p.~105, 1889), who found that freshly prepared surfaces
+of zinc, copper, and brass quickly lost a negative charge
+when exposed to the action of ultra-violet light, while the same
+surfaces retained a positive charge.
+
+\index{Elster and Geitel, electrification produced by glowing bodies}%
+\index{Geitel escape of electricity from illuminated surfaces@\subdashone escape of electricity from illuminated surfaces}%
+\index{Hoor, effect of light on charged metals}%
+The subject was afterwards taken up by Elster and Geitel
+(\textit{Wied.\ Ann.}\ 38, pp.~40, 497, 1889; 41, p.~161, 1890; 42, p.~564, 1891),
+who verified Hoor's result for zinc, but could not detect any loss
+of negative electricity from freshly prepared surfaces of brass
+or copper. They also established the interesting fact that the
+effect is most marked in the case of the electro-positive metals,
+zinc or amalgamated zinc, aluminium, and magnesium. For the
+still more electro-positive metals, potassium and sodium, or
+rather for their amalgams, since the pure metals are difficult to
+work with on account of the tarnishing of their surfaces, they
+found that the effect is so strong that it can readily be
+observed even when the amalgams are enclosed in glass tubes,
+though glass, as is well known, absorbs most of the ultra-violet
+rays. When they succeeded subsequently in working with
+surfaces of potassium and sodium instead of their amalgams,
+they found that these substances are sensitive not merely to
+the ultra-violet rays but even to those emitted by an ordinary
+petroleum lamp (\textit{Wied.\ Ann.}\ 43, p.~225, 1891).
+
+Thus when the surface of some metals is negatively electrified
+and exposed to the action of light, and especially of ultra-violet
+light, we have an exception to the general rule that a charged
+body surrounded by cold air can retain its charge, for an indefinite
+time, provided the charge is not large enough to produce a
+spark. For as Elster and Geitel proved, the smallest negative
+charge rapidly disappears from the illuminated surface.
+
+The order of sensitiveness of metals to this effect is given by
+Elster and Geitel as
+
+\begin{tabular}{@{\indent}l}
+Potassium,\\
+Alloy of Sodium and Potassium,\\
+Sodium,\\
+Amalgams of Rubidium, Potassium, Sodium and Lithium,\\
+Magnesium, Aluminium,\\
+Zinc,\\
+Tin.
+\end{tabular}
+
+It is interesting to note that this is roughly the order of the
+%% -----File: 076.png---Folio 62-------
+metals in Volta's contact electricity series, as each metal is
+positive to the one after it. Elster and Geitel found that the
+effect is too small to be measured in Cadmium, Lead, Copper,
+Iron, Platinum, Mercury, and Carbon. They also found no clear
+indications of it with water. It is well marked, however, in
+phosphorescent substances such as Balmain's luminous paint
+\index{Elster and Geitel, electrification produced by glowing bodies}%
+\index{Geitel and Elster, electrification caused by glowing bodies}%
+(sulphide of calcium), and Elster and Geitel (\textit{Wied.\ Ann.}~44,
+p.~722, 1891) have quite recently shown that it is exhibited by
+Fluor Spar and other phosphorescent minerals.
+
+Another way of observing this effect is to place the illuminated
+body without a charge and in connection with the
+earth in the neighbourhood of a charged body, when the latter
+will lose its charge if it is positively electrified, while it will
+not lose its charge if it is negatively electrified; the positive
+charge induces a negative one on the illuminated body, this
+negative electricity escapes, travels up to and neutralises the
+positive electricity which induced it. When the pressure of the
+gas surrounding the body is less than $1$~mm., the escape of the
+negative electricity from the illuminated surface is considerably
+checked by placing it in a strong magnetic field (Elster and
+Geitel, \textit{Wied.\ Ann.}~41, p.~166, 1890).
+
+
+\Subsection{Discharge of Electricity caused by Glowing Bodies.}
+\index{Incandescent bodies, discharge of electricity by}%
+\index{Incandescent bodies, production of electrification by@\subdashtwo production of electrification by}%
+
+\Article{43} Somewhat similar differences between the discharge of
+positive and negative electricity are observed when the charged
+body, instead of being illuminated, is raised to so high a temperature
+that it becomes luminous itself. Elster and Geitel (\textit{Wied.\
+Ann.}~38, p.~27, 1889) found that when a platinum wire is heated
+to a bright red heat in an atmosphere of air or oxygen at a low
+pressure, a cold metal plate in its neighbourhood discharges
+negative electricity with much greater ease than positive. If, on
+the other hand, a thin platinum wire or carbon-filament is
+heated to incandescence in an atmosphere of hydrogen at a low
+pressure, the cold plate discharges positive electricity more easily
+than negative. Guthrie, who (\textit{Phil.\ Mag.}\ [4]~46, p.~257, 1873)
+was the first to call attention to phenomena of this kind,
+observed that an iron sphere in air when white hot cannot retain
+a charge either of positive or of negative electricity, and
+that as it cools it acquires the power of retaining a negative
+charge before it can retain a positive one. If the sphere
+%% -----File: 077.png---Folio 63-------
+is connected to the earth and held near a charged body, then,
+when the sphere is white hot, the body soon loses its charge
+whether this be positive or negative; when the sphere is
+somewhat colder, the body is discharged if negatively electrified
+but not if positively.
+
+\index{Electrification produced near glowing bodies@\subdashone produced near glowing bodies}%
+\index{Geitel and Elster, electrification caused by glowing bodies}%
+\index{Glowing bodies, discharge of electricity by}%
+\index{Glowing bodies, electrification caused by@\subdashone electrification caused by}%
+The converse problem of the production of electrification by a
+glowing wire has been studied in great detail by Elster and
+Geitel, a summary of whose results is given in \textit{Wied.\ Ann.}~37,
+p.~315, 1889. The conclusions they have come to are that when
+an insulated plate is placed near an incandescent platinum wire,
+the plate becomes positively electrified in air and oxygen, negatively
+electrified in hydrogen. It thus appears that incandescent
+wires discharge most easily the electricity of opposite sign to that
+which they produce on plates placed in their neighbourhood. If
+the incandescence is continued for a long time, then if the wire
+is thin and the pressure low, a plate in the neighbourhood of the
+wire receives a negative charge, whatever be the gas by which
+it is surrounded. Elster and Geitel seem to ascribe this to
+the action of gases driven out of the electrodes. Nahrwold, who
+also observed this effect (\textit{Wied.\ Ann.}~35, 107, 1888), regards it
+as the normal one, and ascribes the positive electrification observed
+when the wire first begins to glow to the action of dust
+in the gas. It is noteworthy that hydrogen, which in Elster
+and Geitel's experiments behaved with platinum electrodes
+oppositely to the other gases, is the only gas in which, according
+to Nahrwold, a platinum wire does not disintegrate when
+heated. With carbon filaments, Elster and Geitel found that
+the neighbouring plate is always negatively electrified, but so
+much gas is given off from these filaments that the interpretation
+of these results is ambiguous.
+
+Elster and Geitel have also observed that the ease with which
+electricity is produced in a plate near a glowing wire is diminished
+if the gas is hydrogen by placing the wire in a magnetic
+field, increased if it is oxygen or air.
+
+\Article{44} The investigations we have just described show clearly
+that metallic surfaces have in general a much greater tendency
+to attract a positive than a negative charge. Thus, for example,
+we have seen that when originally uncharged they become
+positively charged when exposed to the action of ultra-violet
+light, and if charged to begin with, then under the influence of
+%% -----File: 078.png---Folio 64-------
+the light they lose a negative charge much more rapidly than a
+positive one, indeed there seems no evidence to show that there
+is any loss of a small positive charge from this effect.
+
+The phenomena depending on the action of ultra-violet light and
+of incandescent surfaces can be co-ordinated by the conception introduced
+\index{Helmholtz va@Helmholtz, v.\ H., attraction of electricity by different substances}%
+by v.~Helmholtz (\textit{Erhaltung der~Kraft, Wissenschaftliche
+Abhand.}\ vol.~1.\ p.~48), that bodies attract electricity with different
+degrees of intensity. This conception was shown by him to be
+able to explain electrification by friction, and the difference of
+potential produced by the contact of metals. Thus, for example,
+the difference of potential produced by the contact of zinc and
+copper is explained on this hypothesis by saying that the
+positive electricity is attracted more strongly by the zinc than
+it is by the copper.
+
+Instead of considering the specific attraction of different bodies
+for electricity directly, it is equivalent in theory and generally
+more convenient in practice to regard the potential energy
+possessed by a body charged with electricity as consisting of
+two parts, (1)~the part calculated by the ordinary rules of electrostatics,
+and (2)~a part proportional to the charge and equal to~$\sigma Q$,
+where $Q$~is the charge and $\sigma$ a quantity which we shall
+\index{Volta@`Volta, potential'}%
+call the `Volta potential' of the body, and which varies from one
+substance to another.
+
+To investigate the nature of the effects produced by the presence
+of this second term, let us consider the case of two parallel plates
+$A$~and~$B$ made of different metals and connected electrically
+with each other.
+
+Let $Q$ be the charge on the plate~$A$, $-Q$~that on the plate~$B$,
+$\sigma_A$,~$\sigma_B$ the values of the co-efficient~$\sigma$ for the plates $A$~and~$B$
+respectively, then if $C$~is the capacity of the condenser formed
+by the two plates, the potential energy of the system will be
+given by the equation
+\[
+V =\tfrac{1}{2} \frac{Q^2}{C} + \sigma_{A}Q - \sigma_{B}Q.
+\]
+
+The system will be in equilibrium when the potential energy
+is a minimum, i.e.~when $dV/dQ = 0$, or
+\[
+\frac{Q}{C} + \sigma_A - \sigma_B = 0.
+\]
+
+Thus, by the contact of the metals the potential of the plate~$A$
+is raised above that of~$B$ by $\sigma_B-\sigma_A$.
+%% -----File: 079.png---Folio 65-------
+
+It is worthy of notice that on this view the medium separating
+the plates does not affect the value of the potential difference
+between them, however great the value of~$\sigma$ for this medium may
+be, provided that, as in the case of cold air, the medium is
+incapable of receiving a charge of electricity.
+
+The idea of the possession by a charged body of a quantity of
+energy proportional to the first power of the charge is involved
+in the well-used phrase `specific heat of electricity'; for if we
+regard electricity as having a specific heat which varies from one
+substance to another, a body charged with electricity will in \DPtypo{conquence}{consequence}
+of this specific heat possess some energy proportional to
+the charge. The electromotive forces which occur in unequally
+heated bodies may be explained as due to the tendency of the
+electricity to adjust itself so that the potential energy is a
+minimum; if the quantity~$\sigma$ is a function of the temperature,
+the energy will not be a minimum when the body is devoid of
+electrification.
+
+\index{Electrification xeffect of on surface tension@\subdashone effect of on surface tension}%
+The existence of the term~$\sigma Q$ in the expression for the energy
+of a charged body, since the electrification is on the surface,
+makes the energy per unit area of the surface depend upon
+whether the electrification is positive, negative, or zero. Now
+since the apparent surface tension of a liquid is equal to the
+energy per unit area of surface, it may be objected that if
+this view were true the surface tension of such liquids as are
+conductors ought to be changed by electrification, the change
+being in one direction when the electrification is positive and
+in the opposite when it is negative. A short calculation will
+show however that this change in the surface tension is so small
+that it might easily have escaped detection. We have seen that
+$\sigma_B-\sigma_A$ is the potential difference produced by the contact of two
+metals $A$~and~$B$, we know from observation that this difference,
+and therefore presumably $\sigma_A$~and~$\sigma_B$, is of the order of a volt, or
+in electromagnetic units~$10^8$. Now the greatest electrification
+which can exist on the surface without discharge when the metal
+is surrounded by air at the atmospheric pressure is such as to
+produce an electromotive intensity equal approximately to~$10^2$
+in electrostatic measure; thus the greatest surface density is
+in electrostatic units about $10^2/4\pi$, or in electromagnetic units
+$10^{-8}/12\pi$. Hence~$\sigma Q$, the energy of the kind we are considering,
+will at the most be of the order $1/(12\pi)$~ergs per square centimetre.
+%% -----File: 080.png---Folio 66-------
+\index{Surface tension, effect of electrification on}%
+This is so small compared with the energy due to the surface tension
+that it would require very careful observations to detect it.
+
+\Article{45} When a conductor, which does not disintegrate, is surrounded
+by air in its normal state, or by some other dielectric
+incapable of receiving a charge of electricity, the conductor
+cannot get charged, however much the $\sigma$ for the conductor may
+differ from that for the dielectric; for the electricity of opposite
+sign to that which would be left on the conductor has no
+place to which it can go.
+
+The case is however different when the conductor is exposed to
+the action of ultra-violet light, for then, as Lenard and Wolf's
+experiments prove, one or both of the following effects must
+take place: (1)~disintegration of the conductor, (2)~chemical
+changes in the gas in the neighbourhood of the conductor which
+put the gas in a state in which it can receive a charge of electricity.
+If either of these effects takes place it is possible for the
+conductor to be electrified, for the electricity of opposite sign
+to that left on the conductor may go to the disintegrated metal
+or the gas. The experiments hitherto made leave undecided the
+question which of these bodies serves as the refuge of the electricity
+discarded from the metal.
+
+The researches of Hallwachs and Righi on electrification by
+ultra-violet light can be explained on either hypothesis, if we
+assume that $\sigma_1$, the value of~$\sigma$ for the metallic vapour or for the
+dissociated gas, is greater than~$\sigma_2$, the value of~$\sigma$ for the solid
+metal. For when negative electricity~$-Q$ escapes from the metal
+and positive electricity equal to~$+Q$ remains behind, the diminution
+in the part of the potential energy due to the Volta potential
+is $\sigma_1 Q - \sigma_2 Q$ or $(\sigma_1 - \sigma_2)Q$. Thus, since $\sigma_1$~is by hypothesis greater
+than~$\sigma_2$, the departure of the negative electricity from the metal
+will be accompanied by a diminution in the potential energy,
+and will therefore go on until the increase in the ordinary
+potential energy due to the new distribution of electricity is
+sufficient to balance the diminution in the part of the energy
+due to the Volta potential. The positive electrification of the
+plate produced by ultra-violet light can thus be accounted for.
+
+Again, if the metal were initially positively electrified it
+would not be so likely to lose its charge as if it were initially
+charged with negative electricity, for the passage of positive
+electricity from the metal to its vapour or to the dissociated gas
+%% -----File: 081.png---Folio 67-------
+would involve an increase in the energy depending upon the
+Volta potential, and so would be much less likely to occur than
+an escape of negative electricity, which would produce a diminution
+in this energy. We can thus explain the observations of
+Elster and Geitel on the difference in the rates of escape of
+positive and negative electricity from illuminated surfaces. \nblabel{add:1}The
+causes of the electrification by incandescence observed by Elster
+and Geitel~(l.c.)\ are more obscure. Thus if we take the case when a
+plate receives a positive charge in air owing to the presence of
+a neighbouring incandescent platinum wire, the most obvious
+interpretation would be that the incandescence produces electrical
+separation, the wire getting negatively and the adjacent gas
+positively electrified. This view is however open to the very
+serious objection that in the other cases of the electrification of a
+metal in contact with a gas the metal receives the positive charge
+and not the negative one, as it would have to do if the preceding
+explanation were correct.
+
+The plate is exposed to the radiation from the incandescent
+wire and may perhaps under the influence of this radiation become
+a cathode, i.e.\ give out negative electricity and thus become
+positively \DPtypo{electified}{electrified}, just as it would if, as in Hallwach's and
+Righi's experiments, it were exposed to the action of ultra-violet
+light, or as in Crookes' experiment (\artref{40}{Art.~40}) to the
+emanations from a negative electrode. It seems however difficult
+to explain the anomalous behaviour of hydrogen on this
+view, and Nahrwold's discovery of the absence of `spluttering'
+in platinum wires heated to incandescence in an atmosphere of
+hydrogen seems to suggest that the charge on the plate may possibly
+arise in some such way as the following, even though the first
+effect of the incandescence is to produce a positive electrification
+over the wire and a negative one over the adjacent gas. When
+a metallic wire is heated, disintegration may take place in two
+ways, the metal may go off as vapour, or it may be torn off in
+solid lumps or dust. Now there seems to be no reason why $\sigma$
+for these lumps should differ from $\sigma$ for the wire, for both the
+lump and the wire consist of the same substance in the same
+state of aggregation; but if the $\sigma$'s were the same there would
+be no separation of electricity between the two. On the contrary,
+if the wire were charged with positive electricity, the
+lump, when it broke away, would carry positive electricity off
+%% -----File: 082.png---Folio 68-------
+with it. The case is however different when the metal goes
+off as vapour, or when it dissociates the gas in its neighbourhood:
+here the wire and the vapour or gas are in different
+states of aggregation, for which the values of~$\sigma$ are probably
+different, so that there may now be a separation of electricity, the
+wire getting the positive and the vapour or gas the negative.
+
+In air there is such an abundant deposition of platinum on a
+glass tube surrounding an incandescent platinum wire that the
+latter in all probability gives off dust as well as either dissociating
+the surrounding gas or giving off platinum vapour; while
+Nahrwold (\textit{Wied.\ Ann.}~35, 107, 1888) has shown that the deposition
+of platinum is so small in hydrogen that very little can be
+given off as dust in this gas.
+
+Let us now consider what will happen in air. When the
+platinum becomes incandescent there is a separation of electricity,
+the positive remaining on the wire, the negative going to the
+metallic vapour or dissociated gas. Since the wire has got a
+positive charge, any lumps that break away from it will be
+positively electrified. If the positive electricity given by these
+lumps to the plate, which in Elster and Geitel's experiments was
+held above the glowing wire, is greater than the negative charge
+given to it by such vapour or gas as may come in contact with
+it, the charge on the plate will be positive, as in Elster and
+Geitel's experiments. In hydrogen however, where the lumps are
+absent, there is nothing to neutralize the negative electricity on
+the metallic vapour or dissociated gas, so that the charge on the
+plate will, as Elster and Geitel found, be negative.
+
+\nbpagebreak
+\Section{Spark Discharge.}
+\index{Spark, discharge|indexetseq}%
+
+\Subsection{Electric Strength of a Gas.}
+\index{Electric strength@\subdashone strength}%
+
+\Article{46} In \artref{51}{Art.~51} of the first volume of the \textit{Electricity and Magnetism}
+Maxwell defines the \emph{electric strength} of a gas as the
+greatest electromotive intensity it can sustain without discharge
+taking place. This definition suggests that the electric strength
+is a definite specific property of a gas, otherwise the introduction
+of this term would not be of much value. If discharge through
+a gas at a definite pressure and temperature always began when
+the electromotive intensity reached a certain value, then this
+value, which is what Maxwell calls the electric strength of the
+%% -----File: 083.png---Folio 69-------
+gas, would have a perfectly definite meaning. The term `electric
+strength of the gas' would however be misleading if it were
+found to depend on such things, for example, as the materials of
+which the electrodes are made, the state of their surface, their
+shape, size, or distance apart, or on whether the electric field
+was uniform or variable either with regard to time or space. It
+has been found that the `electric strength' does depend upon
+some, perhaps even upon all, of the preceding conditions.
+
+\index{Righi, electrification by light}%
+\index{Spark, length, effect of nature of electrodes on@\subdashone length, effect of nature of electrodes on}%
+\Article{47} Righi (\textit{Nuovo Cimento}, [2]~16, p.~97, 1876) made some
+experiments with electrodes of carbon, bismuth, lead, zinc, tin
+and copper, but found that the substance of which the electrodes
+are made has little effect on the electromotive intensity necessary
+\index{Peace, spark potential|(}%
+for discharge. Mr.~Peace, who made careful experiments in the
+Cavendish Laboratory on this point, could not detect any
+difference in the electromotive intensity required to spark across
+\index{De la Rue and Müller, discharge through gases}%
+\index{Muller@Müller and de la Rue, electric discharge}%
+electrodes made of brass and those made of zinc. De~la~Rue and
+Hugo Müller (\textit{Phil.\ Trans.}~169, Pt.~1.\ p.~93, 1878) came to the
+conclusion that sparks pass more easily between aluminium
+terminals than between terminals of other metals, but that with
+this exception the nature of the electrodes has no influence upon
+the spark length.
+
+\index{Jaumann, discharge facilitated by rapid changes in the potential}%
+Jaumann has shown (\textit{Wien.\ Berichte},~97, p.~765, 1888) that the
+spark discharge is very much facilitated by making small but
+rapid changes in the potential of one of the electrodes.
+
+\Article{48} The reduction by Schuster (\textit{Phil.\ Mag.}\ [5]~29, p.~182, 1890)
+\index{Spark, length, effect of size of electrode on@\subdashone length, effect of size of electrode on}%
+\index{Baille, spark discharge|(}%
+\index{Gaugain, spark discharge}%
+\index{Paschen, spark discharge}%
+of the experiments of Baille, Paschen, and Gaugain on the spark
+discharge shows that with spherical electrodes of different sizes
+($1$~cm., $.5$~cm., and $.25$~cm.\ in radius respectively) the maximum
+electromotive intensity when the spark just passes through air
+at atmospheric pressure varies from $142$ to~$372$, the maximum
+intensity for small spheres being greater than for large ones.
+\index{Schuster, discharge through gases}%
+Schuster sums up the conclusions he draws from these experiments
+as follows, l.~c.\ p.~192:---
+
+(1) `For two similar systems of two equal spheres in which
+only the linear dimensions vary, the breaking-stress is greater
+the greater the curvature of the spheres.'
+
+(2) `If the distance between the spheres is increased, the breaking-stress
+at first diminishes.'
+
+(3) `There is a certain distance for which the breaking-stress
+is a minimum.'
+%% -----File: 084.png---Folio 70-------
+
+We shall find too when we consider the relation between spark
+length and potential difference that the distance between the
+electrodes may have an enormous effect on the electromotive
+intensity required to produce discharge.
+
+The `electric strength' as defined by Maxwell seems to depend
+upon so many extraneous circumstances that there does not
+appear to be any reason for regarding it as an intrinsic property
+of the gas.
+
+\Subsection{Connection between Spark Length and Potential Difference,
+when the Field is approximately uniform.}
+\index{Spark, length, connection between and potential difference@\subdashone length, connection between and potential difference|indexetseq}%
+
+\Article{49} This subject has been investigated by a large number of
+physicists. We have however only space to consider the most
+recent investigations on this subject. Baille (\textit{Annales de Chimie
+et de Physique}, [5]~25, p.~486, 1882) has made an elaborate investigation
+of the potential difference required to produce in air
+at atmospheric pressure sparks of varying lengths, between planes,
+cylinders, and spheres of various diameters. The method he
+used was to charge the conductors between which the sparks
+passed by a Holtz machine, the potential between the electrodes
+being measured by an attracted disc electrometer provided with
+a guard ring: this method is practically the same as that employed
+\index{Kelvin, Lord, spark discharge}%
+by Lord Kelvin (\textit{Reprint of Papers on Electrostatics and
+Magnetism}, p.~247), who in 1860 made the first measurements
+in absolute units of the electromotive intensity required to produce
+a spark.
+
+For very short sparks between two planes Baille (l.c., p.~515)
+found the results given in the following table:---
+\begin{center}
+\tabletextsize
+\begin{tabular}{c|c|c}
+\multicolumn{3}{c}{\normalsize\textit{Potential Difference and Spark Length;}} \\
+\multicolumn{3}{c}{\normalsize(\textit{temperature $15°$ to $20°$\,C, pressure $760~\text{mm}$.\smallskip})}\\
+\hline
+\settowidth{\TmpLen}{in Centimetres.}%
+\parbox[c]{\TmpLen}{\tablespaceup\centering Spark Length\\ in Centimetres.\tablespacedown} &
+\settowidth{\TmpLen}{Potential Difference in}%
+\parbox[c]{\TmpLen}{\centering Potential Difference in\\ Electrostatic Units.} &
+\settowidth{\TmpLen}{Electrostatic Units.}%
+\parbox[c]{\TmpLen}{\centering Surface Density in\\ Electrostatic Units.} \\
+\hline
+\tablespaceup$.0015$ & $1.42$ & $75.4$ \\
+$.0020$ & $1.62$ & $64.5$ \\
+$.0025$ & $1.90$ & $60.5$ \\
+$.0050$ & $2.51$ & $39.9$ \\
+$.0075$ & $2.81$ & $29.8$ \\
+$.0100$ & $3.15$ & $25.1$ \\
+$.0125$ & $3.48$ & $22.1$ \\
+$.0150$ & $3.80$ & $20.1$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+%% -----File: 085.png---Folio 71-------
+
+In another series of experiments where the sparks were slightly
+longer, Baille, p.~515, found the following results:---
+\setlength{\TmpLen}{0.16\linewidth}%
+\begin{center}
+\setlength{\tabcolsep}{0pt}
+\tabletextsize
+\begin{tabular}{c|c|c||c|c|c}
+\multicolumn{6}{c}{\normalsize\textit{Potential Difference and Spark Length.\smallskip}}\\
+\hline
+\parbox[c]{\TmpLen}{\centering\tablespaceup Spark\\ Length.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} &
+\parbox[c]{\TmpLen}{\centering Spark\\ Length.} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} \\
+\hline
+\tablespaceup$.01$ & $\Z3.17$ & $25.2$ & $.08$ & $12.38$ & $12.3$ \\
+$.02$ & $\Z4.51$ & $17.9$ & $.09$ & $13.44$ & $11.9$ \\
+$.03$ & $\Z6.22$ & $16.5$ & $.10$ & $14.67$ & $11.7$ \\
+$.04$ & $\Z7.32$ & $14.6$ & $.11$ & $15.75$ & $11.4$ \\
+$.05$ & $\Z8.71$ & $13.8$ & $.12$ & $16.84$ & $11.1$ \\
+$.06$ & $\Z9.84$ & $13.2$ & $.13$ & $17.94$ & $11.0$ \\
+$.07$ &  $11.20$ & $12.7$ & $.14$ & $19.00$ & $10.8$ \\
+      &          &        & $.15$ & $20.16$ & $10.7$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+For spark lengths between $.025$~cm.\ and $.5$~cm.\ the following
+results were obtained, p.~516, in a different series of experiments:---
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{0pt}
+\begin{tabular}{c|c|c||c|c|c}
+\multicolumn{6}{c}{\normalsize\textit{Potential Difference and Spark Length.\smallskip}}\\
+\hline
+\parbox[c]{\TmpLen}{\centering\tablespaceup Spark\\ Length.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} &
+\parbox[c]{\TmpLen}{\centering Spark\\ Length.} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} \\
+\hline
+\tablespaceup$.025$ & $\Z5.94$ & $18.86$ & $.275$ & $32.69$ & $9.46$ \\
+$.050$ & $\Z8.68$ & $13.76$ & $.300$ & $35.35$ & $9.37$ \\
+$.075$ & $11.87$ & $12.57$ & $.325$ & $37.83$ & $9.25$ \\
+$.100$ & $14.79$ & $11.76$ & $.350$ & $39.95$ & $9.08$ \\
+$.125$ & $17.45$ & $11.06$ & $.375$ & $42.17$ & $8.94$ \\
+$.150$ & $20.29$ & $10.76$ & $.400$ & $44.74$ & $8.90$ \\
+$.175$ & $22.94$ & $10.43$ & $.425$ & $47.30$ & $8.86$ \\
+$.200$ & $25.51$ & $10.15$ & $.450$ & $49.70$ & $8.79$ \\
+$.225$ & $28.17$ & $\Z9.96$ & $.475$ & $52.18$ & $8.75$ \\
+$.250$ & $30.47$ & $\Z9.70$ & $.500$ & $54.48$ & $8.67$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+\sloppy
+For longer sparks Baille, l.c., p.~517, got the numbers given in
+the two following Tables, which represent the results of different
+sets of experiments:---
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{0pt}
+\begin{tabular}{c|c|c||c|c|c}
+\multicolumn{6}{c}{\normalsize\textit{Potential Difference and Spark Length.\smallskip}} \\
+\multicolumn{6}{c}{TABLE (I).\smallskip}\\
+\hline
+\parbox[c]{\TmpLen}{\centering\tablespaceup Spark\\ Length.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} &
+\parbox[c]{\TmpLen}{\centering Spark\\ Length.} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} \\
+\hline
+\tablespaceup$.40$ & $44.80$ & $8.90$ & $.60$ & $63.82$ & $8.47$ \\
+$.45$ & $49.63$ & $8.78$ & $.65$ & $68.75$ & $8.42$ \\
+$.50$ & $54.36$ & $8.65$ & $.70$ & $74.09$ & $8.42$ \\
+$.55$ & $59.09$ & $8.55$ & $.75$ & $79.02$ & $8.39$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+\fussy
+
+\index{Baille, spark discharge|)}%
+%% -----File: 086.png---Folio 72-------
+\index{Liebig, spark potential}%
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{0pt}
+\begin{tabular}{c|c|c||c|c|c}
+\multicolumn{6}{c}{TABLE (II).\smallskip}\\
+\hline
+\parbox[c]{\TmpLen}{\centering\tablespaceup Spark\\ Length.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} &
+\parbox[c]{\TmpLen}{\centering Spark\\ Length.} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Surface\\ Density.} \\
+\hline
+\tablespaceup$.70$ & $73.48$ & $8.84$ & $\Z.90$ & $\Z94.72$ & $8.38$ \\
+$.75$ & $80.13$ & $\DPtypo{3.55}{8.55}$ & $\Z.95$ & $100.16$ & $8.38$ \\
+$.80$ & $84.86$ & $8.40$ & $1.00$  & $105.50$ & $8.39$ \\
+$.85$ & $89.89$ & $8.42$ & & &\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+\Article{50} We may compare with these results those obtained by
+Liebig (\textit{Phil.\ Mag.}\ [5],~24, p.~106, 1887), who used a similar
+method, but whose electrodes were segments of spheres $9.76$~cm.\
+in radius. Liebig's results are as follows:---
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{0pt}
+\begin{tabular}{c|c|c||c|c|c}
+\multicolumn{6}{c}{\normalsize\textit{Potential Difference and Spark Length.\smallskip}}\\
+\hline
+\parbox[c]{\TmpLen}{\centering\tablespaceup Spark Length\\ in centimetres.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Electro- \\motive\\ Intensity.} &
+\parbox[c]{\TmpLen}{\centering Spark\\ Length.} &
+\parbox[c]{\TmpLen}{\centering Potential\\ Difference.} &
+\parbox[c]{\TmpLen}{\centering Electro- \\motive\\ Intensity.} \\
+\hline
+\tablespaceup$.0066$ & $\Z2.630$ & $398.5$ & $\Z.2398$ & $\Z30.622$ & $127.7$ \\
+$.0105$ & $\Z3.357$ & $319.7$ & $\Z.2800$ & $\Z35.196$ & $125.7$ \\
+$.0143$ & $\Z4.017$ & $280.9$ & $\Z.3245$ & $\Z39.816$ & $122.7$ \\
+$.0194$ & $\Z4.573$ & $235.7$ & $\Z.3920$ & $\Z47.001$ & $119.9$ \\
+$.0245$ & $\Z5.057$ & $206.4$ & $\Z.4715$ & $\Z55.165$ & $117.0$ \\
+$.0348$ & $\Z7.190$ & $206.6$ & $\Z.5588$ & $\Z63.703$ & $114.0$ \\
+$.0438$ & $\Z8.863$ & $195.5$ & $\Z.6226$ & $\Z69.980$ & $112.4$ \\
+$.0604$ & $10.866$ & $179.9$ & $\Z.7405$ & $\Z82.195$ & $111.0$ \\
+$.0841$ & $13.548$ & $161.1$ & $\Z.8830$ & $\Z95.540$ & $108.2$ \\
+$.0903$ & $13.816$ & $153.0$ & $\Z.9576$ & $102.463$ & $107.0$ \\
+$.1000$ & $15.000$ & $150.0$ & $1.0672$ & $110.775$ & $103.8$ \\
+$.1520$ & $20.946$ & $137.8$ & $1.1440$ & $117.489$ & $102.7$ \\
+$.1860$ & $24.775$ & $133.2$  & & &\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+The potential difference and electromotive intensity are measured
+in electrostatic units.
+\index{Electromotive intensity, required to produce a spark across a thin layer of gas@\subdashtwo required to produce a spark across a thin layer of gas}%
+
+\bigskip
+\quad
+\includegraphicssideways[!ht]{fig19}{Fig.~19.}
+
+Liebig's results for hydrogen, coal gas and carbonic acid as
+well as air are exhibited graphically in \figureref{fig19}{Fig.~19}, where the nearly
+straight curve represents the relation between potential difference
+and spark length, and the other the relation between electromotive
+intensity and spark length. The abscissae are the spark
+lengths, the ordinates, the potential difference or electromotive
+intensity. It will be seen that Liebig's values for the potential difference
+required to produce a spark of given length are about $8$~per~cent.\
+higher than Baille's. It also appears from any of the preceding
+tables that the electromotive intensity required to spark
+across a layer of air varies very greatly with the thickness
+%% -----File: 087.png---Folio 73-------
+of the layer. Thus from Baille's result we see that the electromotive
+intensity required to spark across a layer $.0015$~cm.\
+thick is about nine times that required to spark across a layer
+$1$~cm.\ thick. The fact that a greater electromotive intensity
+is required to spark across a thin layer of air than a thick one
+\index{Kelvin, Lord, spark discharge}%
+was discovered by Lord Kelvin~(l.c.) in 1860.
+
+%% -----File: 088.png---Folio 74-------
+
+\index{Baille, spark discharge}%
+\Article{51} With regard to the relation between the potential difference~$V$
+and spark length~$l$, Baille deduced from his experiments
+the relation
+\[
+V^2 = 10500 (l + 0.08)l.
+\]
+\index{Chrystal on spark discharge}%
+The agreement between the numbers calculated by this formula
+and those found by experiment is not very close, and Chrystal
+(\textit{Proc.\ Roy.\ Soc.\ Edin.}\ vol.~11. p.~487, 1882) has shown that for
+spark lengths greater than $2$~millimetres the linear relation
+\[
+V = 4.997 + 99.593 l
+\]
+represents Baille's results within experimental errors. This linear
+relation is confirmed by Liebig's results, as the curves, \figureref{fig19}{Fig.~19},
+are nearly straight when the spark length is greater than one
+millimetre.
+
+\index{Foster and Pryson, spark potential}%
+\index{Pryson and Foster, spark potential}%
+Carey Foster and Pryson (\textit{Chemical News},~49, p.~114, 1884)
+found that the linear relation $V = \alpha + \beta l$ was the one which represented
+best the results of their experiments on the discharge
+through air at atmospheric pressure.
+
+\Article{52} When the spark length in air at atmospheric pressure is
+less than about a millimetre, the curve which expresses the relation
+between potential difference and spark length gets concave to
+the axis along which the spark lengths are measured; that is, for
+a given small increase in the spark length the increase in the
+corresponding potential difference is greater when the sparks are
+short than when they are long. For exceedingly short sparks there
+seems to be considerable evidence that when the spark length is
+reduced to a certain critical value there is a point of inflexion in
+the potential difference curve, and that when the spark length is
+reduced below this value the previous concavity is replaced by
+convexity, the curve for very small spark lengths taking somewhat
+the shape of the one in \figureref{fig20}{Fig.~20}. This indicates that the
+potential difference required to produce a spark however short
+cannot be less than a certain finite value, which for air at
+ordinary temperatures is probably between $300$ and $400$~volts.
+If a curve similar to \figureref{fig20}{Fig.~20} represents the relation between
+potential difference and spark length, we see that it would be
+possible under certain conditions to start a spark by pulling two
+plates maintained at a constant potential difference further apart,
+and to stop the spark by pushing the plates nearer together.
+
+\includegraphicsmid{fig20}{Fig.~20.}
+
+\Article{53} At atmospheric pressure the spark length at which the
+potential difference is a minimum must, if such a length exist at
+%% -----File: 089.png---Folio 75-------
+all, be so small, that it would be very difficult to measure the
+spark lengths with sufficient accuracy to investigate this point
+completely; when however the air is at a lower pressure the
+critical spark length is longer, and the investigation of this
+problem easier. The evidence to which I have alluded in \artref{52}{Art.~52}
+comes indirectly from an investigation (which we shall have to
+consider later) made by Mr.~Peace in the Cavendish Laboratory,
+Cambridge. Mr.~Peace's experiments were made with the view
+of finding the relation between the potential difference in air and
+the pressure when the spark length is kept constant, but as experiments
+were made on this relation for sparks of many different
+lengths, they furnish material for drawing the curve expressing
+the relation between potential difference and spark length at
+constant pressure. Such curves are given in \figureref{fig27}{Fig.~27}, and it
+will be seen that at lower pressures they exhibit the peculiarities
+referred to. The discharge took place between very large electrodes,
+one of which was plane and the other a segment of a
+sphere about $20$~cm.\ in radius, and as the difference of potential
+was produced by a large number of storage cells, the equality of
+whose E.~M.~F. was very carefully tested, the measurements of
+the potential difference could be made with great accuracy. It
+must be remembered, however, that the apparatus used was
+designed for the purpose of determining the relation between
+%% -----File: 090.png---Folio 76-------
+potential difference and pressure for constant spark length, and
+not for the relation between potential difference and spark length
+for constant pressure, so that its indications on this point are
+somewhat indirect. The conclusion that with very short sparks
+the potential difference increases as the spark length diminishes
+was, however, borne out to some extent by the observation that
+when the voltage was just not sufficient (i.e.~was about two
+volts too small) to spark across $.002$~of an inch at a pressure of
+$20$~mm.\ of mercury, the same voltage would not send a spark
+between the plates when the distance was reduced to $.001$ or
+even to $.0004$~of an inch. Mr.~Peace also found that when he
+removed the electrodes from the apparatus after sparks had
+passed between them when they were very close together, the
+part of the electrodes most affected by breathing upon them
+formed an annulus at some little distance from the centre, indicating
+that discharge had taken place most freely at distances
+which were slightly greater than the shortest distance between
+the electrodes, which was along the line joining their centres.
+Mr.~Peace has more recently tested this result directly by placing
+two spark gaps in parallel, the electrodes being parallel plane
+plates. One pair of these electrodes were separated by a single
+thickness of thin pieces of glass such as are used for cover
+slips, while the other pair of electrodes were kept at a greater
+distance apart by placing between them two or more of the
+pieces of glass piled one on the top of the other. At atmospheric
+pressure the spark passed across the short gap rather than the
+long one, but when the pressure was reduced the reverse effect
+took place, the spark going across the longer air gap before
+any discharge could be detected across the shorter, and after
+the spark had first passed across the longer path it required in
+some cases an additional potential difference of more than $100$~volts
+to make it go across the shorter as well. When in \artref{170}{Art.~170}
+we consider discharge at very low pressures we shall find that
+\index{Hittorf, discharge through gases}%
+in some experiments of Hittorf's a long spark passed much
+more easily than a very much shorter one between the same
+electrodes; in this case however the electrodes were wires, and
+the field before discharge was not uniform as in the case under
+consideration.
+\index{Peace, spark potential|)}%
+%% -----File: 091.png---Folio 77-------
+
+\Subsection{Discharge when the Electric Field is not uniform.}
+
+\Article{54} In the experiments tabulated above the electrodes were so
+large that the electric field between them might be considered
+as uniform before the spark passed. Baille and Paschen have
+however made some very interesting experiments on the potential
+differences required to spark between spheres small enough to
+make the variations in the electric field considerable. Baille's
+results (\textit{Annales de Chimie et de Physique}~(5), 25, p.~531, 1882)
+are given in the following table, the potential difference being
+measured in absolute electrostatic units:---
+\medskip
+\begin{center}
+\tabletextsize
+\settowidth{\TmpLen}{diameter.}
+\setlength{\tabcolsep}{2pt}
+\begin{tabular}{c|c|c|c|c|c|c|c}
+\multicolumn{8}{c}{\normalsize\textit{Potential Differences: pressure $760$~{\upshape mm.},
+                   temperature $15°$\,to~$20°$\,{\upshape C}.}\smallskip}\\
+\hline
+\parbox[c]{\TmpLen}{\tablespaceup\centering Spark\\Length\\ in cm.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Planes.} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $6$~cm.~in  \\ diameter.} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $3$~cm.~in  \\ diameter.} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $1$~cm.~in  \\ diameter.} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $.6$~cm.~in \\ diameter.} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $.35$~cm.~in\\ diameter.} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $.1$~cm.~in \\ diameter.} \\
+\hline
+$\tablespaceup\Z.05$ & $\Z\Z8.94$ & $\Z\Z8.96$ & $\Z\Z9.18$ & $\Z9.18$ & $\Z9.26$ & $\Z9.30$ & $\Z\mathbf{9.63}$ \\
+$\Z.10$ & $\Z14.70$ & $\Z14.78$ & $\Z14.99$ & $15.25$ & $15.53$ & $16.04$ & $\mathbf{16.10}$ \\
+$\Z.15$ & $\Z20.20$ & $\Z20.31$ & $\Z20.47$ & $21.28$ & $21.24$ & $\mathbf{21.87}$ & $19.58$ \\
+$\Z.20$ & $\Z25.42$ & $\Z25.59$ & $\Z25.95$ & $26.78$ & $26.82$ & $\mathbf{27.13}$ & $21.91$ \\
+$\Z.25$ & $\Z30.38$ & $\Z30.99$ & $\Z31.33$ & $32.10$ & $\mathbf{32.33}$ & $31.96$ & $23.11$ \\
+$\Z.30$ & $\Z35.35$ & $\Z36.12$ & $\Z36.59$ & $37.32$ & $\mathbf{37.38}$ & $36.29$ & $24.12$ \\
+$\Z.35$ & $\Z40.45$ & $\Z41.45$ & $\Z41.47$ & $\mathbf{42.48}$ & $42.16$ & $39.39$ & $25.34$ \\
+$\Z.40$ & $\Z45.28$ & $\Z46.34$ & $\Z46.77$ & $\mathbf{47.62}$ & $46.34$ & $41.77$ & $26.03$ \\
+$\Z.45$ & $\Z50.48$ & $\Z51.46$ & $\Z\mathbf{51.60}$ & $\mathbf{51.56}$ & $50.44$ & $43.76$ & $26.62$\tablespacedown\\
+\hline
+$\tablespaceup\Z.40$ & $\Z44.80$ & $\Z45.00$ & $\Z45.00$ & $\mathbf{45.50}$ & $44.80$ & $41.07$ & $26.58$ \\
+$\Z.45$ & $\Z49.63$ & $\Z50.33$ & $\Z49.63$ & $\mathbf{52.04}$ & $48.42$ & $43.29$ & $28.49$ \\
+$\Z.50$ & $\Z54.35$ & $\Z\mathbf{55.06}$ & $\Z\mathbf{54.96}$ & $\mathbf{54.66}$ & $53.25$ & $47.21$ & $30.00$ \\
+$\Z.60$ & $\Z63.82$ & $\Z\mathbf{65.23}$ & $\Z\mathbf{65.23}$ & $\mathbf{65.23}$ & $59.69$ & $53.75$ & $31.51$ \\
+$\Z.70$ & $\Z74.09$ & $\Z\mathbf{75.40}$ & $\Z73.79$ & $72.28$ & $64.22$ & $56.47$ & $32.92$ \\
+$\Z.80$ & $\Z84.83$ & $\Z\mathbf{87.98}$ & $\Z84.76$ & $77.61$ & $67.75$ & $58.79$ &  $33.82$ \\
+$\Z.90$ & $\Z94.72$ & $\Z\mathbf{97.44}$ & $\Z94.62$ & $80.13$ & $70.56$ & $59.09$ & $34.93$ \\
+$1.00$  & $105.49$ & $\mathbf{112.94}$ & $104.69$ & $83.05$ & $72.38$ & $59.49$ & $36.24$\tablespacedown\\
+\hline
+\end{tabular}
+\end{center}
+
+From this table Baille concludes that for a given length of
+spark between two equal spheres, one charged and insulated
+and the other put to earth, the potential difference varies with
+the diameter of the sphere; starting from the plane the potential
+difference at first increases with the curvature, and attains a
+maximum when the sphere has a certain diameter. This critical
+diameter of the sphere depends upon the spark length, the
+shorter the spark the smaller the critical diameter. In the preceding
+table the maximum potential differences have been
+printed in bolder type.
+%% -----File: 092.png---Folio 78-------
+
+The two parts into which the table is divided by the horizontal
+line correspond to two different sets of experiments.
+
+Paschen's results (\textit{Wied.\ Ann.}~37, p.~79, 1889) are given in the
+following table:---
+\begin{center}
+\tabletextsize
+\settowidth{\TmpLen}{in centimetres.}%
+\begin{tabular}{c|c|c|c}
+\multicolumn{4}{c}{\normalsize\textit{Potential Difference at first Spark: pressure $756~\text{mm.}$}}\\
+\multicolumn{4}{c}{\normalsize\textit{mean temperature $15°$\,C.}\medskip}\\
+\multicolumn{4}{c}{SHORT SPARKS.}\medskip \\
+\hline
+\parbox[c]{\TmpLen}{\tablespaceup\centering Spark Length\\ in centimetres.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $1$~cm.\ radius} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $.5$~cm.\ radius} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $.25$~cm.\ radius} \\
+\hline
+\tablespaceup$.01$ & $\Z3.38$ & $\Z3.42$ & $\Z3.61$ \\
+$.02$ & $\Z5.04$ & $\Z5.18$ & $\Z5.58$ \\
+$.03$ & $\Z6.62$ & $\Z6.87$ & $\Z6.94$ \\
+$.04$ & $\Z8.06$ & $\Z8.22$ & $\Z8.43$ \\
+$.05$ & $\Z9.56$ & $\Z9.75$ & $\Z9.86$ \\
+$.06$ & $10.81$ & $10.87$ & $11.19$ \\
+$.07$ & $11.78$ & $12.14$ & $12.29$ \\
+$.08$ & $13.40$ & $13.59$ & $13.77$ \\
+$.09$ & $14.39$ & $14.70$ & $14.89$ \\
+$.10$ & $15.86$ & $15.97$ & $16.26$ \\
+$.11$ & $16.79$ & $17.08$ & $17.26$ \\
+$.12$ & $18.28$ & $18.42$ & $18.71$ \\
+$.14$ & $20.52$ & $20.78$ & $21.26$\tablespacedown \\
+\hline
+\end{tabular}
+
+\begin{tabular}{c|c|c|c}
+\multicolumn{4}{c}{LONG SPARKS.}\medskip \\
+\hline
+\parbox[c]{\TmpLen}{\tablespaceup\centering Spark Length\\ in centimetres.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $1$~cm.\ radius} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $.5$~cm.\ radius} &
+\parbox[c]{\TmpLen}{\centering Spheres\\ $.25$~cm.\ radius} \\
+\hline
+\tablespaceup$\Z.10$ & $15.96$ & $16.11$ & $\mathbf{16.45}$ \\
+$\Z.15$ & $21.94$ & $22.17$ & $\mathbf{22.59}$ \\
+$\Z.20$ & $27.59$ & $27.87$ & $\mathbf{28.18}$ \\
+$\Z.25$ & $32.96$ & $33.42$ & $\mathbf{33.60}$ \\
+$\Z.30$ & $38.59$ & $\mathbf{39.00}$ & $38.65$ \\
+$\Z.35$ & $43.93$ & $\mathbf{44.32}$ & $43.28$ \\
+$\Z.40$ & $49.17$ & $\mathbf{49.31}$ & $47.64$ \\
+$\Z.45$ & $\mathbf{54.37}$ & $54.18$ & $51.56$ \\
+$\Z.50$ & $\mathbf{59.71}$ & $59.03$ & $54.67$ \\
+$\Z.55$ & $\mathbf{64.60}$ & $63.35$ & $57.27$ \\
+$\Z.60$ & $\mathbf{69.27}$ & $67.80$ & $59.95$ \\
+$\Z.70$ & $\mathbf{78.51}$ & $75.04$ & $63.14$ \\
+$\Z.80$ & $\mathbf{87.76}$ & $81.95$ & $66.39$ \\
+$\Z.90$ & & & $68.65$ \\
+$1.00$ & & & $70.68$ \\
+$1.20$ & & & $74.94$ \\
+$1.50$ & & & $79.42$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+Here the heavy type again denotes the maximum potential
+differences.
+%% -----File: 093.png---Folio 79-------
+
+\medskip
+\includegraphicsmid{fig21}{Fig.~21.}
+
+These results are represented graphically in \figureref{fig21}{Fig.~21}. They
+confirm Baille's conclusion that for a spark of given length the
+potential difference is a maximum when the spheres have a certain
+critical diameter, the critical diameter increasing with the
+length of the spark.
+%% -----File: 094.png---Folio 80-------
+\index{De la Rue and Müller, discharge through gases}%
+\index{Muller@Müller and de la Rue, electric discharge}%
+\index{Schuster, discharge through gases}%
+
+\includegraphicsmid{fig22}{Fig.~22.}
+
+Both Baille's and Paschen's measurements show that when
+the spheres are very small, the potential difference required to
+produce a spark of given length is, if the spark length is not too
+small, much less than the potential difference required to produce
+the same length of sparks between parallel plates. When the
+spark passes between pointed electrodes the potential differences
+are still smaller. This effect is clearly shown in \figureref{fig22}{Fig.~22}, which
+is taken from a paper by De~la~Rue and Hugo Müller (\textit{Phil.\
+Trans.}\ 1878, Pt.~1.\ p.~55), and which contains curves representing
+the relation between potential difference and spark length when
+the electrodes are (i)~two plates, (ii)~two spheres, one $3$~cm.\ in
+radius the other $1.5$~cm.\ in diameter, (iii)~two concentric cylinders,
+(iv)~a plane and a point, (v)~two points. It will be noticed
+that the two points, which give the greatest striking distance
+for long sparks, give the least for short sparks.
+
+\Article{55} If the spark length between parallel plates is taken as
+unity, the spark length corresponding to various potential differences
+for different kinds of electrodes was found by De~la~Rue
+and Müller to be as follows (\textit{Proc.\ Roy.\ Soc.}~36, p.~157,
+1883):---
+\begin{center}
+\tabletextsize
+\settowidth{\TmpLen}{Striking distance for point and plane\quad}%
+\begin{tabular}{@{}l@{}cccccc@{}}
+\parbox[b]{\TmpLen}{\hangindent=1em Number of cells, each cell having\\
+an E.M.F. of $1.03$~volts\mdotfill} & $1000$ & $3000$ & $6000$ & $9000$ & $12,000$ & $15,000$ \\
+\parbox[c]{\TmpLen}{Striking distance for point and plane\quad} & $.60$ & $2.09$ & $3.82$ & $3.89$ & $3.58$ &    $3.30$ \\
+\parbox[c]{\TmpLen}{Striking distance for two points\mdotfill}   & $.84$ & $1.94$ & $4.65$ & $4.65$ & $4.18$ &    $3.68$
+\end{tabular}
+\end{center}
+
+This table would appear to indicate that the ratio of the
+striking distance for pointed electrodes to that of planes attains
+a maximum. It must however be remembered that when the
+sparks are long the conditions are not the same in the two cases;
+in the case of the plates the discharge takes place abruptly,
+while when the electrodes are pointed a brush discharge starts
+long before the spark passes, and materially modifies the conditions.
+
+\Article{56} Schuster (\textit{Phil.\ Mag.}\ [5]~29, p.~182, 1890) has, by the aid of
+Kirchhoff's solution of the problem of the distribution of electricity
+over two spheres, calculated from Baille's and Paschen's
+experiments the maximum electromotive intensity in the field
+when the spark passed. The results for Baille's experiments
+are given in Table~1, for Paschen's in Table~2.
+%% -----File: 095.png---Folio 81-------
+
+\begin{center}
+\setlength{\TmpLen}{0.1\linewidth}
+\tabletextsize
+\setlength{\tabcolsep}{4pt}
+\begin{tabular}{c||c|c|c|c|c|c|c}
+\multicolumn{8}{c}{TABLE 1.\medskip} \\
+\multicolumn{8}{c}{\normalsize\textit{Value of Maximum Electromotive Intensity in Electrostatic Units.}\medskip} \\
+\hline
+\parbox[c]{\TmpLen}{\centering\tablespaceup Spark\\ Length\\ in cm.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Planes} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $6$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $3$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $1$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $.6$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $.35$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $.1$~cm.} \\
+\hline
+$\tablespaceup\Z.05$ & $179$ & $180$ & $186$ & $190$ & $197$ & $206$ & $292$ \\
+$\Z.10$ & $147$ & $149$ & $153$ & $163$ & $176$ & $198$ & $376$ \\
+$\Z.15$ & $135$ & $138$ & $141$ & $157$ & $170$ & $206$ & $425$ \\
+$\Z.20$ & $127$ & $131$ & $137$ & $154$ & $170$ & $219$ & $460$ \\
+$\Z.25$ & $122$ & $127$ & $134$ & $154$ & $180$ & $236$ & $478$ \\
+$\Z.30$ & $118$ & $124$ & $130$ & $156$ & $189$ & $253$ & $494$ \\
+$\Z.35$ & $116$ & $122$ & $129$ & $159$ & $197$ & $263$ & $516$ \\
+$\Z.40$ & $113$ & $122$ & $129$ & $164$ & $204$ & $272$ & $528$ \\
+$\Z.45$ & $112$ & $120$ & $127$ & $166$ & $214$ & $278$ & $540$\tablespacedown \\
+\hline
+$\tablespaceup\Z.40$ & $112$ & $118$ & $124$ & $157$ & $197$ & $268$ & $539$ \\
+$\Z.45$ & $110$ & $119$ & $122$ & $167$ & $206$ & $275$ & $578$ \\
+$\Z.50$ & $109$ & $117$ & $125$ & $166$ & $218$ & $296$ & $608$ \\
+$\Z.60$ & $106$ & $116$ & $125$ & $181$ & $233$ & $327$ & $639$ \\
+$\Z.70$ & $106$ & $117$ & $126$ & $188$ & $234$ & $339$ & $667$ \\
+$\Z.80$ & $106$ & $123$ & $130$ & $192$ & $250$ & $349$ & $685$ \\
+$\Z.90$ & $105$ & $120$ & $132$ & $191$ & $255$ & $349$ & $708$ \\
+$1.00$  & $106$ & $128$ & $133$ & $194$ & $258$ & $349$ & $733$\tablespacedown \\
+\hline
+\end{tabular}
+
+\begin{tabular}{c||c|c|c|c|c|c|c}
+\multicolumn{8}{c}{TABLE 2.\medskip} \\
+\multicolumn{8}{c}{\normalsize\textit{Maximum Electromotive Intensity in Electrostatic Units.}\medskip} \\
+\hline
+\parbox[c]{\TmpLen}{\centering\tablespaceup Spark\\ Length\\ in cm.\tablespacedown} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $2$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $1$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $.5$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spark\\ Length\DPtypo{}{\\ in cm}.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $2$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $1$~cm.} &
+\parbox[c]{\TmpLen}{\centering Spheres,\\ diameter\\ $.5$~cm.} \\
+\hline
+$\tablespaceup.01$ & $336$ & $347$ & $372$ & $\Z.10$ & $166$ & $175$ & $190$ \\
+$.02$ & $258$ & $262$ & $277$ & $\Z.15$ & $155$ & $165$ & $190$ \\
+$.03$ & $224$ & $236$ & $240$ & $\Z.20$ & $148$ & $162$ & $198$ \\
+$.04$ & $206$ & $213$ & $222$ & $\Z.25$ & $145$ & $161$ & $204$ \\
+$.05$ & $194$ & $202$ & $215$ & $\Z.30$ & $143$ & $163$ & $215$ \\
+$.06$ & $184$ & $190$ & $202$ & $\Z.35$ & $143$ & $166$ & $226$ \\
+$.07$ & $175$ & $183$ & $193$ & $\Z.40$ & $142$ & $170$ & $236$ \\
+$.08$ & $172$ & $179$ & $192$ & $\Z.45$ & $142$ & $174$ & $249$ \\
+$.09$ & $165$ & $174$ & $187$ & $\Z.50$ & $144$ & $180$ & $256$ \\
+$.10$ & $164$ & $171$ & $187$ & $\Z.55$ & $145$ & $184$ & $265$ \\
+$.11$ & $160$ & $167$ & $183$ & $\Z.60$ & $145$ & $190$ & $272$ \\
+$.12$ & $159$ & $167$ & $185$ & $\Z.70$ & $148$ & $196$ & $281$ \\
+$.14$ & $154$ & $164$ & $187$ & $\Z.80$ & $151$ & $205$ & $288$ \\
+      &       &       &       & $\Z.90$ &       &       & $293$ \\
+      &       &       &       & $1.00$  &       &       & $301$ \\
+      &       &       &       & $1.20$  &       &       & $312$ \\
+      &       &       &       & $1.50$  &       &       & $327$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+%% -----File: 096.png---Folio 82-------
+
+\index{Gaugain, spark discharge}%
+\Article{57} It will be seen from these tables that the smaller the
+spheres, or in other words the more irregular the electric field,
+the greater the value of the maximum electromotive intensity.
+This is sometimes expressed by saying that the curvature of
+the electrodes increases the electric strength of the gas, and
+Gaugain (\textit{Annales de Chimie et de Physique}, [iv]~8, p.~75, 1866)
+has found that when the spark passes between two coaxial
+cylinders, the maximum value~$R$ of the electromotive intensity
+can be expressed by an equation of the form
+\[
+R = \alpha + \beta r^{-\frac{1}{3}},
+\]
+where $\alpha$~and~$\beta$ are constants and $r$~is the radius of the inner
+cylinder.
+
+\Article{58} The variations in the value of the electromotive intensity
+are so great that they prove that it is not the value of the
+electromotive intensity which primarily determines whether or
+not discharge must take place; and it is probable that the use of
+this quantity as the measure of the electric strength has retarded
+the progress of this subject by withdrawing attention from the
+most important cause of the discharge to this which is probably
+merely secondary.
+
+\index{Electromotive intensity, required to produce a spark in a variable field@\subdashtwo required to produce a spark in a variable field}%
+\index{Spark, potential difference required to produce in a variable field@\subdashone potential difference required to produce in a variable field}%
+\Article{59} The following results taken from Paschen's experiments
+show that when the sparks are not too long the variations in
+the electromotive intensity are very much greater than the variations
+in the potential difference; suggesting that for such
+sparks the potential difference is the most important consideration.
+\begin{center}
+\tabletextsize
+\settowidth{\TmpLen}{Radius of Electrodes}%
+\begin{tabular}{c|l|l|l|@{\;}l}
+\hline
+\parbox[c]{\TmpLen}{\tablespaceup\centering Radius of Electrodes\\ in cm.\tablespacedown} &
+\PadTo{99.9}{1.} & \PadTo{99.9}{.5} & \PadTo{99.9}{.25} & \\
+\hline
+\tablespaceup Potential Difference & $13.4$ & $13.6$ & $13.8$ & \multirow{2}{*}{$\Big\}$ Spark length $.08$~cm.} \\
+Maximum Intensity & $172$ & $179$ & $192$ & \tablespacedown \\
+\tablespaceup Potential Difference & $20.5$ & $20.8$ & $21.3$ & \multirow{2}{*}{$\Big\}$ Spark length $.14$~cm.} \\
+Maximum Intensity & $154$ & $164$ & $187$ & \tablespacedown \\
+\tablespaceup Potential Difference & $49.2$ & $49.3$ & $47.6$ & \multirow{2}{*}{$\Big\}$ Spark length $.40$~cm.} \\
+Maximum Intensity & $142$ & $170$ & $236$ &\tablespacedown  \\
+\tablespaceup Potential Difference & $87.8$ & $81.9$ & $66.4$ & \multirow{2}{*}{$\Big\}$ Spark length $.80$~cm.} \\
+Maximum Intensity & $151$ & $2054$ & $288$ &\tablespacedown  \\
+\hline
+\end{tabular}
+\end{center}
+
+\Article{60} We can explain by the following geometrical illustration
+the two effects produced by the irregularity of the field---the
+diminution in the potential difference, and the increase in the
+maximum electromotive intensity. When a discharge is passing
+%% -----File: 097.png---Folio 83-------
+through gas, we shall see later on, from the consideration of the
+discharge at low pressures, reasons for believing that the distribution
+of potential during discharge may be approximately
+represented by the equation
+\[
+V = \alpha + \beta l,
+\]
+where $\alpha$ and~$\beta$ are constants and $l$~the distance from the
+negative electrode.
+\includegraphicsouter{fig23}{Fig.~23.}
+If the curve representing the distribution
+of potential before discharge cuts the curve representing the
+distribution after discharge, a spark will pass, while if it does
+not cut it, no discharge can take place.
+
+In \figureref{fig23}{Fig.~23}, \smallsanscap{A},~\smallsanscap{B} represent the electrodes, \smallsanscap{CD}~the distribution
+of potential during the discharge.
+If the electric field is uniform
+the curve which represents
+the distribution of potential before
+the spark passes is a straight line
+such as~\smallsanscap{AE}, as the intensity of the
+field increases \smallsanscap{E}~moves higher and
+higher, the first point at which it
+intersects the curve representing the
+distribution of potential after discharge
+being~\smallsanscap{D}. In this case the
+difference of potential between the
+electrodes when the spark passes is~\smallsanscap{BD}, so that the relation
+between the potential difference~$V$ and the spark length~$l$ is
+\[
+V = \alpha + \beta l.
+\]
+When however the electric field is not uniform it is possible
+for the curve representing the potential before discharge to
+intersect the potential curve after discharge, even though the
+difference of potential before discharge is less than~\smallsanscap{BD}.
+will be evident from \figureref{fig23}{Fig.~23}, where the curved line represents
+the distribution of potential in an irregular field. Here we have
+a very rapid change in potential in the neighbourhood of one
+of the electrodes, followed by a comparatively slow rate of change
+midway between them. In this case the curves intersect and a
+discharge would take place, though the difference of potential
+between the electrodes is less than that required for sparking in
+a uniform field. Thus for equal spark lengths the potential
+difference may be less when the field is variable than when it is
+%% -----File: 098.png---Folio 84-------
+uniform. Again, we notice that the slope of this curve in the
+neighbourhood of the electrode~\smallsanscap{A} is steeper than that of a line
+joining \smallsanscap{A}~and~\smallsanscap{D}, in other words the maximum electromotive
+intensity when discharge takes place is greater when the field
+is variable than when it is uniform. Both these results are confirmed
+by Baille's and Paschen's observations.
+
+For a theory of the spark discharge the reader is referred to
+the discussion at the end of this chapter.
+
+\Article{61} It is sometimes said that the reason a thin layer of gas is
+electrically stronger than a thick one is, that a film of condensed
+gas is spread over the surface of the electrodes, and that this film
+is electrically stronger than the free gas. This consideration however,
+\index{Chrystal on spark discharge}%
+as Chrystal (\textit{Proc.\ Roy.\ Soc.\ Edin.},~11, 1881--2, p.~487)
+has pointed out, is quite incapable of explaining the variation in
+electric strength, for it is evident that if this were all that had to
+be taken into account the discharge would pass whenever the
+electromotive intensity was great enough to break through this
+film of condensed gas, so that this intensity would be constant
+when the spark passed whatever the thickness of the layer
+of free gas.
+
+\nbpagebreak
+\Subsection{Connection between Spark Potential and the Pressure of the Gas.}
+
+\index{Pressure, connection between and spark potential@Pressure, connection between and spark potential|indexetseq}%
+\index{Spark, potential effect of pressure on@\subdashone potential effect of pressure on|indexetseq}%
+\index{Critical pressure@`Critical' pressure}%
+\index{Pressure, critical@\subdashone critical}%
+\Article{62} The general nature of this connection is as follows: as
+the pressure of the gas diminishes the difference of potential required
+to produce a spark of given length also diminishes, until
+the pressure falls to a critical value depending upon the length
+of the spark, the nature of the gas, the shape and size of the
+electrodes and of the vessel in which the gas is contained; at
+this pressure the potential difference is a minimum, and any
+further diminution in the pressure is accompanied by an increase
+in the potential difference. The critical pressure varies very
+\index{Peace, spark potential|indexetseq}%
+greatly with the length of the spark; in Mr.~Peace's experiments,
+which we shall consider later, when the spark length was about
+$1/100$~of a millimetre, the critical pressure was that due to about
+$250$~mm.\ of mercury, while for sparks several millimetres long
+the critical pressure was less than that due to $1$~mm.\ of mercury.
+
+\includegraphicsmid[!b]{fig24}{Fig.~24.}
+
+\Article{63} At pressures considerably greater than the critical pressure,
+the curve which represents the relation between potential
+difference and pressure, the spark length being constant, approximates
+to a straight line, or more accurately to a slightly curved
+%% -----File: 099.png---Folio 85-------
+\index{Wolf, effect of pressure on spark potential}%
+hyperbola concave with respect to the axis along which the
+pressures are measured. Thus Wolf, who has determined (\textit{Wied.\
+Ann.}~37.\ 306, 1889) the potential difference required to produce a
+spark through air, hydrogen, carbonic acid, oxygen and nitrogen
+at pressures varying from $1$~to~$5$ atmospheres, found that the
+electromotive intensity,~$y$, required to produce a spark across a
+length of $1$~mm.\ between electrodes $5$~cm.\ in radius when the
+pressure was $x$~atmospheres, could be expressed by the following
+equations:---
+\begin{center}
+\begin{tabular}{p{15em}@{}l}
+For hydrogen \wdotfill      & $y = 65.09x + 62$. \\
+For oxygen \wdotfill        & $y = \PadTo[l]{65.09x}{96.0x} + 44$. \\
+For air \wdotfill           & $y = \PadTo[l]{65.09x}{107x}  + 39$. \\
+For nitrogen \wdotfill      & $y = 120.8x + 50$. \\
+For carbonic acid \wdotfill & $y = 102.2x + 72$.
+\end{tabular}
+\end{center}
+
+\includegraphicsmid[!t]{fig25}{Fig.~25.}
+
+\index{Baille, spark discharge}%
+\index{Macfarlane, spark potential}%
+\index{Paschen, spark discharge}%
+\Article{64} For pressures less than one atmosphere the connection
+between spark length and pressure has been investigated by
+Baille (\textit{Annales de Chimie et de Physique}, [5]~29, p.~181, 1883),
+Macfarlane (\textit{Phil.\ Mag.}\ [5]~10, p.~389, 1880), and Paschen (\textit{Wied.\
+Ann.}~37, p.~69, 1889), who have found that the relation is
+graphically represented by very slightly curved portions of a hyperbola.
+%% -----File: 100.png---Folio 86-------
+Paschen (\textit{l.c.}\ p.~91) made the interesting observation that
+as long as the product of the density and spark length is constant
+the sparking potential is for a considerable range of pressure
+constant for the same gas. This result can also be expressed by
+saying that the sparking potential for a gas can be expressed in
+terms of the ratio of the spark length to the mean free path of
+the molecules of the gas. The curves given in \figureref{fig24}{Fig.~24}, which
+represent for air, hydrogen and carbonic acid the relation between
+the spark potential in electrostatic units as ordinates, and the
+products of the pressure of the gas in centimetres of mercury
+and the spark length in centimetres as abscissæ, seem to show
+that this relation is approximately a linear one.
+
+\includegraphicsouter{fig26}{Fig.~26.}
+
+\Article{65} The preceding experiments were made at pressures much
+greater than the critical pressure. A series of very interesting
+experiments has lately been made by Mr.~Peace in the
+Cavendish Laboratory, Cambridge, on the shape of these curves
+in the neighbourhood of the critical pressure. In these experiments
+the potential difference could be determined with great
+accuracy, as it was produced by a large number of small storage
+%% -----File: 101.png---Folio 87-------
+cells whose E.~M.~F. could very easily be determined. Mr.~Peace's
+curves are represented
+in Figs.~\figureref{fig25}{25}, \figureref{fig26}{26}, \figureref{fig27}{27},~\figureref{fig28}{28}.
+\figureref{fig25}{Fig.~25} represents
+the relation between
+potential difference in
+air and pressure for
+spark lengths varying
+from $.0010$~cm.\ to
+$.2032$~cm. \figureref{fig26}{Fig.~26} represents
+the relation
+between electromotive
+intensity and
+pressure for the same
+spark lengths, and
+\figureref{fig27}{Fig.~27} the relation
+between potential difference
+and spark
+length for a series of
+different pressures:
+the curve representing the relation between electromotive
+intensity and spark length is given in \figureref{fig28}{Fig.~28}. These curves
+%% -----File: 102.png---Folio 88-------
+will be seen to present several points of great interest. In the
+\index{Critical pressure, effect of spark length on@\subdashtwo effect of spark length on}%
+\index{Spark, length, effect of on critical pressure@\subdashone length, effect of on critical pressure}%
+first place, \figureref{fig25}{Fig.~25} shows how much the critical pressure depends
+upon the spark length; this will also be seen from the following
+table:---
+\includegraphicsmid[!t]{fig27}{Fig.~27.}
+\begin{center}
+\tabletextsize
+\begin{tabular}{c|c|c}
+\hline
+\settowidth{\TmpLen}{Spark Length.}%
+\parbox[c]{\TmpLen}{\centering Spark Length.} &
+\settowidth{\TmpLen}{Potential Difference.}%
+\parbox[c]{\TmpLen}{\tablespaceup\centering Minimum\\ Potential Difference.\tablespacedown} &
+\settowidth{\TmpLen}{Critical Pressure.}%
+\parbox[c]{\TmpLen}{\centering Critical Pressure.} \\
+\hline
+\tablespaceup$.0010$~cm.\Z  & $326$~volts. & $250$~mm. \\
+$.00254$~cm. & $330$~volts. & $150$~mm. \\
+$.00508$~cm. & $333$~volts. & $110$~mm. \\
+$.01016$~cm. & $354$~volts. & $\Z55$~mm. \\
+$.02032$~cm. & $370$~volts. & $\Z35$~mm.\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+Thus when the spark length was increased twenty-fold the
+critical pressure was reduced from $250$~mm.\ to~$35$~mm. Another
+very remarkable feature is the small variation in the minimum
+potential difference required to produce the spark. In the preceding
+table there is a very considerable range of pressure, but
+the variation in the potential difference is comparatively small.
+Mr.~Peace too made the interesting observation that he could
+not produce a spark however near he put the electrodes together
+%% -----File: 103.png---Folio 89-------
+or however the pressure was altered, if the potential difference
+was less than something over $300$~volts. Gases in this respect
+seem to resemble electrolytes which require a finite difference of
+potential to produce a steady current through them. This constancy
+in the minimum value of the potential required to produce
+a spark seems additional evidence that the passage of the
+spark is regulated more by the value of the potential difference
+than by that of the electromotive intensity. Another thing to
+be remarked about the curves in \figureref{fig25}{Fig.~25} is the way in which
+they get flatter and flatter as the spark length diminishes: the
+flatness of the curve corresponding to the spark length $.0010$~cm.,
+or $.0004$~inch, is so remarkable that I give the numbers from
+which it was drawn:---
+\begin{center}
+\tabletextsize
+\begin{tabular}{c|c|c}
+\multicolumn{3}{c}{\normalsize\textit{Spark Length} $.00101$~cm.\medskip} \\
+\hline
+\settowidth{\TmpLen}{Pressure in mm.}%
+\parbox[c]{\TmpLen}{\centering\tablespaceup Pressure in mm.\\ of Mercury.\tablespacedown} &
+\settowidth{\TmpLen}{Potential Difference.}%
+\parbox[c]{\TmpLen}{\centering Potential Difference\\ in Volts.} &
+\settowidth{\TmpLen}{Electromotive}%
+\parbox[c]{\TmpLen}{\centering Electromotive\\ Intensity.} \\
+\hline
+$\tablespaceup\Z20$ & $433$ & $1420$ \\
+$\Z30$ & $398$ & $1310$ \\
+$\Z40$ & $380$ & $1245$ \\
+$\Z50$ & $370$ & $1215$ \\
+$\Z60$ & $357$ & $1170$ \\
+$\Z70$ & $353$ & $1160$ \\
+$\Z80$ & $349$ & $1145$ \\
+$\Z90$ & $346$ & $1135$ \\
+$100$ & $343$ & $1125$ \\
+$120$ & $337$ & $1105$ \\
+$140$ & $332$ & $1090$ \\
+$160$ & $330$ & $1085$ \\
+$180$ & $329$ & $1080$ \\
+$200$ & $328$ & $1075$ \\
+$240$ & $326$ & $1070$ \\
+$280$ & $327$ & $1072$ \\
+$300$ & $328$ & $1075$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+\includegraphicsmid{fig28}{Fig.~28.}
+
+The curves representing the relation between potential difference
+and pressure for different lengths of spark cut each other; this
+indicates that at a pressure lower than that where the curves cut
+it requires a greater potential difference to produce the short
+spark than it does the long one. This point has already been
+considered in \artref{53}{Art.~53}.
+
+\Article{66} The connection between the critical pressure and the
+spark length proves that the gas at the critical pressure when
+conveying the electric discharge has a structure of which the
+linear measure of the coarseness is comparable with the spark
+length. This spark length is very much greater than the mean
+%% -----File: 104.png---Folio 90-------
+free path of the molecules, and thus these experiments show that
+a gas conveying electrical discharge possesses a much coarser
+structure than that recognized by the ordinary Kinetic Theory of
+Gases. For the nature of this structure we must refer to the
+general theory of the electrical discharge given at the end of this
+chapter.
+
+\Article{67} Although the magnitude of the critical pressure depends,
+as we have seen, to a very great extent on the distance between
+the electrodes, the actual existence of a critical pressure does not
+seem to depend on the presence of electrodes. In \artref{74}{Art.~74} a
+method is described by which an endless ring discharge can be
+produced in a bulb containing gas at a low pressure; in this case
+the discharge is in the gas throughout the whole of its course,
+and there are no electrodes. If in such an experiment the bulb
+is connected to an air pump it will be found that when the
+pressure of the gas in the bulb is high no discharge at all is
+visible; as however the pressure is reduced a discharge gradually
+appears and increases in brightness until the pressure is reduced
+to a small fraction of a millimetre, when the brightness is a
+maximum; when the pressure is reduced below this value the
+discharge has greater difficulty in passing, it gets dimmer and
+dimmer, and finally stops altogether when the exhaustion is very
+great. This experiment shows that there is a critical pressure
+even when there are no electrodes, but that it is very much lower
+than in an ordinary sized tube when electrodes are used.
+
+\index{Muller@Müller and de la Rue, electric discharge}%
+\index{De la Rue and Müller, discharge through gases}%
+\Article{68} De~la~Rue and Hugo Müller (\textit{Proc.\ Roy.\ Soc.}~35, p.~292,
+1883), using the ordinary discharge with electrodes, found that
+the critical pressure depends on the diameter of the tube in
+which the rarefied gas is confined, the critical pressure getting
+lower as the diameter of the tube is increased.
+
+\Subsection{Potential Difference required to produce Sparks through
+various Gases.}
+\index{Faraday, spark potential through different gases@\subdashone spark potential through different gases}%
+\index{Potential difference required to produce a spark in different gases@\subdashtwo required to produce a spark in different gases}%
+\index{Spark, potential in different gases@\subdashone potential in different gases}%
+
+\Article{69} The potential difference required to send a spark between
+the same electrodes, separated by the same distance, depends, as
+Faraday found, on the nature of the gas surrounding the electrodes:
+thus, for example, the potential difference required to
+produce a spark of given length in hydrogen is much less
+than in air. Measurements of the potential differences required
+to produce discharge through a series of gases have been made
+%% -----File: 105.png---Folio 91-------
+\index{Baille, spark discharge}%
+\index{Liebig, spark potential}%
+\index{Paschen, spark discharge}%
+by, among others, Faraday, Baille (\textit{Annales de Chimie et de
+Physique}, [5]~29, p.~181, 1883), Liebig (\textit{Phil.\ Mag.}\ [5]~24, p.~106,
+1887), Paschen (\textit{Wied.\ Ann.}~37, p.~69, 1889). The results obtained
+by different observers seem to differ very largely. This will be
+seen from the following table, in which Paschen gives the ratio
+of the potential difference required to spark across hydrogen or
+carbonic acid, to the potential difference required to spark across
+a layer of air of the same thickness, the pressure for all the gases
+being $750$~mm.\ of mercury.
+\begin{center}
+\tabletextsize
+\begin{tabular}{c|c|c|c|c|c|c}
+\hline
+\settowidth{\TmpLen}{Spark length in}%
+\multirow{2}{\TmpLen}{\parbox[c]{\TmpLen}{\centering Spark length in\\ centimetres.}} &
+\multicolumn{3}{c|}{\tablespaceup Hydrogen.} & \multicolumn{3}{c}{Carbonic Acid.} \\
+\cline{2-7}
+&\tablespaceup Baille. & Liebig. & Paschen. & Baille. & Liebig. & Paschen.\tablespacedown \\
+\hline
+\tablespaceup$.1$ & $.49$ & $.873$ & $.639$ & $1.67$  & $1.20\Z$ & $1.05\Z$ \\
+$.2$ & $.49$ & $.787$ & $.578$ & $1.24$  & $1.16\Z$ & $\Z.988$ \\
+$.3$ & $.50$ & $.753$ & $.560$ & $\Z.94$ & $1.07\Z$ & $\Z.962$ \\
+$.4$ & $.50$ & $.704$ & $.553$ & $\Z.76$ & $1.03\Z$ & $\Z.930$ \\
+$.5$ & $.50$ & $.670$ & $.548$ &         & $\Z.994$ & $\Z.910$ \\
+$.6$ &       & $.656$ & $.555$ &         & $\Z.974$ & $\Z.940$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+It will be seen that, though the numbers got by different
+observers differ very widely, they all agree in making carbonic
+acid stronger than air for short sparks and weaker than it for
+long. This would indicate that in the formula
+\[
+V = \alpha + \beta l,
+\]
+which gives the spark potential~$V$ in terms of the spark length~$l$,
+$\alpha$~for carbonic acid is greater than $\alpha$~for air, while $\beta$~for carbonic
+acid is less than $\beta$~for air.
+
+It will be seen from \figureref{fig24}{Fig.~24}, which contains Paschen's curves
+showing the relation between potential difference and pressure
+for air, hydrogen and oxygen, that these curves cut each other;
+thus the relation between their `electric strengths' depends to
+a large extent upon the pressure. Liebig's curves for air,
+hydrogen, carbonic oxide and coal gas were given in \figureref{fig19}{Fig.~19}.
+
+\index{Rontgen, discharge through gases@\subdashone discharge through gases}%
+\Article{70} Röntgen (\textit{Göttinger Nachrichten}, 1878, p.~390) arrived at
+the conclusion that the potential difference required to produce
+a spark of given length in different gases was, approximately,
+inversely proportional to the mean free path of the molecules of
+the gas. This approximation, if it exists at all, must be exceedingly
+rough, for we have seen that the relation between the
+potential differences required to spark through different gases
+%% -----File: 106.png---Folio 92-------
+depends on the spark length and the pressure of the gases. If
+the result found by Mr.~Peace for air (\artref{65}{Art.~65}),---that the
+minimum potential difference required to produce a spark varied
+very little with the spark length,---were to hold for other gases,
+there would be much more likelihood of this minimum potential
+difference being connected with some physical or chemical
+property of the gas, than the potential difference required to
+produce a spark of arbitrary length at a pressure chosen at
+random being so connected.
+
+\Article{71} If a permanent gas in a closed vessel be heated up to
+\index{Spark, x effect of temperature on@\subdashtwo effect of temperature on}%
+\index{Temperature, effect of, on spark potential}%
+\index{Cardani, effect of temperature on electric strength of gases}%
+$300°$\,C, the discharge potential does not change (see Cardani,
+\textit{Rend.\ della R.~Acc.\ dei Lincei},~4, p.~44, 1888; J.~J. Thomson,
+\textit{Proc.\ Camb.\ Phil.\ Soc.}, vol.~6, p.~325, 1889): if however the
+vessel be open so that the pressure remains constant, there will
+be a diminution in the discharge potential due to the diminution
+in density. When the temperature gets so high that chemical
+changes such as dissociation take place in the gas the discharge
+potential may fall to zero.
+
+\index{Damp air, potential required to spark through}%
+A great number of experiments have been made on the
+relative `electric strengths' of damp and dry air. The only
+observer who seems to have found any difference is Baille, and
+in his case the difference was so large as to make it probable
+that some of the water vapour had condensed into drops.
+
+
+\Subsection{Phenomena accompanying the Electric Discharge at Low
+Pressures.}
+
+\Article{72} When the discharge passes between metallic electrodes
+sealed into a tube filled with gas at a low pressure, the appearance
+it presents is very complicated: many of the effects observed
+in the tube are however evidently due to the action of the
+electrodes, as the phenomena at the anode are very different
+from those at the cathode; it therefore appears desirable to begin
+the study of the phenomena shown in vacuum tubes by investigating
+the discharge when no electrodes are present.
+
+\index{Discharge electrodeless@\subdashone electrodeless|indexetseq}%
+\index{Electrodeless discharge@Electrodeless discharge|indexetseq}%
+\Article{73} If we wish to produce the endless discharge in a closed
+vessel without electrodes, we must produce in some way or
+another round a closed curve in the vessel an electromotive
+force large enough to break down the insulation of the gas.
+Since, for discharge to take place, the electromotive force round
+a closed curve must be finite, it cannot be produced electrostatically,
+%% -----File: 107.png---Folio 93-------
+we must use the electromotive forces produced by
+electromagnetic induction, and make the closed curve in the
+exhausted vessel practically the secondary of an induction coil.
+As the primary of this induction coil I have used a wire connecting
+the inside and outside coatings of a Leyden jar; when
+the jar is discharged through the wire enormous currents pass
+for a short time backwards and forwards along the wire, the
+currents when the wire is short and the jar small reversing their
+directions millions of times in a second. We thus have here all
+the essentials for producing a very large electromotive force
+round the secondary, viz.~a very intense current in the primary
+and an exceedingly rapid rate of alternation of this current;
+and though the electromotive force only lasts for an exceedingly
+short time, it lasts long enough to produce the discharge through
+the gas and to enable us to study its appearance.
+
+\Article{74} Two convenient methods of producing the discharge are
+shown in \figureref{fig29}{Fig.~29}: in the one on the right two jars are used, the
+outside coatings of which (\smallsanscap{A}~and~\smallsanscap{B}) are connected by a wire in
+which a few turns~\smallsanscap{C} are made; \smallsanscap{C}~forms the primary coil. The
+inside coatings of these jars are connected, one to one terminal~\smallsanscap{E}
+of a Wimshurst electrical machine or of an induction coil, the
+other coating to~\smallsanscap{F}, the other terminal of such a machine. If the
+tubes in which the discharge is to be observed are spherical bulbs,
+they are placed inside the coil~\smallsanscap{C}; if they are endless tubes, they
+are placed just outside it. When the difference of potential between
+\smallsanscap{E}~and~\smallsanscap{F} becomes great enough to spark across~\smallsanscap{EF}, the
+%% -----File: 108.png---Folio 94-------
+jars are discharged and electrical oscillations set up in the wire~\smallsanscap{ACB}.
+The oscillating currents in the primary produce a large
+electromotive intensity in its neighbourhood, sufficient under
+favourable conditions to cause a bright discharge to pass through
+the rarefied gas in the bulb placed inside the coil.
+
+\medskip
+\includegraphicsmid{fig29}{Fig.~29.}
+
+We have described in \artref{26}{Art.~26} the way in which the Faraday
+tubes, which before the spark took place were mainly in the
+glass between the two coatings of the jars, spread through the
+region outside the jars, as soon as the discharge passes, keeping
+their ends on the wire~\smallsanscap{ACB}. They will pass in their journey
+through the bulb in the coil~\smallsanscap{C}, and if they congregate there in
+sufficient numbers the electromotive force will be sufficient to
+cause a discharge to pass through the gas. Anything which
+concentrates the Faraday tubes in the bulb will increase the
+brightness of the discharge through it.
+
+\Article{75} It is necessary to prevent the coil~\smallsanscap{C} getting to a high
+potential before the spark passes, otherwise it may induce a
+negative electrification on the parts of the inside of the glass bulb
+nearest to it and a positive electrification on the parts more
+remote: when the potential of the coil suddenly falls in consequence
+of the passage of the spark, the positive and negative
+electricities will rush together, and in so doing may pass through
+the rarefied gas in the bulb and produce luminosity. This
+luminosity will spread throughout the bulb and will not be
+concentrated in a well-defined ring, as it is when it arises from
+the electromotive force due to the alternating currents passing
+along the wire~\smallsanscap{ACB}. This effect may explain the difference in
+the appearance presented by the discharge in the following experiments,
+where the discharge passes as a bright ring, from that observed
+\index{Hittorf, discharge through gases}%
+by Hittorf (\textit{Wied.\ Ann.}~21, p.~138, 1884), who obtained
+the discharge in a tube by twisting round it a wire connecting
+the two coatings of a Leyden jar: in Hittorf's experiment the
+luminosity seems to have filled the tube and not to have been
+concentrated in a bright ring. To prevent these electrostatic
+effects, due to causes which operate before the electrical oscillations
+in the wires begin, the coil~\smallsanscap{C} is connected to earth, and as
+an additional precaution the discharge tube may be separated
+from the coil by a screen of blotting paper moistened with dilute
+acid. The wet blotting paper is a sufficiently good conductor to
+screen off any purely electrostatic effect, but not a good enough
+%% -----File: 109.png---Folio 95-------
+one to interfere to any appreciable extent with the electromotive
+forces arising from the rapidly alternating currents.
+
+\Article{76} If $C$~is the capacity of the jars, $L$~the coefficient of self-induction
+of the discharging circuit, then if the difference
+of potential between the terminals of the electric machine is
+initially~$V_0$, $\gamma$~the current through the wire at a time~$t$ after
+the spark has passed will (Chap.~IV) be given by the equation
+\[
+\gamma = \frac{CV_0}{(LC)^{\frac{1}{2}}} \sin \frac{t}{(LC)^{\frac{1}{2}}}\;,
+\]
+supposing as a very rough approximation that there is no decay
+either from resistance or radiation in the vibrations.
+
+The rate of variation of the current,~$\dot{\gamma}$, is thus given by the
+equation
+\[
+\dot{\gamma} = \frac{V_0}{L} \cos \frac{t}{(LC)^{\frac{1}{2}}}\;.
+\]
+Thus if $M$~is the coefficient of mutual induction between the
+primary and a secondary circuit, the maximum electromotive
+force round the secondary will be~$MV_0/L$, which for a given
+spark length is independent of the capacity of the jars. But
+though the maximum electromotive force does not depend
+upon the capacity of the jars, the oscillations will last longer
+when the jars have a large capacity than when they have
+a small one, as the energy to begin with is greater; hence,
+though it is possible to get the discharge with jars whose
+capacity is not more than $70$~or~$80$ in electrostatic measure, it is
+not nearly so bright as when larger capacities are used. The best
+number of turns to use in the coil is that which makes $M/L$ a
+maximum. If~$n$ is the number of turns, then $M$~and~$L$ will be
+respectively of the forms~$\beta n$ and $L_0 + \alpha n^2$, where $\alpha$~and~$\beta$ are
+constants and $L_0$~the self-induction of the part of the wire~\smallsanscap{ACB}
+not included in the coil; thus $M/L$~will be of the form
+\[
+\frac{\beta n}{L_0 + \alpha n^2}\;,
+\]
+and this is a maximum when $L_0 = \alpha n^2$, that is when the self-induction
+in the coil is equal to that in the rest of the circuit.
+Though the electromotive force is greatest in this case, in
+practice it is found to be better to sacrifice a little of the
+electromotive force for the sake of prolonging the vibrations;
+%% -----File: 110.png---Folio 96-------
+this can be done by increasing the self-induction of the coil.
+It is thus advisable to use rather more turns in the coil than is
+indicated by the preceding rule.
+
+
+\Subsection{Appearance of the Discharge.}
+
+\Article{77} Let us suppose that a bulb fused on to an air pump is
+placed within the coil~\smallsanscap{C}, and that the jars are kept sparking
+while the bulb is being exhausted. When the pressure is high,
+no discharge at all is to be seen inside the bulb; but when the
+exhaustion has proceeded until the pressure of the air has fallen
+to a millimetre of mercury or thereabouts, a thin thread of reddish
+light is seen going round the bulb in the zone of the coil. As the
+exhaustion proceeds still further, the brightness of this thread
+rapidly increases as well as its thickness; it also changes its colour,
+losing the red tinge and becoming white. Continuing the exhaustion,
+the luminosity attains a maximum and the discharge
+passes as a very bright and well-defined ring. When the pressure
+is still further diminished, the luminosity also diminishes, until
+when an exceedingly good vacuum is reached no discharge at
+all passes. The pressure at which the luminosity is a maximum
+is very much less than the pressure at which the electric
+strength is a minimum in a tube provided with electrodes and
+comparable in size with the size of the bulb; the former pressure
+is in air less than $1/200$~of a millimetre of mercury, while the
+latter is about half a millimetre.
+
+\index{Discharge between electrodes near together, electrodeless, critical pressure for@\subdashtwo critical pressure for}%
+\index{Critical pressure, for electrodeless discharges@\subdashtwo for electrodeless discharges}%
+\Article{78} We see from this result that the difficulty which is
+experienced in getting the discharge to pass through an ordinary
+vacuum tube when the pressure is very low is not altogether
+due to the difficulty of getting the electricity to pass from the
+electrodes into the gas, but that it also occurs in tubes without
+electrodes, though in this case the critical pressure is very much
+lower.
+
+\Article{79} The existence of a critical pressure can also be easily
+shown by putting some mercury in the bulb, and, when the bulb
+has been well exhausted, driving out the remainder of the air by
+heating the mercury and filling the bulb with mercury vapour.
+After this process has been repeated two or three times, the bulb
+should be fused off from the pump when full of mercury vapour.
+It will only be found possible to get a discharge through this bulb
+within a narrow range of temperature, between about $70°$~and~$160°$\,C;
+%% -----File: 111.png---Folio 97-------
+when the bulb is colder than this, the pressure of the
+mercury vapour is too small to allow the discharge to pass;
+when it is hotter, the vapour pressure is too great.
+
+\index{Discharge between electrodes near together, electrodeless, critical pressure for@\subdashtwo critical pressure for}%
+\index{Electrodeless discharge, existence of critical pressure for@\subdashtwo existence of critical pressure for}%
+The critical pressure can also be proved by using the principle
+that a conductor screens off the electromotive intensities due to
+rapidly alternating currents while an insulator does not. For
+this purpose we use two glass bulbs one inside the other, the inner
+bulb containing gas at such a pressure that the discharge can pass
+freely through it. The outer bulb contains nothing but mercury
+and mercury vapour, and is prepared in the way just described.
+If the primary coil is placed round the outer bulb, then, when
+the bulb is cold, the discharge passes through the inner bulb, but
+not through the outer, showing that at this low pressure the conductivity
+of the vapour in the outer bulb is not great enough for
+the vapour to act as an electrical screen to the inner bulb. If,
+however, the outer bulb is warmed, the vapour pressure of the
+mercury increases, and with it the conductivity; a discharge now
+passes through the outer bulb but not through the inner, the
+mercury vapour acting as a screen. When the temperature
+of the outer bulb is still further increased, the pressure of the
+mercury vapour gets so great that it ceases to conduct, and the
+discharge, as at first, passes through the inner bulb but not
+through the outer.
+
+\Article{80} These experiments show that after a certain exhaustion
+has been passed the difficulty of getting a discharge to pass
+through a highly exhausted tube increases as the exhaustion is
+increased. This result is in direct opposition to a theory which
+has found favour with some physicists, viz.~that a vacuum is a
+conductor of electricity. The reason advanced for this belief
+is that when the discharge passes through highly exhausted
+tubes provided with electrodes, the difficulty which it experiences
+in getting through such a tube, though very great, seems to be
+almost as great for a short tube as for a long one; from this it has
+been concluded that the resistance to the discharge is localised
+at the electrodes, and that when once the electricity has succeeded
+in escaping from the electrode it has no difficulty in
+making its way through the rare gas. But although there is no
+doubt that in a highly exhausted tube the rise in potential
+close to the cathode is great compared with the rise in unit
+length of the gas elsewhere, it does not at all follow that the latter
+%% -----File: 112.png---Folio 98-------
+vanishes or that it continually diminishes as the pressure is
+diminished. The experiment we have just described on the bulb
+without electrodes shows that it does not. Numerous other experiments
+of very different kinds point to the conclusion that a
+\index{Vacuum an insulator}%
+\index{Worthington, electric strength of a vacuum}%
+vacuum is not a conductor. Thus Worthington (\textit{Nature},~27, p.~434,
+1883) showed that electrostatic attraction was exerted across the
+best vacuum he could produce, and that a gold-leaf electroscope
+\index{Ayrton and Perry, specific inductive capacity of a `vacuum'}%
+would work inside it. Ayrton and Perry (\textit{Ayrton's Practical
+Electricity}, p.~310) have determined the electrostatic capacity of
+a condenser in a vacuum in which they estimated the pressure to
+be only $.001$~mm.\ of mercury. If the air at this pressure had
+been a good conductor the electrostatic capacity would have been
+infinite, instead of being, as they found, less than at atmospheric
+pressure. Again, if we accept Maxwell's Electromagnetic Theory
+of Light, a vacuum cannot be a conductor or it would be opaque,
+and we should not receive any light from the sun or stars.
+
+\index{Discharge between electrodes near together, electrodeless, difficulty of passing from one medium to another@\subdashtwo difficulty of passing from one medium to another}%
+\index{Electrodeless discharge, difficulty of passing from one medium to another@\subdashtwo difficulty of passing from one medium to another}%
+\index{Electric discharge, passage of across junction of a metal and a gas}%
+\index{Passage of electricity across junction of a metal and a gas}%
+\Article{81} The discharge has considerable difficulty in passing across
+the junction of a metal and rarefied gas. This can easily be shown
+by placing a metal diaphragm across the bulb in which the
+discharge takes place, care being taken that the diaphragm extends
+right up to the surface of the glass. In this case the
+discharge does not cross the metal plate, but forms two separate
+closed circuits, one circuit
+being on one side of the
+diaphragm, the other on
+the other. The nature of
+the discharge is shown in
+\figureref{fig30}{Fig.~30}, in which it is seen
+that it travels through a
+comparatively long distance
+in the
+\includegraphicsouter[15]{fig30}{Fig.~30.}
+rarefied gas to
+avoid the necessity of crossing
+a thin plate of a very
+good conductor. If the
+bulb, instead of merely
+being bisected by one diaphragm,
+is divided into six
+or more regions by a suitable number of diaphragms, it will be
+found a matter of great difficulty to get any discharge at all
+through it. The metal plate in fact behaves in this case almost
+%% -----File: 113.png---Folio 99-------
+exactly like a plate of an insulating substance such as mica,
+which when continuous also breaks the discharge up into as many
+circuits as there are regions formed by the mica diaphragms. When
+however small holes are bored through the mica diaphragms
+the discharge will not be split up into separate circuits, but will
+pass through these holes. By properly choosing the position of
+the holes relative to that of the primary coil, we can get an undivided
+discharge in part of the circuit branching in the neighbourhood
+of the diaphragm into as many separate discharges as
+there are holes through either side
+of the mica plate. The appearance
+presented by the discharge when
+there are two holes on each side
+of the mica plate is shown in
+\figureref{fig31}{Fig.~31}.
+
+\includegraphicsouter[12]{fig31}{Fig.~31.}
+
+\Article{82} A rarefied gas is usually regarded
+\index{Conductivity of rarefied gases}%
+\index{Gases, xhigh conductivity of rarefied@\subdashone high conductivity of rarefied|(}%
+as an exceedingly bad
+conductor, and the experiments of
+many observers, such as those of
+\index{Hittorf, discharge through gases}%
+Hittorf, De~la~Rue and Hugo
+Müller, have shown that when a
+tube provided with electrodes in the usual way and filled
+with such a gas is placed in a circuit round which there is a
+given electromotive force, it produces as great a diminution in
+the intensity of the current as a resistance of several million
+ohms would produce. This great apparent resistance, when the
+pressure of the gas is not too low, is principally due however to
+the difficulty which the discharge has in passing from the electrodes
+into the gas. If we investigate the amount of current sent
+by a given electromotive force round a circuit exclusively confined
+to the rarefied gas, we find that, instead of being exceedingly
+bad conductors, rarefied gases (at not too low a pressure) are, on
+the contrary, surprisingly good ones, having molecular conductivities---that
+is specific conductivities divided by the number of
+molecules in unit volume---enormously greater than those of any
+electrolytes with which we are acquainted.
+
+\Article{83} We cannot avail ourselves of any of the ordinary methods
+of measuring resistances to measure the resistance of rarefied
+gases to these electrodeless discharges; but while the
+very high frequency of the currents through our primary coil
+%% -----File: 114.png---Folio 100-------
+makes the ordinary methods of measuring resistances impracticable,
+it at the same time makes other methods available which
+would be useless if the currents were steady or only varied
+slowly. One such method, which is very easily applied, is based
+on the way in which plates made of conductors screen off the
+action of rapidly alternating currents. If a conducting plate be
+placed between a primary circuit conveying a rapidly alternating
+current and a secondary coil, the electromagnetic action of the
+currents induced in the plate will be opposed to that of the
+currents in the primary, so that the interposition of the plate
+diminishes the intensity of the currents induced in the secondary.
+When we are dealing with currents through the primary with
+frequencies as high as those produced by the discharge of a
+Leyden jar, the thinnest plate of any metal is sufficient to
+entirely screen off the primary from the secondary, and no
+currents at all are produced in the latter when a metal plate
+is interposed between it and the primary; we could not therefore
+use this method conveniently to distinguish between the conductivities
+of different metals. If however instead of a metal plate
+\index{Electrolytes, conductivity of@\subdashone conductivity of}%
+we use a layer of an electrolyte, the conductivity of the electrolyte
+is not sufficient to screen off from the secondary the effect of the
+primary unless the layer is some millimetres in thickness, and
+the worse the conductivity of the electrolyte the thicker will be
+the layer of it required to reduce the action of the primary
+on the secondary to a given fraction of its undisturbed value.
+By comparing the thicknesses of layers of different electrolytes
+which produce the same effect when interposed between the
+primary and the secondary we can, since this thickness is proportional
+to the specific resistance, determine the conductivity
+of electrolytes for very rapidly alternating currents (see J.~J.
+Thomson, \textit{Proc.\ Roy.\ Soc.}~45, p.~269, 1889).
+
+\Article{84} The conductivity of a rarefied gas can on this principle
+be compared with that of an electrolyte in the following way:
+\smallsanscap{A},~\smallsanscap{B},~\smallsanscap{C}, \figureref{fig32}{Fig.~32}, represents the section of a glass vessel shaped
+something like a Bunsen's calorimeter; in the inner portion
+\smallsanscap{ABC} of this vessel, which is exposed to the air, an exhausted
+tube~\smallsanscap{E} is placed. A tube from the outer vessel leads to a
+mercury pump which enables us to alter its pressure at will.
+The primary coil~\smallsanscap{LM} is wound round the outer tube. When
+the air in the outer tube is at atmospheric pressure, a discharge
+%% -----File: 115.png---Folio 101-------
+caused by the action of the primary passes through the tube~\smallsanscap{E};
+but when the pressure of the gas in the outer tube is reduced
+until a discharge passes through it, the discharge in~\smallsanscap{E} stops,
+showing that the currents induced in the gas in the outer vessel
+have been sufficiently intense to neutralise the direct action of the
+primary coil on the tube in~\smallsanscap{E}.
+
+\includegraphicsouter{fig32}{Fig.~32.}
+
+In order to compare the intensity of the currents through the
+rarefied gas with those produced under similar circumstances in
+an electrolyte, the outer vessel~\smallsanscap{ABC}, \figureref{fig32}{Fig.~32}, through which the
+discharge has passed is disconnected from
+the pump, and the portion which has previously
+been occupied by the rarefied gas
+is filled with water, to which sulphuric
+acid is gradually added. Pure water does
+not seem to produce any effect on the
+brightness of the discharge in~\smallsanscap{E}, but as
+more and more sulphuric acid is added to
+the water the discharge in~\smallsanscap{E} gets fainter
+and fainter, until when about $25$\footnotemark~per~cent.\
+\footnotetext{The actual percentage depends on the pressure of the gas as well as on what
+  kind of gas it is; the figures given above refer to an actual experiment.}
+by volume of sulphuric acid has been added
+the effect produced by the electrolyte seems
+to be as nearly as possible the same as that
+produced by the rarefied gas. Thus the
+currents through the rarefied gas must, since they produced
+the same shielding effect, be as intense as those through a
+$25$~per~cent.\ solution of sulphuric acid. The conductivity of
+the gas must therefore be as great as that of the mixture of
+sulphuric acid and water, which is one of the best liquid conductors
+we know. This shielding effect can be produced by the
+rarefied gas when its pressure is as low as that due to $1/100$~mm.\
+of mercury, while the number of molecules of sulphuric acid in a
+$25$~per~cent.\ solution is such as would, if the sulphuric acid were
+in a gaseous state, produce a pressure of about $100$~atmospheres.
+Thus, comparing the conductivity per molecule of the gas and
+of the electrolyte, the molecular conductivity of the gas is about
+seven and a-half million times that of sulphuric acid. The
+relation which the molecular conductivity of the gas bears to
+that of an electrolyte, which produces the same effect in shielding
+%% -----File: 116.png---Folio 102-------
+off the effects of the primary, depends upon the length of spark
+passing between the jars, and so upon the electromotive intensity
+acting on the gas: in other words, conduction through these
+gases does not obey Ohm's law: the conductivity instead of
+being constant increases with the electromotive intensity. This
+is what we should expect if we regard the discharge through
+the gas as due to the splitting up of its molecules: the greater
+the electromotive intensity the greater the number of molecules
+which are split up and which take part in the conduction of the
+electricity.
+
+\includegraphicsouter{fig33}{Fig.~33.}
+
+\Article{85} Another method by which we can prove the great conductivity
+of these rarefied gases at the pressures when they
+conduct best is by measuring the energy absorbed by a secondary
+circuit made of the rarefied gas when placed inside a primary
+circuit conveying a rapidly alternating current. We shall see,
+\chapref{Chapter IV.}{Chapter~IV}, that when a conductor, whose conductivity is comparable
+with that of electrolytes, is placed inside the primary
+coil, the amount of energy absorbed per unit time is proportional
+to the conductivity of the conductor; so that if we measure
+the absorption of energy by equal and similar portions of two
+electrolytes we can find the ratio of their conductivities. In
+the case of these electrodeless discharges we can easily compare
+the absorption of energy by two different secondary circuits
+in the following manner. In the primary circuit connecting
+the outside coatings of two jars, two loops,
+\smallsanscap{A}~and~\smallsanscap{B}, \figureref{fig33}{Fig.~33}, are made, a standard bulb
+is placed in~\smallsanscap{A} and the substance to be
+examined in~\smallsanscap{B}. When a large amount of
+energy is absorbed by the secondary in~\smallsanscap{B},
+the brightness of the discharge through
+the bulb placed in~\smallsanscap{A} is diminished, and
+by observing the brightness of this discharge
+we can estimate whether the absorption
+of energy by two different secondaries
+placed in~\smallsanscap{B} is the same. If, now, an exhausted bulb be placed
+in~\smallsanscap{B}, the brightness of the discharge of the \smallsanscap{A}~bulb is at once
+diminished; indeed it is not difficult so to adjust the spark by
+which the jars are discharged, that a brilliant discharge passes
+in~\smallsanscap{A} when the \smallsanscap{B}~bulb is out of its coil, and no visible discharge
+when it is inside the coil. To compare the absorption of energy
+%% -----File: 117.png---Folio 103-------
+by the rarefied gas with that by an electrolyte we have merely
+to fill the bulb with an electrolyte, and alter the strength of
+the electrolyte until the bulb when filled with it produces the
+same effect as when it contained the rarefied gas. It will be
+found that in order to produce as great an absorption of energy
+as that due to a comparatively inefficient bulb filled with
+rarefied air, a very strong solution of an electrolyte must be
+put into the bulb; while a bulb which is exhausted to the
+pressure at which it produces its maximum effect absorbs a
+greater amount of energy than when filled even with the
+best conducting electrolyte we can obtain. We conclude from
+these experiments that the very large electromotive intensities
+which are produced by the discharge of a Leyden jar can, when
+no electrodes are used, send through a rarefied gas when the
+pressure is not too low much larger currents than the same
+electromotive intensities could send through even the best conducting
+mixture of water and sulphuric acid.
+\index{Gases, xhigh conductivity of rarefied@\subdashone high conductivity of rarefied|)}%
+
+The results just quoted show that the conductivity, if estimated
+per molecule taking part in the discharge, is much higher for
+rare gases than even for metals such as copper or silver.
+
+\Article{86} The large values of the conductivities of these rarefied
+gases when no electrodes are used are in striking contrast
+to the almost infinitesimal values which are obtained when
+electrodes are present. This illustrates the reluctance which the
+discharge has to pass across the junction of a rarefied gas and a
+metal: the experiments described in \artref{81}{Art.~81} are a very direct
+proof of this peculiarity of the discharge. It seems also to be
+indicated, though perhaps not quite so directly, by some experiments
+\index{Dewar and Liveing, effects of metallic dust in discharge}%
+\index{Liveing and Dewar, dust in electric discharge}%
+made by Liveing and Dewar (\textit{Proc.\ Roy.\ Soc.}~48, p.~437,
+1890) on the spectrum of the discharge. They found that the
+spectrum of a discharge passing through a gas which holds in
+suspension a considerable quantity of metallic dust does not show
+any of the lines of the metal. This is what we should expect
+from the experiments described in \artref{81}{Art.~81}, as these show that
+the discharge would take a very round-about course to avoid
+passing through the metal.
+
+\Article{87} There seem some indications that this reluctance of the
+discharge to pass from one substance to another extends also
+to the case when both substances are in the gaseous state, and
+that when the discharge passes through a mixture of two gases
+%% -----File: 118.png---Folio 104-------
+\smallsanscap{A}~and~\smallsanscap{B}, the discharges through~\smallsanscap{A} and through~\smallsanscap{B} respectively
+are in parallel rather than in series: in other words, that the
+polarized chains of molecules, which are formed before the discharge
+passes, consist some of \smallsanscap{A}~molecules and some of \smallsanscap{B}~molecules,
+but that the chains conveying the discharge do not consist
+partly of~\smallsanscap{A} and partly of \smallsanscap{B}~molecules. Thus, if the discharge is
+passing through a mixture of hydrogen and nitrogen, the chains
+in which the molecules split up and along which the electricity
+passes may be either hydrogen chains or nitrogen chains, but not
+chains containing both hydrogen and nitrogen. This seems to be
+indicated by the fact that when the discharge passes through a
+mixture of hydrogen and nitrogen, the spectrum of the discharge
+may, though a considerable quantity of nitrogen is present, show
+nothing but the hydrogen lines.
+
+\index{Crookes on discharge through gases}%
+Crookes' observations on the striations in a mixture of gases
+(\textit{Presidential Address to the Society of Telegraph Engineers},
+1891) seem also to point to the conclusion that the discharges
+through the different gases in the mixture are separate; for he
+found that when several gases are present in the discharge
+tube, different sets of striations, \artref{99}{Art.~99}, are found when the
+discharge passes through the tube, the spectrum of the bright
+portions of the \DPtypo{striae}{striæ}
+in one set showing the
+lines of one, and only
+one, of the gases in the
+mixture; the spectrum
+of another set showing
+the lines of another of
+the gases and so on,
+indicating that the discharges
+through the
+components of the mixture
+are distinct.
+
+\includegraphicsouter{fig34}{Fig.~34.}
+
+\Article{88} When the discharge
+can continue in the same medium all the way it can
+traverse remarkably long distances, even though the greater
+portion of the secondary may be of such a shape as not to
+add anything to the electromotive force acting round it. Thus,
+for example, the discharge will pass through a very long
+secondary, even though the tube of which this secondary is made
+%% -----File: 119.png---Folio 105-------
+\index{Discharge between electrodes near together, electrodeless, action of magnet on@\subdashtwo action of magnet on}%
+is bent up so that the greater part of it is at right angles to the
+electromotive intensity acting upon it. By using square coils
+with several turns for the primaries, I have succeeded in sending
+discharges through tubes of this kind over $12$~feet in length.
+On the other hand, there will be no discharge through a rarefied
+gas if the shape of the tube in which it is contained is such that
+the electromotive force round it is either zero or very small: it
+is impossible, for example, to get a discharge of this kind through
+a tube shaped like the one shown in \figureref{fig34}{Fig.~34}.
+
+\Subsection{Action of a Magnet on the Electrodeless Discharge.}
+\index{Magnets, action of, on electrodeless discharge}%
+\index{Electrodeless discharge, action of magnet on@\subdashtwo action of magnet on}%
+
+\Article{89} A magnet deflects the discharge through a rarefied gas in
+much the same way as it does a flexible wire carrying a current
+which flows in the same direction as the one through the gas. As
+the electrodeless discharges through the rarefied gas are oscillatory,
+they are when under the action of a magnet separated into two
+distinct portions, the magnet driving the discharge in one direction
+one way and that in the opposite direction the opposite way.
+Thus, when a bulb in which the discharge passes as a ring in a
+horizontal plane is placed between the poles of an electromagnet
+arranged so as to produce a horizontal magnetic field, those parts
+of the ring which are at right angles to the lines of magnetic force
+are separated into two portions, one being driven upwards, the
+other downwards. The displacement of the discharge is not
+however the only effect observed when the discharge bulb is
+placed in a magnetic field, for the difficulty which the discharge
+experiences in getting through the rarefied gas is very much increased
+when it has to pass across lines of magnetic force. This
+effect, which is very well marked, can perhaps be most readily
+shown when the discharge passes as a bright ring through a
+spherical bulb. If such a bulb is placed near a strong electromagnet
+it is easy to adjust the length of spark in the primary
+circuit, so that when the magnet is `off' a brilliant discharge
+passes through the bulb, while when the magnet is `on' no
+discharge at all can be detected.
+
+\Article{90} The explanation of this effect would seem to be somewhat
+as follows. The discharge through the rarefied gas does not rise
+to its full intensity quite suddenly, but, as it were, feels its way.
+The gas first breaks down along the line where the electromotive
+intensity is a maximum, and a small discharge takes place along
+%% -----File: 120.png---Folio 106-------
+this line. This discharge produces a supply of dissociated molecules
+along which subsequent discharges can pass with greater
+ease. The gas is thus in an unstable state with regard to the
+discharge, since as soon as any small discharge passes through
+it, it becomes electrically weaker and less able to resist subsequent
+discharges. When, however, the gas is in a magnetic field,
+the magnetic force acting on the discharge produces a mechanical
+force which displaces the molecules taking part in the discharge
+from the line of maximum electromotive intensity; thus subsequent
+discharges will not find it any easier to pass along this line
+in consequence of the passage of the previous discharge. There
+will not therefore be the same unstability in this case as there is
+in the one where the gas is free from the action of the magnetic
+force. A confirmation of this view is afforded by the appearance
+presented by the discharge when the intensity of the magnetic
+field is reduced until the discharge just, but only just, passes when
+the magnetic field is on: in this case the discharge instead of
+passing as a steady fixed ring, flickers about the tube in a very
+undecided way. Unless some displacement of the line of easiest
+discharge is produced by the motion of the dissociated molecules
+under the action of the magnetic force, it is difficult to understand
+why the magnet should displace the discharge at all,
+unless the Hall effect in rarefied gases is very large.
+
+\includegraphicsmid{fig35}{Fig.~35.}
+
+\Article{91} In the preceding case the discharge was retarded because it
+had to flow across the lines of magnetic force, when however
+the lines of magnetic force run along the line of discharge the
+action of the magnet facilitates the discharge instead of retarding
+it. This effect is easily shown by an arrangement of the
+following kind. A square tube~\smallsanscap{ABCD}, \figureref{fig35}{Fig.~35}, is placed outside
+the primary~\smallsanscap{EFGH}, the lower part of the discharge tube
+being situated between the poles \smallsanscap{L},~\smallsanscap{M} of an electromagnet.
+By altering the length of the spark between the jars, the
+electromotive intensity acting on the secondary circuit can be
+adjusted until no discharge passes round the tube~\smallsanscap{ABCD} when
+the magnet is off, whilst a bright discharge occurs as long as
+the magnet is on. The two effects of the magnet on the discharge,
+viz.~the stoppage of the discharge across the lines of
+force and the help given to it along these lines, may be prettily
+illustrated by placing in this experiment an exhausted bulb~\smallsanscap{N}
+inside the primary. The spark length can be adjusted so that
+%% -----File: 121.png---Folio 107-------
+when the magnet is `off' the discharge passes through the bulb
+and not in the square tube; while when the magnet is `on' the
+discharge passes in the square tube and not in the bulb.
+
+\includegraphicsouter{fig36}{Fig.~36.}
+
+\Article{92} The explanation of the longitudinal effect of magnetic
+force is more obscure than that of the transverse effect, it is
+possible however that both are due to the same cause. For if
+the feeble discharge with which we suppose the total discharge
+to begin branches away at all from the main line, these
+branches will, when the magnetic force is parallel to the line of
+discharge, be brought into this line by the action of the magnetic
+force; there will thus be a larger supply of dissociated molecules
+along the main line of discharge, and therefore an easier
+path for subsequent discharges when the magnetic force is acting
+than when it is not.
+
+This action of the magnet is not confined to this kind of
+discharge; in fact I observed it first for a glow discharge,
+which took place more easily from the pole of an electromagnet
+when the magnet was `on' than when it was `off'.
+
+\Article{93} Professor Fitzgerald has suggested that this effect of the
+\index{Fitzgerald, auroras}%
+magnetic field on the discharge may be the cause of the
+streamers which are observed in the aurora, the rare air, since
+\index{Aurora}%
+it is electrically weaker along the lines of magnetic force than
+at right angles to them, transmitting brighter discharges along
+these lines than in any other direction.
+%% -----File: 122.png---Folio 108-------
+
+\Subsection{Electric discharge through rarefied Gases when Electrodes
+are used.}
+
+\Article{94} When the discharge passes between electrodes through a
+\index{Wiedemann, E., on electric discharge}%
+rare gas, the appearance of the discharge at the positive and
+negative electrodes is so strikingly different that the discharge
+loses all appearance of uniformity. \figureref{fig36}{Fig.~36}, which is taken
+from a paper by E.~Wiedemann (\textit{Phil.\ Mag.}~[5], 18,
+p.~35, 1884), represents the appearance presented
+by the discharge when it passes through a gas at
+a pressure comparable with that due to half a
+millimetre of mercury. Beginning at the negative
+electrode~$k$ we meet with the following phenomena.
+A velvety glow runs often in irregular patches over
+the surface of the negative electrode; a wire placed
+inside this glow casts a shadow towards the negative
+\index{Schuster, discharge through gases|(}%
+electrode (Schuster, \textit{Proc.\ Roy.\ Society},~47,
+p.~557, 1890).
+
+\includegraphicsmid{fig37}{Fig.~37.}
+
+Next to this there is a comparatively dark region~$lb$,
+called sometimes `Crookes' space' and sometimes
+\index{Crookes' space}%
+the `first dark space;' the length of this region depends
+\index{Puluj, dark space}%
+\index{Dark space}%
+\index{First dark space}%
+on the density of the gas, it gets longer as
+the density
+diminishes. Puluj's experiments (\textit{Wien.\
+Ber.}\ 81~(2), p.~864, 1880) show that the length
+does not vary directly as the reciprocal of the
+density, in other words, that it is not proportional
+to the mean free path of the molecules.
+
+\includegraphicsmid{fig38}{Fig.~38.}
+
+The luminous boundary~$b$ of this dark space is
+approximately such as could be got by tracing the locus of the
+extremities of normals of constant length drawn from the negative
+electrode: thus, if the electrode is a disc, the luminous
+boundary of the dark space is over a great part of its surface
+%% -----File: 123.png---Folio 109-------
+nearly plane as in \figureref{fig37}{Fig.~37}, which is given by Crookes; while if it
+is a circular ring of wire, the luminous boundary resembles that
+\index{De la Rue and Müller, discharge through gases}%
+shown in \figureref{fig38}{Fig.~38} (De~la~Rue). The length of the dark space also
+depends to some extent on the current passing through the gas,
+an increase of current producing (see Schuster, \textit{Proc.\ Roy.\ Society},
+47, p.~556, 1890) a slight increase in the length of the dark space.
+Some idea of the length of the dark space at different pressures
+may be got from the following table of the results of some experiments
+made by Puluj (\textit{Wien.\ Ber.}\ 81~(2), p.~864, 1880) with
+a cylindrical discharge tube and disc electrodes:---
+\begin{center}
+\begin{tabular}{c|c}
+\settowidth{\TmpLen}{Pressure in millimetres}%
+\parbox[c]{\TmpLen}{\centering Pressure in millimetres\\ of mercury.\tablespacedown} &
+\settowidth{\TmpLen}{Length of dark space}%
+\parbox[c]{\TmpLen}{\centering Length of dark space\\ in air in mm.\tablespacedown} \\
+\hline
+\tablespaceup$\llap{$1$}.46$ & $\Z2.5$ \\
+$.66$ & $\Z4.5$ \\
+$.51$ & $\Z5.8$ \\
+$.30$ & $\Z7.8$ \\
+$.24$ & $\Z9.5$ \\
+$.16$ & $14.0$ \\
+$.12$ & $15.5$ \\
+$.09$ & $19.5$ \\
+$.06$ & $22.0$
+\end{tabular}
+\end{center}
+
+The mean free path of the molecules is very much smaller than
+the length of the dark space; thus at a pressure of $1.46$~mm.\ of
+mercury, the mean free path is only $.04$~mm. Crookes found
+(\textit{Phil.\ Trans.}\ Part~I, 1879, pp.~138--9) that the dark space is
+longer in hydrogen than in air at the same pressure, but that in
+carbonic acid it is considerably shorter.
+
+\index{Crookes on discharge through gases}%
+\index{Dark space, Crookes' theory of@\subdashtwo Crookes' theory of}%
+\Article{95} Crookes' theory of the dark space is that it is the region
+which the negatively electrified particles of gas shot off from
+the cathode (see \artref{108}{Art.~108}) traverse before making an appreciable
+number of collisions with each other, and that the brightly
+luminous boundary of this space is the region where the collisions
+occur, these collisions exciting vibrations in the particles and so
+%% -----File: 124.png---Folio 110-------
+making them luminous. It is an objection, though perhaps not
+a fatal one, to this view, that the thickness of the dark space is
+very much greater than the mean free path of the molecules.
+We shall see later on that if the luminosity is due to gas shot
+from the negative electrode, this gas must be in the atomic
+and not in the molecular condition; in the former condition its
+free path would be greater than the value calculated from the
+ordinary data of the Molecular Theory of Gases, though if we
+take the ordinary view of what constitutes a collision we
+should not expect the difference to be so great as that indicated
+by Puluj's experiments.
+
+\Article{96} The size of the dark space does not seem to be much
+affected by the material of which the negative electrode is made,
+as long as it is metallic. It is however considerably shorter over
+\index{Chree on negative dark space}%
+sulphuric acid electrodes than over aluminium ones (Chree, \textit{Proc.\
+\index{Crookes on discharge through gases}%
+Camb.\ Phil.\ Soc.}~vii, p.~222, 1891). Crookes (\textit{Phil.\ Trans\DPtypo{}{.}}, 1879,
+p.~137) found that if a metallic electrode is partly coated with
+lamp black the dark space is longer over the lamp-blacked
+portion than over the metallic. Lamp black however absorbs
+gases so readily that this effect may be due to a change in the
+gas and not to the change in the electrode. The dark space is also,
+as Crookes has shown (loc.~cit.), independent of the position of
+the positive electrode. When the cathode is a metal wire raised
+to a temperature at which it is incandescent, Hittorf (\textit{Wied.\
+Ann.}~21, p.~112, 1884) has shown that the changes in luminosity
+which with cold electrodes are observed in the neighbourhood of
+the cathode disappear. There is a difference of opinion as to
+whether the dark space exists when the discharge passes through
+\index{Mercury vapour, discharge through}%
+mercury vapour, Crookes maintaining that it does, Schuster
+that it does not.
+
+\Article{97} Adjoining the `dark space' is a luminous space,~$bp$ \figureref{fig36}{Fig.~36},
+called the `negative column,' or sometimes the `negative glow;' the
+\index{Column, negative}%
+\index{Negative glow@\subdashone glow}%
+\index{Negative column}%
+length of this is very variable even though the pressure is constant.
+The spectrum of this part of the discharge exhibits peculiarities
+which are not in general found in that of the other luminous parts
+\index{Goldstein, discharge of electricity through gases|(}%
+of the discharge. Goldstein (\textit{Wied.\ Ann.}~15, p.~280, 1882) however
+has found that when very intense discharges are used, the
+peculiarities in the spectrum, which are usually confined to the
+negative glow, extend to the other parts of the discharge.
+
+\includegraphicsmid{fig39}{Fig.~39.}
+
+\Article{98} The negative glow is independent of the position of the
+\index{Schuster, discharge through gases|)}%
+%% -----File: 125.png---Folio 111-------
+positive electrode; it does not bend round, for example, in a tube
+shaped as in \figureref{fig41}{Fig.~41}, but is formed in the part of the tube
+away from the positive electrode. This glow is stopped by
+any substance, whether a conductor or an insulator, against
+which it strikes. The development of the negative glow is
+also checked when the space round the negative electrode is
+too much restricted by the walls of the discharge tube. Thus
+Hittorf (\textit{Pogg.\ Ann.}~136, p.~202, 1869) found that if the discharge
+took place in a tube shaped like \figureref{fig39}{Fig.~39}, when the wire~$c$
+in the bulb was made the negative electrode, the negative glow
+spread over the whole of its length, while if the wire~$a$ in the
+neck was used as the negative electrode the glow only occurred
+at its tip.
+
+\includegraphicsmid[p]{fig40}{Fig.~40.}
+
+\Article{99} Next after the negative glow comes a second comparatively
+non-luminous space,~$ph$ \figureref{fig36}{Fig.~36}, called the `second negative dark
+\index{Faraday, space@\subdashone space}%
+\index{Negative dark space, second@\subdashone dark space, second}%
+space,' or by some writers the `Faraday space;' this is of very variable
+length and is sometimes entirely absent. Next after this we
+have a luminous column reaching right up to the positive electrode,
+this is called the `positive column.' Its luminosity very often
+\index{Positive column}%
+exhibits remarkable periodic alterations in intensity such as
+those shown in \figureref{fig40}{Fig.~40}, which is taken from a paper by
+\index{De la Rue and Müller, discharge through gases}%
+\index{Muller@Müller and de la Rue, electric discharge}%
+De~la~Rue and Hugo Müller (\textit{Phil.\ Trans.}, 1878, Part~I, p.~155);
+these are called `striations,' or `\DPtypo{striae}{striæ};' under favourable
+\index{Striations}%
+\index{Striations, variation of, with density of gas@\subdashone variation of, with density of gas}%
+circumstances they are exceedingly regular and constitute the
+most striking feature of the discharge. The bright parts of the
+striations are slightly concave to the positive electrode. The
+distance between the bright parts depends upon the pressure of
+the gas and the diameter of the discharge tube. The distance
+increases as the density of the gas diminishes.
+
+According to Goldstein (\textit{Wied.\ Ann.}~15, p.~277, 1882), if $d$~is
+the distance between two striations and $\rho$~the density of the
+gas, $d$~varies as~$\rho^{-n}$, where $n$~is somewhat less than unity. The
+distance between the bright parts of successive striations increases
+%% -----File: 126.png---Folio 112-------
+\index{Positive column, striations in@\subdashtwo striations in}%
+as the diameter of the discharge tube increases, provided
+the striations reach to the sides of the tube. Goldstein (l.~c.)\
+found that the ratio of the values of~$d$ at any two given pressures
+is the same for all tubes. If the discharge takes place
+%% -----File: 127.png---Folio 113-------
+in a tube which is wider in some places than in others, the
+striations are more closely packed in the narrow parts of
+the tube than they are in the wide.
+
+The striations have very often a motion of translation along
+the tube; this motion is quite irregular, being sometimes towards
+the positive electrode and sometimes away from it. This can
+\index{Spottiswoode, on striations}%
+easily be detected by observing, as Spottiswoode did, the discharge
+in a rapidly rotating mirror. These movements of the
+\DPtypo{striae}{striæ} tend to make the striated appearance somewhat indistinct,
+and if the movements are too large may obliterate
+it altogether; thus many discharges which show no appearance
+of striation when examined in the ordinary way, are seen to
+be striated when looked at in a revolving mirror. The difficulty
+of detecting whether a discharge is striated or not is, in consequence
+of the motion of the \DPtypo{striae}{striæ}, very much greater when the
+\DPtypo{striae}{striæ} are near together than when they are far apart, so that it
+is quite possible that discharges are striated at pressures much
+greater than those at which striations are usually observed.
+
+Goldstein, using a tube with moveable electrodes, showed
+(\textit{Wied.\ Ann.}\ 12, p.~273, 1881) that when the cathode is moved
+the \DPtypo{striae}{striæ} move as if they were rigidly connected with it, while
+when the anode is moved the position of the \DPtypo{striae}{striæ} is not affected
+except in so far as they may be obliterated by the anode moving
+past them.
+
+\Article{100} The striations are not confined to any one particular
+method of producing the discharge, they occur equally well
+whether the discharge is produced by an induction coil or by a
+very large number of galvanic cells. They do not, however,
+occur readily in the electrodeless discharge; indeed I have never
+observed them when a considerable interval intervened between
+consecutive sparks. By using an induction coil large enough to
+furnish a supply of electricity sufficient to produce an almost
+continuous torrent of sparks between the jars, I have been able
+to get striations in exhausted bulbs containing hydrogen or
+other gases.
+
+\Article{101} The striations are influenced by the quantity of current
+flowing through the tube; this can easily be shown by putting a
+great external resistance in the circuit, such as a wet string. The
+changes produced by altering the current are complex and irregular:
+there seems to be a certain intensity of current for which the
+%% -----File: 128.png---Folio 114-------
+\index{De la Rue and Müller, discharge through gases}%
+\index{Muller@Müller and de la Rue, electric discharge}%
+steadiness of the striations is a maximum (De~la~Rue and
+Hugo Müller, \textit{Comptes Rendus},~86, p.~1072, 1878). Crookes has
+found (\textit{Presidential Address to the Society of Telegraph Engineers},
+1891) that when the discharge passes through a mixture of different
+gases there is a
+separate set of striations
+for each gas:
+the colour of the
+striations in each
+set being different.
+Crookes proved this
+by observing the spectra
+of the different
+\DPtypo{striae}{striæ}. A full account
+of the different
+coloured striations observed
+in air is given
+by Goldstein (\textit{Wied.\
+Ann.}~12, p.~274, 1881).
+\index{Goldstein, discharge of electricity through gases|)}%
+
+\Article{102} When we
+consider the action
+of a magnet on the
+striated positive column
+we shall see
+reasons for thinking
+that any portion of
+the positive column
+between the bright
+parts of consecutive
+striations constitutes
+a separate discharge,
+and that the discharges
+in the several
+portions do not occur
+simultaneously, but that the one next the anode begins the
+discharge, and the others follow on in order.
+
+\includegraphicsouter{fig41}{Fig.~41.}
+
+\Article{103} The positive column bears a very much more important
+relation to the discharge than either the negative dark space or
+the negative glow. The latter effects are merely local, they do
+not depend upon the position of the positive electrode, nor do they
+%% -----File: 129.png---Folio 115-------
+increase when the length of the discharge tube is increased. The
+positive column, on the other hand, takes the shortest route through
+the gas to the negative electrode. Thus, if, for example, the discharge
+takes place in a tube like \figureref{fig41}{Fig.~41}, the positive column bends
+round the corner so as to get to the negative electrode, while the
+negative glow goes straight down the vertical tube, and is not
+affected by the position of the positive electrode. Again, if the
+length of the tube is increased the size of the negative dark
+space and of the negative glow is not affected, it is only the
+positive column which lengthens out. I have, for example,
+obtained the discharge through a tube $50$~feet long, and this
+tube, with the exception of a few inches next the cathode, was
+entirely filled by the positive column, which was beautifully
+striated. These examples show that it is the positive column
+which really carries the discharge through the gas, and that the
+negative dark space and the negative glow are merely local
+effects, depending on the peculiarities of the transference of
+electricity from a gas to a cathode.
+
+\Article{104} By the use of long discharge tubes such as those mentioned
+\index{Positive column, velocity of@\subdashtwo velocity of}%
+\index{Velocity of positive column@\subdashtwo positive column}%
+\index{Wheatstone, velocity of discharge}%
+\index{Zahn, von, velocity of molecules in electric discharge}%
+above, it is possible to determine the direction in which
+the luminosity in the positive column travels and to measure its
+rate of progression. The first attempt at this seems to have
+been made by Wheatstone, who, in 1835, observed the appearance
+presented in a rotating mirror by the discharge through a
+vacuum tube $6$~feet long; he concluded from his observations
+that the velocity with which the flash went through the tube
+could not have been less than $8 × 10^7$~cm.\ per~second. This
+great velocity is not accompanied by a correspondingly large
+velocity of the luminous molecules, for von~Zahn (\textit{Wied.\ Ann.}~8,
+p.~675, 1879) has shown that the lines of the spectrum of
+the gas in the discharge tube are not displaced by as much as
+$\frac{1}{40}$~of the distance between the D~lines when the line of sight
+is in the direction of the discharge. It follows from this by
+Döppler's principle, that the particles when emitting light are
+not travelling at so great a rate as a mile a second, proving,
+at any rate, that the luminous column does not consist of a
+wind of luminous particles travelling with the velocity of the
+discharge.
+
+\includegraphicsmid{fig42}{Fig.~42.}
+
+\Article{105} Wheatstone's observations only give an inferior limit to
+the velocity of the discharge; they do not afford any information
+%% -----File: 130.png---Folio 116-------
+as to whether the luminous column travels from the anode to the
+cathode or in the opposite direction. To determine this, as well
+as measure the velocity of the luminosity in the positive column,
+I made the following experiment. \smallsanscap{ABCDEFG}, \figureref{fig42}{Fig.~42}, is a glass
+tube about $15$~metres long and $5$~millimetres in diameter, which,
+with the exception of two horizontal pieces of \smallsanscap{BC}~and~\smallsanscap{GH}, is
+covered with lamp black; this tube is exhausted until a current
+can be sent through it from an induction coil. The light from
+the uncovered portions of the tube falls on a rotating mirror~\smallsanscap{MN},
+placed at a distance of about $6$~metres from~\smallsanscap{BC}; the light from~\smallsanscap{GH}
+falls on the rotating mirror directly, that from~\smallsanscap{BC} after
+reflection from the plane mirror~\smallsanscap{P}. The images of the bright
+portions of the tube after reflection from the mirror are viewed
+%% -----File: 131.png---Folio 117-------
+through a telescope, and the mirrors are so arranged that when
+the revolving mirror is stationary the images of the bright portions
+\smallsanscap{GH}~and~\smallsanscap{BC} of the tube appear as portions of the same
+horizontal straight line. The terminals of the long vacuum
+tube are pushed through mercury up the vertical tubes~\smallsanscap{AB~KL}.
+This arrangement was adopted because by running sulphuric
+acid up these tubes the terminals could easily be changed
+from pointed platinum wires to flat liquid surfaces, and the
+effect of very different terminals on the velocity and direction
+of the discharge readily investigated. The bulbs on the tube
+are also useful as receptacles of sulphuric acid, which serves
+to dry the gas left in the tube. The rotating mirror was
+driven at a speed of from $400$ to $500$ revolutions per~second by a
+Gramme machine. It was not found possible to make any
+arrangement work well which would break the primary circuit
+of the induction coil when the mirror was in such a position
+that the images of the luminous portions of the tube would
+be reflected by it into the field of view of the telescope. The
+method finally adopted was to use an independent slow break
+for the coil and look patiently through the telescope at the
+rotating mirror until the break happened to occur just at the
+right moment. When the observations were made in this way
+the observer at the telescope saw, on an average about once in
+four minutes, sharp bright images of the portions \smallsanscap{BC}~and~\smallsanscap{GH}
+of the tube, not sensibly broadened but no longer quite in the
+same straight line. The relative displacement of those images was
+reversed when the poles of the coil were reversed, and also when
+the direction of rotation of the mirror was reversed. This displacement
+of the images of \smallsanscap{BC}~and~\smallsanscap{GH} from the same straight
+line is due to the finite velocity with which the luminosity is
+propagated: for, if the mirror can turn through an appreciable
+angle while the luminosity travels from~\smallsanscap{BC} to~\smallsanscap{GH} or from~\smallsanscap{GH}
+to~\smallsanscap{BC}, these images of \smallsanscap{BC}~and~\smallsanscap{GH}, when seen in the telescope
+after reflection from the revolving mirror, will no longer be in
+the same straight line. If the mirror is turning so that on looking
+through the telescope the images seem to come in at the top
+and go out at the bottom of the field of view, the image of that
+part of the tube at which the luminosity first appears will be
+raised above that of the other part. If we know the rate of
+rotation of the mirror, the vertical displacement of the images
+%% -----File: 132.png---Folio 118-------
+and the distance between \smallsanscap{BC}~and~\smallsanscap{GH}, the rate of propagation of
+the luminosity may be calculated. The displacement of the
+images showed that the luminosity always travelled from the
+positive to the negative electrode. When \smallsanscap{AB}~was the negative
+electrode, the luminous discharge arrived at~\smallsanscap{GH}, a place about
+$25$~feet from the positive electrode, before it reached~\smallsanscap{BC}, which was
+only a few inches from the cathode, and as the interval between
+its appearance at these places was about the same as when
+the current was reversed, we may conclude that when \smallsanscap{AB}~is
+the cathode the luminosity at a place~\smallsanscap{BC}, only a few inches
+from it, has started from the positive electrode and traversed
+a path enormously longer than its distance from the cathode.
+The velocity of the discharge through air at the pressure of
+about $\frac{1}{2}$~a millimetre of mercury in a tube $5$~millimetres in
+diameter was found to be rather more than half the velocity
+of light.
+
+\Article{106} The preceding experiment was repeated with a great
+variety of electrodes; the result, however, was the same whether
+the electrodes were pointed platinum wires, carbon filaments, flat
+surfaces of sulphuric acid, or the one electrode a flat liquid surface
+and the other a sharp-pointed wire. The positive luminosity
+travels from the positive electrode to the negative, even though
+the former is a flat liquid surface and the latter a pointed wire.
+The time taken by the luminosity to travel from~\smallsanscap{BC} to~\smallsanscap{GH} was
+not affected to an appreciable extent by inserting between~\smallsanscap{BC}
+and~\smallsanscap{GH} a number of pellets of mercury, so that the discharge
+had to pass from the gas to the mercury several times in its
+passage between these places: the intensity of the light was
+however very much diminished by the insertion of the mercury.
+
+\Article{107} The preceding results bear out the conclusion which
+\index{Plücker, effect of magnet on discharge}%
+Plücker (\textit{Pogg.\ Ann.}~107, p.~89, 1859) arrived at from the consideration
+of the action of a magnet on the discharge, viz.~that
+the positive column starts from the positive electrode; they also
+confirm the result which Spottiswoode and Moulton (\textit{Phil.\
+Trans.}\ 1879, p.~165) deduced from the consideration of what
+they have termed `relief' effects, that the time taken by the
+negative electricity to leave the cathode is greater than the
+time taken by the positive luminosity to travel over the length
+of the tube.
+%% -----File: 133.png---Folio 119-------
+
+\Subsection{Negative Rays or Molecular Streams.}
+\index{Negative rays@\subdashone rays}%
+\index{Molecular streams}%
+
+\Article{108} Some of the most striking of the phenomena shown by
+the discharge through gases are those which are associated with
+the negative electrode. These effects are most conspicuous at
+\index{Moulton and Spottiswoode, electric discharge}%
+\index{Spottiswoode and Moulton, electric discharge}%
+low pressures, but Spottiswoode and Moulton's experiments
+(\textit{Phil.\ Trans.}\ 1880, pp.~582, 85~\textit{seq.})\ show that they exist over
+a wide range of pressure. The sides of the tube exhibit a
+brilliant phosphorescence, behaving as if something were shot
+out at right angles, or nearly so, to the surface of the cathode,
+which had the power of exciting phosphorescence on any substance
+on which it fell, provided that this substance is
+one which becomes phosphorescent under the action of ultra-violet
+light. The portions of the tube enclosed within the
+surface formed by the normals to the cathode will, when the
+pressure of the gas is low, show a bright green phosphorescence
+if the tube is made of German glass, while the phosphorescence
+will be blue if the tube is made of lead glass. Perhaps the
+easiest way of describing the general features of this effect is
+to say that they are in accordance with Mr.~Crookes' theory, that
+particles of gas are projected with great velocities at right
+angles, or nearly so, to the surface of the cathode, and that these
+particles in a highly exhausted tube strike the glass before they
+have lost much momentum by collision with other molecules,
+and that the bombardment of the glass by these particles is
+intense enough to make it phosphoresce. The following extract
+\index{Priestley@Priestley's \textit{History of Electricity}}%
+from Priestley's \textit{History of Electricity}, p.~294, 1769, is interesting
+\index{Beccaria, phosphorescence}%
+in connection with this view: `Signior Beccaria observed that
+hollow glass vessels, of a certain thinness, exhausted of air, gave
+a light when they were broken in the dark. By a beautiful
+train of experiments, he found, at length, that the luminous
+appearance was not occasioned by the breaking of the glass, but
+by the dashing of the external air against the inside, when it
+was broke. He covered one of these exhausted vessels with
+a receiver, and letting the air suddenly on the outside of it,
+observed the very same light. This he calls his \emph{new invented
+phosphorous}.'
+
+\Article{109} If a screen made either of an insulator or a conductor
+is placed between the electrode and the walls of the tube,
+a shadow of the screen is thrown on the walls of the tube,
+%% -----File: 134.png---Folio 120-------
+\index{Negative rays, shadows cast by@\subdashtwo shadows cast by}%
+\index{Shadows cast by negative rays}%
+the shadow of the screen remaining dark while the glass round
+the shadow phosphoresces brightly. In this way many very
+\index{Crookes on discharge through gases|(}%
+beautiful and brilliant effects have been produced by Mr.~Crookes
+\index{Goldstein, discharge of electricity through gases}%
+and Dr.~Goldstein, the two physicists who have devoted
+most attention to this subject. One of Mr.~Crookes' experiments
+in which the shadow of a Maltese cross is thrown on the walls
+of the tube is illustrated in \figureref{fig43}{Fig.~43}.
+
+\includegraphicsmid{fig43}{Fig.~43.}
+
+\Article{110} As we have already mentioned, the colour of the phosphorescence
+depends on the nature of the phosphorescing substance;
+if this substance is German glass the phosphorescence
+is green, if it is lead glass the phosphorescence is blue. Crookes
+found that bodies phosphorescing under this action of the
+negative electrode give out characteristic band spectra, and he
+has developed this observation into a method of the greatest
+%% -----File: 135.png---Folio 121-------
+importance for the study of the rare earths: for the particulars
+of this line of research we must refer the reader to his papers
+`On Radiant Matter Spectroscopy,' \textit{Phil.\ Trans.}\ 1883, Pt.~III,
+\index{Spectroscopy@`Spectroscopy, Radiant Matter'}%
+\index{Radiant matter}%
+and~1885, Pt.~II.
+
+The way the spectrum is produced is represented in \figureref{fig44}{Fig.~44},
+the substance under examination being placed in a high vacuum
+in the path of the normals to the cathode.
+
+\index{Negative rays, phosphorescence due to@\subdashtwo phosphorescence due to}%
+\index{Phosphorescence, due to magnetic rays}%
+\Article{111} Crookes also found that some substances, when submitted
+for long periods to the action of these rays, undergoes
+remarkable modifications, which seems to suggest that the phosphorescence
+is attended (or caused by?)\ chemical changes slowly
+taking place in the phosphorescent body. He also observed
+that glass which has been phosphorescing for a considerable
+time seems to get tired, and to respond less readily to this
+action of the cathode. Thus, for example, if after the experiment
+in \figureref{fig44}{Fig.~44} has been proceeding for some time the cross is
+shaken down, or a new cathode used whose line of fire does not
+cut the cross, the pattern of the cross will still be seen on the
+glass, but now it will be brighter than the adjacent parts
+instead of darker. The portions outside the pattern of the cross
+have got tired by their long phosphorescence, and respond less
+vigorously to the stimulus than the portions forming the cross
+which were previously shielded. Crookes found this `exhaustion'
+of the glass could survive the melting and reblowing of the bulb.
+
+\includegraphicsmid{fig44}{Fig.~44.}
+
+By using a curved surface for the negative electrode, such
+as a portion of a hollow cylinder or of a spherical shell, this
+effect of the negative rays may be concentrated to such an
+extent that a platinum wire placed at the centre of the cylinder
+or sphere becomes red hot.
+
+\Article{112} The negative rays are deflected by a magnet in the same
+\index{Magnets, action of, on negative rays@\subdashtwo on negative rays}%
+\index{Negative rays, action of a magnet on@\subdashtwo action of a magnet on}%
+way as they would be if they consisted of particles moving away
+from the negative electrode and carrying a charge of negative
+electricity. This deflection is made apparent by the movement
+of the phosphorescence on the glass when a magnet is brought
+near the discharge tube.
+
+On the other hand they are not deflected when a charged
+body is brought near the tube; this does not prove, however,
+that the rays do not consist of electrified particles, for we have
+seen that gas conveying an electric discharge is an extremely
+good conductor, and so would be able to screen the inside of the
+%% -----File: 136.png---Folio 122-------
+tube from any external electrostatic action. Crookes (\textit{Phil.\
+Trans.}\ 1879, Pt.~II, p.~652) has shown, moreover, that two
+\index{Negative rays, repulsion of@\subdashtwo repulsion of}%
+\index{Repulsion, of negative rays@\subdashone of negative rays}%
+pencils of these rays repel each other, as they would do if
+each pencil consisted of particles charged with the same kind
+of electricity. The experiment by which this is shown is
+represented in \figureref{fig45}{Fig.~45}; $a$,~$b$ are metal discs either or both of
+which may be made into cathodes, a diaphragm with two
+openings $d$~and~$e$ is placed in front of the disc, and the path
+of the rays is traced by the phosphorescence they excite in a
+chalked plate inclined at a small angle to their path. When $a$
+is the cathode and $b$ is idle, the rays travel along the path~$df$,
+and when $b$ is the cathode and $a$ idle they travel along the path~$ef$,
+but when $a$~and~$b$ are cathodes simultaneously the paths of
+the rays are $dg$~and~$eh$ respectively, showing that the two
+streams have slightly repelled each other.\nblabel{add:3}
+
+\includegraphicsmid{fig45}{Fig.~45.}
+
+\Article{113} Crookes (\textit{Phil.\ Trans.}\ 1879, Part~II, p.~647) found that
+if a disc connected with an electroscope is placed in the full
+line of fire of these rays it receives a charge of \emph{positive} electricity.
+This is not, however, a proof that these rays do not
+consist of negatively electrified particles, for the experiments
+described in \artref{81}{Art.~81} show that electricity does not pass at
+all readily from a gas to a metal, and the positive electrification
+of the disc may be a secondary effect arising from the same
+cause as the positive electrification of a plate when exposed to
+the action of ultra-violet light. For since the action of these rays
+is the same as that of ultra-violet light in producing phosphorescence
+in the bodies upon which they fall, it seems not
+unlikely that the rays may resemble ultra-violet light still
+further and make any metal plate on which they fall a cathode.
+\index{Crookes on discharge through gases|)}%
+
+\index{Hertz, xnegative rays@\subdashone negative rays}%
+Hertz (\textit{Wied.\ Ann.}\ 19, p.~809, 1883) was unable to discover
+that these rays produced any magnetic effect.
+
+\includegraphicsouter{fig46}{Fig.~46.}
+
+The paths of the negative rays are governed entirely by the
+shape and position of the cathode, they are quite independent
+%% -----File: 137.png---Folio 123-------
+of the shape or position of the anode. Thus, if the cathode
+and anode are placed at one end of an exhausted tube, as in
+\figureref{fig46}{Fig.~46}, the cathode rays will not bend round to the anode,
+but will go straight down the
+tube and make the opposite
+end phosphoresce.
+
+Any part of the tube which
+is made to phosphoresce by
+the action of these rays seems
+to acquire the power of sending out such rays itself, or we may
+express the same thing by saying that the rays are diffusely
+\index{Goldstein, discharge of electricity through gases|(}%
+reflected by the phosphorescent body (Goldstein, \textit{Wied.\ Ann.}\ 15,
+p.~246, 1882). \figureref{fig47}{Fig.~47} represents the appearance presented by
+a bent tube when traversed by such rays, the darkly shaded
+places being the parts of the tube which show phosphorescence.
+
+\includegraphicsmid{fig47}{Fig.~47.}
+
+\Article{114} These rays seem to be emitted by any negative electrode,
+even if this be one made by putting the finger on the glass of
+the tube near the anode. This produces a discharge of negative
+electricity from the glass just underneath the finger, and the
+characteristic green phosphorescence (if the tube is made of
+German glass) appears on the opposite wall of the tube; this
+phosphorescence is deflected by a magnet in exactly the same
+way as if the rays came from a metallic electrode. This experiment
+%% -----File: 138.png---Folio 124-------
+is sufficient to show the inadequacy of a theory that has
+sometimes been advanced to explain the phosphorescence, viz.~that
+the particles shot off from the electrode are not gaseous
+particles, but bits of metal torn from the cathode; the phosphorescence
+being thus due to the disintegration of the negative
+electrode, which is a well-known feature of the discharge in
+vacuum tubes. The preceding experiment shows that this theory
+\index{Crookes on discharge through gases}%
+is not adequate, and Mr.~Crookes has still further disproved
+it by obtaining the characteristic effects in tubes when the
+electrodes were pieces of tinfoil placed \emph{outside} the glass.
+
+\Article{115} Goldstein (\textit{Wied.\ Ann.}\ 11, p.~838, 1880) found that a
+\index{Contraction in discharge-tube produces effects similar to a cathode}%
+\index{Negative electrode, quasi, produced by contraction of tube@\subdashone electrode, quasi, produced by contraction of tube}%
+sudden contraction in the cross section of the discharge tube
+produces on the side towards the anode the same effect as a
+cathode. These quasi-cathodes produced by the contraction of
+the tube are accompanied by all the effects which are observed
+with metallic cathodes, thus we have the dark space, the phosphorescence,
+and the characteristic behaviour of the glow in a
+magnetic field.
+
+\Article{116} Spottiswoode and Moulton (\textit{Phil.\ Trans.}\ 1880, pp.~615--622)
+\index{Moulton and Spottiswoode, electric discharge}%
+\index{Spottiswoode and Moulton, electric discharge}%
+\index{Phosphorescence, due to positive column@\subdashtwo to positive column}%
+have observed a phosphorescence accompanying the positive
+column. They found that in some cases when this strikes the
+gas the latter phosphoresces. They ascribe this phosphorescence
+to a negative discharge called from the sides of the tube by the
+positive electricity in the positive column.
+
+\Subsection{Mechanical Effects produced by the Negative Rays.}
+\index{Mechanical effects due to negative rays}%
+\index{Negative rays, mechanical effects produced by@\subdashtwo mechanical effects produced by}%
+
+\includegraphicsmid[!t]{fig48}{Fig.~48.}
+
+\includegraphicsmid[!b]{fig49}{Fig.~49.}
+
+\Article{117} Mr.~Crookes (\textit{Phil.\ Trans.}\ 1879, Pt.~I, p.~152) has shown
+that when these rays impinge on vanes mounted like those in a
+radiometer the vanes are set in rotation. This can be shown by
+making the axle of the vanes run on rails as in \figureref{fig48}{Fig.~48}. When
+the discharge passes through the tube, the vanes travel from the
+negative to the positive end of the tube. It is not clear, however,
+that this is a purely mechanical effect; it may, as suggested by
+Hittorf, be due to secondary thermal effects making the vanes act
+like those of a radiometer. In another experiment the vanes are
+suspended as in \figureref{fig49}{Fig.~49}, and can be screened from the negative
+rays by the screen $e$; by tilting the tube the vanes can be
+brought wholly or partially out of the shadow of the screen.
+When the vanes are completely out of the shade they do
+not rotate as the bombardment is symmetrical; when, however,
+%% -----File: 139.png---Folio 125-------
+they are half in and half out of the shadow, they rotate
+in the same direction as they would if exposed to a bombardment
+from the negative electrode. The deflection of the negative
+rays by a magnet is well illustrated by this apparatus. Thus, if
+the vanes are placed wholly within the shadow no rotation
+takes place; if, however, the south pole of an electro-magnet is
+brought to \smallsanscap{S}, the shadow is deflected from the former position
+and a part of the vanes is thus exposed to the action of the
+rays; as soon as this takes place the vanes begin to rotate.
+
+\Article{118} The thinnest layer of a solid substance seems absolutely
+\index{Negative rays, opacity of substances to@\subdashtwo opacity of substances to}%
+\index{Opacity of substances to the negative rays,@\subdashone of substances to the negative rays}%
+opaque to these radiations. Thus Goldstein (\textit{Phil.\ Mag.}\ [5]~10,
+p.~177, 1880) found that a thin layer of collodion placed on the
+glass gave a perfectly black shadow, and Crookes (\textit{Phil.\ Trans.}\
+\index{Goldstein, discharge of electricity through gases|)}%
+%% -----File: 140.png---Folio 126-------
+1879, Part~I, p.~151) found that a thin film of quartz, which is
+transparent to ultra-violet light, produced the same effect. This
+last result is of great importance in connection with a theory
+which has received powerful support, viz.~that these `rays' are a
+kind of ethereal vibration having their origin at the cathode. If
+this view were correct we should not expect to find a thin quartz
+plate throwing a perfectly black shadow, as quartz is transparent
+to ultra-violet light. To make the theory agree with the facts
+we have further to assume that no substance has been discovered
+\index{Hertz, xnegative rays@\subdashone negative rays}%
+which is appreciably transparent to these vibrations\footnote
+  {Since the above was written, Hertz (\textit{Wied.\ Ann.}\ 45, p.~28, 1892) has found that
+  thin films of gold leaf do not cast perfectly dark shadows but allow a certain amount of
+  phosphorescence to take place behind them, which cannot be explained by the existence
+  of holes in the film. It seems possible, however, that this is another aspect of
+  the phenomenon observed by Crookes (\artref{113}{Art.~113}) that a metal plate exposed to the full
+  force of these rays becomes a cathode; in Hertz's experiments the films may have been
+  so thin that each side acted like a cathode, and in this case the phosphorescence on
+  the glass would be caused by the film acting like a cathode on its own account.}.
+The sharpness and blackness of these shadows are by far the
+strongest arguments in support of the impact theory of the
+phosphorescence.
+
+
+\Article{119} Though Crookes' theory that the phosphorescence is due
+to the bombardment of the glass by gaseous particles projected
+from the negative electrode is not free from difficulties, it seems
+to cover the facts better than any other theory hitherto advanced.
+On one point, however, it would seem to require a slight modification:
+Crookes always speaks of the \emph{molecules} of the gas receiving
+a negative charge. We have, however (see \artref{3}{Art.~3}), seen
+reasons for thinking that a molecule of a gas is incapable of
+receiving a charge of electricity, and that free electricity must
+be on the \emph{atoms} as distinct from the molecules. If this view is
+right, we must suppose that the gaseous particles projected from
+the negative electrode are atoms and not molecules. This does
+not introduce any additional difficulty into the theory, for in the
+region round the cathode there is a plentiful supply of dissociated
+molecules or atoms; of these, those having a negative
+charge may under the repulsion of the negative electricity on
+the cathode be repelled from it with considerable violence.
+
+\includegraphicsouter{fig50}{Fig.~50.}
+
+\Article{120} An experiment which I made in the course of an investigation
+on discharge without electrodes seems to afford considerable
+evidence that there is such a projection of atoms from the
+%% -----File: 141.png---Folio 127-------
+cathode. The interpretation of this evidence depends upon the
+fact that the presence in a gas of atoms, or the products of a
+previous discharge through the gas, greatly facilitates the passage
+of a subsequent discharge. The experiment is represented in
+\figureref{fig50}{Fig.~50}: the discharge tube~\smallsanscap{A} was fused on to the pump, and
+two terminals~$c$ and~$d$ were fused
+through the glass at an elbow of the
+tube. These terminals were connected
+with an induction coil, and
+the pressure in the discharge tube
+was such that the electrodeless discharge
+would not pass. When the
+induction coil was turned on in such
+a way that $c$ was the negative electrode
+the electrodeless discharge at
+once passed through the tube, but no
+effect at all was produced when $c$
+was positive and $d$ negative.
+
+\Article{121} Assuming with Mr.~Crookes
+that it is the impact of particles driven
+out of the region around the negative
+electrode which produces the
+phosphorescence, it still seems an
+open question whether the luminosity is due to the mechanical
+effect of the impulse or whether the effect is wholly electrical.
+For since these particles are charged, their approach, collision
+with the glass, and retreat, will produce much the same electrical
+effect as if a body close to the glass were very rapidly
+charged with negative electricity and then as rapidly discharged.
+Thus the glass in the neighbourhood of the point of impact of
+one of these particles is exposed to a very rapidly changing
+electric polarization, the effect of which, according to the electromagnetic
+theory of light, would be much the same as if light
+fell on the glass, in which case we know it would phosphoresce.
+
+The sharpness of the shadows cast by these rays shows that the
+phosphorescence cannot be due to what has been called a `lamp
+action' of the particles, each particle acting like a lamp, radiating
+light, and causing the glass to phosphoresce by the light it emits.
+
+\sloppy
+\Article{122} The distance which these particles travel before losing
+their power of affecting the glass is surprising, amounting to a
+%% -----File: 142.png---Folio 128-------
+large multiple of the mean free path of the molecules of the gas
+when in a molecular condition; it is possible, however, that they
+travel together, forming something analogous to the `electrical
+wind,' and that their passage through the gas resembles the
+passage of a mass of air by convection currents rather than a
+process of molecular diffusion. We must remember, too, that
+since atoms are smaller than molecules, the mean free path of a
+gas in the atomic condition would naturally be greater than
+when in the molecular.
+
+\fussy
+\Article{123} Strikingly beautiful as the phenomena connected with
+these `negative rays' are, it seems most probable that the rays
+are merely a local effect, and play but a small part in carrying the
+current through the gas. There are several reasons which lead
+us to come to this conclusion: in the first place, we have seen
+that the great mass of luminosity in the tube starts from the
+anode and travels down the tube with an enormously greater
+velocity then we can assign to these particles; again, this discharge
+seems quite independent of the anode, so that the rays
+may be quite out of the main line of the discharge. The exact
+function of these rays in the discharge is doubtful, it seems just
+possible that they may constitute a return current of gas by
+which the atoms which carry the discharge up to the negative
+electrode are prevented from accumulating in its neighbourhood.
+
+\Article{124} These rays have been used by Spottiswoode and Moulton
+\index{Moulton and Spottiswoode, electric discharge}%
+(\emph{Phil.\ Trans.}\ 1880, p.~627) to determine a point of fundamental
+importance in the theory of the discharge, viz.~the relative magnitudes
+of the following times:---
+
+(1) The period occupied by a discharge.
+
+(2) The time occupied by the discharge of the positive electricity
+from its terminal.
+
+(3) The time occupied by the discharge of the negative electricity
+from its terminal.
+
+(4) The time occupied by molecular streams in leaving a
+negative terminal.
+
+(5) The time occupied by positive electricity in passing along
+the tube.
+
+(6) The time occupied by negative electricity in passing along
+the tube.
+
+(7) The time occupied by the particles composing molecular
+streams in passing along the tube.
+%% -----File: 143.png---Folio 129-------
+
+(8) The time occupied by electricity in passing along a wire
+of the length of the tube.
+
+\includegraphicsmid[!b]{fig51}{Fig.~51.}
+
+The phenomenon which was most extensively used by Spottiswoode
+\index{Negative rays, repulsion of@\subdashtwo repulsion of}%
+\index{Repulsion, of negative rays@\subdashone of negative rays}%
+\index{Spottiswoode and Moulton, electric discharge}%
+and Moulton in investigating the relative magnitude
+of these times was the repulsion of one negative stream by
+another in its neighbourhood. This effect may be illustrated in
+several ways: thus, if the finger or a piece of tin-foil connected
+to earth be placed on the discharge tube, not too far away
+from the anode, the portion of the glass tube immediately
+underneath the finger becomes by induction a cathode and emits
+a negative stream; this stream produces a phosphorescent patch
+on the other side of the tube, diametrically opposite to the
+finger. If two fingers or two pieces of tin-foil are placed on the
+tube two phosphorescent patches appear on the glass, but neither
+of these patches occupies quite the position it would if the other
+patch were away. Another experiment (see Spottiswoode and
+Moulton, \textit{Phil.\ Trans.}\ 1880, Part~II, p.~614) which also illustrates
+the same effect is the following. A tube, \figureref{fig51}{Fig.~51}, was taken, in
+which there was a flat piece of aluminium containing a small
+hole; when the more distant terminal was made negative, a
+bright image~\smallsanscap{A} of the hole appeared on the side of the tube in the
+midst of the shadow cast by the plate. When the tube was
+touched on the side on which this image appeared, but at a
+point on the negative side of the image, it was found that the
+image was splayed out to~\smallsanscap{B}, part of it moving down the tube
+away from the negative terminal. This seems to show that the
+negative electrode formed by the finger pushes away from it
+the rays forming the image. From this case Spottiswoode and
+Moulton reasoned as follows (\textit{Phil.\ Trans.}\ 1880, Part~II, p.~632):
+%% -----File: 144.png---Folio 130-------
+`This image was \emph{splayed out} by the finger being placed on the
+tube. Now a magnet displaced it as a whole without any
+splaying out. This then pointed to a variation in the relative
+strength of the interfering stream and the stream interfered
+with, and such variation must have occurred during the period
+that they were encountering one another, and were moving in
+the ordinary way of such streams, for it showed itself in a variation
+in the extent to which the streams from the negative
+terminal were diverted. We may hence conclude that the time
+requisite for the molecules to move the length of the tube was
+decidedly less than that occupied by the discharge, but was
+sufficiently comparable with it to allow the diminution of
+intensity of the streams from the sides of the tube to make
+itself visible before the streams from the negative terminal experienced
+a similar diminution.'
+
+\Article{125} This may serve as an example of the method used by
+\index{Spottiswoode and Moulton, electric discharge}%
+\index{Times involved in electric discharge}%
+Spottiswoode and Moulton in comparing the time quantities
+enumerated in \artref{124}{Art.~124}. We regret that we have not space to
+describe the ingenious methods by which they brought other
+time quantities into comparison, for these we must refer to their
+paper; we can only quote the final result of their investigation.
+They arrange (l.~c.\ pp.~641--642) the time quantities in groups
+which are in descending order of magnitude, the quantities in
+any group are exceedingly small compared with those in any
+group above them, while the quantities in the same group are
+of the same order of magnitude.
+\begin{olist}
+\item[A.] The interval between two discharges.
+\item[B.] The time occupied by the discharge of the negative electricity
+from its terminal.
+\begin{ilist}
+\item[] The time occupied by negative streams in leaving a negative
+terminal.
+\item[] The time occupied by the particles composing molecular
+streams in passing along the tube.
+\end{ilist}
+\item[C.] The time occupied by positive electricity in passing along
+the tube.
+\begin{ilist}
+\item[] The time occupied by negative electricity in passing along
+the tube.
+\end{ilist}
+\item[D.] The time occupied by positive discharge.
+\begin{ilist}
+\item[] The time required for the formation of positive luminosity
+at the seat of positive discharge.
+%% -----File: 145.png---Folio 131-------
+\item[] The time required for the formation of the dark space at
+the seat of negative discharge.
+\end{ilist}
+\item[E.] The time occupied by either electricity in passing along a
+wire of the length of the tube.
+\end{olist}
+
+The time of a complete discharge is of the order~B.
+
+It will be seen that one of the conclusions given above, viz.~that
+the time required for the positive luminosity to travel the
+length of the tube is very small compared with the time occupied
+by the negative discharge, is confirmed by the experiments with
+the rotating mirror described in \artref{104}{Art.~104}. According to these
+experiments however C~and~E are of the same order.
+
+\Subsection{Action of a Magnet upon the Discharge when Electrodes
+are used.}
+
+\Article{126} The appearance and path of the discharge in a vacuum
+\index{Electric discharge, passage of across junction of a metal and a gas, action of magnet on@\subdashtwo action of magnet on}%
+\index{Magnets, action of, on fdischarge with electrodes@\subdashtwo on discharge with electrodes}%
+tube are affected to a very great extent by the action of magnetic
+force. We may roughly describe the effect produced by
+a magnet by saying that the displacement of the discharge is
+much the same as that of a perfectly flexible wire conveying a
+current in the direction of that through the tube, the position of
+the wire coinciding with the part of the luminous discharge
+under consideration. This statement, which at first sight seems
+to bring the behaviour of the discharge under magnetic force
+into close analogy with that of ordinary currents, is apt, however,
+to obscure an essential difference between the two cases. A
+current through a wire is displaced by a magnetic force because
+the wire itself is displaced, and there is no other path open to
+the current. If, however, the current were flowing through a
+large mass of metal, if, for example, the discharge tube were
+filled with mercury instead of with rarefied gas, there would
+(excluding the Hall effect) be no displacement of the current
+through it. In the case of the rarefied gas, however, we have,
+what we do not have in the metal to any appreciable extent, a
+displacement of the lines of flow through the conductor---the
+rarefied gas. Thus the effects of the magnetic force on currents
+through wires, and on the discharge through a rarefied gas,
+instead of being, as they seem at first sight, the same, are
+apparently opposed to each other.
+
+\Article{127} The explanation which seems the most probable is that
+%% -----File: 146.png---Folio 132-------
+by which we explained the effect of a magnet on the discharge
+without electrodes: viz.~that when an electric discharge has
+passed through a gas, the supply of dissociated molecules, or of
+molecules in a peculiar condition, left behind in the line of the
+discharge, has made that line so much better a conductor than
+the rest of the gas, that when the particles composing it are
+displaced by the action of the magnetic force, the discharge
+continues to pass through them in their displaced position, and
+maintains by its passage the high conductivity of this line of
+particles. On this view the case would be very similar to that
+of a current along a wire, the line of particles along which the
+discharge passes being made by the discharge so much better a
+conductor than the rest of the gas, that the case is analogous to
+a metal wire surrounded by a dielectric.
+
+\includegraphicsouter{fig52}{Fig.~52.}
+
+\Article{128} This view seems to be confirmed by the behaviour of a
+\index{Air-blast, effect of on electric discharge}%
+\index{Electric discharge, qeffect of air blast on@\subdashtwo effect of air blast on}%
+\index{Feddersen, effect of air blast on spark}%
+spark between electrodes when a blast of air is blown across it;
+the spark is deflected by the blast much as a flexible wire
+would be if fastened at the two electrodes. On the preceding
+view the explanation of this would be, that by the passage of the
+spark through the gas, the electric strength of the gas along the
+line of discharge is diminished, partly by the lingering of atoms
+produced by the discharge, partly perhaps by the heat produced
+by the spark. When a blast of air is blowing across the space
+between the electrodes, the electrically weak gas will be carried
+with it, so that the next spark, which
+\includegraphicsmid{fig53}{Fig.~53.}
+will pass through the weak
+gas, will be deflected. Feddersen's observations (\textit{Pogg.\ Ann.}\ 103,
+p.~69, 1858) on the appearance presented
+by a succession of sparks in
+a revolving mirror when a blast of
+air was directed across the electrodes,
+seem to prove conclusively
+that this explanation is the true one,
+for he found that the first spark was quite straight, while the
+successive sparks got, as shown in \figureref{fig52}{Fig.~52}, gradually more
+and more bent by the blast.
+
+\includegraphicstwo{fig54}{Fig.~54.}{fig55}{Fig.~55.}
+
+\Article{129} The effects produced by a magnet show themselves in
+different ways, at different parts of the discharge. Beginning
+\index{Magnets, action of, on negative glow@\subdashtwo on negative glow}%
+\index{Negative glow, action of magnet on@\subdashtwo action of magnet on}%
+\index{Plücker, effect of magnet on discharge}%
+with the negative glow, Plücker (\textit{Pogg.\ Ann.}\ 103, p.~88, 1858),
+who was the first to observe the behaviour of this part of the
+discharge when under the action of a magnet, found that the
+%% -----File: 147.png---Folio 133-------
+appearance of the glow in the magnetic field could be described
+by saying that the negative glow behaved as if it consisted of a
+paramagnetic substance, such as iron filings without weight
+and with perfect freedom of motion. He found that the bright
+boundary of the negative glow coincided with the line of magnetic
+force passing through the extremity of the negative electrode.
+Figs.~\figureref{fig53}{53}, \figureref{fig54}{54},~\figureref{fig55}{55},~\figureref{fig56}{56}, which are taken
+\includegraphicsmid{fig56}{Fig.~56.}
+from Plücker's
+%% -----File: 148.png---Folio 134-------
+\index{Magnets, action of, on negative rays@\subdashtwo on negative rays}%
+\index{Negative rays, phosphorescence due to@\subdashtwo phosphorescence due to}%
+paper, show the shape taken by the glow when placed in the
+magnetic field due to a strong electro-magnet, the tube being
+placed in Figs.~\figureref{fig54}{54},~\figureref{fig55}{55} so that the lines of magnetic force are
+transverse to the line of discharge; while in Figs.\ \figureref{fig53}{53}~and~\figureref{fig56}{56}
+the line of discharge is more or less tangential to the direction
+of the magnetic force.
+
+\includegraphicsmid[!t]{fig57}{Fig.~57.}
+
+\includegraphicsmid[!b]{fig58}{Fig.~58.}
+
+\Article{130} Hittorf (\textit{Pogg.\ Ann.}\ 136, p.~213 \textit{et~seq.}, 1869) found that
+\index{Hittorf, discharge through gases}%
+when the negative rays were subject to the action of magnetic
+force, they were twisted into spirals and sometimes into circular
+rings. In his experiments the negative electrode was fused into
+a small glass tube fused into the discharge tube, the open end of
+the small tube projecting beyond the electrode. The negative
+rays were by this means limited to those which were approximately
+parallel to the axis of the small tube, so that it was easy
+to alter the angle which these rays made with the lines of magnetic
+force either by moving the discharge tube or altering the
+position of the electro-magnet. The discharge tube was shaped
+so that the walls of the tube were at a considerable distance from
+the negative electrode. Hittorf found
+\includegraphicsmid{fig59}{Fig.~59.}
+\includegraphicsmid[b]{fig60}{Fig.~60.}
+that when the direction
+of the negative rays was tangential to the line of magnetic force
+passing through the extremity of the cathode, the rays continued
+%% -----File: 149.png---Folio 135-------
+to travel along this line; that when the rays were initially at
+right angles to the lines of magnetic force they curled up into
+circular rings; and that when the rays were oblique to the direction
+of the magnetic force they were twisted into spirals of which
+two or three turns were visible; the axis of the spiral being
+parallel to the direction of the magnetic force. These effects are
+illustrated in Figs.~\figureref{fig57}{57}, \figureref{fig58}{58},~\figureref{fig59}{59}, and~\figureref{fig60}{60}.
+In Figs.\ \figureref{fig57}{57}~and~\figureref{fig58}{58} the
+rays are at right angles to the lines of magnetic force, while in
+Figs.\ \figureref{fig59}{59}~and~\figureref{fig60}{60} they are oblique to them.
+
+
+\Article{131} This spiral form is the path which would be traversed
+by a negatively charged particle moving away from the cathode.
+To prove this, let us assume that the magnetic field is uniform,
+and that the axis of~$z$ is parallel to the lines of magnetic force.
+Let $e$~be the charge on the particle, $v$~its velocity. Then if we
+regard the particle as a small conducting sphere, the mechanical
+force on it in the magnetic field is, if $v$~is small compared with
+the velocity of light, the same (see \artref{16}{Art.~16}) as that which would
+be exerted on unit length of a wire carrying a current whose
+%% -----File: 150.png---Folio 136-------
+components parallel to the axes of $x$,~$y$,~$z$ are respectively
+\[
+\tfrac{1}{3} ev \frac{dx}{ds}, \quad \tfrac{1}{3} ev \frac{dy}{ds}, \quad \tfrac{1}{3} ev \frac{dz}{ds},
+\]
+where $ds$ is an element of the path of the particle. Thus, if $m$~is
+the mass of the particle, $Z$~the magnetic force, the equations of
+motion of the particle are
+\begin{align*}
+m \frac{d^2 x}{dt^2} &= \tfrac{1}{3} ev Z \frac{dy}{ds}, \Tag{1}\\
+m \frac{d^2 y}{dt^2} &= -\tfrac{1}{3} ev Z \frac{dx}{ds}, \Tag{2}\\
+m \frac{d^2 z}{dt^2} &=  0. \Tag{3}
+\end{align*}
+
+Since the force on the particle is at right angles to its direction
+of motion, the velocity~$v$ of the particle will be constant,
+and since by~(\eqnref{131}{3}) the component of the velocity parallel to the
+axis of~$z$ is constant, the direction of motion of the particle must
+make a constant angle, $\alpha$~say, with the direction of the magnetic
+force. Since $ds/dt$ is constant, equations (\eqnref{131}{1})--(\eqnref{131}{3}) may be written
+\begin{align*}
+mv^2 \frac{d^2 x}{ds^2} &= \tfrac{1}{3} ev Z \frac{dy}{ds},\\
+mv^2 \frac{d^2 y}{ds^2} &= -\tfrac{1}{3} ev Z \frac{dx}{ds},\\
+mv^2 \frac{d^2 z}{ds^2} &=  0.
+\end{align*}
+
+If $\rho$ is the radius of curvature of the path, $\lambda$,~$\mu$,~$\nu$ its direction
+cosines,
+\[
+\frac{d^2 x}{ds^2} = \frac{\lambda}{\rho},\quad \frac{d^2 y}{ds^2} = \frac{\mu}{\rho}, \quad \frac{d^2 z}{ds^2} = \frac{\nu}{\rho}.
+\]
+
+Hence from the preceding equations
+\begin{align*}
+\frac{\lambda}{\rho} &=  \tfrac{1}{3} \frac{Ze}{mv}\, \frac{dy}{ds},\\
+\frac{\mu}{\rho} &= -\tfrac{1}{3} \frac{Ze}{mv}\, \frac{dx}{ds},\\
+\frac{\nu}{\rho} &= 0.
+\end{align*}
+
+Squaring and adding, we get
+\[
+\frac{1}{\rho^2} = \Bigl(\tfrac{1}{3} \frac{Ze}{mv}\Bigr)^2 \left\{\Bigl(\frac{dx}{ds}\Bigr)^2 + \Bigl(\frac{dy}{ds}\Bigr)^2 \right\}.
+\]
+%% -----File: 151.png---Folio 137-------
+
+But
+\[
+\Bigl(\frac{dx}{ds}\Bigr)^2 + \Bigl(\frac{dy}{ds}\Bigr)^2 = \sin^2 \alpha.
+\]
+
+So that
+\[
+\frac{1}{\rho} = \tfrac{1}{3} \frac{Ze}{mv} \sin \alpha.
+\]
+
+Hence the radius of curvature of the path of the particle is
+constant, and since the direction of motion makes a constant
+angle with that of the magnetic force, the path of the particle is
+a helix of which the axis is parallel to the magnetic force; the
+angle of the spiral is the complement of the angle which the
+direction of projection makes with the magnetic force. If $a$~is
+the radius of the cylinder on which the spiral is wound,
+$a = \rho \sin^2 \alpha$, so that
+\[
+a = 3 \frac{mv}{Ze} \sin \alpha.
+\]
+
+If $\alpha = \pi/2$, the spiral degenerates into a circle of which the
+radius is~$3 mv/Ze$.
+
+Let the particle be an atom of hydrogen charged with the
+quantity of electricity which we find always associated with the
+atom of hydrogen in electrolytic phenomena: then since the
+electro-chemical equivalent of hydrogen is about~$10^{-4}$, we have,
+if $N$~is the number of hydrogen atoms in one gramme of that
+substance, $Ne = 10^4$ and $Nm = 1$; hence when the ray is curled up
+into a ring of radius~$a$,
+\[
+a = 10^{-4}\, 3 \frac{v}{Z},
+\]
+or $3v = 10^4 aZ$ in hydrogen.
+
+\Article{132} In one of Hittorf's experiments, that illustrated in
+\figureref{fig60}{Fig.~60}, he estimated the diameter of the ring as less than
+$1$~mm.:\ the gas in this case was air, which is not a simple gas;
+we shall assume, however, that $m/e$~is the same as for oxygen,
+or eight times the value for hydrogen. Putting
+\begin{gather*}
+a = 5 × 10^{-2}, \text{~and~} m/e = 8 × 10^{-4}, \text{~we get} \\
+v = \frac{5}{24}\, 10^2 Z
+\end{gather*}
+
+The value of~$Z$ is not given in Hittorf's paper; we may be sure,
+however, that it was considerably less than~$10^4$, and it follows
+that $v$~must have been less than $2 × 10^5$; this superior limit to
+%% -----File: 152.png---Folio 138-------
+the value of~$v$ is less than six times the velocity of sound.
+Hence the velocity of these particles must be infinitesimal in
+comparison with that of the positive luminosity which, as we
+have seen, is comparable with that of light.
+
+\Article{133} A magnet affects the disposition of the negative glow
+\index{Negative glow, distribution over electrode@\subdashtwo distribution over electrode}%
+\index{Magnets, action of, xdistribution of negative glow over electrodes@\subdashtwo distribution of negative glow over electrodes}%
+\index{Electrode, effect of magnet on distribution of negative glow over}%
+over the surface of the electrode as well as its course through the
+gas. Thus Hittorf (\textit{Pogg.\ Ann.}\ 136, p.~221, 1869) found that
+when the negative electrode is a flat vertical disc, and the discharge
+tube is placed horizontally between the poles of an electromagnet,
+with the disc in an axial plane of the electromagnet;
+the disc is cleared of glow by the magnetic
+force except upon the highest point
+on the side most remote from the positive
+electrode, or the lowest point on the side
+nearest that electrode according to the
+direction of the magnetic force. In
+another experiment Hittorf, using as a
+cathode a metal tube about $1$~cm.\ in
+diameter, found that when the discharge
+tube is placed so that the axis of the
+cathode is at right angles to the line joining
+the poles of the electromagnet the
+cathode is cleared of glow in the neighbourhood
+of the lines where the normals
+are at right angles to the magnetic force.
+These experiments show that the action
+of a magnet on the glow is the same as its
+action on a system of perfectly flexible
+currents whose ends can slide freely over
+the surface of the negative electrode.
+
+\includegraphicsouter{fig61}{Fig.~61.}
+
+\Article{134} The positive column is also deflected
+\index{Magnets, action of, on positive column@\subdashtwo on positive column}%
+\index{Positive column, effect of magnet on@\subdashtwo effect of magnet on}%
+by a magnet in the same way as a perfectly flexible wire
+carrying a current in the direction of that through the discharge
+\index{De la Rive, rotation of electric discharge}%
+tube. This is beautifully illustrated by an experiment due to De~la~Rive
+in which the discharge through a rarefied gas is set in
+continuous rotation by the action of a magnet. The method of
+making this experiment is shown in \figureref{fig61}{Fig.~61}; the two terminals
+$a$~and~$d$ are metal rings separated from each other by an insulating
+tube which fits over a piece of iron resting on one of the
+poles of an electromagnet~$M$. This arrangement is placed in an
+%% -----File: 153.png---Folio 139-------
+egg-shaped vessel from which the air can be exhausted. To
+make the experiment successful it is advisable to introduce a
+small quantity of the vapour of alcohol or turpentine. The terminals
+$a$~and~$d$ are connected with an induction coil, which,
+when the pressure in the vessel is sufficiently reduced, produces a
+discharge through the gas between the terminals $a$~and~$d$, which
+rotates under the magnetic force with considerable velocity. The
+rotation of the discharge through the gas is probably due, as we
+have seen, to the displacement of the particles through which one
+discharge has already passed; the displaced particles form an
+easier path for a subsequent discharge than the original line
+of discharge along which none of the dissociated molecules
+have been left. The new discharge will thus not be along the
+same line as the old one, and the luminous column will therefore
+rotate. We can easily see why a simple gas like hydrogen
+should not show this effect nearly so well as a complicated one
+like the vapour of alcohol or turpentine. For the discharges of
+the induction coil are intermittent, so that to produce this rotation
+the dissociated molecules produced by one discharge must
+persist until the arrival of the subsequent one. Now we should
+expect to find that when a molecule of a stable gas like hydrogen
+is dissociated by the discharge, the recombination of its atoms
+will take place in a much shorter time than similar recombination
+for a complex gas like turpentine vapour; thus we should
+expect the effects of the discharge to be more persistent, and
+therefore the rotation more decided in turpentine vapour than in
+hydrogen.
+
+\Article{135} Crookes (\textit{Phil.\ Trans.}\ 1879, Part~II, p.~657) has produced
+\index{Crookes on discharge through gases}%
+somewhat analogous rotations of the negative rays in a very
+highly exhausted tube. The shape of the tube he employed is
+shown in \figureref{fig62}{Figure~62}. When the discharge went through this tube,
+the neck surrounding the negative pole was covered with two
+or three bright patches which rotated when the tube was placed
+over an electromagnet. Crookes found that the direction of rotation
+was reversed when the magnetic force was reversed, but that
+if the magnetic force were not altered the direction of rotation was
+not affected by reversing the poles of the discharge tube. This
+is what we should expect if we remember that the bright
+spots on the glass are due to the negative rays, and that these
+will be at right angles to the negative electrode; thus the reversal
+%% -----File: 154.png---Folio 140-------
+of the poles of the tube does not reverse the direction of
+these rays; it merely alters their distance from the pole of the
+electromagnet. The curious thing about the rotation was that
+it had the \emph{opposite} direction to that
+which would have been produced by the
+action of a magnet on a current carrying
+electricity in the same direction as that
+carried by the negative rays, showing
+clearly that this rotation is due to some
+secondary effect and not to the primary
+action of the magnetic force on the current.
+
+\includegraphicsouter{fig62}{Fig.~62.}
+
+\Article{136} An experiment due to Goldstein,
+\index{Goldstein, discharge of electricity through gases|(}%
+which may seem inconsistent with the
+view we have taken, viz.~that the deflection
+of the discharge is due to the deflection
+of the line of least electric strength, should
+be mentioned here. Goldstein (\textit{Wied.\
+Ann.}~12, p.~261, 1881) took a large discharge
+tube, $4$~cm.\ wide by $20$~long, the
+electrodes being at opposite ends of the
+tube. A piece of sodium was placed in
+the tube which was then quickly filled
+with dry nitrogen, the tube was then exhausted
+until a discharge passed freely
+through the tube, and the sodium heated
+until any hydrogen it might have contained
+had been driven off. When this
+had been done the tube was refilled with
+nitrogen and then exhausted until the
+positive column filled the tube with a
+reddish purple light. The sodium was then slowly heated until
+its vapour began to come off, when the discharge in the lower part
+of the tube over the sodium became yellow as it passed through
+sodium vapour, while the discharge at the top of the tube remained
+red as the sodium vapour did not extend all the way
+across the tube. The positive discharge was now deflected by
+a magnet and driven to the top of the tube out of the region
+occupied by the sodium vapour, the discharge was now entirely
+red and showed no trace of sodium light. The experiment does
+not seem inconsistent with the view we have advocated, as we
+%% -----File: 155.png---Folio 141-------
+cannot suppose that more than an infinitesimal quantity of
+sodium vapour travelled across the tube under the action of the
+magnetic force, and it does not follow that because we suppose
+the line of discharge to be weakened by the presence of the
+dissociated molecules that these molecules are the only ones
+affected by the discharge; it seems much more probable that
+they serve as nuclei round which the chemical changes which
+transmit the discharge take place.
+
+\Article{137} The striations are affected by magnetic force; in Figs.\
+\figureref{fig53}{53}~and~\figureref{fig56}{56} may be seen the distortion of \DPtypo{striae}{striæ} when the discharge
+\index{Magnets, action of, xstriations@\subdashtwo striations}%
+\index{Striations, effect of magnetic force on@\subdashone effect of magnetic force on}%
+tube is placed in a magnetic field. If the negative
+glow is driven away from the line joining the terminals by
+magnetic force, the positive column lengthens and fills part of
+the space previously occupied by the negative glow; if the positive
+column is striated new \DPtypo{striae}{striæ} appear, so that in this case
+we have a creation of \DPtypo{striae}{striæ} by the action of magnetic force.
+The most remarkable effect of a magnet on the striated discharge,
+\index{Spottiswoode and Moulton, electric discharge}%
+however, is that discovered by Spottiswoode and Fletcher
+Moulton, and Goldstein; Spottiswoode and Moulton (\textit{Phil.\ Trans.}\
+1879, Part~I, p.~205) thus describe the effect: `If a magnet be
+applied to a striated column, it will be found that the column is
+not simply thrown up or down as a whole, as would be the case
+if the discharge passed in direct lines from terminal to terminal,
+threading the \DPtypo{striae}{striæ} in its passage. On the contrary, each stria
+is subjected to a rotation or deformation of exactly the same
+character as would be caused if the stria marked the termination
+of flexible currents radiating from the bright head of the stria
+behind it and terminating in the hazy inner surface of the stria
+in question. An examination of several cases has led the authors
+of this paper to conclude that the currents do thus radiate from
+the bright head of a stria to the inner surface of the next, and
+that there is no direct passage from one terminal of the tube to
+the other.' Goldstein (\textit{Wied.\ Ann.}~11, p.~850, 1880) found that
+the striated column could by the action of magnetic force be
+broken up into a number of bright curves, of the same kind as
+those observed by Hittorf in the negative rays (see \artref{130}{Art.~130}), the
+number of bright curves being the same as the number of \DPtypo{striae}{striæ}
+which had disappeared; each striation was transformed by the
+magnetic force into a separate curve, and these curves were
+separated from each other by dark spaces. We may conclude
+%% -----File: 156.png---Folio 142-------
+\index{Potential distribution of along discharge tube,@\subdashone distribution of along discharge tube}%
+from these experiments that the positive column does not consist
+of a current of electricity traversing the whole of its length
+in the way that such a current would traverse a metal cylinder
+coincident with the positive column, but that it rather consists
+of a number of separate currents, each striation corresponding
+to a current which is to a certain extent independent of those
+which precede or follow. The discharge along the positive
+column might perhaps be roughly illustrated by placing pieces
+of wire equal in length to the \DPtypo{striae}{striæ} and separated by very
+minute air spaces along the line of discharge.
+
+\Article{138} Goldstein found that the rolling up of the \DPtypo{striae}{striæ} by the
+magnetic force was most marked at the end of the positive
+column nearest the negative electrode: the following is a translation
+of Goldstein's description of this process (l.c.\ p.~852). The
+appearance is very characteristic when in the unmagnetized condition
+the negative glow penetrates beyond the first striation into
+the positive column. The end of the negative glow is then further
+from the cathode than the first striation or, even if the rarefaction
+is suitable, than the second or third. Nevertheless the end of the
+negative glow rolls itself under the magnetic action up to the
+cathode in the magnetic curve which passes through the cathode.
+Then separated from this by a dark space follows on the side of
+the anode a curve in which all the rays of the first striation are
+rolled up, then a similar curve for the second striation, and so on.
+
+We shall have occasion to refer to these experiments again in
+the discussion of the theory of the discharge.
+
+\Subsection{On the Distribution of Potential along an Exhausted Tube
+through which an Electric Discharge is passing.}
+
+\Article{139} The changes which take place in the potential as we pass
+along the discharge tube are extremely interesting, as they
+present a remarkable contrast to those which take place along a
+metal wire through which a steady uniform current is passing;
+in this case the potential-gradient is uniform along the wire,
+but changes when the current changes, being by Ohm's law
+proportional to the intensity of the current; in the exhausted
+tube, on the other hand, the potential-gradient varies greatly in
+different parts of the tube, but in the positive column is almost
+independent of the intensity of the current passing through the
+gas. The potentials measured are those of wires immersed in
+\index{Goldstein, discharge of electricity through gases|)}%
+%% -----File: 157.png---Folio 143-------
+the rarefied gas, and the question arises, whether the potentials
+of these wires are constant, as they would be if the wires were
+in a steady current, or whether they are variable, the potentials
+determined in these experiments being the mean values about
+\index{Continuity of current through discharge-tube}%
+which the potentials of the wires fluctuate? This question is
+the same as, whether the current through the gas is continuous
+or intermittent? On this point considerable difference of opinion
+has existed among physicists. There is no doubt that by the
+aid of a battery consisting of a large number of cells a discharge
+can be got, which, if not continuous, has such a high rate of
+intermittence that no unsteadiness can be detected when it is
+observed in a rotating mirror making $100$~revolutions per
+second; this is sufficient to prove that if the intermittence
+exists at all it must be exceedingly rapid. As long, however,
+as the discharge retains the property of requiring a large potential
+difference to exist between the electrodes, this difference
+varying continuously with the pressure, while the latter varies
+from that of an atmosphere to the pressure in the discharge
+tube, we should expect the electrodes to act like condensers
+continually being charged and discharged as they are at atmospheric
+pressures, in other words we should expect the discharge
+to be intermittent. When, however, the discharge passes
+as the `arc discharge,' see \artref{169}{Art.~169}, the potential difference falls
+to a comparatively small value, and it is probable that this discharge
+is much more nearly continuous than the striated one.
+
+It ought also to be remembered that the current through the
+gas may be interrupted even though that through the leads is
+continuous. For since the current through the gas does not
+obey the same laws as when it goes through a metallic conductor,
+the current across a section of the discharge tube need
+not at any specified instant be the same as that across the
+section of one of the leads. The average current must of course
+be the same in the two cases, but only the \emph{average} current
+and not that at any particular instant. To quote an illustration
+\index{Spottiswoode and Moulton, electric discharge}%
+given by Spottiswoode and Moulton, the discharge tube may
+act like the air vessel of a fire engine; all the electricity that
+goes in comes out again, but no longer with the same pulsation.
+The tube may sometimes contain more and sometimes less free
+electricity, and may act as an expansible vessel would act if it
+formed part of the path of an incompressible fluid.
+%% -----File: 158.png---Folio 144-------
+
+The rapidity of the intermittence can to some extent be tested
+by observing whether or not the discharge is deflected by the
+approach of a conductor. When the discharge is intermittent
+and the interval between the discharges so long that the intermittence
+of the discharge can be detected either by the eye or
+by a slowly rotating mirror, the discharge is deflected when a
+conductor is brought near it; when however the intermittence is
+very rapid, the discharge is not affected by the approach of the
+conductor. This effect has been very completely investigated
+\index{Spottiswoode and Moulton, electric discharge}%
+by Spottiswoode and Moulton (\textit{Phil.\ Trans.}\ 1879, Part~I, p.~166;
+1880, Part~II, p.~564).
+
+\Article{140} We shall begin by considering Hittorf's experiments on
+\index{Gradient of potential in discharge tube}%
+\index{Hittorf, discharge through gases}%
+the potential gradient (\textit{Wied.\ Ann.}~20.\ p.~705, 1883). The discharge
+tube, \figureref{fig63}{Fig.~63}, which was $5.5$~cm.\ in diameter and $33.7$~cm.\
+long, had aluminium wires $2$~mm.\ in diameter \DPtypo{$f$ used}{fused} into the
+ends for electrodes, the anode,~$a$, was $2$~cm.\ long, the cathode,~$c$,
+$7$~cm. In addition to the electrodes five aluminium wires,~$b$, $d$,~$e$,
+$f$,~$g$, half a millimetre in diameter, were fused into the tube.
+The difference of potential between any two of these wires could
+be determined by connecting them to the plates of a condenser,
+and then discharging the condenser through a galvanometer.
+The deflection of the galvanometer was proportional to the
+charge in the condenser, which again was proportional to the
+difference of potential between the wires. The discharge was
+produced by means of a large number of cells of Bunsen's
+chromic acid battery, and the intensity of the current was
+varied by inserting in the circuit a tube containing a solution of
+cadmium iodide, which is a very bad conductor. No intermittence
+in the discharge could be detected either by a mirror
+rotating $100$~times a second or a telephone. The tube was filled
+with nitrogen, as this gas has the advantage of not attacking
+the electrodes and of not being absorbed by them so greedily as
+%% -----File: 159.png---Folio 145-------
+hydrogen. The results of some of the measurements are given
+in the following Table, l.~c.\ p.~727:---
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{4pt}
+\settowidth{\TmpLen}{Number fixing the experiment}%
+\begin{tabular}{@{}p{\TmpLen}|c|c|c|c|c|c|c|c@{}}
+\multicolumn{9}{c}{\normalsize\textit{Pressure of Nitrogen $.6$~mm.}\medskip} \\
+\hline
+\tablespaceup Number fixing the experiment & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
+Number of cells \mdotfill & $500$ & $500$ & $500$ & $600$ & $700$ & $800$ & $900$ & $1000$ \\
+\parbox[b]{\TmpLen}{\tabhang Intensity of current in millionths
+of an Ampère \mdotfill}
+ & $244$ & $814$ & $1282$ & $3175$ & $5189$ & $7000$ & $8791$ & $11192$ \\
+\parbox[b]{\TmpLen}{\tabhang Kick of galvanometer due to
+the charging of the condenser
+to the potential difference
+between---} &&&&&&&& \\
+\parbox[c]{\TmpLen}{\hfill $ac$ \mdotfill} & $133$ & $132$ & $133.5$ & $141.5$ & $150$ & $157$ & $165$ & $173$ \\
+\parbox[c]{\TmpLen}{\hfill $ab$ \mdotfill} & $22$ & $22.5$ & $22$ & $21.5$ & $21$ & $21$ & $21$ & $21$ \\
+\parbox[c]{\TmpLen}{\hfill $bd$ \mdotfill} & $14$ & $13$ & $13$ & $12$ & $12.5$ & $12$ & $12$ & $12.25$ \\
+\parbox[c]{\TmpLen}{\hfill $de$ \mdotfill} & $13$ & $13$ & $13$ & $14$ & $14$ & $13.5$ & $12$ & $12.5$ \\
+\parbox[c]{\TmpLen}{\hfill $ae$ \mdotfill} & $52$ & $50$ & $49$ & $47$ & $47$ & $47$ & $47$ & $47$ \\
+\parbox[c]{\TmpLen}{\hfill $fg$ \mdotfill} & --- & $2.25$ & $3$ & $4$ & $3.75$ & $44$ & $3.25$ & $3$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+\medskip
+
+\includegraphicsmid{fig63}{Fig.~63.}
+
+The difference of potential in volts can be approximately got
+by multiplying the galvanometer deflection by~$6$. In experiment~1
+the negative glow covered about $1.5$~cm.\ of the cathode,
+and the positive light extended to~$f$. In experiment~2 the negative
+glow covered $6$~cm.\ of the cathode, and in~3 and the following
+experiments the whole of the cathode. In experiments 1,~2,~3
+the thickness of the negative glow remained the same; in the
+later experiments where the negative glow covered the whole of
+the cathode its thickness increased as the intensity of the current
+increased, and in 7~and~8 it extended to the walls of the tube.
+The table shows that no changes in the potential differences occurred
+until the negative glow began to increase in thickness.
+\index{Positive column, potential gradient in@\subdashtwo potential gradient in}%
+\index{Potential gradient in positive column,@\subdashone gradient in positive column}%
+We see that by far the greatest fall in the potential occurs in the
+immediate neighbourhood of the cathode, the rise in potential
+from the negative electrode to the outside of the negative glow
+being far greater than the rise in all the rest of the tube; we also
+see that the changes which take place when the thickness of the
+negative glow alters take place in this part of the tube, and that
+the potential differences in the positive column are \emph{independent
+of the strength of the current}. The portions $bd$~and~$de$ of the
+positive column, which are very nearly equal in length, have
+also practically the same potential differences; and these are
+each less than that of the portion~$ab$ which contains the anode,
+although the latter portion is considerably shorter. The wires
+$f$\DPtypo{;}{,}~$g$ were in all these experiments in the dark space between the
+%% -----File: 160.png---Folio 146-------
+negative glow and the positive column. The small difference of
+potential between these wires is very noteworthy.
+
+\Article{141} Hittorf also investigated the potential differences for lower
+\index{Potential xgradient in positive column at low pressures@\subdashtwo at low pressures}%
+pressures of the gas than that used in the last experiment;
+for this purpose the tube in \figureref{fig63}{Fig.~63} was not suitable, as the
+negative glow was very much interfered with by the walls of the
+tube, he therefore used a tube shaped like that in \figureref{fig64}{Fig.~64}, which
+was purposely made wide in the region round the negative
+electrode. The diameter of the positive part of the tube was
+$4$~cm., that of the negative $12$~cm. The length of the negative
+electrode was $15$~cm., that of the positive $3$~cm. In this case
+only two wires, $b$~and~$d$, were inserted in the tube. The results
+of experiments with this tube are given in the following
+table:---
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{4pt}
+\settowidth{\TmpLen}{Pressure of the nitrogen in millimetres}%
+\begin{tabular}{@{}p{\TmpLen}|c|c|c|c|c|c@{}}
+\hline
+\tablespaceup Number fixing the experiment & 1 & 2 & 3 & 4 & 5 & 6 \\
+\parbox[b]{\TmpLen}{\tabhang Pressure of the nitrogen in
+millimetres of mercury\mdotfill}
+ & $\Z\Z\llap{.}70$ & $\Z\Z\llap{.}35$ & $\Z\llap{.}175$ & $\Z\llap{.}088$ & $\llap{.}044$ & $\llap{.}022$ \\
+\parbox[b]{\TmpLen}{Number of cells\mdotfill} & $\Z600$ & $\Z600$ & $\Z600$ & $\Z600$ & $600$ & $600$ \\
+\parbox[b]{\TmpLen}{\tabhang Strength of current in millionths
+of an Ampère\mdotfill}
+& $2870$ & $2076$ & $1791$ & $1360$ & $916$ & $488$ \\
+\parbox[c]{\TmpLen}{\tabhang Kick of galvanometer due to
+the charging of the condenser
+to the potential difference
+between---} & & & & & & \\
+\hfill 1 \qquad $ac$\mdotfill & $\Z151\Z$ & $\Z140\Z$ & $\Z145\Z$ & $\Z157\phantom{.0}$   & $\Z168\Z$ & $178\phantom{.00}$ \\
+\hfill 2 \qquad $ab$\mdotfill & $\Z21$    & $\Z15$    & $\Z12$    & $\Z\Z\Z9.5$           & $\Z\Z8$   & $\Z\Z7\phantom{.00}$ \\
+\hfill 3 \qquad $bd$\mdotfill & $\Z30$    & $\Z19$    & $\Z12$    & $\Z\Z\Z8\phantom{.0}$ & $\Z\Z5$   & $\Z\Z4.25$ \\
+\hfill 4 \qquad $ad$\mdotfill & $\Z51$    & $\Z34$    & $\Z24$    & $\Z\Z17.5$            & $\Z13$    & $\Z11.25$\tablespacedown \\
+\hline
+\end{tabular}
+%% -----File: 161.png---Folio 147-------
+
+\bigskip
+\begin{tabular}{@{}p{\TmpLen}|c|c|c|c|c@{}}
+\hline
+\tablespaceup Number fixing the experiment & 7 & 8 & 9 & 10 & 11 \\
+\parbox[b]{\TmpLen}{\tabhang Pressure of the nitrogen in
+millimetres of mercury\mdotfill}
+ & $.011$? & $.0055$? & $.0029$? & $.0014$? & $.0007$? \\
+\parbox[b]{\TmpLen}{Number of cells\mdotfill} & $600$ & $800$ & $1000$ & $1200$ & $1400$ \\
+\parbox[b]{\TmpLen}{\tabhang Strength of current in millionths
+of an Ampère\mdotfill}
+ & $326$ & $610$ & $\Z814$ & $\Z814$ & $1100$ \\
+\parbox[b]{\TmpLen}{\tabhang Kick of galvanometer due to
+the charging of the condenser
+to the potential difference
+between---} & & & & & \\
+\hfill 1 \qquad $ac$\mdotfill & $184$   & $242$ & $298$ & $352$ & $422$ \\
+\hfill 2 \qquad $ab$\mdotfill & $\Z\Z7$ & $\Z\Z7$ & $\Z\Z8$ & $\Z8.5$ & $8.75$ \\
+\hfill 3 \qquad $bd$\mdotfill & $\Z\Z4$ & $\Z\Z4$ & $3.75$ & $\Z2.5$ & $2.25$ \\
+\hfill 4 \qquad $ad$\mdotfill & $\Z11$  & $\Z11\rlap{$.5$}$ & $\Z12$ & $11.5$ & $10.5$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+\includegraphicsmid{fig64}{Fig.~64.}
+
+The negative glow in all these experiments covered the cathode,
+and in all but the first three it extended to the walls of the tube.
+The appearance of the glow at the higher exhaustions is shown
+in \figureref{fig64}{Fig.~64}, where the shaded portions represent the bright parts
+of the discharge; it will be seen from the figure that the positive
+column was striated.
+
+\Article{142} The table shows that at high exhaustions the potential
+difference between the electrodes increases as the density of the
+gas diminishes, but that this increase is confined to the neighbourhood
+of the cathode; the ratio of the change in potential
+near the cathode to that in the rest of the tube increases as
+the pressure of the gas diminishes. The potential difference
+in the positive light diminishes as the pressure is reduced, but
+the diminution in the potential difference is not so rapid as the
+diminution in the pressure. The table seems to suggest that
+the potential gradient in the positive column tends towards a
+constant value which is independent of the density. We must
+remember however that Hittorf's experiments do not give the
+potential difference required to initiate the discharge through
+the gas, but the distribution of potential which accompanies
+the passage of electricity through the gas when the discharge
+has been established for some time, and where there are a
+plentiful supply of dissociated molecules produced by the
+passage of previous discharges. Hittorf found that the number
+of cells which would maintain a discharge after it was once
+started was frequently quite insufficient to initiate it, and
+the gas had to be broken through by a discharge from another
+source.
+
+\Article{143} The experiments described in \artref{79}{Art.~79} on the discharge
+%% -----File: 162.png---Folio 148-------
+without electrodes, when the interval between two discharges
+was long enough to give the gas through which the discharge
+had passed an opportunity of returning to its normal condition
+before the passage of the next discharge, show that even when
+no electrodes are used the electromotive intensity required to
+start the discharge has a minimum value at a particular pressure,
+and that when the pressure is reduced below this value
+the electromotive intensity required for discharge increases.
+
+\Article{144} The supply of dissociated molecules furnished by previous
+discharges also explains another peculiarity of these experiments.
+It will be seen from the table that at a pressure of $.0007$~mm.\
+of mercury, a potential difference which gave a galvanometer
+deflection of~$10.5$, corresponding to about $63$~volts, was all that
+occurred in a length of $12$~cm.\ of the positive light; it does not
+follow however that a potential gradient of about $5$~volts per
+centimetre would be sufficient to \emph{initiate} the discharge even if the
+great change in potential at the cathode were absent. In fact
+the experiments previously described on the discharge without
+electrodes show that it requires a very much greater electromotive
+intensity than this even when the cathode is entirely done
+away with.
+
+The table shows that the potential difference between $a$~and~$b$,
+a space which includes the anode, has at the higher exhaustions
+passed its minimum value and commenced to increase.
+
+\Article{145} Though the potential differences between wires immersed
+in the positive column is independent of the strength of the
+current passing through the tube, yet in such a tube as \figureref{fig63}{Fig.~63}
+the potential differences between wires in the middle of the
+tube may be affected by variations in the current if these variations
+are accompanied by changes in the appearance of the discharge.
+
+Let us suppose, for example, that the tube is filled with nitrogen
+at a pressure of from $2$ to $3$~mm.\ of mercury, then when the intensity
+of the current is very small the tube will appear to be
+dark throughout almost the whole of its length, the positive
+column and negative glow being reduced to mere specks in the
+neighbourhood of the electrodes; when however the intensity of
+the current increases the positive column increases in length, and
+if the increase is great enough to make it envelop two wires
+which were previously in the dark Faraday space, the difference
+%% -----File: 163.png---Folio 149-------
+of potential between these wires will be found to be very much
+greater than when the gas round them was non-luminous. This
+is illustrated for lower pressures by the table in \artref{140}{Art.~140}, which
+shows that the potential gradient between $f$~and~$g$, the wires
+in the dark space between the positive column and the negative
+glow, was very much less than the potential gradient in
+the positive column. It is shown however still more clearly
+in the following set of experiments made with the tube shown
+in \figureref{fig63}{Fig.~63} (l.~c.\ p.~739).
+
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{3pt}
+\settowidth{\TmpLen}{Kick of the galvano-}%
+\begin{tabular} {@{}c|c|c|c|c|c|c|c@{}}
+\multicolumn{8}{c}{\normalsize\textit{Pressure of Nitrogen $3.95$~mm.\ of Mercury.}}\\
+\multicolumn{8}{c}{\normalsize\textit{Temperature $12°$\,C.}\medskip} \\
+\hline
+\parbox[b]{\TmpLen}{\tablespaceup\tabhang Number fixing the
+experiment\mdotfill}           & 1    & 2    & 3   & 4   & 5   & 6   & 7\\
+\parbox[b]{\TmpLen}{Number of cells\mdotfill} & $700$ & $700$ & $700$ & $800$ & $900$ & $1000$ & $1200$ \\
+\parbox[b]{\TmpLen}{\tabhang Intensity of the current
+in millionths
+of an Ampère\mdotfill}         & $1465$ & $2035$ & $2391$ & $2483$ & $2830$ & $3541$ & $5820$ \\
+\parbox[b]{\TmpLen}{\tabhang Kick of the galvano\-meter
+from the
+charge in the condenser
+due to the
+potential difference
+between---} & & & &  & & & \\
+\hfill 1 \qquad $ac$\mdotfill & $166$--$168$ & $175$--$168$ & $190$--$188$ & $212$--$208$ & $238$--$232$ & \PadTo{238$--$232}{255} & $292$--$285$ \\
+\hfill 2 \qquad $ab$\mdotfill & ---      &---      &---      &---      & $63$    & $\Z60$   & \PadTo[l]{99.9}{79} \\
+\hfill 3 \qquad $bd$\mdotfill & $16.5$   & \PadTo[l]{19.9}{18} & $18.5$  & $25$    & $43$    & $\Z61$   & $56.5$ \\
+\hfill 4 \qquad $de$\mdotfill & $17.5$   & \PadTo[l]{19.9}{18} & \PadTo[l]{19.9}{17} & $18$  & $20$   & $\Z26$   & \PadTo[l]{99.9}{62} \\
+\hfill 5 \qquad $fg$\mdotfill & \PadTo[l]{19.9}{10} & $10.5$   & $11.5$    & $12$    & $13$  & $\Z13$   & $12.5$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+\Article{146} In experiments~1--3 the tube was quite dark, except
+quite close to the electrodes; the anode had a thin coating of
+positive light. The negative glow extended in experiment~1 over
+$1$~cm.\ of the cathode, in experiment~2 over $3$~cm., and in experiment~3
+over $3$~cm. In experiment~3 the beginning of a brush
+discharge was discernible at the anode. In consequence of the
+wires being in the dark Faraday space instead of the positive
+column, it will be noticed that the potential difference between
+$b$~and~$d$ is very little greater than in the experiments described
+in \artref{140}{Art.~140}, though the pressure is more than six times
+greater.
+
+\Article{147} In experiment~4 the positive column reached past~$b$;
+it will be seen that the potential difference between $b$~and~$d$
+rose to~$25$, while the differences between $d$~and~$e$ and between
+%% -----File: 164.png---Folio 150-------
+$f$~and~$g$, which were still in the dark, remained unaltered. In
+experiment~5 the positive column reached past the middle of~$bd$;
+the potential difference in~$bd$ rose from~$25$ to~$43$, the potential
+differences between the wires in the dark still being unaltered.
+In experiment~6 the positive light filled the whole space~$ad$;
+the potential difference between $b$~and~$d$ rose to~$61$, and that
+between $d$~and~$e$ also began to rise as $d$~was now in the positive
+column; this difference increased very much in experiment~7,
+when the positive column reached to~$e$.
+
+\Article{148} We now pass on to the effect of an alteration in the
+\index{Cathode, potential fall at}%
+\index{Potential difference at cathode}%
+strength of the current on the potential difference at the cathode.
+We have already remarked that if the negative glow does not
+spread over the whole of the cathode, the only effect of an increase
+in the intensity of the current is to make the negative glow spread
+still further over the cathode, without altering the potential difference.
+Until the glow has covered the electrode, there is, according
+to Hittorf, no considerable increase in temperature at
+the cathode: when however the intensity of the current is increased
+beyond the point at which the whole of the cathode is
+covered by the glow, the temperature of the cathode begins to
+increase; when the current through the gas is very strong, the
+cathode, and sometimes even the anode, becomes white hot. When
+this is the case the character of the discharge changes in a remarkable
+way, all luminosity disappears from the gas, which
+when examined by the spectroscope does not show any trace of
+the lines of its spectrum. The tube with its white hot electrodes
+surrounded by the dark gas presents a remarkable appearance,
+and it is especially to be noted that the electrodes are raised to
+incandescence by a current, which if it passed through them
+when they formed part of a metallic circuit, would hardly make
+them appreciably hot.
+
+Hittorf also found (\textit{Wied.\ Ann.}~21.\ p.~121, 1884) that if in
+a vacuum tube conveying an ordinary luminous discharge, a
+platinum spiral which could be raised by a battery to a white
+heat was placed so as to be in the path of the discharge, the
+latter lost all luminosity in the neighbourhood of the spiral
+when this was white hot. If the spiral was allowed to cool, the
+luminosity appeared again before the spiral had cooled below a
+bright red heat.
+
+\Article{149} For experiments of this kind aluminium electrodes melt
+%% -----File: 165.png---Folio 151-------
+too easily. Hittorf used in most of his experiments iridium
+electrodes, which can be raised to a very high temperature without
+melting. These were raised to a white heat before any
+measurements were made, so as to get rid of any gas they
+might have occluded. The length of the electrodes was $48$~mm.
+The result of some experiments on nitrogen is given in the
+following table (\textit{Wied.\ Ann.}~21, p.~111, 1884); in this, when
+the number of cells is given as $600 × x$, it means that $x$~sets
+of cells, each containing $600$~elements, were connected up in
+parallel.
+
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{3pt}
+\settowidth{\TmpLen}{Pressure of the nitrogen in millimetres}%
+\begin{tabular} {@{}c|c|c|c|c|c@{}}
+\multicolumn{6}{c}{\normalsize\textit{Experiments with Nitrogen.}}\\
+\multicolumn{6}{c}{\normalsize\textit{Iridium electrodes at a distance of $15$~mm.}\medskip}\\
+\hline
+\parbox[b]{\TmpLen}{\tablespaceup Number fixing the experiment} & 1       & 2         & 3         & 4         & 5 \\
+\parbox[b]{\TmpLen}{\tabhang Pressure of the nitrogen in
+millimetres of mercury\mdotfill}              & $19.65$ & $31.9$    & $53.1$    & $53.1$    & $52.4$ \\
+\parbox[b]{\TmpLen}{Number of cells\mdotfill} & $600×3$ & $600×3$   & $600×3$   & $600×4$   & $400×6$ \\
+\parbox[b]{\TmpLen}{Strength of current in Ampères} & $.535$ & $1.225$ & $1.4$ & $2.0$     & $2.1$ \\
+\parbox[b]{\TmpLen}{\tabhang Kick of the galvanometer due to the charge in the condenser
+produced by the potential difference between
+the electrodes\mdotfill} & $75$--$82$   & $25$--$32$     & $25$--$32$     & $15$--$20$     & $17$--$20$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+In the first experiment a reddish-yellow positive column
+stretched at first from the anode to an intensely bright patch on
+the cathode; the cathode however soon became white hot along
+the whole of its length, and then showed no trace of the negative
+glow, nor were any nitrogen lines detected when the region round
+the cathode was examined by the spectroscope. The tip of the
+anode was white hot.
+
+From the second experiment we see that though the density
+of the nitrogen was much greater, the potential difference was
+less than half what it was in the first experiment. This is due
+to the electrodes being hotter in this experiment than in the preceding
+one. In the third experiment only half of the cathode
+was white hot, but the length of the anode which was incandescent
+was greater than in the preceding experiment. In
+the fourth experiment, in which a current of $2$~Ampères passed
+through the gas, the end of the anode was hotter than that of the
+cathode, in fact with this current the anode, though made of
+iridium, began to melt. In the ordinary arc lamp, in which we
+%% -----File: 166.png---Folio 152-------
+have probably a discharge closely resembling that in this experiment,
+the anode is also hotter than the cathode when the
+current is intense.
+
+In this case the gas was quite dark. A very remarkable
+feature shown by it is the smallness of the potential difference
+between the electrodes, not amounting to more than $100$~volts,
+though the gas was at the pressure of $53.1$~millimetres, and the
+distance between the electrodes $15$~mm. When the electrodes
+were cold, the battery power used, about $1200$~volts, was not
+sufficient to break down the gas: the discharge had to be
+started by sending a spark from a Leyden jar through the tube.
+The conduction through the gas in this case is of the same
+character as that described in \artref{169}{Art.~169}.
+
+\Article{150} Hittorf also made experiments on hydrogen and carbonic
+\index{Hittorf, discharge through gases}%
+oxide; the results for hydrogen are given in the following table
+(\textit{Wied.\ Ann.}~21, p.~113, 1884):---
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{3pt}
+\settowidth{\TmpLen}{Kick of the galvanometer due}%
+\begin{tabular}{@{}c|c|c|c|c|c|c@{}}
+\multicolumn{7}{c}{\normalsize\textit{Experiments with Hydrogen.}}\\
+\multicolumn{7}{c}{\normalsize\textit{Distance of the Iridium electrodes $15$~mm.}\medskip}\\
+\hline
+\parbox[b]{\TmpLen}{\tablespaceup\tabhang Number fixing the
+experiment\mdotfill}    & 1  &  2  & 3  & 4  & 5   & 6\\
+\parbox[b]{\TmpLen}{\tabhang Pressure of hydrogen
+in millimetres
+of~Hg\mdotfill}      & $20$  &  $33.8$ & $47.05$ & $47.05$  & $47.05$ & $68.55$\\
+\parbox[b]{\TmpLen}{\tabhang Number of cells\mdotfill} & $400×6$ & $400×6$ & $400×6$ &  $600×4$  &  $800×3$ &  $800×3$\\
+\parbox[b]{\TmpLen}{\tabhang Intensity of current
+in Ampères\mdotfill}  & $.5465$ & $.3415$ &  $.3074$  &  $.9222$  &  $.9905$  &  $.8197$\\
+\parbox[b]{\TmpLen}{\tabhang Kick of the galvanometer due to the
+charge in the condenser produced by the potential difference between
+the electrodes\mdotfill} & $100$ &  $107$--$108$ &  $110$  & $100$--$110$  & $107$--$110$ &  $110$\tablespacedown\\
+\hline
+\end{tabular}
+\end{center}
+
+In experiment~1 the pressure and the current were almost the
+same as for experiment~1, \artref{149}{Art.~149}, in nitrogen; the potential
+difference between the electrodes was however much greater
+in hydrogen than in nitrogen, though the potential difference
+required to \emph{initiate} a discharge in hydrogen is considerably
+less than in nitrogen. In these experiments the potential difference
+between the electrodes for this dark discharge seems
+almost independent of the current and of the density of the
+gas.
+%% -----File: 167.png---Folio 153-------
+
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{3pt}
+\settowidth{\TmpLen}{Kick of the galvanometer due to the charge in the}%
+\begin{tabular}{@{}c|c|c|c|c@{}}
+\multicolumn{5}{c}{\normalsize\textit{Experiments with Carbonic Oxide Gas.}}\\
+\multicolumn{5}{c}{\normalsize\textit{Distance between Iridium electrodes $15$~mm.}\medskip}\\
+\hline
+\parbox[b]{\TmpLen}{\tabhang
+\tablespaceup Number fixing the experiment\mdotfill}    & 1           & 2          & 3        & 4 \\
+\parbox[b]{\TmpLen}{\tabhang
+Pressure of CO in millimetres
+of mercury\mdotfill}                      & $13.1$      & $22.75$    & $51.7$   & $75.85$ \\
+\parbox[b]{\TmpLen}{\tabhang
+Number of cells\mdotfill}                 & $800×3$     & $800×3$    & $800×3$  & $800×3$ \\
+\parbox[b]{\TmpLen}{\tabhang
+Intensity of current in Ampères\mdotfill} & $.8880$     & $.9734$    & $1.3662$ & $1.2978$ \\
+\parbox[b]{\TmpLen}{\tabhang
+Kick of the galvanometer due
+to the charge in the condenser
+produced by the potential
+difference between
+the electrodes\mdotfill}                  & $92$--$100$ & $89$--$92$ & $40$     & $42$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+The great fall in potential, which occurs between experiments 2~and~3
+\index{Cathode, potential fall at}%
+on~CO, was accompanied by a loss of luminosity; in 1~and~2
+there was a little positive blue light at the anode, but in~3
+this had disappeared, and the discharge was quite dark and
+showed in the spectroscope no trace of the carbonic oxide bands.
+
+\includegraphicsouter{fig65}{Fig.~65.}
+
+\Article{151} When repeating these experiments with carbon electrodes
+\index{Hittorf, discharge through gases}%
+instead of iridium ones, Hittorf found that
+with strong currents and at pressures between $10$~mm.\
+and $2$~mm.\ the discharge through hydrogen took a
+very peculiar form, it consisted of ring-shaped \DPtypo{striae}{striæ},
+the insides of which were dark. These rings extended
+through the tubes and encircled both the anode and
+the cathode, as shown in \figureref{fig65}{Fig.~65}.
+
+\Article{152} The preceding experiments show that when
+the electrodes are white hot, the negative glow disappears,
+and the potential difference between the electrodes
+when a current is passing through the gas sinks
+to a fraction of the value it has when the electrodes
+are cold and the negative glow exists. Hittorf (\textit{Wied.\ Ann.}\ 21,
+p.~133) has shown by a direct experiment that when the cathode
+is white hot a very small electromotive force is sufficient to
+maintain the discharge. The arrangement he used is shown in
+\figureref{fig66}{Fig.~66}. A thin carbon filament which serves as a cathode is
+stretched between two conductors~$mn$, and can be raised to
+a white heat by a current passing through it and these conductors;
+the anode~$a$ is vertically below the cathode and remains
+cold. When the pressure was very low, Hittorf found that $1$~cell
+of his battery, equivalent to about $2$~volts, would maintain a
+current between the anode and cathode when they were separated
+%% -----File: 168.png---Folio 154-------
+\index{Potential difference at cathode|indexetseq}%
+by $6$~cm.; in this case the discharge was quite dark. When ten or
+more cells were used a pale bluish light spread over the anode.
+It should be noticed that the single cell does not \emph{start} the current,
+it only maintains it: the current
+has previously to be started
+by the application of a much
+greater potential difference. Hittorf
+generally started the current
+by discharging a Leyden jar
+through the tube. No current at
+all will pass if the poles are reversed
+so that the anode is hot
+and the cathode cold. In these
+experiments it is necessary for
+the cathode to be at a white heat
+for an appreciable current to pass
+between the electrodes; very little
+effect seems to be produced on the
+potential difference at the cathode
+until the latter is hotter than a
+bright red heat. The current
+produced by a given electromotive
+force is greater at higher
+exhaustions than at low ones, but
+Hittorf found he could get appreciable
+effects at pressures up to $9$~or $10$~mm.
+
+\Article{153} In considering the results of experiments in which carbon
+filaments or platinum wires are raised to incandescence, we must
+remember that, as Elster and Geitel have shown (\artref{43}{Art.~43}), electrification
+is produced by the incandescent body, the region around
+which receives a charge of electricity; though whether the carrier
+of this charge is the disintegrated particles of the incandescent
+wire, or the dissociated molecules of the gas itself, is not clear.
+This electrification often makes the interpretation of experiments
+in which incandescent bodies are used ambiguous. Thus for
+example, Hittorf in one experiment (\textit{Wied.\ Ann.}\ 21, p.~137,
+1884) used a U-shaped discharge tube, in one limb of which a
+carbon filament was raised to incandescence; the other limb of
+the tube contained a small gold leaf electroscope; when the
+pressure of the gas in the tube was very low, Hittorf found that
+%% -----File: 169.png---Folio 155-------
+the electroscope would retain a charge of negative electricity but
+immediately lost a positive charge. This experiment does not
+however show conclusively that positive electricity escapes more
+easily than negative from a metal into a gas which is in the condition
+in which it conducts electricity, because the same effect
+would occur if the incandescent carbon filament produced a
+negative electrification in the gas around it.
+
+\includegraphicsouter[22]{fig66}{Fig.~66.}
+
+\Article{154} The way in which the passage of electricity from metal
+to gas, or \emph{vice versâ}, is facilitated by increasing the temperature
+of the metal to the point of incandescence is illustrated by an
+effect observed in the experiments on hot gases described in
+\artref{37}{Art.~37}. It was found that when a current was passing between
+electrodes immersed in a platinum tube at a bright yellow heat
+and containing some gas, such as iodine, which conducts well, the
+current was at once stopped if a large piece of cold platinum foil
+was lowered between the electrodes, although there was a strong
+up-current of gas in the tube which prevented a cold layer of gas
+being formed against the platinum foil: as soon, however, as the
+foil became incandescent the current from one or two Leclanché
+cells passed freely. It would appear, therefore, that even when
+the gas is in the condition in which it conducts electricity freely,
+some of the cathode potential difference will remain as long as the
+cathode itself is not incandescent.
+
+\Article{155} The passage of electricity from a gas to a negative electrode
+seems, as we shall see later, to require something equivalent
+to chemical combination between the charged atoms of the metal
+and the atoms of the gas which carry the discharge; and the
+reason for the disappearance of the fall in potential at the cathode
+when the latter is incandescent is probably due to this combination
+taking place under these circumstances much more easily
+than when the electrode is cold.
+
+\Article{156} Warburg (\textit{Wied.\ Ann.}\ 31, p.~545, 1887: 40\DPtypo{.}{,}\ p.~1, 1890)
+\index{Negative electrode,x potential fall at@\subdashtwo potential fall at}%
+\index{Warburg, potential fall at cathode@\subdashone potential fall at cathode|indexetseq}%
+has made a valuable series of experiments on the circumstances
+which influence the fall of potential at the cathode. He has investigated
+the effect produced on this fall by altering the gas,
+the size and material of the electrodes, and the amount of impurity
+in the gas. Hittorf, as we have seen, had already shown
+that as long as there is room for the negative glow to spread
+over the surface of the cathode, the cathode fall in potential is
+approximately independent of the intensity of the current.
+%% -----File: 170.png---Folio 156-------
+
+In Warburg's experiments, the fall in potential at the cathode,
+by which is meant the potential difference between the cathode
+and a wire at the luminous boundary of the negative glow, was
+measured by a quadrant electrometer. Warburg found that, so
+long as the whole of the cathode was not covered by the negative
+glow, the fall in potential at the cathode was nearly independent
+of the density of the gas: this is shown by the following table
+(l.~c.\ p.~579), in which $E$~represents the potential difference between
+the electrodes, which were made of aluminium, $e$~the potential
+fall at the cathode, $E$~and~$e$ being measured in volts, $p$~the
+pressure of the gas, dry hydrogen, measured in millimetres of mercury,
+$i$~the current through the gas in millionths of an Ampère.
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{2em}
+\begin{tabular}{c|c|c|c}
+\hline
+\tablespaceup $p$. & $e$.  & $E-e$. & $i$.\tablespacedown\\
+\hline
+\tablespaceup $9.5\Z$  & $191$ & $139$ & $6140$ \\
+$6.4\Z$  & $190$ & $103$ & $4740$ \\
+$4.4\Z$  & $190$ & $\Z70$  & $4810$ \\
+$3.0\Z$  & $189$ & $\Z50$  & $2640$ \\
+$1.79$ & $191$ & $\Z40$  & $1730$ \\
+$1.20$ & $192$ & $\Z39$  & $1360$ \\
+$\Z.80$ & $191$ & $\Z39$ & $\Z508$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+This table shows that though the fall in potential in the positive
+light decreased as the pressure diminished, the fall in potential at
+the cathode remained almost constant.
+
+\Article{157} In imperfectly dried nitrogen, which contained also a
+trace of oxygen, the cathode potential difference depended to some
+extent on the metal of which the electrode was made; platinum,
+zinc, and iron electrodes had all practically the same potential
+fall; for copper electrodes the fall was about $3$~per~cent.\ and for
+aluminium electrodes about $15$~per~cent.\ less than for platinum.
+In hydrogen which contained a trace of oxygen, the potential fall
+for platinum, silver, copper, zinc, and steel was practically the
+same, about $300$~volts. In the case of the last three metals,
+however, the value of the cathode potential fall at the beginning
+of the experiment was much lower than $300$~volts, and it was
+not until after long sparking that it rose to its normal value;
+Warburg attributed this to the presence at the beginning of
+the experiment of a thin film of oxide which gradually got
+dissipated by the sparking; he found by direct experiment
+that the potential fall of a purposely oxidised steel electrode
+was less than the value reached by a bright steel electrode after
+%% -----File: 171.png---Folio 157-------
+being used for some time. The potential fall for aluminium and
+magnesium electrodes was about $180$~volts, and was thus considerably
+smaller than for platinum electrodes (cf.\ \artref{47}{Art.~47}); these
+metals, however, are easily oxidised; and as, unlike other metals,
+they do not disintegrate when used as cathodes, the film of oxide
+would not get removed by use.
+
+\Article{158} The fact that a large number of metals give the same
+potential fall, while others give a varying one, seems to indicate
+that this potential fall depends upon whether the electrodes do
+or do not take part in some chemical change occurring at the
+cathode; and the connection between this fall in potential and
+the chemical changes which take place near the cathode seems
+still more clearly shown by the surprisingly large effects produced
+by a small quantity of impurity in the gas. Warburg
+found that the fall of potential at the cathode in nitrogen which
+contained traces both of moisture and oxygen was $260$~volts,
+while the same nitrogen, after being very carefully dried, gave a
+cathode fall of $343$~volts: thus, in this case, a mere trace of
+moisture had diminished the cathode fall by $25$~per~cent., the
+removal of the trace of oxygen produced equally remarkable
+effects, see \artref{160}{Art.~160}. This points clearly to the influence exerted
+by chemical actions at the cathode on the fall of potential in
+that region; since a mere trace of a substance is often sufficient
+to start chemical reactions which would be impossible
+\index{Pringsheim, combination of hydrogen and chlorine}%
+without it: thus, for example, Pringsheim (\textit{Wied.\ Ann.}~32, p.~384,
+1887) found that unless traces of moisture were present, hydrogen
+and chlorine gas would not combine to form hydro-chloric acid
+under the action of sunlight unless it was very intense.
+
+\Article{159} The fall of potential at the cathode seems to be lowered
+as much by a trace of moisture as by a larger quantity, as long
+as the total quantity of moisture in the nitrogen remains small; if,
+however, the amount of aqueous vapour is considerable, the fall in
+potential is greater than for pure nitrogen; thus in a mixture of
+nitrogen and aqueous vapour, in which the pressure due to the
+nitrogen was $3.9$~mm., that due to the aqueous vapour $2.3$~mm.,
+Warburg found that the fall in potential was about $396$~volts, as
+against about $343$~volts for nitrogen containing a trace of oxygen;
+the increase in the fall of potential at the cathode was, however,
+not nearly so great comparatively as the increase in the potential
+differences along the positive column.
+%% -----File: 172.png---Folio 158-------
+
+In hydrogen, Warburg found that a trace of aqueous vapour
+increased the potential difference at the cathode instead of
+diminishing it as in nitrogen.
+
+\Article{160} Warburg (\textit{Wied.\ Ann.}~40, p.~1, 1890) also investigated
+the effects produced by removing from the nitrogen or hydrogen
+any trace of oxygen that might have been present. This was
+done by placing sodium in the discharge tube, and then after the
+other gas had been let into the tube, heating up the sodium,
+which combined with any oxygen there might be in the tube.
+The effect of removing the oxygen from the nitrogen was
+very remarkable: thus, in nitrogen free from oxygen, the fall of
+potential at the cathode when platinum electrodes were used
+was only $232$~volts as against $343$~volts when there was a trace
+of oxygen present; when magnesium electrodes were used the
+fall in potential was $207$~volts; in hydrogen free from oxygen
+the fall of potential was $300$~volts with platinum electrodes, and
+$168$~volts with magnesium electrodes; thus with platinum
+electrodes the potential fall in hydrogen is greater than in
+nitrogen, while with magnesium electrodes it is less.
+
+\Article{161} Warburg also investigated a case in which the conditions
+for chemical change at the cathode were as simple as possible,
+one in which the gas was mercury vapour (with possibly a trace
+of air) and the cathode a mercury surface; he found that the
+negative dark space was present, and that the cathode fall was
+very considerable, amounting to about $340$~volts; this, at the
+pressures used in these experiments between $3.5$~mm.\ and $14.0$~mm.,
+was much greater than the potential difference in a portion
+of the positive light about half as long again as the piece at the
+cathode, for which the potential fall was measured.
+
+\Article{162} In air free from carbonic acid, but containing a little
+moisture, Warburg (\textit{Wied.\ Ann.}~31, p.~559, 1887) found that the
+potential fall was about $340$~volts: this is very nearly the value
+found by Mr.~Peace for the smallest potential difference which
+would send a spark between two parallel plates. When we
+consider the theory of the discharge we shall see that there are
+reasons for concluding that it is impossible to produce a spark
+by a smaller potential difference than the cathode fall of potential
+in the gas through which the spark has to pass.
+
+The researches made by Hittorf on the distribution of potential
+along the tube show, as we have seen, \artref{140}{Art.~140}, that the
+%% -----File: 173.png---Folio 159-------
+potential gradient is by no means constant; to produce the
+changes in this gradient which occur in the neighbourhood of
+the cathode, there must in that region be a quantity of free
+electricity in the tube. Schuster (\textit{Proc.\ Roy.\ Soc.}~47, p.~542, 1890)
+\index{Schuster, discharge through gases}%
+concludes from his measurements of the potential in the neighbourhood
+of the cathode that if $\rho$ is the volume density of the
+free positive electricity at a distance~$x$ from the cathode, $\rho$~varies
+as~$\epsilon^{-\kappa x}$.
+
+\Article{163} The measurements of potential along the positive column
+have been less numerous than those of the negative dark space.
+\index{De la Rue and Müller, discharge through gases}%
+Hittorf, De~la Rue and Hugo Müller concur in finding that the
+potential gradient close to the anode is, though not comparable
+with that at the cathode, greater than that in the middle of the
+tube.
+
+\Article{164} The potential gradient in the positive column is not like
+\index{Potential gradient in positive column,@\subdashone gradient in positive column}%
+the fall in potential at the cathode approximately independent
+of the density, it diminishes as the pressure of the gas diminishes:
+but as the pressure of the gas diminishes, the distance between
+two consecutive striations increases, and though I can find no
+experiments bearing on this point, it would be a matter of great
+interest to know whether or not the potential difference along a
+length of the positive column equal to the distance between two
+striations, where these are regular, is approximately independent
+of the density of the gas.
+
+\Article{165} De~la~Rue and Hugo Müller (\textit{Phil.\ Trans.}\ 1878, Part~I,
+p.~159) measured the potential gradients along a tube in which
+two wide portions were connected by a piece of capillary tubing,
+narrow enough to constrict the \DPtypo{striae}{striæ}; they found the potential
+gradient much greater along the capillary portion than along
+the wide one. Thus the potential difference along $4.25$~inches of
+the positive column in the wide tube, which was about $\frac{15}{16}$~of an
+inch in diameter, was, on an arbitrary scale, $75$, while the
+potential difference along a portion of the positive column,
+which included $2$~inches of the wide tube and $3.75$~inches of the
+capillary tube ($\frac{1}{8}$~of an inch in diameter), was~$138$; the potential
+gradients along the wide and narrow portions are thus in the
+proportion of $1$~to~$1.55$.
+
+\includegraphicsouter{fig67}{Fig.~67.}
+
+In this case the cathode was in the wide part of the tube;
+when the tube round the cathode is so narrow that it restricts
+the negative glow, the increase in the potential difference
+%% -----File: 174.png---Folio 160-------
+at the cathode produced by this restriction makes it very
+much more difficult to get a discharge to pass through the
+narrow tube than through a wider one. An experiment due to
+Hittorf (\textit{Wied.\ Ann.}~21, p.~93, 1884) illustrates this effect in
+a very remarkable way; at a pressure of $.03$~mm.\ of mercury, it
+took $1100$~of his cells to force the discharge through a tube $1$~cm.\
+in diameter, while $300$~cells were sufficient to force it between
+similar electrodes the same distance apart in a tube $11$~cm.\ in
+diameter, filled with the same kind of gas at the same pressure.
+
+\Article{166} When the electrodes are placed so near together that the
+\index{Discharge between electrodes near together|(}%
+\index{Electrodes, discharge between two when close together|(}%
+dark space round the cathode extends
+to the anode, the appearance of
+the discharge is completely changed:
+this is very well shown in an experiment
+\index{Hittorf, discharge through gases}%
+due to Hittorf (\textit{Pogg.\ Ann.}\
+136, p.~213, 1869) represented in \figureref{fig67}{Fig.~67};
+the electrodes were parallel to
+each other, and the pressure of the
+gas in the discharge tube was so low
+that the dark space round the cathode
+extended beyond the anode; the positive
+discharge in this case, instead of
+turning towards the cathode, started
+from the bend in the anode on the
+side furthest away from the cathode,
+and then crept along the surface of
+the glass until it reached the boundary
+of the negative dark space. I observed a similar effect in the
+course of some experiments on the discharge between large
+parallel plates (\textit{Proc.\ Camb.\ Philos.\ Soc.}~5, p.~395, 1886); when the
+pressure of the gas was very small, the positive column, instead
+of passing between the plates, went, as in \figureref{fig68}{Fig.~68}, from the under
+side of the lower plate which was the positive electrode, and
+%% -----File: 175.png---Folio 161-------
+after passing between the glass and the plates reached right up
+to the negative glow, which was above the negative plate: the
+space between the plates was quite dark and free from glow.
+
+\includegraphicsmid{fig68}{Fig.~68.}
+
+Lehmann (\textit{Molekularphysik}, bd.~2, p.~295) has observed with a
+\index{Lehmann, discharge between electrodes close together}%
+microscope the appearance of the discharge passing between
+electrodes of different shapes, placed very close together; they
+exhibit in a very beautiful way the same peculiarities as those
+just described; Lehmann's figures are represented in \figureref{fig69}{Fig.~69}.
+%% -----File: 176.png---Folio 162-------
+
+\includegraphicsmid{fig69}{Fig.~69.}
+
+When the distance between the electrodes is less than the
+thickness of the dark space, it is very difficult to get the discharge
+to pass between them; this is very strikingly illustrated
+by another experiment of Hittorf's (\textit{Wied.\ Ann.}~21, p.~96, 1884)
+\index{Hittorf, discharge through gases}%
+which is represented in \figureref{fig70}{Fig.~70}. The two electrodes were only
+$1$~mm.\ apart, but the regions surrounding them were connected by
+a long spiral tube $3\frac{3}{4}$~m.\ long; in spite of the enormous difference
+between the lengths of the two paths, the discharge, when the
+pressure was very low, all went round through the spiral, and
+\index{Discharge between electrodes near together|)}%
+\index{Electrodes, discharge between two when close together|)}%
+the space between the electrodes remained quite dark.
+
+\Article{167} In cases of this kind the potential difference required
+to produce discharge between two electrodes must be \emph{diminished}
+by increasing the distance between them.  For in Hittorf's
+experiments, the potential difference between the electrodes
+was equal to the potential fall at the cathode, plus the change
+in potential due to the $3\frac{3}{4}$~m.\ of positive light in the spiral,
+while if the shortest distance between the electrodes had
+been increased until it was just greater than the thickness of
+the negative dark space, the potential difference between the
+electrodes when the discharge passed would only have amounted
+to the cathode fall, plus the potential difference due to a short
+positive column instead of to one $3\frac{3}{4}$~metres long, so that the
+potential difference would have been less than when the electrodes
+\index{Peace, spark potential}%
+are nearer together. Peace's experiment described in \artref{53}{\DPtypo{Art.~53.}{Art.~53}}\ is
+a direct proof of the truth of this statement for higher
+pressure, and is free from the objection to which the preceding
+deduction from Hittorf's experiment is liable, that the cathode
+%% -----File: 177.png---Folio 163-------
+fall may not be the same when the discharge starts in the large
+vessel when the negative glow is unrestricted, as it is when the
+discharge passes through the narrow tubes, the walls of which
+constrict the negative glow.
+
+\includegraphicsmid{fig70}{Fig.~70.}
+
+\Article{168} These results explain a peculiar effect which is observed
+when the discharge passes between slightly curved electrodes
+at not too great a distance apart; until the pressure is very
+low the discharge passes across the shortest distance between
+the electrodes, but after a very low pressure is reached the
+discharge leaves the centre of the field, and in order to get a
+longer spark length departs further and further from it as the
+pressure of the gas is reduced.
+
+\Subsection{The Arc Discharge.}
+\index{Arc discharge}%
+
+\Article{169} The `arc discharge,' of which the well-known arc lamp is
+a familiar example, is characterised by the passage of a large
+current and the incandescence of both the terminals, as well as by
+the comparatively small potential difference between them; we
+considered a case of this discharge in \artref{148}{Art.~148}, the gas was, however,
+in that case, at a low pressure; the cases when the gas is at
+higher pressures are of special interest, on account of the extensive
+use made of this form of discharge for lighting purposes.
+
+If the current through a vacuum tube with electrodes is gradually
+\index{Gassiot on electric discharge}%
+increased, the discharge, as Gassiot found in 1863, gradually
+changes from the ordinary type of the vacuum tube discharge
+with the negative space and a striated positive column to the
+arc discharge, in which there is comparatively little difference
+between the appearances at the terminals. The terminals are
+brilliantly incandescent while the gas remains comparatively
+dark, being probably in the state in which it has a large supply
+of dissociated molecules by means of which it can transmit the
+current even though the potential gradient is small.
+
+The connection between spark length, potential difference and
+current in the arc discharge, has been investigated by many
+physicists, who have all found that the potential difference~$V$
+is almost independent of the current and can be expressed by
+the formula
+\[
+V = a + bl,
+\]
+where $l$~is the spark length and $a$~and~$b$ are constants. Ayrton
+%% -----File: 178.png---Folio 164-------
+\index{Ayrton and Perry, xarc discharge@\subdashone arc discharge}%
+and Perry (\textit{Phil.\ Mag.}\ [5]~15, p.~346, 1883), using a formula
+which is identical with the preceding one if the sparks are not
+very short, found that for carbon electrodes $a = 63$~volts and
+$b = 21.6$~volts, if $l$~is measured in centimetres. The value of~$a$
+probably depends on the quality of the carbon of which the electrodes
+are made, as other observers, who have also used carbon
+electrodes, have found considerably smaller values for~$a$. When
+more volatile substances than carbon are used the values of~$a$ are
+smaller, the more volatile the substance the smaller in general
+being the value of~$a$. This is borne out by the following determinations
+\index{Lecher, on the arc discharge}%
+made by Lecher (\textit{Wied.\ Ann.}~33, p.~625, 1888); the
+length~$l$ in these equations is measured in centimetres, and $V$~in
+volts:---
+\begin{center}
+\begin{tabular}{l@{}l}
+Horizontal Carbon Electrodes\mdotfill & $V = 33 + 45l$.\\
+Vertical Carbon Electrodes\mdotfill & $V = 35.5 + 57l$.\\
+Platinum Electrodes, ($.5$ cm.\ in diameter)\quad\mdotfill & $V = 28 + 41l$.\\
+Iron Electrodes, ($.55$ cm.\ in diameter)\mdotfill & $V = 20 + 50l$.\\
+Silver Electrodes, ($.49$ cm.\ in diameter)\mdotfill & $V = 8 + 60l$.
+\end{tabular}
+\end{center}
+
+\Article{170} The form of the expression for~$V$ shows that the potential
+required to maintain the current between two incandescent
+electrodes cannot fall short of a certain minimum value, however
+short the arc may be. The preceding measurements for~$a$ show
+that this potential difference, though small compared with the
+`cathode fall' when the electrodes are cold, is much greater than
+that which Hittorf in his experiments (see \artref{152}{Art.~152}) found
+necessary to maintain a constant current when the cathode was
+incandescent; we must remember, however, that in Lecher's
+experiments the gas was at atmospheric pressure, while in
+Hittorf's the pressure was very low.
+
+\Article{171} Lecher (l.~c.)\ investigated the potential gradient in the
+arc by inserting a spare carbon electrode, and found that it was
+far from uniform: thus when the difference of potential between
+the anode and the cathode was $46$~volts, there was a fall of $36$~volts
+close to the anode, and a smaller fall of ten volts near the
+cathode. The result that the great fall of potential in the arc
+discharge occurs close to the anode is confirmed by an experiment
+\index{Fleming, arc discharge}%
+made by Fleming (\textit{Proc.\ Roy.\ Soc.}~47, p.~123, 1890), in
+which a spare carbon electrode was put into the arc; when this
+electrode was connected with the anode sufficient current went
+%% -----File: 179.png---Folio 165-------
+round the new circuit to ring an electric bell, but when it was
+connected to the cathode the current which went round the
+circuit was not appreciable.
+
+\includegraphicsouter{fig71}{Fig.~71.}
+
+\Article{172} The term in the expression for the potential in \artref{169}{Art.~169},
+which is independent of the length of the arc, and which
+involves an expenditure of energy when electricity travels across
+an infinitesimally small air space, is probably connected with the
+work required to disintegrate the electrodes, since the more
+volatile are the electrodes the smaller is this term.
+
+\Article{173} The disintegration of the electrodes is a very marked
+feature of the arc discharge, and it is not, as in the case when
+small currents pass through
+a highly exhausted gas, confined
+to the negative electrode;
+in fact, when carbon
+electrodes are used, the loss
+in weight of the anode is
+greater than that of the
+cathode, the anode getting
+hollowed out and taking a
+crater-like form.
+
+\Article{174} Perhaps the most
+\index{Arc discharge, with large potential differences@\subdashtwo with large potential differences}%
+interesting examples of the
+arc discharge are those which
+occur when we are able by
+means of transformers to produce
+a great difference of
+potential, say thirty or forty
+thousand volts between two
+electrodes, and also to transmit
+through the arc a very
+considerable current. In this
+case the arc presents the
+appearance illustrated in \figureref{fig71}{Fig.~71}. The discharge, instead of passing
+in a straight line between the electrodes, rises from the
+electrodes in two columns which unite at the top, where striations
+are often seen though these do not appear in the photograph
+from which \figureref{fig71}{Fig.~71} was taken. The vertical columns are
+sometimes from eighteen inches to two feet in length, they flicker
+slowly about and are very easily blown out, a very slight puff of
+%% -----File: 180.png---Folio 166-------
+air being sufficient to extinguish them. The air blast apparently
+breaks the continuity of the belt of dissociated molecules along
+which the current passes, and the current is stopped just as a
+current through a wire would be stopped if the wire were cut.
+The discharge is accompanied by a crackling sound, as if a
+number of minute sparks were passing between portions of the
+arc temporarily separated by very short intervals of space.
+
+\Article{175} The relation between the losses of weight of the anode
+and the cathode in the arc discharge depends however very much
+on the material of which the electrodes are made; thus Matteucci
+\index{Matteuchi arc discharge}%
+(\textit{Comptes Rendus}, 30, p.~201, 1850) found that for copper,
+silver and brass electrodes the cathode lost more than the anode,
+while for iron the loss in weight of the anode was greater than
+that of the cathode.
+
+The electrodes in the arc discharge are at an exceedingly high
+\index{Arc discharge, connection between loss of weight of electrodes and quantity of electricity passing@\subdashtwo connection between loss of weight of electrodes and quantity of electricity passing}%
+temperature, in fact probably the highest temperatures we can
+produce are obtained in this way. With carbon electrodes the
+anode is much hotter than the cathode (compare \artref{149}{Art.~149}).
+Since the temperature of the electrodes is so high, it is probable
+that they are disintegrated partly by the direct action of the
+heat and not wholly by purely electrical processes such as those
+which occur in electrolysis; for this reason, we should not
+expect to find any simple relation between the loss in weight of
+the electrode and the quantity of electricity which has passed
+\index{Grove, on the arc discharge@\subdashone on the arc discharge}%
+through the arc. Grove (\textit{Phil.\ Mag.}\ [3]~16, p.~478, 1840), who
+used a zinc anode sufficiently large for the temperature not to
+rise about its melting point, came to the conclusion that the
+amounts of zinc lost and oxygen absorbed by the electrode were
+chemically equivalent to the oxygen liberated in a voltameter
+\index{Herwig, arc discharge}%
+placed in the circuit. On the other hand, Herwig, (\textit{Pogg.\ Ann.}\
+149, p.~521, 1873), who investigated the relation between the loss
+of weight of a silver electrode in the arc and the amount of
+chemical decomposition in a voltameter placed in the same
+circuit, was however unable to find any simple law connecting
+the two. The brightness of the light given by carbon electrodes
+is much increased by soaking them in a solution of sodium
+sulphate.
+
+\Article{176} The particles projected from the electrodes in the arc
+discharge are presumably charged with electricity, since they
+are deflected by a magnet; thus some of the electricity passing
+%% -----File: 181.png---Folio 167-------
+between the electrodes will be carried by these particles. Comparatively
+few experiments bearing on this point have, however,
+been made on the arc discharge, and we have not the information
+which would enable us to estimate how much of the current is
+carried by the disintegrated electrodes and how much by the gas.
+
+Fleming (\textit{Proc.\ Roy.\ Soc.}~47, p.~123, 1890) has suggested that
+\index{Fleming, arc discharge}%
+\emph{all} the current is carried by particles torn off the electrodes, that
+these particles are projected (chiefly from the cathode) with
+enormous velocities, and that the incandescence of the electrodes
+is due to the heat developed by their bombardment by these
+particles; the hollowing out of the anode is on this theory
+supposed to be due to a kind of sand blast action exerted by the
+particles coming from the negative electrode.
+
+\longpage
+On this theory, if I understand it rightly, the gas by which
+the electrodes are enveloped plays no part in the discharge. I
+do not think that the theory is consistent with Hittorf's and
+Gassiot's observations on the continuity of the arc discharge
+with the ordinary striated discharge produced in a vacuum
+tube through which only a very small current is passing, nor
+does it seem in accordance with what we know about the high
+conductivity of gases which are at a high temperature or through
+which an electric discharge has recently passed.
+
+
+\Subsection{The Heat produced by the Discharge.}
+\index{Heat produced by electric discharge}%
+\index{Discharge, heat produced by@\subdashone heat produced by}%
+\index{Electric discharge, rheat produced by@\subdashtwo heat produced by}%
+
+\Article{177} Though the electric discharge is generally accompanied
+by intense light, the average temperature of the molecules of the
+gas through which it passes is often by no means high. Thus
+\index{Wiedemann, E., on electric discharge}%
+E.~Wiedemann (\textit{Wied.\ Ann.}~6, p.~298, 1879) has found that the
+average temperature of a column of air at a pressure of about
+$3$~mm.\ made luminous by the passage of the discharge can be
+under $100°$\,C\@. As, however, any instrument which we may use
+to measure the temperature of the gas merely measures the
+average temperature of molecules filling a considerable space,
+the fact that this temperature is low does not preclude the
+existence of a small number of molecules moving with velocities
+immensely greater than the mean velocity corresponding to the
+temperature indicated by the thermometer.
+
+On the other hand, the fact that the gas is luminous
+during the discharge does not afford conclusive evidence of the
+%% -----File: 182.png---Folio 168-------
+existence of molecules in a state comparable with that of the
+majority of the molecules in a gas at a very high temperature,
+for mere increase of temperature unaccompanied by chemical
+changes seems to have little effect in increasing the luminosity
+of a gas; thus in one of Hittorf's experiments already mentioned,
+where the temperature of the electrodes was great enough
+to melt iridium, the gas surrounding them when examined by
+the spectroscope did not show any spectroscopic lines. It
+would seem that the interchange of atoms between the molecules
+which probably goes on when the discharge passes through
+the gas is much more effective in making it luminous than mere
+increase in temperature unaccompanied by chemical changes.
+
+\Article{178} Many experiments have been made by G.~and E.~Wiedemann,
+\index{Wiedemann, E., on electric discharge}%
+\index{Wiedemann, G. and E., heat produced by electric discharge}%
+\index{Hittorf, discharge through gases}%
+Hittorf, and others on the distribution along the line of
+discharge of the heat produced by the spark. Hittorf's experiments
+are the easiest to interpret, since by means of a large
+battery he produced through the discharge tube a current which,
+if not absolutely continuous, was so nearly so, that no want of
+continuity could be detected either by a revolving mirror or
+by a telephone; the gas had therefore a much better chance of
+getting into a steady state than if intermittent discharges such
+as those produced by an induction coil had been used.
+
+Hittorf (\textit{Wied.\ Ann.}\ 21, p.~128, 1884) inserted three thermometers
+in the discharge tube, one close to the cathode, another
+in the bright part of the negative glow, and the third in
+the positive column. He found, using small currents and low
+gaseous pressures, that the temperature of the thermometer next
+the cathode was the highest, that of the one in the negative glow
+the next, and that of the one in the positive column the lowest.
+
+The distribution of temperature depends very much upon the
+intensity of the current. Hittorf found that when the strength
+was increased the difference between the temperatures of his
+thermometers increased also. When however the increase in the
+current is so great that the discharge becomes an arc discharge,
+then, at any rate when carbon electrodes are used, the temperature
+at the anode is higher than that of the cathode; with weak
+currents we have seen that it is lower.
+
+E.~Wiedemann (\textit{Wied.\ Ann.}\ 10, p.~225 et~seq., 1880) found that
+the distribution of temperature along the discharge depended on
+the pressure. In his experiments the temperature at the anode
+%% -----File: 183.png---Folio 169-------
+was slightly higher than that at the cathode when the pressure
+was about $26$~mm.\ of mercury, at lower pressures the cathode
+was the hotter, and the difference between the temperatures
+of the cathode and the anode increased as the pressure diminished.
+
+
+\Subsection{Differences between the Phenomena at the Positive and
+Negative Electrodes.}
+\index{Difference between positive and negative discharge}%
+\index{Discharge, electric@\subdashone electric, difference between positive and negative}%
+\index{Electrodes, xdifference between positive and negative@\subdashone difference between positive and negative}%
+\index{Negative qand positive discharges, difference between@\subdashone and positive discharges, difference between}%
+\index{Positive and negative discharge, difference between}%
+
+\Article{179} We have seen already that when the pressure of the gas
+is small the two electrodes present very different appearances,
+there are however many differences between an anode and a
+cathode even at atmospheric pressure.
+
+\includegraphicsmid{fig72}{Fig.~72.}
+
+The appearance of the spark discharge at the two electrodes
+is different. The following figure is from a photograph of the
+spark in air at atmospheric pressure. It will be noticed that
+the sparks seem to reach a definite point on the negative electrode,
+but to spread over a considerable area of the positive.
+Bright dots of light are often to be seen on the positive electrode
+but not on the negative, these are still more striking at lower
+pressures. When the spark is branched as in \figureref{fig73}{Fig.~73}, the
+branches point to the negative electrode.
+
+\includegraphicsmid{fig73}{Fig.~73.}
+
+If the electrodes are not of the same size, the spark length
+for the same potential difference seems to depend upon whether
+the larger or smaller electrode is used as the cathode, though
+it is a disputed question whether this difference exists if the
+spark is not accompanied by some other form of discharge.
+%% -----File: 184.png---Folio 170-------
+Thus, if for example the electrodes are spheres of different sizes,
+Faraday (\textit{Experimental Researches}, §~1480) found that the spark
+\index{Faraday, mdifference between positive and negative discharge@\subdashone difference between positive and negative discharge}%
+length was greater when the smaller sphere was positive than
+when it was negative. We may express this result by saying
+that when the electric field is not uniform the gas does not
+break down so easily when the greatest electromotive intensity
+is at the cathode as it does when it is at the anode.
+
+Macfarlane's measurements (\textit{Phil.\ Mag.}\ [5]~10, p.~403, 1880) of
+\index{Macfarlane, spark potential}%
+the potential difference required to start a discharge between a
+ball and a disc are in accordance with this result, as he found
+that for a given length of spark the potential difference between
+the electrodes was smaller when the ball was positive than when
+it was negative.
+
+\includegraphicstwo{fig74}{Fig.~74.}{fig75}{\hspace*{\stretch{1}}Fig.~75.\hspace*{\stretch{1}}}
+
+\Article{180} De~la~Rue and Hugo Müller (\textit{Phil.\ Trans.}\ 1878, Part~I,
+\index{Muller@Müller and de la Rue, electric discharge}%
+\index{De la Rue and Müller, discharge through gases}%
+p.~55) observed analogous effects in the experiments they made
+with their large chloride of silver battery on the sparking distance
+between a point and a disc. They found that for potential differences
+between $5000$ and $8000$~volts the sparking distance was
+greatest when the point was positive and the disc negative, while
+for smaller potential differences they found that the opposite
+result was true. The appearance of the discharge at the positive
+point they found was different from that at the negative. The
+discharge at the negative point is represented in \figureref{fig74}{Fig.~74}, that at
+the positive in \figureref{fig75}{Fig.~75}.
+
+\Article{181} Wesendonck (\textit{Wied.\ Ann.}\ 38, p.~222, 1889), however,
+\index{Wesendonck, positive and negative discharge}%
+concludes from his experiments that there are no polar differences
+of this kind when the discharge passes entirely as a spark,
+and that the differences which have been observed are due to the
+coexistence of other kinds of discharge such as a brush and glow.
+
+The existence of this kind of discharge would put the gas
+into a condition in which it is electrically weak and thus ill-fitted
+%% -----File: 185.png---Folio 171-------
+to resist the passage of the spark. This explanation does
+not seem inconsistent with Faraday's experiment, for, as we shall
+see in the next paragraph, the negative brush is formed more
+easily than the positive one. Thus if the sparks in his experiments
+only passed when they were preceded by the formation of
+brushes at both the electrodes, it might be produced if the
+greatest electromotive intensity was at the place where the brush
+was formed with the greatest difficulty---the anode---while it
+might not be produced if the smallest intensity was at the anode,
+thus the gas would be electrically weaker in the first case than
+in the second.
+
+\sloppy
+\Article{182} Considerable polar differences seem undoubtedly to occur
+\index{Glow, discharge}%
+in the brush and glow discharges. Thus Faraday (\textit{Experimental
+Researches}, §~1501) found that if two equal spheres were electrified
+until they discharged their electricity by a brush discharge
+\index{Brush discharge}%
+into the air, the discharge occurred at a lower potential for
+the negative ball than for the positive; more electricity thus
+accumulates on the positive ball than on the negative before the
+brush occurs, so that when the positive brush does take place it
+is finer than the negative one.
+
+\includegraphicsmid{fig76}{Fig.~76.}
+
+\fussy
+The brush discharge is also intermittent, and since the positive
+brush requires a greater accumulation of electricity than the
+negative one, the interval between consecutive discharges is
+greater for the positive than for the negative brush.
+
+The positive and negative brushes are represented in \figureref{fig76}{Fig.~76},
+copied from a figure given by Faraday.
+
+In the brush discharge the electricity seems to be carried partly
+\index{Nahrwold, leakage of electricity through air}%
+by particles of metal torn from the electrodes. Nahrwold (\textit{Wied.\
+Ann.}\ 31, p.~473, 1887) has confirmed the conclusion that the
+negative brush is more easily formed than the positive.
+
+Wesendonck (\textit{Wied.\ Ann.}\ 39, p\DPtypo{,}{.}~601, 1890) has shown that when
+\index{Wesendonck, positive and negative discharge}%
+the discharge passes as a glow discharge from a point into air,
+hydrogen, or nitrogen, the potential at which the discharge begins
+is less when the point is negative than when it is positive.
+%% -----File: 186.png---Folio 172-------
+
+\Subsection{Lichtenberg's Figures and Kundt's Dust Figures.}
+\index{Lichtenberg's figures}%
+
+\includegraphicsmid[!t]{fig77}{Fig.~77.}
+
+\includegraphicsouter{fig78}{Fig.~78.}
+
+\Article{183} Very tangible differences between the discharges from
+the positive and negative electrodes at ordinary pressures are
+obtained if we allow the discharge from one or other of the
+electrodes to pass on to a non-conducting plate covered with
+some badly conducting powder. If, for example, we powder
+a plate with a mixture of red minium and yellow sulphur and
+then cause a discharge from a positively electrified point to pass
+to the plate, the sulphur, which by friction against the minium
+is negatively electrified, adheres to the positively electrified parts
+of the plate, and will be found to be
+arranged in a star-like form like that
+represented in \figureref{fig77}{Fig.~77}. If, on
+the other hand, the discharge is taken
+from a negatively electrified body
+the appearance of the minium on
+the plate is that represented in \figureref{fig78}{Fig.~78}.
+These are known as Lichtenberg's
+figures; the positive ones are
+larger than the negative.
+
+If the electrodes are made of very
+bad conductors, such as wood, there
+is no difference between the positive and the negative figures.
+%% -----File: 187.png---Folio 173-------
+
+\Article{184} Very beautiful figures are obtained if a plate of glass
+covered with a non-conducting powder, such as lycopodium, is
+placed on a metal plate, and two wires connected with the poles
+of an induction coil made to touch the powdered surface of the
+glass. When the discharge passes the powder arranges itself in
+patterns which are finely branched and have a moss-like appearance
+at the anode and a more feathery or lichenous appearance at
+the cathode. The accompanying figure is from a paper by Joly
+\index{Joly, discharge figures}%
+(\textit{Proc.\ Roy.\ Soc.}\ 47, p.~84, 1890); the negative electrode is on the
+left.
+
+\includegraphicsmid{fig79}{Fig.~79.}
+
+\Article{185} As Lehmann has remarked (\textit{Molekularphysik}, \DPtypo{b}{bd}.~11,
+\index{Lehmann, xdifference between positive and negative discharge@\subdashone difference between positive and negative discharge}%
+p.~303), the differences between the positive and negative figures
+are what we should expect if the discharge passed as a brush from
+the positive electrode and as a glow from the negative one. He
+has verified by direct observation that this is frequently the case.
+
+A good deal of light is also, I think, thrown on the difference
+between the positive and negative figures by \figureref{fig80}{Fig.~80}, which
+\index{De la Rue and Müller, discharge through gases}%
+\index{Muller@Müller and de la Rue, electric discharge}%
+is given by De~la~Rue and Hugo Müller (\textit{Phil.\ Trans.}\ 1878,
+Part~I, p.~118) as the discharge produced by $11,000$~of their
+%% -----File: 188.png---Folio 174-------
+chloride of silver cells in free air. It will be noticed that there
+is at the negative electrode a continuous discharge superposed
+on the streamers which are the only form of discharge at the
+positive, this continuous discharge will fully account for the
+comparative want of detail in the negative figure.
+
+\includegraphicsmid{fig80}{Fig.~80.}
+
+\Article{186} Kundt's figures are obtained by scattering non-conducting
+powders over a horizontal metal plate, instead of, as in
+Lichtenberg's figures, over a non-conducting one. If the plate
+be shaken after a discharge has passed from a negative point to
+the positive plate, it will be found that the powder will fall from
+every part of the plate except a small circle under the negative
+electrode, where the powder sticks to the plate and forms what
+\index{Dust figures}%
+\index{Kundt, dust figures}%
+is called Kundt's `dust figure.' The dimensions of this circle are
+very variable, ranging in Kundt's original experiments (\textit{Pogg.\
+Ann.}\ 136, p.~612, 1869) from $10$ to $200$~mm.\ in diameter. If the
+point is positive and the plate negative Kundt's figures are only
+formed with great difficulty.
+
+
+\Subsection{Mechanical Effects produced by the Discharge.}
+\index{Discharge, mechanical effects produced by@\subdashone mechanical effects produced by}%
+\index{Electric discharge, smechanical effects produced by@\subdashtwo mechanical effects produced by}%
+\index{Mechanical effects produced by electric discharge@\subdashtwo produced by electric discharge}%
+
+\Article{187} We have already considered the mechanical effects produced
+by the projection of particles from the cathode: many other
+such effects are however produced by the electric discharge. \nblabel{add:4}One
+of the most interesting of these is that described by De~la~Rue
+\index{De la Rue and Müller, discharge through gases}%
+\index{Muller@Müller and de la Rue, electric discharge}%
+and Hugo Müller (\textit{Phil.\ Trans.}\ 1880, p.~86): they found that when
+the discharge from their large chloride of silver battery passed
+through air at the pressure of $53$~mm.\ of mercury, the pressure
+of the air was increased by about $30$~per~cent., and they proved,
+by measuring the temperature, that the increase in pressure
+could not be accounted for by the heat produced by the spark.
+
+This effect can easily be observed if a pressure gauge is
+attached to any ordinary discharge tube, the gas inside being
+most conveniently at a pressure of from $2$ to $10$~mm.\ of mercury.
+At the passage of each spark there is a quick movement of the
+liquid in the gauge as if it had been struck by a blow coming
+from the tube; immediately after the passage of the spark the
+liquid in the gauge springs back to within a short distance of
+its position of equilibrium, and then slowly creeps back the
+rest of the way. This creeping effect is probably due to the
+slow escape of the heat produced by the passage of the spark.
+%% -----File: 189.png---Folio 175-------
+\index{Electric discharge, texpansion due to@\subdashtwo expansion due to}%
+The gauge behaves as if a wave of high pressure rushed through
+the tube when the spark passed.
+
+\Article{188} Meissner, \textit{Abhand.\ der König.\ Gesellschaft, \DPtypo{Gottingen}{Göttingen}}, 16,
+\index{Meissner, expansion due to discharge}%
+p.~98 et~seq., 1871 (who seems to have been the first to observe
+this effect, though in his experiments it was not developed to
+such an extent as in De~la~Rue's and Müller's), found that
+if a tube provided with a gauge were placed between the plates
+of a condenser there was an increase of pressure when the plates
+were charged or discharged, and no effect as long as the charge
+on the condenser remained constant. In this case there was
+no spark between the plates of the condenser, and the effect
+must have been due to the passage through the gas of the
+electricity which, when it was in equilibrium before the spark
+passed, was spread over the glass of the tube.
+
+Meissner observed this effect when the tube was filled with
+oxygen, hydrogen, carbonic acid, and nitrogen, though it was
+very small when the tube was filled with hydrogen.
+
+\includegraphicsmid[!b]{fig81}{Fig.~81.}
+
+\Article{189} The effect seems too great to be accounted for merely
+by the increased statical pressure due to the decomposition
+of the molecules of the gas by the discharge, for in De~la~Rue's
+experiment, where the gas was contained in a large vessel
+and the discharge passed as a narrow thread between the electrodes,
+the pressure was increased by about $30$~per~cent. Now
+if this increase of pressure was due to the splitting up of the
+molecules into atoms it would require about one-third of the
+molecules to be so split up by a discharge which only occupied
+an infinitesimal fraction of the volume of the gas.
+
+\Article{190} It would seem more probable that in this case we had
+something analogous to the driving off of particles from an
+electrified point, as in the ordinary phenomenon of the `electrical
+wind,' or that of the projection of particles from the cathode
+which occurs when the discharge passes through a gas at a very
+low pressure; the difference between this case and the one we
+are considering being that in the latter, since the pressure is
+greater, the molecules shot off from the cathode communicate
+their momentum to the surrounding gas instead of retaining it
+until they strike against the walls of the discharge tube. This
+would have the effect of diminishing the density of the gas in
+the neighbourhood of the line of discharge, and would therefore
+increase the density and pressure in other parts of the tube.
+%% -----File: 190.png---Folio 176-------
+
+\includegraphicsmid{fig82}{Fig.~82.}
+
+\Article{191} Töpler (\textit{Pogg.\ Ann.}\ 134, p.~194, 1868) has investigated by
+\index{Topler@Töpler, disturbance produced by spark}%
+means of a stroboscopic arrangement the disturbance in the air
+produced by the passage of a spark. The following figures taken
+from his paper show the regions when the gas is expanded in the
+neighbourhood of the spark line at successive small intervals
+of time after the passage of the spark. It will be noticed that
+these regions show periodic swellings and contractions as if the
+centres of greatest disturbance were distributed at regular and
+finite intervals along the line of discharge. A similar appearance
+\index{Antolik's figures}%
+was observed by Antolik (\textit{Pogg.\ Ann.}\ 154, p.~14, 1875)
+when the discharge passed over a plate covered with fine powder;
+the powder placed itself in ridges at regular intervals along the
+line of discharge.
+
+\Article{192} This effect is also beautifully illustrated in an experiment
+\index{Discharge, furrows made by@\subdashone furrows made by}%
+\index{Electric discharge, ufurrows made by@\subdashtwo furrows made by}%
+\index{Joly, furrows made by discharge@\subdashone furrows made by discharge}%
+made by Joly (\textit{Proc.\ Roy.\ Soc.}\ 47, p.~78 et~seq., 1890), in which
+the discharge passed from one strip of platinum to another
+between plates of glass placed so close together that they showed
+Newton's rings; it was only with difficulty that the discharge
+could be got through this narrow space at all, it declined to go
+through the centre of the rings, and went out of its way to
+get through the places where the distance between the plates
+was greatest. Where it passed it made furrows on the glass
+at right angles to the line of discharge and separated by regular
+%% -----File: 191.png---Folio 177-------
+intervals; a magnified representation of these is shown in
+\figureref{fig82}{Fig.~82}, taken from Joly's paper. When the air between the
+plates was replaced by hydrogen these furrows had a tendency
+to be more widely separated.
+
+\Article{193} The explosive effects produced by the spark are well
+\index{Hertz, yexplosive effects due to spark@\subdashone explosive effects due to spark}%
+illustrated by an experiment due to Hertz (\textit{Wied.\ Ann.}\ 19,
+p.~87, 1883), in which the anode was placed at the bottom of
+a glass tube with a narrow mouth, while the cathode was placed
+outside the tube and close to the open end. The tube and the
+electrodes were in a bell jar filled with dry air at a pressure of
+$40$--$50$~mm.\ of mercury. When the discharge from a Leyden
+jar charged by an induction-coil passed, the glow accompanying
+it was blown out of the tube and extended several centimetres
+from the open end. In this experiment, as in the well-known
+`electric wind,' the explosive effects seem to be more vigorous
+at the anode than they are at the cathode.
+
+
+\Subsection{Chemical Action of the Electric Discharge.}
+\index{Chemical action of electric discharge}%
+\index{Discharge, chemical action of@\subdashone chemical action of}%
+\index{Electric discharge, vchemical action of@\subdashtwo chemical action of}%
+
+\Article{194} When the electric discharge passes through a gas, it
+produces in the majority of cases perceptible chemical changes,
+though whether these changes are due to the electrical action
+of the spark, or whether they are secondary effects due to a
+great increase of temperature occurring either at the electrodes
+or along the path of the discharge, is very difficult to determine
+when the discharge takes the form of a bright spark.
+%% -----File: 192.png---Folio 178-------
+
+\Article{195} For this reason we shall mainly consider the chemical
+changes produced by those forms of discharge in which the
+thermal effects are as small as possible, though even in these
+cases, since we can only measure the average temperature of a
+large number of molecules, it is always possible to account for
+any chemical effect by supposing that although the average
+temperature is not much increased by the discharge, a small
+number of molecules have their kinetic energy so much increased
+that they can enter into fresh chemical combinations.
+
+The thermal explanation of the chemical changes requires that
+they should be subsequent to, and not contemporaneous with
+the passage of the discharge; on the view adopted in this book
+chemical changes of some kind are necessary before the discharge
+can pass at all, though it by no means follows that the
+chemical changes which are instrumental in carrying the current
+are those which are finally apparent. When electricity passes
+through a liquid electrolyte the substances liberated at the
+electrodes are in consequence of secondary chemical actions
+frequently different from the ions which carry the current.
+
+\includegraphicsmid{fig83}{Fig.~83.}
+
+\Article{196} A very convenient method of producing discharges as free
+as possible from great heat is by using a Siemens' ozonizer, represented
+\index{Siemens, ozonizer}%
+\index{Ozonizer}%
+in \figureref{fig83}{Fig.~83}. Two glass tubes are fused together, and the
+gas through which the discharge takes place circulates between
+them, entering by one of the side tubes and leaving by the other;
+the inside of the inner tube and the outside of the outer are
+coated with tin-foil, and are connected with the poles of an
+%% -----File: 193.png---Folio 179-------
+induction-coil. When the coil is working a quiet discharge
+passes as a series of luminous threads between the surfaces
+of the glass opposed to each other. This form of discharge is
+often called the `silent discharge,' and by French writers \emph{l'effluve
+electrique}.
+
+When air or oxygen is sent through a tube of this kind when
+the coil is working a considerable amount of ozone is produced.
+
+Ozone is not produced by the action of a steady electric field
+on oxygen or air unless the field is intense enough to produce a
+discharge through the gas (see J.~J. Thomson and R.~Threlfall,
+\textit{Proc.\ Roy.\ Soc.}\ 40, p.~340, 1886).
+
+Meissner (\textit{Abhandlungen der König.\ Gesell.\ Göttingen}, 16,
+\index{Meissner, expansion due to discharge}%
+p.~3, 1871) found that ozone was produced in tubes placed
+between the plates of a condenser when the condenser was
+charged or discharged, although no sparks passed between the
+plates, but that no ozone was produced when the charges on
+the plates of the condenser were kept constant. This was probably
+due to the passage through the gas of electricity which
+had distributed itself over the walls of the tube under the inductive
+action of the charged plates of the condenser.
+
+Bichat and Guntz (\textit{Annales de Chimie et de Physique} [6],~19,
+\index{Ozone, production of}%
+\index{Bichat and Guntz, formation of ozone}%
+\index{Guntz and Bichat on the formation of ozone}%
+p.~131, 1890) ascribe the formation of ozone, even by the silent
+discharge, to purely thermal causes. They regard the bright
+thread-like discharge surrounded by the non-luminous gas as a
+column of very hot oxygen surrounded by a cold atmosphere, and
+consider the conditions analogous to those which obtain in a St.~Claire
+Deville `chaud froid' tube, by the aid of which they state
+that Troost and Hautefeuille have produced ozone from oxygen
+without the use of the electric discharge.
+
+\Article{197} By the aid of the silent discharge a great many chemical
+changes are produced, of which the following are given by
+\index{Lehmann, ychemical action of the discharge@\subdashone chemical action of the discharge}%
+Lehmann, \textit{Molekular\-physik}, (bd.~2, p.~328.) Carbonic acid is split
+up by the discharge into carbonic oxide, oxygen, and ozone:
+water vapour into hydrogen and oxygen: when the discharge
+passes through acetylene a solid and a liquid are produced:
+phosphoretted hydrogen yields under similar circumstances a
+solid: methyl hydride gives marsh gas, hydrogen, and an acid:
+nitrous oxide splits up into nitrogen and oxygen: nitric oxide
+into nitrous oxide, nitrogen and oxygen.
+
+A mixture of carbonic acid and marsh gas gives a viscous
+%% -----File: 194.png---Folio 180-------
+fluid; nitrogen partly combines with ammonia: carbonic oxide
+and hydrogen give a solid product: carbonic oxide and marsh
+gas a resinous substance: nitrogen and hydrogen ammonia.
+
+Dextrine, benzine, and sodium absorb nitrogen under the
+influence of the discharge, and enter into chemical combination
+with it. Hydrogen forms with benzine and turpentine resinous
+compounds.
+
+\Article{198} Berthelot (\textit{Annales de Chimie et de Physique}, [5],~10,
+\index{Berthelot, chemical action of electric discharge}%
+p.~55, 1877) has shown that the absorption of nitrogen by
+dextrine takes place under very small electromotive intensities;
+he showed this by connecting the inside and the outside coatings
+of the ozonizer to points at different heights above the surface
+of the ground, and found that this difference of potential, which
+varied in the course of the experiments from $+60$ to $-180$~volts,
+was sufficient to produce in the course of a few weeks an appreciable
+absorption of nitrogen by a solution of dextrine in contact
+with it. The potential differences in these experiments were so
+small, and their rate of variation so slow, that it seems improbable
+that any discharge could have passed through the
+nitrogen, and the experiments suggest that chemical action
+between a gas and a substance with which it is in contact can
+be produced by the action of a variable electric field without
+the passage of electricity through the bulk of the gas. Berthelot
+suggests that plants may, under the influence of atmospheric
+electricity, absorb nitrogen by an action of this kind. This
+suggestion also raises the very important question as to whether
+the chemical changes which accompany the growth of plants can
+have any influence on the development of atmospheric electricity.
+
+\Article{199} We must now consider the relation between the quantity
+\index{Glow, produced by electrodeless discharge@\subdashone produced by electrodeless discharge|(}%
+\index{Phosphorescent glow}%
+of electricity which passes through a gas and the amount of
+chemical action which takes place in consequence. It is necessary
+here to make a distinction, which has been too much
+neglected, between the part of this action which occurs at the
+electrodes and the part which occurs along the length of the
+spark. When a current of electricity passes through a liquid
+electrolyte the only evidence of chemical decomposition is to
+be found at the electrodes. When, however, the electric discharge
+passes through a gas the chemical changes are not confined
+to the electrodes but occur along the line of the discharge
+as well. This is proved by the fact that when the electrodeless
+%% -----File: 195.png---Folio 181-------
+discharge passes through oxygen ozone is produced, as is testified
+by the existence for several seconds after the discharge has
+discharge passes through oxygen ozone is produced, as is testified
+by the existence for several seconds after the discharge has
+passed of a beautiful phosphorescent glow: the same thing is
+also proved by the behaviour of the discharge when it passes
+through acetylene; the first two or three sparks are of a beautiful
+light green colour, while all subsequent discharges are a kind of
+whitish pink, showing that the first two or three sparks have
+decomposed the gas.
+
+\Article{200} Since chemical decomposition is not confined to the electrodes
+its amount must depend upon the length of the spark;
+\index{Perrot, decomposition of steam|indexetseq}%
+\index{Steam, decomposition of by spark}%
+this has been proved by Perrot (\textit{Annales de Chimie et de Physique}
+[3],~61, p.~161, 1861), who compared the amounts of water
+vapour decomposed in the same time in a number of discharge
+tubes placed in series, the spark lengths in the tubes ranging
+from two millimetres to four centimetres; he found that the
+volumes of gas decomposed varied from $2$~c.c.\ to $52$~c.c., and that
+neither the longest nor the shortest spark produced the maximum
+effect. By placing a voltameter in the circuit Perrot found that
+in one of his tubes the amount of water vapour decomposed by
+the sparks was about $20$~times the amount of water decomposed
+in the voltameter. It is evident from this that if we wish to
+arrive at any simple relation between the quantity of electricity
+passing through the gas and the amount of chemical decomposition
+produced we must separate the part of the latter which
+occurs along the length of the spark from that which takes place
+at the electrodes.
+
+\includegraphicsmid{fig84}{Fig.~84.}
+
+\Article{201} This seems to have been done in a remarkable investigation
+made more than thirty years ago by Perrot~(l.c.), which
+does not seem to have attracted the attention it merits, and
+which would well repay repetition. The apparatus used by
+Perrot in his experiments is represented in \figureref{fig84}{Fig.~84} from his
+paper. The spark passed between two platinum wires sealed
+into glass tubes, $c\,f\,g$,~$d\,f\,g$, which they did not touch except at the
+places where they were sealed: the open ends, $c$,~$d$, of these tubes
+were about $2$~mm.\ apart, and the wires terminated inside the
+tubes at a distance of about $2$~mm.\ from the ends. The other
+ends of these tubes were inserted under test tubes~$e\,e$, in which
+the gases which passed up the tubes were collected. The air was
+exhausted from the vessel~\smallsanscap{A} and the water vapour through which
+the discharge passed was obtained by heating the water in the
+%% -----File: 196.png---Folio 182-------
+vessel to about \nblabel{add:5}$90°$\,C.: special precautions were taken to free this %%correction taken from errata
+water from any dissolved gas. The stream of vapour arising
+from this water drove up the tubes the gases produced by the
+passage of the spark; part of these gases was produced along
+the length of the spark, but in this case the hydrogen and
+oxygen would be in chemically equivalent proportions; part of
+the gases driven up the tubes would however be liberated at the
+electrodes, and it is this part only that we could expect to bear
+any simple relation to the quantity of electricity which had
+passed through the gas.
+
+When the sparking had ceased, the gases which had collected in
+the test tubes $e$~and~$e$ were analysed; in the first place they were
+exploded by sending a strong spark through them, this at once
+got rid of the hydrogen and oxygen which existed in chemically
+equivalent proportions and thus got rid of the gas produced
+along the length of the spark. After the explosion the gases
+left in the tubes were the hydrogen or oxygen in excess, together
+with a small quantity of nitrogen, due to a little air which had
+leaked into the vessel in the course of the experiments, or which
+had been absorbed by the water. The results of these analyses
+showed that there was always an excess of oxygen in the test
+tube in connection with the positive electrode, and an excess of
+hydrogen in the test tube connected with the negative electrode,
+and also that the amounts of oxygen and hydrogen in the
+respective tubes were very nearly chemically equivalent to the
+amount of copper deposited from a solution of copper sulphate
+in a voltameter placed in series with the discharge tube.
+%% -----File: 197.png---Folio 183-------
+
+These results are so important that I shall quote one of Perrot's
+experiments in full (l.c.\ pp.~182--3).
+
+Duration of experiment $4$~hours. $8.5$~milligrammes of copper
+deposited in the voltameter from copper sulphate; this amount
+of copper is chemically equivalent to $3$~c.c.\ of hydrogen and $1.5$~c.c.\
+of oxygen at atmospheric pressure.
+
+In the test tube over the negative electrode there were at the
+end of the experiment $37.5$~c.c.\ of gas, after the explosion by
+the spark this was reduced to $3.1$~c.c., so that by far the greater
+part of the gas collected consisted of hydrogen and oxygen in
+chemically equivalent proportions, produced not at the electrodes
+but along the line of the spark. $5.3$~c.c.\ of oxygen were added
+to the original gas, which was again exploded and the contraction
+was $4.5$~c.c.; in the original gas in the test tube there was therefore
+an excess of $3$~c.c.\ of hydrogen and $.1$~c.c.\ of something besides
+hydrogen and oxygen, probably nitrogen. In the test tube over
+the positive electrode there were $35.8$~c.c.\ of gas at the end of the
+experiment, after the explosion by the spark this was reduced
+to $1.6$~c.c. $1.8$~c.c.\ of oxygen were added, but there was no explosion
+when the spark passed; $8.7$~c.c.\ of hydrogen were added
+and the mixture exploded when the spark passed; the contraction
+produced was $9.6$~c.c., showing that the excess of oxygen
+originally present was $1.4$~c.c.\ and that $.2$~c.c.\ of nitrogen were
+mixed with it. Thus the excesses of hydrogen and oxygen in the
+tubes were very nearly chemically equivalent to the amount of
+copper deposited in the voltameter. This is also borne out by
+the following results of other experiments made by Perrot (l.c.\
+p.~183).
+
+2nd~experiment. Duration of experiment $4$~hours. Copper
+deposited in voltameter $6$~milligrammes, chemically equivalent to
+$2.12$~c.c.\ of hydrogen and $1.06$~c.c.\ of oxygen.
+
+Gas in the test tube over the positive electrode $35.10$~c.c.;
+excess of oxygen $.95$~c.c.; nitrogen $.2$~c.c.
+
+Gas in the test tube over the negative electrode $32.40$~c.c.;
+excess of hydrogen $2.10$~c.c.; nitrogen $.1$~c.c.
+
+4th~experiment. Duration of experiment $3$~hours. Copper
+deposited in voltameter $5.5$~milligrammes, chemically equivalent
+to $1.94$~c.c.\ of hydrogen and to $.97$~c.c.\ of oxygen.
+
+Gas in the test tube over the positive electrode $25.10$~c.c.;
+excess of oxygen $.85$~c.c.; nitrogen $.15$~c.c.
+%% -----File: 198.png---Folio 184-------
+
+Gas in the test tube over the negative electrode $27.70$~c.c.;
+excess of hydrogen $1.8$~c.c.; nitrogen $.21$~c.c.
+
+6th~experiment. Duration of experiment $3\tfrac{1}{2}$~hours. Copper
+deposited in voltameter $6$~milligrammes, chemically equivalent to
+$2.12$~c.c.\ of hydrogen and to $1.06$~c.c.\ of oxygen.
+
+Gas in the test tube over the positive electrode $30.20$~c.c.;
+excess of oxygen $.90$~c.c.; nitrogen $.2$~c.c.
+
+Gas in the test tube over the negative pole $32.50$~c.c.; excess
+of hydrogen $2.05$~c.c.; nitrogen $.2$~c.c.
+
+These results seem to prove conclusively (assuming that the
+discharge passed straight between the platinum wires and did
+not pass through a layer of moisture on the sides of the tubes)
+that the conduction through water vapour is produced by chemical
+decomposition, and also that in a molecule of water vapour
+the atoms of hydrogen and oxygen are associated with the same
+electrical charges as they are in liquid electrolytes.
+
+\Article{202} \DPtypo{Anotherway}{Another way} in which the chemical changes which accompany
+the passage of the spark through a gas manifest themselves
+is by the production of a phosphorescent glow, which often lasts
+\index{Phosphorescent glow}%
+\index{Oxygen, glow produced by discharge in}%
+for several seconds after the discharge has ceased. In a great many
+gases this glow does not occur, it is however extremely bright in
+oxygen. A convenient way of producing the glow is to take a
+tube about a metre long filled with oxygen at a low pressure,
+and produce an electrodeless discharge at the middle of the tube.
+From the bright ring produced by the discharge a phosphorescent
+haze will spread through the tube moving sufficiently
+slowly for its motion to be followed by the eye. The haze seems
+to come from the ozone, and the phosphorescence to be due to the
+gradual reconversion of the ozone into oxygen. This view is
+borne out by the fact that if the tube is heated the glow is not
+formed by the discharge, but as soon as the tube is allowed to
+cool down the glow is again produced: thus the glow, like ozone,
+cannot exist at a high temperature.
+
+The spectrum of this glow in oxygen is a continuous one, in
+which, however, a few bright lines can be observed if very high
+dispersive power is used. The glow is also formed in air, though
+not so brightly as in pure oxygen. When electrodes are used it
+seems to form most readily over the negative electrode, especially
+if this is formed of a flat surface of sulphuric acid.
+\index{Glow, produced by electrodeless discharge@\subdashone produced by electrodeless discharge|)}%
+
+I have experimented with a large number of gases in order to
+%% -----File: 199.png---Folio 185-------
+see whether or not the glow was formed when the electrodeless
+discharge passed through them. I have never detected any
+glow in a single gas (as distinct from a mixture) unless that gas
+was one which formed polymeric modifications, but all the gases
+I examined which do polymerize have shown the after-glow.
+The gases in which I have found the glow are oxygen, cyanogen
+(in which it is extremely persistent, though not so bright as in
+oxygen), acetylene, and vinyl chloride, all of which polymerize.
+
+A bulb filled with oxygen seems to retain its power of glowing
+unimpaired, however much it may be sparked through. In bulbs
+filled with the other gases, however, the glow after long sparking
+is not so bright as it was originally. This seems to suggest that
+the polymeric modification produced by the sparking does not
+get completely reconverted into the original form.
+
+\Subsection{Spark facilitated by rapid changes in the intensity of
+the Electric Field.}
+\index{Electric discharge, wfacilitated by rapid changes in the electric field@\subdashtwo facilitated by rapid changes in the electric field}%
+\index{Jaumann, discharge facilitated by rapid changes in the potential}%
+\index{Spark, x effects of rapid alternations in field on@\subdashone effects of rapid alternations in field on}%
+
+\Article{203} Jaumann (\textit{Sitzb.\ d.~Wien Akad.}\ 97, p.~765, 1888) has made
+some interesting experiments on the effect on the spark length of
+small but rapid changes in the electrical condition of the electrodes.
+The arrangement used for these experiments is represented
+in \figureref{fig85}{Fig.~85}, which is taken from Jaumann's paper.
+
+\includegraphicsmid{fig85}{Fig.~85.}
+
+The main current from an electrical machine charged the condenser~\smallsanscap{B},
+while a neighbouring condenser~\smallsanscap{C} could be charged
+through the air-space~\smallsanscap{F}; \smallsanscap{C}~was a small condenser whose capacity
+was only~$.55$ m., while \smallsanscap{B}~was a battery of Leyden Jars whose capacity
+was $1000$~times that of~\smallsanscap{C}. Another circuit connected with the
+%% -----File: 200.png---Folio 186-------
+machine led to a thin wire placed about $5$~mm.\ above a plate~$e$ which
+was connected to the earth. A glow discharge passed between the
+wire and the plate, and the difference of potential between the inside
+and outside coatings of the jar~\smallsanscap{B} was constant and equal to
+about $12$~electrostatic units. When the knobs of the air-break~\smallsanscap{F}
+were pushed suddenly together a spark about $.5$~mm.\ in length was
+produced at~\smallsanscap{F}, and in addition a bright spark $5$~mm.\ long
+jumped across the air space at~$e$ where there was previously
+only a glow. The passage of the spark at~\smallsanscap{F} put the two
+condensers \smallsanscap{B}~and~\smallsanscap{C} into electrical communication, and this was
+equivalent to increasing the capacity of~\smallsanscap{B} by about one part in
+a thousand; this alteration in the capacity produced a corresponding
+diminution in the potential difference between its coatings.
+This disturbance of the electrical equilibrium would give rise to
+small but very rapid oscillations in the potential difference between
+the wire and the plate~$e$, and this variable field seemed able
+to send a spark across~$e$, where when the potential was steady
+nothing but a glow was to be seen.
+
+\Article{204} It thus appears that a gas is electrically weaker under
+oscillating electric fields than under steady ones, for it is not
+apparent why the addition of the capacity of the small condenser
+to that of~\smallsanscap{B} should produce any considerable difference in the
+electromotive intensity at~$e$. It is true that while the discharge
+is oscillating the tubes of electrostatic induction are not distributed
+in the same way as they are when the field is steady, and
+some concentration of these tubes may very likely take place, but
+it does not seem probable that the disturbance produced by so
+small a condenser would be sufficient to account for the large
+effects observed by Jaumann, unless, as he supposes, the gas is
+electrically weaker in variable electric fields.
+
+Another point which might affect the electromotive intensity
+at~$e$ is the following: the comparatively small difference of
+potential between the wire and the plate is partly due to the
+glowing air-space at~$e$ acting as a conductor, this conductivity
+is due to dissociated molecules produced by the discharge, and
+it is likely that this would exhibit what are called `unipolar'
+properties, that is, that its conductivity for a current in one
+direction would not be the same as for one in the opposite.
+Even when the change produced in the distribution of electricity
+is not so great as that due to an actual reversal of the
+%% -----File: 201.png---Folio 187-------
+current it is conceivable that the conductivity of the space at~$e$
+might depend upon the way the electricity was distributed over
+the wire and plate. Thus when this distribution of electricity was
+altered, the air, by becoming a worse conductor, might cause the
+electricity to accumulate on the wire and thus increase the electromotive
+intensity at~$e$. Since, however, there is a condenser of large
+capacity in electrical connection with the wire any increase in
+its electrification would be slow, whereas the spark observed by
+Jaumann seems to have followed that across~\smallsanscap{F} without the lapse
+of any appreciable interval.
+
+\Article{205} The observations of other physicists seem to afford confirmatory
+evidence of the way in which electric discharge is
+facilitated by rapid alterations in the electromotive intensity.
+Thus Meissner (\textit{Abhand.\ der König.\ Gesell.\ Göttingen}, 16, p.~3,
+\index{Meissner, expansion due to discharge}%
+1871; see also \artref{196}{Art.~196}) found that ozone was produced in a
+tube placed between the plates of a condenser when these were
+suddenly charged or discharged, while none was produced when
+the charges on the plates were kept constant; the potential difference
+in this experiment was not sufficient to cause a spark to
+pass between the plates. Again, R.~v.~Helmholtz and Richarz
+\index{Helmholtz vr@Helmholtz, v.\ R., effect of electrification on a steam jet}%
+\index{Richarz@Richarz and R. v.\ Helmholtz, steam jet}%
+(\textit{Wied.\ Ann.}\ 40, p.~161, 1890) using an induction coil that would
+give sparks in air about $4$~inches long, found that when the
+electrodes were separated by about a foot and encased in wet
+linen bags to stop any particles of metal that might be given off
+from them, a steam jet some distance away from the electrodes
+showed very distinct signs of condensation whenever the current
+in the primary of the coil was broken. A steam jet is a very
+sensitive detector of chemical decomposition, free atoms producing
+condensation of the steam even when no particles of dust are
+present.
+
+\bigskip
+\includegraphicsmid{fig86}{Fig.~86.}
+
+If we suppose that the electric field produces a polarized
+arrangement of the molecules of the gas, then considering the
+case when the left-hand electrode is the negative one, the right-hand
+the positive, there will be between the electrodes a chain
+of molecules arranged as in the first line in \figureref{fig86}{Fig.~86}, the
+positively charged atoms being denoted by~$A$, the negatively
+charged ones by~$B$. If the field is now reversed, the molecules
+will be arranged as in the second line in \figureref{fig86}{Fig.~86}. If the reversal
+takes place very slowly, the molecules will reverse their polarity
+by swinging round, but if the rate of reversal is very rapid the
+%% -----File: 202.png---Folio 188-------
+resistance offered by the inertia of the molecules to this rotation
+will give rise to a tendency to produce the reversal of polarity
+of the molecules by chemical decomposition without rotation.
+This may be done by the molecules splitting up and rearranging
+themselves as in the third line of \figureref{fig86}{Fig.~86}.
+
+I have observed the effect of the reversal of the electric field
+when experimenting on the discharge produced in hydrogen at
+low pressures by a battery consisting of a large number of
+storage cells. I found that when the electromotive force was
+insufficient to produce continuous discharge, a momentary discharge
+occurred when the battery was reversed; this discharge
+merely flashed out for an instant, and took place when no
+discharge could be obtained by merely making or breaking the
+circuit without reversing the battery. A momentary discharge,
+however, occurred on making the circuit long before the electromotive
+force was sufficient to maintain a permanent discharge.
+
+\Article{206} Jaumann (l.~c.)\ gives some examples of brushes which
+\index{Brush discharge}%
+are formed at places where the electromotive intensity for steady
+charges is not a maximum. He explains these by supposing
+that the variations in the density of the electricity are more rapid
+%% -----File: 203.png---Folio 189-------
+at some parts of the electrodes than at others, and that \textit{ceteris
+paribus} the discharge takes place most readily at the places
+where the rate of variation of the charge is greatest. Some of
+these brushes are represented in \figureref{fig87}{Fig.~87}, taken from Jaumann.
+
+\bigskip
+\includegraphicsmid{fig87}{Fig.~87.}
+
+\Subsection{Theory of the Electric Discharge.}
+\index{Theory of electric discharge}%
+
+\Article{207} The phenomena attending the electric discharge through
+gases are so beautiful and varied that they have attracted the
+attention of numerous observers. The attention given to
+these phenomena is not, however, due so much to the beauty
+of the experiments, as to the wide-spread conviction that
+there is perhaps no other branch of physics which affords us so
+promising an opportunity of penetrating the secret of electricity;
+for while the passage of this agent through a metal or an electrolyte
+is invisible, that through a gas is accompanied by the
+most brilliantly luminous effects, which in many cases are so
+much influenced by changes in the conditions of the discharge
+as to give us many opportunities of testing any view we may
+take of the nature of electricity, of the electric discharge, and of
+the relation between electricity and matter.
+
+Though the account we have given in this chapter of the discharge
+through gases is very far from complete, it will probably
+have been sufficient to convince the student that the phenomena
+are very complex and very extensive. It is therefore desirable
+to find some working hypothesis by which they can be coordinated:
+the following method of regarding the discharge
+seems to do this to a very considerable extent.
+
+\Article{208} This view is, that the passage of electricity through a gas
+as well as through an electrolyte, and as we hold through a
+metal as well, is accompanied and effected by chemical changes;
+also that `chemical decomposition is not to be considered merely
+as an accidental attendant on the electrical discharge, but as
+an essential feature of the discharge without which it could
+not occur' (\textit{Phil.\ Mag.}\ [5],~15, p.~432, 1883). The nature of the
+chemical changes which accompany the discharge may be roughly
+described as similar to those which on Grotthus' theory of
+\index{Grotthus' chains}%
+electrolysis are supposed to occur in a Grotthus chain. The way
+such chemical changes effect the passage of the electricity has
+been already described in \artref{31}{Art.~31}, when we considered the way
+%% -----File: 204.png---Folio 190-------
+in which a tube of electrostatic induction contracted when in
+a conductor. The shortening of a tube of electrostatic induction
+is equivalent to the passage of electricity through the
+conductor.
+
+In conduction through electrolytes the signs of chemical
+change are so apparent both in the deposition on the electrodes
+of the constituents of the electrolyte and in the close connection,
+expressed by Faraday's Laws, between the quantity of electricity
+transferred through the electrolyte and the amount of chemical
+change produced, that no one can doubt the importance of the
+part played in this case by chemical decomposition in the transmission
+of the electric current.
+
+\Article{209} When electricity passes through gases, though there is
+(with the possible exception of Perrot's experiment, see \artref{200}{Art.~200})
+no one phenomenon whose interpretation is so unequivocal as
+some in electrolysis, yet the consensus of evidence given by the
+very varied phenomena shown by the gaseous discharge seems to
+point strongly to the conclusion that here, as in electrolysis, the
+discharge is accomplished by chemical agency.
+
+Perrot, in 1861, seems to have been the first to suggest that
+the discharge through gases was of an electrolytic nature. In
+\index{Giese, xconduction of electricity through gases@\subdashone conduction of electricity through gases}%
+1882 Giese (\textit{Wied.\ Ann.}\ 17, pp.~1, 236,~519) arrived at the same
+conclusion from the study of the conductivity of flames.
+
+Before applying this view to explain in detail the laws governing
+the electric discharge through gases, it seems desirable to
+mention one or two of the phenomena in which it is most plainly
+suggested.
+
+The experiments bearing most directly on this subject are
+\index{Perrot, decomposition of steam}%
+those made by Perrot on the decomposition of steam by the discharge
+from a Ruhmkorff's coil (see \artref{200}{Art.~200}). Perrot found that
+when the discharge passed through steam there was an excess of
+oxygen given off at the positive pole and an excess of hydrogen
+at the negative, and that these excesses were chemically equivalent
+to each other and to the amount of copper deposited from a
+voltameter containing copper sulphate placed in series with the
+discharge tube. If this result should be confirmed by subsequent
+researches, it would be a direct and unmistakeable proof that the
+passage of electricity through gases, just as much as through
+electrolytes, is effected by chemical means. It would also show
+that the charge of electricity associated with an atom of an
+%% -----File: 205.png---Folio 191-------
+element in a gas is the same as that associated with the same
+atom in an electrolyte.
+
+\Article{210} Again, Grove (\textit{Phil.\ Trans.}\ 1852, Part~I, p.~87) made
+\index{Grove, chemical action of the discharge}%
+nearly forty years ago some experiments which show that the
+chemical action going on at the positive electrode is not the same
+as that at the negative. Grove made the discharge from a
+Ruhmkorff's coil pass between a steel needle and a silver plate,
+the distance between the point of the needle and the plate being
+about $2.5$~mm.; the gas through which the discharge passed was a
+mixture of hydrogen and oxygen at pressures about $2$~cm.\ of
+mercury. When the silver plate was positive and the needle
+negative a patch of oxide was formed on the plate, while if the
+plate were originally negative no oxidation occurred. When
+the silver plate had been oxidised while being used as a positive
+electrode, if the current were reversed so that the plate became
+the negative electrode, the oxide was reduced by the hydrogen
+and the plate became clean. When pure hydrogen was substituted
+for the mixture of hydrogen and oxygen no chemical
+action could be observed on the plate, which was however a
+little roughened by the discharge; if however the plate was
+oxidised to begin with, it rapidly deoxidised in the hydrogen,
+especially when it was connected with the negative pole of
+the coil. Reitlinger and Wächter (\textit{Wied.\ Ann.}\ 12, p.~590, 1881)
+found that the oxidation was very dependent upon the quantity
+of water vapour present; when the gas was thoroughly dried
+very little oxidation took place. The effect may therefore be
+due to the decomposition of the water vapour into hydrogen
+and oxygen, an excess of oxygen going to the positive and
+an excess of hydrogen to the negative pole.
+
+Ludeking (\textit{Phil.\ Mag.}\ [5],~33, p.~521, 1892) has found that
+\index{Ludeking, passage of electricity through steam}%
+when the discharge passes through hydriodic acid gas, iodine is
+deposited on the positive electrode but not on the negative.
+
+\Article{211} Again, chemical changes take place in many gases when
+the electric discharge passes through them. Perhaps the best
+known example of this is the formation of ozone by the silent
+discharge through oxygen. There are however a multitude of
+other instances, thus ammonia, acetylene, phosphoretted hydrogen,
+and indeed most gases of complex chemical constitution are
+decomposed by the spark.
+
+Another fact which also points to the conclusion that the discharge
+%% -----File: 206.png---Folio 192-------
+is accomplished by chemical means is that mentioned in
+\artref{38}{Art.~38}, that the halogens chlorine, bromine, and iodine, which
+are dissociated at high temperatures, and which at such temperatures
+have already undergone the chemical change which we
+regard as preliminary to conduction, have then lost all power of
+insulation and allow electricity to pass through them with ease.
+
+\sloppy
+Then, again, we have the very interesting result discovered by
+R.~v.~Helmholtz (\textit{Wied.\ Ann.}\ 32, p.~1, 1887), that a gas through
+which electricity is passing and one in which chemical changes
+are known to be going on both affect a steam jet in the
+same way.
+
+\fussy
+\Article{212} Again, one of the most striking features of the discharge
+through gases is the way in which one discharge facilitates the
+passage of a second; the result is true whether the discharge
+passes between electrodes or as an endless ring, as in the experiments
+described in \artref{77}{Art.~77}. Closely connected with this effect
+is Hittorf's discovery (\textit{Wied.\ Ann.}~7, p.~614, 1879) that a few
+galvanic cells are able to send a current through gas which is
+conveying the electric discharge. Schuster (\textit{Proceedings Royal
+\index{Schuster, discharge through gases}%
+Soc.}, 42, p.~371, 1887) describes a somewhat similar effect. A
+large discharge tube containing air at a low pressure was
+divided into two partitions by a metal plate with openings
+round the perimeter, which served to screen off from one compartment
+any electrical action occurring in the other, if a
+vigorous discharge passed in one of these compartments, the
+electromotive force of about one quarter of a volt was sufficient
+to send a current through the air in the other.
+
+\sloppy
+Since such electromotive forces would not produce any discharge
+through air in its normal state, these experiments suggest
+that the chemical state of the gas has been altered by the discharge.
+
+\fussy
+\Article{213} We shall now go on to discuss more in detail the consequences
+of the view that dissociation of the molecules of a gas
+always accompanies electric discharge through gases. We notice,
+in the first place, that the separation of one atom from another in
+the molecule of a gas is very unlikely to be produced by the unaided
+agency of the external electric field. Let us take the case
+of a molecule of hydrogen as an example; we suppose that the
+molecule consists of two atoms, one with a positive charge, the
+other with an equal negative one. The most obvious assumption,
+%% -----File: 207.png---Folio 193-------
+which indeed is not an assumption if we accept Perrot's results,
+to make about the magnitude of the charges on the atoms is that
+each is equal in magnitude to that charge which the laws of electrolysis
+show to be associated with an atom of a monovalent
+element. We shall denote this charge by~$e$; it is the one
+molecule of electricity which Maxwell speaks about in Art.~260
+of the \textit{Electricity and Magnetism}.
+
+\index{Molecule, electric field required to decompose}%
+The electrostatic attraction between the atoms is the molecule
+\[
+\frac{e^2}{r^2},
+\]
+where $r$ is the distance between them. If the other molecules of
+hydrogen present do not help to split up the molecule, the force
+tending to pull the atoms apart is
+\[
+2Fe,
+\]
+where $F$ is the external electromotive intensity.
+
+The ratio of the force tending to separate the atoms, to their
+electrostatic attraction, is thus $2 Fr^2/e$; now at atmospheric pressure
+discharge will certainly take place through hydrogen if $F$~in
+electrostatic units is as large as~$100$, while at lower pressures
+a very much smaller value of~$F$ will be all that is required. To
+be on the safe side, however, we shall suppose that $F = 10^2$; then,
+assuming that the electrochemical equivalent of hydrogen is
+$10^{-4}$ and that there are $10^{21}$~molecules per cubic centimetre at
+atmospheric pressure, since the mass of a cubic centimetre of
+hydrogen is $1/11 × 10^3$ of a gramme, $e$~in electromagnetic units
+will be $10^4/11 × 10^{24}$, or $e$~in electrostatic units will be about
+$2.7 × 10^{-11}$ and $r$~is of the order~$10^{-8}$, hence $2Fr^2/e$, the ratio
+under consideration, will be about $1/1.4 × 10^3$; this is so small
+that it shows the separation of the atoms cannot be effected
+by the direct action of the electric field upon them when the
+molecule is not colliding with other molecules. If the atoms in
+a molecule were almost but not quite shaken apart by a collision
+with another molecule, the action of the electric field might be
+sufficient to complete the separation.
+
+The electric field, however, by polarizing the molecules of the
+gas, may undoubtedly exert a much greater effect than it could
+produce by its direct action on a single molecule. When the
+gas is not polarized, the forces exerted on one molecule by its
+neighbours act some in one direction, others in the opposite, so
+%% -----File: 208.png---Folio 194-------
+that the resultant effect is very small; when, however, the
+medium is polarized, order is introduced into the arrangement of
+the molecules, and the inter-molecular forces by all tending in
+the same direction may produce very large effects.
+
+\Article{214} The arrangement of the molecules of a gas in the electric
+field and the tendency of the inter-molecular forces may be illustrated
+to some extent by the aid of a model consisting of a large
+number of similar small magnets suspended by long strings
+attached to their centres. The positive and negative atoms in
+the molecules of the gas are represented by the poles of the
+magnets, and the forces between the molecules by those between
+the magnets. The way the molecules tend to arrange themselves
+in the electric field is represented by the arrangement
+of the magnets in a magnetic field.
+
+The analogy between the model and the gas, though it may
+serve to illustrate the forces between the molecules, is very imperfect,
+as the magnets are almost stationary, while the molecules
+are moving with great rapidity, and the collisions which occur
+in consequence introduce effects which are not represented in the
+model. The magnets, for example, would form long chains
+similar to those formed by iron filings when placed in the
+magnetic field; in the gas, however, though some of the molecules
+would form chains, they would be broken up into short lengths
+by the bombardment of other molecules. The length of these
+chains would depend upon the intensity of the bombardment to
+which they were subjected, that is upon the pressure of the gas;
+the greater the pressure the more intense the bombardment, and
+therefore the shorter the chain.
+
+We shall call these chains of molecules Grotthus' chains,
+because we suppose that when the discharge passes through the
+gas it passes by the agency of these chains, and that the same
+kind of interchange of atoms goes on amongst the molecules of
+these chains as on Grotthus' theory of electrolysis goes on between
+the molecules on a Grotthus' chain in an electrolyte.
+
+The molecules in such a chain tend to pull each other to pieces,
+and the force with which the last atom in the chain is attracted
+to the next atom will be much smaller than the force between
+two atoms in an isolated molecule; this atom will therefore be
+much more easily detached from the chain than it would from a
+single molecule, and thus chemical change, and therefore electric
+%% -----File: 209.png---Folio 195-------
+discharge, will take place much more easily than if the chains
+were absent.
+
+\Article{215} As far as the electrical effects go, it does not matter
+whether the effect of the electric field is merely to arrange
+chains which already exist scattered about in the gas, or
+whether it actually produces new chains; we are more concerned
+with the presence of such chains than with their method of production.
+The existence of a small number of such chains (and it
+only requires a most insignificant fraction of the whole number
+of molecules to be arranged in chains to enable the gas to convey
+the most intense discharge) would have important chemical
+results, as it would greatly increase the ability of the gas to enter
+into chemical combination.
+
+\includegraphicsmid{fig88}{Fig.~88.}
+
+\Article{216} The way in which the electric discharge passes along
+such a chain of molecules is similar to the action in an ordinary
+Grotthus' chain. Thus, let $A_1B_1$,~$A_2B_2$, $A_3B_3$,~\&c., \figureref{fig88}{Fig.~88}, represent
+consecutive molecules in such a chain, the~$A$'s being the
+positive atoms and the~$B$'s the negative. Let one atom,~$A_1$, at the
+end of the chain be close to the positive electrode. Then when
+the chain breaks down the atom~$A_1$ at the end of the chain goes
+to the positive electrode, $B_1$~the other atom in this molecule,
+combining with the negative atom~$A_2$ in the next molecule, $B_2$~combining
+with~$A_3$; the last molecule being left free and serving
+as a new electrode from which a new series of recombinations in
+a consecutive chain originates. There would thus be along the
+line of discharge a series of quasi-electrodes, at any of which the
+products of the decomposition of the gas might appear.
+
+The whole discharge between the electrodes consists on this
+view in a series of non-contemporaneous discharges, these discharges
+travelling consecutively from one chain to the next.
+
+The experiment described in \artref{105}{Art.~105} shows that this discharge
+starts from the positive electrode and travels to the negative
+with a velocity comparable with that of light. The introduction
+of these Grotthus' chains enables us to see how the velocity of the
+\index{Grotthus' chains}%
+discharge can be so great, while the velocity of the individual
+molecules is comparatively small. The smallness of the velocity
+of these molecules has been proved by spectroscopic observations;
+%% -----File: 210.png---Folio 196-------
+many experiments have shown that there is no appreciable displacement
+in the lines of the spectrum of the gas in the discharge
+tube when the discharge is observed end on, while if the molecules
+were moving with even a very small fraction of the velocity
+of light, Döppler's principle shows that there would be a measurable
+displacement of the lines. It does not indeed require spectroscopic
+analysis to prove that the molecules cannot be moving
+with half the velocity of light; if they did it can easily be shown
+that the kinetic energy of the particles carrying the discharge of
+a condenser would have to be greater than the potential energy
+in the condenser before discharge.
+
+When, however, we consider the discharge as passing along
+these Grotthus' chains, since the recombinations of the different
+molecules in the chain go on simultaneously, the electricity will
+pass from one end of the chain to the other in the time required
+for an atom in one molecule to travel to the oppositely charged
+atom in the next molecule in the chain. Thus the velocity of the
+discharge will exceed that of the individual atoms in the proportion
+of the length of the chain to the distance between two
+adjacent atoms in neighbouring molecules. This ratio may be
+very large, and we can understand therefore why the velocity
+of the electric discharge transcends so enormously that of the
+atoms.
+
+\Article{217} We thus see that the consideration of the smallness of the
+electromotive intensity required to produce chemical change or
+discharge, as well as of the enormous velocity with which the
+discharge travels through the gas, has led us to the conclusion
+that a small fraction of the molecules of the gas are held together
+in Grotthus' chains, while the consideration of the method by
+which the discharge passes along these chains indicates that the
+spark through the gas consists of a series of non-contemporaneous
+discharges, the discharge travelling along one chain, then waiting
+for a moment before it passes through the next, and so on.
+It is remarkable that many of the physicists, who have paid the
+greatest attention to the passage of electricity through gases,
+have been driven by their observations to the conclusion that
+the electric discharge is made up of a large number of separate
+discharges. The behaviour of \DPtypo{striae}{striæ} under the action of magnetic
+force is one of the chief reasons for coming to this conclusion. On
+\index{Spottiswoode and Moulton, electric discharge}%
+this point Spottiswoode and Moulton (\textit{Phil.\ Trans.}\ 1879, part~1,
+%% -----File: 211.png---Folio 197-------
+p.~205) say, `If a magnet be applied to a striated column, it will
+be found that the column is not simply thrown up or down as a
+whole, as would be the case if the discharge passed in direct lines
+from terminal to terminal, threading the \DPtypo{striae}{striæ} in its passage. On
+the contrary, each stria is subjected to a rotation or deformation
+of exactly the same character as would be caused if the stria
+marked the termination of flexible currents radiating from the
+bright head of the stria behind it and terminating in the hazy
+inner surface of the stria in question. An examination of several
+cases has led the authors of this paper to conclude that the
+currents do thus radiate from the bright head of a stria to the
+inner surface of the next, and that there is no direct passage
+from one terminal of the tube to the other.'
+
+With regard to the way the discharge takes place, the same
+authors say (\textit{Phil.\ Trans.}\ 1879, part~1, p.~201)---`If, then, we are
+right in supposing that the series of artificially produced hollow
+shells are analogous in their structures and functions to \DPtypo{striae}{striæ}, it
+is not difficult to deduce, from the explanation above given, the
+\textit{modus operandi} of an ordinary striated discharge. The passage of
+each of the intermittent pulses from the bright surface of a stria
+towards the hollow surface of the next may well be supposed, by
+its inductive action, to drive from the next stria a similar pulse,
+which in its turn drives one from the next stria, and so on\ldots.
+The passage of the discharge is due in both cases to an action
+consisting of an independent discharge from one stria to the next,
+and the idea of this action can perhaps be best illustrated by that
+of a line of boys crossing a brook on stepping stones, each boy
+stepping on the stone which the boy in front of him has left.'
+
+Goldstein (\textit{Phil.\ Mag.}\ [5]~10, p.~183, 1880) expresses much the
+\index{Goldstein, discharge of electricity through gases}%
+same opinion. He says: `By numerous comparisons, and taking
+account of all apparently essential phenomena, I have been led to
+the following view:---
+
+`The kathode-light, each bundle of secondary negative light,
+as well as each layer of positive light, represent each a separate
+current by itself, which begins at the part of each structure
+turned towards the kathode, and ends at the end of the negative
+rays or of the stratified structure, without the current flowing in
+one structure propagating itself into the next, without the electricity
+which flows through one also traversing the rest in
+order.
+%% -----File: 212.png---Folio 198-------
+
+`I suspect, then, that as many new points of departure of the
+discharge are present in a length of gas between two electrodes
+as this shows of secondary negative bundles or layers---that as
+according to experiments repeatedly mentioned all the properties
+and actions of the discharge at the kathode are found
+again at the secondary negative light and with each layer of
+positive light, the intimate action is the same with these as it is
+with those.'
+
+\Article{218} Thus, if we regard a stria as a bundle of Grotthus' chains
+in parallel rendered visible, the bright parts of the stria corresponding
+to the ends of the chain, the dull parts to the middle, the
+conclusion of the physicists just quoted are almost identical with
+those we arrived at by the consideration of the chains. We
+therefore regard the stratification of the discharge as evidence of
+the existence of these chains, and suppose that a stria is in fact
+a bundle of Grotthus' chains.
+
+\Article{219} As far as phenomena connected with the electric discharge
+are concerned, the Grotthus' chain is the unit rather than the
+molecule; now the length of this chain is equal to the length of
+a stria, which is very much greater than the diameter of a
+molecule, than the average distance between two molecules,
+or even than the mean free path of a molecule: thus the structure
+of a gas, as far as phenomena connected with the electric discharge
+are concerned, is on a very much coarser scale than its
+structure with reference to such properties as gaseous diffusion
+where the fundamental length is that of the mean free path of
+the molecules.
+
+\Article{220} Peace's discovery that the density---which we shall call
+the critical density---at which the `electric strength' of the gas
+is a minimum depends upon the distance between the electrodes,
+proves that the gas, when in an electric field sufficiently
+intense to produce discharge, possesses a structure whose length
+scale is comparable with the distance between the electrodes
+when these are near enough together to influence the critical
+density. As this distance is very much greater than any of the
+lengths recognized in the ordinary Kinetic Theory of Gases, the
+gas when under the influence of the electric field must have a
+structure very much coarser than that recognized by that theory.
+In our view this structure consists in the formation of Grotthus'
+chains.
+%% -----File: 213.png---Folio 199-------
+
+\Article{221} The striations are only clearly marked within somewhat
+narrow limits of pressure. But it is in accordance with the conclusion
+which all who have studied the spark have arrived at---that
+there is complete continuity between the bright well-defined
+spark which occurs at high pressures and the diffused glow
+which represents the discharge at high exhaustions---to suppose
+that they always exist in the spark discharge, but that at high
+pressures they are so close together that the bright and dark
+parts cease to be separable by the eye.
+
+The view we have taken of the action of the Grotthus' chains
+in propagating the electric discharge, and the connection between
+these chains and the striations, does not require that every discharge
+should be visibly striated; on the contrary, since the
+striations will only be visible when there is great regularity in
+the disposition of these chains, we should expect that it would
+only be under somewhat exceptional circumstances that the conditions
+would be regular enough to give rise to visible striations.
+
+\Article{222} We shall now proceed to consider more in detail the
+application of the preceding ideas to the phenomena of the
+electric discharge. The first case we shall consider is the calculation
+of the potential difference required to produce discharge
+under various conditions.
+
+It is perhaps advisable to begin with the caution that in comparing
+the potential differences required to \emph{produce} discharge
+through a given gas we must be alive to the fact that the condition
+of the gas is altered for a time by the passage of the
+discharge. Thus, when the discharges follow each other so
+rapidly that the interval between two discharges is not sufficiently
+long to allow the gas to return to its original condition
+before the second discharge passes, this discharge is in reality
+passing through a gas whose nature is a function of the electrical
+conditions. Thus, though this gas may be called hydrogen or
+oxygen, it is by no means identical with the gas which was called
+by the same name before the discharge passed through it. When
+the discharges follow each other with great rapidity the supply
+of dissociated molecules left by preceding discharges may be so
+large that the discharge ceases to be disruptive, and is analogous
+to that through a very hot gas whose molecules are dissociated
+by the heat.
+
+The measurements of the potential differences required to send
+%% -----File: 214.png---Folio 200-------
+the first spark through a gas are thus more definite in their
+interpretation than measurements of potential gradients along
+the path of a nearly continuous discharge.
+
+The striations on the preceding view of the discharge may, since
+they are equivalent to a bundle of Grotthus' chains, be regarded as
+forming a series of little electrolytic cells, the beginning and the
+end of a stria corresponding to the electrodes of the cell. Let $F$~be
+the electromotive intensity of the field, $\lambda$~the length of a stria,
+then when unit of electricity passes through the stria the work
+done on it by the electric field is~$F\lambda$. The passage of the electricity
+through the stria is accompanied just as in the case of
+the electrolytic cell, by definite chemical changes, such as the
+decomposition of a certain number of molecules of the gas; thus
+if $w$~is the increase in the potential energy of the gas due to the
+changes which occur when unit of electricity passes through the
+stria, then neglecting the heat produced by the current we have
+by the Conservation of Energy
+\[
+F\lambda = w,
+\]
+or the difference in potential between the beginning and end of a
+stria is equal to~$w$. If the chemical and other changes which
+take place in the consecutive \DPtypo{striae}{striæ} are the same, the potential
+difference due to each will be the same also. There is however
+one stria which is under different conditions from the others,
+viz.~that next the negative electrode, i.e.~the negative dark
+space. For in the body of the gas, the ions set free at an extremity
+of the stria, are set free in close proximity to the ions of opposite
+sign at the extremity of an adjacent stria. In the stria next
+the electrode the ions at one end are set free against a metallic
+surface. The experiments described in the account we have
+already given of the discharge show that the chemical changes
+which take place at the cathode are abnormal; one reason for
+this no doubt is the presence of the metal, which makes many
+chemical changes possible which could not take place if there
+were nothing but gas present. This stria is thus under exceptional
+circumstances and may differ in size and fall of potential
+from the other \DPtypo{striae}{striæ}. Hittorf's experiments, \artref{140}{Art.~140}, show that
+the fall of potential at the cathode is abnormally great. If we call
+this potential fall~$K$ and consider the case of discharge between
+two parallel metal plates; the discharge on this view, starting from
+the positive electrode, goes consecutively across a number~$n$ of
+%% -----File: 215.png---Folio 201-------
+similar \DPtypo{striae}{striæ}, one of which reaches up to the positive electrode,
+the fall of potential across each of these is~$w$; the discharge
+finally crosses the stria in contact with the negative electrode
+in which the fall of potential is~$K$; thus~$V$, the total fall of
+potential as the discharge goes from the positive to the negative
+electrode, is given by the equation
+\[
+V = K + nw.  \Tag{1}
+\]
+If $l$~is the distance between the plates, $\lambda_{0}$~the length of the stria
+next the cathode, $\lambda$~the length of the other stria, then
+\[
+n = \frac{l - \lambda_0}{\lambda}.
+\]
+Substituting this value for~$n$ in~(\eqnref{222}{1}) we get
+\[
+V = \left(K - \frac{w\lambda_0}{\lambda}\right) + \frac{l}{\lambda} w,
+\]
+which may be written
+\[
+V = K' + al. \Tag{2}
+\]
+According to this equation the curve representing the relation
+between potential difference and spark length for constant
+pressure is a straight line which does not pass through the
+origin. The curves we have given from the papers by Paschen
+and Peace show that this is very approximately true. The
+curves show that for air $K'$~would at atmospheric pressure be
+about $600$~volts from Paschen's experiments and about $400$~volts
+from Peace's.
+
+If $R$~is the electromotive intensity required to produce a spark
+of length~$l$ between two parallel infinite plates, then since $R = V/l$
+\[
+R = \frac{K'}{l} + a. \Tag{3}
+\]
+Since $K'$~is positive, the electromotive intensity required to produce
+discharge increases as the length of the spark diminishes;
+in other words, the electric strength of a thin layer of gas is
+greater than that of a thick layer. The electric strength will
+sensibly increase as soon as $K'/l$ \DPtypo{become}{becomes} appreciable in comparison
+with~$a$, this will occur as soon as $l$~ceases to be a very large
+multiple of the length of a stria. Thus the thickness of the layer
+when the `electric strength' begins to vary appreciably is comparable
+with the length of a stria at the pressure at which the
+discharge takes place; this length is very large when compared
+with molecular distances or with the mean free path of the
+%% -----File: 216.png---Folio 202-------
+molecules of the gas; hence we see why the change in the `electric
+strength' of a gas takes place when the spark length is very
+large in comparison with lengths usually recognized in the
+Kinetic Theory of Gases.
+
+According to formula~(\eqnref{222}{3}), the curve representing the relation
+between electromotive intensity and spark length is a rectangular
+hyperbola; this is confirmed by the curves given by Dr.~Liebig
+for air, carbonic acid, oxygen and coal gas (see \figureref{fig19}{Fig.~19}), and by
+those given by Mr.~Peace for air.
+
+\Article{223} The preceding formulæ are not applicable when the distance
+between the electrodes is less than~$\lambda_0$ the length of the
+stria next the cathode. But if the discharge passes through the
+gas and is not carried by metal dust torn from the electrodes we
+can easily see that the electric strength must increase as the
+distance between the electrodes diminishes. For as we have seen,
+the molecules which are active in carrying the discharge are not
+torn in pieces by the direct action of the electric field but by the
+attraction of the neighbouring molecules in the Grotthus' chain.
+Now when we push the electrodes so near together that the
+distance between them is less than the normal length of the
+chain, we take away some of the molecules from the chain
+and so make it more difficult for the molecules which remain to
+split up any particular molecule into atoms, so that in order to
+effect this splitting up we must increase the number of chains in
+the field, in other words, we must increase the electromotive
+intensity.
+
+Peace's curves, \figureref{fig27}{Fig.~27}, showing the relation between the
+potential difference and spark length are exceedingly flat in the
+neighbourhood of the critical spark length. This shows that the
+potential difference required to produce discharge increases very
+slowly at first as the spark length is shortened to less than the
+length of a Grotthus' chain.
+
+We now proceed to consider the relation between the spark
+potential and the pressure. As we have already remarked, the
+length of a Grotthus' chain depends upon the density of the
+gas; the denser the gas the shorter the chain: this is illustrated
+by the way in which the \DPtypo{striae}{striæ} lengthen out when the pressure
+is reduced. The experiments which have been made on the
+connection between the length of a stria and the density of the
+gas are not sufficiently decisive to enable us to formulate the
+%% -----File: 217.png---Folio 203-------
+exact law connecting these two quantities, we shall assume
+however that it is expressed by the equation
+\[
+\lambda = \beta\rho^{-k} ,
+\]
+where $\lambda$~is the length of a stria, $\rho$~the density of the gas, and $\beta$,~$k$
+positive \DPtypo{constant}{constants}.
+
+Equation~(\eqnref{222}{1}) involves $K$~the fall of potential at the cathode
+and $w$~the fall along a stria as well as~$\lambda$. Warburg's experiments
+(\artref{160}{Art.~160}) show that the cathode fall~$K$ is almost independent
+of the pressure, and although no observations have been made
+on the influence of a change in the pressure on the value of~$w$,
+it is not likely that~$w$ any more than~$K$ depends to any great
+extent upon the pressure. If we substitute the preceding value
+of~$\lambda$ in equation~(\eqnref{222}{2}) we get
+\[
+V = K' + \frac{l}{\beta}\rho^kw.
+\]
+Both Paschen's and Peace's experiments show that when the
+spark length is great enough to include several \DPtypo{striae}{striæ} the curve
+representing the relation between the spark potential and density
+for a constant spark length, though very nearly straight, is
+slightly convex to the axis along which the densities are
+measured. This shows that $k$~is slightly, but only slightly,
+greater than unity.
+
+\Article{224} It is interesting to trace the changes which take place
+in the conditions of discharge between two electrodes at a fixed
+distance apart as the pressure of the gas gradually diminishes.
+
+When the pressure is great the \DPtypo{striae}{striæ} are very close together,
+so that if the distance between the electrodes is a millimetre
+or more, a large number of \DPtypo{striae}{striæ} will be crowded in between
+them. As the pressure diminishes the \DPtypo{striae}{striæ} widen out, and
+fewer and fewer of them can find room to squeeze in between
+the electrodes, and as the number of \DPtypo{striae}{striæ} between the electrodes
+diminishes, the potential required to produce a spark diminishes
+also, each stria that is squeezed out corresponding to a
+definite diminution in the spark potential. This diminution
+in potential will go on until the \DPtypo{striae}{striæ} have all been eliminated
+with the exception of one. There can now be no further
+reduction in the number of \DPtypo{striae}{striæ} as the pressure diminishes,
+and the Grotthus' chain which is left, and which is required
+to split up the molecules to allow the discharge to take place,
+%% -----File: 218.png---Folio 204-------
+gets curtailed as the pressure falls by a larger and larger
+fraction of its natural length, and therefore has greater and
+greater difficulty in effecting the decomposition of the molecules,
+so that the electric strength of the gas will now increase as the
+pressure diminishes. There will thus be a density at which
+the electric strength of the gas is a minimum, and that density
+will be the one at which the length of the stria next the cathode
+is equal or nearly equal to the distance between the electrodes.
+Thus the length of a stria at the minimum strength will have
+to be very much less when the electrodes are very near together
+than when they are far apart, and since the stria-length is less
+the density at which the `electric strength' is a minimum will
+be very much greater when the electrodes are near together
+than when they are far apart. This is most strikingly exemplified
+in Mr.~Peace's experiments, for when the distance between the
+electrodes was reduced from $1/5$ to $1/100$~of a millimetre the
+critical pressure was raised from $30$ to $250$~mm.\ of mercury.
+The mean free path of a molecule of air at a pressure of 30~mm.\
+is about $1/400$~of a millimetre.
+
+\Article{225} The existence of a critical pressure, or pressure at which
+the electric strength is a minimum, when the discharge passes
+between electrodes can thus be explained if we recognize the
+formation of Grotthus' chains in the gas, and the theory leads to
+the conclusion which, as we have seen, is in accordance with the
+facts, that the critical pressure depends on the spark length.
+
+\Article{226} We have seen that when the distance between the electrodes
+is less than the length of the stria next the negative
+electrode, the intensity of the field required to produce discharge
+will increase as the distance between the electrodes diminishes.
+Peace's observations show that this increase is so rapid that
+the potential difference between the electrode when the spark
+passes increases when the spark length is diminished, or in other
+words, that the electromotive intensity increases more rapidly
+than the reciprocal of the length of a Grotthus' chain. This
+will explain the remarkable results observed by Hittorf (\artref{170}{Art.~170})
+and Lehmann (\artref{170}{Art.~170}) when the electrodes were placed very
+near together in a gas at a somewhat low pressure. In such cases
+it was found that the discharge instead of passing in the straight
+line between the electrodes took a very roundabout course. To
+explain this, suppose that in the experiment shown in \figureref{fig68}{Fig.~68}
+%% -----File: 219.png---Folio 205-------
+the electrodes are nearer together than the length of the chain
+next the electrode, i.e.~the negative dark space; then if the
+discharge passed along the shortest path between the plates, the
+potential difference required would, by Peace's experiments, considerably
+exceed~$K$, the normal cathode potential fall; if however
+the discharge passed as in the figure along a line of force, whose
+length is greater than the negative dark space, the potential
+difference required would be $K$~plus that due to any small positive
+column which may exist in the discharge. The latter part of
+the potential difference is small compared with~$K$, so that the
+potential difference required to produce discharge along this
+path will only be a little in excess of~$K$, while that required to
+produce discharge along the shortest path would, by Peace's
+experiments, be considerably greater than~$K$, the discharge will
+therefore pass as in the figure in preference to taking the shortest
+path.
+
+\Article{227} Since a term in the expression~(\eqnref{222}{1}) for the potential difference
+required to produce a spark of given length is inversely
+proportional to the length of a stria, anything which diminishes
+the length of a stria will tend to increase this potential difference.
+Now the length of a stria is influenced by the size of the
+discharge tube as soon as the length becomes comparable with
+the diameter of the tube; the narrower the tube the shorter are
+the \DPtypo{striae}{striæ}. Hence we should expect to find that it would require
+a greater potential difference to produce at a given pressure a
+spark through a narrow tube than through a wide one. This
+is confirmed by the experiments made by De~la~Rue and Hugo
+Müller, described in \artref{169}{Art.~169}.
+
+\Article{228} We do not at present know enough about the laws which
+govern the passage of electricity from a gas to a solid, or from
+a solid to a gas, to enable us to account for the difference
+between the appearances presented by the discharge at the cathode
+and anode of a vacuum tube; it may, however, be well to consider
+one or two points which must doubtless influence the behaviour
+of the discharge at the two electrodes.
+
+We have seen (\artref{108}{Art.~108}) that the positive column in the electric
+discharge starts from the positive electrodes, and that with the exception
+of the negative rays, no part of the discharge seems to
+begin at the cathode; we have also seen that the potential differences
+in the neighbourhood of the cathode are much greater than
+%% -----File: 220.png---Folio 206-------
+those near the anode. These results might at first sight seem inconsistent
+with the experiments we have described (\artref{40}{Art.~40}) on the
+electrical effect on metal surfaces of ultra-violet light and incandescence.
+In these experiments we saw that under such influences
+negative electricity escaped with great ease from a metallic
+electrode, while, on the other hand, positive electricity had great
+difficulty in doing so. In the ordinary discharge through gases it
+seems, on the contrary, to be the positive electricity which escapes
+with ease, while the negative only escapes with great difficulty.
+We must remember, however, that the vehicle conveying the electricity
+may not be the same in the two cases. When ultra-violet
+light is incident on a metal plate, there seems to be nothing in
+the phenomena inconsistent with the hypothesis that the negative
+electrification is carried away by the vapour or dust of the
+metal. In the case of vacuum tubes, however, the electricity is
+doubtless conveyed for the most part by the gas and not by
+the metal. In order to get the electricity from the gas into the
+metal, or from the metal into the gas, something equivalent to
+chemical combination must take place between the metal and
+the gas. Some experiments have been made on this point by
+\index{Stanton, escape of electricity from hot metals}%
+Stanton (\textit{Proc.\ Roy.\ Soc.}~47, p.~559, 1890), who found that a
+hot copper or iron rod connected to earth only discharged the
+electricity from a positively electrified conductor in its neighbourhood
+when chemical action was visibly going on over the
+surface of the rod, e.g.~when it was being oxidised in an atmosphere
+of oxygen. When it was covered with a film of oxide it
+did not discharge the adjacent conductor; if when coated with
+oxide it was placed in an atmosphere of hydrogen it discharged
+the electricity as long as it was being deoxidised, but as soon as
+the deoxidation was complete the leakage of the electricity
+stopped. On the other hand, when the conductor was negatively
+electrified, it leaked even when no apparent chemical action was
+taking place. I have myself observed (\textit{Proc.\ Roy.\ Soc.}~49, p.~97,
+1891) that the facility with which electricity passed from a gas
+to a metal was much increased when chemical action took place.
+If this is the case, the question as to the relative ease with which
+the electricity escapes from the two electrodes through a vacuum
+tube, depends upon whether a positively or negatively electrified
+surface more readily enters into chemical combination with the
+adjacent gas, while the sign of the electrification of a metal
+%% -----File: 221.png---Folio 207-------
+surface under the influence of ultra-violet light may, on the
+other hand, depend upon whether the `Volta-potential' (see
+\artref{44}{Art.~44}) for the metal in its solid state is less or greater than
+for the dust or vapour of the metal.
+
+\Article{229} In framing any theory of the difference between the positive
+and negative electrodes, we must remember that at the electrodes
+we have either two different substances or the same substance in
+two different states in contact, and it is in accordance with what we
+know of the electrical effects produced by the contact of different
+substances that the gas in the immediate neighbourhood of the
+electrodes should be polarized, that is, that the molecular tubes
+of induction in the gas should tend to point in a definite
+direction relatively to the outward drawn normals to the
+electrode: let us suppose that the polarization is such that the
+negative ends of the tubes are the nearest to the electrode: we
+may regard the molecules of the gas as being under the influence
+of a couple tending to twist them into this position. If now this
+electrode is the cathode, then before these molecules are available
+for carrying the discharge, they must be twisted right round
+against the action of an opposing couple, so that to produce
+discharge at this electrode the electric field must be strong
+enough to twist the molecules out of their original alignment
+into the opposite one, it must therefore be stronger than in the
+body of the gas where the opposing couple does not exist: a
+polarization of this kind would therefore make the cathode
+potential gradient greater than that in the body of the gas.
+%% -----File: 222.png---Folio 208-------
+
+
+\Chapter{Chapter III.}{Conjugate Functions.}
+
+\index{Christoffel's theorem in conjugate functions}%
+\index{Kirchhoff, on conjugate functions}%
+\index{Love on conjugate functions}%
+\index{Michell, xconjugate functions@\subdashone conjugate functions}%
+\index{Potier, conjugate functions}%
+\index{Schwarz, conjugate functions}%
+\index{Schwarz's transformation}%
+\index{Transformation, Schwarz's}%
+\Article{230} \Firstsc{The} methods given by Maxwell for solving problems in
+Electrostatics by means of Conjugate Functions are somewhat
+indirect, since there is no rule given for determining the proper
+transformation for any particular problem. Success in using
+these methods depends chiefly upon good fortune in guessing the
+suitable transformation. The use of a general theorem in Transformations
+given by Schwarz (\textit{Ueber einige Abbildungsaufgaben},
+Crelle~70, pp.~105--120, 1869), and Christoffel (\textit{Sul problema
+delle temperature stazionarie}, Annali di~Matematica,~I. p.~89,
+1867), enables us to find by a direct process the proper transformations
+for electrostatical problems in two dimensions when
+the lines over which the potential is given are straight. We
+shall now proceed to the discussion of this method which has
+been applied to Electrical problems by Kirchhoff (\textit{Zur Theorie des
+Condensators}, Gesammelte Abhandlungen, p.~101), and by Potier
+(Appendix to the French translation of Maxwell's \textit{Electricity and
+Magnetism}); it has also been applied to Hydrodynamical problems
+by Michell (\textit{On the Theory of Free Stream Lines}, Phil.\
+Trans.\ 1890,~A. p.~389), and Love (\textit{Theory of Discontinuous Fluid
+Motions in two dimensions}, Proc.\ Camb.\ Phil.\ Soc.~7, p.~175,
+1891).
+
+\Article{231} The theorem of Schwarz and Christoffel is that any
+polygon bounded by straight lines in a plane, which we shall
+call the $z$~plane, where $z = x + \iota y$, $x$~and~$y$ being the Cartesian
+coordinates of a point in this plane, can be transformed into the
+axis of~$\xi$ in a plane which we shall call the $t$~plane, where
+$t = \xi + \iota \eta$, $\xi$~and~$\eta$ being the Cartesian coordinates of a point in
+this plane; and that points inside the polygon in the $z$~plane
+%% -----File: 223.png---Folio 209-------
+transform into points on one side of the axis of~$\xi$. The transformation
+which effects this is represented by the equation
+\[
+\frac{dz}{dt} = C(t-t_{1})^{\frac{\alpha_1}{\pi}-1}(t-t_2)^{\frac{\alpha_2}{\pi}-1} \ldots (t-t_r)^{\frac{\alpha_r}{\pi}-1} \ldots (t-t_n)^{\frac{\alpha_n}{\pi}-1}, \Tag{1}
+\]
+where $\alpha_1, \alpha_2, \ldots \alpha_n$ are the internal angles of the polygon in the
+$z$~plane; $t_1, t_2, \ldots t_n$ are real quantities and are the coordinates
+of points on the axis of~$\xi$ corresponding to the angular points of
+the polygon in the $z$~plane.
+
+To prove this proposition, we remark that the argument of
+$dz/dt$, that is the value of~$\theta$ when $dz/dt$ is expressed in the
+form $R\epsilon^{\iota \theta}$ where $R$~is real, remains unchanged as long as $z$
+remains real and does not pass through any one of the values
+$t_1, t_2, \ldots t_n$; in other words, the part of the real axis of~$t$ between
+the points $t_r$~and~$t_{r+1}$ corresponds to a straight line in
+the plane of~$z$.
+
+We must now investigate what happens when $t$~passes through
+one of the points such as~$t_r$ on the axis of~$\xi$. With centre~$t_r$
+describe a small semi-circle~\smallsanscap{BDC} on the positive side of the axis
+of~$\xi$, and consider the change in $dz/dt$ as $t$~passes round~\smallsanscap{BDC}
+from~\smallsanscap{B} to~\smallsanscap{C}.
+
+\bigskip
+\includegraphicsmid{fig89}{Fig.~89.}
+
+Since we suppose $\omega$, the radius of this semi-circle, indefinitely
+small, if any finite change in $dz/dt$ occurs in passing round this
+semi-circle it must arise from the factor $(t-t_r)^{\frac{\alpha_r}{\pi}-1}$.
+
+Now for a point on the semi-circle~\smallsanscap{BDC}
+\begin{gather*}
+t - t_r = \omega \epsilon^{\iota \theta},\\
+(t-t_r)^{\frac{\alpha_r}{\pi}-1} = \omega^{\frac{\alpha_r}{\pi}-1} \epsilon^{\iota\left(\frac{\alpha_r}{\pi}-1\right)\theta},
+\end{gather*}
+hence, since $\theta$~decreases from~$\pi$ to zero as the point travels
+round the semi-circle, the argument of $(t-t_r)^{\frac{\alpha_r}{\pi}-1}$, and therefore
+of~$dz/dt$, is increased by $\pi - \alpha_r$, that is the line corresponding
+to the portion~$t_r\, t_{r+1}$ of the axis of~$\xi$ makes with the line
+corresponding to the portion~$t_{r-1}\, t_r$, the angle $\pi - \alpha_r$; in other
+%% -----File: 224.png---Folio 210-------
+words, the internal angle of the polygon in the $z$~plane at the
+point corresponding to~$t_r$ is~$\alpha_r$.
+
+If we imagine a point to travel along the axis of~$\xi$ in the
+plane of~$t$ from $t = -\infty$ to $t = +\infty$ and then back again from~$+\infty$
+to~$-\infty$ along a semi-circle of infinite radius with its centre
+at the origin of coordinates in the $t$~plane, then, as long as the
+point is on the axis of~$\xi$, the corresponding point in the plane~$z$
+is on one of the sides of the polygon. To find the path in~$z$
+corresponding to the semi-circle in~$t$ we put
+\[
+t = R\epsilon^{\iota\theta},
+\]
+where $R$ is very great and is subsequently made infinite: equation~(\eqnref{231}{1})
+then becomes
+\[
+\frac{dz}{dt} = CR^{\frac{\alpha_1 + \alpha_2 + \ldots \alpha_n}{\pi} - n} \epsilon^{\iota {\left\{\frac{\alpha_1 + \alpha_2 + \ldots \alpha_n}{\pi} - n \right\} \theta}}, \Tag{2}
+\]
+since $R$~is infinite compared with any of the quantities
+$t_1, t_2, \ldots t_n$.
+
+Since along the semi-circle
+\[
+dt = \iota R\epsilon^{\iota\theta}\,d\theta,
+\]
+equation~(\eqnref{231}{2}) becomes
+\begin{DPgather*}
+dz = \iota CR^{\frac{\alpha_1 + \alpha_2 + \ldots \alpha_n}{\pi} - (n-1)} \epsilon^{\iota \left\{\frac{\alpha_1 + \alpha_2 + \ldots \alpha_n}{\pi} - (n-1) \right\} \theta}\,d\theta, \\
+\lintertext{or} z = CR^{\frac{\alpha_1 + \alpha_2 + \ldots \alpha_n}{\pi} - (n-1)} \frac{\epsilon^{\iota \left\{\frac{\alpha_1 + \alpha_2 + \DPtypo{}{\ldots} \alpha_n}{\pi} - (n-1) \right\} \theta}}{\frac{\alpha_1 + \alpha_2 + \ldots \alpha_n}{\pi} - (n-1)}.
+\end{DPgather*}
+
+Thus the path in the $z$~plane corresponding to the semi-circle
+in the plane of~$z$ is a portion of a circle subtending an angle
+$\alpha_1 + \alpha_2 + \ldots \alpha_n - (n - 1)\pi$ at the origin, and whose radius is zero or
+infinite according as
+\[
+\frac{\alpha_1 + \alpha_2 + \ldots \alpha_n}{\pi} - (n-1)
+\]
+is positive or negative.
+
+If this quantity is zero, then equation~(\eqnref{231}{2}) becomes
+\[
+\frac{dz}{dt} = \frac{C}{R\epsilon^{\iota \theta}} = \frac{C}{t},
+\]
+\begin{DPalign*}
+\lintertext{hence} z &= C \log t + A \\
+  &= C \log R + \iota C\theta + A,
+\end{DPalign*}
+where $A$ is the constant of integration.
+%% -----File: 225.png---Folio 211-------
+
+Thus as the point in the $t$~plane moves round the semi-circle
+the point in the $z$~plane will travel over a length~$C\pi$ of a straight
+line parallel to the axis of~$y$ at an infinite distance from the
+origin.
+
+\Article{232} Since by equation~(\eqnref{231}{1}) the value of~$dz/dt$ cannot vanish
+or become infinite for values of~$t$ inside the area bounded by the
+axis of~$\xi$ and the infinite semi-circle, this area can be conformably
+transformed to the area bounded by the polygon in the $z$~plane.
+
+\Article{233} When we wish to transform any given polygon in the $z$~plane
+into the axis of~$\xi$ in the $t$~plane we have the values of
+$\alpha_1, \alpha_2, \ldots \alpha_n$ given. As regards the values of $t_1, t_2, \ldots t_n$ some may
+be arbitrarily assumed while others will have to be determined
+from the dimensions of the polygon. Whatever the values of
+$t_1, t_2, \ldots t_n$, the transformation~(\eqnref{231}{1}) will transform the axis of~$\xi$
+into a polygon whose internal angles have the required values.
+In order that this polygon should be similar to the given one
+we require $n-3$~conditions to be satisfied; hence as regards the
+$n$~quantities $t_1, t_2, \ldots t_n$, the values of $3$~of them may be arbitrarily
+assumed, while the remaining $n-3$ must be determined
+from the dimensions of the polygon in the $z$~plane.
+
+\Article{234} The method of applying the transformation theorem to
+the solution of two dimensional problems in Electrostatics in
+which the boundaries of the conductors are planes, is to take the
+polygon whose sides are the boundaries of the conductors, which
+we shall speak of as the polygon in the $z$~plane, and transform
+it by the Schwarzian transformation into the real axis in a
+new plane, which we shall call the $t$~plane. If $\psi$~represents the
+potential function, $\phi$~the stream function, and $w = \phi + \iota\psi$, the
+condition that $\psi$~is constant over the conductors may be represented
+by a diagram in the $w$~plane consisting of lines parallel
+to the real axis in this plane: we must transform these lines by
+the Schwarzian transformation into the real axis in the $t$~plane.
+Thus corresponding to a point on the real axis in the $t$~plane we
+have a point in the boundary of a conductor in the $z$~plane and
+a point along a line of constant potential in the $w$~plane, and
+we make this potential correspond to the potential of the conductor
+in the electrostatical problem whose solution we require.
+
+In this way we find
+\begin{align*}
+x + \iota y &= f(t),\\
+\phi + \iota\psi &= F(t),
+\end{align*}
+%% -----File: 226.png---Folio 212-------
+\index{Capacity of a semi-infinite plate parallel to an infinite one}%
+where $f$~and~$F$ are known functions; eliminating~$t$ between these
+equations we get
+\[
+\phi + \iota\psi = \chi(x + \iota y),
+\]
+which gives us the solution of our problem.
+
+\Article{235} We shall now proceed to consider the application of this
+method to some special problems. The first case we shall
+consider is the one discussed by Maxwell in Art.~202 of the
+\textit{Electricity and Magnetism}, in which a plate bounded by a
+straight edge and at potential~$V$ is placed above and parallel to
+an infinite plate at zero potential. The diagrams in the $z$~and~$w$
+planes are given in Figs.\ \figureref{fig90}{90}~and~\figureref{fig91}{91} respectively.
+
+\includegraphicsmid{fig90}{Fig.~90.}
+
+\includegraphicsmid{fig91}{Fig.~91.}
+
+The boundary of the $z$~diagram consists of the infinite straight
+line~\smallsanscap{AB}, the two sides of the line~\smallsanscap{CD}, and an arc of a circle
+stretching from $x = -\infty$ on the line~\smallsanscap{AB} to $x = +\infty$ on the line~\smallsanscap{CD}.
+We may assume arbitrarily the values of~$t$ corresponding
+to three corners of the diagram, we shall thus assume $t = -\infty$
+at the point $x = -\infty$ on the line~\smallsanscap{AB}, $t = -1$ at the point
+$x = +\infty$ on the same line, and $t = 0$ at~\smallsanscap{C}. The internal angles
+of the polygon are zero at~\smallsanscap{B} and $2\pi$ at~\smallsanscap{C}; hence by equation~(\eqnref{231}{1}),
+\artref{231}{Art.~231}, the Schwarzian transformation of the diagram in
+the $z$~plane to the real axis of the $t$~plane is
+\[
+\frac{dz}{dt} = C \frac{t}{t+1}. \Tag{3}
+\]
+
+The diagram in the $w$~plane consists of two parallel straight
+lines; the internal angle at~\smallsanscap{G}, the point corresponding to $t = -1$,
+is zero; hence the Schwarzian transformation to the real axis
+of~$t$ is
+\[
+\frac{dw}{dt} = B \frac{1}{t+1}.\Tag{4}
+\]
+%% -----File: 227.png---Folio 213-------
+
+From~(\eqnref{235}{3}) we have
+\[
+z = x + \iota y = C \{t- \log (t+1) + \iota\pi \}, \Tag{5}
+\]
+where the constant has been chosen so as to make $y = 0$ from
+$t = -\infty$ to~$-1$. When $t$~passes through the value~$-1$, the
+value of~$y$ increases by~$C\pi$, so that if $h$~is the distance between
+the plates
+\[
+h = C\pi,
+\]
+hence we have
+\[
+x+\iota y = \frac{h}{\pi} \{t- \log (t+1)+\iota\pi\}. \Tag{6}
+\]
+
+From (\eqnref{235}{4}) we have
+\[
+w = \phi+\iota\psi = B \{\log (t+1)-\iota\pi\};
+\]
+where the constant of integration has been chosen so as to
+make $\psi = 0$ from $t = -\infty$ to $t = -1$. As $t$~passes through the
+value~$-1$, $\psi$~diminishes by~$B\pi$. Hence, as the infinite plate is
+at zero potential and the semi-infinite one at potential~$V$, we
+have
+\begin{DPgather*}
+V = -B\pi, \\
+\lintertext{or} \phi+\iota\psi = -\frac{V}{\pi} \{\log (t+1)-\iota\pi\}. \Tag{7}
+\end{DPgather*}
+
+Eliminating~$t$ from equations (\eqnref{235}{6})~and~(\eqnref{235}{7}), we get
+\[
+x + \iota y = \frac{h}{V} \left\{\phi+\iota\psi - \frac{V}{\pi} \left(1 + \epsilon^{-(\phi+\iota\psi) \frac{\pi}{V}}\right)\right\},
+\]
+which is the transformation given in Maxwell's \textit{Electricity and
+Magnetism}, Art.~202.
+
+For many purposes, however, it is desirable to retain~$t$ in the
+expressions for the coordinates $x$~and~$y$ and for the potential
+and current functions $\psi$~and~$\phi$.
+
+Thus to find the quantity of electricity on a portion of the
+underneath side of the semi-infinite plate, we notice that on this
+side of the plate $t$~ranges from~$-1$ to~$0$, and that at a distance
+from the edge of the plate which is a large multiple of~$h$, $t$~is
+approximately~$-1$. In this case we have by~(\eqnref{235}{6}), if $x$~be the
+distance from the edge of the plate corresponding to~$t$,
+\[
+x = \frac{h}{\pi} \{t-\log(1+t) \},
+\]
+%% -----File: 228.png---Folio 214-------
+or since $t = -1$ approximately
+\[
+\log(t+1) = -\left\{\frac{\pi x}{h} + 1 \right\}.
+\]
+
+The surface density~$\sigma$ of the electricity on a conductor is
+equal to
+\[
+- \frac{1}{4\pi}\, \frac{d\psi}{d\nu},
+\]
+where $d\nu$~is an element of the outward drawn normal to the
+conductor. When, as in the present case, the conductors are
+parallel to the axis of~$x$, $d\nu = ± dy$, the $+$~or~$-$ sign being
+taken according as the outward drawn normal is the positive or
+negative direction of~$y$; i.e.~the positive sign is to be taken at
+the upper surface of the plates, the negative sign at the lower.
+We thus have
+\begin{DPgather*}
+\sigma = \mp \frac{1}{4\pi}\, \frac{d\psi}{dy} = \mp \frac{1}{4\pi}\, \frac{d\phi}{dx}. \\
+\lintertext{Since} \sigma = - \frac{1}{4\pi}\, \frac{d\psi}{d\nu} \\
+\lintertext{and} \frac{d\psi}{d\nu} = \frac{d\phi}{ds},
+\end{DPgather*}
+where $ds$~is an element of the section of the conductor
+\begin{align*}
+\sigma &= - \frac{1}{4\pi}\, \frac{d\phi}{ds} \\
+ &= - \frac{1}{4\pi}\, \frac{d\phi}{dt}\, \frac{dt}{ds}.
+\end{align*}
+The quantity of electricity on a strip of unit depth (the depth
+being measured at right angles to the plane of~$x$,~$y$) is equal to
+\begin{align*}
+\int \sigma ds &= - \frac{1}{4\pi} \int \frac{d\phi}{dt}\, \dfrac{dt}{ds}\,ds \\
+ &= - \frac{1}{4\pi} \{ \phi(t_2)-\phi(t_1) \},
+\end{align*}
+where $t_1$,~$t_2$ are the values of~$t$ at the beginning and end of the
+strip, $t_2$~being algebraically greater than~$t_1$.
+
+The quantity of electricity on the strip of breadth~$x$ is
+equal to
+\[
+\frac{1}{4\pi} \{ \phi_t - \phi_0 \},
+\]
+%% -----File: 229.png---Folio 215-------
+and this by equation~(\eqnref{235}{7}) is equal to
+\begin{gather*}
+- \frac{1}{4\pi}\, \frac{V}{\pi} \log (t+1) \\
+= \frac{V}{4\pi h} \left\{x+\frac{h}{\pi} \right\}.
+\end{gather*}
+
+Thus the quantity of electricity on the lower side of the
+plate is the same as if the density were uniform and equal to
+that on an infinite plate, the breadth of the strip being increased
+by~$h/\pi$. This, however, only represents the electricity on
+the lower side of the plate, there is also a considerable quantity
+of electricity on the top of the plate. To find an expression for
+the quantity of electricity on a strip of breadth~$x$, we notice
+that on the top of the plate $t$~ranges from zero to infinity, and
+that when $x$~is a large multiple of~$h$, $t$~is very large; in this
+case the solution of the equation
+\[
+x = \frac{h}{\pi} \{t-\log (1+t)\}
+\]
+is approximately
+\[
+t = \pi \frac{x}{h} + \log \left\{1+ \frac{\pi x}{h} \right\},
+\]
+and the quantity of electricity on a strip of breadth~$x$ is
+$\dfrac{1}{4\pi} \{ \phi_0 - \phi_t \}$, and thus by equation~(\eqnref{235}{7}) is equal to
+\begin{gather*}
+\frac{V}{4\pi^2} \log (t+1) \\
+= \frac{V}{4\pi^2} \log \left\{1 + \frac{\pi x}{h} + \log \left(1+ \frac{\pi x}{h}\right) \right\}.
+\end{gather*}
+
+Thus the quantity of electricity on an infinitely long strip is
+infinite, though its ratio to the quantity of electricity on the
+lower side of the strip is infinitely small.
+
+The surface density $± d\phi/4\pi\, dx$ of the distribution of electricity
+on the semi-infinite plate is by equations (\eqnref{235}{6})~and~(\eqnref{235}{7}) equal to
+\[
+\mp \frac{V}{4\pi h}\, \frac{1}{t}.
+\]
+On the underneath side of the plate $t$~is very nearly equal to~$-1$
+when the distance from the edge of the plate is a large multiple
+of~$h$, so that in this case the density soon reaches a constant
+%% -----File: 230.png---Folio 216-------
+value. On the upper side of the plate, however, when $x$~is a
+large multiple of~$h$, $t$~is approximately equal to
+\[
+\frac{\pi x}{h},
+\]
+so that the density varies inversely as the distance from the edge
+of the plate.
+
+The capacity of a breadth~$x$ of the upper plate, i.e.~the ratio
+of the charge on both surfaces to~$V$, is
+\[
+\frac{x}{4\pi h} \left[1 + \frac{h}{\pi x} + \frac{h}{\pi x} \log \left\{1 + \frac{\pi x}{h} + \log \left(1+ \frac{\pi x}{h}\right) \right\} \right].
+\]
+
+We see by the principle of images that the distribution of
+electricity on the upper plate is the same as would ensue if,
+instead of the infinite plate at zero potential, we had another
+semi-infinite parallel plate at potential~$-V$, at a distance~$2h$
+below the upper plate, and therefore that in this case the
+capacity of a breadth~$x$, when $x/h$~is large, of either plate is
+approximately
+\[
+\frac{x}{8\pi h} \left[1 + \frac{h}{\pi x} + \frac{h}{\pi x} \log \left\{1 + \frac{\pi x}{h} + \log \left(1+\frac{\pi x}{h}\right) \right\} \right].
+\]
+
+\Article{236} The next case we shall consider is the one discussed by
+\index{Capacity xof a plate between two infinite plates@\subdashone of a plate between two infinite plates}%
+Maxwell in Art.~195, in which a semi-infinite conducting plane is
+placed midway between two parallel infinite conducting planes,
+maintained at zero potential; we shall suppose that the potential
+of the semi-infinite plane is~$V$. The diagrams in the $z$~and~$w$
+planes are given in Figs.\ \figureref{fig92}{92}~and~\figureref{fig93}{93} respectively.
+
+\medskip
+\includegraphicsmid{fig92}{Fig.~92.}
+
+\includegraphicsmid{fig93}{Fig.~93.}
+
+The boundary of the $z$~diagram consists of the infinite line~\smallsanscap{AB},
+the two sides of the semi-infinite line~\smallsanscap{CD}, and the infinite
+%% -----File: 231.png---Folio 217-------
+line~\smallsanscap{EF}. We shall assume $t = 0$ at~\smallsanscap{C}, $t = -\infty$ at the point
+$x = -\infty$ on the line~\smallsanscap{AB}, $t = -1$ at the point $x = +\infty$ on the same
+line, then by symmetry $t = +1$ at the point $x = +\infty$ on the line~\smallsanscap{EF},
+and $t = +\infty$ at the point $x = -\infty$ on the same line. The
+internal angles of the polygon are zero at \smallsanscap{B}~and~\smallsanscap{E}, and $2\pi$ at~\smallsanscap{C},
+hence by equation~(\eqnref{231}{1}) the Schwarzian transformation of the
+diagram in the $z$~plane to the real axis in the $t$~plane is
+\[
+\frac{dz}{dt} = \frac{Ct}{(t+1)(t-1)}. \Tag{8}
+\]
+
+The diagram in the \textit{w}~plane consists of three parallel lines, or
+rather one line and the two sides of another; in \figureref{fig93}{Fig.~93} the upper
+side of the lower line corresponds to the conductor~\smallsanscap{EF}, the lower
+side to the conductor~\smallsanscap{AB}. The internal angles occur at the
+points corresponding to $t = -1$ and to $t = +1$ and are both zero;
+hence the transformation which turns the diagram in the $w$~plane
+to the real axis in the $t$~plane is
+\[
+\frac{dw}{dt} = \frac{B}{(t+1)(t-1)}. \Tag{9}
+\]
+
+From~(\eqnref{236}{8}) we have
+\[
+z = x+\iota y = \tfrac{1}{2} C \{ \log \{t^2-1\}- \iota \pi \}, \Tag{10}
+\]
+where the constant of integration has been determined so
+as to make $x = 0$, $y = 0$ at~\smallsanscap{C}. When $t$~passes through the
+values~$±1$ the value of~$y$ increases by~$-\frac{1}{2} C\pi$, hence if $h$~is the
+distance of the semi-infinite plane from either of the two infinite
+ones we have
+\begin{DPgather*}
+-\tfrac{1}{2} C\pi = h, \\
+\lintertext{or} x + \iota y = \frac{h}{\pi} \{\iota \pi - \log (t^2-1)\}. \Tag{11}
+\end{DPgather*}
+
+From equation~(\eqnref{236}{9}) we have
+\[
+w = \phi+\iota\psi = \frac{V}{\pi} \log \frac{t-1}{t+1}. \Tag{12}
+\]
+
+From this equation we get
+\[
+t^2 - 1 = \frac{4}{ \left(\epsilon^{\frac{1}{2} {\tfrac{\pi}{V}}(\phi+\iota \psi)}-\epsilon^{- \frac{1}{2} \tfrac{\pi}{V} (\phi+\iota\psi)} \right)^2}.
+\]
+%% -----File: 232.png---Folio 218-------
+
+Substituting this value of $t^2 - 1$ in~(\eqnref{236}{11}), we get
+\begin{align*}
+x + \iota y & = \frac{h}{\pi} \left[\iota \pi - 2 \log 2 + 2 \log \left\{\epsilon^{\frac{1}{2} \frac{\pi}{V} (\phi + \iota\psi)} - \epsilon^{-\frac{1}{2} \frac{\pi}{V} (\phi + \iota\psi)}\right\}\right] \\
+& = \frac{h}{\pi} \left[\iota \pi - 2 \log 2 + \log \left\{\epsilon^{\frac{\pi\phi}{V}} + \epsilon^{-\frac{\pi\phi}{V}} - 2 \cos \frac{\pi\psi}{V} \right\}\right. \\
+&  \qquad\qquad \left. {} + 2 \iota \tan^{-1} \left\{\frac{\left(\epsilon^{\frac{1}{2} \frac{\pi\phi}{V}} + \epsilon^{-\frac{1}{2} \frac{\pi\phi}{V}}\right)}{\epsilon^{\frac{1}{2} \frac{\pi\phi}{V}} - \epsilon^{-\frac{1}{2} \frac{\pi\phi}{V}}} \tan \frac{\pi\psi}{2V} \right\} \right],
+\end{align*}
+which is equivalent to the result given in Maxwell, Art.~195.
+
+The quantity of electricity on a portion whose length is~\smallsanscap{CP}
+and breadth unity of the lower side of the plane~\smallsanscap{CD} is
+\[
+\frac{1}{4\pi} \left\{\phi_P - \phi_C\right\}.
+\]
+
+Now $\phi_C = 0$, and when \smallsanscap{CP}~is large compared with~$h$, $t$~is very
+nearly equal to~$-1$, hence if $\text{\smallsanscap{CP}} = x$ we have in this case
+from~(\eqnref{236}{11})
+\[
+x = -\frac{h}{\pi}\{\log 2 + \log (t+1)\},
+\]
+and from~(\eqnref{236}{12})
+\begin{DPalign*}
+\phi_P &= \frac{V}{\pi} \{\log 2 - \log (t+1)\}, \\
+\lintertext{hence} \phi_P &= \frac{V}{\pi} \left\{2 \log 2 + \frac{\pi x}{h}\right\},
+\end{DPalign*}
+and the quantity of electricity on the strip is
+\[
+\frac{V}{4\pi h} x \left\{1 + \frac{2h}{\pi x} \log 2\right\}.
+\]
+
+\index{Capacity xof a plate between two infinite plates@\subdashone of a plate between two infinite plates}%
+That is, it is the same as if the distribution were uniform and
+the same as for two infinite plates with the breadth of the strip
+increased by~$\dfrac{2h}{\pi} \log 2$.
+
+\medskip
+\includegraphicsmid{fig94}{Fig.~94.}
+
+\Article{237} To find the correction for the thickness of the semi-infinite
+plate, we shall solve by the Schwarzian method the problem of
+a semi-infinite plate of finite thickness and rectangular section
+placed midway between two infinite plates. The two infinite
+plates are at zero potential, the semi-infinite one at potential~$V$.
+%% -----File: 233.png---Folio 219-------
+The diagram in the $z$~plane is represented in \figureref{fig94}{Fig.~94}. The
+boundary consists of the infinite line~\smallsanscap{AB}, the semi-infinite line~\smallsanscap{CD},
+the finite line~\smallsanscap{CE}, the semi-infinite line~\smallsanscap{EF} and the infinite
+line~\smallsanscap{GH}. We shall assume $t = -\infty$ at the point on the
+line~\smallsanscap{AB} where $x$~is equal to~$-\infty, t = -1$ at the point on the
+same line where $x = + \infty: t = -a$ at~\smallsanscap{C} $(a < 1)$, $t = +a$ at~\smallsanscap{E},
+$t = + 1$ at the point on the line~\smallsanscap{GH} where $x = + \infty$ and $t = +\infty$
+at the point on the same line where $x = -\infty$. The internal
+angles of the polygon are
+\[
+0 \text{ when } t = ±1,\ \frac{3\pi}{2} \text{ when } t = ±a,
+\]
+hence the transformation which transforms the boundary of the
+$z$~diagram into the real axis of the $t$~plane is
+\begin{align*}
+\frac{dz}{dt} &= \frac{C(t+a)^{\frac{1}{2}} (t-a)^{\frac{1}{2}}}{(t+1)(t-1)} \\
+ &= \frac{C(t^2-a^2)^{\frac{1}{2}}}{t^2-1} \\
+ &= \frac{C}{\{t^2-a^2\}^{\frac{1}{2}}} + \tfrac{1}{2} C (1-a^2) \frac{1}{\{t^2-a^2 \}^{\frac{1}{2}}} \left\{\frac{1}{t-1} - \frac{1}{t+1} \right\}. \Tag{13}
+\end{align*}
+
+The first term on the right-hand side is integrable, and the
+second and third become integrable by the substitutions $u = 1/t-1$
+and $u = 1/t+1$ respectively. Integrating~(\eqnref{237}{13}) we find \\
+\begin{align*}
+z&=C \log \{t+\sqrt{t^2-a^2}\}-C \log \sqrt{-a^2} \\
+&\qquad {}+ \tfrac{1}{2} (1-a^2)^{\frac{1}{2}} C \log \left[ \frac{(t-a^2-\sqrt{1-a^2} \sqrt{t^2-\DPtypo{a}{a^2}})(t+1)}{(t+a^2+\sqrt{1-a^2} \sqrt{t^2-a^2}) (t-1)} \right] , \Tag{14}
+\end{align*}
+where the constant has been chosen so as to make both $x$~and~$y$
+vanish when $t = 0$.
+
+If $2h$~is the thickness of the semi-infinite plate and $2H$~the
+distance between the infinite plates, then when $t$~passes through
+%% -----File: 234.png---Folio 220-------
+the value unity $y$~increases by $H - h$. When $t$~is nearly unity
+we may put
+\[
+t = 1 + R\DPtypo{\iota}{\epsilon}^{\iota\theta},
+\]
+where $R$~is small, and $\theta$~changes from~$\pi$ to zero as $t$~passes through
+unity. When $t$~is approximately~$1$, equation~(\eqnref{237}{13}) becomes
+\[
+\frac{dz}{dt} = \tfrac{1}{2} C (1 - a^2)^{\frac{1}{2}} \frac{1}{t-1},
+\]
+hence the increase in~$z$ as $t$~passes through~$1$ is
+\begin{align*}
+& \tfrac{1}{2} C (1 - a^2)^{\frac{1}{2}} \left[\log R + \iota\theta \right]_\pi^0 \\
+& = - \frac{\iota\pi}{2} C (1 - a^2)^{\frac{1}{2}},
+\end{align*}
+but since the increase in~$z$ when $t$~passes through this value is
+$\iota (H - h)$, we have
+\[
+H - h = - C \frac{\pi}{2} (1 - a^2)^{\frac{1}{2}}.
+\]
+
+When $t$~changes from~$+ \infty$ to~$- \infty$, $z$~diminishes by~$\iota 2 H$; but
+when $t$~is very large, equation~(\eqnref{237}{13}) becomes
+\begin{DPalign*}
+\frac{dz}{dt} & = \frac{C}{t}, \\
+z & = C \log t. \\
+\lintertext{Now} t & = R \epsilon^{\iota\theta},
+\end{DPalign*}
+where $R$~is infinite, and $\theta$~changes from~$0$ to~$\pi$ as $t$~changes from
+$+ \infty$ to~$- \infty$; but as $t$~changes from plus to minus infinity, $z$~increases
+by
+\begin{gather*}
+C \left[\log R + \iota\theta\right]_0^\pi \\
+= \iota C \pi,
+\end{gather*}
+and since the \emph{diminution} in~$z$ is~$\iota 2 H$, we have
+\begin{DPgather*}
+H = -C \frac{\pi}{2}. \\
+\lintertext{Thus} h = H \{1 - \sqrt{1-a^2}\}, \\
+\lintertext{or} a = \sqrt{\frac{h (2H - h)}{H^2}}.
+\end{DPgather*}
+%% -----File: 235.png---Folio 221-------
+
+The diagram in the $w$~plane is the same as in \artref{236}{Art.~236}, hence
+we have
+\[
+\phi+\iota\psi = \frac{V}{\pi} \log\frac{t-1}{t+1}. \Tag{15}
+\]
+
+The quantity of electricity on the portion of the semi-infinite
+plate between~$O$, the point midway between $C$~and~$E$, and~$P$ a
+point on the upper surface of the boundary, is
+\[
+\frac{1}{4\pi} \{\phi_O - \phi_P\}.
+\]
+
+Now at $O$, $t=0$, hence $\phi_O=0$, and if $EP$~is large compared
+with~$H$, $t$~at~\DPtypo{\smallbold{P}}{$P$} is approximately equal to~$1$. In this case we
+find from~(\eqnref{237}{14}), writing $EP=x$,
+\begin{multline*}
+x = C \log \left\{\frac{1+\sqrt{1-a^2}}{a}\right\} + \tfrac{1}{2} C \{1-a^2\}^{\frac{1}{2}} \log \frac{a^2}{2(1-a^2)} \\
+{} + \tfrac{1}{2} C \{1-a^2\}^\frac{1}{2} \log{(t-1)}.
+\end{multline*}
+
+Substituting for $C$~and~$a$ their values in terms of $H$~and~$h$
+we get
+\begin{multline*}
+-\log{(t-1)} = \frac{\pi}{H-h} \left\{x + \frac{H}{\pi} \log \frac{2H-h}{h}\right. \\
+\left.{} + \frac{H-h}{\pi} \log{\frac{h(2H-h)}{2(H-h)^2}}\right\}. \Tag{16}
+\end{multline*}
+
+But from equation~(\eqnref{237}{15})
+\[
+\phi_P = \frac{V}{\pi} \{\log(t-1) - \log2\},
+\]
+since $t$~at~$P$ is approximately equal to~$1$. Hence the quantity
+of electricity on the strip~$OP$ is
+\[
+\frac{V}{4\pi\{H-h\}} \left\{x + \frac{H}{\pi} \log{\frac{2H-h}{h}} + \frac{H-h}{\pi} \log \frac{h(2H-h)}{(H-h)^2} \right\}.
+\]
+Thus the breadth of the strip, which must be added to allow for
+the concentration of the electricity near the boundary, is
+\[
+\frac{H}{\pi} \log \frac{2H-h}{h} + \frac{H-h}{\pi} \log \frac{h(2H-h)}{(H-h)^2}.
+\]
+If $h$~is very small this reduces to
+\[
+\frac{2H}{\pi} \log{2},
+\]
+which was the result obtained in \artref{236}{Art.~236}.
+%% -----File: 236.png---Folio 222-------
+
+The density of the electricity at the point~$x$ on the top of
+the semi-infinite plate is $-\dfrac{1}{4\pi}\, \dfrac{d\phi}{dx}$, now
+\begin{align*}
+\frac{d\phi}{dx} &= \frac{d\phi}{dt}\, \frac{dt}{dx} \\
+ &= \frac{2V}{\pi(t+1)(t-1)} \frac{(t+1)(t-1)}{C(t^2-a^2)^{\frac{1}{2}}} \\
+ &= \frac{V}{\pi C}\, \frac{2}{(t^2-a^2)^{\frac{1}{2}}} \\
+ &= -\frac{V}{H}\, \frac{1}{(t^2-a^2)^{\frac{1}{2}}}.
+\end{align*}
+Hence the density of the electricity on the plate is
+\[
+\frac{V}{4\pi H}\, \frac{1}{(t^2-a^2)^{\frac{1}{2}}}.
+\]
+
+This is infinite at the edges $C$~and~$E$. When $EP$~is a large
+multiple of~$H$, $t=1$ approximately, and the density is
+\begin{DPgather*}
+\frac{V}{4\pi H}\, \frac{1}{\{1-a^2\}^{\frac{1}{2}}}, \\
+\lintertext{or since} (1-a^2)^{\frac{1}{2}}=\frac{H-h}{H},
+\end{DPgather*}
+the density is uniform and equal to
+\[
+\frac{1}{4\pi}\, \frac{V}{H-h}.
+\]
+
+\Article{238} Condensers are sometimes made by placing one cube
+\index{Capacity xof one cube inside another@\subdashone of one cube inside another}%
+inside another; in order to find the capacity of a condenser of
+this kind we shall investigate
+the distribution of electricity on
+a system of conductors such as
+that represented in \figureref{fig95}{Fig.~95},
+where \smallsanscap{ABC}~is maintained at zero
+potential and \smallsanscap{FED}~at potential~$V$.
+
+\includegraphicsouter{fig95}{Fig.~95.}
+
+The diagram in the $z$~plane
+is bounded by the lines \smallsanscap{AB},~\smallsanscap{BC},
+\smallsanscap{DE},~\smallsanscap{EF}; we shall assume that
+$t=-\infty$ at the point on the line~\smallsanscap{AB}
+where $y=+\infty, t=0$ at~\smallsanscap{B}, $t=1$ at the point on~\smallsanscap{BC} where
+%% -----File: 237.png---Folio 223-------
+$x=+\infty$, and $t=a$ at~\smallsanscap{E}, where $a$~is a quantity greater than unity
+which has to be determined by the geometry of the system. The
+internal angles of the polygon in the $z$~plane are $\pi/2$ at~\smallsanscap{B}, zero
+at~\smallsanscap{C}, $3\pi/2$ at~\smallsanscap{E}. The transformation which turns the boundary
+of the $z$~polygon into the real axis in the $t$~plane is by equation~(\eqnref{231}{1})
+expressed by the equation
+\[
+\frac{dz}{dt} = \frac{C(a-t)^{\frac{1}{2}}}{t^{\frac{1}{2}} (1-t)}. \Tag{17}
+\]
+
+The diagram in the $w$~plane consists of the real axis and a line
+parallel to it. The internal angle of the polygon is at $t=1$ and
+is equal to zero, hence the transformation which turns this
+diagram into the real axis of~$t$ is
+\begin{DPgather*}
+\frac{dw}{dt} = \frac{B}{1-t}, \\
+\lintertext{or}  \phi + \iota\psi = \iota V - \frac{V}{\pi} \log(1-t),
+\end{DPgather*}
+since $V$ is the increment in~$\psi$ when $t$~passes through the
+value~$1$.
+
+To integrate~(\eqnref{238}{17}) put
+\[
+t = a\, \frac{u^2}{1+u^2}.
+\]
+
+We have then
+\begin{align*}
+\frac{dz}{du} &= \frac{2Ca}{(1+u^2) \{1-(a-1)u^2\} } \\
+ &= 2C\left\{\frac{1}{1+u^2} + \frac{a-1}{1-(a-1)u^2} \right\}.
+\end{align*}
+
+Hence
+\begin{align*}
+z &= 2C \tan^{-1}u + \sqrt{a-1} C \log \left(\frac{1+\sqrt{a-1}u}{1-\sqrt{a-1} u} \right) \\
+  &= 2C \sin^{-1} \sqrt{\frac{t}{a}} + \sqrt{a-1} C \log \left\{\frac{\sqrt{a-t} + \sqrt{a-1} \surd t}{\sqrt{a-t} - \sqrt{a-1} \surd t} \right\}, \Tag{18}
+\end{align*}
+where the constants have been chosen so as to make $x$~and~$y$
+vanish when $t=0$.
+
+When $t=a$, we have
+\[
+x + \iota y = C\pi + \sqrt{a-1} C\iota \pi.
+\]
+%% -----File: 238.png---Folio 224-------
+
+Hence if $h$~and~$k$ are the coordinates of~$E$ referred to the axes
+\smallsanscap{BC},~\smallsanscap{AB}, we have
+\begin{align*}
+h & = C \pi, \\
+k & = C \sqrt{a-1} \pi.
+\end{align*}
+
+We can also deduce these equations from equation~(\eqnref{238}{17}) by the
+process used to determine the constants in \artref{237}{Art.~237}.
+
+We may write~(\eqnref{238}{18}) in the form
+\[
+x+\iota y = \frac{2h}{\pi} \sin^{-1} \sqrt{\frac{t}{a}} + \frac{k}{\pi} \log \left\{\frac{(\sqrt{a-t} + \sqrt{a-1} \surd{t})^2}{a(1-t)} \right\}. \Tag{19}
+\]
+
+The quantity of electricity on the strip~\smallsanscap{BP}, where \smallsanscap{P}~is a point
+on~\smallsanscap{BC}, is equal to
+\begin{gather*}
+-\frac{1}{4\pi}\{\phi_P-\phi_B\} \\
+=\frac{1}{4\pi}\, \frac{V}{\pi} \log(1-t_P).
+\end{gather*}
+
+Now if \smallsanscap{BP}~is large compared with~$k$, the value of~$t$ at~$P$ is
+approximately unity; from~(\eqnref{238}{19}) we get the more accurate value
+\begin{align*}
+-\log(1-t) &= \frac{\pi}{k} x - \frac{2h}{k} \sin^{-1} \sqrt{\frac{1}{a}} -2\log \left\{2 \sqrt{\frac{a-1}{a}} \right\}, \\
+           &= \pi \frac{x}{k} - \frac{2h}{k} \tan^{-1} \frac{h}{k} - 2\log \frac{2k}{\sqrt{h^2+k^2}}.
+\end{align*}
+
+Hence the quantity of electricity on the strip is
+\[
+-\frac{V}{4\pi k} \left\{x - \frac{2h}{\pi} \tan^{-1} \frac{h}{k} + \frac{2k}{\pi} \log \frac{\sqrt{h^2+k^2}}{2k}\right\}.
+\]
+
+Hence the quantity is the same as if the electricity were distributed
+with the uniform density $-V/4\pi k$ over a strip whose
+breadth was less than~\smallsanscap{BP} by
+\[
+\frac{2h}{\pi} \tan^{-1} \frac{h}{k} - \frac{2k}{\pi} \log \frac{\sqrt{h^2+k^2}}{2k}.
+\]
+
+In the important case when $h=k$, this becomes
+\[
+\frac{h}{2} + \frac{h}{\pi} \log 2.
+\]
+%% -----File: 239.png---Folio 225-------
+
+The surface density of the electricity at any point on~\smallsanscap{BC} or~\smallsanscap{ED}
+is
+\[
+\mp \frac{V}{4\pi^2C} \sqrt{\frac{t}{a-t}},
+\]
+the $-$~or~$+$ sign being taken according as the point is on~\smallsanscap{BC}
+or~\smallsanscap{ED}. This expression vanishes at~\smallsanscap{B} and is infinite at~\smallsanscap{E}.
+
+At~\smallsanscap{P}, a point on~\smallsanscap{BC} at some distance from~\smallsanscap{B}, $t$~is approximately
+unity, so that the surface density is
+\begin{gather*}
+-\frac{V}{4\pi^2C\sqrt{a-1}} \\
+= -\frac{V}{4\pi k}.
+\end{gather*}
+
+This result is of course obvious, but it may be regarded as
+affording a verification of the preceding solution.
+
+\includegraphicsmid{fig96}{Fig.~96.}
+
+\Article{239} Another case of some interest is that represented in
+\figureref{fig96}{Fig.~96}, where we have an infinite plane~\smallsanscap{AB} at potential~$V$ in presence
+of a conductor at zero potential bounded by two semi-infinite
+planes \smallsanscap{CD},~\smallsanscap{DE} at right angles to each other. The diagram in the
+$z$~plane is bounded by the lines~\smallsanscap{AB}, \smallsanscap{CD},~\smallsanscap{DE} and a quadrant of a
+circle whose radius is infinite. We shall assume $t=-\infty$ at the
+point on the line~\smallsanscap{AB} where $x=-\infty$, $t=0$ at the point on the
+same line where $x=+\infty$, $t=1$ at~\smallsanscap{D}. The internal angles of the
+polygon in the $z$~plane are zero at~\smallsanscap{B} and $3\pi/2$ at~\smallsanscap{D}. The transformation
+%% -----File: 240.png---Folio 226-------
+which turns the boundary of the $z$~polygon into the
+real axis in the $t$~plane is therefore, by equation~(\eqnref{231}{1}),
+\[
+\frac{dz}{dt} = \frac{C(1-t)^{\frac{1}{2}}}{t}. \Tag{20}
+\]
+
+The diagram in the $w$~plane consists of two straight lines
+parallel to the real axis, the internal angle being zero at the
+point~$t=0$; hence we have
+\[
+w = \phi + \iota\psi = \frac{V}{\pi} \log t,
+\]
+since the plane~\smallsanscap{AB} is at potential~$V$ and~\smallsanscap{CDE} at potential zero.
+
+Integrating equation~(\eqnref{239}{20}), we find when \DPtypo{$t > 0<,1$}{$0 < t < 1$,}
+\[
+z=x+\iota y=C\left(2\sqrt{1-t} - \log \frac{1+\sqrt{1-t}}{1-\sqrt{1-t}}\right), \Tag{21}
+\]
+where no constant of integration is needed if the origin of
+coordinates is taken at~\smallsanscap{D} where $t=+1$. If $h$~is the distance
+between \smallsanscap{CD}~and~\smallsanscap{AB}, then $z$~increases by~$\iota h$ when $t$~changes sign,
+hence we have by equation~(\eqnref{239}{20}), by the process similar to that
+by which we deduced the constant in \artref{237}{Art.~\DPtypo{(237)}{237}},
+\[
+h = -C\pi;
+\]
+so that (\eqnref{239}{21})~becomes, $0<t<1$,
+\[
+x+\iota y=\frac{h}{\pi} \left\{\log \frac{1+\sqrt{1-t}}{1-\sqrt{1-t}} - 2\sqrt{1-t} \right\}. \Tag{22}
+\]
+
+The quantity of electricity on a strip~\smallsanscap{DP} where~\smallsanscap{P} is a point
+on~\smallsanscap{DC} is
+\[
+\frac{V}{4\pi^2} \log t_P,
+\]
+if $t_P$ is the value of~$t$ at~\smallsanscap{P}. If \smallsanscap{DP}~is large compared with~$h$,
+$t_P$~will be very nearly zero; the value of~$\log t_P$ is then readily
+got by writing~(\eqnref{239}{22}) in the form
+\[
+x+\iota y = \frac{h}{\pi} \{2 \log (1 + \sqrt{1-t})- \log t-2 \sqrt{1-t}\}.
+\]
+
+So that if $x=\text{\smallsanscap{DP}}$, we have approximately,
+\begin{align*}
+-\log t_P &= \frac{\pi}{h} \left\{x - \frac{2h}{\pi} \log 2 + \frac{2h}{\pi} \right\}, \\
+          &= \frac{\pi}{h} \left\{x + \frac{2h}{\pi} (1-\log 2) \right\}.
+\end{align*}
+%% -----File: 241.png---Folio 227-------
+
+Thus the quantity of electricity on~\smallsanscap{DP} is
+\[
+-\frac{V}{4\pi h} \left\{x + \frac{2h}{\pi}(1- \log 2) \right\}.
+\]
+
+We can prove in a similar way that if \smallsanscap{Q} is a point on~\smallsanscap{DE} the
+charge on~\smallsanscap{DQ} is equal to
+\[
+\frac{V}{2\pi^2} \log \left(\frac{\pi \smallbold{DQ}}{2h}\right).
+\]
+
+\Article{240} If the angle~\smallsanscap{CDE}, instead of being equal to~$\pi/2$, were
+equal to~$\pi/n$, the transformation of the diagram in the $z$~plane to
+the real axis of~$t$ could be effected by the relation
+\[
+\frac{dz}{dt} = \frac{C(t-1)^{\frac{n-1}{n}}}{t}.
+\]
+
+\Article{241} We shall now proceed to discuss a problem which enables us
+\index{Guard-ring, distribution of electricity on}%
+to estimate the effect produced by the slit between the guard-ring
+and the plate of a condenser on the capacity of the condenser.
+
+\includegraphicsmid{fig97}{Fig.~97.}
+
+\includegraphicsmid{fig98}{Fig.~98.}
+
+When the plate and the guard-ring are of finite thickness the
+integration of the differential equation between $z$~and~$t$ involves
+the use of Elliptic Functions. In the two limiting cases when the
+thickness of the plate is infinitely small or infinitely great, the
+necessary integrations can however be effected by simpler means.
+
+We shall begin with the case where the thickness of the plate
+is very small, and consider the distribution of electricity on two
+semi-infinite plates separated by a finite interval~$2k$ and placed
+parallel to an infinite plane at the distance~$h$ from it.
+
+We shall suppose that the two semi-infinite plates are at
+the same potential~$V$, and that the infinite plate is at potential
+zero. The diagrams in the $z$~and $w$~planes are represented in
+Figs.\ \figureref{fig97}{97}~and~\figureref{fig98}{98}.
+%% -----File: 242.png---Folio 228-------
+
+The diagram in the $z$~plane is bounded by the infinite straight
+line~\smallsanscap{ED}, the two sides \smallsanscap{AB}~and~\smallsanscap{BC} of the semi-infinite line on
+the right, the two sides \smallsanscap{FG},~\smallsanscap{GH} of the semi-infinite line on
+the left, and a semi-circle of infinite radius. A point traversing
+the straight portion of the boundary might start from~\smallsanscap{A} and
+travel to~\smallsanscap{B} on the upper side of the line on the right, then from
+\smallsanscap{B} to~\smallsanscap{C} along the under side, from~\smallsanscap{D} to~\smallsanscap{E} along the infinite
+straight line, from~\smallsanscap{F} to~\smallsanscap{G} on the under side of the line on the
+left and from~\smallsanscap{G} to~\smallsanscap{H} on the upper side of this line. We shall
+suppose that $t = +\infty$ at~\smallsanscap{A}, $t = +1$ at~\smallsanscap{B}, $t = +a$ ($a<1$) at~\smallsanscap{C},
+$t = -a$ at~\smallsanscap{F}, $t = -1$ at~\smallsanscap{G}, $t = -\infty$ at~\smallsanscap{H}. The internal angles of
+the polygon in the $z$~plane are $2\pi$ at~\smallsanscap{B}, zero at~\smallsanscap{C}, zero at~\smallsanscap{F}, and
+$2\pi$ at~\smallsanscap{G}; hence the transformation which turns the diagram in
+the $z$~plane into the real axis of~$t$ is expressed by the relation
+\[
+\frac{dz}{dt} = C \frac{t^2 - 1}{t^2 - a^2}. \Tag{23}
+\]
+
+The diagram in the $w$~plane consists of two straight lines
+parallel to the real axis and the potential changes by~$V$ when~$t$
+passes through the values $±a$: hence we easily find
+\[
+\phi + \iota\psi = \frac{V}{\pi} \log \frac{t+a}{t-a} + \iota V. \Tag{24}
+\]
+
+We have from equation~(\eqnref{241}{23})
+\[
+z = C \left\{t-\frac{(1-a^2)}{2a} \log \frac{t-a}{t+a} + \frac{(1-a^2)}{2a} \iota\pi \right\}, \Tag{25}
+\]
+where the constant of integration has been chosen so as to make
+$x = 0$, $y = 0$ when $t = 0$. The axis of~$x$ is~\smallsanscap{ED}, the axis of~$y$
+the line at right angles to this passing through the middle
+of~\smallsanscap{GB}.
+
+If $2k$ is the width of the gap and $h$~the vertical distance
+between the plates, $x = k$, $y = h$, when $t = 1$, hence we have
+by~(\eqnref{241}{25})\
+\begin{align*}
+k &= C \left\{1- \frac{(1-a^2)}{2a} \log \frac{1-a}{1+a} \right\}, \\
+h &= C \frac{(1-a^2)}{2a} \pi.
+\end{align*}
+
+Hence $a$ is determined by the equation
+\[
+k = \frac{h}{\pi} \left\{\frac{2a}{1-a^2} + \log \frac{1+a}{1-a}\right\}. \Tag{26}
+\]
+%% -----File: 243.png---Folio 229-------
+
+The quantity of electricity on the lower side of the semi-infinite
+plate between \smallsanscap{B}~and~\smallsanscap{P} is, since $t$~increases from~\smallsanscap{P} to~\smallsanscap{B},
+\[
+\frac{1}{4\pi} \{\phi_P - \phi_B\},
+\]
+or by~(\eqnref{241}{24})
+\[
+\frac{V}{4\pi^2} \left\{\log \frac{t_P+a}{t_P-a} - \log \frac{1+a}{1-a}\right\}.
+\]
+
+But by~(\eqnref{241}{25}) if $\text{\smallsanscap{BP}} = x - k$, we have
+\[
+x - k = C \left[ t_P - 1 - \frac{1-a^2}{2a} \left\{\log \frac{t_P-a}{t_P+a} - \log \frac{1-a}{1+a} \right\} \right].
+\]
+
+Hence if $Q$ is the quantity of electricity on the lower side of
+the plate between \smallsanscap{B}~and~\smallsanscap{P},
+\begin{gather*}
+x - k = C(t_P-1) + \frac{4\pi h}{V} \centerdot Q, \\
+Q = \frac{V}{4\pi h} \{x - k + C(1-t_P)\},
+\end{gather*}
+or since $t_P = a$ approximately, if \smallsanscap{P}~is a considerable distance
+from~\smallsanscap{B}, we have
+\[
+Q = \frac{V}{4\pi h} \{x - k + C(1-a)\}. \Tag{27}
+\]
+
+The quantity of electricity~$Q_1$ on the upper side of the plate,
+from~\smallsanscap{A} to~\smallsanscap{B}, is equal to
+\[
+\frac{1}{4\pi}(\phi_B - \phi_A),
+\]
+or since $t = +\infty$ at~\smallsanscap{A}, and therefore $\phi_A$~vanishes, we have
+\[
+Q_1 = - \frac{V}{4\pi^2} \log \frac{1-a}{1+a}. \Tag{28}
+\]
+
+We can by equation~(\eqnref{241}{26}) easily express $a$ in terms of~$k/h$, when
+this ratio is either very small or very large. We shall begin by
+considering the first case, which is the one that most frequently
+occurs in practice.
+
+We see from~(\eqnref{241}{26}) that when $k/h$ is very small, $a$~is very small
+and is approximately equal to
+\[
+\frac{\pi}{4}\, \frac{k}{h}.
+\]
+%% -----File: 244.png---Folio 230-------
+The corresponding value of~$C$ is~$\frac{1}{2}k$, hence, neglecting~$(k/h)^3$,
+\begin{align*}
+Q &= \frac{V}{4\pi h} \left\{x - \tfrac{1}{2}k - \frac{\pi}{8}\, \frac{k^2}{h} \right\}, \\
+Q_1 &= \frac{V}{4\pi^2}\, 2a \\
+ &= \frac{V}{4\pi h}\, \frac{k}{2}.
+\end{align*}
+
+Hence $Q + Q_1$, the whole quantity of electricity between \smallsanscap{A}~and~\smallsanscap{P},
+is approximately equal to
+\[
+\frac{V}{4\pi h} \left\{x - \frac{\pi}{8}\, \frac{k^2}{h} \right\}.
+\]
+
+Hence the quantity of electricity on the plate of the condenser
+is to the present degree of approximation the same as if the
+electricity were uniformly distributed over the plate with
+the density it would have if the slit were absent, provided
+that the area of the plate is increased by that of a strip whose
+width is
+\[
+k - \frac{\pi}{8}\, \frac{k^2}{h};
+\]
+thus the breadth of the additional strip is very approximately
+half that of the slit.
+
+We pass on now to the case when $h/k$~is very small. We see
+from equation~(\eqnref{241}{26}) that in this case $a$~is very nearly equal to
+unity, the approximate values of $a$~and~$C$ being given by the
+equations
+\begin{align*}
+1 - a &= \frac{h}{\pi k}, \\
+C &= k.
+\end{align*}
+
+Hence by equations (\eqnref{241}{27})~and~(\eqnref{241}{28}) we have
+\begin{align*}
+Q &= \frac{V}{4\pi h} \left\{x - k + \frac{h}{\pi} \right\}, \\
+Q_1 &= \frac{V}{4\pi^2} \log \frac{2\pi k}{h}.
+\end{align*}
+
+So that the total charge $Q + Q_1$ on~\smallsanscap{AP} is equal to
+\[
+\frac{V}{4\pi h} \left[ x - k + \frac{h}{\pi} \left\{1 + \log \frac{2\pi k}{h} \right\} \right],
+\]
+%% -----File: 245.png---Folio 231-------
+\index{Guard-ring, distribution of electricity on}%
+and thus the width of the additional strip is
+\[
+\frac{h}{\pi} \left\{1 + \log {\frac{2\pi k}{h}} \right\}.
+\]
+
+\Article{242} We have hitherto supposed that the potentials of the
+plates \smallsanscap{ABC} and~\smallsanscap{FGH} are the same; we can however easily
+modify the investigation so as to give the solution of the case
+when \smallsanscap{ABC}~is maintained at the potential~$V_1$ and~\smallsanscap{FGH} at the
+potential~$V_2$. The relation between $z$~and~$t$ will not be affected
+by this change, but the relation between $w$~and~$t$ will now be
+represented by the equation
+\[
+\phi+ \iota\psi = \frac{V_2}{\pi} \log(t+a) - \frac{V_1}{\pi} \log(t-a) + \iota V_1.
+\]
+
+The quantity of electricity between \smallsanscap{B}~and~\smallsanscap{P}, a point on the
+lower side of the plate, is
+\[
+\frac{1}{4\pi} \{\phi_P - \phi_B\}.
+\]
+
+Now if \smallsanscap{BP}~is large, $t$~at~\smallsanscap{P} is approximately equal to~$a$, and
+\[
+\phi_P = \frac{V_2}{\pi} \log 2a - \frac{V_1}{\pi} \log(t-a);
+\]
+but by equation~(\eqnref{241}{25}) we have when~$t$ is nearly equal to~$a$,
+\begin{DPgather*}
+-\log (t-a) = \frac{\pi}{h} (x-Ca) - \log 2a, \\
+\lintertext{hence} \phi_P = \frac{V_2 - V_1}{\pi} \log 2a + \frac{V_1}{h} (x-Ca).
+\end{DPgather*}
+
+When $h/k$ is large $a$~is small and approximately equal to~$\pi k/4h$,
+and this equation becomes
+\[
+\phi_P = \frac{V_2 - V_1}{\pi} \log \frac{\pi k}{2h} + \frac{V_1}{h}\, x.
+\]
+
+Since $t = 1$ at~\smallsanscap{B} and $a$~is small, we see that $\phi_B$~is approximately
+equal to $a(V_1 + V_2)/\pi$ or $(V_1 + V_2)k/4h$, hence the quantity
+of electricity between \smallsanscap{B}~and~\smallsanscap{P} is approximately equal to
+\[
+\frac{V_1}{4\pi h}\, x - \frac{V_2-V_1}{4\pi^2} \log \frac{2h}{\pi k} - \frac{V_1+V_2}{4\pi h}\, \frac{k}{4}.
+\]
+
+The charge~$Q_1$ on the upper side of the plate~\smallsanscap{ABC} between a
+%% -----File: 246.png---Folio 232-------
+point~$\text{\smallsanscap{P}}'$ vertically above~\smallsanscap{P} and \smallsanscap{B}~is, since $t$~increases from~\smallsanscap{B} to~$\text{\smallsanscap{P}}'$,
+equal to
+\begin{DPgather*}
+\frac{1}{4\pi}\{\phi_B - \phi_{P'}\}. \\
+\lintertext{Now}   \phi_{P'} = \frac{V_2}{\pi}\log (t_{P'}+ a) - \frac{V_1}{\pi}\log (t_{P'}-a),
+\end{DPgather*}
+which, since $t_{P'}$~is large, may be written as
+\[
+\phi_{P'} = \frac{V_2 - V_1}{\pi}\log t_{P'}.
+\]
+
+When $\text{\smallsanscap{BP}}'$~is large, $t_{P'}$~is large also, and by equation~(\eqnref{241}{25}) is
+approximately equal to~$x/C$, that is to~$2x/k$, thus
+\begin{align*}
+\phi_{P'} = \frac{V_2 - V_1}{\pi}\log \frac{2x}{k}, \\
+\phi_B = (V_1 + V_2)\, \frac{k}{4h};
+\end{align*}
+and therefore $Q_1$, the charge on the upper part of the plate, is
+given by the equation
+\[
+Q_1 = \frac{(V_1 + V_2)}{4\pi}\, \frac{k}{4h} - \frac{(V_2 - V_1)}{4\pi^2}\log \frac{2x}{k};
+\]
+thus $Q + Q_1$, the sum of the charges on the upper and lower
+portions, is given by the equation
+\[
+Q+Q_1=\frac{V_1}{4 \pi h}\, x -\frac{(V_2-V_1)}{4 \pi^2}\left\{\log \frac{2h}{\pi k} + \log \frac{2x}{k}\right\}.
+\]
+
+\medskip
+\includegraphicsmid{fig99}{Fig.~99.}
+
+\Article{243} We shall now proceed to discuss the other extreme case
+\index{Guard-ring, distribution of electricity on}%
+of the guard-ring, that in which the depth of the slit is infinite.
+We shall begin with the case when the guard-ring and the condenser
+plate are at the same potential. The diagram in the $w$~plane
+is the same as that in \artref{239}{Art.~239}, while the diagram in the
+$z$~plane is represented in \figureref{fig99}{Fig.~99}. The boundary of this
+diagram consists of the semi-infinite lines \smallsanscap{AB},~\smallsanscap{BC} at right
+angles to each other, the infinite line~\smallsanscap{DE} parallel to~\smallsanscap{BC}, the
+semi-infinite line~\smallsanscap{FG} which is in the same straight line as~\smallsanscap{BC},
+and the semi-infinite line~\smallsanscap{GH} at right angles to~\smallsanscap{FG}. We shall
+suppose $t = + \infty$ at~\smallsanscap{A}, $t = +1$ at~\smallsanscap{B}, $t = + a$ ($a < 1$) at~\smallsanscap{C}, $t = -a$
+%% -----File: 247.png---Folio 233-------
+at~\smallsanscap{F}, $t = -1$ at~\smallsanscap{G}, and $t = - \infty$ at~\smallsanscap{H}. The internal angles of the
+polygon in the $z$~plane are $3\pi/2$ at~\smallsanscap{B} and~\smallsanscap{G} and zero at~\smallsanscap{C} and~\smallsanscap{F}.
+Thus the transformation which turns the boundary in the $z$~plane
+into the real axis of the $t$~plane is expressed by the equation
+\[
+\frac{dz}{dt} = C \frac{(t_2 - 1)^{\frac{1}{2}}}{t^2-a^2}.
+\]
+
+If we are dealing with the portion of the boundary for which
+$t$~is less than unity, it is more convenient to write this equation
+as
+\begin{align*}
+\frac{dz}{dt} & = C \iota \frac{(1-t^2)^{\frac{1}{2}}}{t^2-a^2} \\
+ & = C \iota \left[\frac{1-a^2}{2a}\, \frac{1}{(1-t^2)^{\frac{1}{2}}} \left\{\frac{1}{t-a} - \frac{1}{t+a}\right\}-\frac{1}{(1-t^2)^{\frac{1}{2}}} \right].
+\end{align*}
+
+Integrating, we find
+\begin{multline*}
+z=-C\iota\left[\frac{\sqrt{1-a^2}}{2a}\log\frac{(1-at+\sqrt{1-a^2}\sqrt{1-t^2})}{(1+at+\sqrt{1-a^2}\sqrt{1-t^2})}\, \frac{(t+a)}{(t-a)} +\sin^{-1} t\right]\\
+  +C \pi\frac{\sqrt{1-a^2}}{2a},
+\end{multline*}
+where the constant of integration has been chosen so as to
+make $x = 0$, $y = 0$ when $t = 0$; \smallsanscap{ED}~is the axis of~$x$, and the axis
+of~$y$ is midway between \smallsanscap{AB}~and~\smallsanscap{GH}. Writing~$D$ for~$-C\iota$, the
+preceding equation takes the form
+\begin{multline*}
+z=D \sin^{-1}t +D\frac{\sqrt{1-a^2}}{2a}\log \frac{(1-at+\sqrt{1-a^2} \sqrt{1-t^2})}{(1+at+\sqrt{1-a^2} \sqrt{1-t^2})}\, \frac{(t+a)}{(t-a)}\\
+  +D\iota\pi\, \frac{\sqrt{1-a^2}}{2a}.
+\Tag{29}
+\end{multline*}
+%% -----File: 248.png---Folio 234-------
+
+Now if $2k$ is the width of the slit, and $h$ the distance of the
+plate of the condenser from the infinite plate, $x = k$, $y = h$ when
+$t = 1$, hence from~(\eqnref{243}{29})
+\begin{DPalign*}
+k &= D \frac{\pi}{2},\\
+h &= D \frac{\pi}{2}\, \frac{\sqrt{1 - a^2}}{a},\\
+\lintertext{or} a^2 &= \frac{k^2}{h^2 + k^2}.
+\end{DPalign*}
+
+The relation between $w$~and~$t$ is the same as in \artref{239}{Art.~239}, and
+we have
+\[
+w = \phi + \iota \psi = \frac{V}{\pi}\log(t + a) - \frac{V}{\pi}\log(t - a) + \iota V.
+\]
+
+The quantity of electricity~$Q$ on the plate of the condenser
+between $\smallsanscap{A}$~and~$\smallsanscap{P}$, a point on~$\smallsanscap{BC}$ at some considerable distance
+from~$\smallsanscap{B}$, is
+\[
+\frac{1}{4 \pi} \{\phi_P - \phi_A\};
+\]
+since $t$ is infinite at the point corresponding to~$\smallsanscap{A}$, we see that $\phi_A$
+is zero, hence
+\begin{align*}
+Q &= \frac{1}{4 \pi} \phi_P\\
+&= \frac{V}{4 \pi^2} \log \frac{(t_P + a)}{(t_P - a)}.
+\end{align*}
+Now the point~\smallsanscap{P} corresponds to a point in the $t$~plane where $t$~is
+very nearly equal to~$a$; hence we have approximately by~(\eqnref{243}{29})
+\begin{DPgather*}
+\begin{aligned}
+\log \frac{t_P + a}{t_P - a} &= \frac{\pi}{h} \left( x - D \sin^{-1} a -\frac{h}{\pi} \log (1 - a^2) \right)\\
+&=\frac{\pi}{h} \left( x - \frac{2k}{\pi} \sin^{-1} \frac{k}{\sqrt{h^2 + k^2}} - \frac{h}{\pi} \log \frac{h^2}{h^2 + k^2} \right).
+\end{aligned} \\
+\lintertext{Thus} Q = \frac{V}{4 \pi h} \left\{x - \frac{2k}{\pi} \sin^{-1} \frac{k}{\sqrt{h^2 + k^2}} -\frac{h}{\pi} \log \frac{h^2}{h^2 + k^2} \right\}.
+\end{DPgather*}
+
+In the case which occurs most frequently in practice, that in
+which $k$~is small compared with~$h$, we have, neglecting~$(k/h)^2$,
+\[
+Q = \frac{V}{4 \pi h}\, x;
+\]
+%% -----File: 249.png---Folio 235-------
+\index{Guard-ring, distribution of electricity on}%
+that is, the quantity of electricity on the plate is the same as if
+the distribution were uniform and the width of the plate were
+increased by half the breadth of the slit.
+
+The quantity of electricity on the face~\smallsanscap{AB} of the slit is equal to
+\[
+\frac{V}{4\pi^2}\log \frac{(1+a)}{(1-a)},
+\]
+or, substituting the value for~$a$ previously found,
+\[
+\frac{V}{4\pi^2}\log\left\{\frac{1+ \dfrac{k}{\sqrt{h^2+k^2}}}{1-\dfrac{k}{\sqrt{h^2+k^2}}}\right\},
+\]
+and this when $k/h$~is small is equal to
+\[
+\frac{V}{4\pi h}\, \frac{2k}{\pi}.
+\]
+Thus $2/\pi$ of the increase in the charge on~\smallsanscap{ABC}, over the value it
+would have if the surface density were uniformly~$V/4\pi h$ on~\smallsanscap{BC},
+is on the side~\smallsanscap{AB} of the slit, and $(\pi-2)/\pi$ is on the face of the
+plate of the condenser.
+
+\Article{244} A slight modification of the preceding solution will
+enable us to find the distribution of electricity on the conductors
+when~\smallsanscap{ABC} and~\smallsanscap{FGH} are no longer at the same potential. If
+$V_1$~is the potential of~\smallsanscap{ABC}, $V_2$~that of~\smallsanscap{FGH}, then the relation
+between $z$~and~$t$ will remain the same as before, while the
+relation between $w$~and~$t$ will now be expressed by the equation
+\begin{DPalign*}
+w =\phi+\iota\psi &=\frac{V_2}{\pi}\log(t+a)-\frac{V_1}{\pi}\log(t-a)+\iota V_1, \\
+\lintertext{\rlap{or}} \phi+\iota\psi & =\frac{V_2-V_1}{\pi}\log{(t+a)}+\frac{V_1}{\pi} \log(t+a)-\frac{V_1}{\pi}\log(t-a)+\iota V_1\rlap{.}
+\end{DPalign*}
+Hence the quantity of electricity on~\smallsanscap{QBP} where \smallsanscap{Q}~is a point on~\smallsanscap{AB}
+at some distance from~\smallsanscap{B} will exceed the quantity that would
+be found from the results of the preceding \artref{243}{Article} by
+\[
+\frac{V_2 - V_1}{4\pi^2}\log \frac{t_P + a}{t_Q + a}.
+\]
+
+Since \smallsanscap{P} is a point on~\smallsanscap{BC} at some distance from~\smallsanscap{B}, $t_P$~is approximately
+equal to~$a$, and since $a$~is small and $t_Q$~large we may
+%% -----File: 250.png---Folio 236-------
+replace $t_Q + a$ by~$t_Q$; making these substitutions the preceding expression
+becomes
+\[
+\frac{V_2 - V_1}{4\pi^2} \log \frac{2a}{t_Q}.  \Tag{30}
+\]
+When $t$ is large, the relation between $z$~and~$t$, which is given by
+the equation
+\[
+\frac{dz}{dt} = C \frac{(t^2-1)^{\frac{1}{2}}}{t^2-a^2},
+\]
+is by integrating this equation found to be
+\begin{multline*}
+x- k + \iota(y-h) \\
+= C \log (t+\sqrt{t^2-1})+\frac{\sqrt{1-a^2}}{2a}\, C\left\{\sin^{-1} \frac{1-at}{t-a} - \sin^{-1}\frac{1+at}{t+a}\right\},
+\end{multline*}
+or substituting for~$C$ (the~$\iota D$ of the preceding \artref{243}{Article}) its value
+$\iota 2k/\pi$, we have
+\begin{multline*}
+x-k+ \iota (y-h) \\
+= \iota \frac{2k}{\pi}\log(t+\sqrt{t^2-1})+\iota\frac{h}{\pi}\left\{\sin^{-1} \frac{1-at}{t-a} - \sin^{-1} \frac{1+at}{t+a}\right\}.
+\end{multline*}
+Hence, when $t$~is large we have approximately
+\[
+\log 2t = \frac{\pi}{2k}\,(y-h).
+\]
+
+Substituting this value for~$\log t_Q$ in the expression~(\eqnref{244}{30}), we find
+that the correction to be applied on account of the difference of
+potential between \smallsanscap{ABC}~and~\smallsanscap{FGH} to the expression given by
+\artref{243}{Art.~243} for the quantity of electricity on~\smallsanscap{QBP} is
+\[
+-\frac{(V_2-V_1)}{4\pi^2}\left\{\log\frac{\sqrt{h^2+k^2}}{4k}+\frac{\pi}{2k}\,(y-h)\right\},
+\]
+where $y-h = \text{\smallsanscap{BQ}}$.
+
+\Article{245} The indirect method given by Maxwell, \textit{Electrostatics},
+Chap.~XII, in which we begin by assuming an arbitrary relation
+between $z$~and~$w$ of the form
+\[
+x + \iota y = F(\phi + \iota\psi),
+\]
+and then proceed to find the problems in electrostatics which
+can be solved by this relation, leads to some interesting results
+when elliptic functions are employed. Thus, let us assume
+\[
+x + \iota y = b\sn(\phi + \iota\psi),   \Tag{31}
+\]
+and suppose that $\phi$~is the potential and $\psi$~the stream function.
+%% -----File: 251.png---Folio 237-------
+Let $k$~be the modulus of the elliptic functions, $2K$~and~$2\iota K'$ the
+real and imaginary periods. Let us trace the equipotential
+surface for which $\phi = K$; we have
+\begin{align*}
+x + \iota y & = b \sn(K+\iota\psi) \\
+& = \frac{b}{\dn(\psi,k')}, \Tag{32}
+\end{align*}
+where $\dn(\psi,k')$ denotes that the modulus of the elliptic function
+is~$k'$, that is $\sqrt{1-k^2}$, and not~$k$. From equation~(\eqnref{245}{32}) we see that
+$y = 0$, and
+\[
+x = \frac{b}{\dn(\psi,k')}.
+\]
+Now $\dn(\psi,k')$ is always positive, its greatest value is unity
+when $\psi = 0$, or an even multiple of~$K'$, its least value is~$k$ when
+$\psi$~is an odd multiple of~$K'$, thus the equation
+\[
+x + \iota y = \frac{b}{\dn(\psi,k')}
+\]
+represents the portion of the axis of~$x$ between $x = b$ and
+$x = b/k$.
+
+If we put $\phi = -K$, we have
+\begin{align*}
+x + \iota y & = b \sn(-K + \iota \psi), \\
+& = - \frac{b}{\dn(\psi,k')};
+\end{align*}
+hence the equipotential surface,~$-K$, consists of the portion of
+the axis of~$x$ between $x = -b$ and $x = -b/k$.
+
+\includegraphicsouter{fig100}{Fig.~100.}
+
+\index{Capacity xof two infinite strips@\subdashone of two infinite strips}%
+Thus the transformation~(\eqnref{245}{31}) solves the
+case of two infinite plane strips \smallsanscap{AB},~\smallsanscap{CD},
+\figureref{fig100}{Fig.~100}, of finite and equal widths,
+$b\, (1 - k)/k$, in one plane placed so that
+their sides are parallel to each other.
+
+In the above investigation the potential difference is~$2K$.
+The quantity of electricity on the top of the strip~\smallsanscap{CD} is equal to
+the difference in the values of~$\psi$ at~\smallsanscap{C} and~\smallsanscap{D} divided by~$4\pi$.
+Now the difference in the values of~$\psi$ at~\smallsanscap{C} and~\smallsanscap{D} is~$K'$, hence the
+quantity of electricity on the top of the strip is
+\[
+\frac{1}{4\pi}\, K'.
+\]
+%% -----File: 252.png---Folio 238-------
+There is an equal quantity of electricity on the bottom of the
+strip, so that the total charge on~\smallsanscap{CD} is
+\[
+\frac{1}{4\pi}\, 2K'.
+\]
+The difference of potential between the strips is~$2K$, hence the
+capacity of the strip per unit length measured parallel to~$z$ is
+\[
+\frac{1}{4\pi}\, \frac{K'}{K}.
+\]
+The modulus~$k$ of the elliptic functions is the ratio of \smallsanscap{BC} to~\smallsanscap{AD},
+that is the ratio of the shortest to the longest distance
+between points in the lines \smallsanscap{AB}~and~\smallsanscap{CD}. The values of $K$~and~$K'$
+for given values of~$k$ are tabulated in Legendre's \textit{Traité des
+Fonctions Elliptiques}: so that with these tables the capacity of
+two strips of any width can be readily found.
+
+When $k$~is small, that is when the breadth of either of the
+strips is large compared with the distance between them, $K$~and~$K'$
+are given approximately by the following equations,
+\begin{align*}
+K & = \frac{\pi}{2}, \\
+K' & = \log(4/k) = \log(4AD/BC).
+\end{align*}
+Hence in this case the capacity is approximately,
+\[
+\frac{1}{2\pi^2}\log(4AD/BC).
+\]
+
+Returning to the general case, if $\sigma$~is the surface density of the
+electricity at the point~$P$ on one of the strips~\smallsanscap{AB}, we have
+\begin{DPgather*}
+\sigma  = \frac{1}{4\pi}\, \frac{d\psi}{dx}; \\
+\lintertext{and since} x  = - \frac{b}{\dn(\psi,k')},\\
+-\frac{dx}{d\psi} = bk'^2 \sn(\psi,k')\cn(\psi,k')/\dn^2(\psi,k')\\
+= \frac{1}{b}\{x^2 - b^2\}^{\frac{1}{2}}\{b^2 - k^2x^2\}^{\frac{1}{2}}\\
+= \frac{k}{b}\sqrt{CP \centerdot DP \centerdot AP \centerdot BP}\;;\\
+\lintertext{hence} \sigma = -\frac{b}{4\pi k}\, \frac{1}{\sqrt{CP \centerdot DP \centerdot AP \centerdot BP}}.
+\end{DPgather*}
+%% -----File: 253.png---Folio 239-------
+
+The solution of the case of two strips at equal and opposite
+potentials, includes that of a strip at potential~$K$ in front of an
+infinite plane at potential zero. The solution of this case can be
+deduced directly from the transformation
+\[
+x + \iota y = b \dn (\phi + \iota\psi),
+\]
+if $\psi$ be taken as the potential and $\phi$~as the stream function.
+
+\bigskip
+\includegraphicsmid{fig101}{Fig.~101.}
+
+\Article{246} \emph{Capacity of a Pile of Plates}, \figureref{fig101}{Fig.~101}. If we put
+\[
+\epsilon^{\frac{x + \iota y}{b}} = \sn(\phi + \iota\psi), \Tag{33}
+\]
+then when $\phi = K$
+\[
+\epsilon^{\frac{x + \iota y}{b}} = \sn (K + \iota\psi) = \frac{1}{\dn(\psi,k')}. \Tag{34}
+\]
+
+Thus, since $\dn (\psi,k')$ is always real and positive,
+\[
+y = 0,\ y = 2\pi b,\ y = 4\pi b, \text{ \&c.},
+\]
+while $x$~varies between the values $x_1$,~$x_2$, where
+\[
+\left.\begin{aligned}
+\epsilon^{\frac{x_1}{b}} & = 1,\\
+\epsilon^{\frac{x_2}{b}} & = \frac{1}{k}.
+\end{aligned}\right\} \Tag{35}
+\]
+
+When $\phi = -K$,
+\[
+\epsilon^{\frac{x + \iota y}{b}} = \sn (-K + \iota\psi) = -\frac{1}{\dn(\psi,k')},
+\]
+hence, since $\dn (\psi,k')$ is always real and positive,
+\[
+y = \pi b,\ y = 3\pi b,\ y = 5\pi b, \text{ \&c.},
+\]
+while $x$~varies between the same values as before. Thus, if in
+equation~(\eqnref{246}{33}) we take $\phi$ to be the potential and $\psi$~the stream
+function, the equation will give the electrical distribution over a
+\index{Capacity xof a pile of plates@\subdashone of a pile of plates}%
+pile of parallel strips of finite width, $x_2 - x_1$, the distance between
+the consecutive strips being~$\pi b$, alternate strips being at the
+same potential. The potential of one set of plates is~$K$, that
+of the other~$-K$.
+%% -----File: 254.png---Folio 240-------
+
+The quantity of electricity on one side of one of the strips per
+unit length parallel to~$z$ is, as in \artref{245}{Art.~245}, equal to $K'/4\pi$, and
+since the charge on either side is the same, the total charge on
+the strips is~$K'/2\pi$. The potential difference is~$2K$, hence the
+capacity of one of the strips per unit length is equal to
+\[
+\frac{K'}{4\pi K}.
+\]
+
+We see from equation~(\eqnref{246}{35}) that
+\[
+k = \epsilon^{-\frac{(x_2-x_1)}{b}};
+\]
+but $x_2-x_1=d$, the breadth of one of the strip, hence
+\[
+k = \epsilon^{-\frac{d}{b}}.
+\]
+Having found~$k$ from this equation, we can by Legendre's Tables
+find the values of $K$~and~$K'$, and hence the capacity of the strips.
+When the breadth of the strips is large compared with the
+distance between them, $d/b$~is large, hence $k$~is small; in this
+case we have approximately
+\begin{align*}
+K &= \frac{\pi}{2},\\
+K' &= \begin{aligned}[t]\log(4/k) &= \log(4\epsilon^{\tfrac{d}{b}})\\
+& = 2 \log 2 + \frac{d}{b},
+\end{aligned}
+\end{align*}
+so that the capacity of one strip is
+\[
+\frac{1}{2\pi^2}\left\{2 \log 2 + \frac{d}{b}\right\}.
+\]
+
+Returning to the general case, the surface density of the
+electricity at a point~$P$ on the positive side of one of the strips~\smallsanscap{AB}
+is equal to
+\[
+\frac{1}{4\pi}\, \frac{d\psi}{dx}.
+\]
+But by equation~(\eqnref{246}{34})
+\[
+\frac{1}{b}\epsilon^{\frac{x}{b}}\, \frac{dx}{d\psi} = k'^2 \sn(\psi,k') \cn(\psi,k')/\dn^2(\psi,k').
+\]
+Substituting the values of
+\[
+\sn(\psi,k'), \quad \cn(\psi,k'), \quad \dn(\psi,k')
+\]
+%% -----File: 255.png---Folio 241-------
+in terms of $\epsilon^{\tfrac{x}{b}}$, we get
+\[
+\frac{d\psi}{dx} = \frac{1}{b}\, \frac{\epsilon^{\tfrac{(x_2 - x_1)}{2b}}}{
+  \Bigl\{
+    {(\epsilon^{\tfrac{(x - x_1)}{b}} - \epsilon^{-\tfrac{(x - x_1)}{b}})
+     (\epsilon^{\tfrac{(x_2 - x)}{b}} - \epsilon^{\tfrac{-(x_2 - x)}{b}})}
+  \Bigr\}^{\frac{1}{2}}}.
+\]
+
+Hence the surface density is equal to
+\[
+\frac{1}{4 \pi b}\, \frac{\epsilon^{\tfrac{AB}{2b}}}{\Bigl\{ (\epsilon^{\tfrac{AP}{b}} - \epsilon^{-\tfrac{AP}{b}})(\epsilon^{\tfrac{BP}{b}} - \epsilon^{-\tfrac{BP}{b}}) \Bigr\}^{\frac{1}{2}}}.
+\]
+
+The distribution of electricity on any one of the plates is
+evidently the same as if the plate were placed midway between
+two infinite parallel plates at potential zero, the distance between
+the two infinite plates being~$2 \pi b$.
+
+\Article{247} \emph{Capacity of a system of $2n$~plates arranged radially and
+\index{Capacity xof a series of radial plates@\subdashone of a series of radial plates}%
+making equal angles with each other, the alternate plates being
+at the same potential, the extremities of the plates lying on two
+coaxial right circular cylinders.} Let us put
+\[
+\left( \frac{x + \iota y}{b} \right)^n = \sn(\phi + \iota \psi),
+\]
+or, transforming to polar coordinates $r$~and~$\theta$,
+\[
+\left( \frac{r}{b} \right)^n \epsilon^{\iota n \theta} = \sn(\phi + \iota \psi).
+\]
+
+Then, as before, we see that when $\phi=K$, $n \theta=0$ or~$2 \pi$, or~$4 \pi$,
+and so on, and when $\phi = -K$, $n \theta = \pi$ or~$3 \pi$, or~$5 \pi$,~\&c.; hence
+this transformation solves the case of $2n$~plates arranged radially,
+making angles~$\pi / n$ with each other, one set of $n$~plates being at
+the potential~$K$, the other set at the potential~$-K$. When
+$\phi=K$, we have
+\[
+\left( \frac{r}{b} \right)^n = \frac{1}{\dn(\psi,k')}.
+\]
+
+Hence if $r_1$~and~$r_2$ are the smallest and greatest distances of
+the edges of a plate from the line to which all the plates converge,
+we have
+\begin{DPalign*}
+\left( \frac{r_1}{b} \right)^n &= 1,\\
+\left( \frac{r_2}{b} \right)^n &= \frac{1}{k},\\
+\lintertext{or} k &= \left( \frac{r_1}{r_2} \right)^n.
+\end{DPalign*}
+%% -----File: 256.png---Folio 242-------
+
+The total charge on both sides of one of the plates is, as before,
+$K'/2\pi$, and since the potential difference is~$2K$ the capacity of
+the plate is~$K'/4\pi K$. When $r_1$~is small compared with~$r_2$, $k$~is
+small, and we have then approximately
+\begin{gather*}
+K = \frac{\pi}{2}, \\
+K' = \log (4/k) = \log 4 + n \log (r_2/r_1).
+\end{gather*}
+Thus the capacity of a plate is in this case approximately
+\[
+\frac{1}{2\pi^2} \,\{\log 4 + n \log (r_2/r_1)\}.
+\]
+
+Returning to the general case, the surface density of the
+electricity on one side of a plate is equal to
+\begin{DPgather*}
+\frac{1}{4\pi}\, \frac{d\psi}{dr}; \\
+\lintertext{but since}  \left(\frac{r}{b}\right)^n = \frac{1}{\dn(\psi,k')}, \\
+\frac{n}{b}\left(\frac{r}{b}\right)^{n-1} \frac{dr}{d\psi} = k'^2\sn(\psi,k')\cn(\psi,k')/\dn^2(\psi,k').
+\end{DPgather*}
+
+Substituting for the elliptic functions their values in terms
+of~$r$, we find when~$\phi=K$
+\[
+\frac{d\psi}{dr} = \frac{nb^nr^{n-1}}{k\{(r^{2n} - {r_1}^{2n})({r_2}^{2n} - r^{2n})\}^{\frac{1}{2}}}.
+\]
+Thus the surface density is equal to
+\[
+\frac{1}{4\pi} \,\frac{{nr_2}^nr^{n-1}}{\{(r^{2n} - {r_1}^{2n})({r_2}^{2n} - r^{2n})\}^{\frac{1}{2}}}.
+\]
+When $n=1$, this case coincides with that discussed in \artref{245}{Art.~245}.
+
+\Article{248} Let us next put
+\[
+x+\iota y = b\cn(\phi + \iota\psi),
+\]
+and take $\psi$~for the potential, and $\phi$~for the stream function.
+Then when $\psi=0$, we have
+\[
+x + \iota y = b\cn\phi,
+\]
+hence $y=0$, and $x$~can have any value between~$±b$: thus the
+%% -----File: 257.png---Folio 243-------
+equipotential surface for which $\psi$~is zero is the portion of the
+axis of~$x$ between $x=-b$, and $x=+b$. When $\psi=K'$,
+\begin{align*}
+x + \iota y &= b \cn(\phi + \iota K')\\
+&= -\frac{b\iota\dn\phi}{k\sn\phi};
+\end{align*}
+hence $x=0$, and $y$~ranges from $+bk'/k$ to~$+\infty$ and from $-bk'/k$
+to~$-\infty$. Hence the section of the equipotential surface for
+which $\psi=K'$ is the portion of the axis of~$y$ included between
+these limits. Thus the section of the conductors over which the
+distribution of electricity is given by this transformation is
+similar to that represented in \figureref{fig102}{Fig.~102}, where the axis of~$x$ is
+\emph{vertical}.
+
+\includegraphicsouter{fig102}{Fig.~102.}
+
+To find the quantity of electricity on~\smallsanscap{AB} we notice that $\phi=0$
+at~\smallsanscap{A} and is equal to~$2K$ at~\smallsanscap{B},
+hence the quantity of
+electricity on one side of~\smallsanscap{AB}
+is equal to~$K/2\pi$, thus the
+total charge on~\smallsanscap{AB} is~$K/\pi$.
+The difference of potential
+between \smallsanscap{AB}~and~\smallsanscap{CD} or~\smallsanscap{EF} is~$K'$, so that the capacity of~\smallsanscap{AB} is
+equal to
+\[
+\frac{1}{\pi}\, \frac{K}{K'}.
+\]
+The modulus~$k$ of the elliptic functions is given by the equation
+\[
+\frac{k'}{k} = \frac{\{1-k^2\}^{\frac{1}{2}}}{k} = \frac{\smallbold{EC}}{\smallbold{AB}}.
+\]
+If~\smallsanscap{AB} is very large compared with~\smallsanscap{EC} then $k$~is very nearly
+unity, and in this case we have
+\begin{align*}
+K &= \log (4/k') = \log (4\smallbold{AB}/\smallbold{EC}),\\
+K'&= \frac{\pi}{2};
+\end{align*}
+so that the capacity of~\smallsanscap{AB} is
+\[
+\frac{2}{\pi^2} \log(4\smallbold{AB}/\smallbold{EC}).
+\]
+
+The surface density of the electricity at a point~$P$ on either
+%% -----File: 258.png---Folio 244-------
+side of~\smallsanscap{AB} is (without any limitation as to the value of~$k$)
+equal to
+\[
+\frac{1}{4\pi}\, \frac{d\phi}{dx},
+\]
+and since $x = b \cn \phi$,
+\begin{align*}
+\frac{dx}{d\phi} &= -b \sn \phi \dn \phi\\
+&= -\frac{k}{b} (b^2 - x^2)^{\frac{1}{2}} \left\{\frac{k'^2}{k^2} b^2 + x^2\right\}^{\frac{1}{2}}\\
+&= -\frac{k}{b} \smallbold{CP} \sqrt{\smallbold{AP} \centerdot \smallbold{BP}};
+\end{align*}
+hence the surface density is equal to
+\[
+- \frac{b}{4\pi k}\, \frac{1}{\smallbold{CP} \sqrt{\smallbold{AP} \centerdot \smallbold{BP}}}.
+\]
+
+\Article{249} We pass on now to consider the transformation
+\[
+\epsilon^{\frac{x+\iota y}{b}} = \cn (\phi + \iota\psi),
+\]
+where $\phi$~is taken as the potential and $\psi$~as the stream function.
+
+Over the equipotential surface for which $\phi = 0$, we have
+\begin{DPgather*}
+\begin{aligned}
+\epsilon^{\frac{x+\iota y}{b}} &= \cn (\iota \psi)\\
+&= \frac{1}{\cn(\psi,k')}.
+\end{aligned} \\
+\lintertext{\indent Hence} y = 0,\quad ±\pi b,\quad ±2\pi b, \ldots;
+\end{DPgather*}
+while $x$~ranges from $0$ to infinity.
+
+For the equipotential surface for which $\phi = K$, we have
+\begin{DPgather*}
+\begin{aligned}
+\epsilon^{\frac{x+\iota y}{b}} &= \cn(K + \iota\psi)\\
+&= -\iota k' \frac{\sn(\psi,k')}{\dn(\psi,k')}.
+\end{aligned} \\
+\lintertext{\indent Hence} y = ±\tfrac{1}{2} \pi b,\quad ±\tfrac{3}{2} \pi b,\quad ±\tfrac{5}{2} \pi b \ldots,
+\end{DPgather*}
+while $x$~ranges from minus infinity to a value~$x_1$ given by the
+equation
+\[
+\epsilon^{\frac{x_1}{b}} = \frac{k'}{k}.
+\]
+
+\includegraphicsmid{fig103}{Fig.~103.}
+
+Thus this transformation gives the distribution of electricity
+%% -----File: 259.png---Folio 245-------
+\index{Capacity xof two piles of plates@\subdashone of two piles of plates|(}%
+on a pile of semi-infinite parallel plates at equal intervals $\pi b$
+apart, maintained at potential zero when in presence of another
+pile of semi-infinite parallel plates at the same distance apart
+maintained at potential~$K$, the planes of the second set of plates
+being midway between those of the first. The second set of
+plates project a distance~$x_1$ into the first set, $x_1$~being given by
+the equation~$\epsilon^{x_1/b} = k'/k$. If the edges of the second set of plates
+are outside the first set, then $x_1$~is negative and numerically
+equal to the distance between the planes containing the ends
+of the two sets of plates. The system of conductors is represented
+in \figureref{fig103}{Fig.~103}.
+
+The quantity of electricity on the two sides of one of the
+plates is~$K'/2\pi$, hence the capacity of such a plate is
+\[
+\frac{K'}{2\pi K}.
+\]
+
+If the ends of the two sets of plates are in the same plane,
+then $x_1 = 0$, and therefore $k' = k$, so that $K' = K$; hence the
+capacity of each plate is in this case~$1/(2\pi)$.
+
+When the plates do not penetrate and are separated by a
+distance which is large compared with the distance between two
+parallel plates, $x_1$~is negative and large compared with~$b$, hence $k'$~is
+small, and therefore $k$~nearly equal to unity; in this case
+\begin{align*}
+K' &= \frac{\pi}{2},\\
+K &= \log(4/k'),\\
+&=\log 4 + \frac{x'}{b},
+\end{align*}
+where $x' = -x_1$.
+
+Thus the capacity of a plate in this case is approximately
+equal to
+\[
+\frac{b}{4(b\log 4 + x')}.
+\]
+The surface density at a point on one of the first set of plates at
+%% -----File: 260.png---Folio 246-------
+a distance~$x$ from the edge is easily shewn by the methods
+previously used to be equal, whatever be the value of~$k$, to
+\[
+-\frac{1}{4\pi kb}\,\frac{\epsilon^{\frac{x}{b}}}{\sqrt{(\epsilon^{\frac{2x}{b}} - 1)(\epsilon^{\frac{2x}{b}} + \epsilon^{\frac{2x_1}{b}})}}.
+\]
+
+\Article{250} The transformation
+\[
+\left(\frac{x + \iota y}{b}\right)^n = \cn(\phi+\iota\psi),
+\]
+with $\phi$~as the potential and $\psi$~as the stream function, gives the
+solution of the case represented in \figureref{fig104}{Fig.~104}; where the $2n$~outer
+planes at potential zero are supposed
+to extend to infinity, the $2n$~inner
+planes at potential~$K$ bisect the angles
+between the outer planes, and $\smallsanscap{OA}=b$.
+
+\includegraphicsouter{fig104}{Fig.~104.}
+
+\index{Capacity xof two series of radial plates@\subdashone of two series of radial plates|(}%
+We can easily prove that in this
+case the quantity of electricity on the
+outer plates is equal to~$nK'/\pi$, so that
+the capacity of the system is equal to
+\[
+\frac{n}{\pi}\, \frac{K'}{K},
+\]
+when the modulus of the elliptic functions is determined by the
+relation
+\[
+\left(\frac{\smallbold{OC}}{\smallbold{OA}}\right)^n = \frac{k'}{k}.
+\]
+
+\Article{251} The transformation
+\[
+x + \iota y = b\dn(\phi + \iota\psi),
+\]
+\index{Capacity xof a strip between two plates@\subdashone of a strip between two plates}%
+where $\phi$~is the potential and $\psi$~the stream function, gives the
+solution of the case represented in \figureref{fig105}{Fig.~105}, in which a finite
+plate is placed in the space between two semi-infinite plates.
+For when $\phi = 0$, we have
+\begin{align*}
+x + \iota y &= b\dn\iota\psi\\
+&= b \frac{\dn(\psi,k')}{\cn(\psi,k')}; \Tag{36}
+\end{align*}
+%% -----File: 261.png---Folio 247-------
+hence $y = 0$, and $x$~ranges from $+b$ to~$+\infty$ and from $-b$ to~$-\infty$,
+thus giving the portions $\smallsanscap{EF}$,~$\smallsanscap{CD}$ of the figure.
+
+\includegraphicsmid{fig105}{Fig.~105.}
+
+When $\phi = K$, we have
+\begin{align*}
+x + \iota y &= b \dn(K + \iota \psi)\\
+&= bk' \frac{\cn(\psi,k')}{\dn(\psi,k')}; \Tag{37}
+\end{align*}
+hence $y = 0$, and $x$~ranges between~$±bk'$, thus giving the portion~$\smallsanscap{AB}$
+of the figure.
+
+The quantity of electricity on the two sides of the plate~$\smallsanscap{AB}$ is
+equal to~$K'/\pi$, hence the capacity of this plate is equal to
+\[
+\frac{1}{\pi}\, \frac{K'}{K},
+\]
+where the modulus~$k$ of the elliptic functions is given by the
+equation
+\[
+k' = \{ 1 - k^2 \}^{\frac{1}{2}} = \smallbold{OA} / \smallbold{OC}.
+\]
+When $\smallsanscap{AC}$~is small compared with~$\smallsanscap{AB}$, $k'$~is nearly equal to unity,
+and $k$~is therefore small, in this case we have approximately
+\begin{align*}
+K &= \frac{\pi}{2},\\
+K' &= \log(4/k)\\
+&= \log 4 + \tfrac{1}{2} \log \frac{\smallbold{OC}^2}{\smallbold{AC} \centerdot \smallbold{BC}};
+\end{align*}
+so that in this case the capacity of the plate~\smallsanscap{AB} is equal to
+\[
+\frac{1}{\pi^2} \left\{\log \frac{\smallbold{OC}^2}{\smallbold{AC} \centerdot \smallbold{BC}} + 2 \log 4 \right\}.
+\]
+
+Returning to the general case, the surface density of the
+electricity on one side of the plate~$\smallsanscap{AB}$ at a point~$\smallsanscap{P}$ is equal to
+\[
+\frac{1}{4 \pi}\, \frac{d\psi}{dx}.
+\]
+Using equation~(\eqnref{251}{37}) we find that this is equal to
+\[
+\frac{b}{4 \pi}\, \frac{1}{\{(b^2 - x^2)(b^2 k'^2 - x^2) \}^{\frac{1}{2}}},
+\]
+which may be written in the form
+\[
+\frac{b}{4 \pi}\, \frac{1}{\{ \smallbold{AP} \centerdot \smallbold{BP} \centerdot \smallbold{CP} \centerdot \smallbold{EP}\}^{\frac{1}{2}}}.
+\]
+%% -----File: 262.png---Folio 248-------
+
+The surface density at a point~$\smallsanscap{Q}$ on~$\smallsanscap{EF}$ may be shown in a
+similar way, using~(\eqnref{251}{36}), to be equal to
+\[
+-\frac{b}{4\pi}\,\frac{1}{\{(x^2 - b^2)(x^2 - b^2k'^2)\}^{\frac{1}{2}}},
+\]
+which is equal to
+\[
+-\frac{b}{4\pi}\,\frac{1}{\{\smallbold{AQ} \centerdot \smallbold{BQ} \centerdot \smallbold{CQ} \centerdot \smallbold{EQ}\}^{\frac{1}{2}}}.
+\]
+
+\Article{252} If we put
+\[
+\epsilon^{\frac{x+\iota y}{b}} = \dn(\phi+\iota\psi),
+\]
+and take as before $\phi$~for the potential and $\psi$~for the stream
+function, then since, when $\phi = 0$,
+\begin{align*}
+\epsilon^{\frac{x+\iota y}{b}} &= \dn(\iota\psi)\\
+&= \frac{\dn(\psi,k')}{\cn(\psi,k')},
+\end{align*}
+we have $y = 0$, $y = ±\pi b$, $y = ±2\pi b \ldots$, while $x$~ranges from~$0$ to~$+\infty$.
+Thus the equipotential surfaces for which $\phi$~vanishes are
+a pile of parallel semi-infinite plates stretching from the axis
+of~$y$ to infinity along the positive direction of~$x$, the distance
+between two adjacent plates being~$\pi b$.
+
+When $\phi = K$, we have
+\begin{align*}
+\epsilon^{\frac{x+\iota y}{b}} &= \dn(K + \iota\psi)\\
+&= k'\frac{\cn(\psi,k')}{\dn(\psi,k')};
+\end{align*}
+thus $y = 0$, $y = ±\pi b$, $y = ±2\pi b \ldots$, while $x$~ranges from~$-\infty$ to~$-x_1$,
+where $x_1$~is given by the equation
+\[
+\epsilon^{-\frac{x_1}{b}} = k'. \Tag{38}
+\]
+\includegraphicsouter{fig106}{Fig.~106.}
+Thus the equipotential surfaces for which $\phi = K$ are a pile of
+parallel semi-infinite plates stretching from~$-\infty$ to a distance $x_1$
+from the previous set of plates. The distance between adjacent
+plates in this set is again~$\pi b$, and the planes of the plates in this
+set are the continuations of those of the plates in the set at
+potential zero. This system of conductors is represented in
+\figureref{fig106}{Fig.~106}.
+%% -----File: 263.png---Folio 249-------
+
+The quantity of electricity on both sides of one of the plates at
+potential zero is~$-K'/2\pi$,
+hence the
+capacity of such a
+plate is
+\[
+\frac{1}{2\pi}\, \frac{K'}{K},
+\]
+the modulus of the elliptic functions being given by equation~(\eqnref{252}{38}).
+
+When the distance between the edges of the two sets of
+plates is large compared with the distance between two adjacent
+parallel plates, then $x_1$~is large compared with~$b$, so that $k'$~is
+small; in this case we have approximately
+\begin{align*}
+K' &= \frac{\pi}{2},\\
+K &= \log(4/k')\\
+&= \log 4 + \frac{x_1}{b};
+\end{align*}
+hence the capacity of a plate is equal to
+\[
+\frac{b}{4(x_1 + b \log 4)}.
+\]
+
+The surface density of the electricity at a point~$P$ on one of
+the planes at potential zero is in the general case easily proved
+to be equal to
+\[
+-\frac{1}{4\pi b}\,\frac{\epsilon^{\frac{x}{b}}}{\left\{(\epsilon^{\frac{2x}{b}} - 1)(\epsilon^{\frac{2x}{b}} - \epsilon^{-\frac{2x_1}{b}})\right\}^{\frac{1}{2}}}.
+\]
+
+\includegraphicsouter{fig107}{Fig.~107.}
+
+\Article{253} The transformation
+\[
+\left(\frac{x + \iota y}{b}\right)^n = \dn(\phi + \iota \psi),
+\]
+where $\phi$~is the potential and $\psi$~the stream function and $n$~a
+positive integer, gives the solution of the case shown in
+\figureref{fig107}{Fig.~107}, when the potential of the outer radial plates is zero
+and that of the inner~$K$. The $2n$~outer plates make equal angles
+with each other and extend to infinity.
+
+The quantity of electricity on both sides of one of the outer
+\index{Capacity xof two piles of plates@\subdashone of two piles of plates|)}%
+\index{Capacity xof two series of radial plates@\subdashone of two series of radial plates|)}%
+%% -----File: 264.png---Folio 250-------
+plates is $-K' / 2 \pi$; since there are $2n$~of these plates the capacity
+of the system is
+\[
+\frac{n}{\pi}\, \frac{K'}{K},
+\]
+the modulus of the Elliptic Functions being given by the equation
+\[
+k' = \{ 1 - k^2 \}^{\frac{1}{2}} = \left( \frac{\smallbold{OA}}{\smallbold{OB}} \right)^n.
+\]
+
+\Article{254} We have only considered those
+applications of elliptic function to electrostatics
+where the expression for the capacity of the electrical
+system proves to be such that it can be readily calculated in any
+special case by the aid of Legendre's Tables. There are many
+other transformations which are of great interest analytically,
+though the want of tables of the special functions involved makes
+them of less interest for experimental purposes than those we
+have considered. Thus, for example, the transformation
+\[
+x + \iota y = Z(\phi + \iota\psi),
+\]
+where $Z$~is the function introduced by Jacobi and defined by
+the equation
+\[
+Z(u) = \int^u \dn^2 u\, du - \frac{E}{K},
+\]
+if $\psi$~is the potential and $\phi$~the stream function, gives the distribution
+of electricity in the important case of a condenser formed
+by two parallel and equal plates of finite breadth.
+%% -----File: 265.png---Folio 251-------
+
+\Chapter{Chapter IV.}{Electrical Waves and Oscillations.}
+
+\Article{255} \Firstsc{The} properties of electrical systems in which the distribution
+of electricity varies periodically and with sufficient rapidity
+to call into play the effects of electric inertia, are so interesting
+and important that they have attracted a very large amount of
+attention ever since the principles which govern them were set
+forth by Maxwell in his \emph{Electricity and Magnetism}. We shall
+in this Chapter consider the theory of such vibrating electrical
+systems, while the \chapref{Chapter V.}{following Chapter} will contain an account of
+some remarkable experiments by which the properties of such
+systems have been exhibited in a very striking way.
+
+\Article{256} We shall begin by writing down the general equations
+which we shall require in discussing the transmission of electric
+disturbances through a field in which both insulators and conductors
+are present.
+
+Let $F$,~$G$,~$H$ be the components of the vector potential parallel
+to the axes of $x$,~$y$,~$z$ respectively, $P$,~$Q$,~$R$ the components of the
+electromotive intensity, and $a$,~$b$,~$c$ those of the magnetic induction
+in the same directions, let $\phi$ be the electrostatic potential,
+$\sigma$~the specific resistance of the conductor, $\mu$~and~$\mu'$ the magnetic
+\DPtypo{permeabilites}{permeabilities} of the conductor and dielectric respectively, and $K$~and~$K'$
+the specific inductive capacities of the conductor and
+dielectric respectively, then we have
+\[
+\left. \begin{aligned}
+P = -\frac{dF}{dt} - \frac{d\phi}{dx},\\
+Q = -\frac{dG}{dt} - \frac{d\phi}{dy},\\
+R = -\frac{dH}{dt} - \frac{d\phi}{dz}.
+\end{aligned}\right\} \Tag{1}
+\]
+%% -----File: 266.png---Folio 252-------
+
+We have also
+\begin{DPalign*}
+&\left.\begin{aligned}
+  \phantom{\frac{da}{dt}}\llap{$a$} &= \frac{dH}{dy} - \frac{dG}{dz},
+\end{aligned}\right.\\
+\lintertext{hence}
+&\left.\begin{aligned}
+  \frac{da}{dt} &= \frac{d}{dy} \frac{dH}{dt} - \frac{d}{dz} \frac{dG}{dt},
+\end{aligned}\right.\\
+\lintertext{\rlap{\raisebox{1.8\baselineskip}{so that }}similarly}
+\rintertext{}
+&\left.\begin{aligned}
+  \frac{da}{dt} &= \frac{dQ}{dz} - \frac{dR}{dy};\\
+  \frac{db}{dt} &= \frac{dR}{dx} - \frac{dP}{dz},\\
+  \frac{dc}{dt} &= \frac{dP}{dy} - \frac{dQ}{dx}.
+\end{aligned}\right\} \eqnlabel{\eqnart.2}\tag*{\llap{(2)}} %[TN:Compensate for difference in how DPalign handles tags]
+\end{DPalign*}
+
+If $\alpha$,~$\beta$,~$\gamma$ are the components of the magnetic force, $u$,~$v$,~$w$
+those of the total current, then (Maxwell's \textit{Electricity and Magnetism},
+Art.~607)
+\[
+\left.
+\begin{aligned}
+4 \pi u &= \frac{d \gamma}{dy} - \frac{d \beta}{dz},\\
+4 \pi v &= \frac{d \alpha}{dz} - \frac{d \gamma}{dx},\\
+4 \pi w &= \frac{d \beta}{dx} - \frac{d \alpha}{dy}.
+\end{aligned} \Tag{3}
+\right\}
+\]
+
+In the metal the total current is the sum of the conduction and
+polarization currents; the conduction current parallel to~$x$ is $P / \sigma$,
+the polarization current $\dfrac{K}{4 \pi} \dfrac{dP}{dt}$, or if $P$ varies as $\epsilon^{\iota pt}$, the
+polarization current is $\dfrac{K}{4 \pi\strut}\iota p \cdot P$. Thus the ratio of the conduction to the
+polarization current is $\dfrac{4 \pi}{K \sigma \iota p}$, and since $\sigma$ in electromagnetic
+measure is of the order $10^4$ for the commoner metals and $K$ in the
+same measure of the order $10^{-21}$, we see that unless the vibrations
+are comparable in rapidity with those of light we may
+neglect the polarization current in the metal in comparison with
+the conduction current. Thus in the conductor we have
+\[
+\frac{4 \pi}{\sigma} P = \frac{d\gamma}{dy} - \frac{d\beta}{dz} = \frac{1}{\mu} \left( \frac{dc}{dy} - \frac{db}{dz} \right),
+\]
+and therefore by (\eqnref{256}{2}) we have, assuming
+\[
+\frac{dP}{dx} + \frac{dQ}{dy} + \frac{dR}{dz} = 0,
+\]
+%% -----File: 267.png---Folio 253-------
+\begin{DPalign*}
+\lintertext{similarly}\left.
+\begin{aligned}
+\nabla^2 P &= \frac{4\pi\mu}{\sigma} \frac{dP}{dt};\\
+\nabla^2 Q &= \frac{4\pi\mu}{\sigma} \frac{dQ}{dt},\\
+\nabla^2 R &= \frac{4\pi\mu}{\sigma} \frac{dR}{dt}.
+\end{aligned}\right\}\Tag{4}
+\end{DPalign*}
+It follows from equation~(\eqnref{256}{2}) that $a$,~$b$,~$c$ satisfy equations of the
+same form.
+
+In the dielectric there is only the polarization current, the component
+of which parallel to~$x$ is $\dfrac{K'}{4\pi} \dfrac{dP}{dt}$; hence in the dielectric
+we have
+\[
+K'\frac{dP}{dt} = \frac{d\gamma}{dy} - \frac{d\beta}{dz} = \frac{1}{\mu'}\left(\frac{dc}{dy} - \frac{db}{dz}\right),
+\]
+and therefore by~(\eqnref{256}{2})
+\begin{DPalign*}
+\lintertext{similarly}\left.
+\begin{aligned}
+\nabla^2 P &= \mu'K'\frac{d^2 P}{dt^2};\\
+\nabla^2 Q &= \mu'K'\frac{d^2 Q}{dt^2},\\
+\nabla^2 R &= \mu'K'\frac{d^2 R}{dt^2}.
+\end{aligned}\right\}\Tag{5}
+\end{DPalign*}
+
+We shall suppose that the effects are periodic and of frequency~$p/2\pi$,
+so that the components of the electromotive intensity, as
+well as of the magnetic induction, will all vary as~$\epsilon^{\iota pt}$ and will
+not explicitly involve the time in any other way. We shall also
+suppose that the electric waves are travelling parallel to the
+axis of~$z$, so that the variables before enumerated will contain
+$\epsilon^{\iota mz}$ as a factor, $m$~being a quantity which it is one of the
+objects of our investigation to determine. With these assumptions
+we see that $d/dt$ may be replaced by~$\iota p$, and $d/dz$ by~$\iota m$.
+
+\Subsection{Alternating Electric Currents in Two Dimensions.}
+\index{Alternating currents, in two dimensions@\subdashtwo in two dimensions}%
+
+\Article{257} The cases relating to alternating currents which are
+of the greatest practical importance are those in which the
+currents flow along metallic wires. As the analysis, however, in
+these cases is somewhat complicated, we shall begin by considering
+the two dimensional problem, as this, though of comparatively
+small practical importance, enables us by the aid of
+simple analysis to illustrate some important properties possessed
+by alternating currents.
+%% -----File: 268.png---Folio 254-------
+
+The case we shall first consider is that of an infinite conducting
+plate bounded by the planes $x = h$, $x = -h$, immersed in
+a dielectric. We shall suppose that plane waves of electromotive
+intensity are advancing through the dielectric, and that these
+waves impinge on the plate. We shall suppose also that the
+waves fall on both sides of the plate and are symmetrical with
+respect to it. These waves when they strike against the plate
+will be reflected from it, so that there will on either side of the
+plate be systems of direct and reflected waves.
+
+Let $P$~and~$R$ denote the components of the electromotive
+intensity parallel to the axes of $x$~and~$z$ respectively, the component
+parallel to the axis of~$y$ vanishing since the case is one
+in two dimensions. Then in the dielectric the part of~$R$ due to
+the direct wave will be of the form
+\[
+B\epsilon^{\iota(mz + lx + pt)},
+\]
+while the part due to the reflected wave will be of the form
+\[
+C\epsilon^{\iota(mz - lx + pt)}.
+\]
+Thus in the dielectric on one side of the plate
+\[
+R = B\epsilon^{\iota(mz + lx + pt)} + C\epsilon^{\iota(mz - lx + pt)}.\Tag{1}
+\]
+If $V$ is the velocity with which electromagnetic disturbances
+are propagated through the dielectric, we have by equation~(\eqnref{256}{5}),
+\artref{256}{Art.~256}, since $\mu'K'= 1/V^2$,
+\begin{DPalign*}
+\frac{d^2 R}{dx^2} + \frac{d^2 R}{dz^2} &= \frac{1}{V^2} \frac{d^2 R}{dt^2},\\
+\lintertext{hence}            l^2 + m^2 &= \frac{p^2}{V^2}.
+\end{DPalign*}
+
+If $\lambda$ is the wave length of the incident wave, $\theta$~the angle
+between the normal to the wave front and the axis of~$x$, we
+have, since
+\begin{gather*}
+p = \frac{2\pi}{\lambda} V,\\
+l = \frac{2\pi}{\lambda}\cos \theta, \quad m = \frac{2\pi}{\lambda} \sin \theta.
+\end{gather*}
+
+Since $Q$ vanishes, we have
+\[
+\frac{dP}{dx} + \frac{dR}{dz} = 0.
+\]
+%% -----File: 269.png---Folio 255-------
+Substituting the value of $R$ from equation~(\eqnref{257}{1}), we find
+\[
+P = -\frac{m}{l} \{B\epsilon^{\iota(mz + lx + pt)} - C\epsilon^{\iota(mz - lx + pt)}\}. \Tag{2}
+\]
+The resultant electromotive intensity in the incident wave is
+\[
+\frac{B}{\cos \theta} \epsilon^{\iota(mz + lx + pt)},
+\]
+in the reflected wave
+\[
+\frac{C}{\cos \theta} \epsilon^{\iota(mz - lx + pt)}.
+\]
+
+Let us now consider the electromotive intensity in the conducting
+plate; in this region we have, by~(\eqnref{256}{4}), \artref{256}{Art.~256}, if $\mu$ is the
+magnetic permeability and $\sigma$~the specific resistance of the plate,
+\begin{DPgather*}
+\frac{d^2 R}{dx^2} + \frac{d^2 R}{dz^2} = \frac{4\pi\mu}{\sigma}\frac{dR}{dt},\\
+\intertext{or, since $R$ varies as $\epsilon^{\iota(mz + pt)}$,}\\
+\frac{d^2 R}{dx^2} = n^2 R,\\
+\lintertext{where} n^2 = m^2 + \frac{4\pi\mu\iota p}{\sigma}.
+\end{DPgather*}
+The solution of this, since the electromotive intensity is symmetrical
+with respect to the plane $x = 0$, is of the form
+\begin{DPgather*}
+R = A(\epsilon^{nx} + \epsilon^{-nx}) \epsilon^{\iota(mz + pt)}, \Tag{3} \\
+\lintertext{and since} \frac{dP}{dx} + \frac{dR}{dz} = 0,\\
+P = -\frac{\iota m}{n} A(\epsilon^{nx} - \epsilon^{-nx}) \epsilon^{\iota (mz + pt)}. \Tag{4}
+\end{DPgather*}
+
+If $a$,~$b$,~$c$ are the components of magnetic induction, then
+\begin{align*}
+\frac{da}{dt} &= \frac{dQ}{dz} - \frac{dR}{dy},\\
+\frac{db}{dt} &= \frac{dR}{dx} - \frac{dP}{dz},\\
+\frac{dc}{dt} &= \frac{dP}{dy} - \frac{dQ}{dx};
+\end{align*}
+hence $a = 0$, $c = 0$, and
+\begin{align*}
+b &= \frac{n^2 - m^2}{\iota pn} A(\epsilon^{nx} - \epsilon^{-nx}) \epsilon^{\iota(mz + pt)} \; \text{in the plate},\Tag{5}\\
+b &= \frac{l^2 + m^2}{lp} (B\epsilon^{\iota lx} - C\epsilon^{-\iota lx)} \epsilon^{\iota(mz + pt)} \; \text{in the dielectric}.\Tag{6}
+\end{align*}
+%% -----File: 270.png---Folio 256-------
+
+We can get the expression for the magnetic force in the
+dielectric very simply by the method given in \artref{9}{Art.~9}. In the
+incident wave the resultant electromotive intensity is
+\[
+\frac{B}{\cos \theta} \epsilon^{\iota(mz + lx + pt)},
+\]
+hence the polarization is
+\[
+\frac{K'}{4 \pi} \frac{B}{\cos \theta} \epsilon^{\iota(mz + lx + pt)},
+\]
+where $K'$ is the specific inductive capacity of the dielectric. The
+Faraday tubes are moving with the velocity~$V$, hence by equations~(\eqnref{9}{4}), \artref{9}{Art.~\DPtypo{(9)}{9}},
+the magnetic force due to their motion is
+\[
+VK'\frac{B}{\cos \theta} \epsilon^{\iota(mz + lx + pt)}.
+\]
+The magnetic induction corresponding to this magnetic force
+is equal, since $\mu'K'$ equals $1/V^2$, to
+\[
+\frac{B}{V \cos \theta} \epsilon^{\iota(mz + lx + pt)},
+\]
+which is the first term on the right in equation~(\eqnref{257}{6}). We may
+show in a similar way that the magnetic force due to the motion
+of the Faraday tubes in the reflected wave is equal to the second
+term on the right in equation~(\eqnref{257}{6}).
+
+We must now consider the conditions which hold at the
+junction of the plate and the dielectric. These may be expressed
+in many different ways: they are, however, when the conductors
+are at rest, equivalent to the conditions that the tangential
+electromotive intensity, and the tangential magnetic force,
+are continuous. Thus when $x = h$ we must have both $R$~and~$b/\mu$
+continuous. The first of these conditions gives
+\[
+A(\epsilon^{nh} + \epsilon^{-nh}) = B \epsilon^{\iota lh} + C \epsilon^{-\iota lh}, \Tag{7}
+\]
+the second
+\[
+\frac{n^2 - m^2}{\mu n} A(\epsilon^{nh} - \epsilon^{-nh}) = \frac{\iota(l^2 + m^2)}{\mu'l} (B \epsilon^{\iota lh} - C \epsilon^{-\iota lh}). \Tag{8}
+\]
+\begin{DPalign*}
+\lintertext{Since} n^2 - m^2 &= \frac{4 \pi \mu \iota  p}{\sigma},\\
+\lintertext{and} \frac{l^2 + m^2}{l} &= \frac{2 \pi}{\lambda \cos \theta},
+\end{DPalign*}
+%% -----File: 271.png---Folio 257-------
+and for all known dielectrics $\mu'$~may without sensible error be
+put equal to unity, equation~(\eqnref{257}{8}) may be written
+\[
+\frac{2p}{\sigma n} A(\epsilon^{nh} - \epsilon^{-nh}) = \frac{1}{\lambda \cos \theta} (B \epsilon^{\iota lh} - C\epsilon^{-\iota lh}). \Tag{9}
+\]
+
+From (\eqnref{257}{7}) and (\eqnref{257}{9}) we get
+\begin{align*}
+A \{ \epsilon^{nh} + \epsilon^{-nh} + \frac{2p \lambda \cos \theta}{\sigma n} (\epsilon^{nh} - \epsilon^{-nh}) \} &= 2B \epsilon^{\iota lh}, \Tag{10}\\
+A \{ \epsilon^{nh} + \epsilon^{-nh} - \frac{2p \lambda \cos \theta}{\sigma n} (\epsilon^{nh} - \epsilon^{-nh}) \} &= 2C \epsilon^{-\iota lh}. \Tag{11}
+\end{align*}
+
+It will be convenient to express $A$,~$B$,~$C$ in terms of the total
+current through the plate. If $w$ is the intensity of the current
+parallel to~$z$ in the plate, $w = R / \sigma$, hence by~(\eqnref{257}{3})
+\[
+w = \frac{A}{\sigma}(\epsilon^{nx}+\epsilon^{-nx}) \epsilon^{\iota(mz + pt)}.
+\]
+If $I_0 \epsilon^{\iota(mz + pt)}$ is the total current passing through unit width
+measured parallel to~$y$, of the plate,
+\begin{DPgather*}
+I_0 \epsilon^{\iota(mz + pt)} = \int_{-h}^{+h} w\, dx;\\
+\lintertext{hence} I_0=\frac{2A}{\sigma n}(\epsilon^{nh}-\epsilon^{-nh}), \Tag{12}\\
+\lintertext{so that} w = \tfrac{1}{2 }n I_0 \frac{(\epsilon^{nx} + \epsilon^{-nx})}{(\epsilon^{nh} - \epsilon^{-nh})} \epsilon^{\iota(mz + pt)}. \Tag{13}
+\end{DPgather*}
+
+Let us now suppose that the frequency of the vibrations is so
+small that $nh$ is a small quantity, this will be the case if
+$h \sqrt{4 \pi \mu p/\sigma}$ is small. When $nh$ and therefore $nx$ is small,
+equation~(\eqnref{257}{13}) becomes approximately
+\[
+w = \frac{I_0}{2h} \epsilon^{\iota(mz + pt)};
+\]
+thus the current in the plate is distributed uniformly across it.
+When $nh$ is small, equations~(\eqnref{257}{12}), (\eqnref{257}{10}) and~(\eqnref{257}{11}) become approximately
+\begin{gather*}
+I_0 = \frac{4Ah}{\sigma},\\
+  \begin{aligned}
+    A (1 + 4 \pi Vh \sigma^{-1} \cos \theta) &= B \epsilon^{\iota lh},\\
+    A (1 - 4 \pi Vh \sigma^{-1} \cos \theta) &= C \epsilon^{-\iota lh}.
+  \end{aligned}
+\end{gather*}
+%% -----File: 272.png---Folio 258-------
+Thus corresponding to the current $I_0 \cos (pt + mz)$ in the plate,
+we find
+\[
+\left.
+\begin{aligned}
+R &= \frac{\sigma I_0}{2h} \cos(p t +mz),\\
+P &= \frac{\sigma I_0 mx}{2h} \sin(pt + mz),\\
+b &= 4 \pi \mu I_0 \frac{x}{2h} \cos(pt + mz)
+\end{aligned}
+\right\}
+\text{ in the plate.}\\
+\]
+Thus, since $mx$~is exceedingly small, we see that the maximum
+electromotive intensity parallel to the boundary of the plate is
+exceedingly large compared with the maximum at right angles
+to it.
+
+In the dielectric we have
+\begin{multline*}
+R = \frac{\sigma I_0}{2h} \cos(pt + mz) \cos l(x - h)\\
+- 2\pi I_0 V \cos \theta \sin(pt + mz)\sin l(x - h),
+\end{multline*}
+\begin{multline*}
+P = \frac{1}{2h} \sigma I_0 \tan \theta \sin(pt + mz) \sin l(x - h)\\
+- 2\pi I_0 V \sin \theta \cos(pt + mz)\cos l(x - h),
+\end{multline*}
+\begin{multline*}
+b = -\frac{\sigma I_0}{2Vh \cos \theta} \sin(pt + mz) \sin l(x - h)\\
++ 2 \pi I_0 \cos(pt + mz) \cos l(x - h).
+\end{multline*}
+
+Thus at the surface of the plate where $x = h$
+\begin{align*}
+R &= \frac{\sigma I_0}{2h} \cos (pt + mz),\\
+P &= -2 \pi I_0 V \sin \theta \cos (pt + mz),\\
+b &= 2 \pi I_0 \cos (pt + mz).
+\end{align*}
+
+Thus at the surface of the plate $P/R = -4 \pi Vh \sigma^{-1} \sin \theta$. If
+the plate is metallic this quantity is exceedingly large unless the
+plate is excessively thin or $\theta$~very small, so that in the dielectric
+the resultant electromotive intensity at the surface of the plate is
+along the normal, this is in striking contrast to the effect inside the
+plate where $P/R$ is very small. The Faraday tubes in the dielectric
+close to the plate are thus at right angles to the plate, while
+in the plate they are parallel to it; hence by \artref{10}{Art.~10} the electric
+momentum in the dielectric close to the plate is parallel to the
+axis of~$z$, or parallel to the plate, while in the plate itself it is
+parallel to the axis of~$x$, or in the direction of motion from the
+outside of the plate to the inside. If $4 \pi Vh \cos \theta = \sigma$, then
+%% -----File: 273.png---Folio 259-------
+$C=0$; in this case there is no reflected wave; the wave reflected
+from one side of the plate is annulled by the direct wave coming
+through the plate from the other side. It is worthy of remark
+that the only one of the quantities we have considered whose
+value either in the interior of the plate or near to the plate in
+the dielectric depends sensibly upon~$\theta$, the direction of motion of
+the incident wave, is the normal electromotive intensity in the
+dielectric and in the plate.
+
+\Article{258} We shall now proceed to discuss the case when $nh$~is
+large. We shall begin by considering the distribution of current
+in the plate. We have by~(\eqnref{257}{13})
+\[
+w=\tfrac{1}{2} I_0 n \frac{(\epsilon^{nx} + \epsilon^{-nx})}{(\epsilon^{nh} + \epsilon^{-nh})}\, \epsilon^{\iota(mz + pt)},
+\]
+and since $nh$~is large this equation may be written as
+\begin{DPalign*}
+w &= \tfrac{1}{2} I_0 n\, \epsilon^{-n(h - x)}\, \epsilon^{\iota(mz + pt)}. \Tag{14}\\
+\lintertext{Now} n^2 &= m^2 + \frac{4 \pi \mu \iota p}{\sigma}\\
+&= \frac{p^2}{V^2} \sin^2 \theta + \frac{4 \pi \mu \iota p}{\sigma}.
+\end{DPalign*}
+Now $p^2/V^2$ is very small compared with $4 \pi \mu p / \sigma$ if the plate
+conducts as well as a metal, unless the vibrations are quicker
+than those of light. When the current makes a million vibrations
+per second $(p^2/V^2) / (4 \pi \mu p / \sigma)$ is approximately $5 × 10^{-16} (\sigma / \mu)$,
+and is thus excessively small unless the resistance is enormously
+greater than that of acidulated water; we may therefore without
+appreciable error write
+\begin{DPgather*}
+n^2 = \frac{4 \pi \mu \iota p}{\sigma},\\
+\lintertext{and} n = \sqrt{2 \pi \mu p / \sigma} (1 + \iota) = n_1(1 + \iota) \text{ say, where}\\
+n_1 = \sqrt{2 \pi \mu p/\sigma}.
+\end{DPgather*}
+
+Substituting this value for~$n$, and taking the real part of~(\eqnref{258}{14}),
+we have
+\[
+w = \sqrt{\pi \mu p / \sigma} I_0 \epsilon^{-n_1(h - x)} \cos \left\{mz - n_1(h - x) + pt +\frac{\pi}{4} \right\}.
+\]
+The presence of the factor $\epsilon^{-n_1(h - x)}$ in this expression shows
+that the current diminishes in geometrical progression as $h - x$
+increases in arithmetical progression, and that it will practically
+%% -----File: 274.png---Folio 260-------
+\index{Alternating currents, flow to surface of conductors@\subdashtwo flow to surface of conductors}%
+\index{Concentration of alternating current on the outside of a conductor}%
+\index{Electric skin@\subdashone `skin'}%
+\index{Skin@`Skin', electrical}%
+vanish as soon as $n_1(h - x)$ is a small multiple of unity. We thus
+get the very interesting result that an alternating current does not
+distribute itself uniformly over the cross-section of the conductor
+through which it is flowing, but concentrates itself towards the
+outside of the conductor. When the vibrations are very rapid
+the currents are practically confined to a thin skin on the outside
+of the conductor. The thickness of this skin will diminish
+as $n_1$~increases; we shall take $1/n_1$ as the measure of its
+thickness.
+
+This inequality in the distribution of alternating currents is
+explicitly stated in Art.~690 of Maxwell's \textit{Electricity and
+Magnetism}, but its importance was not recognised until it was
+brought into prominence and its consequences developed by the
+\index{Heaviside, xconcentration of current@\subdashone concentration of current}%
+\index{Hughes, concentration of alternating current}%
+\index{Rayleigh, Lord, concentration of alternating current@\subdashtwo concentration of alternating current}%
+investigations of Mr.~Heaviside and Lord Rayleigh, and the experiments
+of Professor Hughes.
+
+The amount of this concentration is very remarkable in the
+magnetic metals, even for comparatively slow rates of alternation
+of the current. Let us for example take the case of a current
+making $100$~vibrations per second, and suppose that the plate
+is made of soft iron for which we may put $\mu = 10^3$, $\sigma = 10^4$. In
+this case $p = 2 \pi × 10^2$, and $n_1$ or $\{2 \pi \mu p / \sigma \}^{\frac{1}{2}}$ is approximately
+equal to~$20$; thus at a depth of half a millimetre from the surface
+of such a plate, the maximum intensity of the current will only
+be~$1 / \epsilon$ or $.368$~of its value at the surface. At the depth of $1$~millimetre
+it will only be~$.135$, at $2$~millimetres $.018$, and at $3$~millimetres
+$.0025$, or the $1/400$~part of its value at the exterior.
+Thus in such a plate, with the assigned rate of alternations, the
+currents will practically cease at the depth of about $2$~mm.\ and will
+be reduced to about $1/7$~of their value at the depth of one
+millimetre. Thus in this case the currents, and therefore the
+magnetic force, are confined to a layer not more than $3$~millimetres
+thick.
+
+The thickness of the `skin' for copper is about $13$~times that
+for soft iron.
+
+The preceding results apply to currents making $100$~vibrations
+per second; when we are dealing with such alternating
+currents as are produced by the discharge of a Leyden Jar,
+where there may be millions of alternations per second, the
+thickness of the `skin' in soft iron is often less than the
+hundredth part of a millimetre.
+%% -----File: 275.png---Folio 261-------
+
+\index{Momentum of Faraday tubes}%
+Returning to the determination of $P$,~$R$, and~$b$ for this case,
+we find from equations (\eqnref{257}{3}),~(\eqnref{257}{4}), and~(\eqnref{257}{12}) in the plate
+\begin{align*}
+R &= \{\pi \mu p \sigma \}^{\frac{1}{2}} I_0 \epsilon^{-n_1(h - x)} \cos \left\{mz + n_1(x - h) + pt + \frac{\pi}{4} \right\},\\
+P &= \tfrac{1}{2} m \sigma I_0 \epsilon^{-n_1(h - x)} \cos \left\{mz + n_1(x - h) + pt - \frac{\pi}{2} \right\},\\
+b &= 2 \pi \mu I_0 \epsilon^{-n_1(h - x)} \cos \left\{mz + n_1(x - h) + pt \right\}.
+\end{align*}
+Thus we see that in this case, as well as when $nh$~was small, $P/R$
+is in general very small, so that the resultant electromotive
+intensity is nearly parallel to the surface of the plate.
+
+In the dielectric we have by equations (\eqnref{257}{10}),~(\eqnref{257}{11}), and~(\eqnref{257}{12})
+when $nh$~is large;
+\begin{multline*}
+R = \{\pi \mu p \sigma \}^{\frac{1}{2}} I_0 \cos \left\{mz + pt + \frac{\pi}{4} \right\} \cos l(x - h)\\
+-2 \pi V \cos \theta I_0 \sin(mz + pt) \sin l(x - h),
+\end{multline*}
+\begin{multline*}
+P = \{\pi \mu p \sigma \}^{\frac{1}{2}} \tan \theta I_0 \sin \left\{mz + pt + \frac{\pi}{4} \right\} \sin l(x - h)\\
+-2 \pi V \sin \theta I_0 \cos(mz + pt) \cos l(x - h),
+\end{multline*}
+\begin{multline*}
+b = 2 \pi I_0 \cos(mz + pt) \cos l(x - h)\\
+-\frac{1}{V} \{\pi \mu p \sigma \}^{\frac{1}{2}} \sec \theta I_0 \sin \left( mz + pt + \frac{\pi}{4} \right) \sin l(x - h).
+\end{multline*}
+At the surface of the plate these become
+\begin{align*}
+R &= \{\pi \mu p \sigma \}^{\frac{1}{2}} I_0 \cos \left( mz + pt + \frac{\pi}{4} \right),\\
+P &= -2 \pi V \sin \theta I_0 \cos(mz + pt),\\
+b &= 2 \pi I_0 \cos (mz + pt),
+\end{align*}
+and we see, as before, that in general $P/R$ is very large, so that
+the electromotive intensity near the plate in the dielectric is
+approximately at right angles to it.
+
+Thus, as in the case of the slower vibrations, the momentum is
+tangential in the dielectric and normal in the plate.
+
+If we compare the expressions for the components of the
+electromotive intensity in the dielectric given above with those
+given in the preceding \artref{257}{article}, we see that, except close to the
+plate, they are very approximately the same.
+%% -----File: 276.png---Folio 262-------
+
+\Subsection{Periodic Currents in Cylindrical Conductors, and the Rate
+of Propagation of Electric Disturbances along them.}
+\index{Alternating currents, in wires@\subdashtwo in wires}%
+
+\Article{259} We shall now proceed to consider the case which is most
+easily realized in practice, that in which electrical disturbances
+are propagated along long straight cylindrical wires, such for
+example as telegraph wires or sub-marine cables.
+
+A peculiar feature of electrical problems in which infinitely
+long straight cylinders play a part, is the effect produced by the
+presence of other conductors, even though these are such a long
+way off that it might have appeared at first sight that their
+influence could have been neglected. This is exemplified by the
+well-known formula for the capacity of two coaxial cylinders.
+If $\smallbold{a}$~and~$\smallbold{b}$ are the radii of the two cylinders the capacity per
+unit length is proportional to~$1/\log(\smallbold{b}/\smallbold{a})$. Thus, even though
+the cylinders were so far apart that the radius of the outer
+cylinder was $100$~times that of the inner, yet if the distance
+were further increased until the outer radius was $10,000$~times
+the inner, the capacity of the condenser would be halved, though
+similar changes in the distances between concentric spheres
+would hardly have affected their capacity to an appreciable extent.
+For this reason we shall, though it involves rather more complex
+analysis, suppose that our cylinder is surrounded by conductors,
+and the results we shall obtain will enable us to determine
+when the effects due to the conductors can legitimately be
+neglected.
+
+\Article{260} The case we shall investigate is that of a cylindrical
+metallic wire surrounded by a dielectric, while beyond the
+dielectric we have another conductor; the dielectric is bounded
+by concentric cylinders whose inner and outer radii are $\smallbold{a}$~and~$\smallbold{b}$
+respectively. If $\smallbold{b}/\smallbold{a}$~is a very large quantity, we have a case
+approximating to an aerial telegraph wire, while when $\smallbold{b}/\smallbold{a}$~is
+not large the case becomes that of a sub-marine cable.
+
+In the dielectric between the conductors there are convergent
+and divergent waves of Faraday tubes, the incidence of which on
+the conductors produces the currents through them.
+
+\Article{261} We shall take the axis of the cylinders as the axis of~$z$,
+and suppose that the electric field is symmetrical round this axis;
+then if the components of the electric intensity and magnetic
+%% -----File: 277.png---Folio 263-------
+induction vary as $\epsilon^{\iota(mz + pt)}$, the differential equations by which
+these quantities are determined are of the form
+\[
+\frac{d^2f}{dr^2} + \frac{1}{r}\, \frac{df}{dr} - n^2f = 0,
+\]
+where $r$~denotes the distance of a point from the axis of~$z$. The
+complete solution of this equation is expressed by
+\[
+f = AJ_0(\iota nr) + BK_0(\iota nr)\footnotemark. \Tag{1}
+\]
+
+\footnotetext{Heine, \textit{Kugelfunctionen}, vol.~i.\ p.~189.}%
+\index{Bessel's functions, values of, when variable, is small or large}%
+\index{Functions, Bessel's}%
+Here $J_0(x)$~represents Bessel's function of zero order, and
+\[
+K_0(x) = (C + \log 2 - \log x)J_0(x) + 2 \{J_2(x)-\tfrac{1}{2}J_4(x)
++ \tfrac{1}{3}J_6(x) \ldots \}; \Tag{2}
+\]
+where $C$ is Gauss' constant, and is equal to $.5772157 \ldots \ldots$.
+
+When the real part of~$\iota n$ is finite, $J_0(\iota nr)$~is infinite when $r$~is
+\index{Heine@Heine, \textit{Kügelfunctionen}}%
+infinite (Heine, \textit{Kugelfunctionen}, vol.~i.\ p.~248), so that in any
+region where $r$~may become infinite we must have $A = 0$ in
+equation~(\eqnref{261}{1}). Again, $K_0(\iota nr)$ becomes infinite when $r$~vanishes,
+so that in any region in which $r$~can vanish $B = 0$.
+
+We shall find the following approximate equations very useful
+in our subsequent work.
+
+When $x$~is small
+\begin{DPgather*}
+J_0(\iota x) = 1,\qquad\qquad J_0'(\iota x) = -\tfrac{1}{2}\iota x.\\
+K_0(\iota x) = \log \frac{2 \gamma}{\iota x},\qquad\qquad K_0'(\iota x) = -\frac{1}{\iota x};\\
+\lintertext{where} \log \gamma = .5772157 \ldots,\\
+\lintertext{\rlap{and $J_0'(\iota x)$ is written for}}
+\frac{dJ_0(\iota x)}{d(\iota x)}.
+\end{DPgather*}
+
+When $x$~is very large
+\begin{gather*}
+J_0(\iota x) = \frac{\epsilon^x}{\sqrt{2 \pi x}}; \qquad\qquad J_0'(\iota x) = -\frac{\iota \epsilon^x}{\sqrt{2 \pi x}}.\\
+K_0(\iota x) = \epsilon^{-x} \sqrt{\frac{\pi}{2x}}; \qquad\qquad K_0'(\iota x)=\iota \epsilon^{-x} \sqrt{\frac{\pi}{2x}}.
+\end{gather*}
+(See Heine, \textit{Kugelfunctionen}, vol.~i.\ p.~248).\nbpagebreak[1]
+
+\Article{262} We shall now proceed to apply these results to the investigation
+of the propagation of electric disturbances along the
+%% -----File: 278.png---Folio 264-------
+wire. The axis of the wire is taken as the axis of~$z$; $P$,~$Q$,~$R$ are
+the components of the electromotive intensity parallel to the
+axes of $x$,~$y$,~$z$ respectively; $a$,~$b$,~$c$ are the components of the
+magnetic induction parallel to these axes: $\mu$,~$\sigma$ are respectively
+the magnetic permeability and specific resistance of the wire, $\mu'$,~$\sigma'$
+the values of the same quantities for the external conductor, $K$~is
+the specific inductive capacity of the dielectric between the
+wire and the outer conductor. We shall suppose that the magnetic
+permeability of the dielectric is unity, and that $V$~is the
+velocity of propagation of electromagnetic action through this
+dielectric. We shall begin by considering the equations which
+hold in the dielectric: it is from this region that the Faraday
+tubes come which produce the currents in the conductor. We
+shall assume, as before, that the components of the electromotive
+intensity vary as~$\epsilon^{\iota(mz+pt)}$.
+
+The differential equation satisfied by~$R$, the $z$~component of
+the electromotive intensity in the dielectric, is (\artref{256}{Art.~256})
+\[
+\frac{d^2R}{dx^2} + \frac{d^2R}{dy^2} + \frac{d^2R}{dz^2} = \frac{1}{V^2}\, \frac{d^2R}{dt^2},
+\]
+or, since $R$~varies as~$\epsilon^{\iota(mz+pt)}$,
+\begin{DPgather*}
+\Tag{3} \frac{d^2R}{dx^2} + \frac{d^2R}{dy^2} - k^2R = 0,\\
+\lintertext{where } k^2 = m^2 - \frac{p^2}{V^2}.
+\end{DPgather*}
+
+If we introduce cylindrical coordinates $r, \theta, z$, this equation
+may be written
+\[
+\frac{d^2R}{dr^2} + \frac{1}{r}\, \frac{dR}{dr} + \frac{1}{r^2}\, \frac{d^2R}{d\theta^2} - k^2R = 0;
+\]
+but since the electric field is symmetrical about the axis of~$z$, $R$~is
+independent of~$\theta$, hence this equation becomes
+\[
+\frac{d^2R}{dr^2} + \frac{1}{r}\, \frac{dR}{dr} - k^2R = 0,
+\]
+the solution of which by \artref{261}{Art.~261} is, $C$~and~$D$ being constants,
+\[
+\Tag{4}  R = \{CJ_0 (\iota kr) + DK_0 (\iota kr)\} \epsilon^{\iota(mz+pt)}.
+\]
+
+Both the $J$~and~$K$ functions have to be included, as $r$~can
+neither vanish nor become infinite in the dielectric. This equation
+%% -----File: 279.png---Folio 265-------
+indicates the presence of converging and diverging waves of
+Faraday tubes in the dielectric. If the currents in the wire are
+in planes through the axis of~$z$, and if $S$~is the component of the
+electromotive intensity along~$r$, then
+\[
+P = S \frac{x}{r},\qquad Q = S \frac{y}{r};
+\]
+hence, since $S$~is a function of $r$,~$z$, and~$t$, and not of~$\theta$, we may
+write
+\[
+P = \frac{d\chi}{dx},\qquad  Q = \frac{d\chi}{dy}; \Tag{5}
+\]
+where $\chi$~is a function we proceed to determine. Since $P$~and~$Q$
+satisfy equations of the form
+\begin{DPgather*}
+\lintertext{\rlap{we have}\rlap{\raisebox{-1.8\baselineskip}{\indent But}}}
+\left.\begin{gathered}
+\frac{d^2 P}{dx^2} + \frac{d^2 P}{dy^2} - k^2P = 0,\\
+\frac{d^2 \chi}{dx^2} + \frac{d^2 \chi}{dy^2} - k^2 \chi = 0.\\
+\frac{dP}{dx} + \frac{dQ}{dy} + \frac{dR}{dz} = 0,
+\end{gathered}
+\right\} \Tag{6}
+\end{DPgather*}
+so that by equations (\eqnref{262}{5})~and~(\eqnref{262}{6})
+\[
+k^2 \chi + \frac{dR}{dz} = 0.
+\]
+We thus have the following expressions for $P$,~$Q$,~$R$,
+\[
+\left.
+\begin{aligned}
+P &= -\frac{\iota m}{k^2}\, \frac{d}{dx} \{CJ_0(\iota kr) + DK_0(\iota kr) \} \epsilon^{\iota(mz + pt)},\\
+Q &= -\frac{\iota m}{k^2}\, \frac{d}{dy} \{CJ_0(\iota kr) + DK_0(\iota kr) \} \epsilon^{\iota(mz + pt)},\\
+R &= \phantom{-\frac{\iota m}{k^2}\, \frac{d}{dy}} \{CJ_0(\iota kr) + DK_0(\iota kr) \} \epsilon^{\iota(mz + pt)}.
+\end{aligned}
+\right\} \Tag{7}
+\]
+
+To find $a$,~$b$,~$c$, the components of the magnetic induction, we
+have
+\begin{align*}
+\frac{da}{dt} &= \frac{dQ}{dz} - \frac{dR}{dy},\\
+\frac{db}{dt} &= \frac{dR}{dx} - \frac{dP}{dz},\\
+\frac{dc}{dt} &= \frac{dP}{dy} - \frac{dQ}{dx}.
+\end{align*}
+%% -----File: 280.png---Folio 266-------
+
+From these equations we find
+\[
+\left.
+\begin{aligned}
+a &= \frac{(m^2 - k^2)}{\iota pk^2}\, \frac{d}{dy}  \{CJ_0(\iota kr) + DK_0(\iota kr) \} \epsilon^{\iota(mz + pt)},\\
+b &= - \frac{(m^2 - k^2)}{\iota pk^2}\, \frac{d}{dx} \{CJ_0(\iota kr) + DK_0(\iota kr) \} \epsilon^{\iota(mz + pt)},\\
+c &= 0;
+\end{aligned}
+\right\} \Tag{8}
+\]
+thus the resultant magnetic induction is equal to
+\[
+\frac{m^2 - k^2}{\iota pk^2}\, \frac{d}{dr} \{CJ_0(\iota kr) + DK_0(\iota kr) \} \epsilon^{\iota(mz + pt)},
+\]
+and the lines of magnetic force are circles with their centres
+along the axis of~$z$ and their planes at right angles to it.
+
+We now proceed to consider the wire. The differential equation
+satisfied by~$R$ in the wire is
+\[
+\frac{d^2 R}{dx^2} + \frac{d^2 R}{dy^2} + \frac{d^2  R}{dz^2} = \frac{4 \pi \mu}{\sigma}\, \frac{dR}{dt}.
+\]
+
+Transforming this equation to cylindrical coordinates it becomes,
+since $R$~is independent of~$\theta$,
+\begin{DPgather*}
+\frac{d^2 R}{dr^2} + \frac{1}{r}\, \frac{dR}{dr} - n^2 R = 0,\\
+\lintertext{where, as usual,} n^2 = m^2 + \frac{4 \pi \mu \iota p}{\sigma}.
+\end{DPgather*}
+
+Since $r$~can vanish in the wire, the solution of this equation is
+\[
+R = AJ_0 (\iota nr)\epsilon^{\iota(mz + pt)},
+\]
+where $A$~is a constant.
+
+We can deduce the expressions for $P$~and~$Q$ from~$R$ in the
+same way as for the dielectric, and we find
+\[
+\left.
+\begin{aligned}
+P &= - \frac{\iota m}{n^2}\, A\, \frac{d}{dx} J_0 (\iota nr)\epsilon^{\iota(mz + pt)},\\
+Q &= - \frac{\iota m}{n^2}\, A\, \frac{d}{dy} J_0 (\iota nr)\epsilon^{\iota(mz + pt)},\\
+R &= \phantom{- \frac{\iota m}{n^2}\,}A\, J_0 (\iota nr)\epsilon^{\iota(mz + pt)};
+\end{aligned}
+\right\} \Tag{9}
+\]
+and also
+\[
+\left.
+\begin{aligned}
+a &= \frac{m^2 - n^2}{\iota pn^2}\, A\, \frac{d}{dy} J_0(\iota nr)\epsilon^{\iota(mz + pt)},\\
+b &= - \frac{m^2 - n^2}{\iota pn^2}\, A\, \frac{d}{dx} J_0(\iota nr)\epsilon^{\iota(mz + pt)}, \\
+c &= 0.
+\end{aligned}
+\right\} \Tag{10}
+\]
+%% -----File: 281.png---Folio 267-------
+
+The resultant magnetic induction is at right angles to $r$~and~$z$
+and equal to
+\[
+\frac{m^2-n^2}{\iota pn^2}\, A\, \frac{d}{dr} J_0 (\iota nr)\epsilon^{\iota(mz+pt)}.
+\]
+
+In the outer conductor the differential equations are of the
+same form, but their solution will be expressed by the $K$~functions
+and not by the~$J$'s, since $r$~can be infinite in the outer conductor.
+We find if
+\[
+n'^2 = m^2 + \frac{4\pi\mu'\iota p}{\sigma'},
+\]
+that in the outer conductor, $E$~being a constant,
+\begin{gather*}
+\left. \begin{aligned}
+P &= -\frac{\iota m}{n'^2}\, E\, \frac{d}{dx} K_0 (\iota n'r)\epsilon^{\iota(mz+pt)},\\
+Q &= -\frac{\iota m}{n'^2}\, E\, \frac{d}{dy} K_0 (\iota n'r)\epsilon^{\iota(mz+pt)}, \\
+R &= E\, K_0 (\iota n'r)\epsilon^{\iota(mz+pt)};
+\end{aligned}\right\} \Tag{11} \\
+%
+\left. \begin{aligned}
+a &=  \frac{m^2-n'^2}{\iota pn'^2}\, E\, \frac{d}{dy} K_0 (\iota n'r)\epsilon^{\iota(mz+pt)}, \\
+b &= -\frac{m^2-n'^2}{\iota pn'^2}\, E\, \frac{d}{dx} K_0 (\iota n'r)\epsilon^{\iota(mz+pt)}, \\
+c &= 0.
+\end{aligned}\right\} \Tag{12}
+\end{gather*}
+
+The resultant magnetic induction is equal to
+\[
+\frac{m^2-n'^2}{\iota pn'^2}\, E\, \frac{d}{dr} K_0 (\iota n'r)\epsilon^{\iota(mz+pt)}.
+\]
+
+The boundary conditions at the surfaces of separation of the
+dielectric and the metals are (1)~that the electromotive intensity
+parallel to the surface of separation is continuous, (2)~that the
+magnetic \emph{force} parallel to the surface is also continuous. Hence
+if $\smallbold{a}$,~$\smallbold{b}$ are respectively the inner and outer radii of the layer of
+dielectric, condition~(1) gives
+\[
+\left. \begin{aligned}
+AJ_0 (\iota n\smallbold{a}) &= CJ_0 (\iota k\smallbold{a}) + DK_0 (\iota k\smallbold{a}),\\
+EK_0 (\iota n'\smallbold{b}) &= CJ_0(\iota k\smallbold{b}) + DK_0 (\iota k\smallbold{b}).
+\end{aligned}\right\} \Tag{13}
+\]
+
+\sloppy
+Condition~(2) gives, writing $J_0'(x)$ for~$dJ_0(x)/dx$, and $K_0'(x)$
+for~$dK_0(x)/dx$, and substituting for $m^2-n^2$, $m^2-k^2$, $m^2-n'^2$ the
+values $-4 \pi\mu\iota p/\sigma$, $p^2/V^2$, $-4\pi\mu'\iota p/\sigma'$ respectively,
+\[
+\left. \begin{aligned}
+&\frac{4\pi\iota}{\sigma n}\, AJ_0' (\iota n\smallbold{a}) = - \frac{p}{V^2k} \{CJ_0'(\iota k\smallbold{a}) + DK_0' (\iota k\smallbold{a})\},\\
+&\frac{4\pi\iota}{\sigma'n'}\, EK_0' (\iota n'\smallbold{b}) = - \frac{p}{V^2k} \{CJ_0'(\iota k\smallbold{b}) + DK_0' (\iota k\smallbold{b})\}.
+\end{aligned}\right\} \Tag{14}
+\]
+%% -----File: 282.png---Folio 268-------
+
+\fussy
+Eliminating $A$~and~$E$ from equations (\eqnref{262}{13})~and~(\eqnref{262}{14}), we get
+\begin{multline*}
+C\left(\frac{4\pi\iota}{\sigma n}\, J_0'(\iota n\smallbold{a})\, J_0(\iota k\smallbold{a}) + \frac{p}{V^2k}\, J_0(\iota n\smallbold{a})\, J_0'(\iota k\smallbold{a})\right) \\
+ + D\left(\frac{4\pi\iota}{\sigma n}\, J_0'(\iota n\smallbold{a})\, K_0(\iota k\smallbold{a}) + \frac{p}{V^2k}\, J_0(\iota n\smallbold{a})\, K_0'(\iota k\smallbold{a})\right) = 0,
+\end{multline*}
+\begin{multline*}
+C\left(\frac{4\pi\iota}{\sigma'n'}\, K_0'(\iota n'\smallbold{b})\, J_0(\iota k\smallbold{b})\, + \frac{p}{V^2k}\, K_0(\iota n'\smallbold{b})\, J_0'(\iota k\smallbold{b})\right) \\
+ + D\left(\frac{4\pi\iota}{\sigma'n'}\, K_0'(\iota n'\smallbold{b})\, K_0(\iota k\smallbold{b})\, + \frac{p}{V^2k}\, K_0(\iota n'\smallbold{b})\, K_0'(\iota k\smallbold{b})\right) = 0.
+\end{multline*}
+Eliminating $C$~and~$D$ from these equations, we get
+\begin{multline*}
+\left(\frac{4\pi\iota}{\sigma n}\, J_0'(\iota n\smallbold{a})\, J_0(\iota k\smallbold{a}) + \frac{p}{V^2k}\, J_0(\iota n\smallbold{a})\, J_0'(\iota k\smallbold{a})\right) ×\\
+\left(\frac{4\pi\iota}{\sigma'n'}\, K_0'(\iota n'\smallbold{b})\, K_0(\iota k\smallbold{b}) + \frac{p}{V^2k}\, K_0(\iota n'\smallbold{b})\, K_0'(\iota k\smallbold{b})\right)
+\end{multline*}
+\begin{multline*}
+= \left(\frac{4\pi\iota}{\sigma n}\, J_0'(\iota n\smallbold{a})\, K_0(\iota k\smallbold{a}) + \frac{p}{V^2k}\, J_0(\iota n\smallbold{a})\, K_0'(\iota k\smallbold{a})\right) ×\\
+  \left(\frac{4\pi\iota}{\sigma'n'}\, K_0'(\iota n'\smallbold{b})\, J_0(\iota k\smallbold{b}) + \frac{p}{V^2k}\, K_0(\iota n'\smallbold{b})\, J_0'(\iota k\smallbold{b})\right). \Tag{15}
+\end{multline*}
+
+This equation gives the relation between the wave length
+$2 \pi/m$ along the wire and the frequency $p/2\pi$ of the vibration. To
+simplify this equation, we notice that $k\smallbold{a}$,~$k\smallbold{b}$ are both very small
+quantities, for, as we shall subsequently find, $k$, when the electrical
+waves are very long, is inversely proportional to the wave length,
+while when the waves are short $k$~is small compared with the
+reciprocal of the wave length; we may therefore assume that
+when the waves transmitted along the cable are long compared
+with its radii, $k\smallbold{a}$~and~$k\smallbold{b}$ are very small. But in this case we
+have approximately,
+\begin{gather*}
+J_0(\iota k\smallbold{a}) = 1,\quad J_0(\iota k\smallbold{b}) = 1,\\
+J_0'(\iota k\smallbold{a}) = -\tfrac{1}{2}\iota k\smallbold{a},\quad J_0'(\iota k\smallbold{b}) = -\tfrac{1}{2}\iota k\smallbold{b};\\
+K_0(\iota k\smallbold{a}) = \log \frac{2\gamma}{\iota k\smallbold{a}},\quad K_0(\iota k\smallbold{b}) = \log \frac{2\gamma}{\iota k\smallbold{b}},\\
+K_0'(\iota k\smallbold{a}) = -\frac{1}{\iota k\smallbold{a}},\quad K_0'(\iota k\smallbold{b}) = -\frac{1}{\iota k\smallbold{b}}.
+\end{gather*}
+%% -----File: 283.png---Folio 269-------
+Making these substitutions, equation~(\eqnref{262}{15}) reduces to
+\begin{align*}
+k^2&=-\frac{p}{4\pi V^2}\biggl[\sigma n\Bigl(\frac{1}{\smallbold{a}}+\tfrac{1}{2} k^2 \smallbold{a} \log {\frac{2\gamma}{\iota k\smallbold{b}}}\Bigr)\frac{J_0(\iota n\smallbold{a})}{J'_0(\iota n\smallbold{a})}\\
+&\qquad-\sigma' n'\Bigl(\frac{1}{\smallbold{b}}+\tfrac{1}{2}k^2\smallbold{b} \log {\frac{2\gamma}{\iota k\smallbold{a}}}\Bigr)\frac{K_0(\iota n'\smallbold{b})}{K'_0(\iota n'\smallbold{b})}\\
+&\qquad-\frac{p}{8\pi V^2}\, \frac{\sigma n}{\smallbold{a}}\, \frac{\sigma'n'}{\smallbold{b}}\, (\smallbold{b}^2-\smallbold{a}^2)\, \frac{J_0(\iota n\smallbold{a})K_0(\iota n'\smallbold{b})}{J'_0(\iota n\smallbold{a})K'_0(\iota n'\smallbold{b})}\biggr]\frac{1}{\log(\smallbold{b}/\smallbold{a})}.
+\end{align*}
+
+Now since both $k\smallbold{a}$~and~$k\smallbold{b}$ are very small,
+\[
+k^2 \smallbold{a}^2 \log {\frac{2\gamma}{\iota k\smallbold{b}}},\quad k^2 \smallbold{b}^2 \log{\frac{2\gamma}{\iota k\smallbold{a}}}
+\]
+will be exceedingly small quantities unless $\smallbold{a}$~is so much smaller
+than~$\smallbold{b}$ that $\log(2\gamma/\iota k\smallbold{a})$ is comparable with~$1/k^2\smallbold{b}^2$. This would
+require such a disproportion between $\smallbold{b}$~and~$\smallbold{a}$ as to be scarcely
+realizable in practice on a planet of the size of the earth; we
+may therefore write the preceding equation in the form
+\begin{align*}
+k^2&=-\frac{\iota p^2}{V^2}\biggl[\frac{\mu}{n\smallbold{a}}\, \frac{J_0(\iota n\smallbold{a})}{J'_0(\iota n\smallbold{a})} - \frac{\mu'}{n'\smallbold{b}}\, \frac{K_0(\iota n'\smallbold{b})}{K'_0(\iota n'\smallbold{b})}\\
+&-\tfrac{1}{2} \iota \frac{p^2}{V^2}(\smallbold{b}^2-\smallbold{a}^2)\, \frac{\mu}{n\smallbold{a}}\, \frac{\mu'}{n'\smallbold{b}}\, \frac{J_0(\iota n\smallbold{a})}{J'_0(\iota n\smallbold{a})}\, \frac{K_0(\iota n'\smallbold{b})}{K'_0(\iota n'\smallbold{b})}\biggr] \frac{1}{\log(\smallbold{b}/\smallbold{a})} \cdots, \Tag{16}
+\end{align*}
+where we have put $n^2 = 4\pi \mu \iota p/\sigma$, $n'^2 = 4\pi \mu' \iota p/\sigma'$. We showed in
+\artref{258}{Art.~258} that we were justified in doing this when the electrical
+vibrations are not so rapid as to be comparable in frequency
+with those of light.
+
+We see from~(\eqnref{262}{16}) that $k^2$~is given by an equation of the form
+\[
+k^2 = - \frac{\iota p^2}{V^2}\left(\xi - \eta - \tfrac{1}{2} \frac{\iota p^2}{V^2} (\smallbold{b}^2-\smallbold{a}^2)\xi \eta\right).    \Tag{16*}
+\]
+
+We remark that for all electrical oscillations whose wave lengths
+are large compared with the radii of the cable, $p^2 (\smallbold{b}^2-\smallbold{a}^2)/V^2$ is
+an exceedingly small quantity, since it is of the order $(\smallbold{b}^2-\smallbold{a}^2)/\lambda^2$,
+where $\lambda$~is the length of the electrical wave.
+
+In equation~(\eqnref{262}{16*}) we see that we can neglect the third term
+inside the bracket as long as both $\xi$~and~$\eta$ are small compared
+with $2V^2/p^2(\smallbold{b}^2-\smallbold{a}^2)$.
+\begin{DPgather*}
+\lintertext{\indent Now}     \xi = \frac{\mu}{n\smallbold{a}}\, \frac{J_0(\iota n\smallbold{a})}{J'_0(\iota n\smallbold{a})},
+\end{DPgather*}
+so that the large values of~$\xi$ occur when $n\smallbold{a}$~is small; and in this
+%% -----File: 284.png---Folio 270-------
+case, substituting the approximate values for $J_0$~and~$J_0'$, we see
+that
+\[
+\xi=-\frac{2\mu}{\iota n^2\smallbold{a}^2}=\frac{\sigma}{2\pi p\smallbold{a}^2}=\frac{V^2}{4\pi p^2(\smallbold{b}^2-\smallbold{a}^2)}\, \frac{2\sigma p}{V^2}\, \frac{\smallbold{b}^2-\smallbold{a}^2}{\smallbold{a}^2}.
+\]
+
+Now for cables of practicable dimensions and materials conveying
+oscillations slower than those of light $2\sigma p(\smallbold{b}^2-\smallbold{a}^2)/V^2\smallbold{a}^2$
+is an exceedingly small quantity, so that for such cases $\xi$~is very
+small compared with $2V^2/p^2(\smallbold{b}^2-\smallbold{a}^2)$.
+
+\begin{DPgather*}
+\lintertext{\indent Again,} \eta = \frac{\mu'}{n'\smallbold{b}}\, \frac{K_0(\iota n'\smallbold{b})}{K_0'(\iota n'\smallbold{b})};
+\end{DPgather*}
+the large values of~$\eta$ occur when $n'\smallbold{b}$~is small. Substituting the
+approximate values for $K_0$,~$K_0'$ we find
+\[
+\eta = -\iota\mu' \log \left( \frac{2\gamma}{\iota n'\smallbold{b}}\right).
+\]
+This is very small compared with $\mu'/n'\smallbold{b}$, and may, as in the
+preceding case, be shown for all practicable cases to be very
+small compared with $2 V^2/p^2 (\smallbold{b}^2-\smallbold{a}^2)$. Hence, as both $\xi$~and~$\eta$
+are small compared with this quantity, we may neglect the third
+term inside the bracket in equation~(\eqnref{262}{16}), which thus reduces to
+\[
+k^2=-\frac{\iota p^2}{V^2}\left\{\frac{\mu}{n\smallbold{a}}\, \frac{J_0(\iota n\smallbold{a})}{J'_0(\iota n\smallbold{a})} - \frac{\mu'}{n'\smallbold{b}}\, \frac{K_0(\iota n'\smallbold{b})}{K'_0(\iota n'\smallbold{b})}\right\} \frac{1}{\log{(\smallbold{b}/\smallbold{a})}}. \Tag{17}
+\]
+
+We shall now proceed to deduce from this equation the
+velocity of propagation of electrical oscillations of different
+frequencies.
+
+\Subsection{Slowly Alternating Currents.}
+
+\Article{263} The first case we shall consider is the one where the
+frequency is so small that $n\smallbold{a}$~is a small quantity. In this case,
+since we have approximately
+\[
+J_0 (\iota n\smallbold{a})/J_0' (\iota n\smallbold{a}) = -2/\iota n\smallbold{a},
+\]
+equation~(\eqnref{262}{17}) becomes
+\[
+k^2=-\frac{\iota p^2}{V^2}\left\{\frac{2\iota\mu}{n^2\smallbold{a}^2}-\frac{\mu'}{n'\smallbold{b}}\, \frac{K_0(\iota n'\smallbold{b})}{K'_0(\iota n'\smallbold{b})}\right\} \frac{1}{\log{(\smallbold{b}/\smallbold{a})}}. \Tag{18}
+\]
+
+The first term inside the bracket is very large, for it is
+equal to $2\iota \mu/n^2\smallbold{a}^2$ and $n\smallbold{a}$~is small; the second term in the
+bracket vanishes if $\smallbold{b}$~is infinite, and even if $\smallbold{b}$~is so small that
+$n'\smallbold{b}$~is a small quantity, we see, by substituting the values for
+$K_0$~and~$K_0'$ when the variable is small, that the ratio of the
+%% -----File: 285.png---Folio 271-------
+\index{Alternating currents, of long period@\subdashtwo of long period, velocity along a wire}%
+\index{Propagation velocity of slowly alternating currents along a wire@\subdashone velocity of slowly alternating currents along a wire}%
+\index{Velocity of propagation of slowly alternating currents along wires@\subdashtwo propagation of slowly alternating currents along wires}%
+magnitude of the second term inside the bracket to that of the
+first is approximately equal to
+\[
+\frac{\mu'}{2\mu}\, n^2 \smallbold{a}^2 \log \frac{2\gamma}{\iota n'\smallbold{b}},
+\]
+and thus unless $n'\smallbold{b}$~is exceedingly small compared with~$n\smallbold{a}$ the
+second term may be neglected.
+
+Hence, since $n^2 = 4\pi \mu \iota p/\sigma$, we may write~(\eqnref{263}{18}) in the form
+\[
+k^2 = -\frac{p}{V^2}\, \frac{\iota\sigma}{2\pi \smallbold{a}^2}\, \frac{1}{\log (\smallbold{b}/\smallbold{a})},
+\]
+but $k^2 = m^2 - \dfrac{p^2}{V^2}$, so that
+\[
+m^2 = \frac{p^2}{V^2} \left\{1 - \frac{\iota\sigma}{2\pi p\smallbold{a}^2}\, \frac{1}{\log{(\smallbold{b}/\smallbold{a})}} \right\};
+\]
+we have seen however that the second term in the bracket is
+large compared with unity, so that we have approximately
+\[
+m^2 = - \frac{p}{V^2}\, \frac{\iota\sigma}{2\pi \smallbold{a}^2}\, \frac{1}{\log (\smallbold{b}/\smallbold{a})}.
+\]
+
+{\allowdisplaybreaks
+If $\smallbold{R}$ is the resistance and $\Gamma$~the capacity in electromagnetic
+measure per unit length of the wire, then since
+\begin{DPgather*}
+\smallbold{R} = \frac{\sigma}{\pi \smallbold{a}^2},\quad \Gamma = \frac{1}{2V^2 \log (\smallbold{b}/\smallbold{a})},\\
+\lintertext{we have}  m^2 = -\iota p\smallbold{R}\Gamma, \\
+\lintertext{or}      m = -\{p\smallbold{R}\Gamma\}^{\frac{1}{2}} \left\{\frac{1}{\sqrt{2}} - \frac{\iota}{\sqrt{2}}\right\},
+\end{DPgather*}
+where the sign has been taken so as to make the real part of~$\iota m$
+negative. The reason for this is as follows: if $m = -\alpha + \iota\beta$,
+$\smallbold{R}$,~the electromotive intensity parallel to the axis of the wire,
+will be expressed by terms of the form
+\[
+\cos(-\alpha z+pt)\, \epsilon^{-\beta z}.
+\]
+This represents a vibration whose phases propagated with the
+velocity~$p/\alpha$ in the positive direction of~$z$, and which dies away
+to $1/\epsilon$~of its original value after passing over a distance~$1/\beta$;
+if $\beta$ were negative the disturbance would increase indefinitely
+as it travelled along the wire. Substituting the value of~$\alpha$, we
+see that the velocity of propagation of the phases is
+\[
+\left\{\frac{2p}{\smallbold{R}\Gamma} \right\}^{\frac{1}{2}};
+\]
+%% -----File: 286.png---Folio 272-------
+\index{Alternating currents, of long period@\subdashtwo of long period, rate of decay along a wire}%
+thus the velocity of propagation is directly proportional to the
+square root of the frequency and inversely proportional to the
+square root of the product of the resistance and capacity of
+the wire per unit length.
+}%end \allowdisplaybreaks
+
+\index{Distance alternating currents travel along a wire}%
+\index{Decay, rate of, of slowly alternating currents along a wire}%
+\index{Rate of decay of slowly alternating currents along a wire}%
+The distance to which a disturbance travels before falling to
+$1/\epsilon$~of its original value is, on substituting the value of~$\beta$, seen
+to be
+\[
+\left\{\frac{2}{\smallbold{R}\Gamma p}\right\}^{\frac{1}{2}};
+\]
+thus the distance to which a disturbance travels is inversely
+proportional to the square root of the product of the frequency,
+the resistance, and capacity per unit length.
+
+If we take the case of a cable transmitting telephone messages
+of such a kind that~$2\pi/p$, the period of the electrical vibrations, is
+$1/100$~of a second, then if the copper core is $4$~millimetres in
+diameter and the external radius of the guttapercha covering
+about $2.5$~times that of the core, $\smallbold{R}$~is about $1.3 × 10^{-5}$~Ohms, or
+in absolute measure $1.3 × 10^4$. $\Gamma$~is about $15 × 10^{-22}$. Substituting
+these values for $\smallbold{R}$~and~$\Gamma$, we find that the vibrations will travel
+over about $128$~kilometres before falling to $1/\epsilon$~of their initial
+value. The velocity of propagation of the phases is about
+$80,000$ kilometres per second. If we take an iron telegraph
+wire $4$~mm.\ in diameter, $\smallbold{R}$~is about $9.4 × 10^4$; the capacity of
+such a wire placed $4$~metres above the ground is stated by
+Hagenbach (\textit{Wied. Ann.}~29.\ p.~377, 1886) to be about $10^{-22}$
+per centimetre, hence the distance to which electrical vibrations
+making $100$~vibrations per second would travel before falling to
+$1/\epsilon$~of their original value would be $\{1.3 × 15/9.4\}^{\frac{1}{2}}$, or $1.43$~times
+the distance in the preceding case: thus the messages along the
+aerial wire would travel about half as far again as those along
+the cable, the increased resistance of the iron telegraph wire
+being more than counterbalanced by the smaller electrostatic
+capacity. Since vibrations of different frequencies die away at
+different rates, a message such as a telephone message which is
+made up of vibrations whose frequencies extend over a somewhat
+wide range will lose its character as soon as there is any
+appreciable decay in the vibrations. We see from this investigation
+that the lower the pitch the further will the vibrations
+travel, so that when a piece of music is transmitted along a
+telephone wire the high notes suffer the most.
+%% -----File: 287.png---Folio 273-------
+
+\Article{264} We shall now proceed to consider the expressions when
+$n\smallbold{a}$~is small for the electromotive intensity and magnetic induction
+in the wire and dielectric in terms of the total current
+flowing through any cross section of the wire.
+
+We have seen that
+\begin{DPgather*}
+m = -\{p\smallbold{R}\Gamma\}^{\frac{1}{2}} \left\{\frac{1}{\sqrt{2}} %[TN: surd changed to sqrt for consistency]
+- \frac{\iota}{\sqrt{2}}\right\}, \\
+\lintertext{hence if}   \alpha = \left\{\tfrac{1}{2} p\smallbold{R}\Gamma\right\}^{\frac{1}{2}},
+\end{DPgather*}
+we may suppose that the current through the wire at~$z$ is equal to
+\[
+I_0\epsilon^{-\alpha z} \cos{(-\alpha z+pt)}.
+\]
+\begin{DPgather*}
+\lintertext{This is equal to}  \int_0^\smallbold{a}  \frac{R}{\sigma} 2\pi r\,dr,
+\end{DPgather*}
+so that in this case we find by equation~(\eqnref{262}{9}), since $J_0(\iota nr)$ can be
+replaced by unity as $nr$~is small,
+\[
+A = \frac{\sigma I_0}{\pi \smallbold{a}^2},
+\]
+so that by~(\eqnref{262}{9}) we have approximately
+\[
+R = \frac{\sigma I_0}{\pi \smallbold{a}^2}\, \epsilon^{-\alpha z} \cos{(-\alpha z+pt)}.
+\]
+
+Thus the electromotive intensity, and therefore the current
+parallel to~$z$, is uniformly distributed over the cross-section. The
+electromotive intensity along the radius, $\{P^2 + Q^2\}^{\frac{1}{2}}$, is easily
+found by equation~(\eqnref{262}{9}) to be
+\[
+-\frac{\iota m}{2}\, \frac{\sigma I_0}{\pi \smallbold{a}^2}\, r\epsilon^{-\alpha z}\epsilon^{\iota(-\alpha z+pt)}.
+\]
+
+Substituting the value of~$m$ and taking the real part, we see
+that it is equal to
+\[
+\left\{\frac{p\sigma\Gamma}{\pi \smallbold{a}^2}\right\} \frac{\sigma I_0}{2\pi \smallbold{a}^2}\, r\epsilon^{-\alpha z} \cos{\left(-\alpha z+pt+\frac{\pi}{4}\right)},
+\]
+it is thus very small compared with the intensity along the
+axis of the wire, so that in the wire the Faraday tubes are
+approximately parallel to the axis of the wire.
+
+The magnetic induction in this case reduces approximately to
+\[
+\frac{2\mu I_0}{\smallbold{a}^2}\, r\epsilon^{-\alpha z} \cos{(-\alpha z+pt)}.
+\]
+%% -----File: 288.png---Folio 274-------
+In the dielectric, we have by equations (\eqnref{262}{7}),~(\eqnref{262}{13}), and~(\eqnref{262}{14}),
+assuming that $kr$~is small,
+\[
+R=\frac{\sigma}{\pi \smallbold{a}^2}\, I_0\left\{1-2V^2\Gamma \log{\frac{r}{\alpha}}\right\}\epsilon^{-\alpha z} \cos{(-\alpha z + pt)},
+\]
+since from (\eqnref{262}{13})~and~(\eqnref{262}{14}) $D = 2\Gamma V^2A$.
+
+The electromotive intensity along the radius, $\{P^2 + Q^2\}^{\frac{1}{2}}$, is
+equal to
+\[
+2V^2\{\pi \smallbold{a}^2\Gamma /p\sigma\}^{\frac{1}{2}} \frac{\sigma}{\pi \smallbold{a}^2}\, \frac{1}{r}\, I_0\epsilon^{-\alpha z} \cos{\left(-\alpha z + pt - \frac{\pi}{4}\right)}.
+\]
+In this case the radial electromotive intensity is very large compared
+with the tangential intensity, so that in the dielectric the
+Faraday tubes are approximately at right angles to the wire.
+
+The resultant magnetic induction is equal to
+\[
+\frac{2I_0}{r}\, \epsilon^{-\alpha z} \cos{(-\alpha z+pt)}.
+\]
+
+\Article{265} The interpretation of~(\eqnref{262}{17}) is easy when $n\smallbold{a}$~is very small,
+since in this case the first term inside the bracket is very large
+compared with the second; as $n\smallbold{a}$~increases the discussion of the
+equation becomes more difficult, since the second term in the
+bracket is becoming comparable with the first. It will facilitate
+the discussion of the equation if we consider the march of the
+function $\iota n\smallbold{a}J_0(\iota n\smallbold{a})/J_0'(\iota n\smallbold{a})$. Perhaps the simplest way to do
+this is to expand the function $xJ_0(x)/J_0'(x)$ in powers of~$x$.
+Since $J_0(x)$~is a Bessel's function of zero order, we have
+\[
+J_0''(x) + \frac{1}{x} J_0'(x) + J_0(x) = 0,
+\]
+\begin{DPalign*}
+\lintertext{so that}     \frac{xJ_0(x)}{J_0'(x)} & = -1 - \frac{xJ_0''(x)}{J_0'(x)} \\
+                        &    = -1 - x \frac{d}{dx} \log {J_1(x)},
+\end{DPalign*}
+since $J_0'(x) = -J_1(x)$, $J_1(x)$~being Bessel's function of the first
+order.
+
+Let $0, x_1, x_2, x_3 \ldots$ be the positive roots of the equation
+\begin{DPgather*}
+J_1(x) = 0, \\
+\lintertext{then}  J_1(x) = x\left(1-\frac{x^2}{x_1^2}\right)\left(1-\frac{x^2}{x_2^2}\right)\left(1-\frac{x^2}{x_3^2}\right)\ldots
+\end{DPgather*}
+%% -----File: 289.png---Folio 275-------
+so that
+\[
+\frac{d}{dx} \log {J_1(x)} = \frac{1}{x} - \frac{2x}{x_1^2\left(1-\dfrac{x^2}{x_1^2}\right)} - \frac{2x}{x_2^2\left(1-\dfrac{x^2}{x_2^2}\right)} - \ldots,
+\]
+and therefore
+\begin{multline*}
+                    x \frac{d}{dx} \log {J_1(x)}  = 1 - \frac{2x^2}{x_1^2} \left(1 + \frac{x^2}{x_1^2} + \frac{x^4}{x_1^4} + \ldots\right) \\
+            \shoveright{- \frac{2x^2}{x_2^2} \left(1 + \frac{x^2}{x_2^2} + \frac{x^4}{x_2^4} + \ldots\right) + \ldots} \\
+\shoveleft{\phantom{x \frac{d}{dx} \log {J_1(x)}} = 1 - 2x^2 \left(\frac{1}{x_1^2} + \frac{1}{x_2^2} + \frac{1}{x_3^2} + \ldots\right)} \\
+            \qquad - 2x^4\left(\frac{1}{x_1^4} + \frac{1}{x_2^4} + \frac{1}{x_3^4} + \ldots\right) - \ldots.
+\end{multline*}
+
+Thus if $S_n$~denotes the sum of the reciprocals of the $n$\textsuperscript{th}~powers
+of the roots of the equation
+\[
+J_1(x)/x = 0,
+\]
+we have
+\[
+\frac{xJ_0(x)}{J_0'(x)} = -2 + 2S_2x^2 + 2S_4x^4 + 2S_6x^6 + \ldots.
+\]
+
+Now the equation $J_1(x)/x = 0$, when expanded in powers of~$x$
+is,
+\[
+1 - \frac{x^2}{2\centerdot4} + \frac{x^4}{2\centerdot4\centerdot4\centerdot6} - \frac{x^6}{2\centerdot4\centerdot4\centerdot6\centerdot6\centerdot8} + \ldots = 0.
+\]
+
+Hence, if we calculate $S_2$,~$S_4$, $S_6$~\&c.\ by Newton's Rule, we find
+\begin{gather*}
+S_2 = \frac{1}{8}, \quad S_4 = \frac{1}{12×16}, \quad S_6 = \frac{1}{12×16^2}, \\
+S_8 = \frac{1}{12×15×16^2}, \quad S_{10} = \frac{13}{9×15×16^4};
+\end{gather*}
+hence
+\[
+\frac{xJ_0(x)}{J_0'(x)} = -2 + \frac{x^2}{4} + \frac{x^4}{96} + \frac{x^6}{1536} + \frac{x^8}{23040} + \frac{13x^{10}}{4423680} - \ldots,
+\]
+so that
+\[
+\iota n\smallbold{a}\, \frac{J_0(\iota n\smallbold{a})}{J_0'(\iota n\smallbold{a})} = -2 - \frac{n^2\smallbold{a}^2}{4} + \frac{n^4\smallbold{a}^4}{96} - \frac{n^6\smallbold{a}^6}{1536} + \frac{n^8\smallbold{a}^8}{23040} - \frac{13n^{10}\smallbold{a}^{10}}{4423680},
+\]
+%% -----File: 290.png---Folio 276-------
+and since $n^2 = 4 \pi \mu \iota p / \sigma$ approximately, we have
+\begin{align*}
+\iota n \smallbold{a}\, \frac{J_0(\iota n \smallbold{a})}{J_0'(\iota n \smallbold{a})} &= -2 - \frac{1}{96} (4 \pi \mu p \smallbold{a}^2 / \sigma)^2 + \frac{1}{23040} (4 \pi \mu p \smallbold{a}^2 / \sigma)^4 \ldots \\
+&\phantom{-2} -\iota \left\{\frac{1}{4}(4 \pi \mu p \smallbold{a}^2 / \sigma) - \frac{1}{1536}(4 \pi \mu p\smallbold{a}^2 / \sigma)^3\right. \\
+&\phantom{-2 - \frac{1}{96} 4 \pi} + \left. \frac{13}{4423680} (4 \pi \mu p \smallbold{a}^2 / \sigma)^5 \ldots \right\}. \Tag{19}
+\end{align*}
+
+The values of $\iota n \smallbold{a}\, J_0(\iota n \smallbold{a})/J_0'(\iota n \smallbold{a})$ for a few values of $4 \pi \mu p \smallbold{a}^2 / \sigma$
+are given in the following table:---
+\[
+\begin{array}{c@{\quad}||@{\quad}l}
+\tablespaceup 4\pi\mu p \smallbold{a}^2/\sigma &
+\iota n \smallbold{a} J_0(\iota n \smallbold{a})/J_0'(\iota n \smallbold{a})\tablespacedown\\
+\hline
+\tablespaceup \PadTo[r]{9.9}{.5} &  -2 \{1.001 + .062 \iota \}\\
+
+\PadTo[l]{9.9}{1}  &  -2 \{1.005 + .125 \iota \}\\
+
+1.5 & -2 \{1.012 + .186 \iota \}\\
+
+\PadTo[l]{9.9}{2}  &  -2 \{1.021 + .25 \iota \}\\
+
+2.5 & -2 \{1.032 + .31\iota \}\\
+
+\PadTo[l]{9.9}{3}  &  -2 \{1.045 + .37\iota \}\tablespacedown
+\end{array}
+\]
+From this table we see that even when $4 \pi \mu p \smallbold{a}^2 / \sigma$ is as large as
+unity, we may still as an approximation put
+\[
+\iota n \smallbold{a}\, J_0(\iota n \smallbold{a}) / J_0'(\iota n \smallbold{a})
+\]
+equal to~$-2$, and $k^2$~will continue to be given by~(\eqnref{263}{18}).
+
+\Article{266} We must consider now the relative values of the terms
+inside the bracket in~(\eqnref{263}{18}) when $n \smallbold{a}$~is comparable with unity. In
+the case of aerial telegraph wires it is conceivable that there may
+be cases in which though $n \smallbold{a}$~is not large $n' \smallbold{b}$~may be so; but
+when this is the case we have by \artref{261}{Art.~261}
+\[
+K_0(\iota n' \smallbold{b}) = -\iota K_0'(\iota n'\DPtypo{b}{\smallbold{b}}),
+\]
+so that since $n' \smallbold{b}$~is very large the second term inside the bracket
+in equation~(\eqnref{263}{18}) will be small compared with the first, hence we
+have
+\[
+k^2 = -\frac{p}{V^2}\, \frac{\iota \sigma}{2 \pi \smallbold{a}^2}\, \frac{1}{\log(\smallbold{b} / \smallbold{a})},
+\]
+which is the same value as in \artref{263}{Art.~263}.
+
+In all telegraph cables where the external conductor is
+water, and in all but very elevated telegraph wires where the
+external conductor is wet earth, the value of~$\sigma'$ will so greatly
+exceed that of~$\sigma$ that unless $\smallbold{b}$~is more than a thousand times
+%% -----File: 291.png---Folio 277-------
+\index{Alternating currents, of moderate period@\subdashtwo of moderate period, rate of decay along a wire}%
+\index{Decay, rate of, xof moderately rapid currents along a wire\subdashtwo of moderately rapid currents along a wire}%
+\index{Rate of decay of tmoderately rapid currents@\subdashtwo of moderately rapid currents}%
+as great as~$\smallbold{a}$, $n'\smallbold{b}$~will be very small if the value of~$n\smallbold{a}$ is comparable
+with unity. In this case however by \artref{261}{Art.~261},
+\[
+\frac{K_0(\iota n'\smallbold{b})}{K_0'(\iota n'\smallbold{b})} = -\iota n'\smallbold{b} \log \frac{2\gamma}{\iota n'\smallbold{b}},
+\]
+so that equation~(\eqnref{263}{18}) becomes
+\[
+k^2 = \frac{p^2}{V^2}\, \left\{\frac{2\mu}{n^2\smallbold{a}^2} + \mu' \log \frac{2\gamma}{\iota n'\smallbold{b}}\right\} \frac{1}{\log(\smallbold{b}/\smallbold{a})}.
+\]
+Since $n'\smallbold{b}$~is very small while $n\smallbold{a}$~is comparable with unity, the
+second term inside the brackets will be very large compared
+with the first, hence this equation may be written
+\begin{DPgather*}
+k^2 = \frac{p^2}{V^2} \log {\frac{2\gamma}{\iota n'\smallbold{b}}}\, \frac{\mu'}{\log(\smallbold{b}/\smallbold{a})}; \Tag{20} \\
+\lintertext{or} m^2 = \frac{p^2}{V^2} \left\{1 + \log \frac{2\gamma}{\iota n'\smallbold{b}}\, \frac{\mu'}{\log(\smallbold{b}/\smallbold{a})} \right\}. \\
+\intertext{Thus approximately}
+m^2 = \frac{p^2}{V^2} \log \frac{2\gamma}{\iota n'\smallbold{b}}\, \frac{\mu'}{\log(\smallbold{b}/\smallbold{a})}, \\
+\lintertext{and since} n'^2 = 4\pi\mu'\iota p/\sigma', \\
+m^2 = \tfrac{1}{2} \frac{p^2}{V^2} \left\{\log \frac{\sigma'\gamma^2}{\mu'\pi \smallbold{b}^2p} + \frac{\iota\pi}{2} \right\} \frac{\mu'}{\log(\smallbold{b}/\smallbold{a})},
+\end{DPgather*}
+hence we have approximately
+\[
+m = \frac{1}{\sqrt{2}}\, \frac{p}{V} \left\{\frac{\mu' \log \dfrac{\sigma'\gamma^2}{\mu'\pi \smallbold{b}^2p}}{\log(\smallbold{b}/\smallbold{a})}\right\}^{\frac{1}{2}} \left\{1 + \iota\frac{\pi}{4}\, \frac{1}{\log(\sigma'\gamma^2/\mu'\pi \smallbold{b}^2p)}  \right\}, \Tag{21}
+\]
+where the plus sign has been taken so as to make the real
+part of~$\iota m$ negative. This equation corresponds to a vibration
+\index{Propagation velocity of xmoderately rapid currents along a wire@\subdashtwo of moderately rapid currents along a wire}%
+\index{Velocity of propagation of moderately rapid currents along wires@\subdashtwo propagation of moderately rapid currents along wires}%
+\index{Alternating currents, of moderate period@\subdashtwo of moderate period, velocity along a wire}%
+whose phases are propagated with the velocity
+\[
+V \left\{\frac{\log(\smallbold{b}^2/\smallbold{a}^2)}{\mu'\log(\sigma'\gamma^2/\mu'\pi \smallbold{b}^2p)} \right\}^{\frac{1}{2}},
+\]
+and which fades away to $1/\epsilon$~of its original value after passing
+over a distance
+\index{Distance alternating currents travel along a wire}%
+\[
+\frac{4}{\pi}\, \frac{V}{p\mu'^{\frac{1}{2}}} \{\log(\smallbold{b}^2/\smallbold{a}^2) × \log(\sigma'\gamma^2/\mu'\pi \smallbold{b}^2p)\}^{\frac{1}{2}}.
+\]
+
+This case presents many striking peculiarities. In the first
+%% -----File: 292.png---Folio 278-------
+place we see that to our order of approximation both the velocity
+of propagation of the phases and the rate of decay of the
+vibrations are independent of the resistance of the wire. These
+quantities depend somewhat on the resistance of the external
+conductor, but only to a comparatively small extent even on
+that, as $\sigma'$ only enters their expressions as a logarithm. The
+velocity of propagation of the phases only varies slowly with the
+frequency, as $p$ only occurs in its expression as a logarithm.
+The rate of decay, \mbox{i.\hspace{0.1em}e.}\ the real part of~$\iota m$, is proportional to
+the frequency and thus varies more rapidly with this quantity
+than when $n\smallbold{a}$ is small, as in that case the rate of decay is
+proportional to the square root of the frequency (\artref{263}{Art.~263}).
+We see from the preceding investigation that for sending
+periodic disturbances along a cable, the frequency being such as
+to make $n'\smallbold{b}$ a very small quantity, we do not gain any appreciable
+advantage by making the core of a good conductor
+like copper rather than of an inferior one like iron \emph{unless} the
+conditions are such as to make $n\smallbold{a}$ small compared with
+unity. We see too that the distance to which the disturbance
+travels before it falls to $1/\epsilon$~of its original value increases
+with the resistance of the external conductor. We shall show
+in a subsequent article that the heat produced per second
+in the external conductor is very large compared with that
+produced in the same time in the wire, thus the dissipation of
+energy is controlled by the external conductor and not by the
+wire.
+
+The preceding results will continue true as long as $n'\smallbold{b}$ is
+small, even though the frequency of the electrical vibrations gets
+so large that $n\smallbold{a}/\mu$ is a very large quantity; for when $n\smallbold{a}$ is large
+we have by \artref{261}{Art.~261},
+\[
+J_0'(\iota n\smallbold{a}) = -\iota J_0(\iota n\smallbold{a}),
+\]
+so that equation~(\eqnref{262}{16}) becomes
+\[
+k^2 = \frac{p^2}{V^2} \left\{\frac{\mu}{n\smallbold{a}} + \mu' \log \frac{2\gamma}{\iota n'\smallbold{b}} \right\} \frac{1}{\log(\smallbold{b}/\smallbold{a})}.
+\]
+Since $n\smallbold{a}/\mu$ is large and $n'\smallbold{b}$ small the second term inside the
+bracket is large compared with the first, so that we get the same
+value of~$k^2$ as that given by equation~(\eqnref{266}{20}).
+
+\Article{267} The next case we have to consider is that in which both
+%% -----File: 293.png---Folio 279-------
+\index{Alternating currents, of short period@\subdashtwo of short period, rate of decay along a wire}%
+\index{Alternating currents, of short period@\subdashtwo of short period, velocity of along a wire}%
+\index{Decay, rate of, xof very rapid currents along a wire@\subdashtwo of very rapid currents along a wire}%
+\index{Distance alternating currents travel along a wire}%
+\index{Propagation velocity of yvery rapid currents along a wire@\subdashtwo of very rapid currents along a wire}%
+\index{Rate of decay of very rapid currents@\subdashtwo of very rapid currents}%
+\index{Velocity of propagation of very rapid currents along wires@\subdashtwo propagation of very rapid currents along wires}%
+$n\smallbold{a}$ and $n'\smallbold{b}$ are very large; when this is the case we know by
+\artref{261}{Art.~261} that
+\[
+J'_0 (\iota n\smallbold{a}) = - \iota J_0 (\iota n\smallbold{a}), \quad K'_0 (\iota n'\smallbold{b}) = \iota K_0 (\iota n'\smallbold{b}).
+\]
+Making these substitutions, equation~(\eqnref{262}{17}) becomes
+\begin{DPgather*}
+k^2 = \frac{p^2}{V^2} \left\{\frac{\mu}{n\smallbold{a}} + \frac{\mu'}{n'\smallbold{b}} \right\} \frac{1}{\log(\smallbold{b}/\smallbold{a})}, \Tag{22}\\
+\lintertext{or}
+\begin{aligned}
+  [t]m^2 & = \frac{p^2}{V^2} \left\{1 + \left(\frac{\mu}{n\smallbold{a}} + \frac{\mu'}{n'\smallbold{b}}\right)
+             \frac{1}{\log(\smallbold{b}/\smallbold{a})} \right\} \\
+         & = \frac{p^2}{V^2} \left\{1 - \frac{\iota}{\sqrt {8\pi p}}
+             \left( \sqrt{\frac{\mu \sigma}{\smallbold{a}^2}} + \sqrt{\frac{\mu'\sigma'}{\smallbold{b}^2}} \right)
+             \frac{1}{\log(\smallbold{b}/\smallbold{a})} \right\}
+\end{aligned}
+\end{DPgather*}
+approximately. Since the second term inside the bracket is small
+compared with unity, extracting the square root we have,
+\[
+m = - \frac{p}{V} \left\{1 - \frac{\iota}{\sqrt {32\pi p}}
+    \left( \sqrt{\frac{\mu\sigma}{\smallbold{a}^2}} + \sqrt{\frac{\mu'\sigma'}{\smallbold{b}^2}} \right)
+    \frac{1}{\log(\smallbold{b}/\smallbold{a})} \right\}. \Tag{23}
+\]
+
+This represents a vibration travelling approximately with the
+velocity~$V$ and dying away to $1/\epsilon$~of its initial value after traversing
+a distance
+\[
+4V \sqrt\frac{2\pi}{p} \left\{\sqrt\frac{\mu\sigma}{\smallbold{a}^2} + \sqrt\frac{\mu'\sigma'}{\smallbold{b}^2} \right\}^{-1} \log(\smallbold{b}/\smallbold{a}).
+\]
+
+Since the imaginary part of~$m$ is small compared with the real
+part, the vibration will travel over many wave lengths before
+its amplitude is appreciably reduced. From the expression for
+the rate of decay in this case we see that when the wire is
+surrounded by a very much worse conductor than itself, as is
+practically always the case with cables, the distance to which
+these very rapid oscillations will travel will be governed mainly
+by the outside conductor, and will be almost independent of the
+resistance and permeability of the wire; no appreciable advantage
+therefore would in this case be derived by using a well-conducting
+but expensive material like copper for the wire. In
+aerial wires the decay will be governed by the conductivity of
+the earth rather than by that of the wire, unless the height of
+the wire above the ground, which we may take to be comparable
+with $\smallbold{b}$, is so great that $\mu'\sigma'/\smallbold{b}^2$ is not large compared with $\mu\sigma/\smallbold{a}^2$.
+
+Experiments which confirm the very important conclusion
+that these rapid oscillations travel with the velocity~$V$, that is
+%% -----File: 294.png---Folio 280-------
+with the velocity of light through the dielectric, will be described
+in the \chapref{Chapter V.}{next chapter}.
+
+\Article{268} As rapidly alternating currents are now very extensively
+employed, it will be useful to determine the components of
+the electromotive intensity both in the wire and in the dielectric
+in terms of the total current passing through the wire. Let this
+current at the point~$z$ and time~$t$ be represented by the real
+part of $I_0 \epsilon^{\iota(mz+pt)}$. The line integral of the magnetic force
+taken round any circuit is equal to $4\pi$~times the current through
+that circuit. Now by equation~(\eqnref{262}{10}) the magnetic force at the
+surface of the wire is
+\[
+\frac{4\pi \iota}{\sigma n} A J_0' (\iota n \smallbold{a}) \epsilon^{\iota(mz+pt)}.
+\]
+Since the line integral of this round the surface of the wire is
+equal to $4 \pi I_0 \epsilon^{\iota(mz+pt)}$, we have
+\[
+A = - \frac{\iota\sigma n}{2\pi \smallbold{a}} \frac{I_0}{J'_0(\iota n\smallbold{a})}.
+\]
+Substituting this value for $A$ in equation~(\eqnref{262}{9}), we find that in the
+wire
+\[
+R = - \frac{\iota\sigma n}{2\pi \smallbold{a}} \frac{I_0}{J'_0(\iota n\smallbold{a})} J_0(\iota nr)^{\epsilon^{\iota(mz+pt)}}; \Tag{24}
+\]
+where the real part of the expression on the right-hand side is to
+be taken. When $n\smallbold{a}$ and $nr$ are very large, we have by \artref{261}{Art.~261}
+\[
+J_0'(\iota n\smallbold{a}) = - \iota \frac{\epsilon^{n\smallbold{a}}}{\sqrt{2\pi n\smallbold{a}}},
+    \quad J_0(\iota nr) = \frac{\epsilon^{nr}}{\sqrt{2\pi nr}};
+\]
+substituting these values in~(\eqnref{268}{24}), we find
+\begin{DPgather*}
+R = \left\{\frac{\mu p\sigma}{\pi \smallbold{a}r} \right\}^{\frac{1}{2}}
+    I_0 \epsilon^{-\{2\pi\mu p/\sigma\}^{\frac{1}{2}} (\smallbold{a}-r)} \cos(\psi), \Tag{25} \\
+\lintertext{where} \psi = mz + pt - (2 \pi\mu p/\sigma)^{\frac{1}{2}} (\smallbold{a}-r) + \frac{\pi}{4}.
+\end{DPgather*}
+
+Similarly, we find by equation~(\eqnref{262}{9}) that the radial electromotive
+intensity $(P^2 + Q^2)^{\frac{1}{2}}$ is given by the equation
+\[
+\{P^2 + Q^2\}^{\frac{1}{2}} = - \frac{p}{V} \frac{\sigma I_0}{2\pi \sqrt{\smallbold{a}r}}
+  \epsilon^{-\{2\pi\mu p/\sigma\}^{\frac{1}{2}}(\smallbold{a}-r)} \sin{\left(\psi-\frac{\pi}{4}\right)}. \Tag{26}
+\]
+
+The resultant magnetic force is by equation~(\eqnref{262}{10}) equal to
+\[
+\frac{2}{\sqrt{\smallbold{a}r}} I_0\epsilon^{-\{2\pi\mu p/\sigma\}^{\frac{1}{2}}(\smallbold{a}-r)} \cos{\left(\psi-\frac{\pi}{4}\right)}.
+\]
+%% -----File: 295.png---Folio 281-------
+
+Since all these expressions contain the factor $\epsilon^{-(2\pi\mu p/\sigma)^{\frac{1}{2}} (\smallbold{a}-r)}$,
+\index{Alternating currents, flow to surface of conductors@\subdashtwo flow to surface of conductors}%
+\index{Electric skin@\subdashone `skin'}%
+\index{Skin@`Skin', electrical}%
+we see that the magnitudes of the electromotive intensity and of
+the magnetic force must, since $n\smallbold{a}$---and therefore $(2\pi\mu p/\sigma)^{\frac{1}{2}}\smallbold{a}$---is
+by hypothesis very large, diminish very rapidly as the distance
+from the surface of the wire increases. The maximum values of
+these quantities at the distance $(\sigma/2\pi\mu p)^{\frac{1}{2}}$ from the surface are
+only $1/\epsilon$~of their values at the boundary, and they diminish
+in geometrical progression as the distance from the surface
+increases in arithmetical progression. Thus the currents and
+magnetic forces are, as in \artref{258}{Art.~258}, practically confined to a skin
+on the outside of the wire. We have taken $(\sigma/2\pi\mu p)^{\frac{1}{2}}$ as the
+measure of the thickness of this `skin.' For currents making
+$100$~vibrations per second, the skin for soft iron having a
+magnetic permeability of~$1000$ is about half a millimetre thick,
+for copper it is about thirteen times as great. For currents
+making a million vibrations per second, such as can be produced
+by discharging Leyden jars, the thickness of the skin for soft iron---since
+we know that this substance retains its magnetic properties
+even in these very rapidly alternating magnetic fields (J.~J.
+Thomson, \textit{Phil.\ Mag.}\ Nov.~1891, p.~460)---is about $1/200$~of a
+millimetre, for copper it is about $1/15$~of a millimetre. In these
+cases there is enormous concentration of the current, and since
+the currents produced by the discharge of a Leyden jar, though
+they only last for a short time, are very intense whilst they last,
+the condition of the outer layers of the wires whilst the discharge
+is passing through them is very interesting, as they are conveying
+currents of enormously greater density than would be
+sufficient to melt them if the currents were permanent instead of
+transient.
+
+This concentration of the current, or `throttling' as it is sometimes
+called, produces a great increase in the apparent resistance
+of the wire, since it reduces so largely the area which is available
+for the passage of the current. If in equation~(\eqnref{268}{25}) we put $r = \smallbold{a}$,
+we get maximum value of $R = (\mu p\sigma/\pi \smallbold{a}^2)^{\frac{1}{2}} ×{}$(maximum value of
+the current through the wire), thus we may look upon $(\mu p\sigma/\pi \smallbold{a}^2)^{\frac{1}{2}}$
+as the apparent resistance per unit length of the wire to these
+alternating currents. This resistance increases indefinitely with
+the rate of alternation of the current; we see too that it is
+inversely proportional to the circumference of the wire instead
+of to the area as for steady currents. This is what we should
+%% -----File: 296.png---Folio 282-------
+expect, since the currents are concentrated in the region of the
+circumference. The resistance of the solid wire to these alternating
+currents is the same as that to steady currents of a tube
+of the same material, the outside of the tube coinciding with
+the outside of the wire, and the thickness of the tube being
+$1/\sqrt{2}$~times the thickness of the skin.
+
+We see by comparing equations (\eqnref{268}{25})~and~(\eqnref{268}{26}) that the electromotive
+intensity parallel to the axis of the wire is very large
+compared with the radial electromotive intensity in the wire, so
+that in the wire the Faraday tubes are approximately parallel
+to its axis.
+
+\Article{269} Let us now consider the expressions for the electromotive
+intensities and magnetic force in the dielectric; we find by
+equations (\eqnref{262}{8})~and~(\eqnref{268}{24}), assuming~$k\smallbold{a}$ and~$k\smallbold{b}$ small, $n\smallbold{a}$,~$n'\smallbold{b}$ large,
+\[
+D = 2\iota V^2 k^2 I_0/p.
+\]
+Hence, using~(\eqnref{267}{22}), we have in the dielectric when $kr$~is small,
+\begin{DPgather*}
+R = \left\{\left(\frac{\mu p\sigma}{\pi \smallbold{a}^2}\right)^{\frac{1}{2}}
+   - \left[ \left(\frac{\mu p\sigma}{\pi \smallbold{a}^2}\right)^{\frac{1}{2}}
+          + \left(\frac{\smash{\mu'}p\smash{\sigma'}}{\pi \smallbold{b}^2}\right)^{\frac{1}{2}}\right]
+   \frac{\log r/\smallbold{a}}{\log \smallbold{b}/\smallbold{a}} \right\} I_0 \cos{\phi}, \\
+\lintertext{\rlap{where}} \phi = mz + pt + \frac{\pi}{4},
+\end{DPgather*}
+while the radial electromotive intensity is
+\[
+\frac{2VI_0}{r} \cos{(mz + pt)},
+\]
+and the resultant magnetic force
+\[
+\frac{2I_0}{r} \cos{(mz + pt)}.
+\]
+
+\index{Momentum of Faraday tubes}%
+We see that the maximum value of the radial electromotive
+intensity is very great compared with that of the tangential, so
+that in the dielectric the Faraday tubes are approximately radial.
+The momentum due to these tubes is, by \artref{12}{Art.~12}, at right
+angles both to the tubes and the magnetic force, so that in the
+dielectric it is parallel to the axis of the wire, while in the wire
+itself it is radial. Thus for these rapidly alternating currents
+the momentum in the dielectric follows the wire. The radial
+polarization in the dielectric is $K/4\pi$~times the radial electromotive
+intensity, and since
+\[
+K = 1/V^2,
+\]
+%% -----File: 297.png---Folio 283-------
+it is equal to
+\[
+\frac{I_0}{2\pi Vr} \cos{(mz + pt)}.
+\]
+If the Faraday tubes in the dielectric are moving with velocity~$V$
+at right angles to their length, i.e.~parallel to the wire, the
+magnetic force due to these moving tubes is, by \artref{9}{Art.~9}, at right
+angles both to the direction of motion, i.e.~to the axis of the
+wire, and to the direction of the tubes, i.e.~to the radius, and
+the magnitude of the magnetic force being, by~(\eqnref{9}{4}), \artref{9}{Art\DPtypo{}{.}~9}, $4\pi V$~times
+the polarization, is
+\[
+\frac{2I_0}{r} \cos{(mz + pt)},
+\]
+which is the expression we have already found. Hence we may
+regard the magnetic force in the field as due to the motion
+through it of the radial Faraday tubes, these moving parallel to
+the wire with the velocity with which electromagnetic disturbances
+are propagated through the dielectric.
+
+In the outer conductor when $n'r$~is large
+\begin{DPgather*}
+R = -I_0 \left\{\frac{\mu'p\sigma'}{\pi \smallbold{b}r} \right\}^\frac{1}{2}
+    \epsilon^{-(2\pi\mu'p/\sigma')^{\frac{1}{2}}(r-\smallbold{b})} \cos{\phi'}, \\
+\lintertext{where} \phi' = mz + pt -(2\pi\mu'p/\sigma')^{\frac{1}{2}} + \frac{\pi}{4}.
+\end{DPgather*}
+The radial electromotive intensity is
+\[
+\frac{1}{2\pi}\, \frac{p}{V}\, \frac{\sigma'I_0}{\sqrt{\smallbold{b}r}}
+  \epsilon^{-(2\pi\mu'p/\sigma')^{\frac{1}{2}} (r-\smallbold{b})}
+  \cos{\left(\phi' + \frac{\pi}{4}\right)}.
+\]
+
+The resultant magnetic force is perpendicular to~$r$ and equal to
+\[
+\frac{2I_0}{\sqrt{\smallbold{b}r}}\,
+  \epsilon^{-(2\pi\mu'p/\sigma')^{\frac{1}{2}} (r-\smallbold{b})}
+  \cos{\left(\phi' - \frac{\pi}{4}\right)}.
+\]
+
+We see from these equations that unless $p\sigma'$ is comparable
+with~$V^2$ the tangential electromotive intensity will be large
+compared with the radial.
+
+
+\Subsection{Transmission of Arbitrary Disturbances along Wires.}
+\index{Disturbance, electric, transmission of along a wire}%
+
+\Article{270} Since vibrations with different periods travel at different
+rates, we cannot without further investigation determine the
+rate at which an arbitrary disturbance communicated to a
+%% -----File: 298.png---Folio 284-------
+limited portion of the wire will travel along it. In order to
+deduce an expression which would represent completely the
+way in which an arbitrary disturbance is propagated, we should
+have to make use of the general relation between $m$~and~$p$ given
+by equation~(\eqnref{263}{18}). This relation is however too complicated to
+allow of the necessary integrations being effected. The complication
+arises from the vibrations whose frequencies are so
+great that $2\pi \mu p \smallbold{a}^2 / \sigma$ is no longer a small quantity; such vibrations
+however die away more rapidly than the slower ones,
+so that when the distance from the origin of disturbance is considerable
+the latter are the only vibrations whose effects are felt.
+For such vibrations, we have by \artref{263}{Art.~263}
+\[
+\iota p = -\frac{m^2}{\smallbold{R}\Gamma}.
+\]
+
+{\allowdisplaybreaks
+Hence a term in the expression for~$R$ of the form
+\[
+F(\alpha)\, \epsilon^{-\frac{m^2}{\smallbold{R}\Gamma} t} \cos m (z-\alpha),
+\]
+where $\alpha$ is any constant and $F(\alpha)$~denotes an arbitrary function
+of~$\alpha$, will satisfy the electrical conditions. By Fourier's theorem,
+however,
+\[
+\frac{1}{2\pi} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}
+  F(\alpha)\, \epsilon^{-\frac{m^2}{\smallbold{R}\Gamma} t} \cos m(z-\alpha)\,dm\,d\alpha, \Tag{27}
+\]
+is equal to~$F(z)$ when $t = 0$. Hence this integral, since it
+satisfies the equations of the electric field, will be the expression
+for the disturbance on the wire at~$z$ at the time~$t$ of the disturbance,
+which is equal to~$F(z)$ when $t = 0$. When the disturbance
+is originally confined to a space close to the origin, $F(\alpha)$~vanishes
+unless $\alpha$~is very small; the expression~(\eqnref{270}{27}) becomes in this case
+\begin{DPgather*}
+\smallbold{F} \int_{-\infty}^{+\infty} \epsilon^{-\frac{m^2}{\smallbold{R}\Gamma} t} \cos mz\,dm, \Tag{28} \\
+\lintertext{where} \smallbold{F} = \int F(\alpha)\,d\alpha. \\
+\lintertext{\indent Since}
+\int_{-\infty}^{+\infty} \epsilon^{-\alpha^2x^2} \cos 2\,bx\,dx
+  = \frac{\sqrt\pi}{\alpha}\, \epsilon^{-\frac{b^2}{\alpha^2}},
+\end{DPgather*}
+we see by~(\eqnref{270}{28}) that the disturbance at time~$t$ and place~$z$ will
+be equal to
+\[
+\smallbold{F}\{\pi \smallbold{R} \Gamma/t\}^{\frac{1}{2}} \epsilon^{-z^2 \smallbold{R}\Gamma/4t^2}. \Tag{29}
+\]
+}%end \allowdisplaybreaks
+%% -----File: 299.png---Folio 285-------
+
+Thus at a given point on the wire the disturbance will vary as
+\[
+\frac{1}{\sqrt{t}}\, \epsilon^{-\frac{c}{t}},
+\]
+where $c$ is a constant. The rise and fall of the disturbance with
+the time is represented in \figureref{fig108}{Fig.~108}, where the ordinates represent
+the intensity of the disturbance and the abscissae the time. It
+will be noticed that the disturbance remains very small until $t$
+approaches~$c/4$, when it begins to increase with great rapidity,
+reaching its maximum value when $t = 2c$; when $t$~is greater
+than this the disturbance diminishes, but fades away from its
+maximum value much more slowly than it approached it.
+
+\includegraphicsmid{fig108}{Fig.~108.}
+
+Since the disturbance rises suddenly to its maximum value we
+may with propriety call~$T$, the time which elapses before this
+value is attained at a given point, the time taken by the disturbance
+to travel to that point. We see from~(\eqnref{270}{29}) that
+\[
+T = \tfrac{1}{2} z^2 \smallbold{R}\Gamma. \Tag{30}
+\]
+Thus the time taken by the disturbance to travel a distance~$z$
+is proportional to~$z^2$, it is also proportional to the product of
+the resistance and capacity per unit length.
+%% -----File: 300.png---Folio 286-------
+
+By dividing $z$ by~$T$ we get the so-called `velocity of the
+current along the wire;' this by~(\eqnref{270}{30}) is equal to
+\[
+\frac{2}{z \smallbold{R} \Gamma}. \Tag{31}
+\]
+
+The velocity thus varies inversely as the length of the cable,
+and for short lengths it may be very great. The preceding formula
+would in fact, unless $z$~were greater than $2 / V \smallbold{R} \Gamma$, indicate a
+velocity of propagation greater than~$V$. This however is impossible,
+and the error arises from our using the equation
+$\iota p = -m^2 / \smallbold{R} \Gamma$ instead of the accurate equation~(\eqnref{263}{18}). By our
+approximate equation vibrations of infinite frequency travel
+with infinite velocity, in reality we have seen (\artref{267}{Art.~267}) that
+they travel with the velocity~$V$. These very rapid vibrations
+however die away very quickly, and when we get to a distance
+equal to a small multiple of $2 / V \smallbold{R} \Gamma$ they will practically have
+disappeared, and at such distances we may trust the expressions~(\eqnref{270}{31}).
+
+A considerable number of experiments have been made on the
+time required to transmit messages on both aerial and submarine
+cables; the results of some of these, made on aerial telegraph iron
+wires $4$~mm.\ in diameter, are given in the accompanying table
+\index{Hagenbach, transmission of signals along wires}%
+taken from a paper by Hagenbach (\textit{Wied.\ Ann.}~29.\ p.~377):---
+\begin{center}
+\tabletextsize
+\begin{tabular}{@{}l@{\;}|c|c|c@{}}
+\hline
+\settowidth{\TmpLen}{Plantamour and Hirsch}%
+\parbox[c]{\TmpLen}{\centering Observer.} &
+\settowidth{\TmpLen}{Length of line}
+\parbox[c]{\TmpLen}{\centering Length of line\\ in kilometers.} &
+\settowidth{\TmpLen}{Time taken for message}%
+\parbox[c]{\TmpLen}{\centering Time taken for message\\ to travel ($T$.)} &
+\settowidth{\TmpLen}{$10^{20} T$/(square of}%
+\parbox[c]{\TmpLen}{\tablespaceup\centering $10^{20} T$/(square of\\
+length of line in\\
+centimetres).\tablespacedown}\\
+\hline
+\tablespaceup Fizeau and Gounelle\mdotfill & $314$            & $.003085$    & $313$\\
+Walker\mdotfill              & $885$            & $.02943\Z$   & $376$\\
+Mitchel\mdotfill             & $977$            & $.02128\Z$   & $223$\\
+Gould and Walker\mdotfill    & $\llap{$1$}681$  & $.07255\Z$   & $257$\\
+Guillemin \mdotfill          & $\llap{$1$}004$  & $.028\Z\Z\Z$ & $278$\\
+Plantamour and Hirsch        & $132\rlap{$.6$}$ & $.00895\Z$   & $\llap{$5$}090$\\
+Löwy and Stephan\mdotfill    & $863$            & $.024\Z\Z\Z$ & $322$\\
+Albrecht \mdotfill           & $\llap{$1$}230$  & $.059\Z\Z\Z$ & $390$\\
+Hagenbach\mdotfill           & $284\rlap{$.8$}$ & $.00176\Z$   & $217$\tablespacedown\\
+\hline
+\end{tabular}
+\end{center}
+
+Hagenbach proved by making experiments with lines of
+different lengths that the time taken by a message to travel
+along a line was proportional to the square of the length of the
+line.
+
+If we apply the formula
+\[
+T = \tfrac{1}{2} z^2 R \Gamma
+\]
+%% -----File: 301.png---Folio 287-------
+to Hagenbach's experiment in the above table, where
+\begin{DPalign*}
+z & = 284.8 × 10^5, \\
+R & = \Z\Z9.4 × 10^4, \\
+\lintertext{and (by estimation)}
+\Gamma & = 10^{-22},
+\end{DPalign*}
+we find $T = .0038$, whereas Hagenbach found~$.0017$. The agreement
+is not good, but we must remember that with delicate
+receiving instruments it will be possible to detect the disturbance
+before it reaches its maximum value, so that we should
+expect the observed time to be less than that at which the effect
+is a maximum. In Hagenbach's experiment the line was about
+$4$~times the length which, according to the formula, would have
+made the disturbance travel with the velocity of light, so that
+it would seem to have been long enough to warrant the application
+of a formula which assumes that the shorter waves
+would have become so reduced in amplitude that their effects
+might be neglected.
+
+When the wire is of length~$l$, we know by Fourier's Theorem
+that any initial disturbance~$R$ may be represented by the
+equation
+\begin{multline*}
+R = \Bigl(A_1 \sin{\frac{\pi  z}{l}} + B_1 \cos{\frac{\pi  z}{l}}\Bigr)
+  + \Bigl(A_2 \sin{\frac{2\pi z}{l}} + B_2 \cos{\frac{2\pi z}{l}}\Bigr) \\
+  + \Bigl(A_3 \sin{\frac{3\pi z}{l}} + B_3 \cos{\frac{3\pi z}{l}}\Bigr) + \ldots.
+\end{multline*}
+\begin{DPgather*}
+\lintertext{\indent Since} \iota p = -m^2/\smallbold{R}\Gamma,
+\end{DPgather*}
+the value of~$R$ after a time~$t$ has elapsed will be represented by
+the equation
+\begin{multline*}
+R = \Bigl(A_1 \sin{\frac{\pi  z}{l}} + B_1 \cos{\frac{\pi  z}{l}}\Bigr) \epsilon^{-\frac{\pi^2}{l^2}\,  \frac{t}{\smallbold{R}\Gamma}} \\
+  + \Bigl(A_2 \sin{\frac{2\pi z}{l}} + B_2 \cos{\frac{2\pi z}{l}}\Bigr) \epsilon^{-\frac{4\pi^2}{l^2}\, \frac{t}{\smallbold{R}\Gamma}} + \ldots \\
+  \ldots
+    \Bigl(A_s \sin{\frac{s\pi z}{l}} + B_s \cos{\frac{s\pi z}{l}}\Bigr) \epsilon^{-\frac{s^2\pi^2}{l^2}\, \frac{t}{\smallbold{R}\Gamma}} + \ldots.
+\end{multline*}
+
+For a full discussion of the transmission of signals along
+cables the reader is referred to a series of papers by Lord Kelvin
+\index{Kelvin, Lord, transmission of an electric disturbance along a wire@\subdashtwo transmission of an electric disturbance along a wire}%
+at the beginning of Vol.~II of his Collected Papers.
+%% -----File: 302.png---Folio 288-------
+
+\Subsection{Relation between the External Electromotive Intensity
+and the Current.}
+\index{Current, connection between and external E.M.F.}%
+\index{Electromotive intensity, relation between and current for alternating currents@\subdashtwo relation between and current for alternating currents}%
+\index{Force relation between external electromotive force and alternating current@\subdashone relation between external electromotive force and alternating current}%
+
+\Article{271} We have hitherto only considered the total electromotive
+intensity and have not regarded it as made up of two parts, one
+due to external causes and the other due to the induction of the
+alternating currents in the conductors and dielectric. For some
+purposes, however, it is convenient to separate the electromotive
+intensity into these two parts, and to find the relation between
+the currents and the external electromotive intensity acting on
+the system.
+
+We may conveniently regard the external electromotive
+intensity as arising from an electrostatic potential~$\phi$ satisfying
+the equation $\nabla^2 \phi = 0$. We suppose that, as in the preceding
+investigation, all the variables contain the factor $\epsilon^{\iota(mz+pt)}$. Since
+$\phi$~varies as~$\epsilon^{\iota mz}$, the equation $\nabla^2 \phi = 0$ is equivalent to
+\[
+\frac{d^2\phi}{dr^2} + \frac{1}{r}\, \frac{d\phi}{dr} - m^2\phi = 0.
+\]
+The solution of this is, in the wire
+\[
+\phi = LJ_0(\iota mr)\, \epsilon^{\iota(mz+pt)},
+\]
+in the dielectric
+\[
+\phi = \{MJ_0(\iota mr) + NK_0(\iota mr)\}\,  \epsilon^{\iota(mz+pt)},
+\]
+in the outer conductor
+\[
+\phi = SK_0(\iota mr)\, \epsilon^{\iota(mz+pt)}.
+\]
+If, as before, $\smallbold{a}$~and~$\smallbold{b}$ are the radii of the internal and external
+boundaries of the dielectric, we have, since $\phi$~is continuous,
+\begin{align*}
+LJ_0 (\iota m \smallbold{a}) & = MJ_0 (\iota m \smallbold{a}) + NK_0 (\iota m \smallbold{a}), \\
+SK_0 (\iota m \smallbold{b}) & = MJ_0 (\iota m \smallbold{b}) + NK_0 (\iota m \smallbold{b}).
+\end{align*}
+The excess of the normal electromotive intensity due to the
+electrostatic potential in the dielectric over that in the wire is
+equal to
+\[
+\iota m \{LJ'_{0}(\iota m \smallbold{a})
+       - (MJ'_{0}(\iota m \smallbold{a})
+        + NK'_{0}(\iota m \smallbold{a})\DPtypo{}{)}\}\, \epsilon^{\iota(mz+pt)};
+\]
+substituting the value for $L-M$ in terms of~$N$ from the preceding
+equation, this becomes
+\[
+\iota m \frac{N}{J_0(\iota m \smallbold{a})}\,
+  \{J'_{0}(\iota m \smallbold{a})\, K_0(\iota m \smallbold{a})
+  - J_0(\iota m \smallbold{a})\, K'_{0}(\iota m \smallbold{a})\}.
+\]
+%% -----File: 303.png---Folio 289-------
+\begin{DPgather*}
+\lintertext{Now}
+J'_0 (\iota m \smallbold{a})\, K_0 (\iota m \smallbold{a})
+  - J_0 (\iota m \smallbold{a})\, K'_0 (\iota m \smallbold{a}) = \frac{1}{\iota m \smallbold{a}}, \\
+\lintertext{for let}
+u = J_0'(x) K_0(x) - J_0(x) K_0'(x), \\
+\lintertext{then}
+\frac{du}{dx} = J_0''(x) K_0(x) - J_0(x) K_0''(x), \\
+\lintertext{but}
+J_0''(x) + \frac{1}{x} J_0'(x) - J_0(x) = 0, \\
+K_0''(x) + \frac{1}{x} K_0'(x) - K_0(x) = 0;
+\end{DPgather*}
+substituting the values of $J''_0(x)$, $K''_0(x)$ from these equations,
+we find
+\begin{DPgather*}
+\begin{aligned}
+\frac{du}{dx}
+  &= -\frac{1}{x} \{J_0'(x) K_0(x) - J_0(x) K_0'(x)\} \\
+  &= -\frac{u}{x},
+\end{aligned} \\
+\lintertext{hence} u = \frac{C}{x},
+\end{DPgather*}
+where $C$~is a constant. Substituting from \artref{261}{Art.~261} the values
+for $J_0(x)$, $J_0'(x)$, $K_0(x)$, $K_0'(x)$ when $x$~is very small, we find
+that $C$~is equal to unity.
+
+Thus when $r = \smallbold{a}$, the normal electromotive intensity due to
+the electrostatic potential in the dielectric exceeds that in the
+wire by
+\[
+\frac{N}{\smallbold{a}J_0(\iota m \smallbold{a})}\, \epsilon^{\iota(mz + pt)}.
+\]
+
+Similarly we may show that when $r = \smallbold{b}$ the normal electromotive
+intensity in the dielectric exceeds that in the outer
+conductor by
+\[
+-\frac{M}{\smallbold{b} K_0 (\iota m \smallbold{b})}\, \epsilon^{\iota(mz + pt)}.
+\]
+
+Now the electromotive intensities arising from the induction
+of the currents are continuous, so that the discontinuity in the
+total normal intensity must be equal to the discontinuity in the
+components arising from the electrostatic potential. By equations
+(\eqnref{262}{7}),~(\eqnref{262}{9}),~(\eqnref{262}{14}) the total normal intensity in the dielectric
+at the surface of separation exceeds that in the wire by
+\[
+A \frac{m}{n} J_0' (\iota n \smallbold{a})
+  \left\{\frac{n^2 - m^2}{\mu(k^2 - m^2)} - 1 \right\} \epsilon^{\iota (mz + pt)};
+\]
+%% -----File: 304.png---Folio 290-------
+hence we have
+\[
+A \frac{m}{n} J_0'(\iota n \smallbold{a})
+  \left\{\frac{n^2-m^2}{\mu(k^2-m^2)} - 1 \right\}
+  = \frac{N}{\smallbold{a}J_0(\iota m \smallbold{a})}. \Tag{32}
+\]
+Similarly
+\[
+E \frac{m}{n'} K_0'(\iota n' \smallbold{b})
+  \left\{\frac{(n'^2-m^2)}{\mu'(k^2-m^2)} - 1 \right\}
+  = -\frac{M}{\smallbold{b}K_0(\iota m \smallbold{b})}.  \Tag{33}
+\]
+
+By equations (\eqnref{262}{10})~and~(\eqnref{262}{12})
+\begin{align*}
+& 2\pi \smallbold{a} \frac{n^2-m^2}{\mu p n}\,
+  AJ_0'(\iota n \smallbold{a})\, \epsilon^{\iota(mz + pt)}, \\
+& 2\pi \smallbold{b} \frac{n'^2-m^2}{\mu' p n'}\,
+  EK_0'(\iota n' \smallbold{b})\, \epsilon^{\iota(mz + pt)}
+\end{align*}
+are respectively the line integrals of the magnetic force round
+the circumference of the wire and the inner circumference of
+the outer conductor, hence they are respectively $4\pi$~times the
+current through the wire, and $4\pi$~times the current through
+the wire plus that through the dielectric. Unless however the
+radius of the outer conductor is enormously greater than that of
+the wire, the current through the wire is infinite in comparison
+with that through the dielectric: for the electromotive intensity~$R$
+is of the same order in the wire and in the dielectric; the current
+density in the wire is~$R/\sigma$, that in the dielectric $(K/4\pi)dR/dt$,
+or~$K \iota p R/4\pi$, or~$\iota p R/4\pi V^2$. Now for metals $\sigma$~is of the order~$10^4$;
+and since $V^2$~is $9 × 10^{20}$, we see that even if there are a
+million alternations per second the intensity of the current in
+the wire to that in the dielectric is roughly as $2 × 10^{11}$ is to
+unity; thus, unless the area through which the polarization
+currents flow exceeds that through which the conduction
+currents flow in a ratio which is impracticable in actual experiments,
+we may neglect the polarization currents in comparison
+with the conduction ones, so that
+\[
+\frac{\smallbold{a} (n^2-m^2)}{\mu n}\, AJ_0'(\iota n \smallbold{a})
+  = \frac{\smallbold{b} (n'^2-m^2)}{\mu' n'}\, EK_0'(\iota n' \smallbold{b}).   \Tag{34}
+\]
+
+Returning to equations (\eqnref{271}{32})~and~(\eqnref{271}{33}), we notice that
+\begin{DPgather*}
+(m^2-n^2) / \mu(k^2-m^2),\\
+\lintertext{which is equal to}
+4 \pi \iota V^2/\sigma p,
+\end{DPgather*}
+is very large when $\sigma p$~is small compared with~$V^2$. Now $\sigma$ for
+%% -----File: 305.png---Folio 291-------
+metals is of the order~$10^4$, and $V^2$~is equal to~$9 × 10^{20}$; so that unless
+$p$ is of the order~$10^{16}$ at least, that is unless the vibrations are as
+rapid as those of light, $(m^2-n^2)/\mu(m^2-k^2)$ is exceedingly large.
+Even when the conductivity is no better than that of sea-water,
+where $\sigma$ may be taken to be of the order~$10^{10}$, this quantity will
+be very large unless there are more than a thousand million
+vibrations per second. Hence in equations (\eqnref{271}{32})~and~(\eqnref{271}{33}) we
+may neglect the second terms inside the brackets on the left-hand
+sides, and write
+\[
+\left. \begin{aligned}
+A \frac{m}{n} \frac{(n^2-m^2)}{\mu(k^2-m^2)} J_0'(\iota n \smallbold{a})
+  & = \frac{N}{\smallbold{a}J_0(\iota m \smallbold{a})},  \\
+E \frac{m}{n'} \frac{(n'^2-m^2)}{\mu'(k^2-m^2)} K_0'(\iota n' \smallbold{b})
+  & = - \frac{M}{\smallbold{b}K_0(\iota m \smallbold{b})};
+\end{aligned}\right\} \Tag{35}
+\]
+hence by~(\eqnref{271}{34}), we have
+\[
+\frac{N}{J_0(\iota m \smallbold{a})} = - \frac{M}{K_0(\iota m \smallbold{b})}.  \Tag{36}
+\]
+
+Let $\smallbold{E}$ be the external electromotive intensity parallel to the
+axis of the wire at its surface, then
+\begin{align*}
+\smallbold{E} & = -\iota m \{MJ_0(\iota m \smallbold{a}) + NK_0(\iota m \smallbold{a})\} \epsilon^{\iota(mz+pt)},
+\intertext{or by equation~(\eqnref{271}{36})}
+& = - \iota mN \{K_0(\iota m \smallbold{a}) - K_0(\iota m \smallbold{b}) \} \epsilon^{\iota(mz+pt)}.
+\end{align*}
+
+Since both $m \smallbold{a}$~and~$m\smallbold{b}$ are very small, we have approximately
+by \artref{261}{Art.~261},
+\[
+K_0(\iota m \smallbold{a}) = \log \frac{2\gamma}{\iota m \smallbold{a}}, \quad
+K_0(\iota m \smallbold{b}) = \log \frac{2\gamma}{\iota m \smallbold{b}},
+\]
+hence we have
+\[
+\smallbold{E} = -\iota m N \log(\smallbold{b} / \smallbold{a}) \epsilon^{\iota(mz+pt)},
+\]
+or by equation~(\eqnref{271}{35}), since $J_0(\iota m \smallbold{a}) = 1$,
+\[
+\smallbold{E} = - \frac{\iota m^2 \smallbold{a}}{n}\, \frac{n^2-m^2}{\mu(k^2-m^2)}\,
+  J_0'(\iota n \smallbold{a}) \log(\smallbold{b} / \smallbold{a}) A \epsilon^{\iota(mz+pt)}.
+\]
+
+But by \artref{263}{Art.~263} we have, if $I_0 \epsilon^{\iota(mz+pt)}$ is the total current
+through the wire,
+\[
+I_0 = \frac{2 \pi \smallbold{a} \iota}{\sigma n} A J_0'(\iota n \smallbold{a}),
+\]
+%% -----File: 306.png---Folio 292-------
+\begin{DPalign*}
+\lintertext{hence, since}
+n^2 - m^2 & = 4 \pi \mu \iota p / \sigma, \\
+m^2 - k^2 & = p^2 / V^2 ,
+\end{DPalign*}
+\[
+\smallbold{E} = 2 \iota p \frac{m^2}{\dfrac{p^2}{V^2}}
+  \log(\smallbold{b} / \smallbold{a}) \centerdot I_0 \epsilon^{\iota(mz+pt)}.
+\]
+
+But by equation~(\eqnref{263}{18})
+\[
+m^2 = \frac{p^2}{V^2}
+  \left\{1 - \frac{1}{4 \pi p}
+  \left( \frac{n \sigma}{\smallbold{a}}\, \frac{J_0(\iota n \smallbold{a})}{J_0'(\iota n \smallbold{a})}
+       - \frac{n' \sigma'}{\smallbold{b}}\, \frac{K_0 (\iota n' \smallbold{b})}{K_0'(\iota n' \smallbold{b})} \right)
+  \frac{1}{\log \smallbold{b} / \smallbold{a}}\right\},
+\]
+hence
+\begin{multline*}
+\smallbold{E}
+  = 2 \iota p \left\{\log \frac{\textbf{b}}{\textbf{a}}
+      - \frac{1}{4\pi p} \left( \frac{n \sigma}{\smallbold{a}}\,
+          \frac{J_0(\iota n \smallbold{a})}{J_0'(\iota n \smallbold{a})} \right.\right. \\
+  \left. \left. - \frac{n' \sigma'}{\smallbold{b}}\,
+          \frac{K_0(\iota n' \smallbold{b})}{K_0'(\iota n' \smallbold{b})} \right) \right\}
+  I_0 \epsilon^{\iota(mz+pt)}.\qquad \Tag{37}
+\end{multline*}
+
+\Article{272} Now, as in \artref{263}{Art.~263}, when both $n \smallbold{a}$~and~$n' \smallbold{b}$ are small, the
+last term inside the bracket will be small compared with the
+others; so that we may write equation~(\eqnref{271}{37}) in the form
+\[
+\smallbold{E} = 2 \iota p \left\{\log \frac{\smallbold{b}}{\smallbold{a}} - \frac{1}{4 \pi p}\,
+  \frac{n \sigma}{\smallbold{a}}\,
+  \frac{J_0(\iota n \smallbold{a})}{J_0'(\iota n \smallbold{a})} \right\} I,
+\]
+where $I$ is the total current through the wire and is equal to
+\[
+I_0 \epsilon^{\iota(mz+pt)}.
+\]
+
+From the expressions for $\iota n \smallbold{a} J_0 (\iota n \smallbold{a}) / J_0'(\iota n \smallbold{a})$ given in
+\artref{265}{Art.~265}, we see that we may write this equation
+\begin{multline*}
+\smallbold{E} = 2 \iota p \left\{\log\frac{\smallbold{b}}{\smallbold{a}}
+  + \frac{\sigma}{2 \pi \iota  p \smallbold{a}^2}
+      \left[ 1 + \frac{1}{12 × 16} (4 \pi \mu p \smallbold{a}^2 / \sigma)^2 \right. \right. \\
+  - \frac{1}{12 × 15 × 16^2} (4 \pi \mu p \smallbold{a}^2 / \sigma)^4 + \ldots
+  + \iota \left( \frac{1}{8} (4 \pi \mu p \smallbold{a}^2 / \sigma) \right. \\
+    \left. \left. \left.
+       - \frac{1}{12 × 16^2} (4 \pi \mu p \smallbold{a}^2 / \sigma)^3
+  + \frac{13}{9 × 15 × 16^4} (4 \pi \mu p \smallbold{a}^2 / \sigma)^5 + \ldots\right) \right] \right\} I,
+\end{multline*}
+\begin{DPgather*}
+\lintertext{or}
+\smallbold{E} = \iota p \left\{2 \log\frac{\smallbold{b}}{\smallbold{a}}
+  + \frac{1}{2} \mu - \frac{1}{48}\, \frac{\pi^2 \mu^3 p^2 \smallbold{a}^4}{\sigma^2}
+  + \frac{13}{8640}\, \frac{\pi^4 \mu^5 p^4 \smallbold{a}^8}{\sigma^4} \ldots\right\} I \\
+  + \frac{\sigma}{\pi \smallbold{a}^2}
+    \left\{1 + \frac{1}{12}\, \frac{\pi^2 p^2 \mu^2 \smallbold{a}^4}{\sigma^2}
+  - \frac{1}{180}\, \frac{\pi^4 p^4 \mu^4 \smallbold{a}^8}{\sigma^4} \ldots\right\}I. \Tag{38}
+\end{DPgather*}
+We may write this as
+\begin{DPgather*}
+\smallbold{E} = \smallbold{P} \iota p I + \smallbold{Q}I,  \Tag{39}\\
+%% -----File: 307.png---Folio 293-------
+\lintertext{or since}
+\iota pI = \frac{dI}{dt},\\
+\lintertext{as}
+\smallbold{E} = \smallbold{P}  \frac{dI}{dt} + \smallbold{Q}I.
+\end{DPgather*}
+\index{Alternating currents, expression for self-induction of a single wire@\subdashtwo expression for self-induction of a single wire|(}%
+\index{Alternating currents, expression for `impedance' of a single wire@\subdashtwo expression for `impedance' of a single wire|(}%
+\index{Heaviside, ximpedance@\subdashone impedance}%
+\index{Impedance, expression for@\subdashone expression for|(}%
+\index{Induction, self@\subdashone self, expressions for|(}%
+\index{Self-induction, expression for, for variable currents}%
+
+If $L$ is the coefficient of self-induction and $R$~the resistance of
+a circuit through which a current~$I$ is flowing, we have
+\[
+\text{external electromotive force} = L \frac{dI}{dt} + RI.
+\]
+
+By the analogy of this equation with~(\eqnref{272}{39}) we may call~$\smallbold{P}$ the self-induction
+and $\smallbold{Q}$~the resistance of the cable per unit length for
+these alternating currents. $\smallbold{Q}$~has been called the `impedance' of
+\index{Impedance}%
+unit length of the circuit by Mr.~Heaviside, and this term is
+preferable to resistance as it enables the latter to be used exclusively
+for steady currents.
+
+By comparing (\eqnref{272}{39})~with~(\eqnref{272}{38}), we see that
+\[
+\left.
+\begin{aligned}
+\smallbold{P} &= 2 \log \frac{\smallbold{b}}{\smallbold{a}} + \frac{1}{2}\mu
+  - \frac{1}{48}(\mu^3 p^2 \pi^2 \smallbold{a}^4/\sigma^2)
+  + \frac{13}{8640}(\mu^5 p^4 \pi^4 \smallbold{a}^8 / \sigma^4) - \ldots,\\
+\smallbold{Q} &= \frac{\sigma}{\pi \smallbold{a}^2}
+  \left\{1 + \frac{1}{12}(\mu^2 p^2 \pi^2 \smallbold{a}^4/\sigma^2)
+  - \frac{1}{180}(\mu^4 p^4 \pi^4 \smallbold{a}^8/\sigma^4) + \ldots\right\}.
+\end{aligned}
+\right\}\eqnlabel{\eqnart.40}\nbtag{40}
+\]
+
+These results are the same as those given in equation~(18),
+Art.~690, of Maxwell's \textit{Electricity and Magnetism}, with the
+exception that $\mu$~is put equal to unity in that equation and in it
+$A$~is written instead of~$2 \log (\smallbold{b}/\smallbold{a})$.
+
+We see from these equations that as the rate of alternation
+increases, the impedance increases while the self-induction
+diminishes; both these effects are due to the influence of the rate
+of alternation on the distribution of the current. As the rate of
+alternation increases the current gets more and more concentrated
+towards the surface of the wire; the effective area of the
+wire is thus diminished and the resistance therefore increased.
+On the other hand, the concentration of the current on the surface
+of the wire increases the average distance between the portions
+of the currents in the wire, and diminishes that between the
+currents in the wire and those flowing in the opposite direction
+in the outer conductor; both these effects diminish the self-induction
+of the system of currents.
+
+The expression for~$\smallbold{Q}$ does not to our degree of approximation
+%% -----File: 308.png---Folio 294-------
+involve~$\smallbold{b}$ at all, while $\smallbold{b}$~only enters into the first term
+of the expression for~$\smallbold{P}$, which is independent of the frequency;
+thus, as long as $n\smallbold{a}$ is very small, the presence of the outer
+conductor does not affect the impedance, nor the way in which
+the self-induction varies with the frequency. When $p = 0$ the
+self-induction per unit length is $2 \log(\smallbold{b}/\smallbold{a}) + \frac{1}{2} \mu$. Since $\mu$~for soft
+iron may be as great as~$2000$, the self-induction per unit length
+of straight iron wires will be enormously greater than that of
+wires made of the \DPtypo{non-metallic}{non-magnetic} metals.
+
+\sloppy
+\Article{273} We shall now pass on to the case when $n\smallbold{a}$~is large and
+$n'\smallbold{b}$~small, so that $n\sigma J_0(\iota n\smallbold{a})/p\smallbold{a} J_0'(\iota n\smallbold{a})$ is small compared
+with $n'\sigma'K_0(\iota n'\smallbold{b})/p\smallbold{b} K_0'(\iota n'\smallbold{b})$. These conditions are compatible
+if the specific resistance of the outer conductor is very
+much greater than that of the wire. In this case equation~(\eqnref{271}{37})
+becomes
+\[
+\smallbold{E} = 2\iota p \left\{\log \frac{\smallbold{b}}{\smallbold{a}}
+  + \frac{1}{4\pi p}\, \frac{n'\sigma'}{\smallbold{b}}\,
+    \frac{K_0(\iota n'\smallbold{b})}{K_0'(\iota n'\smallbold{b})} \right\}I.
+\]
+
+\fussy
+Since $n'\smallbold{b}$ is small, we have approximately
+\begin{DPgather*}
+K_0(\iota n'\smallbold{b}) = \log(2\gamma/\iota n'\smallbold{b}), \quad
+K_0'(\iota n'\smallbold{b}) = -1/\iota n'\smallbold{b}; \\
+\lintertext{hence}
+\smallbold{E} = 2\iota p \left\{\log \smallbold{b}/\smallbold{a}
+  + \mu' \log(\gamma/\sqrt{\pi\mu'\smallbold{b}^2 p/\sigma'})
+  - \iota 3\mu' \frac{\pi}{4} \right\}.
+\end{DPgather*}
+Thus the coefficient of self-induction in this case is
+\begin{DPgather*}
+2 \log(\smallbold{b}/\smallbold{a}) + 2\mu'\log(\gamma/\sqrt{\pi\mu'p\smallbold{b}^2/\sigma')}, \\
+\lintertext{\rlap{and the impedance}}
+\tfrac{3}{2} \pi p\mu'.
+\end{DPgather*}
+
+It is worthy of remark that to our order of approximation
+neither the impedance nor the self-induction depends upon the
+resistance of the wire. This is only what we should expect for
+the self-induction, for since $n\smallbold{a}$~is large the currents will all be
+on the surface of the wire; the configuration of the currents has
+thus reached a limit beyond which it is not affected by the
+resistance of the wire. It should be noticed that the conditions
+$n\smallbold{a}$~large and $n'\smallbold{b}$~small make the impedance $\frac{3}{2} \pi p\mu'$ large compared
+with the resistance $\sigma/\pi \smallbold{a}^2$ for steady currents.
+%% -----File: 309.png---Folio 295-------
+
+\Subsection{Very Rapid Currents.}
+
+\Article{274} We must now consider the case where the frequency is so
+great that $n\smallbold{a}$~and~$n'\smallbold{b}$ are very large; in this case, by \artref{261}{Art.~261},
+\[
+J_0' (\iota n\smallbold{a}) = -\iota J_0(\iota n\smallbold{a}), \quad
+K_0' (\iota n'\smallbold{b}) = \iota K_0(\iota n'\smallbold{b}),
+\]
+so that equation~(\eqnref{271}{37}) becomes
+\[
+\smallbold{E} = 2 \iota p \left\{\log \frac{\smallbold{b}}{\smallbold{a}}
+  + \left[ \Bigl( \frac{\sigma \mu}{4 \pi p \smallbold{a}^2} \Bigr)^{\frac{1}{2}}
+  +        \Bigl( \frac{\sigma' \mu'}{4 \pi p \smallbold{b}^2} \Bigr)^{\frac{1}{2}} \right]
+    \Bigl( \frac{1}{\surd{2}} - \frac{\iota}{\surd{2}} \Bigr) \right\} I; \Tag{41}
+\]
+we see from this equation that the self-induction~$\smallbold{P}$ is given by
+the equation
+\[
+\smallbold{P} = 2 \log(\smallbold{b} / \smallbold{a})
+  + (\sigma \mu /2 \pi p \smallbold{a}^2)^{\frac{1}{2}}
+  + (\sigma'\mu'/2 \pi p \smallbold{b}^2)^{\frac{1}{2}},  \Tag{42}
+\]
+and the impedance~$\smallbold{Q}$ by
+\[
+\smallbold{Q}
+  = (\sigma \mu p / 2 \pi \smallbold{a}^2)^{\frac{1}{2}}
+  + (\sigma'\mu'p / 2 \pi \smallbold{b}^2)^{\frac{1}{2}}. \Tag{43}
+\]
+
+In a cable the conductivity of the outer conductor is very
+much less than that of the core, so that $\sigma'/\smallbold{b}^2$ will be large compared
+with~$\sigma/\smallbold{a}^2$; thus the self-induction and impedance of a cable
+are both practically independent of the resistance of the wire
+and depend mainly upon that of the outer conductor. The limiting
+value of the self-induction when the frequency is indefinitely
+increased is $2 \log(\smallbold{b}/\smallbold{a})$; as this does not involve~$\mu$ it is the same
+for iron as for copper wires. The difference between the self-induction
+per unit length of the cable for infinitely slow and
+infinitely rapid vibrations is by equations (\eqnref{272}{40})~and~(\eqnref{274}{42}) equal to~$\mu/2$.
+The impedance of the circuit increases indefinitely with
+the frequency of the alternations.
+
+If we trace the changes in the values of the self-induction
+and impedance as the frequency~$p$ increases, we see from
+Arts.\ \artref{272}{272},~\artref{273}{273},~\artref{274}{274}
+that when this is so small that $n\smallbold{a}$~is a small
+quantity the self-induction decreases and the impedance increases
+by an amount proportional to the square of the
+frequency. When the frequency increases so that $n\smallbold{a}$~is considerable
+while $n'\smallbold{b}$~is small, the self-induction varies very
+slowly with the frequency while the impedance is directly
+proportional to it. When the frequency is so great that both
+$n\smallbold{a}$~and~$n'\smallbold{b}$ are large the self-induction approaches the limit
+$2 \log(\smallbold{b}/\smallbold{a})$, while the impedance is proportional to the square root
+of the frequency.
+\index{Alternating currents, expression for self-induction of a single wire@\subdashtwo expression for self-induction of a single wire|)}%
+\index{Alternating currents, expression for `impedance' of a single wire@\subdashtwo expression for `impedance' of a single wire|)}%
+\index{Impedance, expression for@\subdashone expression for|)}%
+%% -----File: 310.png---Folio 296-------
+
+
+\Subsection{Flat Conductors.}
+\index{Alternating currents, in flat conductors@\subdashtwo in flat conductors}%
+\index{Impedance, for flat conductors@\subdashone for flat conductors}%
+\index{Induction, self, for flat conductors@\subdashtwo for flat conductors}%
+\index{Self-induction, expression for, for xflat conductors@\subdashtwo for flat conductors}%
+
+\Article{275}  In many experiments flat strips of metal in parallel
+planes are used instead of wires, with the view of diminishing
+the self-induction; these are generally arranged so that the
+direct and return currents flow along adjacent and parallel
+strips. When the frequency of the vibrations is very large, the
+positive and negative currents endeavour to get as near together
+as possible, they will thus flow on the surfaces of the strips
+which are nearest each other. If the distance between the
+planes of the strips is small in comparison with their breadth we
+may consider them as a limiting case of the cable, when the
+specific resistance of the wire is the same as that of the outer
+conductor, and when the values of $\smallbold{a}$~and~$\smallbold{b}$ are indefinitely
+great, their difference however remaining finite and equal to the
+distance between the strips. If $I'$~is the current flowing across
+unit width of the strip, then, since with our previous notation $I$
+is the current flowing over the circumference of the cable,
+\[
+I' = I / 2 \pi \smallbold{a}.
+\]
+Since $\smallbold{b} = \smallbold{a} + d$, where $d$~is very small compared with~$\smallbold{a}$,
+\[
+\log \frac{\smallbold{b}}{\smallbold{a}} = \frac{d}{\smallbold{a}} \text{ approximately.}
+\]
+
+Making these substitutions, equation~(\eqnref{274}{41}) becomes
+\[
+\smallbold{E} = 2 \iota p \left\{d + \sqrt{\frac{\mu \sigma}{2 \pi p}} (1 - \iota) \right \} 2 \pi I'.
+\]
+Thus, in this case the self-induction per unit length is
+\[
+4 \pi \left\{d + \sqrt{\frac{\mu \sigma}{2 \pi p}} \right\},
+\]
+and the impedance
+\[
+4\pi \sqrt{\frac{\mu \sigma p}{2 \pi}}.
+\]
+
+\Article{276} Though, as we have just seen, it is possible to regard the
+case of two parallel metal slabs as a particular case of the cable,
+yet inasmuch as the geometry of the particular case is much
+simpler than that of the cable, the case is one where points of
+theory are most conveniently discussed; it is therefore advisable
+\index{Induction, self@\subdashone self, expressions for|)}%
+%% -----File: 311.png---Folio 297-------
+to treat it independently. We shall suppose that we have two
+slabs of the same metal, the adjacent faces of the slabs being
+parallel and separated by the distance~$2h$; we shall take the
+plane parallel to these faces and midway between them as the
+plane of~$yz$, the axis of~$x$ being normal to the faces. We shall
+suppose that all the variable quantities vary as $\epsilon^{\iota(mz+pt)}$ and are
+independent of~$y$. The slabs are supposed also to extend to infinity
+in directions parallel to~$y$ and~$z$ and to be infinitely thick.
+
+Let $\sigma$ be the specific resistance of the slabs, $V$~the velocity of
+propagation of electrodynamic action through the dielectric
+which separates them. Then, using the same notation as before,
+since all the quantities are independent of~$y$, the differential
+equations satisfied by the components of the electromotive
+intensity are by \artref{262}{Art.~262}
+\begin{DPgather*}
+\frac{d^2R}{dx^2} = k^2 R \text{ in the dielectric}, \\
+\lintertext{and}
+\frac{d^2R}{dx^2} = n^2 R
+\end{DPgather*}
+in either of the slabs.
+
+Thus, in the dielectric we may put
+\begin{align*}
+R &= (A \epsilon^{kx} + B \epsilon^{-kx}) \epsilon^{\iota(mz+pt)}, \\
+P &= - \frac{\iota m}{k} (A \epsilon^{kx} - B \epsilon^{-kx})  \epsilon^{\iota(mz+pt)},
+\end{align*}
+in the slab for which $x$~is positive
+\begin{align*}
+R &= C \epsilon^{-nx} \epsilon^{\iota(mz+pt)}, \\
+P &= \frac{\iota m}{n} C \epsilon^{-nx} \epsilon^{\iota(mz+pt)},
+\end{align*}
+and in that for which $x$~is negative
+\begin{align*}
+R &= D \epsilon^{nx} \epsilon^{\iota(mz+pt)}, \\
+P &= -\frac{\iota m}{n} D \epsilon^{nx} \epsilon^{\iota(mz+pt)},
+\end{align*}
+the real part of~$n$ being taken positive in both cases.
+
+Since $R$ is continuous when $x = ±h$, we have
+\[
+\left.\begin{aligned}
+A \epsilon^{kh} + B \epsilon^{-kh} & = C \epsilon^{-nh}, \\
+A \epsilon^{-kh} + B \epsilon^{kh} & = D \epsilon^{-nh}.
+\end{aligned}\right\}\Tag{44}
+\]
+%% -----File: 312.png---Folio 298-------
+
+Since the magnetic force parallel to the surface is continuous,
+we have, if $\mu$ is the magnetic permeability of the slab,
+\begin{gather*}
+\frac{k^2-m^2}{k} (A\epsilon^{kh} - B\epsilon^{-kh}) =  \frac{m^2-n^2}{\mu n} C\epsilon^{-nh}, \\
+\frac{k^2-m^2}{k} (A\epsilon^{-kh} - B\epsilon^{kh}) = -\frac{m^2-n^2}{\mu n} D\epsilon^{-nh}.
+\end{gather*}
+
+Eliminating $C$~and~$D$ by the aid of equations~(\eqnref{276}{44}), we have
+\[
+\left.\begin{aligned}
+A\left(\frac{k^2-m^2}{k} - \frac{m^2-n^2}{\mu n}\right)\epsilon^{kh}  & = B\left(\frac{k^2-m^2}{k} + \frac{m^2-n^2}{\mu n}\right)\epsilon^{-kh}, \\
+A\left(\frac{k^2-m^2}{k} + \frac{m^2-n^2}{\mu n}\right)\epsilon^{-kh} & = B\left(\frac{k^2-m^2}{k} - \frac{m^2-n^2}{\mu n}\right)\epsilon^{kh}.
+\end{aligned}\right\} \Tag{45}
+\]
+From these equations we get
+\[
+A^2 = B^2.
+\]
+
+The solution $A = B$ corresponds to the current flowing in the
+same direction in the two slabs, the other solution corresponds
+to the case when the current flows in one direction in one slab
+and in the opposite direction in the other; it is this case we shall
+proceed to investigate. Putting $A = -B$, equation~(\eqnref{276}{45}) becomes
+\begin{DPgather*}
+\frac{k^2-m^2}{k}(\epsilon^{kh} + \epsilon^{-kh}) +
+\frac{n^2-m^2}{\mu n}(\epsilon^{kh}-\epsilon^{-kh})=0; \Tag{46} \\
+\lintertext{but}
+k^2-m^2 = -p^2/V^2, \\
+n^2-m^2 = 4\pi \mu \iota p/\sigma,
+\end{DPgather*}
+and $kh$ is very small, thus (\eqnref{276}{46})~becomes approximately
+\begin{DPgather*}
+\frac{p^2}{V^2 k} = \frac{4\pi \iota p}{\sigma n}\, kh, \\
+\lintertext{or}
+k^2 = -\frac{p^2}{V^2}\, \frac{\iota n\sigma}{4\pi hp}, \\
+\lintertext{so that}
+m^2 = \frac{p^2}{V^2} \left\{1 - \frac{\iota n\sigma}{4\pi hp} \right\}. \Tag{47}
+\end{DPgather*}
+
+As we have remarked before, $4\pi \mu p/\sigma$ is in the case of metals
+very large compared with~$m^2$, so that we have approximately
+\[
+n^2 = 4\pi \mu \iota p/\sigma,
+\]
+%% -----File: 313.png---Folio 299-------
+and therefore approximately by equation~(\eqnref{276}{47})
+\[
+m^2 = \frac{p^2}{V^2} \left\{1 + \left( \frac{\sigma \mu}{8 \pi h^2 p} \right)^{\frac{1}{2}} (1 - \iota) \right\}.  \Tag{48}
+\]
+
+Thus, if $m = \xi + \iota \eta$, we have
+\begin{gather*}
+\xi^2 - \eta^2 = \frac{p^2}{V^2} \left\{1 + \left( \frac{\sigma \mu}{8 \pi h^2 p} \right)^{\frac{1}{2}} \right\},\\
+2 \xi \eta = - \frac{p^2}{V^2} \left( \frac{\sigma \mu}{8 \pi h^2 p} \right)^{\frac{1}{2}}.
+\end{gather*}
+
+But if $\omega$~is the velocity with which the phases are propagated
+along the slab, $\omega^2 = p^2 / \xi^2$, so that we have
+\[
+\frac{1}{\omega^2} = \frac{\eta^2}{p^2}
+  + \frac{1}{V^2} \left\{1 + \left( \frac{\sigma \mu}{8 \pi h^2 p} \right)^{\frac{1}{2}} \right\},
+\]
+thus $1 / \omega^2$ is never less than~$1/V^2$, or $\omega$~is never greater than~$V$,
+so that the velocity of propagation of the phases along the slab
+can never exceed the rate at which electrodynamic action travels
+through the dielectric.
+
+If the frequency is so high that $\sigma \mu / 8 \pi h^2 p$ is small, then we
+have by equation~(\eqnref{276}{48})
+\[
+m = - \frac{p}{V} \left\{1 - \frac{\iota}{2} \left( \frac{\sigma \mu}{8 \pi h^2 p} \right)^{\frac{1}{2}} \right\} \text{ approximately}.
+\]
+
+This equation represents a disturbance propagated with the
+velocity~$V$, whose amplitude fades away to $1/\epsilon$~of its original
+value after traversing a distance
+\[
+2Vh \left\{\frac{8 \pi p}{\sigma \mu} \right\}^{\frac{1}{2}}.
+\]
+
+If the frequency is so low that $\sigma \mu / 8 \pi h^2 p$ is large, then we
+have approximately by equation~(\eqnref{276}{48})
+\begin{DPgather*}
+m = -\frac{p}{V} \left\{\frac{\sigma \mu}{4 \pi h^2 p} \right\}^{\frac{1}{4}}
+  \left( \cos{\frac{\pi}{8}} - \iota \sin{\frac{\pi}{8}} \right), \\
+\lintertext{or}
+m = -\frac{p}{V} \left\{\frac{\sigma \mu}{4 \pi h^2 p} \right\}^{\frac{1}{4}}
+  \left( .92-\iota .38 \right).
+\end{DPgather*}
+This corresponds to a vibration propagated with the velocity
+\[
+1.08 V \{4 \pi h^2 p / \sigma \mu \}^{\frac{1}{4}},
+\]
+%% -----File: 314.png---Folio 300-------
+and fading away to $1/\epsilon$~of its original amplitude after traversing
+a distance
+\[
+2.6V \{4 \pi h^2 / \mu \sigma p^3 \}^{\frac{1}{4}}.
+\]
+
+If the total current through a slab per unit width is represented
+by the real part of $I_0 \epsilon^{\iota (mz+pt)}$, then, when the frequency
+is so great that $\sigma \mu / 8 \pi h^2 p$ is a small quantity and therefore the
+real part of~$m$ large compared with the imaginary part, we
+have since
+\begin{DPgather*}
+I_0 \epsilon^{\iota (mz+pt)}
+  = \int_h^\infty \frac{R}{\sigma}\,dx
+  = \frac{C}{\sigma n} \epsilon^{-nh} \epsilon^{\iota (mz+pt)}, \\
+C = \sigma n \epsilon^{nh} I_0; \\
+\lintertext{hence by~(\eqnref{276}{44})}
+A = -B = \frac{\sigma n I_0}{2kh}.
+\end{DPgather*}
+We have therefore in the dielectric
+\begin{DPgather*}
+\begin{aligned}
+R &= \sigma I_0 \sqrt{2} n' (x/h) \cos \left( mz + pt + \frac{\pi}{4} \right),\\
+P &= 4 \pi m I_0 (V^2/p) \cos(mz + pt),\\
+b &= -4 \pi I_0 \cos(mz + pt),
+\end{aligned} \\
+\lintertext{where}
+n' = \{2 \pi \mu p/ \sigma \}^{\frac{1}{2}}.
+\end{DPgather*}
+
+In the metal slab we have on the side where $x$~is positive,
+\begin{align*}
+R &= \sigma I_0 \DPtypo{\surd{2}}{\sqrt{2}} n' \epsilon^{-n'(x-h)} \cos \left( mz+pt-n'(x-h) + \frac{\pi}{4} \right),\\
+P &= -\sigma I_0 m \epsilon^{-n'(x-h)} \sin(mz+pt-n'(x-h)),\\
+b &= -4 \pi \mu I_0 \epsilon^{-n'(x-h)} \cos(mz+pt-n'(x-h)).
+\end{align*}
+
+We see from these equations that $P/R$ is very large in the
+dielectric and very small in the metal slab, thus the Faraday
+tubes are at right angles to the conductor in the dielectric and
+parallel to it in the metal slab.
+
+
+\Subsection{Mechanical Force between the Slabs.}
+\index{Attraction between flat conductors conveying variable currents}%
+\index{Force between flat conductors conveying alternating currents@\subdashone between flat conductors conveying alternating currents}%
+\index{Mechanical force xbetween flat conductors conveying alternating currents@\subdashtwo between flat conductors conveying alternating currents}%
+
+\Article{277} This may be regarded as consisting of two parts, (1)~an
+attractive force, due to the attraction of the positive electricity
+of one slab on the negative of the other, (2)~a repulsive force,
+due to the repulsion between the positive currents in one slab
+and the negative in the other. To calculate the first force we
+notice that since $V^2 / p \sigma$ is very large, the value of~$P$ in the conductor
+%% -----File: 315.png---Folio 301-------
+is very small compared with the value in the dielectric,
+and may without appreciable error be neglected; hence if~$e$ is the
+surface density of the electricity on the slab and $K$~the specific
+inductive capacity of the dielectric,
+\[
+4 \pi e = -K 4 \pi m (V^2/p) I_0 \cos(mz + pt).
+\]
+The force on the slab per unit area is equal to~$Pe/2$; substituting
+the values of $P$~and~$e$ this becomes
+\[
+2 \pi K m^2 (V^4 / p^2) I_0^2 \cos^2 (mz + pt).
+\]
+
+The force due to the repulsion between the currents in the
+slabs per unit volume is equal to the product of the magnetic
+induction~$b$ into~$w$, the intensity of the current parallel to~$z$.
+Since
+\[
+4 \pi \mu w = - \frac{db}{dx},
+\]
+the force per unit volume is equal to
+\[
+-\frac{1}{8 \pi \mu}\, \frac{db^2}{dx},
+\]
+hence the repulsive force per unit area of the surface of the slab
+\begin{align*}
+&= - \int_h^\infty \frac{1}{8 \pi \mu}\, \frac{db^2}{dx}\,dx \\
+&= \frac{1}{8 \pi \mu} \left( b^2 \right)_{x = h} = 2 \pi \mu I_0^2 \cos^2 (mz + pt).
+\end{align*}
+When the alternations are so rapid that the vibrations travel
+with the velocity of light
+\[
+V^2 m^2 = p^2,
+\]
+and since $K = 1/V^2$, the attraction between the slabs is equal to
+\[
+2 \pi I_0^2 \cos^2 (mz + pt),
+\]
+while the repulsion is
+\[
+2 \pi \mu I_0^2 \cos^2 (mz + pt),
+\]
+hence the resultant repulsion is equal to
+\[
+2 \pi (\mu - 1) I_0^2 \cos^2 (mz + pt).
+\]
+
+If the slabs are non-magnetic $\mu = 1$, so that for these very
+rapid vibrations the electrostatic attraction just counterbalances
+the electromagnetic repulsion. Mr.~Boys (\textit{Phil.\ Mag.}\ [5],~31, p.~44,
+1891) found that the mechanical forces between two conductors
+carrying very rapidly alternating currents was too small to be
+%% -----File: 316.png---Folio 302-------
+detected, even by the marvellously sensitive methods for measuring
+small forces which he has perfected, and which would have
+enabled him to detect forces comparable in magnitude with
+those due to the electrostatic charges or to the repulsion between
+the currents.
+
+
+\Subsection{Propagation of Longitudinal Waves of Magnetic
+Induction along Wires.}
+\index{Longitudinal waves of magnetic induction along wires}%
+\index{Magnetic yinduction@\subdashone induction, longitudinal waves of, along wires}%
+
+\Article{278} In the preceding investigations the current has been
+along the wire and the lines of magnetic force have formed a
+series of co-axial circles, the axis of these circles being that of
+the wire. Another case, however, of considerable practical importance
+is when these relations of the magnetic force and current
+are interchanged, the current flowing in circles round the axis of
+the wire while the magnetic force is mainly along it. This
+condition might be realized by surrounding a portion of the wire
+by a short co-axial solenoid, then if alternating currents are sent
+through this solenoid periodic magnetic forces parallel to the
+wire will be started. We shall in this article investigate the
+laws which govern the transmission of such forces along the wire.
+The problem has important applications to the construction of
+transformers; in some of these the primary coil is wound round
+one part of a closed magnetic circuit, the secondary round
+another. This arrangement will not be efficient if there is any
+considerable leakage of the lines of magnetic force between the
+primary and the secondary. We should infer from general considerations
+that the magnetic leakage would increase with the
+rate of alternation of the current through the primary. For let us
+suppose that an alternating current passes through an insulated
+ring imbedded in a cylinder of soft iron surrounded by air, the
+straight axis of the ring coinciding with the axis of the cylinder.
+The variations in the intensity of the current through this ring
+will induce other currents in the iron in its neighbourhood; the
+magnetic action of these currents will, on the whole, cause the component
+of the magnetic force along the axis of the cylinder to be
+less and the radial component greater than if the current through
+the ring were steady; in which case there are no currents in the
+iron. Thus the effect of the changes in the intensity of the current
+through the primary will be to squeeze as it were the lines of
+%% -----File: 317.png---Folio 303-------
+magnetic force out of the iron and make them complete their circuit
+through the air. Thus when the field is changing quickly, the
+lines of magnetic force, instead of taking a long path through the
+medium of high permeability, will take a short path, even though
+the greater part of it is through a medium of low permeability
+such as air. The case is quite analogous to the difference between
+the path of a steady current and that of a rapidly alternating
+one. A steady current flows along the path of least resistance,
+a rapidly alternating one along the path with least self-induction.
+Thus, for example, if we have two wires in parallel, one very
+long but made of such highly conducting material that the
+total resistance is small, the other wire short but of such a nature
+that the resistance is large, then when the current is steady
+by far the greater part of it will travel along the long wire;
+if however the current is a rapidly alternating one, the greater
+part of it will travel along the short wire because the self-induction
+is smaller than for the long wire, and for these
+rapidly alternating currents the resistance is a secondary consideration.
+
+In the magnetic problem the iron corresponds to the good
+conductor, the air to the bad one. When the field is steady
+the lines of force prefer to take a long path through the iron
+rather than a short one through the air; they will thus tend to
+keep within the iron; when however the magnetic field is a very
+rapidly alternating one, the paths of the lines of force will tend
+to be as short as possible, whatever the material through which
+they pass. The lines of force will thus in this case leave the
+iron and complete their circuit through the air.
+
+We shall consider the case of a right circular soft iron cylinder
+where the lines of magnetic force are in planes through the axis
+taken as that of $z$, the corresponding system of currents flowing
+round circles whose axis is that of the cylinder. The cylinder
+is surrounded by a dielectric which extends to infinity. Let $a$,~$b$,~$c$
+be the components of the magnetic induction parallel to the axes
+of $x$,~$y$,~$z$ respectively; then, since the component of the magnetic
+induction in the $xy$~plane is at right angles to the axis of the
+cylinder, we may put
+\[
+a = \frac{d \chi}{dx}, \quad b = \frac{d \chi}{dy}.
+\]
+
+Let us suppose that $a$,~$b$,~$c$ all vary as $\epsilon^{\iota (mz + pt)}$.
+%% -----File: 318.png---Folio 304-------
+
+Now in the iron cylinder $a$,~$b$,~$c$ all satisfy differential equations
+of the form
+\begin{DPgather*}
+\frac{d^2c}{dx^2} + \frac{d^2c}{dy^2} = n^2c, \\
+\lintertext{where} n^2 = m^2 + 4\pi \mu \iota p/\sigma,
+\end{DPgather*}
+$\mu$ being the magnetic permeability and~$\sigma$ the specific resistance of
+the cylinder.
+
+In the dielectric outside the cylinder the differential equation
+satisfied by the components of the magnetic induction is of the
+form
+\begin{DPgather*}
+\frac{d^2c}{dx^2} + \frac{d^2c}{dy^2} = k^2c, \\
+\lintertext{where} k^2 = m^2 - \frac{p^2}{V^2},
+\end{DPgather*}
+and $V$ is the velocity with which electromagnetic disturbances
+are propagated through the dielectric.
+
+We have also
+\[
+\frac{da}{dx} + \frac{db}{dy} + \frac{dc}{dz} = 0.
+\]
+
+The solution of these equations is easily seen to be, in the
+iron cylinder,
+\begin{align*}
+c & = AJ_0(\iota nr)\epsilon^{\iota (mz+pt)}, \\
+a & = -\frac{\iota m}{n^2} A \frac{d}{dx} J_0(\iota nr)\epsilon^{\iota (mz+pt)}, \\
+b & = -\frac{\iota m}{n^2} A \frac{d}{dy} J_0(\iota nr)\epsilon^{\iota (mz+pt)},
+\end{align*}
+while in the dielectric, since $r$~can become infinite,
+\begin{align*}
+c & = CK_0(\iota kr)\epsilon^{\iota (mz+pt)}, \\
+a & = -\frac{\iota m}{k^2} C \frac{d}{dx} K_0(\iota kr)\epsilon^{\iota (mz+pt)}, \\
+b & = -\frac{\iota m}{k^2} C \frac{d}{dy} K_0(\iota kr)\epsilon^{\iota (mz+pt)}.
+\end{align*}
+
+Let $\smallbold{a}$ be the radius of the cylinder, then when $r = \smallbold{a}$ the
+tangential magnetic force in the cylinder is equal to that in the
+dielectric, hence
+\[
+\frac{A}{\mu} J_0(\iota n\smallbold{a}) = CK_0(\iota k\smallbold{a});
+\]
+%% -----File: 319.png---Folio 305-------
+since the radial magnetic induction is continuous, we have
+\[
+\frac{m}{n} AJ_0'(\iota n\smallbold{a}) = \frac{m}{k} CK_0'(\iota k\smallbold{a}).
+\]
+Eliminating $A$ and~$C$ from these equations, we get
+\[
+\frac{\iota n\smallbold{a}}{\mu}\, \frac{J_0(\iota n\smallbold{a})}{J_0'(\iota n\smallbold{a})}
+  = \iota k\smallbold{a} \frac{K_0(\iota k\smallbold{a})}{K_0'(\iota k\smallbold{a})}, \Tag{49}
+\]
+an equation which will enable us to find~$m$ when $p$~is known.
+
+Let us begin with the case when the frequency of the alternations
+is small enough to allow of the currents being nearly
+uniformly distributed over the cross-section of the cylinder. In
+this case we have approximately
+\[
+J_0(\iota n\smallbold{a})= 1, \quad
+J_0'(\iota n\smallbold{a}) = -\tfrac{1}{2} \iota n\smallbold{a},
+\]
+so that equation~(\eqnref{278}{49}) becomes
+\[
+-\frac{2}{\mu} = \iota k\smallbold{a} \frac{K_0(\iota k\smallbold{a})}{K_0'(\iota k\smallbold{a})}.   \Tag{50}
+\]
+
+Since for soft iron $2/\mu$ is a small quantity, the right-hand side
+of this equation and therefore $k\smallbold{a}$ must be small; but in this case
+we have approximately
+\begin{align*}
+K_0(\iota k\smallbold{a}) & = \log(2\gamma/\iota k\smallbold{a}), \\
+K_0'(\iota k\smallbold{a}) & = -\frac{1}{\iota k\smallbold{a}},
+\end{align*}
+so that~(\eqnref{278}{50}) becomes
+\[
+-\frac{2}{\mu} = k^2 \smallbold{a}^2 \log(2\gamma/\iota k\smallbold{a}).   \Tag{51}
+\]
+To solve this equation consider the solution of
+\[
+x \log x = -y,
+\]
+when $y$~is small. If $x= -y/\log y$, then
+\[
+x \log x = -y \left\{1 + \frac{\log \log(1/y)}{\log(1/y)} \right\},
+\]
+but when $y$~is small $\log \log(1/y)$ is small compared with~$\log(1/y)$,
+so that an approximate solution of the equation is
+\[
+x = -y/\log y.
+\]
+%% -----File: 320.png---Folio 306-------
+
+If we apply this result to equation~(\eqnref{278}{51}), we find that the
+approximate solution of that equation is
+\[
+k^2 = - \frac{4}{\smallbold{a}^{2} \mu}\, \frac{1}{\log(\mu\gamma^2)}.
+\]
+\begin{DPgather*}
+\lintertext{\indent Now} k^2 = m^2 - \frac{p^2}{V^2},
+\end{DPgather*}
+and since the value we have just found for~$k$ is in any practicable
+case very large compared with~$p^2/V^2$, we see that $k^2=m^2$ approximately,
+so that
+\[
+m = \frac{\iota 2}{\smallbold{a}} \left\{\frac{1}{\mu} \frac{1}{\log(\mu\gamma^2)}\right\}^{\frac{1}{2}}.
+\]
+
+Thus since in the expression for~$c$ there is the factor
+\[
+\epsilon^{\iota mz} \text{ or }
+\epsilon^{-\frac{2}{\smallbold{a}} z
+  \left\{\frac{1}{\mu \log(\mu\gamma^2)} \right\}^{\frac{1}{2} }},
+\]
+we see that the magnetic force will die away to $1/\epsilon$~of its value
+at a distance
+\[
+\tfrac{1}{2} \smallbold{a} \{\mu \log(\mu\gamma^2)\}^{\frac{1}{2}}
+\]
+from its origin.
+
+\Article{279} In the last case the current was uniformly distributed
+over the cross-section. We can investigate the effect
+of the concentration of the current at the boundary of the
+cylinder by supposing that~$n\smallbold{a}$ is large compared with unity
+though small compared with~$\mu$. In this case, since approximately
+\[
+J_0' (\iota n \smallbold{a}) = -\iota J_0(\iota n \smallbold{a}),
+\]
+equation~(\eqnref{278}{49}) becomes
+\[
+-\frac{n \smallbold{a}}{\mu}
+  = \iota k \smallbold{a} \frac{K_0(\iota k \smallbold{a})}{K_0'(\iota k \smallbold{a})}.
+\]
+Since the left-hand side of this equation is small, $\iota k \smallbold{a}$~is also small,
+so that by \artref{261}{Art.~261} we may write this equation as
+\[
+-\frac{n \smallbold{a}}{\mu} = k^2 \smallbold{a}^2 \log(2\gamma/\iota k \smallbold{a}). \Tag{52}
+\]
+This equation gives a value for~$k^2$ which is very large compared
+with~$p^2/V^2$, so that approximately $m = k$. We also see that $k$ or
+$m$ is small compared with~$n$, we may therefore put
+\[
+n = \{4\pi\mu\iota p/ \sigma\}^{\frac{1}{2}}.
+\]
+%% -----File: 321.png---Folio 307-------
+Thus equation~(\eqnref{279}{52}) becomes
+\[
+k^2 \smallbold{a}^2 \log \frac{\iota k\smallbold{a}}{2\gamma}
+  = \left\{\frac{4\pi p\smallbold{a}^2}{\mu \sigma}\right\}^{\tfrac{1}{2}}\,
+    \epsilon^{\tfrac{\iota \pi}{4}},
+\]
+or putting $\iota k\smallbold{a} / 2\gamma = q$,
+\[
+q^2 \log q^2
+  = -\frac{1}{2\gamma^{2}} \left\{\frac{4\pi p\smallbold{a}^2}{\mu \sigma}\right\}^{\frac{1}{2}}\,
+     \epsilon^{\tfrac{\iota \pi}{4} }.
+\]
+
+To solve this equation put $q^2 = w\epsilon^{\iota \psi }$; equating real and imaginary
+parts, we get
+\begin{align*}
+w \log w \cos \psi - w \psi \sin \psi &= -\frac{1}{\gamma^2} \left\{\frac{\pi p\smallbold{a}^2}{2\mu \sigma} \right\}^{\frac{1}{2}},\\
+w \log w \sin \psi + w \psi \cos \psi &= -\frac{1}{\gamma^2} \left\{\dfrac{\pi p\smallbold{a}^2}{2\mu \sigma} \right\}^{\frac{1}{2}}.
+\end{align*}
+Since $w$~is very small, the terms in~$\log w$ are much the most
+important; an approximate solution of these equations is, therefore,
+since the solution of $x \log x = -y$, is $x = -y/\log y$,
+\begin{align*}
+w &= - \frac{\dfrac{1}{\gamma^2}
+  \left\{\dfrac{\pi p \smallbold{a}^2}{\mu \sigma} \right\}^{\frac{1}{2}}}
+        {\log \dfrac{1}{\gamma^2}
+  \left\{\dfrac{\pi p \smallbold{a}^2}{\mu \sigma} \right\}^{\frac{1}{2}}},\\
+\psi &= \frac{\pi}{4}.
+\end{align*}
+Hence, since $k = m$ and $k^2\smallbold{a}^2 = -4\gamma^2 w \epsilon^{\dfrac{\iota \pi}{4}}$, we find
+\begin{align*}
+m\smallbold{a}
+  &= 2\gamma \sqrt{-w} \left\{\cos\frac{\pi}{8} + \iota \sin\frac{\pi}{8}\right\}\\
+  &= 2\gamma w^{\frac{1}{2}} \left(\cos \frac{5\pi}{8} + \iota \sin \frac{5\pi}{8}\right).
+\end{align*}
+
+Thus, since in the expression for~$c$ there is the factor~$\epsilon^{\iota mz}$, we
+see that $c$~will fade away to $1/\epsilon$~of its initial value at a distance
+from the origin equal to
+\[
+\frac{\smallbold{a}}{2\gamma w^{\frac{1}{2}}} \cosec \frac{5\pi}{8},
+\]
+or substituting the value of~$w$ just found,
+\[
+\frac{\smallbold{a}}{2} \cosec \frac{5\pi}{8}
+  \biggl\{\frac{\mu \sigma}{\pi p\smallbold{a}^2} \biggr\}
+  \biggl\{\log \gamma^2 \Bigl(\frac{\mu \sigma}{\pi p\smallbold{a}^2}\Bigr)^{\tfrac{1}{2}} \biggr\}^{\tfrac{1}{2}}.
+\]
+
+This distance is much shorter than the corresponding one
+%% -----File: 322.png---Folio 308-------
+when the current was uniformly distributed over the cross-section
+of the wire, and the important factor varies as~$\mu^{\frac{1}{4}}$
+instead of~$\mu^{\frac{1}{2}}$. Thus the leakage of the lines of magnetic force
+out of the iron cylinder is much greater when the alternations
+are rapid than when they are slow. This is in accordance with
+the conclusion we came to from general reasoning at the
+beginning of \artref{278}{Art.~278}.
+
+The result of this investigation points strongly to the advisability
+of very fine lamination of the core of a transformer,
+so as to get a uniform distribution of magnetic force over
+the iron and thus avoid magnetic leakage. There are many
+other advantages gained by fine lamination, of which one, more
+important than the effect we are considering, is the diminution
+in the quantity of heat dissipated by eddy currents. We shall
+proceed to consider in the next \artref{280}{article} the dissipation of energy
+by the currents in the wire.
+
+
+\Subsection{Dissipation of Energy by the Heat produced by
+Alternating Currents.}
+\index{Energy, transfer of}%
+
+\Article{280} A great deal of light is thrown on the laws which govern
+the decay of currents in conductors by the consideration of the
+circumstances which affect the amount of heat produced in unit
+time by these currents. As we have obtained the expressions
+for these currents we could determine their heating effect by
+direct integration; we shall however proceed by a different
+method for the sake of introducing a very important theorem
+\index{Poynting's theorem}%
+due to Professor Poynting, and given by him in his paper `On
+the Transfer of Energy in the Electromagnetic Field,' \textit{Phil.\
+\index{Transfer of energy}%
+Trans.}\ 1884, Part~II, p.~343. The theorem is that
+\begin{multline*}
+\frac{K}{4 \pi} \iiint \left(
+  P\, \frac{dP}{dt} + Q\, \frac{dQ}{dt} + R\, \frac{dR}{dt}
+\right) dx\,dy\,dz\\
+  \shoveleft{\qquad + \frac{\mu}{4 \pi} \iiint \left(
+  \alpha\, \frac{d\alpha}{dt} + \beta\, \frac{d\beta}{dt} + \gamma\, \frac{d\gamma}{dt}
+\right) dx\,dy\,dz}\\
+  \shoveleft{\qquad + \iiint ( X \dot{x} + Y \dot{y} + Z \dot{z} )\,dx\,dy\,dz
+  + \iiint ( Pp + Qq + Rr )\,dx\,dy\,dz}\\
+  = \frac{1}{4 \pi} \iint
+    \{l ( R' \beta - Q' \gamma )
+     + m ( P' \gamma - R' \alpha )
+     + n ( Q' \alpha - P' \beta ) \}\,dS,
+\end{multline*}
+%% -----File: 323.png---Folio 309-------
+where the volume integrals on the left-hand side are taken
+throughout the volume contained by the closed surface~$S$, of
+which $dS$~is an element and $l$,~$m$,~$n$ the direction cosines of the
+normal drawn outwards.
+
+$P$,~$Q$,~$R$ are the components of the electromotive intensity.
+
+$\alpha$,~$\beta$,~$\gamma$ those of the magnetic force.
+
+$X$,~$Y$,~$Z$ those of the mechanical force acting on the body in
+consequence of the passage of currents through it.
+
+$\dot{x}$,~$\dot{y}$,~$\dot{z}$ the components of the velocity of a point in the
+body.
+
+$p$,~$q$,~$r$ the components of the conduction currents.
+
+$P'$,~$Q'$,~$R'$ the parts of the components of the electromotive
+intensity which do not depend upon the motion of the
+body.
+
+$K$~the specific inductive capacity and $\mu$~the magnetic permeability.
+
+The following proof of this theorem is taken almost verbatim
+from Professor Poynting's paper. Let $u$,~$v$,~$w$ be the components
+of the total current, which is the sum of the polarization and
+conduction currents; we have, since the components of the former
+are respectively
+\begin{gather*}
+\frac{K}{4 \pi}\, \frac{dP}{dt}, \quad
+\frac{K}{4 \pi}\, \frac{dQ}{dt}, \quad
+\frac{K}{4 \pi}\, \frac{dR}{dt},\\
+\frac{K}{4 \pi}\, \frac{dP}{dt} = u - p,\\
+\frac{K}{4 \pi}\, \frac{dQ}{dt} = v - q,\\
+\frac{K}{4 \pi}\, \frac{dR}{dt} = w - r.
+\end{gather*}
+Hence
+\begin{multline*}
+\frac{K}{4 \pi} \iiint \left(
+  P\, \frac{dP}{dt} + Q\, \frac{dQ}{dt} + R\, \frac{dR}{dt}
+\right) dx\,dy\,dz\\
+  = \iiint \{P(u - p) + Q(v - q) + R(w - r) \}\,dx\,dy\,dz\\
+  = \iiint ( Pu + Qv + Rw )\, dx\,dy\,dz
+  - \iiint ( Pp + Qq + Rr )\, dx\,dy\,dz. \Tag{53}
+\end{multline*}
+%% -----File: 324.png---Folio 310-------
+
+Now (Maxwell's \textit{Electricity and Magnetism}, Vol.~II, Art.~598),
+\begin{align*}
+P & = c\dot{y} - b\dot{z} - \frac{dF}{dt} - \frac{d\psi}{dx}
+    = c\dot{y} - b\dot{z} + P',\\
+Q & = a\dot{z} - c\dot{x} - \frac{dG}{dt} - \frac{d\psi}{dy}
+    = a\dot{z} - c\dot{x} + Q',\\
+R & = b\dot{x} - a\dot{y} - \frac{dH}{dt} - \frac{d\psi}{dz}
+    = b\dot{x} - a\dot{y} + R',
+\end{align*}
+where $P'$,~$Q'$,~$R'$ are the parts of $P$,~$Q$,~$R$ which do not contain
+the velocities.
+
+Thus
+\begin{align*}
+Pu&+Qv+Rw\\
+&= (c\dot{y}-b\dot{z})u + (a\dot{z}-c\dot{x})v + (b\dot{x}-a\dot{y})w + P'u + Q'v + R'w,\\
+&= -\{(vc-wb)\dot{x} + (wa-uc)\dot{y} + (ub-va)\dot{z} \} + P'u + Q'v + R'w,\\
+&= -\{X\dot{x} + Y\dot{y} + Z\dot{z} \} + P'u + Q'v + R'w;
+\end{align*}
+where $X$,~$Y$,~$Z$ are the components of the mechanical force per
+unit volume (Maxwell, Vol.~II, Art.~603).
+
+Substituting this value for $Pu + Qv + Rw$ in~(\eqnref{280}{53}) and transposing,
+we obtain
+\begin{multline*}
+\frac{K}{4 \pi} \iiint \left(
+    P\, \frac{dP}{dt} + Q\, \frac{dQ}{dt} + R\, \frac{dR}{dt}
+  \right)\,dx\,dy\,dz\\
+  + \iiint ( X\dot{x} + Y\dot{y} + Z\dot{z} )\, dx\,dy\,dz
+  + \iiint ( Pp + Qq + Rr )\, dx\,dy\,dz\\
+  = \iiint (P'u + Q'v + R'w ) \, dx\,dy\,dz. \Tag{54}
+\end{multline*}
+\begin{DPalign*}
+\lintertext{\indent Now}
+4\pi u & = \frac{d\gamma}{dy} - \frac{d\beta}{dz},\\
+4\pi v & = \frac{d\alpha}{dz} - \frac{d\gamma}{dx},\\
+4\pi w & = \frac{d\beta}{dx} - \frac{d\alpha}{dy}.
+\end{DPalign*}
+Substituting these values for $u$,~$v$,~$w$ in the right-hand side of
+equation~(\eqnref{280}{54}), that side of the equation becomes
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}
+\frac{1}{4 \pi} \iiint \left\{
+    P' \left( \frac{d\gamma}{dy} - \frac{d\beta}{dz} \right)
+  + Q' \left( \frac{d\alpha}{dz} - \frac{d\gamma}{dx} \right)
+  + R' \left( \frac{d\beta}{dx}  - \frac{d\alpha}{dy} \right) \right\} dx\,dy\,dz\\
+  \shoveleft{\qquad = \frac{1}{4 \pi} \iiint \left[
+    \left\{R'\frac{d\beta}{dx} - Q'\frac{d\gamma}{dx} \right\}
+  + \left\{P'\frac{d\gamma}{dy} - R'\frac{d\alpha}{dy} \right\}\right.}\\
+   + \left.\left\{Q'\frac{d\alpha}{dz} - P'\frac{d\beta}{dz} \right\} \right] dx\,dy\,dz.
+\end{multline*}
+}
+%% -----File: 325.png---Folio 311-------
+Integrating by parts, we find that the expression is equal
+to
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}
+  \frac{1}{4 \pi} \iint ( R'\beta - Q'\gamma ) \, dy\,dz
++ \frac{1}{4 \pi} \iint ( P'\gamma - R'\alpha )\, dx\,dz\\
+\shoveleft{\qquad + \frac{1}{4 \pi} \iint ( Q'\alpha - P'\beta ) \, dx\,dy}\\
+- \frac{1}{4 \pi} \iiint \left(
+    \beta\frac{dR'}{dx} - \gamma\frac{dQ'}{dx}
+  + \gamma\frac{dP'}{dy} - \alpha\frac{dR'}{dy}
+  + \alpha\frac{dQ'}{dz} - \beta\frac{dP'}{dz}
+  \right) dx\,dy\,dz,
+\end{multline*}
+}
+the double integrals being taken over the closed surface. This
+expression may be written as
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}
+\frac{1}{4 \pi} \iint \{
+    l(R'\beta - Q'\gamma)
+  + m(P'\gamma - R'\alpha)
+  + n(Q'\alpha - P'\beta) \}\, dS\\
+  - \frac{1}{4 \pi} \iiint \left\{
+    \alpha \left( \frac{dQ'}{dz} - \frac{dR'}{dy} \right)
+  + \beta  \left( \frac{dR'}{dx} - \frac{dP'}{dz} \right) \right.\\
+  + \gamma \left. \left( \frac{dP'}{dy} - \frac{dQ'}{dx} \right) \right\} dx\,dy\,dz,
+\end{multline*}
+}
+where $dS$~is an element of the surface and $l$,~$m$,~$n$ are the
+direction cosines of the normal to the surface drawn outwards.\nbpagebreak[1]
+\begin{DPalign*}
+\lintertext{\indent But}
+\frac{dQ'}{dz} - \frac{dR'}{dy}
+  &= \frac{d}{dt} \left( \frac{dH}{dy} - \frac{dG}{dz} \right)\\
+  &= \frac{da}{dt} = \mu\, \frac{d\alpha}{dt}.
+\end{DPalign*}
+\begin{DPalign*}
+\lintertext{Similarly}
+\frac{dR'}{dx} - \frac{dP'}{dz}
+  & = \frac{db}{dt} = \mu\, \frac{d\beta}{dt},\\
+\frac{dP'}{dy} - \frac{dQ'}{dx}
+  & = \frac{dc}{dt} = \mu\, \frac{d\gamma}{dt}.
+\end{DPalign*}
+
+Hence we see that the right-hand side of~(\eqnref{280}{54}) is equal
+to
+\begin{multline*}
+\frac{1}{4 \pi} \iint \{
+  l( R'\beta - Q'\gamma ) + m( P'\gamma - R'\alpha ) + n( Q'\alpha - P'\beta ) \}\,dS\\
+  - \frac{\mu}{4 \pi} \iiint \left(
+    \alpha\frac{d\alpha}{dt}
+  + \beta\frac{d\beta}{dt}
+  + \gamma\frac{d\gamma}{dt} \right) dx\,dy\,dz.
+\end{multline*}
+%% -----File: 326.png---Folio 312-------
+
+Transposing the last term to the other side of the equation,
+we get
+\begin{multline*}
+\frac{K}{4\pi}  \iiint \left(
+  P\, \frac{dP}{dt} + Q\, \frac{dQ}{dt} + R\, \frac{dR}{dt}
+  \right) dx\,dy\,dz\\
+  + \frac{\mu}{4 \pi} \iiint \left(
+  \alpha\, \frac{d\alpha}{dt} + \beta\, \frac{d\beta}{dt} + \gamma\, \frac{d\gamma}{dt}
+  \right) dx\,dy\,dz \\
+  + \iiint ( X\dot{x} + Y\dot{y} + Z\dot{z} )\, dx\, dy\, dz
+  + \iiint ( Pp + Qq + Rr )\,\DPtypo{dz}{dx}\,dy\,dz\\
+  = \frac{1}{4 \pi} \iiint  \{
+    l ( R' \beta - Q' \gamma )
+  + m ( P' \gamma - R' \alpha )
+  + n ( Q' \alpha - P' \beta ) \}\,dS, \Tag{55}
+\end{multline*}
+which is the theorem we set out to prove.
+
+Now the electrostatic energy inside the closed surface is
+(Maxwell, Art.~631)
+\begin{DPgather*}
+\frac{1}{2} \iiint ( Pf + Qg + Rh )\,dx\,dy\,dz,\\
+\lintertext{or since}
+f = \frac {K}{4 \pi} P, \qquad
+g = \frac {K}{4 \pi} Q, \qquad
+h = \frac {K}{4 \pi} R, \\
+\frac{K}{8 \pi} \iiint ( P^2 + Q^2 + R^2 )\,dx\,dy\,dz.
+\end{DPgather*}
+The electromagnetic energy inside the same surface is (Maxwell,
+Art.~635)
+\begin{DPalign*}
+& \frac{1}{8 \pi} \iiint  ( a \alpha + b \beta + c \gamma )\,dx\,dy\,dz,\\
+& \lintertext{or}
+  \frac{\mu}{8 \pi} \iiint ( \alpha^2 + \beta^2 + \gamma^2 )\,dx\,dy\,dz.
+\end{DPalign*}
+
+Thus the first two integrals on the left-hand side of equation~(\eqnref{280}{55})
+express the gain per second in electric and magnetic energy.
+The third integral expresses the work done per second by the
+mechanical forces. The fourth integral expresses the energy
+transformed per second in the conductor into heat, chemical
+energy, and so on. Thus the left-hand side expresses the total
+gain in energy per second within the closed surface, and equation~(\eqnref{280}{55})
+expresses that this gain in energy may be regarded as
+coming across the bounding surface, the amount crossing that
+surface per second being expressed by the right-hand side of
+that equation.
+%% -----File: 327.png---Folio 313-------
+
+Thus we may regard the change in the energy inside the
+closed surface as due to the transference of energy across that
+surface; the energy moving at right angles both to~$\smallbold{H}$, the
+resultant magnetic force, and to~$\smallbold{E}$, the resultant of $P'$,~$Q'$,~$R'$. The
+amount of energy which in unit time crosses unit area at right
+angles to the direction of the energy flow is $\smallbold{H}\smallbold{E} \sin \theta/4\pi$,
+where~$\theta$ is the angle between $\smallbold{H}$~and~$\smallbold{E}$. The direction of the
+energy flow is related to those of $\smallbold{H}$~and~$\smallbold{E}$ in such a way
+that the rotation of a positive screw from~$\smallbold{E}$ to~$\smallbold{H}$ would be
+accompanied by a translation in the direction of the flow of
+energy.
+
+Equation~(\eqnref{280}{55}) justifies us in asserting that we shall arrive at
+correct results as to the changes in the distribution of energy in
+the field if we regard the energy as flowing in accordance with
+the laws just enunciated: it does not however justify us in
+asserting that the flow of energy at any point \emph{must} be that given
+by these laws, for we can find an indefinite number of quantities
+$u_s$,~$v_s$,~$w_s$ of the dimensions of flow of energy which satisfy the
+condition
+\[
+\iint  (l u_s + mv_s + nw_s)\,dS = 0,
+\]
+where the integration is extended over any closed surface.
+Hence, we see that if the components of the flow of energy were
+\begin{align*}
+R'\beta - Q'\gamma + \tsum u_s & \text{ instead of } R'\beta - Q'\gamma,\\
+P'\gamma - R'\alpha + \tsum v_s & \text{ instead of } P'\gamma- R'\alpha,\\
+Q'\alpha - P'\beta  + \tsum w_s & \text{ instead of } Q'\alpha- P'\beta,
+\end{align*}
+the changes in the distribution of energy would still be those
+which actually take place.
+
+Though Professor Poynting's investigation does not give a
+unique solution of the problem of finding the flow of energy at
+any point in the electromagnetic field, it is yet of great value, as
+the solution which it does give is simple and one that readily
+enables us to form a consistent and vivid representation of the
+changes in the distribution of energy which are going on in any
+actual case that we may have under consideration. Several applications
+of this theorem are given by Professor Poynting in the
+paper already quoted, to which we refer the reader. We shall
+now proceed to apply it to the determination of the rate of heat
+production in wires at rest traversed by alternating currents.
+%% -----File: 328.png---Folio 314-------
+
+\Article{281} Since the currents are periodic, $P^2$,~$Q^2$,~$R^2$, $\alpha^2$,~$\beta^2$,~$\gamma^2$ will
+be of the form
+\[
+A + B \cos(2pt + \theta),
+\]
+where $A$~and~$B$ do not involve the time; hence the first two
+integrals on the left-hand side of equation~(\eqnref{280}{55}) will be multiplied
+by factors which, as far as they involve~$t$, will be of the form
+$\sin (2pt + \theta)$; hence, if we consider the mean value of these terms
+over a time involving a great many oscillations of the currents,
+they may be neglected: the gain or loss of energy represented
+by these terms is periodic, and at the end of a period the energy
+is the same as at the beginning. The third term on the left-hand
+side vanishes in our case because the wires are at rest, and
+since $\dot{x}$,~$\dot{y}$,~$\dot{z}$ vanish $P'$,~$Q'$,~$R'$ become identical with $P$,~$Q$,~$R$.
+
+Thus when the effects are periodic we see that equation~(\eqnref{280}{55})
+leads to the result that the mean value with respect to the
+time of
+\[
+\iiint (Pp + Qq + Rr)\,dx\,dy\,dz
+\]
+is equal to that of
+\[
+\frac{1}{4 \pi} \iint \{l(R\beta-Q\gamma)+m(P\gamma-R\alpha)+n(Q\alpha-P\beta)\}\,dS.
+\]
+The first of these expressions is, however, the mean rate of heat
+production, and in the case of a wire whose electrical state is
+symmetrical with respect to its axis, the value of the quantity
+under the sign of integration is the same at each point of the
+circumference of a circle whose plane is at right angles to the
+axis of the wire; hence in this case we have the result:
+
+The mean rate of heat production per unit length of the wire
+is equal to the mean value of
+\[
+\tfrac{1}{2}\smallbold{a}\ \text{(tangential electromotive intensity)} ×
+\text{(tangential magnetic force)},\eqnlabel{\eqnart.56}\nbtag{56}
+\]
+$\smallbold{a}$, as before, being the radius of the wire.
+
+\Article{282} Let us apply this result to find the rate of heat production
+\index{Heat produced by electric discharge, in wires carrying alternating currents@\subdashtwo in wires carrying alternating currents}%
+in the wire and in the outer conductor of a cable when
+the current is parallel to the axis of the wire. By the methods
+of \artref{268}{Art.~268}, we see that if the total current through the wire at
+the point~$z$ is equal to the real part of
+\[
+I_0\epsilon^{\iota(mz+pt)},
+\]
+%% -----File: 329.png---Folio 315-------
+or if $m = -\alpha + \iota \beta$, to
+\[
+I_0 \epsilon^{-\beta z} \cos (-\alpha z + pt),
+\]
+then, \artref{268}{Art.~268}, equation~(\eqnref{268}{24}), the electromotive intensity~$R$ in
+the wire parallel to the axis of~$z$ is equal to the real part of
+\[
+- \frac{\iota \sigma n}{2 \pi \smallbold{a}}\,
+  \frac{J_0 (\iota n r)}{J_0' (\iota n \smallbold{a})}\,
+  I_0 \epsilon^{-\beta z} \epsilon^{\iota (-\alpha z + pt)}. \Tag{57}
+\]
+If we neglect the polarization currents in the dielectric in
+comparison with the conduction currents through the wire, then
+the line integral of the magnetic force round the inner surface
+of the outer conductor must equal $4 \pi I_0 \epsilon^{\iota (mz + pt)}$; using this
+principle we see that~$E$ in equation~(\eqnref{262}{11}), \artref{262}{Art.~262}, equals
+$ -\iota n' \sigma' I_0/2 \pi \smallbold{b} K_0' (\iota n' \smallbold{b})$, and hence the electromotive intensity
+parallel to~$z$ in the outer conductor is equal to the real part of
+\[
+- \frac{\iota \sigma' n'}{2 \pi \smallbold{b}}\,
+  \frac{K_0 (\iota n' r)}{K_0' (\iota n' \smallbold{b})}\,
+  I_0 \epsilon^{-\beta z} \epsilon^{\iota(-\alpha z + pt)}, \Tag{58}
+\]
+the notation being the same as in \artref{262}{Art.~262}.
+
+The tangential magnetic force at the surface of the wire is
+(\artref{262}{Art.~262})
+\[
+\frac{2 I_0}{\smallbold{a}} \epsilon^{-\beta z}  \cos (-\alpha z + pt), \Tag{59}
+\]
+while that at the surface of the outer conductor is, if we neglect
+the polarization currents in the dielectric in comparison with
+the conduction currents through the wire,
+\[
+\frac{2 I_0}{\smallbold{b}} \epsilon^{-\beta z} \cos (-\alpha z + pt).      \Tag{60}
+\]
+
+Let us now consider the case when the rate of alternation of
+the current is so slow that both $n\smallbold{a}$~and~$n'\smallbold{b}$ are small quantities.
+When $n\smallbold{a}$~is small $J_0'(\iota n\smallbold{a}) = -\iota n\smallbold{a}/2$, while $J_0(\iota n \smallbold{a}) = 1$ approximately;
+hence, putting $r = \smallbold{a}$ in~(\eqnref{282}{57}), we find that the
+tangential electromotive intensity is
+\[
+\frac{\sigma}{\pi \smallbold{a}^2}\, I_0 \epsilon^{-\beta z} \cos (-\alpha z + pt).
+\]
+Hence by (\eqnref{281}{56})~and~(\eqnref{282}{59}) the rate of heat production in the wire is
+equal to the mean value of
+\[
+\frac{\sigma}{\pi \smallbold{a}^2}\, I_0^2 \epsilon^{-2\beta z} \cos^2 (-\alpha z + pt).
+\]
+%% -----File: 330.png---Folio 316-------
+\begin{DPgather*}
+\lintertext{that is to} \frac{\sigma}{2 \pi \smallbold{a}^2}\, I_0^2 \epsilon^{-2 \beta z}.
+\end{DPgather*}
+
+Let us now consider the rate of heat production in the outer
+conductor; since $n'\smallbold{b}$~is very small, we have approximately
+\[
+K_0 (\iota n' \smallbold{b}) = \log (2 \gamma/\iota n' \smallbold{b}), \quad
+K_0' (\iota n' \smallbold{b}) = -1/\iota n' \smallbold{b}.
+\]
+Making these substitutions in~(\eqnref{282}{58}), we see that the tangential
+electromotive intensity at the surface of the outer conductor is
+equal to the real part of
+\[
+- \frac{\sigma' n'^2}{2 \pi} \log ( 2 \gamma / \iota n' \smallbold{b})\,
+  I_0 \epsilon^{-\beta z} \epsilon^{\iota(-\alpha z + pt)},
+\]
+and since $n'^2 = 4 \pi \mu' \iota p / \sigma'$, the real part of this expression is
+\begin{multline*}
+2 \mu' p \log (\gamma \sqrt{\sigma' / \pi \mu' p \smallbold{b}^2})
+  I_0 \epsilon^{-\beta z} \sin (-\alpha z + pt) \\
+  -\tfrac{3}{2} \pi \mu' p I_0 \epsilon^{-\beta z} \cos(-\alpha z + pt).
+\end{multline*}
+
+Hence by (\eqnref{281}{56})~and~(\eqnref{282}{60}) the rate of heat production in the
+outer conductor is equal to
+\[
+\tfrac{3}{4} \pi \mu' p I_0^2 \epsilon^{-2 \beta z},
+\]
+since the mean value with respect to the time of
+\[
+\sin (-\alpha z + pt) \cos (-\alpha z + pt)
+\]
+is zero. Thus, when $n'\smallbold{b}$~is small, the rate of heat production in
+the outer conductor is independent both of the radius and specific
+resistance of that conductor. The ratio of the heat produced in
+unit time in the wire to that produced in the outer conductor is
+thus $2 \sigma / 3 \pi^2 \smallbold{a}^2 \mu' p$, which is very large since we have assumed that
+$n^2 \smallbold{a}^2$, i.e.~$4 \pi \mu p \smallbold{a}^2 / \sigma$, is a small quantity; in this case, therefore, by
+far the larger proportion of the heat is produced in the wire.
+This explains the result found in \artref{263}{Art.~263} that the rate of decay
+of the vibrations is nearly independent of the resistance of the
+outer conductor and depends almost wholly upon that of the
+wire.
+
+\Article{283} When the frequency is so great that $n\smallbold{a}$~is large though
+$n'\smallbold{b}$~is still small, then $J_0(\iota n\smallbold{a}) = \iota J_0'(\iota n\smallbold{a})$, so that by~(\eqnref{282}{57}) the
+tangential electromotive intensity at the surface of the wire is
+equal to the real part of
+\[
+\frac{\sigma}{2 \pi \smallbold{a}} \{4 \pi \mu \iota p / \sigma \}^\frac{1}{2}\,
+  I_0 \epsilon^{-\beta z} \epsilon^{\iota (-\alpha z + pt)},
+\]
+%% -----File: 331.png---Folio 317-------
+\index{Heat produced by electric discharge, in wires carrying alternating currents@\subdashtwo in wires carrying alternating currents}%
+which is equal to
+\[
+\frac{\sigma}{2 \pi \smallbold{a}} \{2 \pi \mu p / \sigma\}^{\frac{1}{2}}
+  I_0 \epsilon^{-\beta z} \{\cos (-\alpha z+pt)-\sin(-\alpha z+pt)\}.
+\]
+Hence by (\eqnref{281}{56})~and~(\eqnref{282}{59}) the mean rate of heat production in the
+wire is equal to
+\[
+\frac{\sigma}{4 \pi \smallbold{a}} ( 2 \pi \mu p/ \sigma)^{\frac{1}{2}}
+  I_0^{2} \epsilon^{-2 \beta z.}
+\]
+Since $n'\smallbold{b}$~is supposed to be small the rate of heat production
+in the outer conductor is as before
+\[
+\frac{3 \pi}{4} \mu' p I_0^{2} \epsilon^{-2 \beta z},
+\]
+hence the ratio of the amount of heat produced in unit time in
+the wire to that produced in the outer conductor is
+\[
+\frac{\mu}{\mu'}
+  \left\{\frac{2 \sigma}{9 \pi^3 p \mu \smallbold{a}^2} \right\}^{\frac{1}{2}}.
+\]
+Thus, since $n^2 \smallbold{a}^2$ and so~$4 \pi p \mu \smallbold{a}^2 / \sigma$ is very large by hypothesis, we
+see that unless $\mu / \mu'$~is very large this ratio will be very small; in
+other words the greater part of the heat is produced in the
+outer conductor; this is in accordance with the result obtained
+in \artref{266}{Art.~266}, which showed that the rate of decay of the vibrations
+was independent of the resistance of the wire.
+
+\Article{284} When the frequency is so high that both $n \smallbold{a}$~and~$n' \smallbold{b}$
+are large, then the expression for the heat produced in the wire
+is that just found. To find the heat produced in the outer conductor
+we have, when $n' \smallbold{b}$~is very large,
+\[
+K_0 ( \iota n' \smallbold{b})=-\iota K_0' ( \iota n' \smallbold{b} );
+\]
+hence by~(\eqnref{282}{58}) the tangential electromotive intensity in the outer
+conductor is equal to the real part of
+\[
+-\frac{\sigma'}{2 \pi \smallbold{b}} ( 4 \pi \mu' \iota p / \sigma')^{\frac{1}{2}}
+  I_0 \epsilon^{-\beta z} \epsilon^{\iota (-\alpha z+pt)},
+\]
+which is equal to
+\[
+-\frac{\sigma'}{2 \pi \smallbold{b}} ( 2 \pi \mu' p / \sigma' )^{\frac{1}{2}}
+  I_0 \epsilon^{-\beta z} \{\cos ( -\alpha z+pt) - \sin ( - \alpha z+pt)\}.
+\]
+Hence by (\eqnref{281}{56})~and~(\eqnref{282}{60}) the mean rate of heat production in the
+outer conductor is
+\[
+\frac{\sigma'}{4 \pi \smallbold{b}} \{2 \pi \mu' p / \sigma' \}^{\frac{1}{2}}
+  I_0^{2} \epsilon^{-2 \beta z}.
+\]
+%% -----File: 332.png---Folio 318-------
+\index{Alternating currents, heat produced in wire traversed by@\subdashtwo heat produced in wire traversed by}%
+\index{Heat produced by electric discharge, in wires carrying alternating currents@\subdashtwo in wires carrying alternating currents}%
+
+Thus the ratio of the heat produced in unit time in the wire
+to that produced in the same time in the outer conductor is
+\[
+\Bigl\{\frac{\mu \sigma}{\smallbold{a}^2} \Bigr\}^{\frac{1}{2}} \bigg/
+\Bigl\{\frac{\mu' \sigma'}{\smallbold{b}^2} \Bigr\}^{\frac{1}{2}},
+\]
+so that if, as is generally the case in cables, $\sigma'$~is very much
+greater than~$\sigma$, by far the larger part of the heat will be produced
+in the outer conductor.
+
+\Subsection{Heat produced by Foucault Currents in a Transformer.}
+\index{Foucault currents, heat produced by, in a transformer}%
+\index{Heat produced by electric discharge, by Foucault currents in a transformer@\subdashtwo by Foucault currents in a transformer}%
+\index{Transformer, heat produced in}%
+
+\Article{285} We shall now proceed to consider the case discussed
+in \artref{278}{Art.~278}, where the lines of magnetic force are in planes
+through the axis of the wire, the currents flowing in circles in
+planes at right angles to this axis. This case is one which is of
+great practical importance, as the conditions approximate to
+those which obtain in the soft iron cylindrical core of an induction
+coil or a transformer; in this case the windings of the
+primary coil are in planes at right angles to the axis of the iron
+cylinder, while the lines of magnetic force due to the primary
+coil are in planes passing through this axis. When a variable
+current is passing through the primary coil, currents are induced
+which heat the core and the heat thus produced is wasted as
+far as the production of useful work is concerned; it is thus a
+matter of importance to investigate the laws which govern its
+development, so that the apparatus may be designed in such a
+way as to reduce this waste to a minimum. We shall suppose
+that the magnetic force parallel to the axis at the surface of the
+wire is represented by the real part of
+\[
+H \epsilon^{\iota (mz + pt)},
+\]
+or if $m = \alpha + \iota \beta$, by
+\[
+H \epsilon^{-\beta z} \cos (\alpha z + pt).
+\]
+The magnetic force at the surface of the cylinder is the most
+convenient quantity in which to express the rate of heat production,
+for it is due entirely to the external field and is
+not, when the field is uniform, affected by the currents in the
+wire itself.
+
+Using the notation of \artref{278}{Art.~278} we see by the results of that
+article that in the wire
+\[
+c = A J_0 (\iota n r) \epsilon^{\iota (mz + pt)}.
+\]
+%% -----File: 333.png---Folio 319-------
+
+The tangential electromotive intensity~$\Theta$ is given by the
+equation
+\begin{DPgather*}
+\frac{dc}{dt} = -\frac{1}{r} \frac{d}{dr} (r \Theta); \\
+\lintertext{hence} \Theta = \frac{p}{n} A J_0' (\iota n r) \epsilon^{\iota (mz+pt)};
+\end{DPgather*}
+but since at the surface of the wire, $c$~is equal to the real part of
+\[
+\mu H \epsilon^{-\beta z} \epsilon^{\iota (\alpha z + pt)} ,
+\]
+we see that at the surface $\Theta ={}$real part of
+\[
+\frac{p}{n}\, \frac{J_0' (\iota n \smallbold{a})}{J_0 (\iota n \smallbold{a})}
+  \mu H \epsilon^{-\beta z} \epsilon^{\iota (\alpha z + pt)}.\Tag{61}
+\]
+
+Let us first take the case when the radius of the wire is
+so small that $n \smallbold{a}$~is small; in this case we have, since approximately
+\begin{DPalign*}
+J_0(x) & = 1 - \frac{x^2}{2^2} + \frac{x^4}{2^{2} 4^2},\\
+J_0'(x) & = -\tfrac{1}{2} x \left\{1 - \frac{x^2}{8} \right\}, \\
+\lintertext{and} n^2 & = 4 \pi \mu \iota p /  \sigma,
+\end{DPalign*}
+\begin{multline*}
+\Theta = \text{real part of} \\
+\shoveleft{-\tfrac{1}{2} \iota \smallbold{a} p
+  \left\{1 - \frac{\pi \mu \iota p \smallbold{a}^2}{2 \sigma} \right\}
+  \mu H \epsilon^{-\beta z} \epsilon^{\iota (\alpha z + pt)}}\\
+  = \tfrac{1}{2} \mu \smallbold{a} p H \epsilon^{-\beta z} \sin(\alpha z + pt)
+  - \frac{1}{4 \sigma} \pi \mu^2 p^2 \smallbold{a}^3 H \epsilon^{-\beta z} \cos (\alpha z + pt).
+\end{multline*}
+
+But by equation~(\eqnref{281}{56}) the rate of heat production in the
+wire per unit length is equal to the mean value
+\[
+-\tfrac{1}{2} \smallbold{a} \Theta H \epsilon^{-\beta z} \cos (\alpha z + pt);
+\]
+where the minus sign has been taken because (\artref{280}{Art.~280}) $\Theta H$~is
+proportional to the rate of flow of energy in the direction of
+translation of a right-handed screw twisting from~$\Theta$ to~$H$; in
+this case this direction is radially \emph{outwards}.
+
+Thus the rate of heat production in the wire is
+\[
+\frac{1}{16 \sigma} \pi \mu^2 p^2 \smallbold{a}^4 H^2 \epsilon^{-2 \beta z},
+\]
+and is thus proportional to the conductivity, so that good
+conductors will in this case absorb more energy than bad ones.
+%% -----File: 334.png---Folio 320-------
+
+Let us now apply this result to find the energy absorbed
+in the core of a transformer or induction coil. We shall
+suppose that the core consists of iron wire of circular section,
+the wires being insulated from each other by the coating of
+rust with which they are covered. We shall consider the case
+when the magnetic force due to the primary coil is uniform both
+along the axis of the coil and over its cross-section. When the
+external magnetic force is uniform along~$z$, the axis of a wire, the
+currents induced in the wire by the variation of the magnetic
+force flow in circles whose planes are at right angles to~$z$, and
+the intensities of the currents are independent of the value of~$z$.
+Under these circumstances the currents in the wire do not give
+rise to any magnetic force outside it. The magnetic force
+outside the wires will thus be due entirely to the primary coil,
+and as this magnetic force is uniform over the cross-section
+it will be the same for each of the wires, so that we can
+apply the preceding investigation to the wires separately.
+In order to use the whole of the iron, the magnetic force must
+be approximately uniformly distributed over the cross-section
+of the wires; for this to be the case $n \smallbold{a}$~must be small, as we
+have seen that when $n \smallbold{a}$~is large the magnetic force is confined
+to a thin skin round each wire. For soft iron, for which
+we may put $\mu = 10^3$, $\sigma = 10^4$, the condition that $n \smallbold{a}$~is small
+implies that when the primary current makes one hundred
+alternations per second, the radius of the wire should not be
+more than half a millimetre. If now the total cross-section of
+the iron is kept constant so as to keep the magnetic induction
+through the core constant, we have, if $\smallbold{N}$~is the number of wires,
+$\smallbold{A}$~the total cross-section of the iron,
+\[
+\smallbold{N} \pi \smallbold{a}^2 = \smallbold{A}.
+\]
+
+The heat produced in all the wires per unit length of core in
+one second is, if $H$~is the maximum magnetic force due to the coil,
+\begin{DPgather*}
+\frac{\smallbold{N}}{16 \sigma} \pi  \mu^2  p^2  \smallbold{a}^4  H^2, \\
+\lintertext{or} \frac{\smallbold{A}^2}{16 \pi \sigma \smallbold{N}} \mu^2 p^2 H^2,
+\end{DPgather*}
+and is thus \emph{inversely proportional to the number of wires}. We
+may therefore diminish the waste of energy due to the heat
+%% -----File: 335.png---Folio 321-------
+produced by the induced currents in the wires by increasing the
+number of wires in the core. We thus arrive at the practical rule
+that to diminish the waste of work by eddy currents the core
+should be made up of as fine wire as possible. In many transformers
+the iron core is built up of thin plates instead of wires;
+when this is the case the advantage of a fine sub-division of the
+core is even more striking than for wires, for we can easily
+prove that the work wasted by eddy currents is inversely proportional
+to the square of the number of plates (see J.~J. Thomson,
+\textit{Electrician}, 28, p.~599, 1892).
+
+If $\gamma$~is the current flowing through the primary coil and $N$~the
+number of turns of this coil per centimetre, then
+\begin{DPgather*}
+H \cos p t= 4 \pi N \gamma, \\
+\lintertext{and} \tfrac{1}{2} H^2 = 16 \pi^2 N^2\, (\text{mean value of~$\gamma^2$}),
+\end{DPgather*}
+thus in the case of a cylindrical core of radius~$\smallbold{a}$ the heat produced
+in one second in a length~$l$ of the core will be
+\[
+2 \pi^3 \mu^2 p^2 \smallbold{a}^4 N^2 l\, (\text{mean value of~$\gamma^2$}) / \sigma.
+\]
+
+If $\smallbold{Q}$ is the \emph{impedance} of a circuit (\artref{272}{Art.~272}) the heat produced
+in unit time is equal to
+\[
+\smallbold{Q}\,(\text{mean value of~$\gamma^2$});
+\]
+thus the core will increase the impedance of the primary coil by
+\[
+2 \pi^3 \mu^2 p^2 \smallbold{a}^4 N^2l / \sigma.
+\]
+
+\Article{286} Let us now consider the case when $n \smallbold{a}$~is large; here we
+have
+\begin{DPgather*}
+J_0'(\iota n \smallbold{a}) = -\iota J_0(\iota n \smallbold{a}), \\
+\lintertext{and since} n^2 = 4 \pi \mu \iota p / \sigma,
+\end{DPgather*}
+we see by~(\eqnref{285}{61}), putting $\alpha$~and~$\beta$ equal to zero, that
+\begin{gather*}
+\Theta = \text{real part of}\\
+- \sqrt{\frac{p \mu \sigma}{4 \pi}}\,
+  \epsilon^{\frac{\iota \pi}{4}} H \epsilon^{\iota pt}\\
+  = - \sqrt{\frac{p \mu \sigma}{8 \pi}}\, H \{\cos pt - \sin pt\}.
+\end{gather*}
+
+But by equation~(\eqnref{281}{56}) the rate of heat production per unit
+length is equal to the mean value of
+\[
+- \tfrac{1}{2} \smallbold{a} \Theta H \cos pt,
+\]
+and is thus equal to
+\[
+\frac{1}{8} \sqrt{\frac{p \mu \sigma}{\pi 2}}\, \smallbold{a} H^2.
+\]
+%% -----File: 336.png---Folio 322-------
+
+We can show, as before, that this corresponds to an increase
+in the impedance of the primary circuit equal to
+\[
+4 \pi^2 l N^2 \{p\mu\sigma/2\pi\}^{\frac{1}{2}} \smallbold{a}.
+\]
+
+In this case the heat produced is proportional to the square
+root of the specific resistance of the core, so the worse the conductivity
+of the core the greater the amount of heat produced by
+eddy currents, whereas in the case when $n\smallbold{a}$~was small, the
+greater the conductivity of the core the greater was the loss due
+to heating.
+
+When $n\smallbold{a}$~is large, the heat produced varies as the circumference
+of the core instead of, as in the previous case, as the square
+of the area; it also varies much more slowly with the frequency
+and magnetic permeability. This is due to the fact that when
+$n\smallbold{a}$~is large the currents are not uniformly distributed over the
+core but confined to a thin layer on the outside, the thickness of
+this layer diminishing as the magnetic permeability or the frequency
+increases; thus, though an increase in $\mu$~or~$p$ may be
+accompanied by an increase in the intensity of the currents, it
+will also be attended by a diminution in the area over which
+the currents are spread, and thus the effect on the heat produced
+of the increase in $p$~or~$\mu$ will not be so great as in the previous
+case when $n\smallbold{a}$~was small, and when no limitation in the area
+over which the current was spread accompanied an increase in
+the frequency or magnetic permeability.
+
+If we compare the absorption of energy when $n\smallbold{a}$~is large by
+\index{Alternating magnetic force, behaviour of iron under@\subdashone magnetic force, behaviour of iron under}%
+\index{Magnetic yproperties of iron in rapidly alternating fields@\subdashone properties of iron in rapidly alternating fields}%
+\index{Iron, magnetic properties of@\subdashone magnetic properties of, under rapidly alternating currents}%
+cores of iron and copper of the same size subject to alternating
+currents of the same frequency, we find---since for iron $\mu$~may
+be taken as~$10^3$ and~$\sigma$ as~$10^4$, while for copper $\mu = 1$, $\sigma = 1600$,---that
+the absorption of energy by the iron core is between~$70$ and
+$80$~times that by the copper. The greater absorption by the iron
+can be very easily shown by an experiment of the kind figured
+in \artref{85}{Art.~85}, in which two coils are placed in the circuit connecting
+the outer coatings of two Leyden Jars; in one of these coils
+an exhausted bulb is placed, while the core in which the heat
+produced is to be measured is placed in the other. When the
+oscillating current produced by the discharge of the jars passes
+through the coils a brilliant discharge passes through the exhausted
+bulb in~$\smallbold{A}$, if the coil~$\smallbold{B}$ is empty or if it contains a
+copper cylinder; if however an iron cylinder of the same size
+%% -----File: 337.png---Folio 323-------
+replaces the copper one, the discharge in the bulb is at once
+extinguished, showing that the iron cylinder has absorbed a
+great deal more energy than the copper one. This experiment
+also shows that iron retains its magnetic properties even when
+the forces to which it is exposed are reversed, as in this experiment,
+millions of times in a second.
+
+\Article{287} Another remarkable result is that though a cylinder or
+\index{Heat produced by electric discharge, by currents induced in a tube@\subdashtwo by currents induced in a tube}%
+\index{Tube, heat produced in under variable magnetic field}%
+tube of a non-magnetic metal does not stop the discharge in the
+bulb in~$\smallbold{A}$, yet if a piece of glass tubing of the same size is
+coated with thin tinfoil or Dutch metal, or if it has a film of
+silver deposited upon it, it will check the discharge very decidedly.
+We are thus led to the somewhat unexpected result that a
+thin layer of metal when exposed to very rapidly alternating
+currents may absorb more energy than a thick layer. The
+following investigation affords the explanation of this, and shows
+that there is a certain thickness for which the heat produced is
+a maximum. This result can easily be verified by the arrangement
+just described, for if an excessively thin film of silver is
+deposited on a beaker very little effect is produced on the discharge
+in the bulb placed in~$\smallbold{A}$, but if successive layers
+of very thin tinfoil are wrapped round the beaker over the
+silver film the brightness of the discharge in~$\smallbold{A}$ at first rapidly
+diminishes, it however soon increases again, and when a few
+layers of tinfoil have been wrapped round the beaker the
+discharge becomes almost as bright as if the beaker were
+away.
+
+To investigate the theory of this effect we shall calculate the
+energy absorbed by a metal tube of circular cross-section, when
+placed inside a primary coil whose windings are in planes at
+right angles to the axis of the tube; this coil is supposed to be
+long, and uniformly wound, so that the distribution of magnetic
+force and current is the same in all planes at right angles to
+its axis. We shall use the same notation as before; the only
+symbols which it is necessary to define again are $\smallbold{a}$~and~$\smallbold{b}$,
+which are respectively the internal and external radius of the
+tube, and $V$~the velocity with which electromagnetic action
+is propagated through the dielectric inside the tube. The
+magnetic force outside the tube is represented by the real part
+of~$H \epsilon^{\iota pt}$, and this force is due entirely to the currents in the
+primary coil.
+%% -----File: 338.png---Folio 324-------
+
+Then $\gamma$, the magnetic force parallel to the axis of the tube,
+may be (\artref{262}{Art.~262}) expressed by the following equations,
+\begin{align*}
+\gamma = AJ_0 (\iota kr)\epsilon^{\iota pt} \text{ in the dielectric inside the tube}, \\
+\gamma = \{BJ_0 (\iota nr)
+       + CK_0 (\iota nr)\}\epsilon^{\iota pt} \text{ in the tube itself}.
+\end{align*}
+
+Here $k^2 = -p^2 /V^2$, $n^2 = 4\pi\mu\iota p/\sigma$, thus they represent the quantities
+represented by the same symbols in previous investigations,
+if in these we put~$m = 0$.
+
+Let $\smallbold{I}$~denote the tangential current at right angle to~$r$ and the
+axis of the cylinder, then
+\[
+4\pi \smallbold{I} = -\frac{d\gamma}{dr},
+\]
+if $\Theta$~is the tangential electromotive intensity in the same direction,
+then in the dielectric
+\begin{DPalign*}
+\smallbold{I} & = \frac{K}{4\pi}\, \frac{d\Theta}{dt} \\
+& = \frac{K}{4\pi}\, \iota p\Theta, \\
+\lintertext{so that}  \Theta & = -\frac{V^2}{\iota p}\, \frac{d\gamma}{dr},
+\end{DPalign*}
+since $1/K = V^2$.
+
+\begin{DPalign*}
+\lintertext{\indent In the tube} \Theta & = \sigma \smallbold{I} \\
+& = -\frac{\sigma}{4\pi}\, \frac{d\gamma}{dr}.
+\end{DPalign*}
+
+Since $\gamma$~is continuous, we have
+\[
+AJ_0 (\iota k\smallbold{a}) = BJ_0 (\iota n\smallbold{a}) + CK_0 (\iota n\smallbold{a}),
+\]
+and since $\Theta$~is continuous, we have
+\[
+\frac{V^2k}{p} AJ_0'(\iota k\smallbold{a})
+  = \frac{\sigma\iota n}{4\pi} \{BJ_0'(\iota n\smallbold{a}) + CK_0'(\iota n\smallbold{a})\}.
+\]
+
+Since $\iota k = p/V$, $\iota k\smallbold{a}$~will be very small, hence we may put
+\[
+J_0 (\iota k\smallbold{a}) = 1, \quad
+J_0' (\iota k\smallbold{a}) = -\tfrac{1}{2} \iota k\smallbold{a}.
+\]
+Making these substitutions and remembering that
+\[
+J_0 (\iota n\smallbold{a})K_0' (\iota n\smallbold{a})
+  - J_0' (\iota n\smallbold{a})K_0 (\iota n\smallbold{a}) = -\frac{1}{\iota n\smallbold{a}},
+\]
+we find
+\begin{align*}
+B & = -A \{K_0'(\iota n\smallbold{a}) + \frac{\iota n\smallbold{a}}{2\mu} K_0(\iota n\smallbold{a})\}\iota n\smallbold{a}, \\
+C & =  A \{J_0'(\iota n\smallbold{a}) + \frac{\iota n\smallbold{a}}{2\mu} J_0(\iota n\smallbold{a})\}\iota n\smallbold{a}.
+\end{align*}
+%% -----File: 339.png---Folio 325-------
+
+To determine~$A$ we have the condition that when
+\begin{DPgather*}
+r = \smallbold{b}, \quad \gamma = H \epsilon^{\iota p t}, \\
+\lintertext{hence}
+H \epsilon^{\iota p t}
+  = \{BJ_0(\iota n\smallbold{b}) + CK_0(\iota n\smallbold{b})\} \epsilon^{\iota p t}.
+\end{DPgather*}
+In order to find the heat produced in the tube we require the
+value of~$\Theta$ when~$r=\smallbold{b}$; but here
+\[
+\Theta = -\frac{\sigma \iota n}{4 \pi}
+  \{BJ_0'(\iota n \smallbold{b}) + CK_0'(\iota n \smallbold{b})\} \epsilon^{\iota p t}.
+\]
+
+{Eliminating $B$~and~$C$ from these equations, we find
+\begin{multline*}
+-\Theta = \text{real part of } H \epsilon^{\iota pt} \frac{\sigma \iota n}{4 \pi} × \\
+\text{\footnotesize$\frac{\left\{J_0'(\iota n\smallbold{a}) K_0'(\iota n\smallbold{b})
+           - J_0'(\iota n\smallbold{b}) K_0'(\iota n\smallbold{a})
+  + \frac{\iota n\smallbold{a} }{2\mu}
+   [J_0(\iota n\smallbold{a}) K_0'(\iota n\smallbold{b})
+  - J_0'(\iota n\smallbold{b}) K_0(\iota n\smallbold{a})]\right\}}
+     {J_0'(\iota n\smallbold{a}) K_0 (\iota n\smallbold{b})
+     - J_0 (\iota n\smallbold{b}) K_0'(\iota n\smallbold{a})
+  + \frac{\iota n\smallbold{a} }{2\mu}
+   [J_0 (\iota n\smallbold{a}) K_0 (\iota n\smallbold{b})
+  - J_0 (\iota n\smallbold{b}) K_0 (\iota n\smallbold{a})]}$}.
+\end{multline*}
+}
+
+The effect we are considering is one which is observed when
+the rate of alternation of the current is very high, so that both
+$n\smallbold{a}$~and~$n\smallbold{b}$ are very large; but when this is the case
+\begin{align*}
+J_0(\iota n\smallbold{a}) & = \frac{\epsilon^{n\smallbold{a}}}{\sqrt{2 \pi n \smallbold{a}}},   &
+J_0'(\iota n\smallbold{a}) & = -\frac{\iota \epsilon^{n\smallbold{a}}}{\sqrt{2 \pi n \smallbold{a}}},\\
+%
+K_0(\iota n\smallbold{a}) & = \epsilon^{-n\smallbold{a}} \sqrt{\frac{\pi}{2 n \smallbold{a}}},  &
+K_0'(\iota n \smallbold{a}) & = \iota \epsilon^{-n\smallbold{a}} \sqrt{\frac{\pi}{2 n \smallbold{a}}},\\
+%
+J_0(\iota n\smallbold{b}) & = \frac{\epsilon^{n\smallbold{b}}}{\sqrt{2 \pi n \smallbold{b}}},   &
+J_0'(\iota n \smallbold{b}) & = -\frac{\iota \epsilon^{n\smallbold{b}}}{\sqrt{2 \pi n \smallbold{b}}},\\
+%
+K_0(\iota n\smallbold{b}) & = \epsilon^{-n\smallbold{b}} \sqrt{\frac{\pi}{2 n \smallbold{b}}},  &
+K_0'(\iota n \smallbold{b}) & = \iota \epsilon^{-n\smallbold{b}} \sqrt{\frac{\pi}{2 n \smallbold{b}}};
+\end{align*}
+making these substitutions and writing~$h$ for $\smallbold{b}-\smallbold{a}$, we find
+\[
+-\Theta = \text{real part of} \quad
+\frac{\sigma n}{4 \pi}\,
+  \frac{\epsilon^{nh} - \epsilon^{-nh} + \dfrac{n\smallbold{a}}{2\mu}\left(\epsilon^{nh} + \epsilon^{-nh}\right)}
+       {\epsilon^{nh} + \epsilon^{-nh} + \dfrac{n\smallbold{a}}{2\mu}\left(\epsilon^{nh} - \epsilon^{-nh}\right)}\,
+  H \epsilon^{\iota p t}. \Tag{62}
+\]
+
+Now since $n \smallbold{a}$~is very large, $n\smallbold{a}/\mu$~is also very large for the
+non-magnetic metals, and even for the magnetic metals if the
+frequency of the currents in the primary is exceedingly large;
+%% -----File: 340.png---Folio 326-------
+but when this is the case, then, unless $h$~is so small that $n^2 \smallbold{a} h / \mu$~is
+no longer large, we may write equation~(\eqnref{287}{62}) as
+\[
+-\Theta = \text{real part of} \quad
+\frac{\sigma n}{4 \pi}\,
+  \frac{\epsilon^{nh} + \epsilon^{-nh}}
+       {\epsilon^{nh} - \epsilon^{-nh}}\, H \epsilon^{\iota pt}. \Tag{63}
+\]
+
+Since $n=\{4 \pi \mu \iota p / \sigma \}^{\frac{1}{2}}$, we may write
+$n=n_1(1+\iota)$, where $n_1=\{2 \pi \mu p / \sigma\}^{\frac{1}{2}}$, and equation~(\eqnref{287}{63}) becomes
+\begin{multline*}
+\Theta = -\frac{\sigma n_1}{4 \pi}\,
+  \frac{\epsilon^{2n_1h} - \epsilon^{-2n_1h} + 2\sin 2n_1h}
+       {\epsilon^{2n_1h} + \epsilon^{-2n_1h} - 2\cos 2n_1h}\, H \cos pt \\
+  + \frac{\sigma n_1}{4 \pi}\,
+  \frac{\epsilon^{2n_1h} - \epsilon^{-2n_1h} - 2\sin 2n_1h}
+       {\epsilon^{2n_1h} + \epsilon^{-2n_1h} - 2\cos 2n_1h}\, H \sin pt. \Tag{64}
+\end{multline*}
+
+In calculating the part of the energy flowing into the tube
+which is converted into heat, we need only consider the part
+which flows across the outer surface of the tube, because the
+energy flowing across the inner surface is equal to that which
+flows into the dielectric inside the tube, and since there is no
+dissipation of energy in this region the average of the flow of
+energy across the inner surface of the tube must vanish. Hence
+the amount of heat produced in unit time in the tube is by equation~(\eqnref{271}{36})
+equal to the mean value of
+\[
+-\tfrac{1}{2} \smallbold{b} \Theta H \cos pt,
+\]
+where the $-$~sign has been taken because the translatory motion
+of a right-handed screw twisting from~$\Theta$ to~$H$ is radially outwards;
+this by~(\eqnref{287}{64}) is equal to
+\[
+\frac{\sigma n_1 \smallbold{b}}{16 \pi}\,
+  \frac{(\epsilon^{2n_1h} - \epsilon^{-2n_1h} + 2\sin 2n_1h)}
+       {\epsilon^{2n_1h} + \epsilon^{-2n_1h} - 2\cos 2n_1h}\, H^2;
+\]
+when $n_1h$~is very large this is equal to
+\[
+\frac{\sigma n_1 \smallbold{b}}{16 \pi}\, H^2,
+\]
+which is (\artref{286}{Art.~286}), as it ought to be, the same as for a solid
+cylinder of radius~$\smallbold{b}$.
+
+When $h$~is small and $n^2 \smallbold{a} h / \mu$ not large we must take into
+account terms which we have neglected in arriving at the
+preceding expression.
+%% -----File: 341.png---Folio 327-------
+
+In this case, we find from~(\eqnref{287}{62}) that
+\[
+\Theta
+  = -\frac{(\pi p^2 \smallbold{a}^2 h / \sigma)H\cos pt}
+          {1 + 4 \pi^2 p^2 \smallbold{a}^2 h^2/ \sigma^2}
+  + \tfrac{1}{2}
+     \frac{p \smallbold{a} H \sin pt}
+          {1 + 4 \pi^2 p^2 \smallbold{a}^2 h^2/ \sigma^2},  \Tag{65}
+\]
+so that the rate of heat production is
+\[
+\tfrac{1}{4}
+  \frac{(\pi p^2 \smallbold{a}^2 \smallbold{b} h/  \sigma) H^2}
+       {1 + 4 \pi^2 p^2 \smallbold{a}^2 h^2/\sigma^2}.
+\]
+Thus it vanishes when $h=0$, and is a maximum when
+\[
+h =  \frac{\sigma}{2\pi \smallbold{a} p};
+\]
+the rate of heat production is then
+\[
+\tfrac{1}{16} p \smallbold{b} \smallbold{a} H^2,
+\]
+and bears to the rate when the tube is solid the ratio
+\[
+\frac{\pi p \smallbold{a}}{n_1 \sigma} : 1,
+\]
+which is equal to~$n_1 \smallbold{a} / 2 \mu$.
+
+Since $n_1 \smallbold{a} / \mu$~is very large the heat produced in a tube of this
+thickness is very much greater than that produced in a solid
+cylinder.
+
+Let us take the case of a tin tube whose internal radius is
+$3$~cm.\ surrounded by a primary coil conveying a current making
+a hundred thousand vibrations per second, then since in this
+case
+\[
+\sigma = 1.3 × 10^4, \quad \smallbold{a}=3, \quad p = 2 \pi × 10^5, \quad \mu=1,
+\]
+the thickness which gives the maximum heat production is
+about $1/90$~of a millimetre, and the heat produced is about $26$~times
+as much as would be produced in a solid tin cylinder of
+the same radius as the tube.
+
+We see from equation~(\eqnref{287}{65}) that the amplitude of~$\Theta$ diminishes
+as the thickness of the plate increases, but that when the plate
+is indefinitely thin the phases of the tangential electromotive
+intensity and of the tangential magnetic force differ by a quarter-period;
+the product of these quantities will thus be proportional
+to~$\sin 2 pt$, and as the mean value of this vanishes there is no
+energy converted into heat in the tube. As the thickness of
+the tube increases the amplitude of~$\Theta$ diminishes, but the phase
+of~$\Theta$ gets more nearly into unison with that of~$H$. We may
+regard~$\Theta$ as made up of two oscillations, one being in the same
+phase as~$H$ while the phase of the other differs from that of~$H$ by
+%% -----File: 342.png---Folio 328-------
+a quarter-period. The amplitude of the second component
+diminishes as the thickness of the tube increases, while that of
+the first reaches a maximum when $h = \sigma/2\pi \smallbold{a}p$.
+
+In the investigation of the heat produced when $h$~is small,
+$n \smallbold{a} /\mu$~has been assumed large. We can however easily show
+that unless this is the case the heat produced in a thin tube
+will not exceed that produced in a solid cylinder.
+
+
+\Subsection{Vibrations of Electrical Systems.}
+\index{Electrical vibrations|indexetseq}%
+\index{Vibrations of electrical systems|indexetseq}%
+
+\Article{288} If the distribution of electricity on a system in electrical
+equilibrium is suddenly disturbed, the electricity will redistribute
+itself so as to tend to go back to the distribution it
+had when in electrical equilibrium; to effect this redistribution
+electric currents will be started. The currents possess kinetic
+energy which is obtained at the expense of the potential energy
+of the original distribution of electricity; this kinetic energy
+will go on increasing until the distribution of electricity is the
+same as it was in the state from which it was displaced. As this
+state is one of equilibrium its potential energy is a minimum.
+The kinetic energy which the system has acquired will carry it
+through this state, and the system will go on losing kinetic and
+reacquiring potential energy until the kinetic energy has all
+disappeared. The system will then retrace its steps, and if there
+is no dissipation of energy will again regain the distribution
+of electricity from which it started. The distribution of electricity
+on the system will thus oscillate backwards and forwards;
+we shall in the following articles endeavour to calculate
+the time taken by such oscillations for some of the simpler
+electrical systems.
+
+
+\Subsection{Electrical Oscillations when Two Equal Spheres are connected
+by a Wire\protect\footnotemark.}
+\footnotetext{See J.~J. Thomson, \textit{Proc.\ Lond.\ Math.\ Soc.}~19, p.~542, 1888.}
+\index{Time of vibration of two spheres connected by a wire@\subdashtwo vibration of two spheres connected by a wire}%
+
+\Article{289} The first case we shall consider is that of two equal
+spheres, or any two bodies possessing equal electric capacities,
+connected by a straight wire. This case can be solved at once
+by means of the analysis given at the beginning of this chapter.
+
+Let us take the point on the wire midway between the
+spheres as the origin of coordinates, and the axis of the wire as
+%% -----File: 343.png---Folio 329-------
+the axis of~$z$. We shall suppose that the electrostatic potential
+has equal and opposite values at points on the wire equidistant
+from the origin and on opposite sides of it. Then using the
+same notation as in \artref{271}{Art.~271}, we may put
+\begin{gather*}
+\phi = L(\epsilon^{\iota mz} - \epsilon^{-\iota mz})\,
+  J_0(\iota mr)\epsilon^{\iota pt}, \text{ in the wire}, \\
+  = L(\epsilon^{\iota mz} - \epsilon^{-\iota mz}) \epsilon^{\iota pt}
+\end{gather*}
+approximately, since $mr$~will be very small. Thus~$\smallbold{E}$, the external
+electromotive intensity parallel to the wire, is equal to
+\[
+-\iota mL(\epsilon^{\iota mz} + \epsilon^{-\iota mz})\epsilon^{\iota pt}.
+\]
+
+If $2l$~is the length of the wire, then the potential of the sphere
+at the end~$z=l$, will be
+\[
+2\iota L\sin ml \epsilon^{\iota pt}.
+\]
+
+If $C$~is the capacity of the sphere at one end of the wire, the
+quantity of electricity on the sphere is
+\[
+2\iota CL\sin ml \epsilon^{\iota pt},
+\]
+and this increases at the rate
+\[
+-2CpL\sin ml \epsilon^{\iota pt}.
+\]
+Now the increase in the charge of the sphere must equal the
+current flowing through the wire at the point~$z=l$, hence if $I$~denotes
+this current, we have
+\[
+I = -2CpL\sin ml\epsilon^{\iota pt},
+\]
+but by equation~(\eqnref{272}{39}) of \artref{272}{Art.~272} we have
+\[
+\smallbold{E} = (\iota p \smallbold{P} + \smallbold{Q})I,
+\]
+whence substituting the values for~$\smallbold{E}$ and~$I$ when~$z=l$, we get
+\begin{DPalign*}
+-2 \iota mL\cos ml\epsilon^{\iota pt}
+  & = -(\iota p \smallbold{P} + \smallbold{Q})2CpL\sin ml\epsilon^{\iota pt}, \\
+\lintertext{or} m\cot ml
+  & = -\iota p(\iota p \smallbold{P} + \smallbold{Q})C. \Tag{66}
+\end{DPalign*}
+
+\Article{290} Let us first consider the case when the wave length of
+electrical vibrations is very much longer than the wire; here $ml$~is
+very small, so that equation~(\eqnref{289}{66}) becomes
+\[
+\frac{1}{l} = -\iota p(\iota p \smallbold{P} + \smallbold{Q})C. \Tag{67}
+\]
+
+The values of $\smallbold{P}$~and~$\smallbold{Q}$, the self-induction and impedance of
+the wire, are given in equation~(\eqnref{272}{40}) of \artref{272}{Art.~272}; they depend upon
+the frequency of the electrical vibrations. When this is so
+%% -----File: 344.png---Folio 330-------
+slow that $n \smallbold{a}$~is a small quantity, $\smallbold{a}$~being the radius of the wire,
+then approximately
+\begin{align*}
+\smallbold{P} & = \frac{L}{2l}, \\
+\smallbold{Q} & = \frac{R}{2l},
+\end{align*}
+where $L$~is the coefficient of self-induction and $R$~the resistance
+of the whole wire for steady currents.
+
+Substituting these values in~(\eqnref{290}{67}), we get
+\begin{DPgather*}
+(\iota p)^2 L + \iota pR + \frac{2}{C} = 0, \\
+\lintertext{or} \iota p = -\frac{R}{2L} ± \iota \sqrt{\frac{2}{CL} - \frac{R^2}{4L^2}}. \Tag{68}
+\end{DPgather*}
+
+Since the various quantities which fix the state of the electric
+field contain $\epsilon^{\iota pt}$ as a factor, we see that when $8L>CR^2$ these
+quantities will be proportional to
+\[
+\epsilon^{-\frac{R}{2L} t}
+  \cos \left\{\left(\frac{2}{CL} - \frac{R^2}{4L^2}\right)^{\frac{1}{2}} t + \alpha \right\},
+\]
+where $\alpha$~is a constant.
+
+This represents an oscillation whose period is
+\[
+2\pi \bigg/ \left\{\frac{2}{CL} - \frac{R^2}{4L^2} \right\}^{\frac{1}{2}},
+\]
+and whose amplitude dies away to $1/\epsilon$~of its original value after
+the time $2L/R$.
+
+Thus, if $2/CL$ is greater than~$R^2/4L^2$, that is if $R^2$~is less than~$8 L/C$,
+the charges on the spheres will undergo oscillations like
+those performed by a pendulum in a resisting medium.
+
+\sloppy
+Suppose, for example, that the electrical connection between
+\index{Oscillatory discharge}%
+the spheres is broken, and let one sphere~$A$ be charged with
+positive, the other sphere~$B$ with an equal quantity of negative
+electricity; if now the electrical connection between the spheres
+is restored, the positive charge on~$A$ and the negative on~$B$ will
+diminish until after a time both spheres are free from electrification.
+They will not however remain in this state, for negative
+electricity will begin to appear on~$A$, positive on~$B$, and these
+charges will increase in amount until (neglecting the resistance of
+the circuit connecting the spheres) the charges on $A$~and~$B$ appear
+to be interchanged, there being now on~$A$ the same quantity of
+%% -----File: 345.png---Folio 331-------
+negative electricity as there was initially on~$B$, while the charge
+on~$B$ is the same as that originally on~$A$. When the negative
+charge on~$A$ has reached this value it begins to decrease, and after
+a time both spheres are again free from electrification. After this
+positive electricity begins to reappear on~$A$, and increases until the
+charge on~$A$ is the same as it was to begin with; this positive
+charge then decreases, vanishes, and is replaced by a negative
+one as before. The system thus behaves as if the charges
+vibrated backwards and forwards between the spheres. The
+changes which take place in the electrical charges on the spheres
+are of course accompanied by currents in the wire, these currents
+flowing sometimes in one direction, sometimes in the opposite.
+
+\fussy
+When the circuit has a finite resistance the amplitude of the
+\index{Condenser, discharge of}%
+\index{Discharge, of a condenser@\subdashone of a condenser}%
+\index{Leyden jar, oscillatory discharge of@\subdashtwo oscillatory discharge of|indexetseq}%
+oscillations gradually diminishes, while if the resistance is greater
+than~$(8 L/C)^{\frac{1}{2}}$ there will not be any vibrations at all, but the
+charges will subside to zero without ever changing sign; in this
+case the current in the connecting wire is always in one direction.
+
+\Article{291} If we assume that the wave length of the electrical
+vibrations is so great that the current may be regarded as uniform
+all along the wire, and that the vibrations are so slow that the
+current is uniformly distributed across the wire, the discharge of
+a condenser can easily be investigated by the following method,
+which is due to Lord Kelvin (\textit{Phil.\ Mag.}~[4], 5, p.~393, 1853).
+\index{Kelvin, Lord, xoscillatory discharge@\subdashtwo oscillatory discharge}%
+Let $Q$~be the quantity of electricity on one of the plates of a
+condenser whose capacity is~$C'$ and whose plates, like those of a
+Leyden Jar, are supposed to be close together; also let $R$~be
+the resistance and $L$~the coefficient of self-induction for steady
+currents of the wire connecting the plates. The electromotive
+force tending to increase~$Q$ is~$-Q/C'$; of this $R\,dQ/dt$ is required
+to overcome the resistance and $L\,d^2Q/dt^2$ to overcome the inertia
+of the circuit; hence we have
+\[
+L\, \frac{d^2Q}{dt^2} + R\, \frac{dQ}{dt} + \frac{Q}{C'} = 0. \Tag{69}
+\]
+
+The solution of this equation is, if
+\begin{gather*}
+\frac{1}{C'L} > \frac{R^2}{4L^2}, \\
+Q = A \epsilon^{-\frac{R}{2L} t}
+  \cos \left\{\left( \frac{1}{C'L} - \frac{R^2}{4L^2}\right)^{\frac{1}{2}} t + \beta \right\},
+\end{gather*}
+where $A$~and~$\beta$ are arbitrary constants.
+%% -----File: 346.png---Folio 332-------
+
+In this case we have an oscillatory discharge whose frequency
+is equal to
+\begin{DPgather*}
+\left( \frac{1}{C'L} - \frac{R^2}{4L^2} \right)^{\frac{1}{2}}. \\
+\lintertext{When}  \frac{1}{C'L} < \frac{R^2}{4L^2},
+\end{DPgather*}
+the solution of equation~(\eqnref{291}{69}) is
+\[
+Q = \epsilon^{-\frac{R}{2L} t}
+  \left\{A \epsilon^{\left(\frac{R^2}{4L^2} - \frac{1}{C'L}\right)^{\frac{1}{2}} t}
+        + B \epsilon^{-\left(\frac{R^2}{4L^2} - \frac{1}{C'L}\right)^{\frac{1}{2}} t} \right\},
+\]
+where $A$~and~$B$ are arbitrary constants. In this case the discharge
+is not oscillatory.
+
+To compare the results of this investigation with those of the
+previous one, we must remember that the capacities which occur
+in the two investigations are measured in somewhat different
+ways. The capacity~$C$ in the first investigation is the ratio of
+the charge on the condenser to~$\phi$ its potential; in the second
+investigation $C'$~is the ratio of the charge to~$2\phi$, the difference
+between the potentials of the plates, so that to compare the
+results we must put $C' = C/2$; if we do this the results given
+by the two investigations are identical.
+
+\Article{292} The existence of electrical vibrations seems to have
+\index{Henry, on electrical vibrations}%
+been first suspected by Dr.~Joseph Henry in 1842 from some experiments
+he made on the magnetization of needles placed in a
+coil in circuit with a wire which connected the inside to the
+outside coating of a Leyden Jar. He says (\textit{Scientific Writings}
+of Joseph Henry, Vol.~I, p.~201, Washington, 1886): `This
+anomaly which has remained so long unexplained, and which at
+first sight appears at variance with all our theoretical ideas of
+the connection of electricity and magnetism, was after considerable
+study satisfactorily referred by the author to an action of the
+discharge of the Leyden jar which had never before been recognised.
+The discharge, whatever may be its nature, is not correctly
+represented (employing for simplicity the theory of Franklin)
+by the simple transfer of an imponderable fluid from one side of
+the jar to the other, the phenomenon requires us to admit the
+existence of a principal discharge in one direction, and then
+several reflex actions backward and forward, each more feeble
+than the preceding, until the equilibrium is obtained. All the
+facts are shown to be in accordance with this hypothesis, and a
+ready explanation is afforded by it of a number of phenomena
+%% -----File: 347.png---Folio 333-------
+which are to be found in the older works on electricity but
+which have until this time remained unexplained.'
+
+In 1853, Lord Kelvin published (\textit{Phil.\ Mag.}\ [4],~5, p.~393,
+\index{Kelvin, Lord, xoscillatory discharge@\subdashtwo oscillatory discharge}%
+\index{Electrical vibrations, Lord Kelvin on@\subdashtwo Lord Kelvin on}%
+\index{Vibrations of electrical systems, Lord Kelvin on@\subdashtwo electrical systems, Lord Kelvin on}%
+1853) the results we have just given in \artref{291}{Art.~291}, thus proving
+by the laws of electrical action that electrical vibrations must
+be produced when a Leyden Jar is short circuited by a wire of
+not too great resistance.
+
+From 1857 to 1862, Feddersen (\textit{Pogg.\ Ann.}\ 103, p.~69, 1858;
+\index{Feddersen, electrical vibrations@\subdashone electrical vibrations}%
+\index{Electrical vibrations, Feddersen on@\subdashtwo Feddersen on}%
+\index{Vibrations of electrical systems, Feddersen on@\subdashtwo electrical systems, Feddersen on}%
+108, p.~497, 1859; 112, p.~452, 1861; 113, p.~437, 1861; 116,
+p.~132, 1862) published accounts of some beautiful experiments
+by which he demonstrated the oscillatory character of the jar
+discharge. His method consisted in putting an air break in the
+wire circuit joining the two coatings of the jar. When the current
+through this wire is near its maximum intensity a spark passes
+across the circuit, but when the current is near its minimum
+value the electromotive force is not sufficient to spark across the
+air break, which at these periods therefore is not luminous.
+Thus the image of the air space formed by reflection from a
+rotating mirror will be drawn out into a series of bright and
+dark spaces, the interval between two dark spaces depending of
+course on the speed of the mirror and the frequency of the
+electrical vibrations. Feddersen observed this appearance of the
+image of the air space, and he proved that the oscillatory
+character of the discharge was destroyed by putting a large resistance
+in circuit with the air space, by showing that in this case
+the image of the air space was a broad band of light gradually
+fading away in intensity instead of a series of bright and dark
+spaces. This experiment, which is a very beautiful one, can be
+repeated without difficulty. To excite the vibrations the coatings
+of the jar should be connected to the terminals of an induction
+coil or an electric machine. It is advisable to use a large jar
+with its coatings connected by as long a wire as possible. By
+connecting the coatings of the jar by a circuit with very large
+\index{Electrical vibrations, Lodge on@\subdashtwo Lodge on}%
+\index{Lodge, electrical vibration}%
+\index{Vibrations of electrical systems, Lodge on@\subdashtwo electrical systems, Lodge on}%
+self-induction, Dr.~Oliver Lodge (\textit{Modern Views of Electricity},
+p.~377) has produced such slow electrical vibrations that the
+sounds generated by the successive discharges form a musical
+note.
+
+\Article{293} In the course of the investigation in \artref{290}{Art.~290} we have
+made two assumptions, (1)~that $ml$~is small, (2)~that $n \smallbold{a}$~is also
+small, which implies that the currents are uniformly distributed
+%% -----File: 348.png---Folio 334-------
+across the section of the discharging circuit. This condition is
+however very rarely fulfilled, as the electrical oscillations which
+are produced by the discharge of a condenser are in general so
+rapid that the currents in the discharging circuit fly to the
+outside of the wire instead of distributing themselves uniformly
+across it; when the currents do this, however, the resistance of
+the circuit depends on the frequency of the electrical vibrations,
+and the investigation of \artref{290}{Art.~290} has to be modified. Before
+proceeding to the discussion of this case we shall write down the
+conditions which must hold when the preceding investigation is
+applicable.
+
+In the first place, $ml$~is to be small; now by \artref{263}{Art.~263} we
+have when $n \smallbold{a}$~is small,
+\begin{multline*}
+m^2 = -\iota p\, \text{(resistance of unit length of the wire)} ×\\
+\text{(capacity of unit length of wire)},
+\end{multline*}
+\begin{DPgather*}
+\lintertext{hence} m^2 l^2 = - \tfrac{1}{2} \iota p R l \Gamma,
+\end{DPgather*}
+where, as before, $R$~is the resistance of the whole of the discharging
+circuit, while $\Gamma$~is the capacity of unit length of the wire.
+
+But by equation~(\eqnref{290}{68}) when the discharge is oscillatory, we
+have
+\[
+\iota p = -\frac{R}{2 L}
+  ± \iota \left\{\frac{2}{LC} - \frac{R^2}{4 L^2} \right\}^{\frac{1}{2}};
+\]
+thus the modulus of~$\iota p$ is equal to
+\[
+\left\{\frac{2}{LC} \right\}^{\frac{1}{2}},
+\]
+hence, when $ml$~is small,
+\[
+\frac{R \Gamma l}{\sqrt{CL}}
+\]
+must be small.
+
+The other condition is that $n \smallbold{a}$~is small, which since
+\[
+n^2 = m^2 + \frac{4 \pi \mu \iota p}{\sigma},
+\]
+and $ml$~is also small, is equivalent to the condition that
+\[
+4 \pi \mu \iota p \smallbold{a}^2 / \sigma
+\]
+should be small. Since the modulus of~$\iota p$ is equal to~$\{2/LC \}^{\frac{1}{2}}$,
+we see that if $n^2 \smallbold{a}^2$~is small,
+\[
+4 \pi \mu \smallbold{a}^2 \{2 / LC \}^{\frac{1}{2}} / \sigma
+\]
+%% -----File: 349.png---Folio 335-------
+\index{Condenser, discharge of}%
+must be small. The capacity~$C$ which occurs in this expression
+is measured in electromagnetic units, its value in such measure is
+only~$1/V^2$ (where `$V$'~is the ratio of the units and $V^2 = 9 × 10^{20}$)
+of its value in electrostatic measure. Thus the expression which
+has to be small to ensure the condition we are considering,
+contains the large factor $3 × 10^{10}$, so that to fulfil this condition
+the capacity and self-induction of the circuit must be very large
+when the discharging circuit consists of metal wire of customary
+dimensions. Thus, to take an example, suppose two spheres
+each one metre in radius are connected by a copper wire
+$1$~millimetre in diameter. In this case
+\[
+C = 1/9 × 10^{18}, \quad \sigma = 1600, \quad \smallbold{a} = .05, \quad \mu = 1,
+\]
+substituting these values we find that to ensure $n \smallbold{a}$~being small,
+the self-induction of the circuit must be comparable with the
+enormously large value~$10^{11}$, which is comparable with the self-induction
+of a coil with $10,000$ turns of wire, the coil being
+about half a metre in diameter.
+
+The result of this example is sufficient to show that it is only
+when the self-induction of the circuit or the capacity of the
+condenser is exceptionally large that a theory based on the
+assumption that $n \smallbold{a}$~is a small quantity is applicable, it is therefore
+important to consider the case where $n \smallbold{a}$~is large and the
+currents in the discharging circuit are on the surface of the
+wire.
+
+\Article{294} The theory of this case is given in \artref{274}{Art.~274}, and we
+see from equations (\eqnref{274}{42})~and~(\eqnref{274}{43}) of that Article that when the
+frequency of the vibrations is so great that $n \smallbold{a}$~and~$n' \smallbold{b}$ (using
+the notation of \artref{274}{Art.~274}, and supposing that the wire connecting
+the spheres is a cable whose external radius is~$\smallbold{b}$) are large
+quantities, equation~(\eqnref{289}{66}) of \artref{289}{Art.~289} becomes
+\[
+m \cot ml = -2\iota p \left\{
+     \iota p \log{\smallbold{b} / \smallbold{a}}
+  + (\iota p)^{\frac{1}{2}} (\mu \sigma / 4 \pi \smallbold{a}^2)^{\frac{1}{2}}\right.
+  \left. + (\iota p)^{\frac{1}{2}} (\mu'\sigma'/ 4 \pi \smallbold{b}^2)^{\frac{1}{2}} \right\} C.
+\]
+
+Retaining the condition that $ml$~is small, which will be the case
+when the wave length of the electrical vibrations is very much
+greater than the length of the discharging circuit, this equation
+becomes
+\[
+\frac{1}{Cl} = -\iota p \left[ \iota p 2 \log( \smallbold{b} / \smallbold{a})
+  + (\iota p)^{\frac{1}{2}} 2 \left\{( \mu \sigma / 4 \pi \smallbold{a}^2 )^{\frac{1}{2}}
+  + ( \DPtypo{\mu'/sigma'}{\mu' \sigma'/} 4 \pi \smallbold{b}^2 )^{\frac{1}{2}} \right\} \right],
+\]
+%% -----File: 350.png---Folio 336-------
+which we shall write as
+\[
+\frac{2}{C} = -\iota p \left\{\iota pL' + 2(\iota p)^{\frac{1}{2}} S \right\}, \Tag{70}
+\]
+where $L'$ is the coefficient of self-induction of the discharging
+circuit for infinitely rapid alternating currents, and $S$~is written
+for
+\begin{DPgather*}
+\left\{(\sigma \mu /4\pi \smallbold{a}^2)^{\frac{1}{2}}
+      + (\sigma'\mu'/4\pi \smallbold{b}^2)^{\frac{1}{2}} \right\} 2l. \\
+\lintertext{By \artref{274}{Art.~274},}    L' = L - \mu l,
+\end{DPgather*}
+where $L$~is the self-induction of the circuit for steady currents.
+
+If we write~$x$ for~$\iota p$, equation~(\eqnref{294}{70}) becomes
+\begin{DPgather*}
+x^2 L'+ 2x^{\frac{3}{2}} S + \frac{2}{C} = 0, \\
+%
+\lintertext{hence}
+\left(x^2 L' + \frac{2}{C}\right)^2 = 4x^3 S^2, \\
+%
+\lintertext{or}
+L'^2 x^4 - 4S^2 x^3 + 4 \frac{L'}{C} x^2 + \frac{4}{C^2} = 0, \Tag{71}
+\end{DPgather*}
+a biquadratic equation to determine~$x$.
+
+If electrical oscillations take place the roots of this equation
+must be imaginary.
+
+From the theory of the biquadratic equation (Burnside and
+Panton, \textit{Theory of Equations}, §~68)
+\[
+ax^4 + 4bx^3 + 6cx^2 + 4dx + e = 0,
+\]
+we know that if
+\begin{gather*}
+H = ac - b^2, \qquad I = ae - 4bd + 3c^2, \qquad G = a^2 d - 3abc + 2b^3, \\
+J = ace + 2bcd - ad^2 - eb^2 - c^3, \qquad \Delta = I^3 - 27J^2;
+\end{gather*}
+the condition that the roots of the biquadratic are all imaginary
+is that $\Delta$~should be positive as well as one of the two following
+quantities $H$~and $a^2I - 12H^2$.
+
+Dividing equation~(\eqnref{294}{71}) by~$L'^2$, we see that for equation~(\eqnref{294}{71})
+\begin{gather*}
+H = \frac{2}{3}\, \frac{1}{L'C} - \frac{S}{L'^4}, \qquad I = \frac{16}{3}\, \frac{1}{L'^2 C^2}, \\
+G = 2 \frac{S^2}{L'^3 C} \left\{1 - \frac{CS^4}{L'^3} \right\}, \qquad
+J = \frac{64}{27}\, \frac{1}{L'^3 C^3} \left\{1 - \frac{27}{16}\, \frac{CS^4}{L'^3} \right\}.
+\end{gather*}
+Hence we see that $a^2 I - 12H^2$ and~$\Delta$ are both positive, if
+\[
+S^4 < 32L'^3/27C,
+\]
+that is if
+\[
+16l^4 \left\{(\sigma \mu /4\pi \smallbold{a}^2)^{\frac{1}{2}}
+           +  (\sigma'\mu'/4\pi \smallbold{b}^2)^{\frac{1}{2}} \right\}^4 < 32L'^3/27C,
+\]
+%% -----File: 351.png---Folio 337-------
+which is the condition that the system should execute electrical
+vibrations.
+
+When the spheres are connected by a free wire and not by a
+cable $\sigma'/ \smallbold{b}^2$ vanishes, and the condition that the system should
+oscillate reduces to
+\[
+l^2 ( \sigma \mu / \pi \smallbold{a}^2)^2 < 32 L'^{3} / 27 l^2 C.
+\]
+
+The results given by Ferrari's method for solving biquadratic
+equations are too complicated to be of much practical value in
+determining the roots of equation~(\eqnref{294}{71}), neither, since the roots
+are imaginary, can we apply the very convenient method known
+as Horner's method to determine the numerical value of these
+roots to any required accuracy.
+
+\Article{295} For the purpose of analysing the nature of the electrical
+oscillations it is convenient to consider separately the real and
+imaginary parts of~$\iota p$, the~$x$ of equation~(\eqnref{294}{71}). The real part,
+supposed negative, determines the rate at which the electrical
+vibrations die away, while the imaginary part gives the period
+of these vibrations. We shall now proceed to show how equation~(\eqnref{294}{71})
+can be treated so as to admit of the real and imaginary
+parts of~$x$ being separately determined by Horner's method.
+
+If we put
+\[
+\xi = x - \frac{S^2}{L'^{2}},
+\]
+equation~(\eqnref{294}{71}) becomes
+\[
+\xi^4 + 6 H \xi^2 + 4 G \xi + I - 3 H^2 = 0, \Tag{72}
+\]
+where $H$,~$G$,~$I$ are the quantities whose values we have just
+written down. Since the coefficient of~$\xi^3$ in this equation
+vanishes and since its roots are by hypothesis complex, we see
+that the real part of one pair of roots will be positive, that of the
+other pair negative: the pair of roots whose real parts are
+negative are those which correspond to the solution of the
+electrical problem. For if the real part of~$\xi$ were positive the
+real part of~$\iota p$ would also be positive, so that such a root would
+correspond to an electrical vibration whose amplitude increased
+indefinitely with the time.
+
+The roots of equation~(\eqnref{295}{72}) will be of the form
+\[
+x_1 + \iota y_1, \quad
+x_1 - \iota y_1, \quad
+-x_1 + \iota y_2, \quad
+-x_1 - \iota y_2.
+\]
+We shall now proceed to show how~$x_1$ may be uniquely determined.
+Since $6H$,~$-4G$,~$I - 3H^2$ are respectively the sums of
+%% -----File: 352.png---Folio 338-------
+the products of the roots of equation~(\eqnref{295}{72}) two and two, three
+and three, and all together, we have
+\begin{DPalign*}
+y_1^2 + y_2^2 - 2x_1^2 &= 6H, \Tag{73} \\
+x_1 (y_1^2 - y_2^2) &= 2G, \Tag{74} \\
+(x_1^2 + y_1^2)(x_1^2 + y_2^2) &= I - 3H^2,\\
+\lintertext{or} x_1^4 + x_1^2(y_1^2 + y_2^2)
+  + \tfrac{1}{4} \{(y_1^2 + y_2^2)^2
+  &- (y_1^2 - y_2^2)^2 \DPtypo{}{\}} = I - 3H^2.
+\end{DPalign*}
+Eliminating $y_1^2 + y_2^2$ and $y_1^2 - y_2^2$ by equations (\eqnref{295}{73})~and~(\eqnref{295}{74}),
+we get
+\begin{align*}
+4x_1^4 + 12Hx_1^2 + (12H^2 - I)- \frac{G^2}{x_1^2} &= 0,\\
+\intertext{or putting $x_1^2 = \eta$,}
+4\eta^3 + 12H\eta^2 +(12H^2 - I)\eta - G^2 &= 0. \Tag{75}
+\end{align*}
+
+Since the last term of this expression is negative there is at
+least one positive real root of this equation, and since the values
+given for $H$~and~$I$ show that when $\Delta$~is positive $12H^2 - I$ is essentially
+negative, we see by Fourier's rule that there is only one
+such root. But since $x_1$~is real the value of~$\eta$ will be positive, so
+that the root we are seeking will be the unique positive real root
+of equation~(\eqnref{295}{75}), which can easily be determined by Horner's
+method. The value of~$x_1$ is equal to minus the square root of
+this root, and knowing~$x_1$ we can find~$y_1^2 / 4\pi^2$, the square of the
+corresponding frequency uniquely from equations (\eqnref{295}{73})~and~(\eqnref{295}{74}).
+We can in this way in any special case determine with ease
+the logarithmic decrement and the frequency of the vibrations.
+
+\Article{296} If in equation~(\eqnref{295}{75}) we substitute the values of $G$,~$H$, and~$I$,
+and write
+\[
+L' C \eta = \zeta, \qquad  C S^4 / L'^3 = q,
+\]
+that equation becomes
+\[
+\zeta^3 + 2 \zeta^2(1 - \tfrac{3}{2} q) - \zeta(3q^2 - 4q) - q(1 - q)^2 = 0.
+\]
+We can by successive approximations expand~$\zeta$ in terms of~$q$,
+and thus when $CS^4 / L'^3$ is small approximate to the value of~$\zeta$.
+The first term in this expansion is
+\begin{DPgather*}
+\zeta = (q / 2)^{\frac{1}{2}},\\
+\lintertext{or since} L'Cx_1^2 = \zeta,\\
+x_1 = -\frac{S}{2^{\frac{1}{4}} C^{\frac{1}{4}} L'^{\frac{5}{4}}}.
+\end{DPgather*}
+%% -----File: 353.png---Folio 339-------
+
+The corresponding value of~$y_{1}^{2}$ determined by equations (\eqnref{295}{73})~and~(\eqnref{295}{74})
+is, retaining only the lowest power of~$q$, approximately,
+\begin{DPgather*}
+y_{1}^{2} = \frac{2}{L'C}
+  \left\{1 - \frac{2^{\frac{1}{4}} SC^{\frac{1}{4}}}{L'^{\frac{3}{4}}} \right\}. \\
+\lintertext{Now}
+S = \left\{\DPtypo{}{(}
+      \mu \sigma / 4 \pi \smallbold{a}^{2})^{\frac{1}{2}}
+  + ( \mu'\sigma'/ 4 \pi \smallbold{b}^{2})^{\frac{1}{2}} \right\} 2l, \\
+\lintertext{\rlap{and, approximately,}} y_{1}^{2} = \frac{2}{L' C}, \\
+\lintertext{and} x_{1} = - \frac{S y^{\frac{1}{2}}}{2^{\frac{1}{2}} L},
+\end{DPgather*}
+hence we see,
+\[
+x_{1} = - \frac{l}{L'}
+  \left\{( \mu \sigma y_{1} / 2 \pi \smallbold{a}^{2} )^{\frac{1}{2}}
+        + ( \mu'\sigma'y_{1} / 2 \pi \smallbold{b}^{2} )^{\frac{1}{2}}\right\}.
+\]
+
+But by \artref{274}{Art.~274}, the quantity enclosed in brackets is equal to~$\smallbold{Q}$,
+the impedance of unit length of the circuit when the frequency
+of vibration is~$y_{1} / 2 \pi$; thus we have
+\[
+x_{1} = - \frac{Q}{2 L'},
+\]
+where $Q$~is the impedance of the whole circuit.
+
+\begin{DPalign*}
+\lintertext{\indent Since}
+\iota p &= x_{1} + \iota y_{1} + \frac{S^{2}}{L'^{2}}\\
+        &= x_{1} \left\{1 - (2q)^{\frac{1}{4}} \right\} + \iota y_{1} ,
+\end{DPalign*}
+the real part of~$\iota p$ differs from~$x_{1}$ by a quantity involving~$q$.
+Neglecting this term, we see that the expression for the amplitude
+of the vibrations contains the factor~$\epsilon^{-\frac{Q}{2L'} t}$. Comparing this
+with the factor~$\epsilon^{-\frac{R}{2L} t}$, which occurs when the oscillations are so
+slow that the current is uniformly distributed over the cross-section
+of the discharging wire, we find that to our order of approximation
+we may for quick vibrations use a similar formula
+for the decay of the amplitude to that which holds for slow
+vibrations, provided we use the impedance instead of the resistance,
+and the coefficient of self-induction for infinitely rapid
+vibrations instead of that for infinitely slow ones. This result is,
+however, only true when~$CS^{4}/L'^{3}$ is a small quantity. Now if
+the external conductor is so far away that $\mu' \sigma' / \smallbold{b}^{2}$ is small compared
+with~$\mu \sigma / \smallbold{a}^{2} $, then
+\[
+S^{4} = \left\{2 l ( \mu \sigma / 4 \pi \smallbold{a}^{2} )^{\frac{1}{2}}\right\}^4
+  = \tfrac{1}{4} l^{2} \mu^{2} R^{2},
+\]
+%% -----File: 354.png---Folio 340-------
+where $R$~is the resistance of the whole circuit to steady currents.
+Substituting this value for~$S^{4}$ we see that the condition that
+$CS^4/L'^3$ is a small quantity is that $C l^2 \mu^2 R^2 / 4 L'^3$ should be small.
+When this is the case we see that, neglecting the effect of the
+external conductor,
+\[
+x_1 = -\frac{l}{L'} ( \mu \sigma y_1 / 2 \pi \smallbold{a}^2 )^{\frac{1}{2}}~.
+\]
+
+Since $x_1$~is proportional to~$\mu^{\frac{1}{2}}$, the rate of decay of the vibrations
+\index{Iron, xdecay of electromagnetic waves in@\subdashone decay of electromagnetic waves in}%
+\index{Trowbridge, decay of vibrations along iron wires}%
+will be greater when the discharging wire is made of
+iron than when it is made of a non-magnetic metal of the same
+resistance. This has been observed by Trowbridge (\textit{Phil.\ Mag.}\
+[5],~32, p.~504, 1891).
+
+\Article{297} We have assumed in the preceding work that the
+length of the electrical wave is great compared with that of the
+wire; we have by equation~(\eqnref{289}{66})
+\[
+m \cot {ml} = -\iota p \{\iota p \smallbold{P} + \smallbold{Q} \} C.
+\]
+When the frequency is very high, $\iota p \smallbold{P}$~will be very large compared
+with~$\smallbold{Q}$, hence this equation may be written as
+\[
+m \cot {ml} = p^2 \smallbold{P} C.
+\]
+Now if $V$~is the velocity of light in the dielectric, $p = Vm$, hence
+we have
+\[
+\frac{\cot{ml}}{ml} = \frac{V^{2} 2 \smallbold{P} l C}{2 l^2}.
+\]
+Now $2 \smallbold{P} l$ is equal to~$L'$, the self-induction of the discharging
+circuit for infinitely rapid vibrations, and $V^2 C$~is equal to the
+electrostatic measure of the capacity of the sphere which we
+shall denote by~$[C]$, hence the preceding equation may be
+written as
+\[
+\frac{\cot {ml}}{ml} = \frac{L'[C]}{2l^2}.
+\]
+
+Thus, if $L'[C]/ 2 l^2$~is very large, $ml$~will be very small; if, on
+the other hand, $L'[C]/ 2 l^2$~is very small, $\cot{ml}$~will be very small,
+or $ml = (2j + 1) \dfrac{\pi}{2}$ approximately, where $j$~is an integer. Since
+$2 \pi / m$~is the length of the electrical wave the latter will equal
+$4l,~4l/3, 4l/5\ldots$, or the half-wave length will be an odd submultiple
+of the length of the discharging wire. We are limited
+by our investigation to the odd submultiple because we have
+assumed that the current in the discharging wire is symmetrical
+%% -----File: 355.png---Folio 341-------
+about the middle point of that wire. If we abandon
+this assumption we find that the half-wave length may be any
+submultiple of the length of the wire. The frequencies of the
+vibrations are thus independent of the capacity at the end of the
+wire provided this is small enough to make $L[C]/2l^2$ small. In
+this case the vibrations are determined merely by the condition
+that the current in the discharging wire should vanish at its
+extremities.
+
+
+\Subsection{Vibrations along Wires in Multiple Arc.}
+\index{Vibrations along wires in multiple arc}%
+\index{Multiple arc, electrical vibrations along wires in}%
+
+\Article{298} When the capacities of the conductors at the ends of a
+single wire are very small, we have seen that the gravest
+electrical vibration has for its wave length twice the length of
+the wire and that the other vibrations are harmonics of this.
+We shall now investigate the periods of vibration of the system
+when the two conductors of small capacity are connected by two
+or more wires in parallel. The first case we shall consider is
+the one represented by \figureref{fig109}{Fig.~109}, where in the connection between
+the points $A$~and~$F$ we have the loop~$BCED$.
+
+\includegraphicsmid{fig109}{Fig.~109.}
+
+We proved in \artref{272}{Art.~272} that the relation between the current~$I$
+and the external electromotive intensity~$\smallbold{E}$ is expressed by the
+equation
+\[
+\smallbold{E} = \{\iota p \smallbold{P} + \smallbold{Q} \} I.
+\]
+Where, when as in this case the vibrations are rapid enough to
+make $n \smallbold{a}$~large, the term~$\iota p \smallbold{P}$ is much larger than~$\smallbold{Q}$, we may
+therefore for our purpose write this equation as
+\[
+\smallbold{E} = \iota p \smallbold{P} I,  \Tag{76}
+\]
+where $\smallbold{P}$~is the coefficient of self-induction of unit length of
+the wire for infinitely rapid vibrations.
+
+Let the position of a point on~$AB$ be fixed by the length~$s_1$
+measured along~$AB$ from~$A$, that of one on~$BCE$ by the length~$s_2$
+measured from~$B$, that of one on~$BDE$ by~$s_3$ measured also
+from~$B$, and of one on~$EF$ by~$s_4$ measured from~$E$. Let $l_1$,~$l_2$, $l_3$,~$l_4$
+%% -----File: 356.png---Folio 342-------
+denote the lengths~$AB$, $BCE$,~$BDE$, and~$EF$ respectively, and
+let $P_1$,~$P_2$, $P_3$,~$P_4$ denote the self-induction per unit length of
+these wires. Let $\phi$~denote the electrostatic potential, then the
+external electromotive intensity along a wire is~$-d \phi / ds$, and
+as this is proportional to the current it must vanish at the
+ends $A$,~$F$ of the wire if the capacity there is, as we suppose,
+very small.
+
+Hence along~$AB$ we may write, if $p / 2 \pi$ is the frequency,
+\begin{DPgather*}
+\phi = a \cos {m s_1} \cos {pt}, \\
+\lintertext{along $BCE$}
+\phi = (a \cos{ms_2} \cos{m l_1} + b \sin{ms_2}) \cos{pt}, \\
+\lintertext{along $BDE$}
+\phi = (a \cos{ms_3} \cos{m l_1} + c \sin{ms_3}) \cos{pt}, \\
+\lintertext{and along $EF$}
+\phi = d \cos{m} (s_4- l_4) \cos{pt}.
+\end{DPgather*}
+
+Equating the expressions for the potential at~$E$, we have
+\[
+\left.\begin{aligned}
+a \cos{ml_2} \cos{ml_1} + b \sin{ml_2} &= d \cos{ml_4},\\
+a \cos{ml_3} \cos{ml_1} + c \sin{ml_3} &= d \cos{ml_4}.
+\end{aligned}\right\} \Tag{77}
+\]
+
+The current flowing along~$AB$ at~$B$ must equal the sum of the
+currents flowing along $BCE$,~$BDE$, hence by~(\eqnref{298}{76}) we have
+\[
+\frac{a \sin{ml_1}}{P_1} = -\frac{b}{P_2} - \frac{c}{P_3}. \Tag{78}
+\]
+
+Again, the current along~$EF$ at~$E$ must equal the sum of the
+currents flowing along $BCE$,~$BDE$, hence we have
+\begin{multline*}
+\frac{d \sin{ml_4}}{P_4}
+  = \frac{b \cos{ml_2}}{P_2} - \frac{a \sin{ml_2} \cos{ml_1}}{P_2}
+  + \frac{c \cos{ml_3}}{P_3} \\
+  - \frac{a \sin{ml_3} \cos{ml_1}}{P_3}. \Tag{79}
+\end{multline*}
+
+We get from equations (\eqnref{298}{77})~and~(\eqnref{298}{78})
+\begin{multline*}
+a \left\{\frac{\sin{ml_1}}{P_1}
+  - \frac{\cot{ml_2} \cos{ml_1}}{P_2}
+  - \frac{\cot{ml_3} \cos{ml_4}}{P_3} \right\}\\
+  = -d \cos{ml_4} \left\{\frac{\operatorname{cosec}{ml_2}}{P_2}
+                        + \frac{\operatorname{cosec}{ml_3}}{P_3} \right\}.
+\end{multline*}
+
+From equations (\eqnref{298}{77})~and~(\eqnref{298}{79}) we get
+\begin{multline*}
+d \left\{\frac{\sin{ml_4}}{P_4}
+  - \frac{\cot{ml_2} \cos{ml_4}}{P_2}
+  - \frac{\cot{ml_3} \cos{ml_4}}{P_3} \right\}\\
+  = -a \cos{ml_1} \left\{\frac{\operatorname{cosec}{ml_2}}{P_2}
+                        + \frac{\operatorname{cosec}{ml_3}}{P_3} \right\}.
+\end{multline*}
+%% -----File: 357.png---Folio 343-------
+
+Eliminating $a$~and~$d$ from these equations, we get
+\begin{multline*}
+\left\{\frac{\tan{ml_1}}{P_1} - \frac{\cot{ml_2}}{P_2} - \frac{\cot{ml_3}}{P_3} \right\}
+\left\{\frac{\tan{ml_4}}{P_4} - \frac{\cot{ml_2}}{P_2} - \frac{\cot{ml_3}}{P_3} \right\} \\
+  = \left\{\frac{\operatorname{cosec}{ml_2}}{P_2}
+          + \frac{\operatorname{cosec}{ml_3}}{P_3} \right\}^2. \Tag{80}
+\end{multline*}
+
+If $AB$~and~$EF$ are equal lengths of the same kind of wire,
+$l_1 = l_4$, and $P_1 = P_4$, and~(\eqnref{298}{80}) reduces to the simple form
+\[
+\frac{\tan{ml_1}}{P_1} - \frac{\cot{ml_2}}{P_2} - \frac{\cot{ml_3}}{P_3}
+  = ± \left\{\frac{\operatorname{cosec}{ml_2}}{P_2}
+            + \frac{\operatorname{cosec}{ml_3}}{P_3} \right\};
+\]
+taking the upper sign, we have
+\[
+\frac{\tan{ml_1}}{P_1}
+  = \frac{\cot{\frac{1}{2} ml_2}}{P_2}
+  + \frac{\cot{\frac{1}{2} ml_3}}{P_3}, \Tag{81}
+\]
+if we take the lower sign, we have
+\[
+\frac{\tan{ml_1}}{P_1}
+  = - \left\{\frac{\tan{\frac{1}{2} ml_2}}{P_2}
+            + \frac{\tan{\frac{1}{2} ml_3}}{P_3} \right\}. \Tag{82}
+\]
+
+Since $m = 2 \pi / \lambda$, where $\lambda$~is the wave length, these equations
+determine the wave lengths of the electrical vibrations.
+
+If all the wires have the same radius, $P_1 = P_2 = P_3$, and equations
+(\eqnref{298}{81})~and~(\eqnref{298}{82}) become respectively
+\begin{DPgather*}
+\tan{2 \pi \frac{l_1}{\lambda}}
+  = \cot{\left( \pi \frac{l_2}{\lambda} \right)}
+  + \cot{\left( \pi \frac{l_3}{\lambda} \right)}, \Tag{81*} \\
+\lintertext{and}
+\tan{2 \pi \frac{l_1}{\lambda}}
+  + \tan{\pi \frac{l_2}{\lambda}}
+  + \tan{\pi \frac{l_3}{\lambda}} = 0. \Tag{82*}
+\end{DPgather*}
+
+From these equations we can determine the effect on the
+period of an alteration in the length of one of the wires.
+Suppose that the length of~$BDE$ is increased by~$\delta l_3$, and let~$\delta \lambda$
+be the corresponding increase in~$\lambda$, then from~(81*)
+\[
+\frac{\delta \lambda}{\lambda}
+  \left\{l_1 \sec^2{\frac{2 \pi l_1}{\lambda}}
+  + \tfrac{1}{2} l_2 \operatorname{cosec}^2{\frac{\pi l_2}{\lambda}}
+  + \tfrac{1}{2} l_3 \operatorname{cosec}^2{\frac{\pi l_3}{\lambda}} \right\}
+  = \tfrac{1}{2} \delta l_3 \operatorname{cosec}^2{\frac{\pi l_3}{\lambda}}.
+\]
+We see from this equation that $\delta \lambda$~and~$\delta l_3$ are of the same sign,
+so that an increase in~$l_3$ increases the wave length.
+
+If we take equation~(\eqnref{298}{82*}), we have
+\[
+\frac{\delta \lambda}{\lambda}
+  \left\{l_1 \sec^2{\frac{2 \pi l_1}{\lambda}}
+  + \tfrac{1}{2} l_2 \sec^2{\frac{\pi l_2}{\lambda}}
+  + \tfrac{1}{2} l_3 \sec^2{\frac{\pi l_3}{\lambda}} \right\}
+  = \tfrac{1}{2} \delta l_3 \sec^2{\frac{\pi l_3}{\lambda}},
+\]
+hence, in this case also, an increase in~$l_3$ increases~$\lambda$. If~$l_3$ is
+%% -----File: 358.png---Folio 344-------
+infinite the wave length is $4 l_1 + 2 l_2$ and its submultiples, as we
+diminish~$l_3$ the wave length shortens, hence we see that the
+effect of introducing an alternative path is to shorten the wave
+lengths of all the vibrations. The shortening of the wave length
+goes on until $l_3$~vanishes, when the wave length of the gravest
+vibration is~$4 l_1$.
+
+\Article{299} The currents through the wires $BCE$~and~$BDE$ are at~$B$
+in the proportion of
+\[
+\frac{\cot{\frac{1}{2}ml_2}}{P_2} \text{ to } \frac{\cot{\frac{1}{2}ml_3}}{P_3},
+\]
+if we take the vibrations corresponding to equation~(\eqnref{298}{81}), and in
+the proportion of
+\[
+\frac{\tan{\frac{1}{2}ml_2}}{P_2} \text{ to } \frac{\tan{\frac{1}{2}ml_3}}{P_3},
+\]
+for the vibration given by~(\eqnref{298}{82}).
+
+We can prove by the method of \artref{298}{Art.~298} that if we have
+$n$~wires between $B$~and~$F$, and if $AB = EF$,
+\begin{multline*}
+\frac{\tan{m l_1}}{P_1} - \frac{\cot{m l_2}}{P_2} - \frac{\cot{m l_3}}{P_3} - \ldots \\
+  = ± \left\{\frac{\operatorname{cosec}{m l_2}}{P_2}
+            + \frac{\operatorname{cosec}{m l_3}}{P_3}
+            + \frac{\operatorname{cosec}{m l_4}}{P_4} + \ldots \right\}.
+\end{multline*}
+It follows from this equation that if any of the wires are
+shortened the wave lengths of the vibrations are also shortened.
+
+
+\Section{Electrical Oscillations on Cylinders.}
+\index{Cylinder, electrical oscillations on}%
+\index{Electrical vibrations, on cylinders@\subdashtwo on cylinders}%
+\index{Oscillations, electrical, on cylinders}%
+\index{Time of vibration of electricity on a cylinder@\subdashtwo vibration of electricity on a cylinder}%
+
+\Subsection{Periods of Vibration of Electricity on the Cylindrical Cavity
+inside a Conductor.}
+
+\Article{300} If on the surface of a cylindrical cavity inside a
+conductor an irregular distribution of electricity is produced,
+then on the removal of the cause producing this irregularity,
+currents of electricity will flow from one part of the cylinder to
+another to restore the electrical equilibrium, electrical vibrations
+will thus be started whose periods we now proceed to
+investigate.
+
+Take the axis of the cylinder as the axis of~$z$, and suppose
+that initially the distribution of electricity is the same on all
+sections at right angles to the axis of the cylinder; it will
+evidently remain so, and the currents which restore the electrical
+%% -----File: 359.png---Folio 345-------
+distribution to equilibrium will be at right angles to the axis
+of~$z$.
+
+If $c$~is the magnetic induction parallel to~$z$, then in the cavity
+filled with the dielectric $c$~satisfies the differential equation
+\[
+\frac{d^2 c}{dx^2} + \frac{d^2 c}{dy^2} = \frac{1}{V^2}\, \frac{d^2 c}{dt^2},
+\]
+where $V$~is the velocity of propagation of electrodynamic action
+through the dielectric.
+
+In the conductor $c$~satisfies the equation
+\[
+\frac{d^2 c}{dx^2} + \frac{d^2 c}{dy^2} = \frac{4 \pi \mu}{\sigma}\, \frac{dc}{dt},
+\]
+where $\sigma$~is the specific resistance and $\mu$~the magnetic permeability
+of the substance.
+
+Transform these equations to polar coordinates $r$~and~$\theta$, and
+suppose that $c$~varies as $\cos{s \theta \epsilon^{\iota p t}}$; making these assumptions, the
+differential equation satisfied by~$c$ in the dielectric is
+\[
+\frac{d^2 c}{dr^2} + \frac{1}{r}\, \frac{dc}{dr}
+  + c \left( \frac{p^2}{V^2} - \frac{s^2}{r^2} \right) = 0,
+\]
+the solution of which is
+\[
+c = A \cos s \theta J_s \left( \frac{p}{V} r \right) \epsilon^{\iota p t},
+\]
+where $J_s$~denotes the internal Bessel's function of the $s$\textsuperscript{th}~order.
+
+The differential equation satisfied by~$c$ in the conductor is
+\[
+\frac{d^2 c}{dr^2} + \frac{1}{r}\, \frac{dc}{dr}
+  + \left\{- \frac{4 \pi \mu \iota p}{\sigma} - \frac{s^2}{r^2} \right\} c = 0.
+\]
+Let $n^2 = 4 \pi \mu \iota p / \sigma$, then the solution of this equation is
+\[
+c = B \cos s \theta K_s ( \iota n r ) \epsilon^{\iota p t},
+\]
+where $K_s$ denotes the external Bessel's function of the $s$\textsuperscript{th}~order.
+
+Since the magnetic force parallel to the surface of the cylinder
+is continuous, we have if $\smallbold{a}$~denotes the radius of the cylindrical
+cavity
+\[
+A J_s \left( \frac{p}{V} \smallbold{a} \right) = \frac{B}{\mu} K_s ( \iota n \smallbold{a} ). \Tag{83}
+\]
+
+The electromotive intensity at right angles to~$r$ is also continuous.
+Now the current at right angles to $r$~and~$z$ is
+\[
+- dc / 4 \pi \mu\, dr,
+\]
+%% -----File: 360.png---Folio 346-------
+hence in the conductor the electromotive intensity perpendicular
+to $r$~and~$z$ is $-\sigma dc/4 \pi \mu\, dr$. In the dielectric the current is equal
+to the rate of increase of the electric displacement, i.e.~to $\iota p$~times
+the electric displacement or to $\iota p K / 4 \pi$ times the electromotive
+intensity; we see that in the dielectric the electromotive
+intensity perpendicular to~$r$ is $-\dfrac{1}{K \iota p}\, \dfrac{dc}{dr}$, hence we have
+\[
+A \frac{4 \pi}{K \iota p}\, \frac{p}{V}  J'_s \left( \frac{p}{V} \smallbold{a} \right)
+  = \frac{B}{\mu} \iota n \sigma K'_s ( \iota n \smallbold{a} ).  \Tag{84}
+\]
+
+Eliminating $A$~and~$B$ from (\eqnref{300}{83})~and~(\eqnref{300}{84}), we get
+\[
+\frac{4 \pi}{K V}\,
+  \frac{J'_s \left( \dfrac{p}{V} \smallbold{a} \right)}
+       {J_s  \left( \dfrac{p}{V} \smallbold{a} \right)}
+  = - \sigma n \frac{K'_s(\iota n \smallbold{a})}{K_s(\iota n \smallbold{a})}.  \Tag{85}
+\]
+Now $K = \dfrac{1}{V^2}$ and $\sigma = \dfrac{4 \pi \mu \iota p}{n^2}$, so that~(\eqnref{300}{85}) may be written
+\[
+\frac{V}{p \smallbold{a}}\,
+  \frac{J'_s \left( \dfrac{p}{V} \smallbold{a} \right)}
+       {J_s  \left( \dfrac{p}{V} \smallbold{a} \right)}
+  = \frac{\mu}{\iota n \smallbold{a}}\,
+    \frac{K'_s(\iota n \smallbold{a})}{K_s(\iota n \smallbold{a})}.  \Tag{86}
+\]
+
+Now the wave length of the electrical vibrations will be comparable
+with the diameter of the cylinder, and the value of~$p$
+corresponding to this will be sufficient to make~$n\smallbold{a}$ exceedingly
+large, but when $n \smallbold{a}$~is very large we have (Heine, \textit{Kugelfunctionen},
+vol.~i.\ p.~248)
+\[
+K_s (\iota n \smallbold{a})
+  = (-\iota)^{s} \epsilon^{-n \smallbold{a}}
+    \sqrt{\frac{\pi}{2 n \smallbold{a}}}\quad \text{approximately},
+\]
+hence $K'_s (\iota n \smallbold{a}) = \iota K_s(\iota n \smallbold{a})$; thus the right-hand side of~(\eqnref{300}{86})
+will be exceedingly small, and an approximate solution of this
+equation will be
+\[
+J_s' \left( \frac{V}{p} \smallbold{a} \right) = 0.
+\]
+This signifies that the tangential electromotive intensity vanishes
+at the surface of the cylinder, or that the tubes of electrostatic
+induction cut its surface at right angles. The roots of the
+equation
+\[
+J_s'(x) = 0,
+\]
+%% -----File: 361.png---Folio 347-------
+for~$s = 1$, $2$,~$3$, are given in the following table taken from Lord
+Rayleigh's \textit{Theory of Sound}, Vol.~II, p.~266:---
+\begin{center}
+\tabletextsize
+\begin{tabular}{|*{3}{c|}}
+\hline
+\tablespaceup s = 1   & s = 2 &  s = 3 \tablespacedown\\
+\hline
+\tablespaceup\Z1.841 & 3.054 & \Z4.201 \\
+\Z5.332 & 6.705 & \Z8.015 \\
+\Z8.536 & 9.965 & 11.344  \\
+11.706  &       &         \\
+14.864  &       &         \\
+18.016  &       &       \tablespacedown  \\
+\hline
+\end{tabular}
+\end{center}
+
+\index{Cylinder, electrical oscillations on}%
+\index{Time of vibration of electricity on a cylinder@\subdashtwo vibration of electricity on a cylinder}%
+Thus, when $s = 1$, the gravest period of the electrical vibrations
+is given by the equation
+\[
+\frac{p}{V} \smallbold{a} = 1.841,
+\]
+or the wave length of the vibration $2 \pi V/p = .543 × 2 \pi \smallbold{a}$, and is
+thus more than half the circumference of the cylinder. In this
+case, as far as our approximations go, there is no decay of the
+vibrations, though if we took into account the right-hand side
+of~(\eqnref{300}{86}) we should find there was a small imaginary term in the
+expression for~$p$, which would indicate a gradual fading away of
+the vibrations. If it were not for the resistance of the conductor
+the oscillations would last for ever, as there is no radiation of
+energy away from the cylinder. The magnetic force vanishes in
+the conductor except just in the neighbourhood of the cavity,
+and the magnetic waves emitted by one portion of the walls of
+the cavity will be reflected from another portion, so that no
+energy escapes.
+
+
+\Subsection{Metal Cylinder surrounded by a Dielectric.}
+
+\Article{301} In this case the waves starting from one portion of the
+cylinder travel away through the dielectric and carry energy
+with them, so that the vibrations will die away independently
+of the resistance of the conductor.
+
+Using the same notation as before, we have in the conducting
+cylinder
+\[
+c = A \cos{s \theta J_s (\iota n r) \epsilon^{\iota pt}},
+\]
+and in the surrounding dielectric
+\[
+c = B \cos{s \theta K_s \left( \frac{p}{V} r \right) \epsilon^{\iota pt}}.
+\]
+%% -----File: 362.png---Folio 348-------
+
+\index{Functions, Bessel's}%
+Since the magnetic force parallel to~$z$ is continuous, we have
+\[
+\frac{A}{\mu} J_s (\iota n \smallbold{a}) = B K_s \left( \frac{p}{V} \smallbold{a} \right).
+\]
+Since the electromotive intensity perpendicular to~$r$ is continuous,
+we have
+\[
+\frac{A}{\mu} \iota n \sigma J'_s(\iota n \smallbold{a})
+  = B \frac{4 \pi}{K \iota p}\, \frac{p}{V} K_s' \left( \frac{p}{V} \smallbold{a} \right).
+\]
+
+Eliminating $A$~and~$B$ from these equations, we get
+\begin{DPgather*}
+\iota n \sigma \frac{J'_s(\iota n \smallbold{a})}{J_s(\iota n \smallbold{a})}
+  = \frac{4 \pi}{K \iota V}\,
+  \frac{K'_s \left( \dfrac{p}{V} \smallbold{a} \right) }
+       {K_s  \left( \dfrac{p}{V} \smallbold{a} \right) }, \\
+\lintertext{or}
+\frac{1}{\iota n \smallbold{a}}\, \frac{J'_s(\iota n \smallbold{a})}{J_s(\iota n \smallbold{a})}
+  = \frac{V}{\mu p \smallbold{a}}\,
+  \frac{K'_s \left( \dfrac{p}{V} \smallbold{a} \right) }
+       {K_s  \left( \dfrac{p}{V} \smallbold{a} \right) }. \Tag{87}
+\end{DPgather*}
+
+Now, as before, $n \smallbold{a}$~will be large, and therefore
+\[
+J_s(\iota n \smallbold{a})
+  = \frac{\iota^s \epsilon^{n \smallbold{a}}}{\sqrt{2 \pi n \smallbold{a}}}\quad \text{approximately},
+\]
+hence $J_s'(\iota n \smallbold{a}) = - \iota J_s(\iota n \smallbold{a})$, and the left-hand side of equation~(\eqnref{301}{87})
+is very small, so that the approximate form of~(\eqnref{301}{87}) will be
+\[
+K'_s \left( \frac{p}{V} \smallbold{a} \right) = 0, \Tag{88}
+\]
+which again signifies that the electromotive intensity tangential
+to the cylinder vanishes at its surface.
+
+In order to calculate the approximate values of the roots of
+the equation $K'_s(x) = 0$, it is most convenient to use the expression
+for~$K_s(x)$ which proceeds by powers of~$1/x$. This
+series is expressed by the equation
+\begin{multline*}
+K_s(x) = C \frac{\epsilon^{-\iota x}}{(\iota x)^{\frac{1}{2}}}
+  \left\{1 - \frac{(1^2 - 4s^2)}{8 \iota x}
+          + \frac{(1^2-4s^2)(3^2 - 4s^2)}{1\centerdot2(8 \iota x)^2} \right.\\
+  - \left. \frac{(1^2 - 4s^2)(3^2 - 4s^2)(5^2 - 4s^2)}{1\centerdot2\centerdot3 (8 \iota x)^3} + \ldots\right\},
+\end{multline*}
+where $C$~is a constant (see Lord Rayleigh, \textit{Theory of Sound},
+Vol.~II, p.~271).
+%% -----File: 363.png---Folio 349-------
+
+When $s = 1$,
+\[
+K_1(x) = C \frac{\epsilon^{-\iota x}}{(\iota x)^{\frac{1}{2}}}
+  \left\{1 + \frac{3}{8 \iota x} - \frac{15}{2(8 \iota x)^2} + \frac{105}{2(8 \iota x)^3} - \ldots \right\}.
+\]
+Thus
+\[
+K_1'(x) = - \iota C \frac{\epsilon^{-\iota x}}{(\iota x)^{\frac{1}{2}}}
+  \left\{1 + \frac{7}{8 \iota x} + \frac{57}{128 (\iota x)^2} - \frac{195}{1024 (\iota x)^3}\ldots \right\}.
+\]
+
+\index{Decay, xof electrical oscillations on cylinders@\subdashone of electrical oscillations on cylinders}%
+\index{Rate of decay of yoscillation on cylinders@\subdashtwo of oscillation on cylinders}%
+\index{Vibrations d, decay of, on cylinders@\subdashtwo on cylinders}%
+To approximate to the roots of the equation $K_1'(x) = 0$, put
+$\iota x = y$, and equate the first four terms inside the bracket to zero;
+we get
+\[
+y^3 + \frac{7}{8} y^2 + \frac{57}{128} y - \frac{195}{1024} = 0,
+\]
+a cubic equation to determine~$y$. One root of this equation is
+real and positive, the other two are imaginary; if~$\alpha$ is the
+positive root, $\beta ± \iota \gamma$~the two imaginary roots, then we have
+\begin{align*}
+\alpha + 2 \beta & = -\frac{7}{8}, \\
+2 \beta \alpha + \beta^2 + \gamma^2 & = \frac{57}{128}, \\
+\alpha(\beta^2 + \gamma^2) & = \frac{195}{1024}.
+\end{align*}
+
+We find by the rules for the solution of numerical equations
+that $\alpha = .26$ approximately, hence
+\[
+\beta = - .56, \qquad  \gamma = ± .64.
+\]
+
+These roots are however not large enough for the approximation
+to be close to the accurate values.
+
+Hence from equation~(\eqnref{301}{88}), we see that when $s = 1$,
+\begin{DPgather*}
+\frac{\iota p}{V} \smallbold{a} = - .56 ± \iota .64, \\
+\lintertext{or} \iota p = (- .56 ± \iota .64) \frac{V}{\smallbold{a}}.
+\end{DPgather*}
+This represents a vibration whose period is $3.1 \pi \smallbold{a} / V$, and whose
+amplitude fades away to $1 / \epsilon$~of its original value after a time
+$1.8 \smallbold{a} / V$.
+
+The radiation of energy away from the \DPtypo{sphere}{cylinder} in this case is
+so rapid that the vibrations are practically dead beat; thus after
+one complete vibration the amplitude is only~$\epsilon^{-1.74 \pi}$, or about
+one two hundred and fiftieth part of its value at the beginning
+of the oscillation.
+%% -----File: 364.png---Folio 350-------
+
+\Article{302} If we consider the state of the field at a considerable
+\index{Cylinder, field of force round oscillating@\subdashone field of force round oscillating}%
+\index{Faradayx tubes, disposition of round vibrating cylinder@\subdashtwo disposition of round vibrating cylinder}%
+distance from the cylinder and only retain in each expression
+the lowest power of~$1/r$, we find that the magnetic induction~$c$,
+the tangential and radial components $\Theta$~and~$R$ of the electric
+polarization in the dielectric, may be consistently represented by
+the following equations:
+\begin{DPgather*}
+c = \cos{\theta} \frac{1}{r^{\frac{1}{2}}}
+  \epsilon^{- .56 \left( \frac{Vt - r}{\smallbold{a}} \right)}
+         \cos .64 \left( \frac{Vt - r}{\smallbold{a}} \right), \\
+\lintertext{since} K\, \frac{d \Theta}{dt} = - \frac{1}{\mu}\, \frac{dc}{dr}, \\
+\intertext{we have}
+\Theta = \frac{\cos {\theta}}{K \mu V}\, \frac{1}{r^{\frac{1}{2}}}
+  \epsilon^{- .56 \left( \frac{Vt - r}{\smallbold{a}} \right)}
+         \cos .64 \left( \frac{Vt - r}{\smallbold{a}} \right), \\
+\lintertext{and since}
+K\, \frac{dR}{dt} = \frac{1}{\mu r}\, \frac{dc}{d \theta},
+\end{DPgather*}
+we have
+\begin{multline*}
+R = \frac{\sin \theta}{K \mu V}\, \frac{\smallbold{a}}{r^{\frac{3}{2}}}\,
+  1.34 \epsilon^{- .56 \left( \frac{Vt - r}{\smallbold{a}} \right)} \\
+  \left\{.56 \cos .64 \left( \frac{Vt - r}{\smallbold{a}} \right)
+        - .64 \sin .64 \left( \frac{Vt - r}{\smallbold{a}} \right) \right\}.
+\end{multline*}
+
+Thus $R$~vanishes at all points on a series of cylinders concentric
+with the original one whose radii satisfy the equation
+\[
+\cot{.64} \left( \frac{Vt - r}{\smallbold{a}} \right) = 1.13,
+\]
+the distance between the consecutive cylinders in this series is
+\[
+1.57 \pi \smallbold{a}.
+\]
+The Faraday tubes between two such cylinders form closed
+curves, all cutting at right angles the cylinder for which
+\[
+\Theta = 0, \quad \text{or} \quad
+\cos{.64} \left( \frac{Vt - r}{\smallbold{a}} \right) = 0.
+\]
+The closed Faraday tubes move away from the cylinder and
+are the vehicles by which the energy of the cylinder radiates
+into space. The axes of the Faraday tubes, i.e.~the lines of
+electromotive intensity between two cylinders at which $R = 0$,
+are represented in \figureref{fig110}{Fig.~110}.
+
+\includegraphicsmid{fig110}{Fig.~110.}
+
+The genesis of these closed endless tubes from the unclosed
+ones, which originally stretched from one point to another of the
+%% -----File: 365.png---Folio 351-------
+cylinder, which we may suppose to have been electrified initially
+so that the surface density was proportional to~$\sin \theta$, is shown in
+\figureref{fig111}{Fig.~111}.
+
+\includegraphicsmid[!t]{fig111}{Fig.~111.}
+
+The lines represent the changes in shape in a Faraday tube
+which originally stretched from a positively to a negatively electrified
+place on the cylinder. The outer line~\smallsanscap{A} represents the original
+position of the tube; when the equilibrium is disturbed some of
+the tubes inside this one will soon run into the cylinder, and the
+lateral repulsion they exerted on the tube under consideration
+will be removed; the outside lateral pressure on this tube will
+%% -----File: 366.png---Folio 352-------
+\index{Decay, xof currents and magnetic force in cylinders@\subdashone of currents and magnetic force in cylinders}%
+\index{Electric currents, decay of, in cylinders}%
+\index{Magnetic zforce@\subdashone force, decay of in cylinders}%
+\index{Rate of decay of xcurrents in cylinders@\subdashtwo of currents in cylinders}%
+now overpower the inside pressure and will produce the indentation
+shown in the second position~\smallsanscap{B} of the tube; this indentation
+increases until the two sides of the tube meet as in the third
+position~\smallsanscap{C} of the tube; when this takes place the tube breaks up,
+the outer part~\smallsanscap{D} travelling out into space and forming one of the
+closed tubes shown in \figureref{fig111}{Fig.~111}, while the inner part~\smallsanscap{E} runs into
+the cylinder.
+
+
+\Subsection{Decay of Magnetic Force in a Metal Cylinder.}
+
+\Article{303} In addition to the very rapid oscillations we have just
+investigated there are other and slower changes which may
+occur in the electrical state of the cylinder. Thus, for example,
+a uniform magnetic field parallel to the axis of the cylinder
+might suddenly be removed; the alteration in the magnetic force
+would then induce currents in the cylinder whose magnetic action
+would tend to maintain the original state of the magnetic field,
+so that the field instead of sinking abruptly to zero would die
+away gradually. The rate at which the state of the system
+changes with the time in cases like this is exceedingly slow
+compared with the rate of change we have just investigated.
+Using the same notation as in the preceding investigation, it
+will be slow enough to make $p \smallbold{a}/V$ an exceedingly small quantity;
+when however $p \smallbold{a}/V$ is very small, $K'_s (p \smallbold{a}/V)$ is exceedingly
+large compared with $K_s(p \smallbold{a}/V)$, since (Heine, \textit{Kugelfunctionen},
+vol.~i.\ p.~237) $K_s(\theta)$ is equal to
+\[
+(-2\theta )^s  \frac{d^s K_0(\theta)}{(d\theta^2)^s};
+\]
+thus since when $\theta$ is small $K_0(\theta)$~is proportional to~$\log \theta_1$,
+$K_s(p \smallbold{a}/V)$~is proportional to~$(V/p \smallbold{a})^s$, and $K'_s (p \smallbold{a}/V)$ to~$(V/p \smallbold{a})^{s+1}$;
+hence the right-hand side of equation~(\eqnref{301}{87}) is exceedingly
+large, so that an approximate solution of that equation
+will be
+\[
+J_s(\iota n \smallbold{a}) = 0.
+\]
+
+We notice that this condition makes the normal electromotive
+intensity at the surface of the cylinder vanish, while it will be
+remembered that for the very rapid oscillations the tangential
+electromotive intensity vanished. As the normal intensity
+vanishes there is no electrification on the surface of the cylinder
+in this case.
+
+The equation $J_s(x) = 0$ has an infinite number of roots all
+%% -----File: 367.png---Folio 353-------
+\index{Rayleigh@Rayleigh, Lord, \textit{Theory of Sound}}%
+real, the smaller values of which from $s = 0$ to $s = 5$ are given
+in the following table, taken from Lord Rayleigh's \textit{Theory of
+Sound}, Vol.~I, p.~274.
+\begin{center}
+\tabletextsize
+\begin{tabular}{|*{6}{c|}}
+\hline
+\tablespaceup$s = 0  $ & $s = 1  $ & $s = 2  $ & $s = 3  $ & $s = 4  $ & $s = 5$\tablespacedown   \\
+\hline
+\tablespaceup$\Z2.404$ & $\Z3.832$ & $\Z5.135$ & $\Z6.379$ & $\Z7.586$ & $\Z8.780$  \\
+$\Z5.520$ & $\Z7.016$ & $\Z8.417$ & $\Z9.760$ & $11.064 $ & $12.339 $  \\
+$\Z8.654$ & $10.173 $ & $11.620 $ & $13.017 $ & $14.373 $ & $15.700 $  \\
+$11.792 $ & $13.323 $ & $14.796 $ & $16.224 $ & $17.616 $ & $18.982 $  \\
+$14.931 $ & $16.470 $ & $17.960 $ & $19.410 $ & $20.827 $ & $22.220 $  \\
+$18.071 $ & $19.616 $ & $21.117 $ & $22.583 $ & $24.018 $ & $25.431 $  \\
+$21.212 $ & $22.760 $ & $24.270 $ & $25.749 $ & $27.200 $ & $28.628 $  \\
+$24.353 $ & $25.903 $ & $27.421 $ & $28.909 $ & $30.371 $ & $31.813 $  \\
+$37.494 $ & $29.047 $ & $30.571 $ & $32.050 $ & $33.512 $ & $34.983 $\tablespacedown  \\
+\hline
+\end{tabular}
+\end{center}
+
+This table may be supplemented by the aid of the theorem that
+\index{Bessel's functions, xroots of@\subdashtwo roots of}%
+\index{Functions, Bessel's}%
+the large roots of the equation got by equating a Bessel's function
+to zero form approximately an arithmetical progression whose
+common difference is~$\pi$.
+
+If $x_q$~denotes a root of the equation
+\[
+J_s(x) = 0,
+\]
+then since $p$~is given by the equation
+\begin{DPgather*}
+J_s(\iota n \smallbold{a}) = 0, \\
+\lintertext{where} n^2 = \frac{4 \pi \mu \iota p}{\sigma},
+\end{DPgather*}
+we see that~$p_q$, the corresponding value of~$p$, is given by the
+equation
+\[
+-\iota p_q = \frac{\sigma}{4\pi \smallbold{a}^2 \mu}\, x_q^2.
+\]
+
+Thus, since $\iota p_q$~is real and negative, the system simply fades
+away to its position of equilibrium and does not oscillate about it.
+
+The term in~$c$ which was initially expressed by
+\[
+A \cos s \theta J_s \left( x_q \frac{r}{\smallbold{a}} \right),
+\]
+will after the lapse of a time~$t$ have diminished to
+\[
+A \cos s \theta J_s \left( x_q \frac{r}{\smallbold{a}} \right)
+  \epsilon^{-\frac{\sigma}{4 \pi \smallbold{a}^2 \mu} x_q^2 t}.
+\]
+If we call $T$ the time which must elapse before the term sinks
+to $1/\epsilon$~of its original value, the `time modulus' of the term, then,
+since
+\[
+T = \frac{4 \pi \smallbold{a}^2 \mu}{\sigma x_q^2},
+\]
+%% -----File: 368.png---Folio 354-------
+we see that the time modulus is inversely proportional to the
+resistance of unit length of the cylinder and directly proportional
+to the magnetic permeability. Since $\mu / \sigma$ for iron is larger
+than it is for copper, the magnetic force will fade away more
+slowly in an iron cylinder than in a copper one.
+
+\Article{304} A case of great interest, which can be solved without
+difficulty by the preceding equations, is the one where a cylinder
+is placed in a uniform magnetic field which is suddenly
+annihilated, the lines of magnetic force being originally parallel
+to the axis of the cylinder. We may imagine, for example, that
+the cylinder is placed inside a long straight solenoid, the current
+through which is suddenly broken.
+
+Since in this case everything is symmetrical about the axis
+of the cylinder, $s = 0$, and the values of~$\iota p$ are therefore
+\[
+-(2.404)^2 \frac{\sigma}{4 \pi \smallbold{a}^2 \mu}, \qquad
+-(5.520)^2 \frac{\sigma}{4 \pi \smallbold{a}^2 \mu}, \text{ \&c.}
+\]
+
+Now we know from the theory of Bessel's functions that any
+function of~$r$ can for values of~$r$ between $0$~and~$\smallbold{a}$ be expanded
+in the form
+\[
+A_1 J_0 \left( x_1 \frac{r}{\smallbold{a}} \right) +
+A_2 J_0 \left( x_2 \frac{r}{\smallbold{a}} \right) +
+A_3 J_0 \left( x_3 \frac{r}{\smallbold{a}} \right) + \ldots,
+\]
+where $x_1, x_2, x_3 \ldots$ are the roots of the equation
+\[
+J_0(x) = 0.
+\]
+
+Thus, initially
+\[
+c = A_1 J_0 \left( x_1 \frac{r}{\smallbold{a}} \right)
+  + A_2 J_0 \left( x_2 \frac{r}{\smallbold{a}} \right)
+  + A_3 J_0 \left( x_3 \frac{r}{\smallbold{a}} \right) + \ldots,
+\]
+hence the value of~$c$ after a time~$t$ will be given by the equation
+\[
+c = A_1 J_0 \left( x_1 \frac{r}{\smallbold{a}} \right)
+      \epsilon^{-\frac{\sigma}{4 \pi \smallbold{a}^2 \mu} x_1^2 t}
+  + A_2 J_0 \left( x_2 \frac{r}{\smallbold{a}} \right)
+      \epsilon^{-\frac{\sigma}{4 \pi \smallbold{a}^2 \mu} x_2^2 t} + \ldots,
+\]
+so that all we have to do is to find the coefficients $A_1, A_2,
+A_3\ldots$.
+
+We shall suppose that initially $c$~was uniform over the section
+of the cylinder and equal to~$c_0$.
+
+Then, since
+\[
+\int_0^{\smallbold{a}} r
+  J_0 \left( x_p \frac{r}{\smallbold{a}} \right)
+  J_0 \left( x_q \frac{r}{\smallbold{a}} \right) dr = 0
+\]
+%% -----File: 369.png---Folio 355-------
+when $p$~and~$q$ are different, we see that
+\[
+c_0 \int_0^{\smallbold{a}} rJ_0 \left( x_q \frac{r}{\smallbold{a}} \right) dr
+  = A_q \int_0^{\smallbold{a}} rJ_0^2 \left( x_q \frac{r}{\smallbold{a}} \right) dr.
+\]
+
+Now, since
+\begin{gather*}
+J_0''(x) + \frac{1}{x} J_0'(x) + J_0(x) = 0,\\
+xJ_0(x) = -\frac{d}{dx} (x J_0'(x)),
+\end{gather*}
+\begin{DPalign*}
+\lintertext{hence} \int_0^{\smallbold{a}} r J_0 \left( x_q \frac{r}{\smallbold{a}} \right) dr
+  &= -\frac{\smallbold{a}^2}{x_q} J_0'(x_q) \\
+  &=  \frac{\smallbold{a}^2}{x_q} J_1(x_q).
+\end{DPalign*}
+
+Again, since
+\[
+J_0''(x) + \frac{1}{x} J_0'(x) + J_0(x) = 0,
+\]
+we have, multiplying by~$2x^2 J_0'(x)$,
+\begin{DPgather*}
+\frac{d}{dx} \{x^2 J_0'^2 (x) + x^2 J_0^2(x) \} = 2 x J_0^2 (x), \\
+\lintertext{hence} x^2 \{J_0'^2 (x) + J_0^2(x) \} = 2 \int_0^x x J_0^2(x)\,dx.
+\end{DPgather*}
+
+\begin{DPalign*}
+\lintertext{\indent Thus, since} J_0(x_q) &= 0,\\
+\int_0^{\smallbold{a}} r J_0^2 \left( x_q \frac{r}{\smallbold{a}} \right) dr
+  &= \frac{1}{2} \smallbold{a}^2 J_0'^2(x_q)\\
+  &= \frac{1}{2} x^2 J_1^2 (x_q).
+\end{DPalign*}
+
+Hence, we see that
+\[
+A_q = \frac{2c_0}{x_q J_1(x_q)},
+\]
+and therefore
+\[
+c = 2 c_0 \tsum \frac{1}{x_q}\,
+    \frac{J_0 \left( x_q \dfrac{r}{\smallbold{a}} \right)}{J_1(x_q)}\,
+    \epsilon^{- \frac{\sigma}{4\pi \smallbold{a}^2 \mu} x_q^2 t}.
+\]
+
+We see from this equation that immediately after the magnetic
+force is removed $c$~vanishes at the surface of the cylinder; also,
+since the terms in the expression for~$c$ corresponding to the
+large roots of the equation $J_0(x) = 0$ die away more quickly
+than those corresponding to the smaller roots, $c$~will ultimately
+%% -----File: 370.png---Folio 356-------
+be very approximately represented by the first term in the
+preceding expression; hence we have, since
+\begin{gather*}
+J_1(2.404) = .519, \\
+c = 1.6c_0J_0 \left (2.404 \frac{r}{\smallbold{a}}\right )
+  \epsilon^{-\frac{\sigma} {4\pi \smallbold{a}^2\mu} 5.78t}.
+\end{gather*}
+
+This expression is a maximum when $r = 0$ and gradually dies
+away to zero when~$r = \smallbold{a}$, thus the lines of magnetic force fade
+away most quickly at the surface of the cylinder and linger
+longest at the centre.
+
+The time modulus for the first term is $4\pi \smallbold{a}^2\mu / 5.78\sigma$. For a
+copper rod $1$~cm.\ in radius for which $\sigma = 1600$, this is about
+$1/736$~of a second; for an iron rod of the same radius for which
+$\mu = 1000$, $\sigma = 10^4$, it is about $2/9$~of a second.
+
+\Article{305} The intensity of the current is $-\dfrac{1}{4\pi \mu}\, \dfrac{dc}{dr}$, hence at a distance~$r$
+from the axis of the cylinder the intensity is
+\[
+\frac{c_0}{2\pi \mu \smallbold{a}}
+  \tsum \frac{J_1 \left (x_q\dfrac{r}{\smallbold{a}} \right ) }{J_1(x_q)}\,
+  \epsilon^{-\frac{\sigma }{4\pi \smallbold{a}^2\mu} {x_q}^2t}.
+\]
+
+Since at the instant the magnetic force is destroyed, $c$~is
+constant over the cross-section of the cylinder, the intensity of
+the current when~$t = 0$ will vanish except at the surface of the
+cylinder, where, as the above equation shows, it is infinite. After
+some time has elapsed the intensity of the current will be
+adequately represented by the first term of the series, i.e.~by
+\[
+\frac{c_0}{2\pi \mu \smallbold{a}}\,
+  \frac{J_1\left(2.404 \dfrac{r}{\smallbold{a}}\right)}{.52}\,
+  \epsilon^{-\frac{\sigma }{4\pi \smallbold{a}^2\mu} 5.78t}.
+\]
+This vanishes at the axis of the cylinder and, as we see from
+tables for~$J_1(x)$ (Lord Rayleigh, \textit{Theory of Sound}, vol.~I, p.~265),
+\index{Rayleigh@Rayleigh, Lord, \textit{Theory of Sound}}%
+attains a maximum when $2.404 \dfrac{r}{\smallbold{a}} = 1.841$, or at a distance from
+the axis about $3/4$~the radius of the cylinder.
+
+The following table, taken from the paper by Prof.\ Lamb on
+\index{Lamb, decay of currents in cylinders}%
+this subject (\textit{Proc.\ Lond.\ Math.\ Soc.}~XV, p.~143), gives the value
+of the total induction through the cylinder, and the electromotive
+force round a circuit embracing the cylinder for a series
+of values of~$t/\tau$, where $\tau = 4\pi \mu \smallbold{a}^2/\sigma$:---
+%% -----File: 371.png---Folio 357-------
+\begin{center}
+\tabletextsize
+\begin{tabular}{|@{\qquad}c@{\qquad}|c|c|}
+\hline
+$t/\tau$ &
+\settowidth{\TmpLen}{Total Induction}%
+\parbox[c]{\TmpLen}{\centering Total Induction} &
+\settowidth{\TmpLen}{Electromotive}%
+\parbox[c]{\TmpLen}{\tablespaceup\centering Electromotive\\ force$/\sigma c_0$\tablespacedown} \\
+\hline
+\tablespaceup $.00$          & \llap{$1$}$.0000$ & infinite \\
+ $.02$          & $.7014$           & $1.7332$\\
+ $.04$          & $.5904$           & $1.1430$\\
+ $.06$          & $.5105$           & $\Z.8789$\\
+ $.08$          & $.4470$           & $\Z.7195$\\
+ $.10$          & $.3941$           & $\Z.6089$\\
+ $.20$          & $.2178$           & $\Z.3168$\\
+ $.30$          & $.1220$           & $\Z.1765$\\
+ $.40$          & $.0684$           & $\Z.0989$\\
+ $.50$          & $.0384$           & $\Z.0555$\\
+ $.60$          & $.0215$           & $\Z.0311$\\
+ $.70$          & $.0121$           & $\Z.0174$\\
+ $.80$          & $.0068$           & $\Z.0098$\\
+ $.90$          & $.0038$           & $\Z.0055$\\
+\llap{$1$}.00 & $.0021$           & $\Z.0031$\tablespacedown\\
+\hline
+\end{tabular}
+\end{center}
+
+
+\Subsection{Rate of Decay of Currents and Magnetic Force in infinite
+Cylinders when the Currents are Longitudinal and the Magnetic
+Force Transversal.}
+
+\Article{306} We have already considered this problem in the special
+case when the currents are symmetrically distributed through
+the cylinder in \artref{262}{Art.~262}; we shall now consider the case when the
+currents are not the same in all planes through the axis.
+
+Let $w$~be the intensity of the current parallel to the axis of
+the cylinder, then (\artref{256}{Art.~256}) in the cylinder~$w$ satisfies the
+differential equation
+\[
+\frac{d^2w}{dx^2} + \frac{d^2w}{dy^2} = \frac{4 \pi \mu}{\sigma}\, \frac{dw}{dt}.
+\]
+If $w'$~denotes the rate of increase in the electric displacement
+parallel to~$z$ in the dielectric surrounding the cylinder, then,
+since $w'$~is equal to $\dfrac{K}{4 \pi}\, \dfrac{dZ}{dt}$, where $Z$~is the electromotive intensity
+parallel to~$z$ the axis of the cylinder, $w'$~satisfies the equation
+\[
+\frac{d^2w'}{dx^2} + \frac{d^2w'}{dy^2} = \frac{1}{V^2}\, \frac{d^2w'}{dt^2}.
+\]
+
+Let us suppose that $w$~varies as $\cos s \theta \epsilon^{\iota p t}$, then transforming to
+cylindrical coordinates $r$,~$\theta$, the equation satisfied by~$w$ in the
+cylinder becomes
+\[
+\frac{d^2w}{dr^2} + \frac{1}{r}\, \frac{dw}{dr}
+  + \left\{- \frac{4 \pi \mu \iota p}{\sigma} - \frac{s^2}{r^2} \right\} w = 0,
+\]
+the solution of which is
+\[
+w = A \cos s \theta \epsilon^{\iota p t} J_s(\iota n r),
+\]
+%% -----File: 372.png---Folio 358-------
+\begin{DPgather*}
+\lintertext{where} n^2 = \frac{4 \pi \mu \iota p}{\sigma};
+\end{DPgather*}
+while in the dielectric we have
+\[
+\frac{d^2w'}{dr^2} + \frac{1}{r}\, \frac{dw'}{dr}
+  + \left( \frac{p^2}{V^2} - \frac{s^2}{r^2} \right) w' = 0,
+\]
+the solution of which is
+\[
+w' = B \cos s \theta \epsilon^{\iota p t} K_s \left( \frac{p}{V} r \right).
+\]
+
+The electromotive intensity~$Z$, parallel to the axis of the
+cylinder, is equal to~$\sigma w$ in the cylinder and to $4 \pi w' / K \iota p$ in the
+dielectric. At the surface of the cylinder $r = \smallbold{a}$ these must be
+equal, hence we have
+\[
+\sigma A J_s(\iota n \smallbold{a})
+  = \frac{4 \pi}{K \iota p} B K_s \left( \frac{p}{V} \smallbold{a} \right). \Tag{89}
+\]
+
+If $\Theta$~is the magnetic induction at right angles to~$r$, then
+\[
+\frac{d \Theta}{dt} = \frac{dZ}{dr},
+\]
+or, since $\Theta$~varies as~$\epsilon^{\iota p t}$,
+\[
+\Theta = \frac{1}{\iota p}\, \frac{dZ}{dr}.
+\]
+
+Thus, in the cylinder
+\begin{align*}
+\Theta &= \frac{\sigma}{\iota p}\, \frac{dw}{dr} \\
+  &= \frac{\sigma}{\iota p}\, \iota n A J_s'(\iota n \smallbold{a}), \text{ at the surface}.
+\end{align*}
+
+In the dielectric
+\[
+\Theta = \frac{1}{\iota p}\, \frac{4 \pi}{K \iota p}\, \frac{p}{V}\,
+  B K_s' \left( \frac{p}{V} \smallbold{a} \right), \text{ at the surface}.
+\]
+
+Since the magnetic force parallel to the surface is continuous,
+we have
+\begin{DPgather*}
+\left( \frac{1}{\mu} \Theta \right) \text{ in the cylinder}
+  = \Theta \text{ in the dielectric}, \\
+\lintertext{hence}
+\frac{\sigma}{\mu} \iota n A J_s' ( \iota n \smallbold{a} )
+  = \frac{4 \pi}{K \iota p}\, \frac{p}{V} BK_s'\left( \frac{p}{V} \smallbold{a} \right). \Tag{90}
+\end{DPgather*}
+
+Eliminating $A$~and~$B$ from (\eqnref{306}{89})~and~(\eqnref{306}{90}), we have
+\[
+\frac{\iota n \smallbold{a} J_s'(\iota n \smallbold{a})}{\mu J_s (\iota n \smallbold{a})}
+  = \frac{p \smallbold{a}}{V}\,
+    \frac{K_s' \left( \dfrac{p}{V} \smallbold{a} \right)}
+         {K_s  \left( \dfrac{p}{V} \smallbold{a} \right)}. \Tag{91}
+\]
+%% -----File: 373.png---Folio 359-------
+
+In this case $p \smallbold{a} / V$ is very small, so that (\artref{303}{Art.~303}) $K_s \left( \dfrac{p}{V} \smallbold{a} \right)$ is
+approximately proportional to $\left( \dfrac{p}{V} \smallbold{a} \right)^{-s}$, and thus
+\[
+\frac{p}{V}\, \frac{K_s' \left( \dfrac{p}{V} \smallbold{a} \right)}
+                   {K_s  \left( \dfrac{p}{V} \smallbold{a} \right)}
+  = -\frac{s}{\smallbold{a}}, \text{ approximately};
+\]
+hence equation~(\eqnref{306}{91}) becomes
+\[
+\iota n \smallbold{a} J_s'(\iota n \smallbold{a}) + s \mu J_s(\iota n \smallbold{a}) = 0. \Tag{92}
+\]
+
+Bessel's functions, however, satisfy the relation
+\[
+J_s'(\iota n \smallbold{a}) + \frac{s}{\iota n \smallbold{a}} J_s(\iota n \smallbold{a})
+  = J_{s-1}(\iota n \smallbold{a}),
+\]
+so that (\eqnref{306}{92})~may be written
+\[
+s(\mu-1)J_s(\iota n \smallbold{a}) + \iota n \smallbold{a} J_{s-1}(\iota n \smallbold{a}) = 0.
+\]
+
+For non-magnetic substances $\mu = 1$, so that this equation
+reduces to
+\[
+J_{s-1}(\iota n \smallbold{a}) = 0.
+\]
+The magnetic induction along the radius is equal to
+\[
+-\frac{\sigma}{\iota p}\, \frac{1}{r}\, \frac{dw}{d \theta};
+\]
+at right angles to the radius it is equal to
+\[
+\frac{\sigma}{\iota p}\, \frac{dw}{dr}.
+\]
+
+\Article{307} Let us consider the case when $s = 1$. For a non-magnetic
+cylinder $n$~will be given by the equation
+\[
+J_0(\iota n \smallbold{a}) = 0;
+\]
+thus the values of~$\iota p$ will be the same as those in \artref{304}{Art.~304}, and
+we may put
+\[
+w = \cos{\theta}
+  \left\{A_1 J_1 \left( x_1 \frac{r}{\smallbold{a}} \right) \epsilon^{-\frac{\sigma}{4 \pi \smallbold{a}^2} x_1^2 t} \right.
+  \left. + A_2 J_1 \left( x_2 \frac{r}{\smallbold{a}} \right) \epsilon^{-\frac{\sigma}{4 \pi \smallbold{a}^2} x_2^2 t} + \ldots \right\}, \Tag{93}
+\]
+where $x_1$,~$x_2$ are the values $2.404, 5.520\ldots$, which are the roots
+of the equation
+\[
+J_0(x) = 0.
+\]
+%% -----File: 374.png---Folio 360-------
+The magnetic force along the radius is therefore
+\begin{multline*}
+-\frac{4 \pi \smallbold{a}^2 \sin{\theta}}{r}
+  \left\{\frac{1}{x_1^2} A_1 J_1 \left( x_1 \frac{r}{\smallbold{a}} \right) \epsilon^{-\frac{\sigma}{4 \pi \smallbold{a}^2} x_1^2 t} \right. \\
+\left.  + \frac{1}{x_2^2} A_2 J_1 \left( x_2 \frac{r}{\smallbold{a}} \right) \epsilon^{-\frac{\sigma}{4 \pi \smallbold{a}^2} x_2^2 t} + \ldots \right\}. \Tag{94}
+\end{multline*}
+
+If originally the magnetic force is parallel to~$y$ and equal
+to~$H$, the radial component of the magnetic force is~$H \sin{\theta}$;
+hence, if we determine $A_1$,~$A_2$ so that when $t = 0$ the expression~(\eqnref{307}{94})
+is equal to~$H \sin{\theta}$, then equation~(\eqnref{307}{93}) will give the currents
+generated by the annihilation of a uniform magnetic field
+parallel to~$y$.
+
+\begin{DPgather*}
+\lintertext{\indent Since} J_1(x) = - J_0'(x), \\
+\int_0^{\smallbold{a}} r^2 J_1 \left( x_p \frac{r}{\smallbold{a}} \right) dr
+  = - \int_0^{\smallbold{a}} r^2 J_0' \left( x_p \frac{r}{\smallbold{a}} \right) dr.
+\end{DPgather*}
+Integrating by parts and remembering that $J_0(x_p) = 0$, we see
+that each of these integrals equals
+\[
+\frac{2 \smallbold{a}}{x_p} \int_0^{\smallbold{a}} r J_0 \left( x_p \frac{r}{\smallbold{a}} \right) dr,
+\]
+which is equal to
+\begin{DPgather*}
+\frac{2 \smallbold{a}^3}{x_p^2} J_1(x_p), \Tag{95} \\
+\lintertext{since} \frac{d}{dx} \{x J_0'(x) \} = - x J_0(x).
+\end{DPgather*}
+Again, since
+\[
+\frac{d^2 J_1(x)}{dx^2} + \frac{1}{x}\, \frac{d J_1(x)}{dx}
+  + \left(1 - \frac{1}{x^2} \right) J_1(x) = 0,
+\]
+multiplying by
+\[
+2 x^2 \frac{d J_1}{dx},
+\]
+we get
+\[
+\frac{d}{dx} \left\{x^2 J_1'^2(x)
+  + x^2 \left( 1 - \frac{1}{x^2} \right) J_1^2(x) \right\} = 2 x J_1^2(x).
+\]
+
+Hence
+\[
+\int_0^{\xi} x J_1^2(x)\,dx
+  = \frac{1}{2} \xi^2 \left\{J_1'^2(\xi) + \left( 1 - \frac{1}{\xi^2} \right) J_1^2(\xi) \right\}.
+\]
+
+\begin{DPalign*}
+\lintertext{\indent Thus, since} \xi J_1'(\xi) + J_1(\xi) & = \xi J_0(\xi), \\
+\lintertext{we have if} J_0(\xi) & = 0, \\
+\int_0^{\xi} x J_1^2(x)\,dx & = \frac{1}{2} \xi^2 J_1^2(\xi).
+\end{DPalign*}
+%% -----File: 375.png---Folio 361-------
+Hence, when $x_p$~is a root of
+\begin{gather*}
+J_{0}(x) = 0, \\
+\int_0^\smallbold{a} r J_1^2  \left(x_p \frac{r}{\smallbold{a}}\right)\,dr = \frac{1}{2} \smallbold{a}^2  J_1^2(x_p).          \Tag{96}
+\end{gather*}
+
+Now by~(\eqnref{307}{94})
+\[
+H = - \; \frac{4\pi \smallbold{a}^2}{r} \left\{\frac{A_1}{x_1^2}  J_1 \left(x_1 \frac{r}{\smallbold{a}}\right) + \frac{A_2}{x_2^2}  J_1 \left(x_2  \frac{r}{\smallbold{a}}\right) + \ldots\right\},
+\]
+so that
+\[
+H \int_0^\smallbold{a} r^2  J_1 \left(x_p \frac{r}{\smallbold{a}}\right)\,dr = - \frac{4\pi \smallbold{a}^2}{x_p^2} A_p \int_0^\smallbold{a} r  J_1^2  \left(x_p \frac{r}{\smallbold{a}}\right)\,dr.
+\]
+Hence by~(\eqnref{307}{95}) and~(\eqnref{307}{96})
+\[
+A_p=-\frac{H}{\pi \smallbold{a} J_1(x_p)}.
+\]
+
+Thus by~(\eqnref{307}{93}), the currents produced by the annihilation of
+a magnetic field~$H$ parallel to~$y$ are given by the equation
+\[
+w = - \frac{H \cos \theta}{\pi \smallbold{a}} \tsum \frac{J_1 \left(x_p \dfrac{r}{\smallbold{a}}\right)}{J_1 (x_p)} \, \epsilon^{- \frac{\sigma x_p^2}{4\pi \smallbold{a}^2}  t}.
+\]
+Thus the currents vanish at the axis of the cylinder; when
+$t = 0$ they are infinite at the surface and zero elsewhere.
+
+When, as in the case of iron, $\mu$~is very large, the equation~(\eqnref{306}{92})
+becomes approximately
+\[
+J_s(\iota n \smallbold{a}) = 0.
+\]
+
+The solution in this case can be worked out on the same lines
+as the preceding one; for the results of this investigation we
+\index{Lamb, decay of currents in cylinders}%
+refer the reader to a paper by Prof.\ H.~Lamb (\textit{Proc.\ Lond.\ Math.\
+Soc.}\ XV, p.~270).
+
+
+\Subsection{Electrical Oscillations on a Spherical Conductor.}
+\index{Electrical vibrations, on spheres@\subdashtwo on spheres}%
+\index{Oscillations, electrical, on spheres@\subdashtwo on spheres}%
+\index{Sphere, xelectrical oscillations on@\subdashone electrical oscillations on}%
+
+\Article{308} The equations satisfied in the electromagnetic field by
+the components of the magnetic induction, or of the electromotive
+intensity, when these quantities vary as $\epsilon^{\iota p t}$, are, denoting
+any one of them by F, of the form
+\[
+\frac{d^2 F}{dx^2} + \frac{d^2 F}{dy^2} + \frac{d^2 F}{dz^2} = - \lambda^2 F,     \Tag{97}
+\]
+where in an insulator $\lambda^2 = p^2 / V^2$, $V$~being the velocity of propagation
+of electrodynamic action through the dielectric, and
+in a conductor, whose specific resistance is~$\sigma$ and magnetic
+permeability~$\mu$,
+\[
+\lambda^2 = - 4 \pi \mu \iota p / \sigma.
+\]
+%% -----File: 376.png---Folio 362-------
+
+In treating problems about spheres and spherical waves it is
+convenient to express~$F$ as the sum of terms of the form
+\[
+f(r) Y_n,
+\]
+where $f(r)$ is a function of the distance from the centre, and
+$Y_n$~a surface spherical harmonic function of the $n$\textsuperscript{th}~order.
+Transforming~(\eqnref{308}{97}) to polar coordinates, we find that~$f(r)$ satisfies
+the differential equation
+\[
+\frac{d^2f}{dr^2} + \frac{2}{r} \frac{df}{dr} + \left(\lambda^2 - \frac{n(n + 1)}{r^2}\right)f = 0.
+\]
+
+We can easily verify by substitution that the solution of this
+equation is, writing~$\rho$ for~$\lambda r$,
+\[
+f(r) = \rho^n \left\{\frac{1}{\rho} \frac{d}{d\rho} \right\}^n \left(\frac{A\epsilon^{\iota\rho} + B\epsilon^{-\iota\rho}}{\rho}\right),
+\]
+where $A$ and $B$ are arbitrary constants; particular solutions of
+this equation are thus
+\begin{align*}
+f(r) & = \rho^n \left\{\frac{1}{\rho} \frac{d}{d\rho} \right\}^n \frac{\sin \rho}{\rho}, \Tag[alpha]{$\alpha$} \\
+f(r) & = \rho^n \left\{\frac{1}{\rho} \frac{d}{d\rho} \right\}^n \frac{\cos \rho}{\rho}, \Tag[beta]{$\beta$} \\
+f(r) & = \rho^n \left\{\frac{1}{\rho} \frac{d}{d\rho} \right\}^n \left\{\frac{\epsilon^{-\iota \rho}}{\rho} \right\}, \Tag[gamma]{$\gamma$} \\
+f(r) & = \rho^n \left\{\frac{1}{\rho} \frac{d}{d\rho} \right\}^n \left\{\frac{\epsilon^{\iota \rho}}{\rho}\right\}. \Tag[delta]{$\delta$}
+\end{align*}
+
+The first of these solutions is the only one which does not
+become infinite when~$\rho$ vanishes, so that it is the solution we must
+choose in any region where~$\rho$ can vanish; in the case of the
+sphere it is the function which must be used inside the sphere;
+we shall denote it by~$S_n(\rho)$.
+
+Outside the sphere, where~$\rho$ cannot vanish, the choice of the
+function must be governed by other considerations. If we are
+considering wave motions, then, since the solution~($\gamma$) will contain
+the factor $\epsilon^{\iota(pt - \rho)}$, it will correspond to a wave diverging from
+the sphere; the solution~($\delta$), which contains the factor $\epsilon^{\iota(pt + \rho)}$,
+corresponds to waves converging on the sphere; the solutions~($\alpha$)
+and~($\beta$) correspond to a combination of convergent and divergent
+waves; thus, where there is no reflection we must take~($\gamma$)
+if the waves are divergent, ($\delta$)~if they are convergent. In other
+%% -----File: 377.png---Folio 363-------
+cases we find that~$\lambda$ is complex and of the form $p + \iota q$; in this
+case~($\alpha$) and~($\beta$) will be infinite at an infinite distance from the
+origin, while of the two solutions~\DPtypo{$f$}{($\gamma$)} and~\DPtypo{$\delta$}{($\delta$)} one will be infinite,
+the other zero, we must take the solution which vanishes when
+$\rho$~is infinite. We shall denote~($\gamma$) by $E_n^-(\rho)$, ($\delta$)~by $E_n^+(\rho)$, and
+when, as we shall sometimes do, we leave the question as to
+which of the two we shall take unsettled until we have determined~$\lambda$,
+we shall use the expression $E_n(\rho)$, which thus denotes
+one or other of~($\gamma$) and~($\delta$).
+
+When there is no reflection, the solution of~(\eqnref{308}{97}) is thus expressed
+by
+\begin{align*}
+& S_n(\rho) Y_n\; \epsilon^{\iota pt}\ \text{inside the sphere}, \\
+& E_n(\rho) Y_n\; \epsilon^{\iota pt}\ \text{outside the sphere}.
+\end{align*}
+
+In particular when $Y_n$ is the zonal harmonic~$Q_n$, the solutions
+are
+\[
+S_n(\rho) Q_n \epsilon^{\iota pt}, \quad E_n(\rho) Q_n \epsilon^{\iota pt}.
+\]
+
+When $Y_n$ is the first tesseral harmonic, the solutions are
+\begin{gather*}
+\frac{x}{r} S_n(\rho) \frac{dQ_n}{d\mu} \epsilon^{\iota pt}, \quad \frac{x}{r} E_n(\rho) \frac{dQ_n}{d\mu} \epsilon^{\iota pt}, \\
+\frac{y}{r} S_n(\rho) \frac{dQ_n}{d\mu} \epsilon^{\iota pt}, \quad \frac{y}{r} E_n(\rho) \frac{dQ_n}{d\mu} \epsilon^{\iota pt},
+\end{gather*}
+where $\mu = \cos \theta$, $\theta$~being the colatitude of the intersection of the
+radius with the surface of the sphere.
+
+\Article{309} We shall now proceed to prove those properties of the
+functions~$S_n$ and~$E_n$ which we shall require for the subsequent
+investigations. The reader who desires further information
+about these interesting functions can derive it from the following
+sources:---
+
+\index{Stokes, zon the functions `S' and `E'@\subdashone on the functions `S' and `E'}%
+Stokes, `On the Communication of Vibration from a Vibrating
+Body to the Surrounding Gas,' \textit{Phil.\ Trans.}\ 1868, p.~447.
+
+\index{Rayleigh@Rayleigh, Lord, \textit{Theory of Sound}}%
+Rayleigh, `Theory of Sound,' Vol.~II, Chap.~XVII.
+
+C.~Niven, `On the Conduction of Heat in Ellipsoids of Revolution,'
+\textit{Phil.\ Trans.}\ Part~I, 1880, p.~117.
+
+\index{Niven, C., on the functions `S' and `E'}%
+C.~Niven, `On the Induction of Electric Currents in Infinite
+Plates and Spherical Shells,' \textit{Phil.\ Trans.}\ Part~II, 1881, p.~307.
+
+\index{Lamb, on the functions `S' and `E'@\subdashone on the functions `S' and `E'}%
+H.~Lamb, `On the Vibrations of an Elastic Sphere,' and `On
+the Oscillations of a Viscous Spheroid,' \textit{Proc.\ Lond.\ Math.\ Soc.},
+13, pp.~51, 189.
+
+H.~Lamb, `On Electrical Motions in a Spherical Conductor,'
+\textit{Phil.\ Trans.}\ Part~II, 1883, p.~519.
+%% -----File: 378.png---Folio 364-------
+
+\index{Helmholtz vb@\subdashtwo on the functions `S' and `E'}%
+V.~Helmholtz, `Wissenschaftliche Abhandlungen,' Vol.~I, p.~320.
+
+\index{Heine@Heine, \textit{Kügelfunctionen}}%
+Heine, `Kugelfunctionen,' Vol.~I, p.~140.
+
+The following propositions are for brevity expressed only for
+the~$S_n$ functions, since, however, their proof only depends upon
+the differential equations satisfied by these functions they are
+equally true for the functions~$\beta$, $\gamma$, $\delta$.
+
+Since
+\index{Functions, z `S' and `E'@\subdashone `S' and `E'}%
+\begin{DPgather*}
+S_n(\rho) = \rho^n \left\{\frac{1}{\rho} \frac{d}{d\rho} \right\}^n \frac{\sin \rho}{\rho}, \\
+\frac{1}{\rho} \frac{d}{d\rho} \left\{\frac{S_{n-1}}{\rho^{n-1}} \right\} = \frac{S_n}{\rho^n}, \\
+\lintertext{or} \rho \frac{dS_{n-1}}{d\rho} - (n-1) S_{n-1} = \rho S_n, \Tag{98}
+\end{DPgather*}
+and therefore
+\[
+\rho \frac{dS_n}{d\rho} - nS_n = \rho S_{n+1}. \Tag{99}
+\]
+
+Multiply~(\eqnref{309}{98}) by $\rho^n$ and differentiate with respect to~$\rho$, and
+we get
+\[
+\frac{d^2}{d\rho^2} S_{n-1} + \frac{2}{\rho} \frac{dS_{n-1}}{d\rho} - \frac{n(n-1) S_{n-1}}{\rho^2} = (n+1) \frac{S_n}{\rho} + \frac{dS_n}{d\rho}.
+\]
+
+But
+\[
+\frac{d^2 S_{n-1}}{d\rho^2} + \frac{2}{\rho} \frac{dS_{n-1}}{d\rho} + \left(1 - \frac{n(n-1)}{\rho^2}\right) S_{n-1} = 0,
+\]
+hence
+\[
+-\rho S_{n-1} = (n+1) S_n + \rho \frac{dS_n}{d\rho}. \Tag{100}
+\]
+
+From~(\eqnref{309}{99}) and~(\eqnref{309}{100}), we get
+\begin{DPalign*}
+(2n + 1) S_n + \rho (S_{n-1} + S_{n+1}) = 0, & \Tag{101} \\
+\lintertext{and}  (2n + 1) \frac{dS_n}{d\rho} = (n+1)S_{n+1} - nS_{n-1}. &
+\end{DPalign*}
+
+Again, since
+\begin{align*}
+& \frac{d^2}{dr^2} S_n(\lambda r) + \frac{2}{r} \frac{d}{dr} S_n(\lambda r) +\left(\lambda^2 - \frac{n(n+1)}{r^2}\right) S_n(\lambda r) = 0, \\
+\intertext{and}
+& \frac{d^2}{dr^2} S_n(\lambda' r) + \frac{2}{r} \frac{d}{dr} S_n(\lambda' r) +\left(\lambda'^2 - \frac{n(n+1)}{r^2}\right) S_n(\lambda' r) = 0,
+\end{align*}
+we have
+\begin{multline*}
+r^2 \left\{S_n(\lambda' r) \frac{d^2}{dr^2} S_n(\lambda r) - S_n(\lambda r) \frac{d^2}{dr^2} S_n(\lambda' r) \right\} \\
++ 2r \left\{S_n(\lambda' r) \frac{d}{dr} S_n(\lambda r) - S_n(\lambda r) \frac{d}{dr} S_n(\lambda' r) \right\} \\
+= (\lambda'^2 - \lambda^2) r^2 S_n(\lambda r) S_n(\lambda' r),
+\end{multline*}
+%% -----File: 379.png---Folio 365-------
+and hence
+\begin{multline*}
+\int_a^b r^2 S_n(\lambda r) S_n (\lambda'r)\, dr \\
+= \frac{1}{\lambda'^2 - \lambda^2} \left\{r^2 S_n (\lambda'r) \frac{d}{dr} S_n (\lambda r) - r^2 S_n(\lambda r) \frac{d}{dr} S_n (\lambda'r) \right\}_a^b, \Tag{102}
+\end{multline*}
+so that if $\lambda$, $\lambda'$ satisfy the equations
+\begin{DPgather*}
+a^2 \left\{S_n (\lambda'a) \frac{d}{da} S_n (\lambda a) - S_n (\lambda a) \frac{d}{da} S_n (\lambda'a) \right\} = 0,\\
+b^2 \left\{S_n (\lambda'b) \frac{d}{db} S_n (\lambda b) - S_n (\lambda b) \frac{d}{db} S_n (\lambda'b) \right\} = 0,\\
+\lintertext{then}
+\int_a^b r^2 S_n (\lambda r) S_n (\lambda'r)\, dr = 0.
+\end{DPgather*}
+
+Proceeding to the limit $\lambda' = \lambda$, we get from (\eqnref{309}{102})
+\[
+\int_a^b r^2 S_n^2 (\lambda r)\, dr = -\frac{1}{2 \lambda^2} \left[ r^2 S_n (\lambda r) \frac{d}{dr} \left\{r \frac{d S_n(\lambda r)}{dr} \right\} \right. \\
+- \left. r^3 \left\{\frac{d S_n (\lambda r)}{dr} \right\}^2 \right]_a^b.
+\]
+
+{\allowdisplaybreaks
+The following table of the values of the first four of the $S$ and
+$E$ functions will be found useful for the subsequent work:---
+\begin{align*}
+S_0(x) =& \frac{\sin x}{x},\\
+S_1(x) =& \frac{\cos x}{x} - \frac{\sin x}{x^2},\\
+S_2(x) =& -\frac{\sin x}{x} - \frac{3 \cos x}{x^2} + \frac{3 \sin x}{x^3},\\
+S_3(x) =& - \frac{\cos x}{x} + \frac{6 \sin x}{x^2} + \frac{15 \cos x}{x^3} - \frac{15 \sin x}{x^4}.\\
+. \qquad &. \qquad . \qquad . \qquad . \qquad . \qquad . \qquad . \qquad . \\
+&E_0^-(x) =   \frac{\epsilon^{-\iota x}}{x},\\
+&E_1^-(x) = - \frac{\epsilon^{-\iota x}}{x} \left(\iota + \frac{1}{x} \right),\\
+&E_2^-(x) = - \frac{\epsilon^{-\iota x}}{x} \left( 1 - \frac{3 \iota}{x} - \frac{3}{x^2} \right),\\
+&E_3^-(x) =  \frac{\epsilon^{-\iota x}}{x} \left( \iota + \frac{6}{x} - \frac{15 \iota}{x^2} - \frac{15}{x^3} \right).\\
+ &\quad . \qquad . \qquad . \qquad . \qquad . \qquad . \qquad .
+\end{align*}
+The values of $E^+$ can be got from those of $E^-$ by changing the
+sign of $\iota$.
+}%end \allowdisplaybreaks
+%% -----File: 380.png---Folio 366-------
+
+\Article{310} We shall now proceed to the study of the oscillations
+of a distribution of electricity over the surface of a sphere. Let
+us suppose that a distribution of electricity whose surface
+density is proportional to a \emph{zonal} harmonic of the $n^{\text{th}}$ order is
+produced over the surface of the sphere, and that the cause producing
+this distribution is suddenly removed; then, since this
+distribution cannot be in equilibrium unless under the influence
+of external forces, electric currents will start off to equalize it,
+and electrical vibrations will be started whose period it is the
+object of the following investigation to determine.
+
+Since the currents obviously flow in planes through the axis
+of the zonal harmonic, which we shall take for the axis of $z$,
+there is no electromotive force round a circuit in a plane at
+right angles to this axis; and since the electromotive force
+round a circuit is equal to the rate of diminution in the number
+of lines of magnetic force passing through it, we see that in this
+case, since the motion is periodic, there can be no lines of
+magnetic force at right angles to such a circuit; in other words,
+the magnetic force parallel to the axis of $z$ vanishes. Again,
+taking a small closed circuit at right angles to a radius of the
+sphere, we see that the electromotive force round this circuit,
+and therefore the magnetic force at right angles to it, vanish;
+hence the magnetic force has no component along the radius,
+and is thus at right angles to both the axis of $z$ and the radius,
+so that the lines of magnetic force are a series of small circles
+with the axis of the harmonic for axis.
+
+Hence, if $a$, $b$, $c$ denote the components of magnetic induction
+parallel to the axes of $x$, $y$, $z$ respectively, we may put
+\begin{align*}
+a &= y\chi(r, \mu),\\
+b &= -x\chi(r, \mu),\\
+c &= 0,
+\end{align*}
+where $\chi(r, \mu)$ denotes some function of $r$ and $\mu$. Comparing
+this with the results of \artref{308}{Art.~308}, we see that inside the sphere
+\begin{equation}
+\left.
+\begin{aligned}
+a &= A \frac{y}{r} S_n (\lambda'r) \frac{dQ_n}{d\mu} \epsilon^{\iota p t},\\
+b &= -A \frac{x}{r} S_n (\lambda'r) \frac{dQ_n}{d\mu} \epsilon^{\iota p t},\\
+c &= 0,
+\end{aligned} \right\} \Tag{103}
+\end{equation}
+where $\lambda'^2 = - 4\pi\mu\iota p/\sigma$, and $A$ is a constant.
+%% -----File: 381.png---Folio 367-------
+
+Outside the sphere,
+\[
+\left.
+\begin{aligned}
+a &= B \frac{y}{r} E_n (\lambda r) \frac{dQ_n}{d \mu} \epsilon^{\iota pt},\\
+b &= -B \frac{x}{r} E_n (\lambda r) \frac{dQ_n}{d \mu} \epsilon^{\iota pt},\\
+c &= 0,
+\end{aligned}
+\right\} \Tag{104}
+\]
+where $\lambda = p/V$, and $B$ is a constant.
+
+Since the tangential magnetic force is continuous, we have if
+$\smallbold{a}$ is the radius of the sphere,
+\[
+\frac{A}{\mu} S_n (\lambda' \smallbold{a}) = B E_n (\lambda \smallbold{a}). \Tag{105}
+\]
+To get another surface condition we notice that the electromotive
+intensity parallel to the surface of the sphere is continuous.
+Now the total current through any area is equal to
+$1 / 4 \pi$ times the line integral of the magnetic force round that
+area, hence, taking as the area under consideration an elementary
+one $dr \; r \,\sin \theta\, d\phi$, whose sides are respectively parallel to an
+element of radius and to an element of a parallel of latitude, we
+find, if $q$ is the current in a meridian plane at right angles to the
+radius,
+\[
+4 \pi q = \frac{1}{r} \frac{d}{dr} (\gamma r),
+\]
+where $\gamma$ is the resultant magnetic force which acts tangentially
+to a parallel of latitude.
+
+The electromotive intensity parallel to $q$ is, in the conductor
+\[
+\sigma q,
+\]
+and in the dielectric
+\[
+\frac{4 \pi}{\iota pK} q.
+\]
+Hence, since this is continuous, we have
+\[
+\frac{A \sigma}{\mu} \frac{d}{d \smallbold{a}} (\smallbold{a}S_n(\lambda'\smallbold{a}))
+= \frac{B 4 \pi}{\iota pK} \frac{d}{d\smallbold{a}} \{\smallbold{a}E_n(\lambda \smallbold{a}) \}. \Tag{106}
+\]
+Eliminating $A$ and $B$ from equations (\eqnref{310}{105}) and (\eqnref{310}{106}), we get
+\[
+\frac{\sigma \dfrac{d}{d\smallbold{a}} \{\smallbold{a}S_n(\lambda'\smallbold{a})\}}{S_n(\lambda'\smallbold{a})}
+= \frac{4\pi}{\iota pK} \frac{\dfrac{d}{d\smallbold{a}} \{\smallbold{a}E_n(\lambda \smallbold{a})\}}{E_n(\lambda \smallbold{a})}. \Tag{107}
+\]
+\Article{311} The oscillations of the surface electrification about the
+%% -----File: 382.png---Folio 368-------
+state of uniform distribution are extremely rapid as the wave
+length must be comparable with the radius of the sphere. For
+such rapid vibrations as these however $\lambda'\smallbold{a}$, or
+$\{-4 \pi \mu \iota p / \sigma \}^{\frac{1}{2}} \smallbold{a}$,
+is very large, but when this is the case, we see from the equation
+\[
+S_n(\rho) = \rho^n \left\{\frac{1}{\rho} \frac{d}{d\rho} \right\}^n \frac{\sin \rho}{\rho},
+\]
+that $S_n' (\lambda'\smallbold{a})$ is approximately equal to $\pm \iota S_n(\lambda'\smallbold{a})$, so that the
+left-hand side of equation (\eqnref{310}{107}) is of the order
+\[
+\sigma \smallbold{a} \sqrt{-\frac{4 \pi \mu \iota p}{\sigma}},
+\]
+and thus, since $1/K = V^2$,
+\[
+\frac{\dfrac{d}{d\smallbold{a}} \{\smallbold{a} E_n(\lambda \smallbold{a}) \}}{E_n(\lambda \smallbold{a})}
+\]
+is of the order
+\[
+\frac{1}{4 \pi V^2} p\sigma \smallbold{a}\; \sqrt{-\frac{4 \pi \mu \iota p}{\sigma}}, \quad
+\text{or} \quad \sqrt{\frac{\sigma}{\smallbold{a}V}},
+\]
+since $p$ is comparable with $V/\smallbold{a}$.
+
+This, when the sphere conducts as well as iron or copper, is
+extremely small unless $\smallbold{a}$ is less than the wave length of sodium
+light, while for a perfect conductor it absolutely vanishes, hence
+equation (\eqnref{310}{107}) is very approximately equivalent to
+\[
+\frac{d}{d \smallbold{a}} \{\smallbold{a}E_n(\lambda \smallbold{a}) \} = 0. \Tag{108}
+\]
+This, by the relation (\eqnref{309}{101}), may be written
+\[
+E_{n+1}(\lambda \smallbold{a}) - \frac{n+1}{n} E_{n-1}(\lambda \smallbold{a}) = 0,
+\]
+which is the form given in my paper on `Electrical Oscillations,'
+\textit{Proc. Lond. Math. Soc}.~XV, p.~197.
+
+This condition makes the tangential electromotive intensity
+vanish, so that the lines of electrostatic induction are always at
+right angles to the surface of the sphere.
+
+\Article{312} In order to show that the equations (\eqnref{310}{103}) and (\eqnref{310}{104}) in
+\index{Sphere, yperiod of these oscillations@\subdashone period of these oscillations}%
+\index{Time of vibration of electricity on a sphere@\subdashtwo vibration of electricity on a sphere}%
+the preceding \artref{310}{article} correspond to a distribution of electricity
+over the surface of the sphere represented by a zonal harmonic
+$Q_n$ of the $n^{\text{th}}$ order, we only need to show that the current along
+the radius vector varies as $Q_n$, for the difference between the
+%% -----File: 383.png---Folio 369-------
+radial currents in the sphere and in the dielectric is proportional
+to the rate of variation of the surface density of the electricity
+on the sphere, and therefore, since the surface density varies as
+$\epsilon^{\iota p t}$, it will be proportional to the radial current.
+
+Consider a small area at right angles to the radius, and apply
+the principle that $4\pi$ times the current through this area is
+equal to the line integral of the magnetic force round it, we get,
+if $\mathtt{P}$ is the current along the radius and $\mu = \cos \theta$,
+\[
+4\pi \mathtt{P} = \frac{1}{r} \frac{d}{d\mu} (\gamma \sin{\theta}), \Tag{109}
+\]
+where $\gamma$, as before, is the resultant magnetic force which acts
+along a tangent to a parallel of latitude.
+
+By equation (\eqnref{310}{103}), $\gamma$ is proportional to
+\[
+\sin{\theta} \frac{d Q_n}{d \mu},
+\]
+so that $\mathtt{P}$ is proportional to
+\begin{DPgather*}
+\frac{d}{d\mu} \left\{\sin^2{\theta}\; \frac{dQ_n}{d\mu} \right\};\\
+\lintertext{but}
+\frac{d}{d\mu} \left\{\sin^2{\theta}\; \frac{dQ_n}{d\mu} \right\} + n(n+1)Q_n = 0,
+\end{DPgather*}
+hence $\mathtt{P}$, and therefore the surface density, is proportional to $Q_n$.
+
+We shall now consider in more detail the case $n = 1$.\\
+We have
+\[
+\frac{p^2}{V^2} = \lambda^2.
+\]
+
+We shall take as the solution of the equation $p^2/V^2 = \lambda^2$
+\[
+\frac{p}{V} = \lambda,
+\]
+and we shall take $E_n^- (\lambda r)$ as our solution, as this corresponds to
+a wave diverging from the sphere. Thus, equation (\eqnref{311}{108}) becomes
+\[
+\frac{d}{d \smallbold{a}} \{\smallbold{a} E_1^-(\lambda \smallbold{a}) \} = 0,
+\]
+or substituting for $E_1^- (\lambda \smallbold{a})$ the value given in \artref{309}{Art.~309},
+\begin{DPgather*}
+\epsilon^{-\iota\lambda \smallbold{a}} \left\{\frac{1}{(\lambda \smallbold{a})^2} + \frac{\iota}{\lambda \smallbold{a}} - 1 \right\} = 0,\\
+\lintertext{or}
+(\lambda \smallbold{a})^2 - \iota \lambda \smallbold{a} = 1,\\
+\lambda \smallbold{a} = \frac{\iota}{2} \pm \frac{\sqrt{3}}{2}.
+\end{DPgather*}
+%% -----File: 384.png---Folio 370-------
+\begin{DPgather*}
+\lintertext{Hence} p = \frac{V}{\smallbold{a}} \left\{\frac{\iota}{2} + \frac{\sqrt{3}}{2} \right\},
+\end{DPgather*}
+taking the positive sign since the wave is divergent.
+
+\index{Rate of decay of yoscillation on spheres@\subdashtwo of oscillation on spheres}%
+\index{Decay, xof electrical oscillations on spheres@\subdashone of electrical oscillations on spheres}%
+\index{Vibrations d, decay of, on spheres@\subdashtwo on spheres}%
+\index{Faradayx tubes, disposition of round vibrating sphere@\subdashtwo disposition of round vibrating sphere}%
+\index{Sphere, zfield of force round vibrating@\subdashone field of force round vibrating}%
+Hence, the time of vibration is $4 \pi \smallbold{a} / \sqrt{3}V$, and the wave
+length $4 \pi \smallbold{a} / \sqrt{3}$. The amplitude of the vibration falls to $1 / \epsilon$ of
+its original value after a time $2 \smallbold{a} / V$, that is after the time taken
+by light to pass across a diameter of the sphere. In the time
+occupied by one complete vibration the amplitude falls to $\epsilon^{-\dfrac{2 \pi}{\sqrt{3}}}$,
+or about $1/35$ of its original value, thus the vibrations will
+hardly make a complete oscillation before they become practically
+extinguished. This very rapid extinction of the vibrations is
+independent of the resistance of the conductor and is due to the
+emission of radiant energy by the sphere. Whenever these
+electrical vibrations can radiate freely they die away with
+immense rapidity and are practically dead beat.
+
+If we substitute this value of $\lambda$ in the expressions for the
+magnetic force and electromotive intensity in the dielectric, we
+shall find that the following values satisfy the conditions of the
+problem. If $\gamma$ is the resultant magnetic force, acting at right
+angle to the meridional plane,
+\[
+\gamma = \frac{\sin \theta \smallbold{a}}{r} \left\{1 - \frac{\smallbold{a}}{r} + \frac{\smallbold{a}^2}{r^2} \right\}^{\frac{1}{2}} \epsilon^{-\dfrac{(Vt-r)}{2 \smallbold{a}}} \cos(\phi + \delta),
+\]
+\begin{DPalign*}
+\lintertext{where} \phi &= \frac{\sqrt{3}}{2 \smallbold{a}} (Vt - r),\\
+\tan \delta &= \frac{r - \smallbold{a}}{r + \smallbold{a}} \tan{\frac{\pi}{3}}.
+\end{DPalign*}
+
+If $\Theta$ is the electromotive intensity at right angles to $r$ in the
+meridional plane, $K$ the specific inductive capacity of the dielectric
+surrounding the sphere, then by \artref{310}{Art.~310}
+\begin{DPgather*}
+K \Theta = \frac{\sin \theta}{V r} \smallbold{a} \left( 1 - \frac{\smallbold{a}}{r} \right) \left\{1 + \frac{\smallbold{a}}{r} + \frac{\smallbold{a}^2}{r^2} \right\}^{\frac{1}{2}} \epsilon^{-\dfrac{(Vt-r)}{2 \smallbold{a}}} \cos(\phi + \delta'),\\
+\lintertext{\rlap{where}} \tan \delta' = \frac{\sin{\dfrac{\pi}{3}}}{\cos{\dfrac{\pi}{3}} + \dfrac{\smallbold{a}}{r}},
+\end{DPgather*}
+and $V$ is the velocity of propagation of electromagnetic action
+%% -----File: 385.png---Folio 371-------
+through the dielectric. Close to the surface of the sphere $\delta = 0$,
+$\delta'= \pi / 6$, thus $\gamma$ and $\Theta$ differ in phase by $\pi / 6$. At a large
+distance from the sphere
+\[
+\delta = \delta'
+\]
+so that $\Theta$ and $\gamma$ are in the same phase, and we have
+\[
+V K \Theta = \gamma = \frac{\sin \theta \smallbold{a}}{r} \epsilon^{-\dfrac{1}{2 \smallbold{a}} (V t-r)} \cos \left( \phi + \frac{\pi}{3} \right).
+\]
+
+The radial electromotive intensity $P$ is, by equation \DPtypo{\eqnref{312}{109}}{(\eqnref{312}{109})}, given
+by the equation
+\[
+K P = \frac{2 \cos \theta \smallbold{a}^2}{V r^2} \left\{1 - \frac{\smallbold{a}}{r} + \frac{\smallbold{a}^2}{r^2} \right\}^{\frac{1}{2}} \epsilon^{-\dfrac{(Vt-r)}{2 \smallbold{a}}} \sin{\left( \phi + \delta - \frac{\pi}{6} \right)}.
+\]
+
+Thus at a great distance from the sphere $P$ varies as $\smallbold{a}^2/r^2$,
+while $\Theta$ only varies as $\smallbold{a}/r$, thus the electromotive intensity is
+very approximately tangential. The general character of the
+lines of electrostatic induction is similar to that in the case of
+the cylinder shown in \figureref{fig110}{Fig.~110}.
+
+\Article{313} The time of vibration of the electricity about the distribution
+represented by the second zonal harmonic is given by
+a cubic equation, whose imaginary roots I find to be
+\[
+\iota \lambda \smallbold{a} = -.7 \pm 1.8 \iota.
+\]
+
+The rate of these vibrations is more than twice as fast as
+those about the first harmonic distribution; the rate of decay of
+these vibrations, though absolutely greater than in that case, is
+not increased in so great a ratio as the frequency, so that the
+system will make more vibrations before falling to a given
+fraction of its original value than before.
+
+The time of vibration of the electricity about the distribution
+represented by the third zonal harmonic is given by a biquadratic
+equation whose roots are imaginary, and given by
+\begin{align*}
+\iota \lambda \smallbold{a} &= -.85 \pm 2.76 \iota,\\
+\iota \lambda \smallbold{a} &= -2.15 \pm .8 \iota.
+\end{align*}
+
+The quicker of these vibrations is more than three times
+faster than that about the first zonal harmonic, and there will be
+many more vibrations before the disturbance sinks to a given
+fraction of its original value. The slower vibration is of nearly
+the same period as that about the first harmonic, but it fades
+away much more rapidly than even that vibration.
+%% -----File: 386.png---Folio 372-------
+
+The vibrations about distributions of electricity represented
+by the higher harmonics thus tend to get quicker as the degree
+of the harmonic increases, and more vibrations are made before
+the disturbance sinks into insignificance.
+
+\Article{314} We have seen in \artref{16}{Art.~16} that a charged sphere when
+moving uniformly produces the same magnetic field as an
+element of current at its centre. If the sphere is oscillating
+instead of moving uniformly, we may prove (J.~J. Thomson,
+\textit{Phil.\ Mag.}~[5], 28, p.~1, 1889) that if the period of its oscillations
+is large compared with that of a distribution of electricity over
+the surface of the sphere, the vibrating sphere produces the
+same magnetic field as an alternating current of the same period.
+Waves of electromotive intensity carrying energy with them
+travel through the dielectric, so that in this case the energy of
+the sphere travels into space far away from the sphere. When,
+however, the period of vibration of the sphere is less than
+that of the electricity over its surface, the electromotive intensity
+and the magnetic force diminish very rapidly as we
+recede from the sphere, the magnetic field being practically
+confined to the inside of the sphere, so that in this case the
+energy of the moving sphere remains in its immediate neighbourhood.
+
+We may compare the behaviour of the electrified sphere with
+that of a string of particles of equal mass placed at equal
+intervals along a tightly stretched string; if one of the particles,
+say one of the end ones, is agitated and made to vibrate more
+slowly than the natural period of the system, the disturbance
+will travel as a wave motion along the string of particles, and
+the energy given to the particle at the end will be carried far
+away from that particle; if however the particle which is
+agitated is made to vibrate more quickly than the natural
+period of vibration of the system, the disturbance of the adjacent
+particles will diminish in geometrical progression, and
+the energy will practically be confined to within a short
+distance of the disturbed particle. This case possesses additional
+interest since it was used by Sir G.~G. Stokes to explain
+fluorescence.
+
+\Article{315} To consider more closely the effect of reflection let us
+\index{Sphere, zvibrations of concentric spheres@\subdashone vibrations of concentric spheres}%
+take the case of two concentric spherical conductors of radius
+$\smallbold{a}$~and~$\smallbold{b}$ respectively. Then in the dielectric between the
+%% -----File: 387.png---Folio 373-------
+spheres, the components of magnetic induction are given by
+\begin{align*}
+a &=  \frac{y}{r} \left\{BE_n^+(\lambda r) + CE_n^-(\lambda r) \right\}  \frac{dQ_n}{d\mu},\\
+b &= -\frac{x}{r} \left\{BE_n^+(\lambda r) + CE_n^-(\lambda r) \right\}  \frac{dQ_n}{d\mu},\\
+c &= 0.
+\end{align*}
+
+We may show, as in \artref{311}{Art.~311}, that if the spheres are metallic
+and not excessively small the electromotive intensity parallel to
+the surface of the spheres vanishes when $r = \smallbold{a}$ and when $r = \smallbold{b}$;
+thus we have\nblabel{eqnp:373}
+\begin{align*}
+0 &= B \frac{d}{d\smallbold{a}} \{\smallbold{a}E_n^+(\lambda \smallbold{a})\}
+   + C \frac{d}{d\smallbold{a}} \{\smallbold{a}E_n^{\DPtypo{-1}{-}}(\lambda \smallbold{a})\},\\
+0 &= B \frac{d}{d\smallbold{b}} \{\smallbold{b}E_n\DPtypo{}{^+}(\lambda \smallbold{b})\}
+   + C \frac{d}{d\smallbold{b}} \{\smallbold{b}E_n^-(\lambda \smallbold{b})\}.
+\end{align*}
+
+Eliminating \DPtypo{$A$~and~$B$}{$B$~and~$C$}, we have
+\[
+\frac{d}{d\smallbold{a}} \{\smallbold{a}E_n^+(\lambda \smallbold{a})\} \,
+\frac{d}{d\smallbold{b}} \{\smallbold{b}E_n^-(\lambda \smallbold{b})\} =
+\frac{d}{d\smallbold{a}} \{\smallbold{a}E_n^-(\lambda \smallbold{a})\} \,
+\frac{d}{d\smallbold{b}} \{\smallbold{b}E_n^+(\lambda \smallbold{b})\}.
+\]
+When $n = 1$, this becomes
+\[
+\tan \lambda \{\smallbold{b} - \smallbold{a} \}
+  = \lambda \frac{\left\{\dfrac{1}{\smallbold{a}} - \dfrac{1}{\smallbold{b}} \right\}
+                   \left\{\lambda^2 + \dfrac{1}{\smallbold{ab}} \right\}}
+                 {\left( \dfrac{1}{\smallbold{a}^2} - \lambda^2 \right)
+                  \left( \dfrac{1}{\smallbold{b}^2} - \lambda^2 \right)
+                       + \dfrac{\lambda^2}{\smallbold{ab}}}.
+\Tag{110}
+\]
+The roots of this equation are real, so that in this case there is
+no decay of the vibrations apart from that arising from the
+resistance of the conductors.
+
+If $\smallbold{a}$ is very small compared with~$\smallbold{b}$, this equation reduces to
+\[
+\tan \lambda \smallbold{b} = \frac{\lambda \smallbold{b}}{1-\lambda^2 \smallbold{b}^2}.
+\]
+The least root of this equation other than $\lambda = 0$, I find by the
+method of trial and error to be $\lambda \smallbold{b} = 2.744$.
+
+This case is that of the vibration of a spherical shell excited
+by some cause inside, here there is no radiation of the energy
+into space, the electrical waves keep passing backwards and
+forwards from one part of the surface of the sphere to another.
+
+The wave length in this case is $2\pi \smallbold{b}/2.744$ or~$2.29\smallbold{b}$, and is
+therefore less than the wave length, $4\pi \smallbold{b}/\sqrt{3}$, of the oscillations
+which would occur if the vibrations radiated off into space: this
+is an example of the general principle in the theory of vibrations
+that when dissipation of energy takes place either from friction,
+%% -----File: 388.png---Folio 374-------
+electrical resistance, or radiation, the time of vibration is increased.
+
+In this case, since the radius of the inner sphere is made to
+vanish in the limit, the magnetic force inside the sphere whose
+radius is~$\smallbold{b}$ must be expressed by that function of~$r$ which does
+not become infinite when $r$~is zero, i.e.\ by $S_n (\lambda r)$. In the case
+when $n = 1$, the components $a$,~$b$,~$c$ of the magnetic induction
+are given by
+\begin{align*}
+a &=  \tsum B \frac{y}{r} S_1 (\lambda r) \epsilon^{\iota pt},\\
+b &= -\tsum B \frac{x}{r} S_1 (\lambda r) \epsilon^{\iota pt},\\
+c &= 0;
+\end{align*}
+where the summation extends over all values of~$\lambda$ which satisfy
+the equation
+\[
+\tan \lambda \smallbold{b} = \frac{\lambda \smallbold{b}}{1 - \lambda^2 \smallbold{b}^2}.
+\]
+
+Let us consider the case when only the gravest vibration is
+excited. Let $e$~be the surface density of the electricity, then it
+will be given by an equation of the form
+\[
+e = C \cos \theta \cos pt;
+\]
+where $p = V\lambda_1$, $\lambda_1$~being equal to~$2.744 / \smallbold{b}$.
+
+By equation~(\eqnref{312}{109}) the normal displacement current~$\smallbold{P}$ is given
+by the equation
+\[
+4 \pi \smallbold{P}
+  = \frac{1}{r}\, \frac{d}{d\centerdot\cos \theta} \{\sin\theta \{a^2 + b^2 \}^{\frac{1}{2}} \}.
+\]
+In this case
+\begin{DPgather*}
+\left.
+\begin{aligned}
+a &=  \frac{y}{r} BS_1(\lambda_1 r)\epsilon^{\iota pt},\\
+b &= -\frac{x}{r} BS_1(\lambda_1 r)\epsilon^{\iota pt},
+\end{aligned}
+\right\}
+\Tag{111} \\
+\lintertext{so that}
+4 \pi \smallbold{P} = - \frac{2}{r} B \cos \theta S_1 (\lambda_1 r) \epsilon^{\iota pt}.
+\end{DPgather*}
+When $r = \smallbold{b}$ the normal displacement current $= de / dt$, hence
+\[
+-4 \pi C \cos \theta p \sin pt
+  = - \frac{2}{\smallbold{b}} B \cos \theta S_1 (\lambda_1 \smallbold{b}) \epsilon^{\iota pt}.
+\]
+
+Substituting this value of $B \epsilon^{\iota pt}$ in~(\eqnref{315}{111}), we have
+\begin{align*}
+a &=  \frac{y}{r} 2 \pi \smallbold{b} p \sin pt C \frac{S_1(\lambda_1r)}{S_1(\lambda_1 \smallbold{b})},\\
+b &= -\frac{x}{r} 2 \pi \smallbold{b} p \sin pt C \frac{S_1(\lambda_1r)}{S_1(\lambda_1 \smallbold{b})},\\
+c &= 0.
+\end{align*}
+%% -----File: 389.png---Folio 375-------
+
+At the surface of the sphere the maximum intensity of the
+magnetic force is
+\begin{DPgather*}
+2 \pi \smallbold{b} p C \sin \theta,\\
+\lintertext{or since}
+\smallbold{b} p = V \lambda_1 \smallbold{b},\\
+\lintertext{and}
+\lambda_1 \smallbold{b} = 2.744,
+\end{DPgather*}
+the maximum magnetic force is
+\[
+2 \pi × 2.744 V C \sin \theta.
+\]
+
+For air at atmospheric pressure~$VC$ may be as large as~$25$
+without the electricity escaping; taking this value of~$VC$, the
+maximum value of the magnetic force will be
+\[
+431 \sin \theta;
+\]
+this indicates a very intense magnetic field, which however
+would be difficult to detect on account of its very rapid rate of
+reversal.
+
+
+\Subsection{Electrical Oscillations on Two Concentric Spheres of
+nearly equal radius.}
+
+\Article{316} When $d$, the difference between the radii $\smallbold{a}$~and~$\smallbold{b}$, is very
+small compared with $\smallbold{a}$~or~$\smallbold{b}$, equation~(\eqnref{315}{110}) becomes
+\[
+\tan \lambda d
+  = \frac{\lambda d ( 1 + \lambda^2 \smallbold{a}^2)}
+         {\lambda^4 \smallbold{a}^4 - \lambda^2 \smallbold{a}^2 + 1}.
+\Tag{112}
+\]
+
+There will be one root of this equation corresponding to a
+vibration whose wave length is comparable with~$\smallbold{a}$, and other
+roots corresponding to wave lengths comparable with~$d$.
+
+When the wave length is comparable with~$\smallbold{a}$, $\lambda$~is comparable
+with~$1 / \smallbold{a}$, so that in this case $\lambda d$~is very small; when this is the
+case $( \tan \lambda d ) / \lambda d = 1$, and equation~(\eqnref{316}{112}) becomes approximately
+\begin{DPgather*}
+1 = \frac{1 + \lambda^2 \smallbold{a}^2}
+         {\lambda^4 \smallbold{a}^4 - \lambda^2 \smallbold{a}^2 + 1},\\
+\lintertext{or}
+\lambda \smallbold{a} = \sqrt{2}.
+\end{DPgather*}
+The wave length $2 \pi / \lambda$ is thus equal to $\pi \sqrt{2}$~times the radius of
+the sphere.
+
+In this case, since the distance between the spheres is very
+small compared with the wave length, the tangential electromotive
+intensity, since it vanishes at the surface of both spheres,
+will remain very small throughout the space between them; the
+electromotive intensity will thus be very nearly radial between
+the spheres, and the places nearest each other on the two spheres
+%% -----File: 390.png---Folio 376-------
+will have opposite electrical charges. The tubes of electrostatic
+induction are radial, and moving at right angles to themselves
+traverse during a complete oscillation a distance comparable
+with the circumference of one of the spheres.
+
+When the wave length is comparable with the distance
+between the spheres, $\lambda$~is comparable with~$1/d$, and $\lambda \smallbold{a}$~is therefore
+very large. The denominator of the right-hand side of
+equation~(\eqnref{316}{112}), since it involves~$(\lambda \smallbold{a})^4$, will be exceedingly large
+compared with the numerator, and this side of the equation will
+be exceedingly small, so that an approximate solution of it is
+\begin{DPgather*}
+\tan \lambda d = 0, \\
+\lintertext{or}
+\lambda d = n \pi,
+\end{DPgather*}
+where $n$~is an integer.
+
+The wave length $2 \pi / \lambda = 2 d / n$. Hence, the length of the
+longest wave is~$2 d$, and there are harmonics whose wave lengths
+are $d$,~$2d/3$, $2d/4, \ldots$.
+
+When $\lambda \smallbold{a}$ is very large, the equation on p.~\pageref{eqnp:373}
+\[
+\frac{d}{dr} \{B r E_1^+(\lambda r) + C r E_1^-(\lambda r) \}_{r=\smallbold{a}} = 0,
+\]
+is equivalent to
+\[
+B \epsilon^{\iota \lambda \smallbold{a}} + C \epsilon^{-\iota \lambda \smallbold{a}} = 0.
+\]
+Hence we may put, introducing a new constant~$A$,
+\begin{align*}
+B &= A \epsilon^{-\iota \lambda \smallbold{a}},\\
+C &= -A \epsilon^{\iota \lambda \smallbold{a}}.
+\end{align*}
+
+The resultant magnetic force in the dielectric is equal to
+\[
+\{B E_1^+ ( \lambda r ) + C E_1^- ( \lambda r ) \} \sin \theta \epsilon^{\iota p t},
+\]
+or substituting the preceding values of $B$~and~$C$ and retaining
+only the lowest powers of~$1 / \lambda r$,
+\begin{DPgather*}
+\frac{A \iota}{\lambda r}
+  \left\{\epsilon^{\iota \lambda (r-\smallbold{a})}
+        + \epsilon^{-\iota \lambda (r-\smallbold{a})} \right\}
+  \sin \theta \epsilon^{\iota p t}, \\
+\lintertext{or}
+2 \frac{A\iota}{\lambda r} \cos \lambda (r - \smallbold{a})
+  \sin \theta \epsilon^{\iota p t}.
+\end{DPgather*}
+
+The tangential electromotive intensity is therefore, by \artref{310}{Art.~310},
+\[
+2 A \frac{V}{\lambda r} \sin \lambda (r-\smallbold{a}) \sin \theta \epsilon^{\iota p t},
+\]
+while the normal intensity is
+\[
+4 \frac{AV}{\lambda^2 r^2} \cos \lambda (r-\smallbold{a}) \cos \theta \epsilon^{\iota p t},
+\]
+%% -----File: 391.png---Folio 377-------
+and is thus, except just at the surface of the spheres, very small
+compared with the tangential electromotive intensity. The
+normal intensity changes sign as we go from~$r = \smallbold{a}$ to~$r = \smallbold{b}$, so
+that the electrification on the portions of the spheres opposite to
+each other is of the same sign. In this case the lines of electromotive
+intensity are approximately tangential; during the
+vibrations they move backwards and forwards across the short
+space between the spheres. The case of two parallel planes can
+be regarded as the limit of that of the two spheres, and the
+preceding work shows that the wave length of the vibrations
+will either be a sub-multiple of twice the distance between the
+planes, or else a length comparable with the dimensions of the
+plane at right angles to their common normal.
+
+If we arrange two metal surfaces, say two silvered glass
+plates, so that, as in the experiment for showing Newton's rings,
+the distance between the plates is comparable with the wave
+length of the luminous rays, care being taken to insulate one
+plate from the other, then one of the possible modes of electrical
+vibration will have a wave length comparable with that of the
+luminous rays, and so might be expected to affect a photographic
+plate. These vibrations would doubtless be exceedingly difficult
+to excite, on account of the difficulty of getting any lines of
+induction to run down between the plates before discharge took
+place, but this would to some extent be counterbalanced by the
+fact that the photographic method would enable us to detect
+vibrations of exceedingly small intensity.
+
+
+\Subsection{On the Decay of Electric Currents in Conducting Spheres.}
+\index{Decay, xof currents and magnetic force in spheres@\subdashone of currents and magnetic force in spheres}%
+\index{Electric currents, xdecay of@\subdashtwo decay of, in spheres}%
+\index{Magnetic zforce, decay of in spheres@\subdashtwo of in spheres}%
+\index{Rate of decay of xcurrents in spheres@\subdashtwo of currents in spheres}%
+\index{Sphere, zzdecay of electric currents in@\subdashone decay of electric currents in}%
+
+\Article{317} The analysis we have used to determine the electrical
+oscillations on spheres will also enable us to determine the rate
+at which a system of currents started in the sphere will decay if
+left to themselves. Let us first consider the case when, as in
+the preceding investigation, the lines of magnetic force are
+circles with a diameter of the sphere for their common axis.
+Using the same notation as before, when there is only a single
+sphere of radius~$\smallbold{a}$ in the field, we have by equation~(\eqnref{310}{107})
+\[
+\frac{\sigma \dfrac{d}{d \smallbold{a}} \{\smallbold{a} S_n(\lambda' \smallbold{a})\}}
+     {S_n(\lambda' \smallbold{a})}
+  = \frac{4\pi}{K\iota p}\,
+    \frac{\dfrac{d}{d \smallbold{a}} \{\smallbold{a} E_n(\lambda \smallbold{a})\}}
+         {E_n(\lambda \smallbold{a})}.
+\Tag{113}
+\]
+%% -----File: 392.png---Folio 378-------
+
+The rate at which the system of currents decay is infinitesimal
+in comparison with the rate at which a distribution of electricity
+over the surface changes, so that~$\lambda \smallbold{a}$ or~$p \smallbold{a}/V$ will in this
+case be exceedingly small: but when $\lambda \smallbold{a}$~is very small
+\begin{gather*}
+E_n(\lambda \smallbold{a})
+  = (-1)^n 1\centerdot3\centerdot5 \ldots (2n-1)\centerdot
+    \frac{\epsilon^{±\lambda \smallbold{a}}}{\smallbold{a}^{n+1}}, \\
+\frac{d}{d \smallbold{a}} \{\smallbold{a}E_n(\lambda \smallbold{a}) \}
+  = (-1)^{n+1} 1\centerdot3\centerdot5 \ldots (2n-1)\centerdot n
+    \frac{\epsilon^{±\lambda \smallbold{a}}}{\smallbold{a}^{n+1}},
+\end{gather*}
+so that the right-hand side of~(\eqnref{317}{113}) is equal to
+\begin{DPgather*}
+-\frac{4 \pi n}{K \iota p}, \\
+\lintertext{or}
+-\frac{4 \pi n V^2}{\iota p}. \\
+\lintertext{Thus}
+\frac{S_n(\lambda' \smallbold{a})}
+     {\dfrac{d}{d \smallbold{a}} \{\smallbold{a}S_n(\lambda' \smallbold{a}) \}}
+  = -\frac{\iota p \sigma}{4 \pi n V^2}.
+\end{DPgather*}
+
+Now, since $V^2 = 9 × 10^{20}$ and $\sigma$~for copper is about~$1600$, the
+right-hand side of this equation is excessively small, so that it
+reduces to
+\[
+S_n (\lambda' \smallbold{a}) = 0.
+\]
+
+When $n = 1$, since
+\[
+S_1(\lambda' \smallbold{a})
+  = \frac{\cos \lambda' \smallbold{a}}{\lambda'\smallbold{a}}
+  - \frac{\sin \lambda' \smallbold{a}}{\lambda'^2 \smallbold{a}^2},
+\]
+$\lambda'$~is given by the equation
+\[
+\tan \lambda' \smallbold{a} = \lambda' \smallbold{a};
+\]
+the roots of which are approximately
+\[
+\lambda' \smallbold{a} = 1.4303 \pi,\quad 2.4590\pi, \quad 3.4709\pi \dots.
+\]
+
+The roots of the equation
+\[
+S_2(\lambda' \smallbold{a})= 0
+\]
+are approximately
+\[
+\lambda' \smallbold{a} = 1.8346 \pi, \quad 2.8950 \pi, \quad 3.9225 \pi.
+\]
+(See Prof.\ H.~Lamb, `Electrical Motions on Spherical Conductors,'
+\index{Lamb, decay of currents in spheres@\subdashone decay of currents in spheres}%
+\textit{Phil.\ Trans.}\ Pt.~II, p.~530.\ 1883.)
+
+The value of~$\iota p$ corresponding to any value of~$\lambda'$ is given by
+the equation
+\[
+\iota p = -\frac{\sigma \lambda'^2}{4 \pi \mu}.
+\]
+
+The time factors in the expressions for the currents will be of
+%% -----File: 393.png---Folio 379-------
+the form $\epsilon^{-\frac{\sigma\lambda'^2}{4\pi\mu} t}$. The most persistent type of current will be
+that corresponding to the smallest value of~$\lambda'$, i.e.\
+\[
+\lambda' = 1.4303 \pi / \smallbold{a}.
+\]
+The time required for a current of this type to sink to $1/\epsilon$~of its
+original value in a copper sphere when $\sigma = 1600$ is $.000379 \smallbold{a}^2$
+seconds; for an iron sphere when $\mu = 1000$, $\sigma = 10^4$, it is $.0622 \smallbold{a}^2$
+seconds, thus the currents will be much more persistent in the
+iron sphere than in the copper one. The persistence of the
+vibrations is proportional to the square of the radius of the
+sphere, thus for very large spheres the rate of decay will be exceedingly
+slow; for example, it would take nearly $5$~million years
+for currents of this type to sink to $1/\epsilon$~of their original value in
+a copper sphere as large as the earth.
+
+Since $S_n(\lambda' \smallbold{a}) = 0$, we see from~(\eqnref{310}{105}) that $B = 0$, and therefore
+that the magnetic force is zero everywhere outside the
+sphere. Hence, since these currents produce no magnetic effect
+outside the sphere they cannot be excited by any external
+magnetic influence. The current at right angles to the radius
+inside the sphere is by \artref{310}{Art.~310}
+\[
+\frac{\sin \theta}{4 \pi \mu r}\,
+  \frac{d}{dr} \{A r S_n ( \lambda' r ) \}
+  \frac{dQ_n}{d\mu}\, \epsilon^{\iota p t},
+\]
+or in particular, when $n = 1$
+\[
+\frac{\sin \theta}{4 \pi \mu r}\,
+  \frac{d}{dr} \{r S_1 ( \lambda' r) \}\, \epsilon^{\iota p t}.
+\]
+
+Now $\dfrac{d}{dr} \{r S_1 ( \lambda' r ) \}$ vanishes when $\lambda' r = 2.744$, hence the
+tangential current will vanish when
+\begin{align*}
+r &= \frac{2.744}{1.4303 \pi} \smallbold{a}\\
+  &= .601 \smallbold{a};
+\end{align*}
+thus there is a concentric spherical surface over which the
+current of this type is entirely radial.
+
+The magnetic force vanishes at the surface and at the centre,
+and as we travel along a radius attains, when $n = 1$, a maximum
+when $r$~satisfies the equation
+\[
+\frac{d}{dr} S_1 (\lambda' r) = 0.
+\]
+%% -----File: 394.png---Folio 380-------
+
+The smallest root of this equation is
+\begin{DPgather*}
+\lambda' r = .662 \pi, \\
+\lintertext{where}
+r = \frac{.662}{1.4303} \smallbold{a} = .462 \smallbold{a}.
+\end{DPgather*}
+
+\includegraphicsmid{fig112}{Fig.~112.}
+
+This is nearer the centre of the sphere than the place where
+the tangential current vanishes. The lines of flow of the current
+in a meridional section of the sphere when $\lambda' \smallbold{a} = 2.4590 \pi$ are
+given in \figureref{fig112}{Fig.~112}, which is taken from the paper by Professor
+Lamb already quoted (p.~378).
+
+
+\Subsection{Rate of Decay of Currents flowing in Circles which have a
+Diameter of the Sphere as a Common Axis.}
+\index{Decay, xof currents and magnetic force in spheres@\subdashone of currents and magnetic force in spheres}%
+\index{Electric currents, xdecay of@\subdashtwo decay of, in spheres}%
+\index{Magnetic zforce, decay of in spheres@\subdashtwo of in spheres}%
+\index{Rate of decay of xcurrents in spheres@\subdashtwo of currents in spheres}%
+\index{Sphere, zzdecay of electric currents in@\subdashone decay of electric currents in}%
+
+\Article{318} In this case the lines of flow of the current are coincident
+with the lines of magnetic force of the last example and \textit{vice versa}.
+
+Let $P$,~$Q$,~$R$ denote the components of electromotive intensity,
+then in the sphere we have
+\[
+\left.\begin{aligned}
+P & =  A \frac{y}{r}\, S_n(\lambda'r) \frac{d Q_n}{d \mu}\, \epsilon^{\iota p t},\\
+Q & = -A \frac{x}{r}\, S_n(\lambda'r) \frac{d Q_n}{d \mu}\, \epsilon^{\iota p t},\\
+R & = 0;
+\end{aligned}\right\}
+\Tag{114}
+\]
+%% -----File: 395.png---Folio 381-------
+while in the dielectric surrounding the sphere, we have
+\[
+\left.
+\begin{aligned}
+P & =  B \frac{y}{r}\, E_n(\lambda r) \frac{dQ_n}{d\mu}\, \epsilon^{\iota pt},\\
+Q & = -B \frac{x}{r}\, E_n(\lambda r) \frac{dQ_n}{d\mu}\, \epsilon^{\iota pt},\\
+R & = 0.
+\end{aligned}
+\right\}
+\Tag{115}
+\]
+
+Since the electromotive intensity tangential to the sphere is
+continuous, we have, if $\smallbold{a}$~is the radius of the sphere,
+\[
+A S_n(\lambda' \smallbold{a}) = B E_n(\lambda \smallbold{a}).
+\Tag{116}
+\]
+
+If $\omega$~is the magnetic induction tangentially to a meridian, then,
+since the line integral of the electromotive intensity round a
+circuit is equal to the rate of diminution of the number of lines
+of magnetic induction passing through it,
+\[
+\frac{d\omega}{dt}
+  = \frac{1}{r}\, \frac{d}{dr} \left\{r \{P^2 + Q^2 \}^{\frac{1}{2}} \right\}.
+\]
+Since the tangential magnetic force is continuous, we have at the
+surface
+\[
+\left( \frac{\omega}{\mu} \right) \text{ in the sphere}
+  = \omega \text{ in the dielectric}.
+\]
+Hence
+\[
+\frac{A}{\mu}\, \frac{d}{d \smallbold{a}} \{\smallbold{a} S_n(\lambda' \smallbold{a} ) \}
+  = B\, \frac{d}{d \smallbold{a}} \{\smallbold{a} E_n(\lambda \smallbold{a}) \}.
+\Tag{117}
+\]
+
+Eliminating $A$ and~$B$ from equations (\eqnref{318}{116})~and~(\eqnref{318}{117}), we get
+\[
+\mu \frac{S_n(\lambda' \smallbold{a})}
+         {\dfrac{d}{d \smallbold{a}} \{\smallbold{a}S_n(\lambda' \smallbold{a}) \}}
+  = \frac{E_n(\lambda \smallbold{a})}
+         {\dfrac{d}{d \smallbold{a}} \{\smallbold{a}E_n(\lambda \smallbold{a})\}}.
+\Tag{118}
+\]
+
+In this case the currents and magnetic forces change so slowly
+that $\lambda \smallbold{a}$ or~$p \smallbold{a}/V$ is an exceedingly small quantity, but when this
+is the case we have proved \artref{317}{Art.~317}, that approximately
+\[
+\frac{E_n(\lambda \smallbold{a})}
+     {\dfrac{d}{d \smallbold{a}} \{\smallbold{a}E_n(\lambda \smallbold{a}) \}} = -\frac{1}{n},
+\]
+so that equation~(\eqnref{318}{118}) becomes
+\[
+n \mu S_n(\lambda' \smallbold{a})
+  + \frac{d}{d \smallbold{a}} \{\smallbold{a}S_n(\lambda' \smallbold{a}) \} = 0.
+\Tag{119}
+\]
+But by equation~(\eqnref{309}{100}), \artref{309}{Art.~309},
+\[
+\smallbold{a} \frac{d}{d \smallbold{a}} S_n(\lambda' \smallbold{a})
+  + (n+1)S_n(\lambda' \smallbold{a})
+  = - \lambda' \smallbold{a} S_{n-1}(\lambda' \smallbold{a}),
+\]
+%% -----File: 396.png---Folio 382-------
+hence (\eqnref{318}{119}) may be written
+\[
+n(\mu-1)S_n(\lambda' \smallbold{a})-\lambda' \smallbold{a} S_{n-1}(\lambda' \smallbold{a}) = 0.
+\Tag{120}
+\]
+
+For non-magnetic metals for which $\mu = 1$ this reduces to
+\[
+S_{n-1} (\lambda' \smallbold{a}) = 0,
+\]
+while for iron, for which $\mu$~is very great, the equation approximates
+very closely to
+\[
+S_n (\lambda' \smallbold{a}) = 0.
+\]
+
+The smaller roots of the equation
+\[
+S_n (x) = 0,
+\]
+when $n = 0$, $1$,~$2$, are given below;
+\begin{align*}
+n =& 0, \quad x = \pi, \quad 2\pi, \quad 3\pi, \dotso;\\
+n =& 1, \quad x = 1.4303\pi,\quad  2.4590\pi, \quad 3.4709\pi;\\
+n =& 2, \quad x = 1.8346\pi, \quad 2.8950\pi, \quad 3.9225\pi.
+\end{align*}
+
+Thus for a copper sphere for which $\sigma = 1600$, the time the
+currents of the most permanent type, i.e.~those corresponding to
+the root $\lambda' \smallbold{a} = \pi$, take to fall to $1 / \epsilon$~of their original value is
+$.000775 \smallbold{a}^2$ seconds, which for a copper sphere as large as the
+earth is ten million years. These numbers are given by Prof.\
+\index{Lamb, decay of currents in spheres@\subdashone decay of currents in spheres}%
+Horace Lamb in the paper on `Electrical Motion on a Spherical
+Conductor,' \textit{Phil.\ Trans.}\ 1883, Part~II.
+
+\Article{319} As the magnetic force outside the sphere does not
+vanish in this case, this distribution of currents produces an
+external magnetic field, and conversely, such a distribution
+could be induced by changes in such a field. We have supposed
+that the currents are symmetrical about an axis, but
+by superposing distributions symmetrical about different axes
+we could get the most general distribution of this type of
+current. The most general distribution of this type would
+however be such that the lines of current flow are on concentric
+spherical surfaces, it is only distributions of this kind
+which can be excited in a sphere by variations in the external
+magnetic field.
+
+We can prove without difficulty that whenever radial currents
+exist in a sphere the magnetic force outside vanishes, provided
+displacement currents in the dielectric are neglected.
+
+Let $u$,~$v$,~$w$ be the components of the current inside the sphere,
+they will, omitting the time factor, be given by equations of the
+%% -----File: 397.png---Folio 383-------
+form
+\begin{align*}
+u & = S_n (\lambda'r) Y', \\
+v & = S_n (\lambda'r) Y'', \\
+w & = S_n (\lambda'r) Y''',
+\end{align*}
+where $Y'$, $Y''$, $Y'''$ are surface harmonics of the $n$\textsuperscript{th}~order.
+
+The radial current is
+\[
+S_n(\lambda'r)\left(\frac{x}{r}\, Y' + \frac{y}{r}\, Y'' + \frac{z}{r}\, Y'''\right),
+\]
+at the surface of the sphere the radial current must vanish, i.e.\
+\[
+S_n (\lambda'\smallbold{a})
+  \left\{\frac{x}{\smallbold{a}}\, Y' + \frac{y}{\smallbold{a}}\, Y'' + \frac{z}{\smallbold{a}}\, Y'''\right\}.
+\]
+Now the second factor is a function merely of the angular
+coordinates, and if it vanished there would not be any radial
+currents at any point in the sphere, hence, on the hypothesis that
+there are radial currents in the sphere, we must have
+\[
+S_n (\lambda'\smallbold{a}) = 0,
+\]
+i.e.~$u$, $v$,~$w$ all vanish on the surface of the sphere. But if there
+are no currents on the surface the electromotive intensity must
+vanish over the surface, and hence also the radial magnetic induction;
+for the rate of change of the radial induction through a small
+area on the surface of the sphere is equal to the electromotive
+force round that area. But neglecting the displacement current
+in the dielectric the magnetic force outside the sphere will be
+derived from a potential; hence, since the radial magnetic force
+vanishes over the sphere $r = \smallbold{a}$, and over $r = \infty$, and since the
+space between the two is acyclic, the magnetic force must
+vanish everywhere in the region between them. Thus the
+presence of radial currents in the sphere requires the magnetic
+force due to the currents to be entirely confined to the inside
+of the sphere.
+
+\Article{320} Returning to the case where the system is symmetrical
+about an axis, we see from equation~(\eqnref{318}{120}) that if the sphere is an
+iron one, $\lambda'$~is given approximately by the equation
+\[
+S_n (\lambda'\smallbold{a}) = 0.
+\]
+
+Hence, by equation~(\eqnref{318}{114}) the electromotive intensity, and
+therefore the currents, vanish over the surface of the sphere.
+Since the currents also vanish at the centre, they must attain a
+maximum at some intermediate position; the distance~$r$ of this
+%% -----File: 398.png---Folio 384-------
+position from the centre of the sphere is given by the equation
+\[
+\frac{d}{dr} S_n(\lambda'r) = 0;
+\]
+if $n = 1$, a root of this equation is
+\begin{DPalign*}
+\lambda'r &= .663\pi, \\
+\lintertext{and since}
+\lambda'\smallbold{a} &= 1.4303\pi, \\
+\lintertext{we have}
+r &= .463\smallbold{a}.
+\end{DPalign*}
+
+
+\Subsection{Currents induced in a Uniform Sphere by the sudden
+destruction of a Uniform Magnetic Field.}
+
+\Article{321} We shall now apply the results we have just obtained to
+find the currents produced in a sphere placed in a uniform
+magnetic field which is suddenly destroyed; this problem was
+\index{Lamb, decay of currents in spheres@\subdashone decay of currents in spheres}%
+solved by Lamb (\textit{Proc.\ Lond.\ Math.\ Soc.}~15, p.~139, 1884). The
+currents will evidently flow in circles having the diameter of the
+sphere which is parallel to the magnetic force for axis.
+
+If $H$ is the intensity of the original field at a great distance
+from the sphere, the lines of force being parallel to~$z$, then inside
+the sphere the magnetic induction will be parallel to~$z$, and will
+be equal to $3\mu H/(\mu + 2)$. The radial component will thus be proportional
+to $\cos \theta$. If $\rho$~be the normal component of magnetic
+induction, $a$,~$b$,~$c$ the components parallel to the axes of $x$,~$y$,~$z$
+respectively, then
+\begin{align*}
+r\rho & = xa + yb + zc, \\
+\nabla^2(r\rho)
+  & = x\nabla^2a + y\nabla^2b + z\nabla^2c
+    + 2\left\{\frac{da}{dx} + \frac{db}{dy} + \frac{dc}{dz}\right\} \\
+  & = -\lambda'^2r\rho,
+\end{align*}
+\begin{DPgather*}
+\lintertext{since}
+\nabla^2a = -\lambda'^2a, \\
+\lintertext{and}
+\frac{da}{dx} + \frac{db}{dy} + \frac{dc}{dz} = 0. \\
+\lintertext{Hence, by~(\eqnref{308}{97}),}
+r\rho = C \cos \theta S_1 (\lambda'r)\epsilon^{\iota pt}
+\Tag{121}
+\end{DPgather*}
+where, by~(\eqnref{318}{119}), $\lambda'$~is given by the equation
+\begin{DPalign*}
+(\mu + 1) S_1 (\lambda'\smallbold{a})
+  + \smallbold{a} \frac{dS_1 (\lambda'\smallbold{a})}{d\smallbold{a}} & = 0, \\
+\lintertext{or by~(\eqnref{318}{120})}
+(\mu - 1) S_1 (\lambda'\smallbold{a})
+  - \lambda'\smallbold{a} S_0 (\lambda'\smallbold{a}) = 0.
+\Tag{122}
+\end{DPalign*}
+
+When the sphere is non-magnetic $\mu = 1$, and the values of~$\lambda'$
+are given by
+\[
+S_0 (\lambda'\smallbold{a}) = 0,
+\]
+%% -----File: 399.png---Folio 385-------
+\begin{DPgather*}
+\lintertext{or}
+\frac{\sin \lambda' \smallbold{a}}{\lambda' \smallbold{a}} = 0, \\
+\lintertext{hence}
+\lambda' = \frac{p \pi}{\smallbold{a}}, \text{ where $p$~is an integer}.
+\end{DPgather*}
+
+When $\mu$~is very large, $\lambda'$~is given approximately by the equation
+\begin{DPgather*}
+S_1 (\lambda' \smallbold{a}) = 0, \\
+\lintertext{or}
+\tan\lambda' \smallbold{a} = \lambda' \smallbold{a}.
+\end{DPgather*}
+The roots of this equation are given in \artref{317}{Art.~317}.
+
+We shall for the present not make any assumption as to the
+magnitude of~$\mu$, but suppose that $\lambda_1, \lambda_2 \ldots$ are the values of~$\lambda'$
+which satisfy~(\eqnref{321}{122}). The value of~$\iota p$ corresponding to~$\lambda_s$ is
+$-\sigma \lambda_s^2 / 4 \pi \mu$, hence by~(\eqnref{321}{121}) we have
+\[
+r \rho = \cos \theta \Bigl\{
+    C_1 S_1(\lambda_1 r)\, \epsilon^{-\frac{\sigma \lambda_1^2}{4 \pi \mu} t}
+  + C_2 S_1(\lambda_2 r)\, \epsilon^{-\frac{\sigma \lambda_2^2}{4 \pi \mu} t}
+  + \ldots \Bigr\}.
+\]
+To determine $C_1, C_2 \ldots$ we have the condition that when $t = 0$,
+\[
+r \rho = 3 r \cos \theta \frac{\mu H}{\mu + 2},
+\]
+hence, for all values of~$r$ between~$0$ and~$\smallbold{a}$, we have
+\[
+\frac{3 \mu H r}{\mu+2} = C_1 S_1(\lambda_1 r) + C_2 S_1(\lambda_2 r) +\ldots.
+\Tag{123}
+\]
+
+Now by \artref{309}{Art.~309}, if $\lambda_p$,~$\lambda_q$ are different roots of~(\eqnref{321}{122})
+\[
+\int_0^{\smallbold{a}} r^2 S_1(\lambda_p r)S_1(\lambda_q r)\,dr = 0,
+\]
+while
+\begin{multline*}
+\int_0^{\smallbold{a}} r^2 S_1^2(\lambda_p r)\,dr
+  = -\frac{1}{2}\, \frac{\smallbold{a}^2}{\lambda_p^2}
+     \left\{S_1(\lambda_p \smallbold{a}) \frac{d}{d \smallbold{a}}
+       \left\{\smallbold{a}\, \frac{dS_1(\lambda_p \smallbold{a})}{d \smallbold{a}} \right\} \right.\\
+       \left. - \smallbold{a} \left( \frac{d S_1(\lambda_p \smallbold{a})}{d \smallbold{a}} \right)^2 \right\}.
+\Tag{124}
+\end{multline*}
+
+But
+\begin{DPgather*}
+\frac{d^2}{d \smallbold{a}^2} S_1(\lambda_p \smallbold{a})
+  + \frac{2}{\smallbold{a}}\, \frac{d}{d \smallbold{a}} S_1(\lambda_p \smallbold{a})
+  + \left( \lambda_p^2 - \frac{2}{\smallbold{a}^2} \right) S_1(\lambda_p \smallbold{a}) = 0, \\
+\lintertext{and}
+(\mu+1)S_1(\lambda_p \smallbold{a})
+  + \smallbold{a}\, \frac{d}{d \smallbold{a}} S_1(\lambda_p \smallbold{a}) = 0.
+\end{DPgather*}
+
+Substituting in~(\eqnref{321}{124}) the values of $\dfrac{d^2}{d \smallbold{a}^2} S_1 (\lambda_p \smallbold{a})$ and $\dfrac{d}{d \smallbold{a}} S_1 (\lambda_p \smallbold{a})$
+given by these equations, we get
+\[
+\int_0^{\smallbold{a}} r^2 S_1^2(\lambda_p r)\,dr
+  = \frac{\smallbold{a}}{2\lambda_p^2}\, S_1^2(\lambda_p \smallbold{a})
+    \{\lambda_p^2 \smallbold{a}^2 + (\mu+2)(\mu-1) \}.
+\]
+%% -----File: 400.png---Folio 386-------
+
+Hence, multiplying both sides of~(\eqnref{321}{123}) by $r^2 S_1 (\lambda_p r)$ and integrating
+from~$0$ to~$\smallbold{a}$, we get
+\begin{multline*}
+\frac{3 \mu H}{\mu+2} \int_0^{\smallbold{a}} r^3 S_1(\lambda_p r)\,dr \\
+  = \frac{1}{2}\, \frac{\smallbold{a}C_p}{\lambda_p^2}\, S_1^2(\lambda_p \smallbold{a})
+    \{\lambda_p^2 \smallbold{a}^2 + (\mu+2)(\mu-1) \}.
+\Tag{125}
+\end{multline*}
+
+To find the integral on the left-hand side, we notice
+\[
+r^3\, \frac{d^2 S_1(\lambda_p r)}{dr^2}
+  + 2 r^2\, \frac{d}{dr} S_1(\lambda_p r)
+  - 2 r S_1(\lambda_p r) + \lambda_p^2 r^3 S_1(\lambda_p r) = 0,
+\]
+or
+\[
+\frac{d}{dr} \left\{r^3\, \frac{d}{dr} S_1(\lambda_p r) \right\}
+  - \frac{d}{dr} \{r^2 S_1(\lambda_p r) \} + \lambda_p^2 r^3 S_1(\lambda_p r) = 0;
+\]
+hence, integrating from~$0$ to~$\smallbold{a}$
+\[
+\smallbold{a}^3\, \frac{d}{d \smallbold{a}} S_1(\lambda_p \smallbold{a})
+  - \smallbold{a}^2 S_1(\lambda_p \smallbold{a})
+  + \lambda_p^2 \int_0^{\smallbold{a}} r^3 S_1(\lambda_p r)\,dr = 0,
+\]
+which by the use of~(\eqnref{321}{122}) reduces to
+\[
+\int_0^{\smallbold{a}} r^3 S_1(\lambda_p r)\,dr
+  = \frac{\smallbold{a}^2(\mu+2)}{\lambda_p^2}\, S_1(\lambda_p \smallbold{a}).
+\]
+
+Hence, from~(\eqnref{321}{125}) we get
+\[
+C_p = \frac{6\mu H \smallbold{a}}{S_1(\lambda_p \smallbold{a})}
+  ÷ \{\lambda_p^2 \smallbold{a}^2 + (\mu+2)(\mu-1) \}.
+\]
+
+\Article{322} When the sphere is non-magnetic $\mu = 1$, and therefore
+\[
+C_p = \frac{6H}{\smallbold{a} S_1(\lambda_p \smallbold{a})\lambda_p^2}.
+\]
+
+In this case $\lambda_p = \dfrac{p \pi}{\smallbold{a}}$, and therefore
+\begin{align*}
+\lambda_p^2 S_1(\lambda_p \smallbold{a})
+  & = \lambda_p\, \frac{\cos \lambda_p \smallbold{a}}{\smallbold{a}}
+    - \frac{\sin \lambda_p \smallbold{a}}{\smallbold{a}^2} \\
+  & = (-1)^p\, \frac{p \pi}{\smallbold{a}^2}.
+\end{align*}
+
+Thus the normal magnetic induction
+\[
+= \frac{6H\cos \theta \centerdot \smallbold{a}^3}{r\pi^3}
+  \tsum_{p=1}^{p=\infty} (-1)^p\, \frac{1}{p^3}
+    \left\{\frac{p \pi}{\smallbold{a} r} \cos p \frac{\pi r}{\smallbold{a}}
+                       - \frac{1}{r^2} \sin p \frac{\pi r}{\smallbold{a}} \right\}
+    \epsilon^{-\frac{p^2 \pi\sigma}{4 \smallbold{a}^2} t}.
+\]
+When $r = \smallbold{a}$, this equals
+\[
+\frac{6H\cos \theta}{\pi^2}
+  \tsum_{p=1}^{p=\infty} \frac{1}{p^2}\, \epsilon^{-\frac{p^2\pi\sigma}{4 \smallbold{a}^2} t}.
+\]
+
+This summation could be expressed as a theta function, but as
+%% -----File: 401.png---Folio 387-------
+the series converges very rapidly it is more convenient to leave
+it in its present form.
+
+Since we neglect the polarization currents outside the sphere,
+the magnetic force in that region is derivable from a potential,
+hence we find that the radial magnetic force is
+\[
+\frac{6 H \cos \theta}{\pi^2}\, \frac{\smallbold{a}^3}{r^3}
+  \tsum_{p=1}^{p=\infty} \frac{1}{p^2}\, \epsilon^{-\frac{p^2 \pi\sigma}{4 \smallbold{a}^2} t}.
+\]
+
+The magnetic force at right angles to the radius is
+\[
+\frac{3H \sin \theta \smallbold{a}^3}{\pi^2r^3}
+  \tsum_{p=1}^{p=\infty} \frac{1}{p^2}\, \epsilon^{-\frac{p^2 \pi \sigma}{4 \smallbold{a}^2} t}.
+\]
+
+The sphere produces the same effect at an external point as a
+small magnet whose moment is
+\[
+\frac{3 H \smallbold{a}^3}{\pi^2}
+  \tsum_{p=1}^{p=\infty} \frac{1}{p^2}\, \epsilon^{-\frac{p^2 \pi \sigma}{4 \smallbold{a}^2} t}.
+\]
+
+\Article{323} When $\mu$ is very great
+\[
+C_p = \frac{6 H \smallbold{a}}{\mu S_1(\lambda_p \smallbold{a})}.
+\]
+
+Hence, the normal magnetic force at the surface of the sphere is
+\[
+\frac{6H}{\mu^2} \cos \theta \tsum \epsilon^{-\frac{\lambda p^2 \sigma}{4 \pi \mu} t}.
+\]
+Outside the sphere the magnetic force is the same as that due to
+a magnet whose moment is
+\[
+\frac{3 H \smallbold{a}^3}{\mu} \tsum \epsilon^{-\frac{\lambda p^2 \sigma}{4 \pi \mu} t},
+\]
+placed at its centre. These results are given by Lamb (\textit{l.c.}).
+
+Thus the magnetic effects of the currents induced in a soft
+iron sphere are less than those which would be produced by a
+copper sphere of the same size placed in the same field. This is
+due to the changes of magnetic force proceeding more slowly in
+the iron sphere on account of its greater self-induction; as the
+changes in magnetic force are slower, the electromotive forces,
+and therefore the currents, will be smaller.
+
+Since $S_1 (\lambda_p \smallbold{a}) = 0$ when $\mu$~is large, the currents on the surface
+of the sphere vanish, and the currents congregate towards the
+middle of the sphere.
+%% -----File: 402.png---Folio 388-------
+
+\Chapter{Chapter V.}{Experiments on Electromagnetic Waves.}
+\index{Electromagnetic waves@\subdashone waves}%
+\index{Hertz, zelectromagnetic waves@\subdashone electromagnetic waves|indexetseq}%
+\index{Vibrator, Electrical}%
+\index{Waves, electromagnetic}%
+\index{Waves, electromagnetic, production of@\subdashtwo production of}%
+
+\Article{324} \Firstsc{Professor Hertz} has recently described a series of experiments
+which show that waves of electromotive and magnetic force
+are present in the dielectric medium surrounding an electrical
+system which is executing very rapid electrical vibrations. A
+complete account of these will be found in his book \textit{Ausbreitung
+der elektrischen Kraft}, Leipzig, 1892. The vibrations which
+Hertz used in his investigations are of the type of those which
+occur when the inner and outer coatings of a charged Leyden
+jar are put in electrical connection. The time of vibration of
+such a system when the resistance of the discharging circuit
+may be neglected is, as we saw in \artref{296}{Art.~296}, approximately
+equal to $2 \pi \sqrt{LC}$, where $L$~is the coefficient of self-induction of
+the discharging circuit for infinitely rapid vibrations and $C$~is
+the capacity of the jar in electromagnetic measure. If $\smallbold{C}$~is the
+capacity of the jar in electrostatic measure, then, since $C = \smallbold{C}/V^2$,
+where $V$~is the ratio of the electromagnetic unit of electricity
+to the electrostatic unit, the time of vibration is equal to
+$2 \pi \sqrt{L \smallbold{C}} / V$. But since $V$~is equal to the velocity of propagation
+of electrodynamic action through air, the distance the disturbance
+will travel in the time occupied by a complete oscillation,
+in other words the wave length in air of these vibrations,
+will be~$2 \pi \sqrt{L \smallbold{C}}$. By using electrical systems which had very
+small capacities and coefficients of self-induction Hertz succeeded
+in bringing the wave length down to a few metres.
+
+\includegraphicsouter{fig113}{Fig.~113.}
+
+\Article{325} The electrical vibrator which Hertz used in his earlier
+experiments (\textit{Wied.\ Ann}.~34, pp.~155, 551,~609, 1888) is represented
+in \figureref{fig113}{Figure~113}.
+
+\smallsanscap{A}~and~\smallsanscap{B} are square zinc plates whose sides are $40$~cm.\ long,
+copper wires \smallsanscap{C}~and~\smallsanscap{D} each about $30$~cm.\ long are soldered to
+the plates, these wires terminate in brass balls \smallsanscap{E}~and~\smallsanscap{F}. To
+%% -----File: 403.png---Folio 389-------
+ensure the success of the experiments it is necessary that these
+balls should be exceedingly brightly and smoothly polished,
+and inasmuch as the passage of the sparks from one ball to the
+other across the air space~\smallsanscap{EF} roughens the balls by tearing
+particles of metal from them, it is necessary to keep repolishing
+the balls at short intervals during the course of the experiment.
+It is also advisable to keep the air space~\smallsanscap{EF} shaded from the
+light from any sparks that
+may be passing in the neighbourhood.
+In order to excite
+electrical vibrations in
+this system the extremities
+of an induction coil are connected
+with \smallsanscap{C}~and~\smallsanscap{D} respectively. When the coil is in action
+it produces so great a difference of potential between the balls
+\smallsanscap{E}~and~\smallsanscap{F} that the electric strength of the air is overcome, sparks
+pass across the air gap which thus becomes a conductor; the two
+plates \smallsanscap{A}~and~\smallsanscap{B} are now connected by a conducting circuit, and
+the charges on the plates oscillate backwards and forwards from
+one plate to another just as in the case of the Leyden jar.
+
+\Article{326} As these oscillations are exceedingly rapid they will
+not be excited unless the electric strength of the air gap breaks
+down suddenly; if it breaks down so gradually that instead of
+a spark suddenly rushing across the gap we have an almost
+continuous glow or brush discharge, hardly any vibrations will
+be excited. A parallel case to this is that of the vibrations of a
+simple pendulum, if the bob of such a pendulum is pulled out
+from the vertical by a string and the string is suddenly cut the
+pendulum will oscillate; if however the string instead of breaking
+suddenly gives way gradually, the bob of the pendulum will
+merely sink to its position of equilibrium and no vibrations will
+be excited. It is this which makes it necessary to keep the balls
+\smallsanscap{E}~and~\smallsanscap{F} well polished, if they are rough there will in all likelihood
+be sharp points upon them from which the electricity will
+gradually escape, the constraint of the system will then give way
+gradually instead of suddenly and no vibrations will be excited.
+
+The necessity of shielding the air gap from light coming from
+other sparks is due to a similar reason. Ultra-violet light in which
+these sparks abound possesses, as we saw in \artref{39}{Art.~39}, the property
+of producing a gradual discharge of electricity from the negative
+%% -----File: 404.png---Folio 390-------
+terminal, so that unless this light is shielded off there will be a
+tendency to produce a gradual and therefore non-effective discharge
+instead of an abrupt and therefore effective one.
+
+\Article{327} The presence of the coil does not, as the following calculation
+of the period of the compound system shows, affect the
+time of vibration to more than an infinitesimal extent, if, as is
+practically always the case, the coefficient of self-induction of the
+secondary of the coil is almost infinite in comparison with that
+of the vibrator.
+
+Let $L$~be the coefficient of self-induction of the vibrator~$AB$,
+$C$~its capacity, $L'$~the coefficient of self-induction of the secondary
+of the coil, $M$~the coefficient of mutual induction between this
+coil and the vibrator, $x$~the quantity of electricity at any time on
+either plate of the condenser, $\dot{y}$~the current in the vibrator, $\dot{z}$~that
+through the secondary of the coil.
+\begin{DPalign*}
+\lintertext{Then we have}
+\dot{x} & = \dot{y} + \dot{z} \\
+\lintertext{or}
+x & = y + z.
+\end{DPalign*}
+
+The Kinetic energy of the currents is
+\[
+\tfrac{1}{2} L \dot{y}^2 + \tfrac{1}{2} L' \dot{z}^2 + M \dot{y} \dot{z}.
+\]
+
+The potential energy is
+\[
+\tfrac{1}{2}\, \frac{x^2}{C} \quad \text{or} \quad \tfrac{1}{2}\, \frac{(y+z)^2}{C}.
+\]
+
+Hence, if we neglect the resistance of the circuit, we have by
+Lagrange's equations
+\begin{align*}
+L y'' + M z'' + \frac{y+z}{C} &= 0, \\
+L' z'' + M y'' + \frac{y+z}{C} &= 0.
+\end{align*}
+
+Thus if $x$~and~$y$ each vary as~$\epsilon^{\iota p t}$, we have
+\begin{align*}
+y \left( \frac{1}{C} - L p^2 \right) + z \left( \frac{1}{C} - M p^2 \right) &= 0, \\
+z \left( \frac{1}{C} - L'p^2 \right) + y \left( \frac{1}{C} - M p^2 \right) &= 0.
+\end{align*}
+
+Eliminating $y$~and~$z$ we get
+\begin{DPgather*}
+\left( \frac{1}{C} - L p^2 \right) \left( \frac{1}{C} - L' p^2 \right)
+  = \left( \frac{1}{C} - M p^2 \right)^2, \\
+\lintertext{or}
+p^2 = \frac{1}{CL} \left\{1 + \frac{L}{L'} - \frac{2M}{L'} \right\} \bigg/ \left( 1 - \frac{M^2}{LL'} \right).
+\end{DPgather*}
+%% -----File: 405.png---Folio 391-------
+
+But for a circuit as short as a Hertzian vibrator $L/L'$ and
+$M/L'$ will be exceedingly small, so that we have as before
+\[
+p^2 = \frac{1}{CL}.
+\]
+
+
+\Subsection{The Resonator.}
+\index{Resonator}%
+
+\includegraphicsouter{fig114}{Fig.~114.}
+
+\Article{328} When the electrical oscillations are taking place in the
+vibrator the space around it will be the seat of electric and
+magnetic intensities. Hertz found that he could detect these
+by means of an instrument which is called the Resonator. It
+consists of a piece of copper wire bent into a circle; the ends of
+the wire, which are placed very near together, are furnished with
+two balls or a ball and a point, these are connected by an insulating
+screw, so that the distance between them admits of very
+fine adjustment. A resonator without the
+screw adjustment is shown in \figureref{fig114}{Fig.~114}.
+With a vibrator having the dimensions of
+the one in \artref{325}{Art.~325}, Hertz used a resonator
+$35$~cm.\ in radius.
+
+\Article{329} When the resonator was held near
+the vibrator Hertz found that sparks passed
+across the air space in the resonator and
+that the length of the air space across which
+the sparks would pass varied with the position
+of the resonator. This variation was found by Hertz to
+be of the following kind:
+
+Let the vibrator be placed so that its axis, the line~\smallsanscap{EF}, \figureref{fig113}{Fig.~113},
+is horizontal; let the horizontal line which bisects this axis at
+right angles, i.e.~which passes through the middle point of the air
+space~\smallsanscap{EF}, be called the base line. Then, when the resonator is
+placed so that its centre is on the base line and its plane at right
+angles to that line, Hertz found that sparks pass readily in the
+resonator when its air space is either vertically above or
+vertically below its centre, but that they cease entirely when
+the resonator is turned in its own plane round its centre until
+the air space is in the horizontal plane through that point. Thus
+the sparks are bright when the line joining the ends of the
+resonator is parallel to the axis of the vibrator and vanish when
+it is at right angles to this axis. In intermediate positions of the
+air gap faint sparks pass between the terminals of the resonator.
+%% -----File: 406.png---Folio 392-------
+
+When the centre of the resonator is in the base line and its
+plane at right angles to the axis of the vibrator no sparks pass,
+whatever may be the position of the air space.
+
+When the centre of the resonator is in the base line and its
+plane horizontal the sparks are strongest when the air space
+is nearest to the vibrator, and as the resonator turns about its
+centre in its own plane the length of the sparks diminishes as
+the air space recedes from the vibrator and is a minimum when
+the air gap is at its maximum distance from the axis of the
+vibrator. They do not however vanish in this case for any
+position of the air space.
+
+\Article{330} In the preceding experiments the length of the sparks
+changes as the resonator rotates in its own plane about its
+centre. Since rotation is not accompanied by any change in the
+number of lines of magnetic force passing through the resonator
+circuit, it follows that we cannot estimate the tendency to spark
+across the air gap by calculating by Faraday's rule the electromotive
+force round the circuit from the diminution in the
+number of lines of magnetic force passing through it.
+
+\Article{331} The effects on the spark length are, however, easily
+explained if we consider the arrangement of the Faraday tubes
+radiating from the vibrator. The tendency to spark will be
+proportional to the number of tubes which stretch across the
+air gap; these tubes may fall directly on the air gap or they
+may be collected by the wire of the resonator and thrown on the
+air gap, the resonator acting as a kind of trap for Faraday tubes.
+
+\includegraphicsmid{fig115}{Fig.~115.}
+
+Let us first consider the case when the centre of the resonator
+is on, and its plane at right angles to, the base line, then in the
+neighbourhood of the base line the Faraday tubes are approximately
+parallel to the axis of the vibrator, and their direction of
+motion is parallel to the base line; thus the Faraday tubes are
+parallel to the plane of the resonator and are moving at right
+angles to it. When they strike against the wire of the resonator
+they will split up into separate pieces as in \figureref{fig115}{Fig.~115}, which
+represents a tube moving up to and across the resonator, and after
+passing the cross-section of the wire of the resonator will join
+again and go on as if they had not been interrupted. The
+resonator will thus not catch Faraday tubes and throw them in
+the air gap, and therefore the tendency to spark across the gap
+will be due only to those tubes which fall directly upon it. When
+%% -----File: 407.png---Folio 393-------
+the air gap is parallel to the tubes, i.e.~when it is at the
+highest or lowest point of the resonator, some of the tubes will
+be caught and will stretch across the gap and thus tend to
+produce a spark. When, however, the gap is at right angles to
+the tubes, i.e.~when it is in the horizontal plane through the
+centre of the resonator, the tubes will pass right through it.
+None of them will stretch across the gap and there will be consequently
+no tendency to spark.
+
+When the plane of the resonator is at right angles to the
+axis of the vibrator, the tubes when they meet the wire of
+the resonator are, as in the last case, travelling at right angles
+to it, so that the wire of the resonator will not collect the
+tubes and throw them into the air gap. In this case the air
+gap is always at right angles to the tubes, which will therefore
+pass right through it, and none of them will stretch across the
+gap. Thus in this case there is no tendency to spark whatever
+may be the position of the air space.
+
+Let us now consider the case when the centre of the resonator
+is on the base line and its plane horizontal. In this case, as we
+see by the figures \figureref{fig116}{Fig.~116}, Faraday tubes will be caught by the
+wire of the resonator and thrown into the air gap wherever that
+may be; thus, whatever the position of the gap, Faraday tubes
+will stretch across it, and there will be a tendency to spark.
+When the gap is as near as possible to the vibrator the Faraday
+tubes which strike against the resonator will break and a portion
+%% -----File: 408.png---Folio 394-------
+of them will stretch right across the gap. When however the
+gap is a considerable distance from this position the tubes which
+stretch across it are due to the bending together of two portions
+of the tubes broken by previously striking against the resonator,
+the end of one of the portions having travelled along one side
+of the resonator while the end of the other has travelled along
+the other side,~(\textit{a}); these portions bend together across the gap,
+(\textit{b})~and~(\textit{c}); then break up again, one long straight tube travelling
+outwards, the other shorter one running into the gap, as in~(\textit{d})
+\figureref{fig116}{Fig.~116}. The portion connecting the two sides of the gap
+diverges more from the shortest distance between the terminals
+than in the case where the air gap is as near to the vibrator as
+possible, the field in \figureref{fig116}{Fig.~116} will not therefore be so concentrated
+%% -----File: 409.png---Folio 395-------
+round the gap, so that there will be less tendency to spark,
+though this tendency will still remain finite.
+
+\includegraphicsmid{fig116}{Fig.~116.}
+
+\Subsection{Resonance.}
+\index{Resonance}%
+
+\Article{332} Hitherto we have said nothing as to the effect produced
+by the size of the resonator on the brightness of the sparks,
+this effect is however often very great, especially when we are
+using condensers with fairly large capacities which can execute
+several vibrations before the radiation of their energy reduces
+the amplitude of the vibration to insignificance.
+
+The cause of this effect is that the resonator is itself an
+electrical system with a definite period of vibration of its own,
+hence if we use a resonator the period of whose free vibration
+is equal to that of the vibrator, the efforts of the vibrator to
+produce a spark in the resonator will accumulate, and we may
+be able as the result of this accumulation to get a spark which
+would not have been produced if the resonator had not been in
+tune with the vibrator. The case is analogous to the one in
+which a vibrating tuning fork sets another of the same pitch
+in vibration, though it does not produce any appreciable effect
+on another of slightly different pitch.
+
+\includegraphicsmid{fig117}{Fig.~117.}
+
+\Article{333} Professor Oliver Lodge (\textit{Nature}, Feb.~20, 1890, vol.~41,
+\index{Lodge, xelectrical resonance@\subdashone electrical resonance}%
+p.~368) has described an experiment which shows very beautifully
+the effect of electric resonance. \smallsanscap{A}~and~\smallsanscap{B}, \figureref{fig117}{Fig.~117}, represent two
+Leyden jars whose inner and outer coatings are connected by
+a wire bent so as to include a considerable area. The circuit
+connecting the coatings of one of these jars,~\smallsanscap{A}, contains an air
+break. Electrical oscillations are started in this jar by connecting
+the two coatings with the poles of an electrical machine.
+%% -----File: 410.png---Folio 396-------
+The circuit connecting the coatings of the other jar,~\smallsanscap{B}, is provided
+with a sliding piece by means of which the self-induction
+of the discharging circuit, and therefore the time of an electrical
+oscillation of the jar, can be adjusted. The inner and outer
+coatings of this jar are put almost but not quite into electrical
+contact by means of a piece of tin-foil bent over the lip of the
+jar. The jars are placed face to face so that the circuits connecting
+their coatings are parallel to each other, and approximately
+at right angles to the line joining the centre of the
+circuits. When the electrical machine is in action sparks pass
+across the air break in the circuit in~\smallsanscap{A}, and by moving the
+slider in~\smallsanscap{B} about it is possible to find a position for it in which
+sparks pass by means of the tin-foil from one coating of the jar
+to the other; as soon however as the slider is moved from this
+position the sparks cease.
+
+Resonance effects are most clearly marked in cases of this
+kind, where the system which is vibrating electrically has considerable
+capacity, since in such cases several complete oscillations
+have to take place before the radiation of energy from
+the system has greatly diminished the amplitude of the vibrations.
+When the capacity is small, the energy radiates so
+quickly that only a small number of vibrations have any appreciable
+amplitude; there are thus only a small number of impulses
+acting on the resonator, and even if the effects of these
+few conspire, the resonance cannot be expected to be very
+marked. In the case of the vibrating sphere we saw (\artref{312}{Art.~312})
+that for vibrations about the distribution represented by the
+first harmonic the amplitude of the second vibration is only about
+$1/35$~of that of the first, in such a case as this the system is
+practically dead-beat, and there can be no appreciable resonance
+or interference effects.
+
+The Hertzian vibrator is one in which, as we can see by
+considering the disposition of the Faraday tubes just before
+the spark passes across the air, there will be very considerable
+radiation of energy. Many of the tubes stretch from one plate
+of the vibrator to the other, and when the insulation of the air
+space breaks down, closed Faraday tubes will break off from
+these in the same way as they did from the cylinder; see \figureref{fig14}{Fig.~14}.
+These closed tubes will move off from the vibrator with the
+velocity of light, and will carry the energy of the vibrator away
+%% -----File: 411.png---Folio 397-------
+\index{Decay, xof vibrations in Hertz's vibrator@\subdashone of vibrations in Hertz's vibrator}%
+\index{Rate of decay of yoscillation in xHertz's vibrator@\subdashtwo of oscillation in Hertz's vibrator}%
+\index{Vibrations d@Vibrations, decay of, in Hertz's vibrator}%
+with them. In consequence of this radiation the decay of the
+oscillations in the vibrator will be very rapid, indeed we should
+expect the rate of decay to be comparable with its value in the
+case of the vibrations of electricity over the surfaces of spheres
+or cylinders, where the Faraday tubes which originally stretched
+from one part to another of the electrified conductor emit closed
+tubes which radiate into space in the same way as the similar
+tubes in the case of the Hertzian vibrator: we have seen, however,
+that for spheres and cylinders the decay of vibration is so
+rapid that they may almost be regarded as dead-beat. We
+should expect a somewhat similar result for the oscillations of
+the Hertzian vibrator.
+
+\Article{334} On the other hand, the disposition of the Faraday tubes
+shows us that the electrical vibrations of the resonator will
+be much more persistent. In this case
+the Faraday tubes will stretch from
+side to side across the inside of the
+resonator as in \figureref{fig118}{Fig.~118}, and these tubes
+will oscillate backwards and forwards
+inside the resonator; they will have
+no tendency to form closed
+\includegraphicsouter{fig118}{Fig.~118.}
+curves, and
+consequently there will be little or no
+radiation of energy. In this case the
+decay of the vibrations will be chiefly
+due to the resistance of the resonator,
+as in the corresponding cases of oscillations
+in the electrical distribution over spherical or cylindrical
+cavities in a mass of metal, which are discussed in Arts.\ \artref{315}{315}~and~\artref{300}{300}.
+
+\Article{335} The rate at which the vibrations die away for a
+vibrator and resonator of dimensions not very different from
+those used by Hertz has been measured by Bjerknes (\textit{Wied.\ Ann.}~44,
+\index{Bjerknes, decay of vibrations}%
+p.~74, 1891), who found that in the vibrator the oscillations
+died away to $1/\epsilon$~of their original value after a time~$T/.26$, where
+$T$~is the time of oscillation of the vibrator. This rate of decay,
+though not so rapid as for spheres and cylinders, is still very
+rapid, as the amplitude of the tenth swing is about $1/14$~of
+that of the first. The amplitudes of the successive vibrations are
+represented graphically in \figureref{fig119}{Fig.~119}, which is taken from Bjerknes'
+paper.
+%% -----File: 412.png---Folio 398-------
+
+\includegraphicsmid{fig119}{Fig.~119.}
+
+The time taken by the vibrations in the resonator to fade
+away to $1/\epsilon$~of their original value was found by Bjerknes to be
+$T'/.002$ or~$500 T'$, where $T'$~is the time of the electrical oscillation
+of the resonator; thus the resonator will make more than
+$1000$ complete oscillations before the amplitude of the vibration
+falls to $1/10$~of its original value. The very slow rate of decay
+of these oscillations confirms the conclusion we arrived at from
+the consideration of the Faraday tubes, that there was little
+or no radiation of energy in this case. The rate of decay of the
+vibrations in the resonator compares favourably with that of
+pendulums or tuning-forks, and is in striking contrast to the
+very rapid fading away of the oscillations of the vibrator.
+These experiments show that, as the theory led us to expect, we
+must regard the vibrator as a system having a remarkably large
+logarithmic decrement, the resonator as one having a remarkably
+small one.
+
+
+\Subsection{Reflection of Electromagnetic Waves from a Metal Plate.}
+\index{Electromagnetic waves, reflection of@\subdashtwo reflection of}%
+\index{Waves, electromagnetic, reflection of@\subdashtwo reflection of}%
+\index{Reflection of electromagnetic waves}%
+
+\Article{336} We shall now proceed to describe the experiments by
+which Hertz succeeded in demonstrating, by means of the
+vibrator and resonator described in Arts.\ \artref{325}{325}~and~\artref{328}{328}, the
+existence in the dielectric of waves of electromotive intensity
+and magnetic force (\textit{Wied.\ Ann.}~34, p.~610, 1888).
+
+The experiments were made in a large room about $15$~metres
+long, $14$~broad, and $6$~high. The vibrator was placed $2$~m.\ from
+one of the main walls, in such a position that its axis was
+vertical and its base line at right angles to the wall. At all
+points along the base line the electromotive intensity is vertical,
+being parallel to the axis of the vibrator. At the further end
+of the room a piece of sheet zinc $4$~metres by~$2$ was placed
+%% -----File: 413.png---Folio 399-------
+vertically against the wall, its plane being thus at right angles
+to the base line of the vibrator. The zinc plate was connected
+to earth by means of the gas and water pipes. In one set of
+experiments the centre of the resonator was on and its plane at
+right angles to the base line. When it is in this position the
+Faraday tubes from the vibrator strike the wire of the resonator
+at right angles; the resonator therefore does not catch the
+tubes and throw them into the air gap, and the spark will be
+due to the tubes which fall directly upon the air gap. Thus, as
+might be expected, the sparks vanish when the gap is at the
+highest or lowest point of the resonator, when the tubes are at
+right angles to the direction in which the sparks would pass, and
+the sparks are brightest when the air gap is in the horizontal
+plane through the base line, when the incident tubes are parallel
+to the sparks.
+
+\Article{337} Let the air gap be kept in this plane, and the resonator
+moved about, its centre remaining on the base line, and its plane
+at right angles to it. When the resonator is quite close to the
+zinc plate no sparks pass across the air space; feeble sparks,
+however, begin to pass as soon as the resonator is moved a short
+distance away from the plate. They increase rapidly in brightness
+as the resonator is moved away from the plate until the distance
+between the two is about $1.8$~m., when the brightness of the
+sparks is a maximum. When the distance is still further increased
+the brightness of the sparks diminishes, and vanishes
+again at a distance of about $4$~metres from the zinc plate, after
+which it begins to increase, and attains another maximum,
+and so on. Thus the sparks exhibit a remarkable periodic
+character, similar to that which occurs when stationary vibrations
+are produced by the reflection of wave motion from a
+surface at right angles to the direction of propagation of the
+motion.
+
+\Article{338} Let the resonator now be placed so that its plane is the
+vertical one through the base line, the air gap being at the
+highest or lowest point; in this position the Faraday tubes
+which fall directly on the air gap are at right angles to the
+sparks, so that the latter are due entirely to the Faraday tubes
+collected by the resonator and thrown into the air gap.
+
+When the resonator is in this position and close to the reflecting
+plate sparks pass freely. As the resonator recedes from the
+%% -----File: 414.png---Folio 400-------
+plate the sparks diminish and vanish when its distance from
+the plate is about $1.8$~metres, the place at which they were a
+maximum when the resonator was at right angles to the base
+line; after the resonator passes through this position the sparks
+increase and attain a maximum $4$~metres from the plate, the
+place where, with the other position of the resonator, they were
+a minimum; when the resonator is removed still further from
+the plate the sparks diminish, then vanish, and so on. The
+sparks in this case show a periodicity of the same wave length
+as when the resonator was in its former position, the places of
+minimum intensity for the sparks in one position of the
+resonator corresponding to those of maximum intensity in the
+other.
+
+\Article{339} If the zinc reflecting plate is mounted on a movable
+frame work so that it can be placed behind the resonator and
+removed at will, its effect can be very clearly shown by the
+following experiments:---
+
+Hold the resonator in the position it had in the last experiment
+at some distance from the vibrator and observe the sparks,
+the zinc plate being placed on one side out of action: then place
+the reflector immediately behind the resonator, the sparks will
+increase in brightness; now push the reflector back, and at
+about $2$~metres from the resonator the sparks will stop. On
+pushing it still further back the sparks will increase again, and
+when the reflector is about $4$~metres away they will be a little
+brighter than when it was absent altogether.
+
+\Article{340} Hertz only used one size of resonator, which was
+selected so as to be in tune with the vibrator. Sarasin and
+\index{De la Rive and Sarasin, experiments on electromagnetic waves}%
+\index{Sarasin and De la Rive, reflection of electromagnetic waves}%
+\index{Waves, electromagnetic, Sarasin's and de la Rive's experiments on@\subdashtwo Sarasin's and de la Rive's experiments on}%
+De~la~Rive (\textit{Comptes Rendus}, March~31, 1891), who repeated this
+experiment with vibrators and resonators of various sizes, found
+however that the apparent wave length of the vibrations, that
+is twice the distance between two adjacent places where the
+sparks vanish, depended entirely upon the size of the resonator,
+and not at all upon that of the vibrator. The following table
+contains the results of their experiments; $\lambda$~denotes the wave
+length, a `loop' means a place where the sparks are at their
+maximum brightness when the resonator is held in the first
+position, a `node' a place where the brightness is a minimum.
+The line beginning `$1/4\;\lambda$~wire' relates to another series of experiments
+which we shall consider subsequently. It is included here
+%% -----File: 415.png---Folio 401-------
+to avoid the repetition of the table. The distances of the loops
+and nodes are measured in metres from the reflecting surface.
+\begin{center}
+\tabletextsize
+\setlength\tabcolsep{4pt}
+\begin{tabular}{|c|c|c|c|c|c|}
+\hline
+\settowidth{\TmpLen}{resonator circle ($D$).}%
+\parbox[c]{\TmpLen}{\centering%
+  Diameter of\\ resonator circle ($D$).} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering\tablespaceup%
+  $1$ metre,\\ stout wire\\$1$ cm.\ in\\ diameter.\tablespacedown} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.75$ m.\\stout wire.} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.50$ m.\\stout wire.} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.35$ m.\\stout wire.} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.35$ m.\\ fine wire\\ $2$~mm.\ in \\ diameter.} \\
+\hline
+1st Loop\mdotfill & $2.11$ & $1.60$ & $1.11$ & $\Z.76$ & $\Z.75$\\
+\hline
+1st Node\mdotfill & $4.14$ & $3.01$ &        &  $1.49$ & $1.51$\\
+\hline
+2nd Loop\mdotfill &        &        &        &  $2.30$ & $2.37$\\
+\hline
+2nd Node\mdotfill &        &        &        &  $3.04$ & $3.10$\\
+\hline
+3rd Loop\mdotfill &        &        &        &         &       \\
+\hline
+3rd Node\mdotfill &        &        &        &         &       \\
+\hline
+\tablespaceup$\frac{1}{4} \lambda$ air\mdotfill
+                 & $2.03$ & $1.41$ & $1.11$ & $\Z.76$ & $\Z.80$ \\
+\tablespaceup$\frac{1}{4} \lambda$ wire\mdotfill
+                 & $1.92$ & $1.48$ & $\Z.98$ & $\Z.73$ &        \\
+\tablespaceup$2D$\mdotfill     & $2.00$ & $1.50$ & $1.00$ & $\Z.70$ & $\Z.70$ \\
+\hline
+\hline
+\settowidth{\TmpLen}{resonator circle ($D$).}%
+\parbox[c]{\TmpLen}{\centering\tablespaceup%
+  Diameter of\\ resonator circle ($D$).\tablespacedown} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.25$~m.\\ stout wire} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.25$ m.\\ fine wire.} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.20$~m.\\ stout wire.} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.20$ m.\\ fine wire.} &
+\settowidth{\TmpLen}{stout wire.}%
+\parbox[c]{\TmpLen}{\centering%
+  $.10$ m.\\ stout wire.} \\
+\hline
+1st Loop\mdotfill & $\Z.46$ & $\Z.54$ & $\Z.39$ & $\Z.42$ & $\Z.21$\\
+\hline
+1st Node\mdotfill & $\Z.94$ &  $1.17$ & $\Z.80$ & $\Z.93$ & $\Z.41$\\
+\hline
+2nd Loop\mdotfill &  $1.63$ &  $1.89$ &  $1.24$ &  $1.55$ & $\Z.59$\\
+\hline
+2nd Node\mdotfill &  $2.15$ &  $2.40$ &  $1.69$ &  $2.05$ & $.79$  \\
+\hline
+3rd Loop\mdotfill &  $2.71$ &         &         &  $2.46$ & $\Z.96$\\
+\hline
+3rd Node\mdotfill &  $3.14$ &         &         &         &        \\
+\hline
+\tablespaceup$\frac{1}{4} \lambda$ air\mdotfill
+                 &         & $\Z.60$ & $\Z.43$ & $\Z.51$ & $\Z.19$\\
+\tablespaceup$\frac{1}{4} \lambda$ wire\mdotfill
+                 &         & $\Z.56$ &         & $\Z.45$ &        \\
+\tablespaceup$2D$\mdotfill     &         & $\Z.50$ & $\Z.40$ & $\Z.40$ & $\Z.20$\\
+\hline
+\end{tabular}
+\end{center}
+
+The most natural interpretation of Hertz's original experiment
+was to suppose that the vibrator emitted waves of electromotive
+intensity which, by interference with the waves reflected from the
+zinc plate, produced standing waves in the region between the
+vibrator and the reflector, the places in these waves where the
+electromotive intensity was a maximum being where the sparks
+were brightest when the resonator was held in the first position.
+%% -----File: 416.png---Folio 402-------
+
+Sarasin's and De~la~Rive's discovery of the influence of the
+size of the resonator on the positions of maximum sparking, and
+the independence of these positions on the period of the vibrator,
+compels us, if we retain this explanation, to suppose that any
+electrical vibrator gives out vibrations of all periods, emitting as
+it were a continuous electric spectrum.
+
+\Article{340*} This hypothesis appears most improbable, and a more
+satisfactory explanation seems to be afforded by means of the
+fact that the oscillations of the vibrator die away with great
+rapidity, while those of the resonator are extremely persistent.
+Let us consider what would happen in the extreme case when
+the oscillations in the vibrator are absolutely dead-beat. Here
+an electric impulse starts from the vibrator; on its way to the
+reflector it strikes against the resonator and sets it in electrical
+vibration; the impulse then travels up to the plate and is
+reflected, the electromotive intensity in the impulse being reversed
+by reflection; after reflection the impulse again strikes
+the resonator, which has maintained the vibrations started by
+the first impact. If when the reflected impulse reaches the resonator
+the phase of the vibrations of the latter is opposite to the
+phase when the impulse passed it on its way to the reflector, the
+electromotive intensity across the air gap due to the direct and
+reflected impulses will conspire, so that if the resonator is held
+in the first position a bright spark will be produced. Now the
+reflected impulse will strike the resonator the second time when
+its vibration is in the opposite phase to that which it had just
+after the first impact if the time which has elapsed between the
+two impacts is equal to half the time of a complete electrical
+oscillation of the resonator. The impulse travels at the rate at
+which electromagnetic action is propagated; hence, if the distance
+travelled by the impulse between the two impacts is equal
+to half the wave length of the free electrical vibrations of the
+resonator, that is, if the distance of the resonator from the reflecting
+plane is equal to one quarter of the wave length of this
+vibration, the direct and reflected waves will conspire. If the
+path travelled by the impulse between the two impacts is equal
+to a wave length, the electromotive intensity at the air gap
+due to the incident impulse will be equal and opposite to
+that due to the reflected one; so that there will in this case,
+in which the resonator is half a wave length away from the
+%% -----File: 417.png---Folio 403-------
+reflector, be no tendency to spark when the resonator is held
+in this position.
+
+Thus we see that on this view the distances from the reflecting
+plane of the places where the sparks have their maximum
+brightness will depend entirely upon the size of the resonator,
+and not upon that of the vibrator. This, as we have seen, was
+found by Sarasin and De~la~Rive to be a very marked feature in
+their experiments. We have assumed in this explanation that
+the vibrator does not vibrate. Bjerknes' experiments (l.c.)\ show
+that though the vibrations die away very rapidly they are not
+absolutely dead-beat. The existence of a small number of oscillations
+in the vibrator will cause the effects to be more vivid with
+a resonator in tune with it than with any other resonator. Since,
+however, the rate of decay of the vibrator is infinitely rapid
+compared with that of the resonator, the positions in which the
+sparks are brightest will depend much more upon the time of
+oscillation of the resonator than upon that of the vibrator.
+
+\Article{341} We have still to explain why the places at which the
+sparks were a maximum when the resonator was in the first
+position (i.e.~with its plane at right angles to the base line) were the
+places where the sparks vanished when the vibrator was in the
+second position (i.e.~with its plane containing the base line and
+the axis of the vibrator). When the resonator is in the first
+position the sparks are wholly due to the Faraday tubes which
+fall directly upon the air gap, hence the sparks will be a
+maximum when the state of the resonator corresponds to the
+incidence upon it of Faraday tubes from the vibrator of the
+same kind as those which reach it after reflection from the zinc
+plate. When the resonator is in the second position, having
+the line joining the terminals of the air gap at right angles to
+the axis of the vibrator, the sparks are due entirely to the
+Faraday tubes collected by the resonator and thrown into the
+air gap, and there would be no tendency to spark in the case
+just mentioned. For when two Faraday tubes of the same
+kind moving in opposite directions strike against opposite
+sides of the resonator, the tubes thrown into the air gap are
+of opposite signs, and thus do not produce any tendency to
+spark. When the resonator is in this position the maximum
+sparks will be produced when the positive tubes strike against
+one side of the resonator, the negative tubes against the other;
+%% -----File: 418.png---Folio 404-------
+the tubes thrown into the air gap will then be of the same sign
+and their efforts to produce a spark will conspire: if however
+the resonator had been held in the first position the positive
+tubes would have counterbalanced the negative ones, and there
+would not have been any tendency to spark.
+
+\Article{342} There is one result of Sarasin's and De~la~Rive's
+experiments which it is difficult to reconcile with theory. As
+will be seen from the table they found that the wave length of
+the vibration was equal to $8$~times the diameter of the resonator;
+theory would lead us to expect that the circumference of the
+resonator should be half a wave length, since, until the sparks
+pass, the current in the resonator will vanish at each end of the
+resonator, as we may neglect the capacity of the knobs. Thus
+there will be a node at each end of the resonator, and we should
+expect the wave length to be $2 \pi$~times the diameter instead of
+$8$~times, as found by Sarasin and De~la~Rive.
+
+
+\Subsection{Parabolic Mirrors.}
+\index{Mirrors, parabolic, for electromagnetic waves}%
+\index{Parabolic mirrors for electromagnetic waves}%
+
+\Article{343} If the vibrator is placed in the focal line of a parabolic
+cylinder, and if it is of such a kind that the Faraday tubes it
+emits are parallel to the focal line, then the waves emitted by
+the vibrator will, if the laws of reflection of these waves are the
+same as for light, after reflection from the cylinder emerge as
+a parallel beam and will therefore not diminish in intensity as
+they recede from the mirror; if such a beam falls on another
+parabolic mirror whose axis (i.e.~the axis of its cross-section) is
+parallel to the beam, it will be brought to a focus on the focal
+line of the second mirror. For these reasons the use of parabolic
+mirrors facilitates very much many experiments on electromagnetic
+waves.
+
+\includegraphicsmid{fig120}{Fig.~120.}
+
+The parabolic mirrors used by Hertz were made of sheet zinc,
+and their focal length was about $12.5$~cm. The vibrator which
+was placed in the focal line of one of the mirrors consisted of two
+equal brass cylinders placed so that their axes were coincident
+with each other and with the focal line; the length of each of
+the cylinders was $12$~cm.\ and the diameter $3$~cm., their sparking
+ends being rounded and well polished. The resonator, which was
+placed in the focal line of an equal parabolic mirror, consisted of
+two pieces of wire, each had a straight piece $50$~cm.\ long, and was
+then bent round at right angles so as to pass through the back
+%% -----File: 419.png---Folio 405-------
+of the mirror, the length of this bent piece being $15$~cm. The
+ends which came through the mirror were connected with a
+spark micrometer and the sparks were observed from behind the
+mirror. The mirrors are represented in \figureref{fig120}{Fig.~120}.
+
+
+\Subsection{Electric Screening.}
+\index{Electric screening@\subdashone screening}%
+\index{Screening, electric}%
+
+\Article{344} If the mirrors are placed about $6$~or $7$~feet apart in such
+a way that they face each other and have their axes coincident,
+then when the vibrator is in action vigorous sparks will be
+observed in the resonator. If a screen of sheet zinc about $2$~m.\
+high by $1$~broad is placed between the mirrors the sparks in the
+resonator will immediately cease; they will also cease if a paste-board
+screen covered with gold-leaf or tin-foil is placed between
+the mirrors; the interposition of a non-conductor, such as a
+wooden door, will not however produce any effect. We thus see
+that a very thin metallic plate acts as a perfect screen and is
+absolutely opaque to electrical oscillations, while on the other
+hand a non-conductor allows these radiations to pass through
+quite freely. The human body is a sufficiently good conductor
+to produce considerable screening when interposed between the
+vibrator and resonator.
+
+\Article{345} If wire be wound round a large rectangular framework
+in such a way that the turns of wire are parallel to one pair of
+sides of the frame, and if this is interposed between the mirrors,
+it will stop the sparks when the wires are vertical and thus
+parallel to the Faraday tubes emitted from the resonator; the
+sparks however will begin again if the framework is turned
+through a right angle so that the wires are at right angles to the
+Faraday tubes.
+%% -----File: 420.png---Folio 406-------
+
+\Subsection{Reflection of Electric Waves.}
+\index{Electromagnetic waves, reflection of from grating@\subdashtwo reflection of from grating}%
+\index{Grating, reflection of electromagnetic waves from}%
+\index{Reflection of electromagnetic waves from a grating@\subdashtwo electromagnetic waves from a grating}%
+\index{Waves, electromagnetic, reflection of from grating@\subdashtwo reflection of from grating}%
+
+\Article{346} To show the reflection of these waves place the mirrors
+side by side, so that their openings look in the same directions
+and their axes converge at a point distant about $3$~m.\ from the
+mirrors. No sparks can be detected at the resonator when the
+vibrator is in action. If, however, we place at the point of
+intersection of the axes of the mirrors a metal plate about $2$~m.\
+square at right angles to the line which bisects the angle between
+the axes of the mirrors, sparks will appear at the resonator;
+they will however disappear if the metal plate is twisted through
+about~$15°$ on either side. This experiment shows that these waves
+are reflected and that, approximately at any rate, the angle of
+incidence is equal to the angle of reflection.
+
+If the framework wound with wire is substituted for the
+metal plate sparks will appear when the wires are vertical and
+so parallel to the Faraday tubes, while the sparks will disappear
+if the framework is turned round until the wires are horizontal.
+Thus this framework reflects but does not transmit Faraday
+tubes parallel to the wires, while it transmits but does not
+reflect Faraday tubes at right angles to them. It behaves in
+fact towards the electrical waves very much as a plate of tourmaline
+does to light waves.
+
+\Subsection{Refraction of Electric Waves.}
+\index{Electromagnetic waves, refraction of@\subdashtwo refraction of}%
+\index{Refraction of electromagnetic waves}%
+\index{Waves, electromagnetic, refraction of from grating@\subdashtwo refraction of from grating}%
+
+\Article{347} To show the refraction of these waves Hertz used a large
+prism made of pitch; it was about $1.5$~metres in height, had a
+refracting angle of~$30°$, and a slant side of $1.2$~metres. When
+the electric waves from the vibrator passed through this prism
+the sparks in the resonator were not excited when the axes of
+the two mirrors were parallel, but they were produced when
+the axis of the mirror of the resonator made a suitable angle
+with that of the vibrator. When the system was adjusted for
+minimum deviation the sparks were most vigorous in the resonator
+when the axis of its mirror made an angle of~$22°$  with
+that of the vibrator. This shows that the refractive index for
+pitch is~$1.69$ for these long electrical waves.
+
+
+\Subsection{Angle of Polarization.}
+\index{Electromagnetic waves, xangle of polarization of@\subdashtwo angle of polarization of}%
+\index{Polarization angle of for electromagnetic waves,@\subdashone angle of for electromagnetic waves}%
+
+\Article{348} When light polarized in a plane at right angles to that
+of incidence falls upon a plate of refracting substance and the
+%% -----File: 421.png---Folio 407-------
+normal to the wave front makes with the normal to the surface
+an angle $\tan^{-1} \mu$ where $\mu$~is the refractive index, all the light is
+refracted and none reflected.
+
+Trouton (\textit{Nature}, February\DPtypo{,}{} 21, 1889) has observed a similar
+\index{Trouton, angle of polarization for electromagnetic waves}%
+effect with these electrical vibrations. From a wall $3$~feet thick
+reflections were obtained when the vibrator, and therefore the
+Faraday tubes, were perpendicular to the plane of incidence,
+while there was no reflection when the vibrator was turned
+through a right angle so that the Faraday tubes were in
+the plane of incidence. This experiment proves that in the
+Electromagnetic Theory of Light the Faraday tubes and the
+electric polarization are at right angles to the plane of polarization.
+
+\index{Electromagnetic waves, xtheory of reflection of from insulators@\subdashtwo theory of reflection of from insulators}%
+\index{Waves, electromagnetic, theory of reflection of from insulators@\subdashtwo theory of reflection of from insulators}%
+Before proceeding to describe some other interesting experiments
+of Mr.~Trouton's on the reflection of these waves from
+slabs of dielectrics, we shall investigate the theory of these
+phenomena on Maxwell's Theory.
+
+\Article{349} Let us suppose that plane waves are incident on a plate
+of dielectric bounded by parallel planes, let the plane of the
+paper be taken as that of incidence and of~$xy$, let the plate be
+bounded by the parallel planes $x = 0$, $x = -h$, the wave being incident
+on the plane $x = 0$. We shall first take the case when the
+polarization and Faraday tubes are at right angles to the plane
+of incidence. Let the electromotive intensity in the incident
+wave be represented by the real part of
+\[
+A\, \epsilon^{\iota(ax+by+pt)};
+\]
+if $i$~is the angle of incidence, $\lambda$~the wave length of the vibrations,
+$V$~their velocity of propagation,
+\[
+a = \frac{2 \pi}{\lambda} \cos i,\quad
+b = \frac{2 \pi}{\lambda} \sin i,\quad
+p = \frac{2 \pi}{\lambda} V.
+\]
+
+Let the intensity in the reflected wave be represented by the
+real part of
+\[
+A'\,\epsilon^{\iota(-ax+by+pt)}.
+\]
+
+The coefficient of~$y$ in the exponential in the reflected wave
+must be the same as that in the incident wave, otherwise the
+ratio of the reflected to the incident light would depend upon
+the portion of the plate on which the light fell. The coefficient
+of~$x$ in the expression for the reflected wave can only differ in
+sign from that in the incident wave: for if $E$~is the electromotive
+%% -----File: 422.png---Folio 408-------
+intensity in either the incident or reflected wave, we
+have
+\[
+\frac{d^{2} E}{dx^{2}} + \frac{d^{2} E}{dy^{2}} = \frac{1}{V^{2}}\, \frac{d^{2} E}{dt^{2}},
+\]
+hence the sum of the squares of the coefficients of $x$~and~$y$ must
+be the same for the incident and reflected waves, and since the
+\DPtypo{cofficients}{coefficients} of~$y$ are the same the coefficients of~$x$ can only differ in
+sign. If $E_1$,~$E_2$,~$E_3$ are the total electromotive intensities at
+right angles to the plane of incidence in the air, in the plate, and
+in the air on the further side of the plate, we may put
+\begin{align*}
+E_1 &= A\, \epsilon^{\iota(ax+by+pt)}  + A'\, \epsilon^{\iota(-ax+by+pt)},\\
+E_2 &= B\, \epsilon^{\iota(a'x+by+pt)} + B'\, \epsilon^{\iota(-a'x+by+pt)},\\
+E_3 &= C\, \epsilon^{\iota(ax+by+pt)},
+\end{align*}
+\begin{DPgather*}
+\lintertext{where}
+a'^{2} + b^{2} = \frac{p^{2}}{V'^{2}},
+\end{DPgather*}
+$V'$ being the velocity with which electromagnetic action travels
+through the plate. The real parts of the preceding expressions
+only are to be taken.
+
+Since the electromotive intensity is continuous when $x = 0$
+and when $x = -h$, we have
+\begin{align*}
+A + A' &= B + B', \Tag{1}\\
+C\, \epsilon^{-\iota a h}
+  &= B\, \epsilon^{-\iota a'h} + B'\, \epsilon^{\iota a' h}. \Tag{2}
+\end{align*}
+
+Since there is no accumulation of Faraday tubes on the surface
+of the plate the normal flow of these tubes in the air must equal
+that in the dielectric. Let $K$ be the specific inductive capacity of
+the plate, that of air being taken as unity, then in the air just
+above the plate the normal flow of tubes towards the plate is
+\[
+\frac{1}{4 \pi} (A-A')V \cos{i} \epsilon^{\iota(by+pt)},
+\]
+the normal flow of tubes in the plate away from the surface
+$x = 0$ is
+\[
+\frac{K}{4 \pi} (B-B')V' \cos{r} \epsilon^{\iota(by+pt)},
+\]
+where $r$~is the angle of refraction. Since these must be equal we
+have
+\[
+(A - A')V \cos i = K(B - B')V' \cos r.
+\Tag{3}
+\]
+The corresponding condition when $x = -h$ gives
+\[
+C\, \epsilon^{-\iota ah}\, V \cos i
+  = K \left(B\,  \epsilon^{-\iota a'h}
+          - B'\, \epsilon^{\iota a'h}\right)\, V' \cos r.
+\Tag{4}
+\]
+%% -----File: 423.png---Folio 409-------
+
+Equations (\eqnref{349}{3})~and~(\eqnref{349}{4}) are equivalent to the condition that the
+tangential magnetic force is continuous.
+
+Solving equations (\eqnref{349}{1}),~(\eqnref{349}{2}),~(\eqnref{349}{3}),~(\eqnref{349}{4}), we get
+\[
+\left.
+\begin{aligned}
+A' &= -A(K^{2} V'^{2} \cos^{2}r - V^{2} \cos^{2}i)(\epsilon^{\iota a'h} - \epsilon^{-\iota a'h}) ÷ \Delta,\\
+B  &= 2AV \cos i (KV' \cos r + V \cos i)\epsilon^{\iota a'h} ÷ \Delta,\\
+B' &= 2AV \cos i (KV' \cos r - V \cos i)\epsilon^{-\iota ah} ÷ \Delta,\\
+C  &= 4AKVV' \cos i \cos r \epsilon^{\iota ah} ÷ \Delta,
+\end{aligned}
+\right\}
+\Tag{5}
+\]
+where
+\begin{multline*}
+\Delta = (K^{2}V'^{2} \cos^{2}r + V^{2} cos^{2}i)
+  (\epsilon^{\iota a'h} - \epsilon^{-\iota a'h})\\
+  + 2KVV' \cos i \cos r (\epsilon^{\iota a'h} + \epsilon^{-\iota a'h}).
+\end{multline*}
+
+Thus, corresponding to the incident wave of electromotive
+intensity
+\[
+\cos \frac{2 \pi}{\lambda} (x \cos i + y \sin i + Vt),
+\]
+there will be a reflected wave represented by
+\begin{multline*}
+-(K^{2} V'^{2} \cos^{2}r - V^{2} \cos^{2}i)
+  \sin \left( \frac{2 \pi}{\lambda'} h \cos r \right) × \\
+  \cos \left[ \frac{2 \pi}{\lambda} (-x \cos i + y \sin i + Vt)
+    + \frac{\pi}{2} - \vartheta \right] ÷ D,
+\end{multline*}
+where $\lambda'$ is the wave length in the plate.
+\begin{multline*}
+D^{2} = (K^{2} V'^{2} \cos^{2}r + V^{2} \cos^{2}i)^2
+  \sin^{2} \left( \frac{2 \pi}{\lambda'} h \cos r \right) \\
+  + 4K^{2} V'^{2} V^{2} \cos^{2}i \cos^{2}r
+    \cos^{2} \left( \frac{2 \pi}{\lambda'} h \cos r \right),
+\end{multline*}
+\begin{DPgather*}
+\lintertext{and}
+\tan \vartheta = \frac{K^{2} V'^{2} \cos^{2}r + V^{2} \cos^{2}i}{2KVV' \cos i \cos r}
+  \tan \left( \frac{2 \pi}{\lambda'} h \cos r \right).
+\end{DPgather*}
+The waves in the plate will be
+\begin{multline*}
+V \cos i(KV' \cos r + V \cos i) × \\
+  \cos \left[ \frac{2 \pi}{\lambda'} \bigl((x + h) \cos r + y \sin r + V't\bigr) - \vartheta \right] ÷ D,
+\end{multline*}
+and
+\begin{multline*}
+V \cos i(KV' \cos r - V \cos i) × \\
+  \cos \left[ \frac{2 \pi}{\lambda'} \bigl(-(x + h) \cos r + y \sin r + V't\bigr) - \vartheta \right] ÷ D;
+\end{multline*}
+while the wave emerging from the plate will be
+\[
+2KVV' \cos i \cos r \cos \left[ \frac{2 \pi}{\lambda} \bigl((x + h) \cos i + y \sin i + Vt\bigr) - \vartheta \right] ÷ D.
+\]
+%% -----File: 424.png---Folio 410-------
+
+Thus we see that when $2\pi h \cos r/\lambda'$ is very small the reflected
+wave vanishes; this is what we should have expected, as it must
+require a slab whose thickness is at least comparable with the
+wave length in the slab to produce any appreciable reflection.
+When the reflecting surface is too thin we get a result analogous
+to the blackness of very thin soap films. Trouton has
+verified that there is no reflection of the electrical waves from
+window-glass unless this is covered with moisture.
+
+The expression for the amplitude for the reflected wave shows
+that this will vanish not merely when $2\pi h \cos r/\lambda'$ vanishes but
+also when this is a multiple of~$\pi$. Trouton used as the dielectric
+plate a wall built of paraffin bricks, a method which enabled him
+to try the effect of altering the thickness of the plate; he found
+that after reaching the thickness at which the reflected wave
+became sensible, by making the wall still thicker the reflected
+wave could be diminished so that its effects were insensible.
+The case is exactly analogous to that of Newton's rings, where
+we have darkness whenever $2h \cos r$~is a multiple of a wave
+length of the light in the plate.
+
+There will be a critical angle in this case if the solution of the
+equation
+\[
+K^{2} V'^{2} \cos^{2} r - V^{2} \cos^{2} i = 0
+\Tag{6}
+\]
+is real. If the plate is non-magnetic the magnetic permeability
+is unity, and we have
+\[
+K = \frac{V^{2}}{V'^{2}} = \frac{\sin^{2}i}{\sin^{2}r},
+\]
+so equation~(\eqnref{349}{6}) becomes
+\[
+\cot^{2} r - \cot^{2} i = 0,
+\]
+an equation which cannot be satisfied, so that there is no critical
+angle in this case. This result would not however be true if it
+were possible to find a magnetic substance which was transparent
+to electric waves; for if $\mu'$~is the magnetic permeability
+of the substance, we have
+\[
+\mu' K = \frac{V^{2}}{V'^{2}},
+\]
+so that equation~(\eqnref{349}{6}) becomes
+\begin{DPalign*}
+\frac{\cot^{2} r}{\mu'^{2}} & = \cot^{2} i, \\
+\lintertext{or}
+\frac{\cot r}{\mu'} & = \cot i.
+\end{DPalign*}
+\begin{DPgather*}
+\lintertext{\indent Since}
+\sqrt{\mu'K} \sin r = \sin i
+\end{DPgather*}
+%% -----File: 425.png---Folio 411-------
+we may transform this equation to
+\[
+\sin^{2}i
+  = \frac{\mu'^{2} - \mu'K}{\mu'^{2}-1}
+  = \frac{\mu'(\mu' - K)}{\mu'^{2}-1};
+\]
+hence if $i$~is real, $\mu'$~must be greater than~$K$. No substance is
+known which fulfils the conditions of being transparent and
+having the magnetic permeability greater than the specific
+inductive capacity, which are the conditions for the existence of
+a polarizing angle when the Faraday tubes are at right angles
+to the plane of incidence.
+
+When the plane is infinitely thick, we see that
+\[
+A' = - \frac{K V' \cos r - V \cos i}{K V' \cos r + V \cos i}\, A,
+\]
+or if the magnetic permeability is unity,
+\[
+A' = - \frac{\sin(i - r)}{\sin(i + r)}\, A,
+\]
+which is analogous to the expression obtained by Fresnel for
+the amplitude of the reflected ray when the incident light is
+polarized in the plane of incidence.
+
+\Article{350} In the preceding investigation the Faraday tubes were at
+right angles to the plane of incidence, we shall now consider the
+case when they are in that plane: they are also of course in the
+planes at right angles to the direction of propagation of the
+several waves.
+
+Let the electromotive intensity at right angles to the incident
+ray be
+\[
+A\, \epsilon^{\iota(ax + by + pt)},
+\]
+that at right angles to the reflected ray
+\[
+A'\, \epsilon^{\iota(-ax + by + pt)}.
+\]
+
+Let the electromotive intensity at right angles to the ray
+which travels in the same sense as the incident one through
+the plate of dielectric, i.e.~in a direction in which $x$~diminishes, be
+\[
+B\, \epsilon^{\iota(a'x + by + pt)},
+\]
+while that at right angles to the ray travelling in a direction in
+which $x$~increases is represented by
+\[
+B'\, \epsilon^{\iota(-a'x + by + pt)}.
+\]
+The electromotive intensity at right angles to the ray emerging
+from the plate is
+\[
+C\, \epsilon^{\iota(ax + by + pt)}
+\]
+%% -----File: 426.png---Folio 412-------
+
+The conditions at the boundary are (1)~that the electromotive
+intensity parallel to the surface of the plate is continuous;
+(2)~that the electric polarization at right angles to the plate is
+also continuous.
+
+Hence if $i$~is the angle of incidence, $r$~that of refraction, the
+boundary conditions at the surface $x = 0$ of the plate give
+\[
+\left.
+\begin{aligned}
+(A-A') \cos i &= (B-B') \cos r,\\
+(A+A') \sin i &= K(B+B') \sin r,
+\end{aligned}
+\right\}
+\Tag{7}
+\]
+where $K$~is the specific inductive capacity of the plate.
+
+The boundary conditions at the lower surface of the plate
+give
+\[
+\left.
+\begin{aligned}
+C\, \epsilon^{-\iota ah} \cos i &=  (B\, \epsilon^{-\iota a'h} - B'\, \epsilon^{\iota a'h}) \cos r, \\
+C\, \epsilon^{-\iota ah} \sin i &= K(B\, \epsilon^{-\iota a'h} + B'\, \epsilon^{\iota a'h}) \sin r.
+\end{aligned}
+\right\}
+\Tag{8}
+\]
+
+Solving equations (\eqnref{350}{7})~and~(\eqnref{350}{8}) we get
+\begin{align*}
+A' &= A(K^{2} \tan^{2} r - tan^{2} i)(\epsilon^{\iota a'h} - \epsilon^{-\iota a'h}) ÷ \Delta',\\
+B\phantom{'}
+   &=  2 A(\sin i / \cos r)(K \tan r + \tan i) \epsilon^{\iota a'h} ÷ \Delta',\\
+B' &= -2 A(\sin i / \cos r)(K \tan r - \tan i) \epsilon^{-\iota a'h} ÷ \Delta',\\
+C \phantom{'}
+   &= 4 A K \tan i \tan r \epsilon^{\iota ah} ÷ \Delta',
+\end{align*}
+where
+\[
+\Delta' = (K^{2} \tan^{2} r + \tan^{2} i)(\epsilon^{\iota a'h} - \epsilon^{-\iota a'h})
+  + 2 K \tan i \tan r (\epsilon^{\iota a'h} + \epsilon^{-\iota a'h}).
+\]
+From these equations we see that if the incident wave is equal to
+\[
+\cos \frac{2 \pi}{\lambda} (x \cos i + y \sin i + Vt)
+\]
+the reflected wave will be
+\begin{multline*}
+(K^{2} \tan^{2} r - \tan^{2} i)
+  \sin \left( \frac{2 \pi}{\lambda'} h \cos r \right) × \\
+  \cos \left[ \frac{2 \pi}{\lambda} (-x \cos i + y \sin i + Vt) + \frac{\pi}{2} - \theta \right] ÷ D';
+\end{multline*}
+the waves in the plate will be represented by
+\begin{multline*}
+(\sin i / \cos r)(K \tan r + \tan i) × \\
+  \cos \left[ \frac{2 \pi}{\lambda'} \bigl((x + h) \cos r + y \sin r + V't\bigr) - \theta \right] ÷ D',
+\end{multline*}
+and
+\begin{multline*}
+-(\sin i / \cos r)(K \tan r - \tan i) × \\
+  \cos \left[ \frac{2 \pi}{\lambda'} \bigl(-(x+h) \cos r + y \sin r + V't\bigr) - \theta \right] ÷ D'
+\end{multline*}
+%% -----File: 427.png---Folio 413-------
+respectively, while the emergent wave is
+\[
+2K \tan i \tan r \cos\left[\frac{2}{\pi}{\lambda} \bigl((x + h) \cos i + y \sin i + Vt\bigr) - \theta\right] ÷ D',
+\]
+where
+\begin{multline*}
+D'^2 = (K^2 \tan^2 r + \tan^2 i)^2
+  \sin^2 \left(\frac{2\pi}{\lambda'} h \cos r\right) \\
+  + 4K^2 \tan^2 r \tan^2 i \cos^2\left(\frac{2\pi}{\lambda'} h \cos r\right),
+\end{multline*}
+\begin{DPgather*}
+\lintertext{and}
+\tan \theta = \frac{K^2 \tan^2r + \tan^2i}{2K \tan r \tan i}
+  \tan\left(\frac{2\pi}{\lambda'} h \cos r\right).
+\end{DPgather*}
+
+From these expressions we see that, as before, there is no
+reflected wave when $h$~is very small compared with~$\lambda'$ and when
+$h \cos r$ is a multiple of~$\lambda'/2$; these results are the same whether the
+Faraday tubes are in or at right angles to the plane of incidence.
+We see now, however, that in addition to this the reflected wave
+vanishes, whatever the thickness of the plate, when $K \tan r = \tan i$,
+or since $\sqrt{\mu'K} \sin r = \sin i$ where $\mu'$~is the magnetic permeability,
+the reflected wave vanishes when
+\[
+\tan^2 i = \frac{K (K-\mu')}{\mu' K - 1};
+\]
+if the plate is non-magnetic $\mu'= 1$, and we have
+\[
+\tan i = \sqrt{K}.
+\]
+
+When $K \tan r = \tan i$ the reflected wave and one of the waves
+in the plate vanish; the electromotive intensity in the other wave
+in the plate is equal to
+\[
+\sqrt{\frac{\mu'}{K}} \cos \frac{2\pi}{\lambda'} (x \cos r + y \sin r + V't),
+\]
+and the emergent wave is
+\[
+\cos \frac{2\pi}{\lambda} \left((x + h) \cos i + y \sin i + Vt - \frac{h\lambda}{\lambda'} \cos r\right).
+\]
+The intensity of all these waves are independent of the thickness
+of the plate.
+
+If the plate is infinitely thick we must put $B' = 0$ in equations~(\eqnref{350}{7});
+doing this we find from these equations that
+\begin{align*}
+A' & = A\, \frac{(K \tan r - \tan i)}{K \tan r + \tan i}, \\
+B\phantom{'} & = A\, \frac{\sin 2i}{\sin i \cos r + K \cos i \sin r}.
+\end{align*}
+%% -----File: 428.png---Folio 414-------
+
+If the plate is made of a non-magnetic material $K = \sin^2 i / \sin^2 r$,
+and in this case we have
+\begin{align*}
+A' & = A\, \frac{\tan(i - r)}{\tan(i + r)}, \\
+B  & = 4A\, \frac{\sin r \cos i}{\sin 2i + \sin 2r}.
+\end{align*}
+
+
+\Subsection{Reflection from a Metal Plate.}
+\index{Electromagnetic waves, xtheory of reflection of from metals@\subdashtwo theory of reflection of from metals}%
+\index{Waves, electromagnetic, theory of reflection of from metals@\subdashtwo theory of reflection of from metals}%
+
+\Article{351} The very important case when the plate is made of a metal
+instead of an insulator can be solved in a similar way. The
+expressions for the electromotive intensities in the various media
+will be of the same type as before; in the case of metallic
+reflection however the quantity~$a'$, which occurs in the expression
+for the electromotive intensity in the plate, will no longer be
+real. In a conductor whose specific resistance is~$\sigma$ the electromotive
+intensity will satisfy a differential equation of the form
+\[
+\frac{d^2 E}{dx^2} + \frac{d^2 E}{dy^2} = \frac{4\pi\mu}{\sigma}\, \frac{dE}{dt},
+\]
+or, since $E$~varies as~$\epsilon^{\iota pt}$,
+\[
+\frac{d^2 E}{dx^2} + \frac{d^2 E}{dy^2} = \frac{4\pi\mu\iota p}{\sigma}\, E.
+\]
+
+Hence, since in the metal plate $E$~varies as~$\epsilon^{\iota(± a'x + by + pt)}$, we
+see that
+\[
+a'^2 + b^2 = -4\pi\mu\iota p / \sigma.
+\Tag{9}
+\]
+
+To compare the magnitude of the terms in this equation, let us
+suppose that we are dealing with a wave whose wave length is
+$10^q$~centimetres. Then since $2\pi/p$~is the time of a vibration, if
+$V$~is the velocity of propagation of electromagnetic action in air,
+\[
+V 2\pi / p = \lambda,
+\]
+but $V$~is equal to $3 × 10^{10}$, hence
+\[
+p = 6\pi 10^{10-q}.
+\]
+
+If the plate is made of zinc $\sigma$~is about~$10^4$, so that the modulus
+of $4\pi\mu\iota p / \sigma$ is about $24\pi^2 10^{6-q}$. Now $b^2$~is less than $4\pi^2 / \lambda^2$,
+i.e.~$4\pi^2 × 10^{-2q}$, hence the ratio of the modulus of $4\pi\mu\iota p / \sigma$ to
+$b^2$ is of the order $6 × 10^{6+q}$, and is therefore exceedingly large
+unless $q$~is less than~$-6$, that is, unless the wave length of the
+electrical oscillation is much less than that of green light. Thus
+for waves appreciably longer than this we may for a zinc plate
+%% -----File: 429.png---Folio 415-------
+neglect~$b^{2}$ in equation~(\eqnref{351}{9}), which then becomes
+\begin{DPalign*}
+a'^2 & = -4 \pi \mu \iota p / \sigma, \\
+\lintertext{or}
+a'\phantom{^2} & = ± \sqrt{2 \pi \mu p / \sigma} (1-\iota),
+\end{DPalign*}
+thus $a'$~is exceedingly large compared with~$a$.
+
+We shall first consider the case when the Faraday tubes are
+at right angles to the plane of incidence as in \artref{349}{Art.~349}. The
+condition that the electromotive intensity parallel to the surface
+of the plate is continuous will still be true, but since there is no
+real angle of refraction in metals it is convenient to recognize
+the second condition of that article as expressing the condition
+that the tangential magnetic force is continuous. The tangential
+magnetic force is parallel to~$y$ and is equal to
+\[
+\frac{1}{\mu \iota p}\, \frac{dE}{dx},
+\]
+where $\mu$~is the magnetic permeability. By means of this and the
+previous condition we find, using the notation of \artref{349}{Art.~\DPtypo{(349)}{349}},
+\[
+\left.
+\begin{aligned}
+&A' = -A(a'^{2} / \mu^{2} - a^{2})(\epsilon^{\iota h a'} - \epsilon^{-\iota h a'}) ÷ D, \\
+&B\phantom{'} = 2 A a(a' / \mu + a)\epsilon^{\iota h a'} ÷ D, \\
+&B' = 2 A a(a' / \mu - a)\epsilon^{-\iota h a'} ÷ D, \\
+&C\phantom{'} = A 4 a(a' / \mu)\epsilon^{\iota h a} ÷ D, \\
+&\text{where} \\
+& D\phantom{'} = (a'^{2} / \mu^{2} + a^{2})(\epsilon^{\iota ha'} - \epsilon^{-\iota ha'})
+  + 2a(a' / \mu)(\epsilon^{\iota ha'} + \epsilon^{-\iota ha'}).
+\end{aligned}
+\right\}
+\Tag{10}
+\]
+
+Since $\epsilon^{\iota(a'x + by + pt)}$ represents a wave travelling in the plate in
+the direction of the incident wave, i.e.~so that $x$~is increasingly
+negative; the real part of~$\iota a'$ must be positive, otherwise the
+amplitude of the wave would continually increase as the wave
+travelled onwards; hence if $ha'$~is very large, equations~(\eqnref{351}{10})
+become approximately, remembering that $a'/a$~is also very large,
+\begin{align*}
+A' &= -A,\\
+B\phantom{'} &= \frac{2 a \mu}{a'}\, A,\\
+C\phantom{'} &= B' = 0.
+\end{align*}
+
+Hence in this case there is complete reflection from the metal
+plate, and since $A' + A = 0$ we see that the electromotive intensity
+vanishes at the surface of the plate, and since $C = 0$
+there is no electromotive intensity on the far side of the plate.
+%% -----File: 430.png---Folio 416-------
+The condition that the plate should act as a perfect reflector or,
+which is the same thing, as a perfect screen, is that $\{4 \pi \mu p h^{2} / \sigma\}^{\frac{1}{2}}$
+should be large. In the case of zinc plates the value of this
+quantity for vibrations whose wave length is $10^{q}$~centimetres is
+equal to $1.5 × 10^{4-q/2} h$, so that for waves $1$~metre long it is equal
+to $1500 h$; thus, if $h$~were as great as $\frac{1}{15}$~of a millimetre, $a'h$
+would be equal to~$10$, and since $\epsilon^{10}$~is very large the reflection
+in this case would be practically perfect. We see from this
+result the reason why gold-leaf and tin-foil are able to reflect
+these very rapid oscillations almost completely. If however the
+conductor is an electrolyte $\sigma$~may be of the order~$10^{10}$, so that
+$a'h$~will now be only~$1.5 h$ for waves $1$~metre in length, in this case
+it will require a slab of electrolyte several millimetres in thickness
+to produce complete reflection. We shall consider a little
+more fully the wave emergent from the metallic plate. We
+have by equations~(\eqnref{351}{10})
+\[
+C = \frac{4 A a a'\, \epsilon^{\iota h a}}
+         {\mu \{(a'^{2} / \mu^{2} + a^{2}) (\epsilon^{\iota ha'} - \epsilon^{-\iota ha'})
+                            + (2 a a'/ \mu) (\epsilon^{\iota ha'} + \epsilon^{-\iota ha'}) \} }.
+\Tag{11}
+\]
+
+If $ha'$~is very small this may be written
+\[
+C = \frac{2Aaa'\, \epsilon^{\iota h a}}
+         {\mu \{(a'^{2} / \mu^{2} + a^{2}) h a' \iota + (2 a a' / \mu) \} },
+\]
+or, since $a'^{2} / \mu^{2}$ is very large compared with~$a^{2}$,
+\begin{align*}
+C &= \frac{A\, \epsilon^{\iota h a}}{1 + \dfrac{\iota h a'^{2}}{2 \mu a} + \frac{1}{2} \iota \mu h a} \\
+  &= \frac{A\, \epsilon^{\iota h a}}{1 + (2 \pi V h / \sigma) + \frac{1}{2} \iota \mu h a}.
+\end{align*}
+
+Thus, corresponding to the incident wave
+\[
+\cos \frac{2 \pi}{\lambda} (x + V t),
+\]
+we have, since $h a$ is very small, an emergent wave
+\begin{DPgather*}
+\frac{1}{1 + (2 \pi h V / \sigma)} \cos \frac{2 \pi}{\lambda} (x + h' + V t), \\
+\lintertext{where}
+h' = h \left\{1 - \tfrac{1}{2}\, \frac{\mu}{1 + 2 \pi h V / \sigma} \right\}.
+\end{DPgather*}
+
+Since $V$~is equal to $3 × 10^{10}$ and $\sigma$~for electrolytes is rarely
+greater than~$10^{9}$, we see that for very moderate thicknesses
+$(2 \pi h V / \sigma)$ will be large compared with unity, so that the expression
+%% -----File: 431.png---Folio 417-------
+for the emergent wave becomes
+\[
+\frac{1}{(2 \pi h V / \sigma)} \cos \frac{2 \pi}{\lambda} (x + h + Vt).
+\]
+
+The thickness of the conducting material which, when interposed
+in the path of the wave, produces a given diminution in
+the electric intensity is thus proportional to the specific resistance
+of the material; this result has been applied to measure the
+specific resistance of electrolytes under very rapidly alternating
+\index{Electrolytes, under rapidly alternating currents@\subdashone under rapidly alternating currents}%
+currents (see J.~J. Thomson, \textit{Proc.\ Roy.\ Soc.}~45, p.~269, 1889).
+
+The preceding investigation applies to the case when the
+Faraday tubes are at right angles to the plane of incidence, the
+same results will apply when the Faraday tubes are in the plane
+of incidence: the proof of these results for this case we shall
+however leave as an exercise for the student.
+
+
+\Subsection{Reflection of Light from Metals.}
+\index{Light, xreflection of from metals@\subdashone reflection of from metals}%
+\index{Metals, xreflection of light from@\subdashone reflection of light from}%
+\index{Reflection of light from metals@\subdashtwo light from metals}%
+
+\Article{352} The assumption that $a'/a$~is very large is legitimate when
+we are dealing with waves as long as those produced by Hertz's
+apparatus, it ceases however to be so when the length of the
+wave is as small as it is in the electrical vibrations we call
+light. We shall therefore consider separately the theory of the
+reflection of such waves from metallic surfaces. With the view
+of making our equations more general we shall not in this case
+neglect the effects of the polarization currents in the metal; when
+we include these, the components of the magnetic force and
+electromotive intensity in the metal satisfy differential equations
+of the form
+\[
+\mu K'\, \frac{d^{2}f}{dt^{2}} + \frac{4 \pi \mu}{\sigma}\, \frac{df}{dt}
+  = \frac{d^{2}f}{dx^{2}} + \frac{d^{2}f}{dy^{2}} + \frac{d^{2}f}{dz^{2}}.
+\Tag{1}
+\]
+See Maxwell's \textit{Electricity and Magnetism}, Art.~783; here $K'$~is
+the specific inductive capacity of the metal.
+
+\Article{353} Let us first consider the case when the incident wave is
+polarized in the plane of incidence, which we take as the plane
+of~$xy$, the reflecting surface being given by the equation $x = 0$.
+In this case the electromotive intensity~$Z$ is parallel to the axis
+of~$z$; let the incident wave be
+\begin{DPgather*}
+Z = \epsilon^{\iota(ax + by + pt)},\\
+\intertext{the reflected wave}
+Z = A\, \epsilon^{\iota(-ax + by + pt)},\\
+%% -----File: 432.png---Folio 418-------
+\lintertext{where}
+a^{2} + b^{2} = Kp^{2},
+\Tag{2}
+\end{DPgather*}
+$K$~being the specific inductive capacity of the dielectric, and
+the magnetic permeability of this dielectric being assumed to be
+unity.
+
+Let the wave in the metal be given by the equation
+\begin{DPgather*}
+Z = B\, \epsilon^{\iota(a'x + by + pt)}, \\
+\lintertext{where}
+a'^{2} + b^{2} = p^{2} \mu K' - \frac{4 \pi \mu \iota p}{\sigma}.
+\Tag{3}
+\end{DPgather*}
+
+Thus in the dielectric we have
+\begin{DPgather*}
+Z = \epsilon^{\iota(ax + by + pt)} + A\, \epsilon^{\iota(-ax + by + pt)}, \\
+\lintertext{and in the metal}
+Z = B\, \epsilon^{\iota(a'x + by + pt)}.
+\end{DPgather*}
+
+Since $Z$, the electromotive intensity, is continuous when $x = 0$,
+we have
+\[
+1 + A = B.
+\]
+
+By equation~(\eqnref{256}{2}) of \artref{256}{Art.~256} the magnetic induction parallel to~$y$
+is equal to
+\[
+\frac{1}{\iota p}\, \frac{dZ}{dx},
+\]
+and since the magnetic force parallel to~$y$ is continuous when
+$x = 0$, we have
+\[
+a(1-A) = \frac{a'}{\mu} B.
+\]
+
+From these equations we find
+\[
+A = \frac{1 - \dfrac{a'}{\mu a}}{1 + \dfrac{a'}{\mu a}}.
+\Tag[3s]{3*}
+\]
+
+Let us for the present confine our attention to the non-magnetic
+metals for which $\mu = 1$, in this case the preceding
+equation becomes
+\[
+A = \frac{1 - \dfrac{a'}{a}}{1 + \dfrac{a'}{a}}.
+\]
+
+The expression given by Fresnel for the amplitude of the wave
+reflected from a transparent substance is of exactly the same
+form as this result, the only difference being that for a transparent
+substance $a'$~is real, while in the case of metals it is
+complex.
+%% -----File: 433.png---Folio 419-------
+
+Now for transparent substances the relation between $a'$~and~$a$
+is
+\[
+\frac{a'^{2} + b^{2}}{a^{2} + b^{2}} = \mu'^{2},
+\]
+where $\mu'$~is the refractive index of the substance.
+
+In the case of metals however the relation between $a'$~and~$a$ is
+\[
+\frac{a'^{2} + b^{2}}{a^{2} + b^{2}}
+  = \mu \frac{K'}{K} - \frac{4 \pi \mu \iota}{K p \sigma}
+  = R^{2}\, \epsilon^{2 \iota\alpha}, \text{ say},
+\Tag{4}
+\]
+which is of exactly the same form as the preceding, with $R\, \epsilon^{\iota \alpha}$
+written instead of~$\mu'$, the refractive index of the transparent
+substance.
+
+Thus, if in Fresnel's formula for the reflected light we suppose
+that the refractive index is complex and equal to~$R\, \epsilon^{\iota \alpha}$, where $R$
+and~$\alpha$ are defined by equation~(\eqnref{353}{4}), we shall arrive at the results
+given by the preceding theory of the reflection of light by
+metals.
+
+\Article{354} Let us now consider the case when the plane of polarization
+is perpendicular to the plane of incidence; in this case the
+electromotive intensity is in the plane of incidence and the
+magnetic force~$\gamma$ at right angles to it. If the incident wave is
+expressed by the equation
+\[
+\gamma = \epsilon^{\iota(ax + by + pt)},
+\]
+then in the dielectric we may put
+\[
+\gamma = \epsilon^{\iota(ax + by + pt)} + A'\, \epsilon^{\iota(-ax + by + pt)},
+\]
+while in the metal we have
+\[
+\gamma = B'\, \epsilon^{\iota(a'x + by + pt)}.
+\]
+
+Since the magnetic force parallel to the surface is continuous,
+we have
+\[
+1 + A' = B'.
+\Tag{5}
+\]
+
+The other boundary condition we shall employ is that~$Q$, the
+tangential electromotive intensity parallel to the axis of~$y$, is
+continuous. Now if $g$~is the electric polarization parallel to~$y$,
+and $v$~the conduction current in the same direction, then in the
+dielectric above the metal
+\[
+4 \pi \frac{dg}{dt} = -\frac{d \gamma}{dx},
+\]
+or since
+\[
+g = \frac{K}{4 \pi} Q = \frac{a^{2} + b^{2}}{4 \pi p^{2}} Q
+\]
+%% -----File: 434.png---Folio 420-------
+by equation~(\eqnref{353}{2}) we have
+\[
+\frac{\iota(a^{2} + b^{2})}{p} Q = - \frac{d \gamma}{dx}.
+\]
+
+In the metal
+\begin{DPgather*}
+4 \pi \frac{dg}{dt} + 4 \pi v = -\frac{d \gamma}{dx}, \\
+\lintertext{or}
+\left( K' \iota p + \frac{4 \pi}{\sigma} \right) Q = -\frac{d \gamma}{dx},
+\end{DPgather*}
+this by equation~(\eqnref{353}{3}) becomes
+\[
+\frac{\iota}{p \mu} (a'^{2} + b^{2}) Q = -\frac{d \gamma}{dx};
+\]
+hence, since $Q$~is continuous when $x = 0$, we have
+\[
+\frac{a}{a^{2} + b^{2}} (1 - A') = \frac{\mu a'}{(a'^{2} + b^{2})} B'.
+\Tag{6}
+\]
+
+Equations (\eqnref{354}{5})~and~(\eqnref{354}{6}) give
+\[
+A' = \frac{1 - \mu \dfrac{a'}{a}\, \dfrac{a^{2} + b^{2}}{a'^{2} + b^{2}}}
+          {1 + \mu \dfrac{a'}{a}\, \dfrac{a^{2} + b^{2}}{a'^{2} + b^{2}}},
+\]
+which is again, for non-magnetic metals for which $\mu = 1$, of the
+same form as Fresnel's expression for the amplitude of the
+reflected wave from a transparent substance. So that in this
+case, as in the previous one, we see that we can get the results
+of this theory of metallic reflection by substituting in Fresnel's
+expression a complex quantity for the refractive index.
+
+\Article{355} This result leads to a difficulty similar to the one which
+was pointed out by Lord Rayleigh (\textit{Phil.\ Mag.}~[4], 43, p.~321, 1872)
+\index{Rayleigh, Lord, metallic reflection@\subdashtwo metallic reflection}%
+in the theory of metallic reflection on the elastic solid theory of
+light. The result of substituting in Fresnel's expressions a complex
+quantity for the refractive index has been compared with the
+\index{Eisenlohr, metallic reflection}%
+result of experiments on metallic reflection by Eisenlohr (\textit{Pogg.\
+Ann.}~104, p.~368, 1858) and Drude (\textit{Wied.\ Ann.}~39, p.~481, 1890).
+\index{Drude on metallic reflection}%
+The latter writer finds that if the real part of $R^{2}\, \epsilon^{2 \iota \alpha}$, the
+quantity which for metals replaces the square of the refractive
+index for transparent substances, is written as $n^{2} (1 - k^{2})$, the
+imaginary part as $-2 \iota n^{2} k$; then $n$~and~$k$ have the following
+values, where the accented letters refer to the values for red
+light, the unaccented to sodium light.
+%% -----File: 435.png---Folio 421-------
+\begin{center}
+\index{Refractive@`Refractive Indices' of metals}%
+\tabletextsize
+\settowidth{\TmpLen}{Copper-Nickel alloy\qquad}
+\begin{tabular}{| l@{}|*{4}{>{\quad}c<{\quad}|}}
+\hline
+\parbox{\TmpLen}{\bigskip$~$\bigskip}& $n$ & $n'$ & $k$ & $k'$ \\
+\hline
+\tablespaceup Bismuth\mdotfill &$1.90\Z$  &$2.07\Z$  &$\Z1.93$   &$\Z1.90$ \\
+Lead, pure\mdotfill          &$2.01\Z$  &$1.97\Z$  &$\Z1.73$   &$\Z1.74$ \\
+Lead, impure\mdotfill        &$1.97\Z$  &          &$\Z1.74$   &       \\
+Mercury, pure\mdotfill       &$1.73\Z$  &$1.87\Z$  &$\Z2.87$   &$\Z2.78$ \\
+Mercury, impure\mdotfill     &$1.55\Z$  &          &$\Z3.14$   &       \\
+Platinum, pure\mdotfill      &$2.06\Z$  &$2.16\Z$  &$\Z2.06$   &$\Z2.06$ \\
+Platinum, impure\mdotfill    &$2.15\Z$  &          &$\Z1.92$   &       \\
+Gold, pure\mdotfill          &$\Z.366$  &$\Z.306$  &$\Z7.71$   &$10.2\Z$ \\
+Gold, impure\mdotfill        &$\Z.570$  &          &$\Z5.31$   &       \\
+Antimony\mdotfill            &$3.04\Z$  &$3.17\Z$  &$\Z1.63$   &$\Z1.56$ \\
+Tin, solid\mdotfill          &$1.48\Z$  &$1.66\Z$  &$\Z3.55$   &$\Z3.30$ \\
+Tin, liquid\mdotfill         &$2.10\Z$  &          &$\Z2.15$   &       \\
+Cadmium\mdotfill             &$1.13\Z$  &$1.31\Z$  &$\Z4.43$   &$\Z4.05$ \\
+Silver\mdotfill              &$\Z.181$  &$\Z.203$  &$20.3\Z$   &$19.5\Z$ \\
+Zinc\mdotfill                &$2.12\Z$  &$2.36\Z$  &$\Z2.60$   &$\Z2.34$ \\
+Copper, pure\mdotfill        &$\Z.641$  &$\Z.580$  &$\Z4.09$   &$\Z5.24$ \\
+Copper, impure\mdotfill      &$\Z.686$  &          &$\Z3.85$   &       \\
+Copper-Nickel alloy\mdotfill &$1.55\Z$  &          &$\Z2.14$   &       \\
+Nickel\mdotfill              &$1.79\Z$  &$1.89\Z$  &$\Z1.86$   &$\Z1.88$ \\
+Iron\mdotfill                &$2.36\Z$  &          &$\Z1.36$   &       \\
+Steel\mdotfill               &$2.41\Z$  &$2.62\Z$  &$\Z1.38$   &$\Z1.32$ \\
+Aluminium\mdotfill           &$1.44\Z$  &$1.62\Z$  &$\Z3.63$   &$\Z3.36$ \\
+Magnesium\mdotfill           &$\Z.37\Z$ &$\Z.40\Z$ &$11.8\Z$   &$11.5\Z$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+It will be seen that for all these metals without exception the
+value of~$k$ is greater than unity, so that the real part of $R^{2}\, \epsilon^{2 \iota \alpha}$
+or $n^{2}(1-k^{2})$ is negative. Equation~(\eqnref{353}{4}), \artref{353}{Art.~353}, shows, however,
+that the real part of $R^{2}\, \epsilon^{2 \iota \alpha}$ is equal to~$\mu K'/K$, an essentially
+positive quantity. This shows that the electromagnetic theory
+of metallic reflection is not general enough to cover the facts.
+In this respect, however, it is in no worse position than any
+other existing theory of light, while it possesses the advantage
+over other theories of explaining why metals are opaque.
+
+\Article{356} The direction in which to look for an improvement of
+the theory seems pretty obvious. The preceding table shows
+how rapidly the effects vary with the frequency of the light
+vibrations; they are in this respect analogous to the effects of
+`anomalous dispersion' (see Glazebrook, \textit{Report on Optical
+\index{Glazebrook@Glazebrook, \textit{Report on Optical Theories}}%
+Theories, B.~A. Report}, 1885), which have been accounted for by
+assuming that the molecules of the substance through which the
+light passes have free periods of vibration comparable with the
+frequency of the light vibrations. The energy absorbed by such
+molecules is then a function of the frequency of the light
+vibrations, and the optical character of the medium cannot be
+fixed by one or two constants, such as the specific inductive
+%% -----File: 436.png---Folio 422-------
+capacity or the specific resistance; we require to know in
+addition the free periods of the molecules.
+
+\Article{357} We now return to the case of the magnetic metals; the
+question arises whether or not these substances retain their
+magnetic properties under magnetic forces which oscillate as
+rapidly as those in a wave of light. We have seen (\artref{286}{Art.~286})
+that iron retains its magnetic properties when the magnetic
+forces make about one million vibrations per second; in the
+light waves, however, the magnetic forces are vibrating more
+than five hundred million times faster than this, and the only
+means we have of testing whether magnetic substances retain
+their properties under such circumstances is to examine the light
+reflected from or transmitted through such bodies. When we
+do this, however, we labour under the disadvantage that, as the
+preceding investigation shows, the theory of metallic reflection
+is incomplete, so that the conclusions we may come to as the
+results of this theory are not conclusive. Such evidence as we
+have, however, tends to show that iron does not retain its
+magnetic properties under such rapidly alternating magnetic
+forces. An example of such evidence is furnished by equation~(\hyperref[eqn:353.3s]{3*}),
+\artref{353}{Art.~353}. We see from that equation that if $\mu$~for light waves
+in iron were very large, the intensity of the light reflected from
+iron would be very nearly the same as that of the incident
+light, in other words iron would have a very high reflecting
+power. The reverse, however, seems to be true; thus Drude
+(\textit{Wied.\ Ann}.\ 39, p.~549, 1890) gives the following numbers as
+representing the reflective powers of some metals for yellow
+light:---
+\begin{center}
+\setlength{\tabcolsep}{12pt}
+\begin{tabular}{c c  c  c  c  c }
+\tabletextsize Silver. &
+\tabletextsize Gold.   &
+\tabletextsize Copper. &
+\tabletextsize Iron.   &
+\tabletextsize Steel.  &
+\tabletextsize Nickel. \\
+$95.3$    &$85.1$  &$73.2$    &$56.1$  &$58.5$   &$62.0$
+\end{tabular}
+\end{center}
+Rubens (\textit{Wied.\ Ann.}~37, p.~265, 1889) gives for the same metals
+\index{Rubens, metallic reflection}%
+the following numbers:---
+\begin{center}
+\setlength{\tabcolsep}{12pt}
+\begin{tabular}{c c c c c }
+\tabletextsize Silver. &
+\tabletextsize Gold.   &
+\tabletextsize Copper. &
+\tabletextsize Iron.   &
+\tabletextsize Nickel. \\
+$90.3$     &$71.1$   &$70.0$     &$56.1$   &$62.1$
+\end{tabular}
+\end{center}
+The near agreement of the numbers found by these two experimenters
+seems to show that the smallness of the reflection observed
+from iron could not be due to any accidental cause such as
+want of polish. Another reason for believing that iron does
+not manifest magnetic properties under the action of light
+waves, is that there is nothing exceptional in the position of
+%% -----File: 437.png---Folio 423-------
+iron with respect to the optical constants of metals in the table
+given in \artref{353}{Art.~353}. The theory of metallic reflection is however
+so far from accounting for the facts that we cannot attach much
+weight to considerations based on it. The only conclusion we
+can come to is the negative one, that there is no evidence to
+show that iron does retain its magnetic properties for the light
+vibrations.
+
+\Subsection{The change in Phase produced by the Transmission of Light
+through thin Films of Metal.}
+\index{Films, transmission of light through}%
+\index{Kundt, transmission of light through thin films@\subdashone transmission of light through thin films}%
+\index{Light, xtransmission of through thin films@\subdashone transmission of through thin films}%
+\index{Metals, xtransmissions of light through thin films of@\subdashone transmissions of light through thin films of}%
+\index{Quincke, transmission of light through thin films}%
+
+\Article{358} Quincke (\textit{Pogg.\DPtypo{,}{}\ Ann.}\ 120, p.~599, 1863) investigated the
+change in phase produced when light passed through thin silver
+plates, and found that in many cases the phase was accelerated,
+the effect being the same as if the velocity of light through
+silver was greater than that through air. Kundt (\textit{Phil.\ Mag.}\
+[5], 26, p.~1, 1888), in a most beautiful series of experiments,
+measured the deviation of a ray passing through a small metal
+\emph{prism}, and found that when the prism was made of silver,
+gold, or copper, the deviation was towards the thin end. With
+platinum, nickel, bismuth, and iron prisms the deviation was,
+on the other hand, towards the thick end. We can readily find
+on the electromagnetic theory of light the change in phase produced
+when the light passes through a thin film of metal. The
+equation~(\eqnref{351}{11}) of \artref{351}{Art.~351} shows, that if the incident wave (supposed
+for simplicity to be travelling at right angles to the film)
+is represented by
+\[
+\epsilon^{\iota(ax+pt)},
+\]
+the emergent wave will be
+\[
+\frac{4a(a'/\mu)\, \epsilon^{\iota ha}\, \epsilon^{\iota(ax+pt)}}
+     {(a'^2/\mu^2 + a^2)(\epsilon^{\iota ha'} - \epsilon^{-\iota ha')}
+            + 2a(a'/\mu)(\epsilon^{\iota ha'} + \epsilon^{-\iota ha')}},
+\]
+or if the film is so thin that $ha'$ is a small quantity, the emergent
+wave is equal to
+\[
+\frac{\epsilon^{\iota ha}\, \epsilon^{\iota(ax+pt)}}
+     {1 + \frac{1}{2}\, \dfrac{\iota h\mu}{a} \left(\dfrac{a'^2}{\mu^2} + a^2\right)}.
+\]
+
+Now, since in this case $b = 0$, we have by equation~(\eqnref{353}{4}) of \artref{353}{Art.~353}
+$\dfrac{a'^2}{a^2} = R^2\, \epsilon^{2\iota\alpha}$, hence the emergent wave is equal to
+\[
+\frac{\epsilon^{\iota ha}\, \epsilon^{\iota(ax+pt)}}
+     {1 + \frac{1}{2} \iota h\mu a \left\{\dfrac{R^2\, \epsilon^{2\iota\alpha}}{\mu^2} + 1\right\}},
+\]
+%% -----File: 438.png---Folio 424-------
+or, neglecting squares and higher powers of~$h$, this is equal to
+\begin{multline*}
+\epsilon^{\frac{1}{2} haR^2 \sin 2\alpha\mu^{-1}}\,
+\epsilon^{\iota ha}\,
+\epsilon^{-\frac{1}{2} \iota h\mu a (1 + R^2 \cos 2\alpha/\mu^2)}\,
+\epsilon^{\iota(ax+pt)} \\
+= \epsilon^{\frac{1}{2} haR^2 \sin 2\alpha\mu^{-1}}\,
+\epsilon^{\iota ha \left(1 - \frac{\mu}{2} \{1 + R^2 \cos 2\alpha/\mu^2\}\right)}\,
+\epsilon^{\iota(ax+pt)},
+\end{multline*}
+hence the acceleration of phase expressed as a length is equal to
+\[
+h\left(1 - \frac{\mu}{2} \{1 + R^2 \cos 2\alpha/\mu^2\}\right),
+\]
+or for non-magnetic substances to
+\[
+\tfrac{1}{2} h(1 - R^2 \cos 2\alpha).
+\]
+In the interpretation of this result we are beset with difficulties,
+whether we take $R^2 \cos 2\alpha$ as determined by the electromagnetic
+theory, or whether we take it as given by Drude's experiments.
+In the former case $R^2 \cos 2\alpha$ is positive, so that the acceleration
+cannot be greater than~$h/2$, or the apparent speed of light
+through the metal cannot be greater than twice that through
+air; this is not in accordance with Kundt's experiments on silver
+and gold. If, on the other hand, we take Drude's values for
+$R^2 \cos 2\alpha$, since these are negative for all metals, the apparent
+velocity of light through a film of any metal ought to be more
+than double that through air; this again is not in accordance with
+Kundt's observations, according to which the apparent velocity
+of light through films of metals other than gold, silver, or
+copper is less than that through air. We might have anticipated
+that such a discrepancy would arise, for we have assumed in
+deducing the expression for the transmitted ray that the electromotive
+intensity parallel to the surface of the metal is continuous.
+Now if we suppose that the light vibrations have
+periods comparable with periods of the molecules of the metal,
+the electromotive intensity in the metal will arise from two
+causes. The first is due to magnetic induction, this will be continuous
+with that due to the same cause in the air; the second is
+due to the reaction of the molecules of the metal on the medium
+conveying the light. Now there does not seem to be any reason
+to assume that this part of the electromotive intensity should
+be continuous as we pass from the air which does not exhibit
+anomalous dispersion to the metal which does. The electromotive
+intensity parallel to the boundary is thus probably
+discontinuous, and we could not therefore expect a formula
+obtained by the condition that this intensity was continuous to
+be in accordance with experiment.
+%% -----File: 439.png---Folio 425-------
+
+\Section{Reflection of Electromagnetic Waves from Wires.}
+
+\Subsection{Reflection from a Grating.}
+\index{Grating, reflection of electromagnetic waves from}%
+\index{Reflection of electromagnetic waves from a grating@\subdashtwo electromagnetic waves from a grating}%
+
+\Article{359} We shall now consider the reflection of electromagnetic
+waves from a grating consisting of similar and parallel metallic
+wires, whose cross-sections we leave for the present indeterminate,
+arranged at equal intervals, the axes of all the wires being
+in one plane, which we shall take as the plane of~$yz$, the axis of~$z$
+being parallel to the wires: the distance between the axes of
+two adjacent wires is~$a$. We shall suppose that a wave in which
+the electromotive intensity is parallel to the wires, and whose
+front is parallel to the plane of the grating, falls upon the wires.
+The electromotive intensity in the incident wave may be represented
+by the real part of $A\, \epsilon^{\frac{\iota 2 \pi}{\lambda} (Vt+x)}$, $x$~being measured from
+the plane of the grating towards the advancing wave. The
+incidence of this wave will induce currents in the wires, and
+these currents will themselves produce electromotive intensities
+parallel to~$z$ in the region surrounding them; these intensities
+will evidently be expressed by a periodic function of~$y$ of such
+a character that when $y$~is increased by~$a$ the value of the
+function remains unchanged. If we make the axis of~$z$ coincide
+with the axis of one of the wires, the electromotive intensity
+will evidently be an even function of~$y$. Thus~$E_2$, the electromotive
+intensity due to the currents in the wire, will be given by
+an equation of the form
+\[
+E_2 =\tsum A_m \cos \frac{2 m \pi y}{a}\, \epsilon^{\iota n x}\, \epsilon^{\frac{\iota 2 \pi}{\lambda} Vt},
+\]
+where $m$~is an integer.
+
+Since the electromotive intensity satisfies the equation
+\[
+\frac{d^{2} E}{dx^{2}} + \frac{d^{2} E}{dy^{2}}
+  = \frac{1}{V^{2}}\, \frac{d^{2} E}{dt^{2}},
+\]
+we have
+\[
+n^{2}  =  - \frac{4 \pi^{2} m^{2}}{a^{2}}  +  \frac{4 \pi^{2}}{\lambda^{2}}.
+\]
+
+We shall assume that the distance between the wires of the
+grating is very small compared with the length of the wave;
+thus, unless $m$~is zero, the first term on the right-hand side of the
+above equation will be very large compared with the second, so
+%% -----File: 440.png---Folio 426-------
+that when $m$~is not zero we may put
+\[
+n =  ± \frac{\iota 2 \pi m}{a},
+\]
+while when $m$ is zero
+\[
+n = -\frac{2 \pi}{\lambda},
+\]
+the minus sign being taken so as to represent a wave diverging
+from the wires. Substituting these values we find that when
+$x$~is positive,
+\[
+E_2 = A_0\, \epsilon^{\frac{\iota 2 \pi}{\lambda}\bigl(Vt-(x+\alpha)\bigr)}
+  + \tsum_{m=1}^{m=\infty} A_m\, \epsilon^{-\frac{2\pi m}{a} x}
+    \cos{\frac{2 \pi m y}{a}}\, \epsilon^{\frac{\iota 2 \pi}{\lambda} Vt},
+\]
+where $\alpha$~is a constant.
+
+When the rate of alternation is so rapid that the waves are
+only a few metres in length the electromotive intensity at the surface
+of the metal wire must vanish, see Arts.\ \artref{300}{300}~and~\artref{301}{301}; hence
+if $E_1$~is the electromotive intensity in the incident wave, $E_1 + E_2$
+must vanish at the surface of the wire. Near the grating however
+$x/\lambda$~will be small; hence we may put, writing
+\[
+A_m' \cos\frac{2 \pi}{\lambda} Vt + B_m' \sin\frac{2 \pi}{\lambda} Vt
+\quad \text{for} \quad
+A_m\, \epsilon^{\iota 2 \pi Vt/\lambda},
+\]
+\begin{multline*}
+E_1 + E_2 = (A +A_0) \cos{\frac{2 \pi}{\lambda} Vt}
+  + \bigl(A_0(x + \alpha) - Ax\bigr) \frac{2 \pi}{\lambda} \sin{\frac{2 \pi}{\lambda}} Vt \\
+  + \tsum \epsilon^{-\frac{2 \pi m x}{a}} \cos{\frac{2\pi m y}{a}}
+    \left( A_m' \cos{\frac{2 \pi}{\lambda}} Vt
+         + B_m' \sin{\frac{2 \pi}{\lambda}} Vt \right).
+\end{multline*}
+
+Now in Maxwell's \textit{Electricity and Magnetism}, Vol.~i. Art.~203,
+it is shown that the expression
+\[
+C \log \left\{1 - 2\epsilon^{-\frac{2 \pi x}{a}} \cos{\frac{2 \pi y}{a}} + \epsilon^{-\frac{4 \pi x}{a}} \right\} + Dx,
+\]
+where $C$~and~$D$ are constants, is constant over a series of equidistant
+parallel wires, whose axes are at a distance~$a$ apart and
+whose cross-section is approximately circular. The logarithm
+can be expanded in the form
+\[
+-2 C \tsum \frac{1}{m}\, \epsilon^{-\frac{2 m \pi x}{a}} \cos{\frac{2 m \pi y}{a}}.
+\]
+
+Now in the expression for $E_1 + E_2$ put
+\[
+A + A_0 = 0,\quad  A_m = 0,\quad  B_m = -\frac{2 C}{m},
+\]
+%% -----File: 441.png---Folio 427-------
+then
+\begin{multline*}
+E_1 + E_2 = A \cos{\frac{2 \pi}{\lambda}} (Vt + x)
+          - A \cos{\frac{2 \pi}{\lambda}} \bigl(Vt - (x+\alpha)\bigr)\\
+  + C \log \left( 1 - 2 \epsilon^{-\frac{2 \pi x}{a}} \cos{\frac{2 \pi y}{a}} + \epsilon^{-\frac{4 \pi x}{a}} \right) \sin{\frac{2 \pi}{\lambda}} Vt,
+\end{multline*}
+hence near the grating where $x/\lambda$~is small
+\begin{multline*}
+E_1 + E_2 = \sin{\frac{2 \pi}{\lambda}} Vt
+  \left\{-A \frac{2 \pi x}{\lambda} - A \frac{2 \pi}{\lambda} (x+\alpha) \right.\\
+  \left. + C \log \left( 1 - 2 \epsilon^{-\frac{2 \pi x}{a}} \cos{\frac{2 \pi y}{a}} + \epsilon^{-\frac{4 \pi x}{a}} \right) \right\},
+\end{multline*}
+and we see by Maxwell's result that the quantity inside the
+bracket has a constant value over the surface of the wires;
+hence, if we make this value zero, we shall have satisfied the
+conditions of the problem. Let $2c$~be the diameter of any one
+of the wires in the plane of the grating, then when $x = 0$ and
+$y = c$ the expression inside the bracket must vanish, hence
+\[
+-A \frac{2 \pi}{\lambda} \alpha + C \log 4 \sin^{2} \frac{\pi c}{a} = 0.
+\]
+
+To find another relation between $A$,~$C$, and~$\alpha$ we must consider
+the equation to the cross-section of the wire at the origin, viz.,
+\[
+-A \frac{2 \pi}{\lambda} (2x + \alpha)
+  + C \log \left( 1 - 2\epsilon^{-\frac{2 \pi x}{a}} \cos{\frac{2 \pi y}{a}} + \epsilon^{-\frac{4 \pi x}{a}} \right) = 0,
+\]
+or substituting for~$C$ its value in terms of~$A$,
+\[
+\left( \frac{2 x}{\alpha} + 1 \right) \log \left\{4 \sin^{2} \frac{\pi c}{a} \right\}
+  = \log \left( 1 - 2 \epsilon^{-\frac{2 \pi x}{a}} \cos \frac{2 \pi y}{a} + \epsilon^{-\frac{4 \pi x}{a}} \right).
+\Tag{1}
+\]
+
+If $d$~is the value of~$x$ when $y = 0$,
+\[
+\alpha = 2 d \frac{\log 2 \sin \dfrac{\pi c}{a}}{\log\left\{\dfrac{1-\epsilon^{-\frac{2 \pi d}{a}}}{2 \sin \dfrac{\pi c}{a}} \right\} }.
+\Tag{2}
+\]
+
+When $c = d$, this equation becomes, since $c/a$~is small,
+\[
+\alpha = -\frac{2 a}{\pi} \log 2 \sin \frac{\pi c}{a}.
+\]
+
+The expression for~$E_2$ consists of two parts, one of which is
+\[
+-A\, \epsilon^{\frac{\iota 2 \pi}{\lambda}(Vt - (x+\alpha))},
+\]
+%% -----File: 442.png---Folio 428-------
+which represents a reflected wave equal in intensity to the incident
+one, but whose phase is changed by reflection by $\left(\frac{1}{2} \lambda -\alpha\right)$,
+where~$\alpha$ is given by~(\eqnref{359}{2}) and depends upon the size of the wires
+and their distance apart. The other part of the expression for~$E_2$
+is
+\[
+C \log \left( 1 - 2 \epsilon^{-\frac{2 \pi x}{a}} \cos \frac{2 \pi y}{a} + \epsilon^{-\frac{4 \pi x}{a}} \right).
+\]
+This is inappreciable at a distance from the grating $4$~or $5$~times
+the distance between the wires, hence the reflection, at some
+distance from the grating, is the same, except for the alteration
+in phase as from a continuous metallic surface.
+
+\Article{360} If the electromotive intensity had been at right angles to
+the wires the reflection would have been very small; thus a
+grating of this kind will act like a polariscope, changing either
+by reflection or transmission an \DPtypo{unpolarised}{unpolarized} set of electrical
+vibrations into a \DPtypo{polarised}{polarized} one. When used to produce \DPtypo{polarisation}{polarization}
+by transmission we may regard it as the electrical analogue
+of a plate of tourmaline crystal.
+
+\Subsection{Scattering of Electromagnetic Waves by a Metallic Wire.}
+\index{Waves, electromagnetic, scattering of from cylinders@\subdashtwo scattering of from cylinders}%
+\index{Cylinder, scattering of electromagnetic waves by@\subdashone scattering of electromagnetic waves by}%
+\index{Electromagnetic waves, xscattering of by a cylinder@\subdashtwo scattering of by a cylinder}%
+\index{Light, zscattering of by cylinders@\subdashone scattering of by cylinders}%
+\index{Scattering of electromagnetic waves by a cylinder}%
+
+\Article{361} The scattering produced when a train of plane electromagnetic
+waves impinges on an infinitely long metal cylinder,
+whose axis is at right angles to the direction of propagation of
+the waves and whose diameter is small compared with the wave
+length, can easily be found as follows:---
+
+We shall begin with the case where the electromotive intensity
+in the incident wave is parallel to the axis of the cylinder, which
+we take as the axis of~$z$; the axis of~$x$ being at right angles to
+the fronts of the incident waves.
+
+Let $\lambda$~be the wave length, then $E_1$, the electromotive intensity
+in the incident waves, may be represented by the equation
+\[
+E_1 = \epsilon^{\frac{\iota 2 \pi}{\lambda}(Vt+x)},
+\]
+where the real part of the right-hand side is to be taken. The
+positive direction of~$x$ is opposite to that in which the waves are
+travelling. In the neighbourhood of the cylinder $x/\lambda$~is small,
+so that we may put
+\[
+E_1 = \epsilon^{\frac{\iota 2 \pi}{\lambda} Vt} \left( 1+\iota x \frac{2 \pi}{\lambda} \right)
+\]
+%% -----File: 443.png---Folio 429-------
+approximately, or if $r$~and~$\theta$ are the polar coordinates of the point
+where the intensity is~$E_1$,
+\[
+E_1 = \epsilon^{\frac{\iota 2 \pi}{\lambda} Vt} \left( 1+\iota \frac{2 \pi}{\lambda} r \cos \theta \right).
+\]
+
+Let $E_2$~be the electromotive intensity due to the currents
+induced in the cylinder, then $E_2$~satisfies the differential equation
+\begin{align*}
+\frac{d^{2}E_2}{dr^{2}}
+  + \frac{1}{r}\, \frac{d E_2}{dr}
+  + \frac{1}{r^{2}}\, \frac{d^{2} E_2}{d\theta^{2}}
+  &= \frac{1}{V^{2}}\, \frac{d^{2} E_2}{dt^{2}} \\
+  &= -\frac{4 \pi^{2}}{\lambda^{2}}\, E_2,
+\end{align*}
+or if $E_2$~varies as~$\cos n\theta$,
+\[
+\frac{d^{2} E_2}{dr^{2}}
+  + \frac{1}{r}\, \frac{d E_2}{dr}
+  + \left( \frac{4 \pi^{2}}{\lambda^{2}} - \frac{n^{2}}{r^{2}} \right) E_2 = 0.
+\]
+
+The solution of which outside the cylinder is
+\[
+E_2 = A_n \cos{n\theta} K_n \left( \frac{2 \pi}{\lambda} r \right) \epsilon^{\frac{\iota 2 \pi}{\lambda} Vt},
+\]
+where $K_n$~represents the `external' Bessel's function of the $n$\textsuperscript{th}~order.
+
+Thus
+\begin{multline*}
+E_2 = \left\{A_0 K_0 \left( \frac{2 \pi}{\lambda} r \right)
+  + A_1 \cos \theta K_1 \left( \frac{2 \pi}{\lambda} r \right) \right. \\
+   \left. + A_2 \cos 2  \theta K_2 \left( \frac{2 \pi}{\lambda} r \right) + \ldots \right\} \epsilon^{\frac{\iota 2\pi}{\lambda} Vt}.
+\end{multline*}
+
+Now since the cylinder is a good conductor, the total tangential
+electromotive intensity must vanish over its surface, see Arts.\ \artref{300}{300}~and~\artref{301}{301}.
+Hence if $c$~is the radius of the cylinder, $E_1 + E_2 = 0$
+when $r = c$; from this condition we get
+\[
+A_0 = -\frac{1}{K_0 \left( \dfrac{2 \pi}{\lambda} c \right) }, \quad
+A_1 = -\frac{\iota 2 \pi c}{\lambda K_1 \left( \dfrac{2 \pi}{\lambda} c \right) }, \quad
+A_2 = A_3 = \ldots = 0.
+\]
+
+Thus
+\[
+E_2 = \left\{-\frac{K_0 \left( \dfrac{2 \pi}{\lambda} r \right)}
+                    {K_0 \left( \dfrac{2 \pi}{\lambda} c \right)}
+  - \frac{\iota 2 \pi c}{\lambda} \cos \theta
+    \frac{K_1 \left( \dfrac{2 \pi}{\lambda} r \right)}
+         {K_1 \left( \dfrac{2 \pi}{\lambda} c \right)} \right\}
+  \epsilon^{\frac{\iota 2 \pi}{\lambda} Vt}.
+\]
+
+\Article{362} Let us first consider the effect of the cylinder on the
+lines of magnetic force in its neighbourhood. If $\alpha$,~$\beta$ are the
+%% -----File: 444.png---Folio 430-------
+components of the magnetic force parallel to the axes of $x$~and~$y$
+respectively, $E$~the total electromotive intensity, then
+\begin{align*}
+\frac{dE}{dx} &= \frac{d\beta}{dt} = \iota \frac{2\pi}{\lambda} V \beta, \\
+\frac{dE}{dy} &= -\frac{d\alpha}{dt} = - \iota \frac{2\pi}{\lambda} V \alpha.
+\end{align*}
+
+Thus the direction of the magnetic force will be tangential to
+the curves over which $E$~is constant, the equations to the lines of
+magnetic force in the neighbourhood of the cylinder are therefore
+\[
+\left\{\left(
+  1 - \frac{K_0 \left(\dfrac{2\pi}{\lambda} r\right)}
+           {K_0 \left(\dfrac{2\pi}{\lambda} c\right)}\right)
+  + \iota \frac{2\pi}{\lambda} \cos{\theta}
+    \left( r - c \frac{K_1 \left(\dfrac{2\pi}{\lambda} r\right)}
+                      {K_1 \left(\dfrac{2\pi}{\lambda} c\right)}\right) \right\}
+  \epsilon^{\frac{\iota 2 \pi}{\lambda}Vt} = C,
+\]
+where $C$~is independent of $r$~and~$\theta$.
+
+Now $2\pi c/\lambda$~is by hypothesis very small, and when $x$~is small
+then, by \artref{261}{Art.~261}, the values of $K_0$~and~$K_1$ are given approximately
+by the equations
+\begin{align*}
+K_0(x) &= \log (2\gamma/x), \\
+K_1(x) &= - K_0'(x) = \frac{1}{x},
+\end{align*}
+where $\gamma$~is Euler's constant and $\log \gamma$ is equal to~$.5772157$.
+
+In the neighbourhood of the cylinder $r/\lambda$~is small as well as~$c/\lambda$,
+so that in this region the equations to the lines of magnetic
+force are, approximately,
+\[
+\frac{\log{(r/c)}}{\log{(\gamma\lambda/\pi c)} } \cos{\frac{2\pi}{\lambda} Vt}
+  + \frac{2\pi}{\lambda} \cos{\theta \frac{(c^2 - r^2)}{r}} \sin{\frac{2\pi}{\lambda} Vt} = C.
+\]
+
+In this expression the coefficient of $\cos (2\pi Vt/\lambda)$ is very large
+compared with that of $\sin (2\pi Vt/\lambda)$, so that unless $2\pi Vt / \lambda$ is an
+odd multiple of~$\pi/2$, that is, unless the intensity in the incident
+wave at the axis of the cylinder vanishes, the equations to the
+lines of magnetic force are
+\[
+\log{(c/r)} = \text{a constant},
+\]
+so that these lines are circles concentric with the cylinders.
+
+When $2\pi Vt/\lambda$~is an odd multiple of~$\pi/2$, the lines of magnetic
+force are given by the equation
+\[
+\cos \theta \frac{(c^2 - r^2)}{r} = C,
+\]
+%% -----File: 445.png---Folio 431-------
+or in Cartesian coordinates
+\[
+x \left\{c^2 - (x^2 + y^2) \right\} = C (x^2 + y^2);
+\]
+these curves are shown in \figureref{fig121}{Fig.~121}.
+
+\includegraphicsmid{fig121}{Fig.~121.}
+
+\Article{363} Since the direction of motion of the Faraday tubes is at
+right angles to themselves and to the magnetic force, when the
+lines of magnetic force near the cylinder are circles, these tubes
+will, in the neighbourhood of the cylinder, move radially, the
+positive tubes (i.e.~those parallel to the tubes in the incident
+wave) moving inwards, the negative ones outwards. In the
+special case where the electromotive intensity vanishes at the
+axis of the cylinder, the incident wave throws tubes of one sign
+into the half of the cylinder in front, where $x$~is positive, and
+tubes of opposite sign into the half in the rear, where $x$~is
+negative; in this case, if the positive tubes in the neighbourhood
+of the cylinder are moving radially inwards in front, they are
+moving radially outwards in the rear and \textit{vice versâ}; there are
+in this case but few tubes near the equatorial plane, and the
+motion of these is no longer radial.
+
+\Article{364} When the distance from the cylinder is large compared
+with the wave length, we have
+\begin{align*}
+K_0 \left(\frac{2\pi}{\lambda} r \right) &=  \tfrac{1}{2} \iota^{\frac{1}{2}} \frac{\epsilon^{-\iota 2 \pi r / \lambda}}{(r/\lambda)^{\frac{1}{2}}}, \\
+K_1 \left(\frac{2\pi}{\lambda} r \right) &= -\tfrac{1}{2} \iota^{\frac{3}{2}} \frac{\epsilon^{-\iota 2 \pi r / \lambda}}{(r/\lambda)^{\frac{1}{2}}}.
+\end{align*}
+%% -----File: 446.png---Folio 432-------
+
+Thus in the wave `scattered' by the cylinder
+\[
+E_2 = - \frac{\epsilon^{-\iota \frac{2\pi}{\lambda} \left(r-Vt-\frac{\lambda}{8}\right)}}
+             {2(r/\lambda)^{\frac{1}{2}}}
+  \left\{\frac{1}{\log(\gamma\pi\lambda/c)} + \frac{4\pi^2c^2}{\lambda^2} \cos \theta\right\}.
+\]
+
+Thus in this case, as we should expect, the part of the scattered
+wave which is independent of the azimuth is very much larger
+than the part which varies with~$\theta$, so that there is no direction in
+which the intensity of the scattered light vanishes. In this respect
+the metal cylinder resembles one made of a non-conductor, the
+effect of which on a train of waves has been investigated by Lord
+Rayleigh (\textit{Phil.\ Mag.}\ [5], 12, p.~98, 1881): there are however
+\index{Rayleigh, Lord, scattering of light by fine particles@\subdashtwo scattering of light by fine particles}%
+some important differences between the two cases; in the first
+place we see that since $c$~occurs in the leading term only as a
+logarithm, the amount of light scattered by the cylinder changes
+very slowly with the dimensions of the cylinder, while in the
+light scattered from a dielectric cylinder the electromotive intensity
+in the scattered wave is proportional to the area of the
+cross-section of the cylinder. Again, when the cylinder is a good
+conductor the electromotive intensity in the scattered wave, if
+we regard the logarithmic term as approximately constant, varies
+as~$\lambda^{\frac{1}{2}}$ and so increases with the wave length, while when the
+cylinder is an insulator the electromotive intensity varies as~$\lambda^{-\frac{3}{2}}$,
+so that the scattering \emph{decreases} rapidly as the length of the wave
+increases. The most interesting case of this kind is when the
+wave incident on the cylinder is a wave of light; in this case the
+theory indicates that the light scattered by the metallic cylinder
+would be slightly reddish, while that from the insulating cylinder
+would be distinctly blue; the blue in the latter case would be
+much more decided than the red of the previous one, since the
+variation of the intensity of the scattered light with the wave
+length is much more rapid when the cylinder is an insulator
+than when it is a good conductor.
+
+\Article{365} We shall now proceed to consider the case when the
+electromotive intensity in the incident wave is at right angles to
+the axis of the cylinder. This case is of more interest than the
+preceding because the general features of the results obtained
+will apply to the scattering of light by particles limited in every
+direction; it is thus representative of the scattering by small
+particles in general, while the peculiarities of the case discussed
+%% -----File: 447.png---Folio 433-------
+in the preceding \artref{364}{article} were due to the cylindrical shape of the
+obstacle. The only case to which the results of this article would
+not be applicable without further investigation is that in which
+the particles are highly magnetic, and we shall find that even
+this case constitutes no exception since our results do not involve
+the magnetic permeability of the cylinder.
+
+As the electromotive intensity is at right angles to the axis of
+the cylinder, the magnetic force will be parallel to the axis.
+
+Let the magnetic force~$H_1$ in the incident wave be expressed
+by the equation
+\[
+H_1 = \epsilon^{\frac{\iota 2\pi}{\lambda} (Vt+x)}.
+\]
+When $x$ which is equal to $r \cos \theta$ is small compared with~$\lambda$, this
+is approximately
+\[
+H_1 = \epsilon^{\frac{\iota 2\pi}{\lambda} Vt}
+  \left\{1 - \frac{\pi^2}{\lambda^2} r^2
+         + \frac{\iota 2\pi}{\lambda} r \cos \theta
+         - \frac{\pi^2}{\lambda^2} r^2 \cos 2\theta\right\}.
+\]
+
+Since $H$, the magnetic force, satisfies the differential equation
+\[
+\frac{d^2H}{dx^2} + \frac{d^2H}{dy^2} = \frac{1}{V^2}\, \frac{d^2H}{dt^2},
+\]
+the magnetic force~$H_2$ due to the currents induced in the cylinder
+may be expressed by the equation
+\[
+H_2 = \epsilon^{\frac{\iota 2\pi}{\lambda} Vt}
+  \left\{A_0K_0\left(\frac{2\pi}{\lambda} r\right)
+        + A_1 \cos  \theta K_1\left(\frac{2\pi}{\lambda} r\right)
+        + A_2 \cos 2\theta K_2\left(\frac{2\pi}{\lambda} r\right)\right\},
+\]
+where $A_0$,~$A_1$ and~$A_2$ are arbitrary constants.
+
+The condition to be satisfied at the boundary of the cylinder
+is that the tangential electromotive intensity at its surface should
+vanish. In this case we have, however,
+\[
+\frac{d}{dr}(H_1 + H_2) = 4\pi \text{ (intensity of current at right angles to~$r$)}.
+\]
+
+The current in the dielectric is a polarization current, and
+if $E$~is the tangential electromotive intensity, the intensity of
+this current at right angles to~$r$ is
+\[
+\frac{K}{4\pi}\, \frac{dE}{dt},
+\]
+which is equal to
+\[
+\frac{K}{4\pi}\, \frac{\iota 2\pi}{\lambda}\, VE.
+\]
+
+Thus the condition that $E$~should vanish at the surface is
+%% -----File: 448.png---Folio 434-------
+equivalent to the condition that
+\[
+\frac{d}{dr} (H_1 + H_2) = 0
+\]
+when $r = c$, $c$~being the radius of the cylinder.
+
+From this condition we get
+\begin{align*}
+-2c \frac{\pi^2}{\lambda^2} + A_0 \frac{d}{dc} K_0 \left(\frac{2\pi}{\lambda} c\right) &= 0, \\
+ \frac{\iota 2\pi}{\lambda} + A_1 \frac{d}{dc} K_1 \left(\frac{2\pi}{\lambda} c\right) &= 0, \\
+-2c \frac{\pi^2}{\lambda^2} + A_2 \frac{d}{dc} K_2 \left(\frac{2\pi}{\lambda} c\right) &= 0.
+\end{align*}
+
+Since $2\pi c/\lambda$ is very small and therefore approximately
+\begin{align*}
+K_0 \left(\frac{2\pi}{\lambda} c\right) &= \log\left(2\gamma\bigg{/}\frac{2\pi c}{\lambda}\right), \\
+K_1 \left(\frac{2\pi}{\lambda} c\right) &= \frac{\lambda}{2\pi c}, \\
+K_2 \left(\frac{2\pi}{\lambda} c\right) &= \frac{\lambda^2}{2\pi^2c^2},
+\end{align*}
+we get
+\begin{align*}
+A_0 &= -2\pi^2 \frac{c^2}{\lambda^2}, \\
+A_1 &= \iota 4\pi^2 \frac{c^2}{\lambda^2}, \\
+A_2 &= -2\pi^4 \frac{c^4}{\lambda^4}.
+\end{align*}
+
+Thus the magnetic force due to the currents induced in the
+cylinder is given by the equation
+\begin{multline*}
+H_2 = 2\pi^2 \frac{c^2}{\lambda^2}\, \epsilon^{\frac{\iota 2\pi}{\lambda} Vt}
+  \biggl\{-K_0\Bigl(\frac{2\pi}{\lambda} r\Bigr)
+         + 2\iota \cos \theta K_1\Bigl(\frac{2\pi}{\lambda} r\Bigr)\\
+         - \frac{\pi^2c^2}{\lambda^2} \cos 2\theta K_2 \Bigl(\frac{2\pi}{\lambda} r\Bigr)\biggr\}.
+\end{multline*}
+
+\Article{366} To draw the lines of electromotive intensity, we notice
+that if $ds$~is an element of a curve in the dielectric, $d(H_1 + H_2)/ds$
+is proportional to the electromotive intensity at right angles
+to~$ds$, so that the lines of electromotive intensity will be the
+lines
+\[
+H_1 + H_2 = \text{a constant}.
+\]
+%% -----File: 449.png---Folio 435-------
+
+When $r/\lambda$ is small, this condition leads to the equation
+\begin{multline*}
+\epsilon^{\frac{\iota 2\pi}{\lambda} Vt}
+  \left[1 - \frac{\pi^2}{\lambda^2} r^2
+    - \frac{2\pi^2c^2}{\lambda^2} K_0\left(\frac{2\pi}{\lambda} r\right)
+    + \frac{2\iota\pi}{\lambda} \cos \theta
+        \left\{r + \frac{2\pi}{\lambda} c^2 K_1 \left(\frac{2\pi}{\lambda} r\right)\right\} \right.  \\
+  \left. - \frac{\pi^2}{\lambda^2} \cos 2\theta
+    \left\{r^2 + \frac{2\pi^2c^4}{\lambda^2} K_2 \left(\frac{2\pi}{\lambda} r\right)\right\}\right] = C,
+\end{multline*}
+where $C$~is a constant.
+
+Substituting the approximate values of $K_0$,~$K_1$ and~$K_2$ this
+becomes
+\begin{multline*}
+\epsilon^{\frac{\iota 2\pi}{\lambda} Vt}
+  \left[1 - \frac{\pi^2}{\lambda^2} r^2
+    + \frac{2\pi^2c^2}{\lambda^2} \log(\pi r/\gamma\lambda)
+    + \frac{2\iota\pi}{\lambda} \cos \theta \frac{(r^2+c^2)}{r}  \right. \\
+    \left. - \frac{\pi^2}{\lambda^2} \cos 2\theta \left(r^2 + \frac{c^4}{r^2}\right)\right] = C.
+\end{multline*}
+Except when $\epsilon^{\frac{\iota 2\pi}{\lambda} Vt/\lambda}$ is wholly real, i.e.~except when the rate
+of variation of the magnetic force in the incident wave at the axis
+of the cylinder vanishes, by far the most important term is that
+which contains $\cos \theta$, so that the equations to the lines of electromotive
+intensity are
+\[
+\frac{c^2 + r^2}{r} \cos \theta = \text{a constant} = C', \text{ say.}
+\]
+
+\includegraphicsmid{fig122}{Fig.~122.}
+
+The lines of electromotive intensity are represented in \figureref{fig122}{Fig.~122}.
+
+At the times when $\epsilon^{\iota 2\pi Vt/\lambda}$ is wholly real, the lines are approximately
+circles concentric with the cross-section of the
+cylinder, since in this case the term involving the logarithm is
+the most important of the variable terms.
+%% -----File: 450.png---Folio 436-------
+
+\Article{367} When $r$~is large compared with~$\lambda$, we find by introducing
+the values of the $K$~functions when the argument is very
+large, viz.\
+\begin{gather*}
+\begin{aligned}
+K_0(x) & =  \iota^{\frac{1}{2}} \left( \frac{\pi}{2 x} \right)^{\frac{1}{2}} \epsilon^{-\iota x},\\
+K_1(x) & = -\iota^{\frac{3}{2}} \left( \frac{\pi}{2 x} \right)^{\frac{1}{2}} \epsilon^{-\iota x},
+\end{aligned} \\
+H_2 = -\frac{\pi^{2} c^{2}}{r^{\frac{1}{2}}\lambda^{\frac{3}{2}}}\,
+  \epsilon^{\frac{\iota 2 \pi}{\lambda}\left( Vt-r+ \frac{\lambda}{8} \right) }
+  (1 + 2 \cos \theta),
+\end{gather*}
+retaining only the lowest powers of~$c/\lambda$.
+
+Thus the magnetic force in the scattered wave vanishes when
+$2 \cos \theta = -1$, or in a direction making an angle of~$120°$ with the
+incident ray. When the wave is scattered by an insulating
+cylinder Lord Rayleigh (l.c.)\ found that the magnetic intensity
+in the scattered ray was expressed by a similar formula \emph{with the
+exception that the factor $(1 + 2 \cos \theta)$ was replaced by $\cos \theta$}. Thus,
+if we take the case where the incident wave is a luminous one,
+the scattered light will vanish in the direction of the electric
+displacement when the particles are insulators, while it will
+vanish in a direction making an angle of~$30°$ with this direction
+if the particles are metallic. If the incident light is not
+polarized, then with metallic particles the scattered light will
+be completely polarized in a direction making~$120°$ with the
+direction of propagation of the incident light, while if the particles
+are insulators the direction in which the polarization is
+complete is at right angles to the direction of the incident light.
+The observations of Tyndall, Brücke, Stokes, and Lord Rayleigh
+afford abundant proof of the truth of the last statement: but no
+experiments seem to have been published on the results of the
+reflection of light from small metallic particles.
+
+\Article{368} The preceding results have also an important application
+to the consideration of the influence of the size of the reflector on
+the intensity of reflected electromagnetic waves. When the
+electromotive intensity is parallel to the axis of the cylinder, the
+most important term in the expression for the reflected wave
+only involves the radius of the cylinder as a logarithm, it will
+thus only vary slowly with the radius, so that in this case the
+size of the cylinder is of comparatively little importance: hence
+we may conclude that we shall get good reflection if the length
+%% -----File: 451.png---Folio 437-------
+of the reflector measured in the direction of the electromotive
+intensity is considerable, whatever may be the breadth of the
+reflector at right angles to the electromotive intensity. On the
+other hand, when the electromotive intensity is at right angles
+to the axis of the cylinder, the electromotive intensity in the
+scattered wave increases as the square of the radius of the
+cylinder, so that in this case the size of the reflector is all important.
+These results are confirmed by Trouton's experiments
+\index{Trouton, influence of size of reflector on Hertz's experiments@\subdashone influence of size of reflector on Hertz's experiments}%
+on `The Influence the Size of the Reflector exerts in Hertz's
+Experiment,' \textit{Phil.\ Mag.}~[5], 32, p.~80, 1891.
+
+\Subsection{On the Scattering of Electric Waves by Metallic Spheres.}
+\index{Electromagnetic waves, xscattering of by a metal sphere@\subdashtwo scattering of by a metal sphere}%
+\index{Light, zscattering of by metallic spheres@\subdashtwo of by metallic spheres}%
+\index{Scattering of electromagnetic waves by a sphere@\subdashone of electromagnetic waves by a sphere}%
+\index{Sphere, zzscattering of light by@\subdashone scattering of light by}%
+\index{Waves, electromagnetic, scattering of from spheres@\subdashtwo scattering of from spheres}%
+
+\Article{369} We shall proceed to discuss in some detail the problem
+of the incidence of a plane electric wave upon a metal sphere\footnotemark.
+\index{Michell, plane electromagnetic waves}%
+  \footnotetext{The scattering by an insulating sphere is discussed by Lord Rayleigh (\textit{Phil.\ Mag.}~12,
+  p.~98, 1881). The incidence of a plane wave on a sphere was the subject of a dissertation
+  sent in to Trinity College, Cambridge, by Professor Michell in 1890. I do not know
+  of any papers which discuss the special problem of the scattering by \emph{metal} spheres.}
+
+If $\alpha$,~$\beta$,~$\gamma$; $f$,~$g$,~$h$ are respectively the components of the
+magnetic force and of the polarization in the dielectric which
+are radiated from the sphere, then if $\psi$~stands for any one of
+these quantities it satisfies a differential equation of the form
+\[
+\frac{d^{2} \psi}{dx^{2}} + \frac{d^{2} \psi}{dy^{2}} + \frac{d^{2} \psi}{dz^{2}}
+  = \frac{1}{V^{2}}\, \frac{d^{2} \psi}{dt^{2}},
+\Tag{1}
+\]
+where $V$~is the velocity with which electric action is propagated
+through the dielectric surrounding the sphere. If $\lambda$~is the wave
+length of the disturbance incident upon the sphere, then the
+components of magnetic induction and of electric polarization
+will all vary as $\epsilon^{\frac{\iota 2 \pi}{\lambda} Vt}$; thus $V^{-2} d^{2} \psi / dt^{2}$ may be replaced by
+$-4 \pi^{2} \psi / \lambda^{2}$, so that writing~$k$ for~$2 \pi / \lambda$, equation~(\eqnref{369}{1}) may be
+written
+\[
+\frac{d^{2} \psi}{dx^{2}} + \frac{d^{2} \psi}{dy^{2}} + \frac{d^{2} \psi}{dz^{2}} + k^{2} \psi = 0,
+\]
+a solution of which is by \artref{308}{Art.~308},
+\[
+\psi = \epsilon^{\iota kVt} \tsum f_n(kr)S_n,
+\]
+where $r$~is the distance from the centre of the sphere. Since the
+waves of magnetic force and dielectric polarization are radiating
+outwards from the sphere
+\[
+f_n(kr) = \left( \frac{1}{kr}\, \frac{d}{d(kr)} \right)^{n}
+  \frac{\epsilon^{-\iota kr}}{kr},
+\]
+%% -----File: 452.png---Folio 438-------
+$S_n$~is a solid spherical harmonic of degree~$n$. It should be noted
+that $f_n (kr)$ of this article is $(kr)^{-n} f(kr)$ of \artref{308}{\DPtypo{article}{Article}~308}.
+
+\Article{370} We shall now prove a theorem due to Professor Lamb
+\index{Lambx@Lamb's theorem}%
+(\textit{Proc.\ Lond.\ Math.\ Soc.}~13, p.~189, 1881), that if $\alpha$,~$\beta$,~$\gamma$ satisfy
+equations of the form~(\eqnref{369}{1}), and if
+\[
+\frac{d\alpha}{dx} + \frac{d\beta}{dy} + \frac{d\gamma}{dz} = 0;
+\]
+then the most general solution of these equations is given by
+\[
+\left.
+  \begin{aligned}
+    \alpha = \tsum \left\{(n+1)f_{n-1} (kr) \frac{d\omega_n}{dx} - nk^2r^{2n+3} f_{n+1} (kr) \frac{d}{dx}\, \frac{\omega_n}{r^{2n+1}}\right\}\qquad \\
+    + \tsum f_n (kr)\left(y \frac{d}{dz} - z \frac{d}{dy}\right)\omega'_n,\\
+    \beta = \tsum \left\{(n+1)f_{n-1} (kr) \frac{d\omega_n}{dy} - nk^2r^{2n+3} f_{n+1} (kr) \frac{d}{dy}\, \frac{\omega_n}{r^{2n+1}}\right\}\qquad \\
+    + \tsum f_n (kr)\left(z \frac{d}{dx} - x \frac{d}{dz}\right)\omega'_n,\\
+    \gamma = \tsum \left\{(n+1)f_{n-1} (kr) \frac{d\omega_n}{dz} - nk^2r^{2n+3} f_{n+1} (kr) \frac{d}{dz}\, \frac{\omega_n}{r^{2n+1}}\right\}\qquad \\
+    + \tsum f_n (kr)\left(x \frac{d}{dy} - y \frac{d}{dx}\right)\omega'_n.\\
+  \end{aligned}
+\right\}
+\Tag{2}
+\]
+where $\omega_n$,~$\omega'_n$ represent arbitrary solid spherical harmonics of
+degree~$n$.
+
+Since
+\[
+\frac{d\omega_n}{dx}, \qquad
+\frac{d}{dx}\, \frac{\omega_n}{r^{2n+1}}, \qquad
+\left(y \frac{d}{dz} - z \frac{d}{dy}\right)\omega'_n
+\]
+are solid spherical harmonics of degrees $(n - 1)$,~$-(n + 1)$,~$n$ respectively,
+we see that the expression given for~$\alpha$ satisfies the
+differential equation~(\eqnref{369}{1}); similarly this equation is satisfied by
+the values of $\beta$~and~$\gamma$.
+
+Let us now find the value of $d\alpha/dx + d\beta/dy + d\gamma/dz$; we notice
+that the terms involving~$\omega'_n$ vanish identically, and since
+\[
+\nabla^2(\omega_n) = 0, \quad
+\nabla^2 \frac{\omega_n}{r^{2n+1}} = 0,
+\]
+we have
+\begin{multline*}
+\frac{d\alpha}{dx} + \frac{d\beta}{dy} + \frac{d\gamma}{dz}
+  = \tsum (n+1) \frac{k}{r} f'_{n-1} (kr)\left\{x \frac{d}{dx} + y \frac{d}{dy} + z \frac{d}{dz}\right\}\omega_n  \\
+  - \tsum [nk^3r^{2n+2} f'_{n+1} (kr) + n(2n+3)k^2r^{2n+1} f_{n+1} (kr)] × \\
+    \left\{x \frac{d}{dx} + y \frac{d}{dy} + z \frac{d}{dz}\right\} \frac{\omega_n}{r^{2n+1}} \\
+  = \tsum n \centerdot n+1 \centerdot \frac{k}{r}
+    \{f'_{n-1} (kr) + k^2r^2f'_{n+1} (kr) + (2n+3)krf_{n+1} (kr)\} \omega_n.
+\end{multline*}
+%% -----File: 453.png---Folio 439-------
+
+Now
+\[
+f_n (kr) = \left( \frac{1}{kr}\, \frac{d}{d(kr)} \right)^{n} \frac{\epsilon^{-\iota kr}}{kr},
+\]
+hence
+\[
+f'_{n-1} (kr) = kr\, f_n (kr).
+\Tag{3}
+\]
+We have also
+\[
+f''_n (kr) + \frac{2(n+1)}{kr}\, f'_n (kr) + f_n (kr) = 0,
+\]
+which may be written as
+\begin{align*}
+\frac{d}{d(kr)} \{kr\, f'_n (kr) + (2n+1) f_n (kr) \}
+  &= -kr\, f_n (kr)\\
+  &= -f'_{n-1} (kr) \text{ by (\eqnref{370}{3})};
+\end{align*}
+hence, since the constant of integration must vanish since all the~$f$'s
+involve~$\epsilon^{-\iota kr}$,
+\[
+kr\, f'_n (kr) + (2n+1) f_n (kr) = - f_{n-1} (kr),
+\Tag{4}
+\]
+and by~(\eqnref{309}{101}), \artref{308}{Art.~\DPtypo{308}{309}},
+\[
+(2n+1) f_n (kr) = -\{f_{n-1} (kr) + k^{2} r^{2}\, f_{n+1} (kr) \}.
+\Tag{5}
+\]
+Writing $(n+1)$ for~$n$ in~(\eqnref{370}{4}), we have
+\begin{align*}
+kr\, f'_{n+1} (kr) + (2n+3) f_{n+1} (kr) &= -f_n (kr)\\
+  &= -\frac{f'_{n-1} (kr)}{kr}.
+\Tag{6}
+\end{align*}
+From this equation we see that
+\[
+\frac{d \alpha}{dx} + \frac{d \beta}{dy} + \frac{d \gamma}{dz} = 0.
+\]
+
+To prove that equation~(\eqnref{370}{2}) gives the most general expressions
+for $\alpha$,~$\beta$,~$\gamma$, we notice that the values of $\alpha$,~$\beta$ may be written
+\[
+\left.\begin{aligned}
+\alpha = \tsum f_{n} (kr) \left[ \left\{(n+2) \frac{d \omega_{n+1}}{dx}
+  - (n-1) k^{2} r^{2n+1} \frac{d}{dx}\, \frac{\omega_{n-1}}{r^{2n-1}} \right\} \right. \\
+  + \left. \left( y \frac{d}{dz} - z \frac{d}{dy} \right) \omega'_{n} \right], \\
+\beta = \tsum f_n (kr) \left[ \left\{(n+2) \frac{d \omega_{n+1}}{dy}
+  - (n-1) k^{2} r^{2n+1} \frac{d}{dy}\, \frac{\omega_{n-1}}{r^{2n-1}} \right\} \right. \\
+  + \left. \left( z \frac{d}{dx} - x \frac{d}{dz} \right) \omega'_{n} \right].
+\end{aligned}\right\}
+\Tag{7}
+\]
+
+The most general expressions for $\alpha$,~$\beta$, when they represent
+radiation outwards from the sphere, may however, \artref{308}{Art.~308}, be
+expressed in the form
+\[
+\left.\begin{aligned}
+\alpha & = \tsum f_{n} (kr) U_{n},\\
+\beta & = \tsum f_{n} (kr) V_{n},
+\end{aligned} \right\}
+\Tag{8}
+\]
+where $U_{n}$,~$V_{n}$ are solid spherical harmonics of degree~$n$. Since
+%% -----File: 454.png---Folio 440-------
+$\omega_n$~and~$\omega'_n$ are arbitrary, we may determine them so as to make
+the values of $\alpha$~and~$\beta$ given by~(\eqnref{370}{7}) agree with those given by~(\eqnref{370}{8}).
+Thus (\eqnref{370}{7})~are sufficiently general expressions for $\alpha$,~$\beta$, and
+when $\alpha$~and~$\beta$ are given $\gamma$~follows from the equation
+\[
+\frac{d\alpha}{dx} +  \frac{d\beta}{dy} + \frac{d\gamma}{dz} = 0.
+\]
+
+\Article{371} If $\alpha$,~$\beta$,~$\gamma$ represent the components of the magnetic
+force, $f$,~$g$,~$h$ the components of the electric polarization are, in a
+dielectric, given by the equations
+\begin{align*}
+4\pi \frac{df}{dt} & = \frac{d\gamma}{dy} - \frac{d\beta}{dz}, \\
+4\pi \frac{dg}{dt} & = \frac{d\alpha}{dz} - \frac{d\gamma}{dx}, \\
+4\pi \frac{dh}{dt} & = \frac{d\beta}{dx} - \frac{d\alpha}{dy}.
+\end{align*}
+
+Taking the values of $\beta$~and~$\gamma$ given in~(\eqnref{370}{2}), we see that the
+term in $4\pi df/dt$ involving~$\omega_n$ is equal to
+\begin{multline*}
+\left\{(n+1) \frac{k}{r} f'_{n-1}(kr) - nk^3 rf'_{n+1}(kr) - n(2n+3) k^2 f_{n+1}(kr) \right\} × \\
+  \left\{y \frac{d\omega_n}{dz} - z \frac{d\omega_n}{dy} \right\},
+\end{multline*}
+and this by equations (\eqnref{370}{4})~and~(\eqnref{370}{6}) is equal to
+\[
+(2n+1)k^2 f_n(kr) \left\{y \frac{d\omega_n}{dz} - z \frac{d\omega_n}{dy} \right\}.
+\]
+
+Let us now consider the term in~$4\pi df/dt$ involving~$\omega_n'$; this
+equals
+\begin{multline*}
+f_n(kr) \left\{-2 \frac{d}{dx} - \left(x \frac{d}{dx} + y \frac{d}{dy} + z \frac{d}{dz}\right) \frac{d}{dx}\right\} \omega'_n \\
+  + f'_n(kr)k \left( x \frac{d}{dr} - r \frac{d}{dx} \right) \omega'_n \\
+  = -(n+1) f_n(kr) \frac{d\omega'_n}{dx} + rk\, f'_n(kr) \frac{nx}{r^2} \omega'_n - kr\, f'_n(kr) \frac{d\omega'_n}{dx},
+\end{multline*}
+this by equations (\eqnref{370}{4})~and~(\eqnref{370}{6}) equals
+\begin{multline*}
+- \frac{n+1}{2n+1} \left\{\bigl(k^2 r^2 f_{n+1}(kr) + f_{n-1}(kr)\bigr) - k^2 r^2 f_{n+1}(kr) \right\} \frac{d\omega'_n}{dx} \\
+  + rk\, f'_n(kr) \frac{nx}{r^2} \omega'_n \\
+  = \frac{1}{(2n+1)} \left\{(n+1) f_{n-1}(kr) \frac{d\omega'_n}{dx} - nk^2 r^{2n+3} f_{n+1}(kr) \frac{d}{dx}\, \frac{\omega'_n}{r^{2n+1}} \right\}.
+\end{multline*}
+%% -----File: 455.png---Folio 441-------
+
+Thus if $\alpha$,~$\beta$,~$\gamma$ are given by~(\eqnref{370}{2}), then we have
+\begin{multline*}
+4\pi \frac{df}{dt} = \tsum \frac{1}{2n+1} \left\{(n+1) f_{n-1} (kr) \frac{d\omega'_n}{dx}
+  - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dx}\, \frac{\omega'_n}{r^{2n+1}}\right\} \\
+  + \tsum (2n+1) k^2 f_n (kr) \left(y \frac{d\omega_n}{dz} - z \frac{d\omega_n}{dy}\right),
+\end{multline*}
+\begin{multline*}
+4\pi \frac{dg}{dt} = \tsum \frac{1}{2n+1} \left\{(n+1) f_{n-1} (kr) \frac{d\omega'_n}{dy}
+  - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dy}\, \frac{\omega'_n}{r^{2n+1}}\right\} \\
+  + \tsum (2n+1) k^2 f_n (kr) \left(z \frac{d\omega_n}{dx} - x \frac{d\omega_n}{dz}\right),
+\end{multline*}
+\begin{multline*}
+4\pi \frac{dh}{dt} = \tsum \frac{1}{2n+1} \left\{(n+1) f_{n-1} (kr) \frac{d\omega'_n}{dz}
+  - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dz}\, \frac{\omega'_n}{r^{2n+1}}\right\} \\
+  + \tsum (2n+1) k^2 f_n (kr) \left(x \frac{d\omega_n}{dy} - y \frac{d\omega_n}{dx}\right).
+\end{multline*}
+
+\Article{372} In the plane electrical wave incident on the sphere, let
+us suppose that the electric polarization~$h_0$ in the wave front
+is parallel to~$z$ and expressed by the equation
+\[
+h_0 = \epsilon^{\frac{\iota 2\pi}{\lambda} (Vt+x)} = \epsilon^{\iota k(Vt+x)},
+\]
+where the axis of~$x$ is at right angles to the wave front.
+
+We have to expand~$h_0$ in the form
+\[
+\epsilon^{\iota kVt} \tsum A_n Q_n,
+\]
+where $Q_n$~is a zonal harmonic of degree~$n$ whose axis is the axis
+of~$x$ and $A_n$~is a function of~$r$ which we have to determine.
+
+Since
+\[
+\epsilon^{\iota  kx} = \tsum A_n Q_n,
+\]
+and since it satisfies the equation
+\[
+\frac{d^2\psi}{dx^2} + \frac{d^2\psi}{dy^2} + \frac{d^2\psi}{dz^2} + k^2\psi = 0,
+\]
+and is finite when $r = 0$, we see by \artref{308}{Art.~308} that
+\[
+A_n = A_n' S_n (kr)
+    = A_n' (kr)^n \left\{\frac{1}{kr}\, \frac{d}{d(kr)}\right\}^n \frac{\sin kr}{kr},
+\]
+where $A_n'$~is independent of~$r$.
+
+\begin{DPgather*}
+\lintertext{\indent Since}
+\frac{\sin kr}{kr} = 1 - \frac{k^2 r^2}{3!} + \frac{k^4 r^4}{5!} - \ldots,
+\end{DPgather*}
+we see that when $kr$~is very small
+\[
+A_n = (-1)^n A_n'\, \frac{(kr)^n}{(2n+1)(2n-1) \ldots 1}.
+\Tag{9}
+\]
+%% -----File: 456.png---Folio 442-------
+
+But if $x/r = \mu$, we have
+\begin{gather*}
+\epsilon^{\iota k r \mu} = \tsum A_n Q_n, \\
+\therefore \int_{-1}^{+1} \epsilon^{\iota k r \mu} Q_n\,d\mu
+  = A_n \int_{-1}^{+1} Q_n^2\,d\mu = \frac{2 A_n}{2n + 1}.
+\end{gather*}
+
+The lowest power of~$kr$ on the left-hand side of this equation
+is the~$n$\textsuperscript{th}, the coefficient of this is equal to
+\[
+\frac{\iota^n}{\nbfactorial{n}} \int_{-1}^{+1} \mu^n Q_n\,d\mu
+  = \frac{2 \iota^n}{(2n + 1)(2n - 1)(2n - 3) \ldots 1};
+\]
+hence when $kr$~is small we have
+\[
+\frac{2 \iota^n (kr)^n}{(2n + 1)(2n - 1) \ldots 1} = \frac{2 A_n}{2n + 1}
+\]
+Comparing this equation with~(\eqnref{372}{9}) we see that
+\begin{DPalign*}
+A_n' & = \frac{(2n + 1)}{\iota^n}, \\
+A_n  & = \frac{2n + 1}{\iota^n} S_n(kr), \\
+\lintertext{so that}
+\epsilon^{\iota k r \mu} &= \tsum \frac{2n + 1}{\iota^n} S_n(kr) Q_n.
+\end{DPalign*}
+This expression is given by Lord Rayleigh (\textit{Theory of Sound},
+\index{Rayleigh@Rayleigh, Lord, \textit{Theory of Sound}}%
+ii.\ p.~239).
+
+By equation~(\eqnref{309}{101}) of \DPtypo{\artref{308}{Art.~308}}{\artref{309}{Art.~309}} we have
+\[
+\frac{A_{n-1}}{2n - 1} - \frac{A_{n+1}}{2n + 3} = \frac{1}{\iota k r} A_n.
+\Tag{10}
+\]
+This can also be proved directly thus,
+\begin{align*}
+\frac{A_{n-1}}{2n - 1}
+  & = \tfrac{1}{2} \int_{-1}^{+1} \epsilon^{\iota k r \mu} Q_{n-1}\,d\mu, \\
+\frac{A_{n+1}}{2n + 3}
+  & = \tfrac{1}{2} \int_{-1}^{+1} \epsilon^{\iota k r \mu} Q_{n+1}\,d\mu,
+\end{align*}
+\begin{multline*}
+\frac{A_{n-1}}{2n - 1} - \frac{A_{n+1}}{2n + 3}
+  = \left[\tfrac{1}{2} \frac{\epsilon^{\iota k r \mu}}{\iota k r}(Q_{n-1} - Q_{n+1})\right]_{-1}^{+1} \\
+    - \tfrac{1}{2} \frac{1}{\iota k r} \int_{-1}^{+1} \epsilon^{\iota k r \mu} \left(\frac{dQ_{n-1}}{d\mu} - \frac{dQ_{n+1}}{d\mu}\right) d\mu.
+\end{multline*}
+The terms within square brackets vanish, and since
+\[
+\frac{dQ_{n-1}}{d\mu} - \frac{dQ_{n+1}}{d\mu} = -(2n + 1) Q_n,
+\]
+%% -----File: 457.png---Folio 443-------
+we have
+\begin{align*}
+\frac{A_{n-1}}{2n-1} - \frac{A_{n+1}}{2n+3}
+  &= \frac{1}{2\iota kr} \int_{-1}^{+1} (2n+1)\epsilon^{\iota kr\mu} Q_n\,d\mu \\
+  &= \frac{A_n}{\iota kr}.
+\end{align*}
+
+\Article{373} It will be convenient to collect together the results we
+have obtained.
+
+In the incident wave,
+\[
+f_0 = 0, \qquad g_0 = 0, \qquad
+h_0 = \epsilon^{\iota kVt} \tsum \frac{2n+1}{\iota^n} Q_n S_n (kr),
+\]
+and therefore by \artref{9}{Art.~9},
+\[
+\alpha_0 = 0, \qquad \gamma_0 = 0, \qquad
+\beta_0 = 4\pi h_0 V = 4\pi V\, \epsilon^{\iota kVt} \tsum \frac{2n+1}{\iota^n} Q_n S_n (kr).
+\]
+For the wave scattered by the sphere, omitting the time factor,
+we have since $d/dt = \iota kV$
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}
+4\pi\iota kVf = \tsum \frac{1}{2n+1}
+  \left\{(n+1)f_{n-1} (kr) \frac{d\omega'_n}{dx} - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dx}\, \frac{\omega'_n}{r^{2n+1}}\right\} \\
+  + \tsum(2n+1)k^2 f_n (kr)\left(y \frac{d\omega_n}{dz} - z \frac{d\omega_n}{dy}\right),
+\end{multline*}
+\begin{multline*}
+4\pi\iota kVf = \tsum \frac{1}{2n+1}
+  \left\{(n+1)f_{n-1} (kr) \frac{d\omega'_n}{dy} - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dy}\, \frac{\omega'_n}{r^{2n+1}}\right\} \\
+  + \tsum(2n+1)k^2 f_n (kr)\left(z \frac{d\omega_n}{dx} - x \frac{d\omega_n}{dz}\right),
+\end{multline*}
+\begin{multline*}
+4\pi\iota kVf = \tsum \frac{1}{2n+1}
+  \left\{(n+1)f_{n-1} (kr) \frac{d\omega'_n}{dz} - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dz}\, \frac{\omega'_n}{r^{2n+1}}\right\} \\
+  + \tsum(2n+1)k^2 f_n (kr)\left(x \frac{d\omega_n}{dy} - y \frac{d\omega_n}{dx}\right).
+\end{multline*}
+\begin{multline*}
+\alpha = \tsum\left\{(n+1)f_{n-1} (kr) \frac{d\omega_n}{dx}
+  - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dx}\, \frac{\omega_n}{r^{2n+1}}\right\} \\
+  + \tsum f_n (kr)\left(y \frac{d\omega'_n}{dz} - z \frac{d\omega'_n}{dy}\right),
+\end{multline*}
+\begin{multline*}
+\beta = \tsum\left\{(n+1)f_{n-1} (kr) \frac{d\omega_n}{dy}
+  - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dy}\, \frac{\omega_n}{r^{2n+1}}\right\} \\
+  + \tsum f_n (kr)\left(z \frac{d\omega'_n}{dx} - x \frac{d\omega'_n}{dz}\right),
+\end{multline*}
+\begin{multline*}
+\gamma = \tsum\left\{(n+1)f_{n-1} (kr) \frac{d\omega_n}{dz}
+  - nk^2 r^{2n+3} f_{n+1} (kr) \frac{d}{dz}\, \frac{\omega_n}{r^{2n+1}}\right\} \\
+  + \tsum f_n (kr)\left(x \frac{d\omega'_n}{dy} - y \frac{d\omega'_n}{dx}\right).
+\end{multline*}
+}
+%% -----File: 458.png---Folio 444-------
+
+\Article{374} To determine $\omega_n$,~$\omega_n'$, we shall assume that the sphere is
+a perfect conductor and therefore that the electromotive intensity,
+and therefore the electric polarization, is at right angles to the
+sphere. This condition is satisfied whatever the resistance of
+the sphere if the frequency is so great that $kV\mu a^2 /\sigma$ is large;
+$a$~being the radius of the sphere, $\sigma$~its specific resistance, and $\mu$~its
+magnetic permeability. If $R$~is the normal electromotive
+polarization, $\Theta$~that along a tangent to a meridian, $\Phi$~that along
+a parallel of latitude, then the condition
+\[
+\frac{df}{dx} + \frac{dg}{dy} + \frac{dh}{dz} = 0
+\]
+is equivalent to
+\[
+\frac{d}{dr} (r^2 R)
+  + \frac{1}{\sin \theta}\, \frac{d}{d\theta} (r \sin \theta\Theta)
+  + \frac{1}{\sin \theta}\, \frac{d}{d\phi} (r\Phi) = 0;
+\]
+but since $\Theta$~and~$\Phi$ vanish all over the sphere, this, if $a$~is the
+radius of the sphere, gives the condition
+\[
+\frac{d}{dr} (r^2 R) = 0 \text{ when } r = a.
+\]
+
+\begin{DPgather*}
+\lintertext{\indent Now} rR = x(f + f_0) + y(g + g_0) + z(h + h_0);
+\end{DPgather*}
+but
+\begin{align*}
+4\pi\iota kV(xf + yg + zh)
+  &= \tsum \frac{n\centerdot(n+1)}{2n+1} \bigl(f_{n-1} (kr) + k^2r^2f_{n+1}(kr)\bigr)\omega_n \\
+  &= - \tsum n \centerdot n+1 \centerdot f_n(kr)\omega_n', \text{ by equation~(\eqnref{370}{6})}, \\
+xf_0 + yg_0 + zh_0
+  &= z\tsum A_n Q_n, \text{ omitting the time factor}.
+\end{align*}
+
+But if $r$,~$\theta$,~$\phi$ are the polar coordinates of the point whose
+Cartesian coordinates are $x$,~$y$,~$z$,
+\begin{DPalign*}
+z & = r \sin \theta \sin \phi, \\
+\lintertext{and}
+Q_n & = \frac{1}{2n+1} \left\{\frac{dQ_{n+1}}{d\mu} - \frac{dQ_{n-1}}{d\mu}\right\};
+\end{DPalign*}
+hence, if $\omega_n' = r^n Y_n'$ where $Y_n'$~is a surface harmonic of degree~$n$,
+the condition
+\[
+\frac{d}{dr} (r^2 R) = 0 \text{ when } r = a
+\]
+becomes
+\begin{multline*}
+\frac{1}{4\pi\iota KV} \tsum n \centerdot (n+1)Y_n'\, \frac{d}{da} (a^{n+1} f_n(ka))
+  = \sin \theta \sin \phi \tsum Q_n\, \frac{d}{da} (a^2A_n) \\
+  = \sin \theta \sin \phi \tsum \frac{dQ_n}{d\mu}\,
+    \frac{d}{da} \left\{\frac{a^2A_{n-1}}{2n-1} - \frac{a^2 A_{n+1}}{2n+3}\right\};
+\end{multline*}
+%% -----File: 459.png---Folio 445-------
+but $\sin \theta \sin \phi \dfrac{dQ_n}{d\mu}$ is a surface harmonic of \DPtypo{degrees}{degree}~$n$, hence
+\[
+Y_n' = \frac{\sin \theta \sin \phi \dfrac{dQ_n}{d\mu}\,
+              \dfrac{d}{da} \left\{\dfrac{a^2 A_{n-1}}{2n-1}
+            - \dfrac{a^2 A_{n+1}}{2n+3}\right\} }
+            {\dfrac{n\centerdot n+1}{4\pi\iota kV} \centerdot
+              \dfrac{d}{da} \bigl(a^{n+1} f_n (ka)\bigr) } ,
+\]
+or by~(\eqnref{372}{10})
+\[
+Y_n' = 4\pi V \sin \theta \sin \phi \frac{dQ_n}{d\mu}\,
+  \frac{\dfrac{d(aA_n)}{da} }
+       {n \centerdot n+1 \centerdot \dfrac{d}{da} \bigl(a^{n+1} f_n (ka)\bigr) },
+\]
+and $\omega_n' = r^n Y_n'$.
+
+\Article{375} We now proceed to find~$\omega_n$. The line integral of the
+electromotive intensity taken round any closed curve is equal to
+the rate of diminution of the number of lines of magnetic induction
+passing through it: if we take as our closed curve one
+drawn on the surface of the sphere, we see, since the tangential
+electromotive intensity over the surface of the sphere vanishes,
+that the rate of diminution of the normal magnetic induction
+also vanishes; this condition, since the induction varies harmonically,
+is equivalent to the condition that the normal magnetic
+induction vanishes over the surface of the surface; hence when
+$r = a$, we have
+\[
+x(\alpha + \alpha_0) + y(\beta+\beta_0) + z(\gamma+\gamma_0) = 0.
+\Tag{1}
+\]
+But when $r = a$,
+\begin{align*}
+x\alpha + y\beta + z\gamma
+  &= \tsum n \centerdot (n+1)(2n+1)f_n (ka) \omega_n, \\
+x\alpha_0 + y\beta_0 + z\gamma_0
+  &= 4\pi Vy \tsum A_n Q_n \\
+  &= 4\pi Va \sin \theta \cos \phi
+     \tsum\left(\frac{A_{n-1}}{2n-1} - \frac{A_{n+1}}{2n+3}\right) \frac{dQ_n}{d\mu} \\
+  &= \frac{4\pi V}{\iota k} \sin \theta \cos \phi \tsum A_n \frac{dQ_n}{d\mu}.
+\end{align*}
+
+Let $\omega_n = r^n Y_n$, where $Y_n$~is a surface harmonic of degree~$n$.
+Then we have
+\[
+Y_n = \frac{4\pi Va^{-n}}{n \centerdot n + 1 \centerdot 2n + 1 \centerdot \iota k}
+  \sin \theta \cos \phi \frac{dQ_n}{d\mu}\, \frac{A_n}{f_n(ka)}.
+\]
+
+\Article{376} Substituting the values just found for $\omega_n$,~$\omega_n'$, we find
+that the values of $f$,~$g$,~$h$, $\alpha$,~$\beta$,~$\gamma$ in the wave scattered by the
+%% -----File: 460.png---Folio 446-------
+sphere are, omitting the time factor, given by the equations
+{\setlength{\multlinegap}{0pt}\footnotesize
+\begin{multline*}
+f = \tsum \frac{1}{n \centerdot n+1 \centerdot}
+  \frac{1}{\iota^n}\, \frac{1}{\iota k}
+  \biggl\{(n+1)f_{n-1}(kr)\, \frac{d}{dx} \left( r^n \sin \theta \sin \phi \frac{dQ_n}{d\mu} \right) \\
+    - nk^2 r^{2n+3} f_{n+1}(kr)\,
+     \frac{d}{dx} \Biggl(\frac{\sin \theta \sin \phi \dfrac{dQ_n}{d\mu}} {r^{n+1}}\Biggr) \biggr\}
+  \frac{\dfrac{d}{da} \bigl(aS_n(ka)\bigr) }
+       {\dfrac{d}{da} \bigl(a^{n+1} f_n (ka)\bigr)} \\
+  - \tsum \frac{2n+1}{\iota^n a^n \centerdot n \centerdot n+1 \centerdot}
+    \cdot \frac{S_n(ka)}{f_n(ka)}\, f_n(kr) \left(y \frac{d}{dz} - z \frac{d}{dy}\right)
+    \left(r^n \sin \theta \cos \phi \frac{dQ_n}{d\mu}\right),
+\end{multline*}
+\begin{multline*}
+g = \tsum \frac{1}{n \centerdot n+1} \cdot \frac{1}{\iota^n}\, \frac{1}{\iota k}
+  \biggl\{(n+1)f_{n-1}(kr)\, \frac{d}{dy} \left( r^n \sin \theta \sin \phi \frac{dQ_n}{d\mu} \right) \\
+    - nk^2 r^{2n+3} f_{n+1}(kr)\,
+    \frac{d}{dy} \Biggl(\frac{\sin \theta \sin \phi \dfrac{dQ_n}{d\mu}}{r^{n+1}}\Biggr) \biggr\}
+  \frac{\dfrac{d}{da} \bigl(aS_n(ka)\bigr) }
+       {\dfrac{d}{da} \bigl(a^{n+1} f_n (ka)\bigr)} \\
+  - \tsum \frac{2n+1}{\iota^n a^n \centerdot n \centerdot n+1 \centerdot}
+    \cdot \frac{S_n(ka)}{f_n(ka)}\, f_n(kr)  \left(z \frac{d}{dx} - x \frac{d}{dz}\right)
+    \left(r^n \sin \theta \cos \phi \frac{dQ_n}{d\mu}\right),
+\end{multline*}
+\begin{multline*}
+h = \tsum \frac{1}{n \centerdot n+1 \centerdot}
+  \frac{1}{\iota^n}\, \frac{1}{\iota k}
+  \biggl\{(n+1)f_{n-1}(kr)\, \frac{d}{dz} \left( r^n \sin \theta \sin \phi \frac{dQ_n}{d\mu} \right) \\
+  - nk^2 r^{2n+3} f_{n+1}(kr)\, \frac{d}{dz} \Biggl(\frac{\sin \theta \sin \phi \dfrac{dQ_n}{d\mu}}{r^{n+1}}\Biggr) \biggr\}
+    \frac{\dfrac{d}{da} \bigl(aS_n(ka)\bigr) }
+         {\dfrac{d}{da} \bigl(a^{n+1} f_n (ka)\bigr)} \\
+  - \tsum \frac{2n+1}{\iota^n a^n \centerdot n \centerdot n+1 \centerdot}
+    \cdot \frac{S_n(ka)}{f_n(ka)}\, f_n(kr) \left(x \frac{d}{dy} - y \frac{d}{dz}\right)
+    \left(r^n \sin \theta \cos \phi \frac{dQ_n}{d\mu}\right);
+\end{multline*}
+\begin{multline*}
+\alpha = 4\pi V \tsum \frac{a^{-n}}{n \centerdot n+1 \centerdot}
+  \frac{1}{\iota^n \iota k}\, \frac{S_n(ka)}{f_n(ka)}
+  \biggl\{(n+1)f_{n-1}(kr)\, \frac{d}{dx} \left(r^n \sin \theta \cos \phi \frac{dQ_n}{d\mu}\right) \\
+  - nk^2 f_{n+1}(kr) r^{2n+3}\, \frac{d}{dx} \Biggl(\frac{\sin \theta \cos \phi \dfrac{dQ_n}{d\mu}} {r^{n+1}}\Biggr) \biggr\} \\
+  + 4\pi V \tsum \cdot \frac{2n+1}{n \centerdot n+1 \centerdot}
+    \cdot \frac{1}{\iota^n}\,
+    \frac{\dfrac{d}{da} \bigl(aS_n(ka)\bigr) }
+         {\dfrac{d}{da} \bigl(a^{n+1} f_n(ka)\bigr) }
+    f_n(kr)  \left(y \frac{d}{dz} - z \frac{d}{dy}\right)
+    \left(r^n \sin \theta \sin \phi \frac{dQ_n}{d\mu}\right)
+\end{multline*}
+%% -----File: 461.png---Folio 447-------
+\begin{multline*}
+\beta = 4\pi V \tsum \frac{a^{-n}}{n \centerdot n+1 \centerdot}
+  \frac{1}{\iota^n \centerdot \iota k}\,
+  \frac{S_n(ka)}{f_n(ka)}
+  \biggl\{(n+1)f_{n-1}(kr)\, \frac{d}{dy} \left(r^n \sin \theta \cos \phi \frac{dQ_n}{d\mu}\right) \\
+  - nk^2 f_{n+1}(kr) r^{2n+3}\,
+    \frac{d}{dy} \Biggl(\frac{\sin \theta \cos \phi \dfrac{dQ_n}{d\mu}} {r^{n+1}}\Biggr) \biggr\} \\
+  + 4\pi V \tsum \cdot \frac{2n+1}{n \centerdot n+1 \centerdot}
+  \cdot \frac{1}{\iota^n}\,
+  \frac{\dfrac{d}{da} \bigl(aS_n(ka)\bigr) }
+       {\dfrac{d}{da} \bigl(a^{n+1} f_n(ka)\bigr) } f_n(kr)  \left(z \frac{d}{dx} - x \frac{d}{dz}\right)
+  \left(r^n \sin \theta \sin \phi \frac{dQ_n}{d\mu}\right),
+\end{multline*}
+\begin{multline*}
+\gamma = 4\pi V \tsum \frac{a^{-n}}{n \centerdot n+1}
+  \frac{1}{\iota^n \iota k}\, \frac{S_n(ka)}{f_n(ka)}
+  \biggl\{(n+1)f_{n-1}(kr)\, \frac{d}{dz} \left(r^n \sin \theta \cos \phi \frac{dQ_n}{d\mu}\right) \\
+  - nk^2 f_{n+1}(kr) r^{2n+3}\, \frac{d}{dz} \Biggl(\frac{\sin \theta \cos \phi \dfrac{dQ_n}{d\mu}}{r^{n+1}}\Biggr) \biggr\} \\
+  + 4\pi V \tsum \frac{2n+1}{n \centerdot n+1} \cdot \frac{1}{\iota^n}\,
+  \frac{\dfrac{d}{da} \bigl(aS_n(ka)\bigr) }
+       {\dfrac{d}{da} \bigl(a^{n+1} f_n(ka)\bigr) }\DPtypo{}{f_n(kr)}  \left(x \frac{d}{dy} - y \frac{d}{dx}\right)
+  \left(r^n \sin \theta \sin \phi \frac{dQ_n}{d\mu}\right).
+\end{multline*}
+}
+
+\Article{377} These expressions give the solution of the problem of
+the scattering of a plane wave by a sphere of any size. The
+particular case when the radius of the sphere is very small compared
+with the wave length of the incident wave is of great
+importance. In this case $ka$~is very small, and the approximate
+values of $S_n (ka)$, $f_n (ka)$ are, \artref{308}{Art.~308}, expressed by the equations
+\begin{align*}
+S_n(ka) &= \frac{(-1)^n (ka)^n}{2n+1 \centerdot 2n-1 \ldots 1}, \\
+f_n(ka) &= (-1)^n 2n-1 \centerdot 2n-3 \ldots 1 \frac{\epsilon^{-\iota ka}} {(ka)^{2n+1}}.
+\end{align*}
+
+Substituting these values in the preceding equations and retaining
+only the lowest powers of~$ka$, we find, omitting the
+time factor,
+\begin{align*}
+&\left.
+\begin{aligned}
+f &= k^5 a^3\, \epsilon^{\iota ka} f_2 (kr)xz + \frac{1}{2\iota}\, k^4 a^3\, \epsilon^{\iota ka} f_1 (kr)z, \\
+g &= k^5 a^3\, \epsilon^{\iota ka} f_2 (kr)yz, \\
+h &= \tfrac{1}{3} k^3 a^3\, \epsilon^{\iota ka}
+     \{2f_0(kr) + k^2(3z^2-r^2)f_2 (kr)\}
+   - \frac{1}{2\iota}\, k^4 a^3\, \epsilon^{\iota ka} f_1 (kr)x;
+\end{aligned}
+\right.\\
+%% -----File: 462.png---Folio 448-------
+&\left.\begin{aligned}
+\alpha & = -4\pi V \tfrac{1}{2} k^5 a^3\, \epsilon^{\iota ka} f_2 (kr)xy
+  + 4\pi V \iota k^4 a^3\, \epsilon^{\iota ka} f_1(kr)y. \\
+\beta & =
+  \begin{aligned}[t]
+  -4 \pi V \frac{1}{6} k^3a^3\, \epsilon^{\iota ka} \{2f_0(kr)
+  + k^2(3y^2 - r^2)f_2(kr)\} \\
+  -4 \pi V \iota k^4 a^3\, \epsilon^{\iota ka} f_1(kr)x,
+  \end{aligned} \\
+\gamma & = -4 \pi V \frac{1}{2} k^5 a^3\, \epsilon^{\iota ka} f_2(kr)yz.
+\end{aligned}
+\right\}
+\end{align*}
+At a distance from the sphere, which is large compared with the
+wave length, $kr$~is very large; we then have approximately
+\[
+f_2(kr) = - \frac{\epsilon^{-\iota kr}}{k^3 r^3}, \qquad
+f_1(kr) = - \frac{\iota\, \epsilon^{-\iota kr}}{k^2 r^2}, \qquad
+f_0(kr) =   \frac{\epsilon^{-\iota kr}}{kr}.
+\]
+
+Substituting their value and introducing the time factor, we get
+\[
+\left.\begin{aligned}
+f &= -\epsilon^{\iota k \bigl(Vt-(r-a)\bigr)} \frac{k^2a^3}{r} \left(\frac{xz}{r^2} + \tfrac{1}{2}\frac{z}{r}\right), \\
+g &= -\epsilon^{\iota k \bigl(Vt-(r-a)\bigr)} \frac{k^2a^3}{r}\, \frac{yz}{r^2}, \\
+h &=  \epsilon^{\iota k \bigl(Vt-(r-a)\bigr)} \frac{k^2a^3}{r} \left(1 - \frac{z^2}{r^2} + \tfrac{1}{2} \frac{x}{r}\right);
+\end{aligned} \right\}
+\]
+\[
+\left.\begin{aligned}
+\alpha &= 4 \pi V\, \epsilon^{\iota k\bigl(Vt-(r-a)\bigr)} \frac{k^2a^3}{r} \left\{\tfrac{1}{2} \frac{xy}{r^2} + \frac{y}{r}\right\}, \\
+\beta  &= 4 \pi V\, \epsilon^{\iota k\bigl(Vt-(r-a)\bigr)} \frac{k^2a^3}{r} \left\{\frac{y^2-r^2}{2r^2} - \frac{x}{r}\right\}, \\
+\gamma &= 4 \pi V\, \epsilon^{\iota k\bigl(Vt-(r-a)\bigr)} \frac{k^2a^3}{r}\, \tfrac{1}{2} yz.
+\end{aligned} \right\}
+\]
+From these expressions we see that
+\[
+xf + yg + zh = 0, \qquad x\alpha + x\beta + z\gamma = 0;
+\]
+so that both the electric polarization and the magnetic induction
+are at right angles to the radius. We have also
+\[
+f\alpha + g\beta + h\gamma = 0,
+\]
+so that the electric polarization is at right angles to the magnetic
+induction. Taking the real part of the preceding expressions,
+we find
+\begin{gather*}
+f^2 + g^2 + h^2 = \cos^2 k\bigl(Vt-(r-a)\bigr)
+  \frac{k^4a^6}{r^2} \left\{\left(\frac{x}{r} + \tfrac{1}{2}\right)^2 + \tfrac{3}{4} \frac{y^2}{r^2} \right\}, \\
+\alpha^2 + \beta^2 + \gamma^2 = (4 \pi V)^2 \cos^2 k \bigl(Vt-(r-a)\bigr)
+  \frac{k^4a^6}{r^2} \left\{\left(\frac{x}{r} + \tfrac{1}{2}\right)^2 + \tfrac{3}{4} \frac{y^2}{r^2} \right\}.
+\end{gather*}
+
+Thus we see that the resultant magnetic induction is equal to
+$4 \pi V$~times the resultant electric displacement. We could have
+%% -----File: 463.png---Folio 449-------
+deduced this result directly from \artref{9}{Art.~9}, since the Faraday
+tubes are moving outwards at right angles to themselves with
+the velocity~$V$.
+
+\Article{378} We see from the expressions for the resultant electric
+polarization and the magnetic force that at the places where the
+scattered wave vanishes
+\[
+x/r = -\tfrac{1}{2}, \qquad y = 0.
+\]
+Thus the scattered light produced by the incidence of a plane
+polarized wave vanishes in the plane through the centre at right
+angles to the magnetic induction in the incident wave along
+a line, making an angle of~$120°$ with the radius to the point
+at which the wave first strikes the sphere, and it does not
+vanish in any direction other than this. Thus if non-polarized
+waves of light or of electric displacement are incident upon a
+sphere, whose radius is small compared with the wave length of
+the incident vibration, the direction in which the scattered light
+is plane polarized will be inclined at an angle of~$120°$ to the
+direction of the incident light. The scattering of light by small
+metallic spheres thus follows laws which are quite different from
+those which hold when the scattering is produced by non-conducting
+particles. In the latter case (see Lord Rayleigh, \textit{Phil.\ Mag.}\
+[5], 12, p.~81, 1881), when a ray of plane polarized light falls
+\index{Rayleigh, Lord, scattering of light by fine particles@\subdashtwo scattering of light by fine particles}%
+upon a small sphere, the scattered light vanishes at all points
+in the plane normal to the magnetic induction, where the radius
+vector makes an angle of~$90°$, and not~$120°$, with the direction
+of the incident light. Thus, when non-polarized light falls upon
+a small non-conducting sphere, the scattered light will be completely
+polarized at any point in a plane through the centre of
+the sphere at right angles to the direction of the incident light.
+When the light is scattered by a conducting sphere, the points at
+which the light is completely polarized are on the surface of
+a cone whose axis is the direction of propagation of the incident
+light and whose semi-vertical angle is~$120°$. The Faraday tubes
+given off by the conducting sphere form two sets of closed
+curves, which are separated by the surface of this cone. The
+momentum of these tubes being at right angles both to the
+magnetic induction and the electric polarization is radial, so
+that the energy emitted by the conducting sphere is, when we
+are considering a point whose distance from the centre is a large
+%% -----File: 464.png---Folio 450-------
+number of wave lengths, travelling radially outwards from the
+sphere.
+
+At a point close to the sphere $kr$~is very small, so that we
+have approximately
+\[
+f_0(kr) =  \frac{\epsilon^{-\iota kr}}{kr}, \qquad
+f_1(kr) = -\frac{\epsilon^{-\iota kr}}{k^3r^3}, \qquad
+f_2(kr) =  \frac{3\epsilon^{-\iota kr}}{k^5r^5}.
+\]
+
+Substituting these values in the expressions in \artref{377}{Art.~377}, we find
+that the components of the total electric polarization and magnetic
+induction, i.e.~the polarization and induction scattered from the
+sphere plus that due to the incident wave, are given approximately
+by the equations
+\begin{align*}
+f &= \frac{3a^3}{r^5}\, xz \cos kVt, \\
+g &= \frac{3a^3}{r^5}\, yz \cos kVt, \\
+h &= \left\{\frac{a^3}{r^5} (3z^2 - r^2) + 1\right\} \cos kVt;
+\end{align*}
+\begin{align*}
+\alpha &= -6\pi V \frac{a^3}{r^5}\, xy \cos kVt, \\
+\beta  &= -2\pi V \left\{\frac{a^3}{r^5} (3y^2 - r^2) - 2 \right\} \cos kVt, \\
+\gamma &= -6\pi V \frac{a^3}{r^5}\, yz \cos kVt.
+\end{align*}
+Thus when $r = a$,
+\begin{gather*}
+f = \frac{3xz}{a^2} \cos kVt, \qquad
+g = \frac{3yz}{a^2} \cos kVt, \qquad
+h = \frac{3z^2}{a^2} \cos kVt; \\
+%
+\alpha = -6\pi V \frac{xy}{a^2} \cos kVt, \qquad
+\beta  =  6\pi V \frac{(x^2 + z^2)}{a^2} \cos kVt, \\
+\gamma = -6\pi V \frac{yz}{a^2} \cos kVt.
+\end{gather*}
+Thus at the surface of the sphere the resultant electric polarization
+is radial and proportional to~$z$; there is thus a distribution
+of electricity over the sphere whose surface density varies as the
+distance of the point on the sphere from a plane through its
+centre parallel to the plane of polarization of the incident wave,---the
+plane of polarization being the plane at right angles to
+the electric polarization.
+%% -----File: 465.png---Folio 451-------
+
+The magnetic induction at the surface of the sphere is tangential
+to the sphere and equal to
+\[
+6\pi V \frac{1}{a} \{x^2 + z^2\}^\frac{1}{2} \cos kVt;
+\]
+it is thus proportional to the distance of a point on the surface
+of the sphere from the diameter of the sphere parallel to the
+magnetic force in the incident wave. The lines of magnetic force
+on the sphere are great circles all passing through this diameter.
+
+Since the electric polarization is radial and the magnetic
+induction is tangential, the momentum due to the Faraday tubes
+which is at right angles to each of these quantities is tangential.
+The direction of the momentum is tangential to a series of small
+circles on the sphere whose planes are at right angles to the
+diameter of the sphere parallel to the magnetic induction in the
+incident wave.
+
+
+\Subsection{Waves along Wires.}
+\index{Electromagnetic waves, yalong wires@\subdashtwo along wires}%
+\index{Propagation velocity of zelectromagnetic waves along a wire@\subdashtwo of electromagnetic waves along a wire}%
+\index{Velocity of xelectromagnetic waves along wires@\subdashtwo electromagnetic waves along wires}%
+\index{Waves, electromagnetic, along wires@\subdashtwo along wires}%
+\index{Wires, electromagnetic waves along}%
+
+\Article{379} If the electric potential at one end of a wire be made to
+vary harmonically so as at any time to be represented by $\cos pt$,
+the electromotive intensity, as we proceed along the wire, will be
+a harmonic function of the distance from the end of the wire; if
+the wave length of this harmonic distribution is~$\lambda$, the velocity of
+propagation of the disturbance along the wire is defined to be
+$\lambda p/2\pi$. This velocity ought, if Maxwell's theory is true, to
+be equal to~$V$, the velocity with which electrodynamic disturbances
+are propagated through air (see \artref{267}{Art.~267}). Indeed on this
+theory the effects observed do in reality travel through the air
+even though the wire is present, so that the introduction of the
+wire does not materially alter the physical conditions. The
+electrical vibrations considered in this chapter are all of very
+high frequency, being produced by the discharge of condensers
+through short discharging circuits. In this case (see \artref{269}{Art.~269})
+the electromotive intensity in the region around the wire is at
+right angles to it, and we may suppose that the phenomena near
+the wire are due to radial Faraday tubes, with their ends on the
+wire travelling along it with the velocity of light.
+
+\Article{380} Considerable interest attaches to some experiments
+made by Hertz, which seemed to indicate that the velocity along
+the wire was considerably less than that through the air; and
+though later experiments have shown that this conclusion is
+%% -----File: 466.png---Folio 452-------
+erroneous, and that, as Maxwell's theory indicates, the two
+velocities are identical, Hertz's experiments are of great interest
+both from the methods used and the points they illustrate.
+
+\includegraphicsmid{fig123}{Fig.~123.}
+
+In these experiments Hertz (\textit{Wied.\ Ann.}\ 34, p.~551, 1888)
+used the vibrator described in \artref{325}{Art.~325}. This was placed in a
+vertical plane; behind and parallel to one of the metal plates~$A$,
+and insulated from it, was a metal plate~$B$ of equal area (see
+\figureref{fig123}{Fig.~123}). A long wire was soldered to~$B$ and bent round so
+as to come in front of the vibrator and lie in the vertical plane of
+symmetry of the vibrator about a foot above the base line. The
+wire, which was above sixty metres long, was taken through
+a window, and was kept as far as possible from walls,~\&c., so as
+to avoid disturbances arising from reflected waves. In the first
+set of experiments the free end of the wire was insulated. The
+resonator used was the circular coil of wire $35$~cm.\ in radius previously
+described. When the plane of the resonator was at right
+angles to the axis of the vibrator, the electromotive intensity due
+to the vibrator (apart from the action of the wire) did not (\artref{331}{Art.~331})
+produce any tendency to spark in the resonator, so that the
+sparks in this position of the resonator must have been entirely
+due to the disturbance produced by the wire. To observe the
+effects due to the wire, the resonator was turned round in its own
+plane until the air gap was at the highest point, and therefore
+parallel to the wire. When the resonator was moved along the
+wire the following effects were observed. At the free end of
+the wire (which was insulated) the sparks in the resonator were
+extremely small, as the resonator was moved towards the
+vibrator the sparks increased and attained a maximum; they then
+%% -----File: 467.png---Folio 453-------
+decreased again until they almost vanished. If we call such a
+place a node, then, as the resonator moved along the wire, such
+nodes were found to occur at approximately equal intervals.
+
+\Article{381} Similar periodic effects were observed when the plane of
+the resonator was at right angles to the wire, the air gap being
+vertical; in such a position there would have been no sparks
+unless the wire had been present. On moving the resonator
+along the wire the brightness of the sparks changed in a periodic
+way: the positions however in which the sparks were brightest
+with the resonator in this position were those in which they had
+been dullest when the resonator was in its previous position.
+
+This result is what we should expect from theoretical considerations.
+For when the resonator is in the first position, with
+its plane passing through the wire, the air gap is placed
+parallel to the wire. Now the Faraday tubes travelling along the
+wire are, as we saw \artref{269}{Art.~269}, at right angles to it and therefore to
+the air gap: thus the tubes which fall directly on the air gap
+do not tend to produce a spark; the sparks must be due to the
+tubes collected by the resonator and thrown by it into the air
+gap. The tubes which travel with their ends on the wire will
+be reflected from the insulated extremity of it, so that there will
+be tubes travelling in opposite directions along the wire; incident
+tubes travelling from the vibrator to the free end of the wire, and
+reflected tubes travelling back from the free end to the vibrator.
+
+Let us now consider what will happen when the vibrator is
+in such a position as that represented in \figureref{fig124}{Fig.~124}. The tube
+thrown into the air gap by a positive tube, such as~$CD$
+proceeding from the vibrator, will be of opposite sign to that
+thrown by a positive tube, such as~$AB$ proceeding from the free
+end: thus in this position of the vibrator the positive tubes
+%% -----File: 468.png---Folio 454-------
+moving in opposite directions will neutralize each other's effects
+in producing sparks, though they increase the resultant electromotive
+intensity: thus, in this case, at the places where the
+electromotive intensity is greatest there will be no sparks in the
+resonator, for this maximum intensity will be due to two sets of
+tubes of the same sign, one set moving in one direction, the other
+in the opposite.
+
+\includegraphicsmid{fig124}{Fig.~124.}
+
+Since the free end of the wire has little or no capacity, no
+electricity can accumulate there, so that when one set of positive
+tubes arrives at the free end from the vibrator an equal number of
+positive tubes must start from the free end and move towards the
+vibrator; thus at the free end we have equal numbers of positive
+(or negative) tubes travelling in opposite directions. We should
+expect therefore that no sparks would be produced when the
+resonator was placed close to the free end; this, as we have seen,
+was found by Hertz to be the case.
+
+When however the resonator is placed in the second position,
+with its plane at right angles to the wire, the conditions are very
+different; for the tubes which though they strike the resonator yet
+miss the air gap, are not hampered by the resonator in their
+passage through it; thus the resonator does not in this case collect
+tubes and throw them into the air gap. The sparks are now entirely
+due to the tubes which strike the air gap itself, and thus will
+be brightest at those points on the wire where the electromotive
+intensity is a maximum, while at such places, as we have seen,
+the sparks vanish when the resonator is in the former position.
+
+\Article{382} Hertz found that when the wire was cut at a node the
+nodes in the portion of the wire which remained were not
+altered in position, but that they were displaced when the wire
+was cut at any place other than a node.
+
+Hertz also found that the distance between the nodes was independent
+of the diameter of the wire and of the material of
+which it was made, and that in particular the positions of the nodes
+were not affected by substituting an iron wire for a copper one.
+
+The distance between the nodes is half the wave length along
+the wire; thus, if we know the period of the electrical vibrations
+of the system we can determine the velocity of propagation along
+the wire. Hertz, by using the formula $2\pi \sqrt{LC}$  for the wave
+length of the vibrations emitted by a condenser of capacity~$C$,
+whose plates are connected by a discharging circuit whose coefficient
+%% -----File: 469.png---Folio 455-------
+of self-induction is~$L$, came to the conclusion that the
+velocity of propagation along the wire was only about $2/3$~of
+that through the dielectric; there are however many difficulties
+and doubtful points in the theoretical calculation of the period of
+vibration of such a system as Hertz's.
+
+\Article{383} Before discussing these we shall consider another method
+which Hertz used to compare directly the velocity of propagation
+along a wire with that through the air.
+
+In this method interference was produced in the following way
+between the waves travelling out from the vibrator through
+the air and those travelling along the wire. The free end of
+the wire was put to earth so as to get rid of reflected waves along
+the wire, and as there were no metallic reflectors in the way of
+the waves proceeding directly through the air from the vibrator,
+the only reflected waves of this kind must have come from the
+floors or walls of the room; we shall assume for the present that
+there were no reflected air waves. The resonator was placed so
+that the air gap was at the highest point and vertically under
+the wire, and the plane of the resonator could rotate about a
+vertical axis passing through the middle of the air gap. When
+the plane of the resonator was at right angles to the wire,
+the waves proceeding along the latter had no tendency to produce
+a spark; any sparks that passed across the resonator must
+have been entirely due to the waves travelling from the vibrator
+through the air independently of the wire. In Hertz's experiments
+when the resonator was in this position the sparks were
+about $2$~mm.\ long. On the other hand, when the resonator was
+twisted about the axis so that its plane passed through the wire
+and was at right angles to the axis of the vibrator, the direct waves
+through the air from the vibrator would have no tendency to produce
+sparks; which in this case must have been entirely due to the
+waves travelling along the wire. In Hertz's experiments when
+the resonator was in this position the sparks were again about
+$2$~mm.\ long. When the resonator was in a position intermediate
+between these two, the sparks were due to the combined action of
+the waves travelling along the wire and those coming directly
+through the air. In such a case the brightness of the sparks
+would, in general, change when the plane of the vibrator was
+twisted through a considerable angle. If now the fronts of the two
+sets of waves were parallel and moving forward with the same
+%% -----File: 470.png---Folio 456-------
+velocity, then the effect of turning the plane of the vibrator
+through a definite angle in a definite direction would be the
+same at all points on the wire: if however the two waves were
+travelling at different rates the effect of turning the resonator
+would vary as it is moved from place to place along the wire.
+
+\Article{384} To prove this, let the electromotive intensity in the air
+gap due to the wave travelling along the wire be
+\[
+A \cos \frac{2\pi}{\lambda} (Vt - z),
+\]
+when the plane of the resonator passes through the wire; here
+the wire is taken as the axis of~$z$, and $\lambda$~is the wave length of the
+waves travelling along it. Then, when the plane of the resonator
+is twisted through an angle~$\phi$ from this position, the electromotive
+intensity in the air gap due to the wire waves will be
+\[
+A \cos\phi \cos \frac{2\pi}{\lambda} (Vt - z),
+\]
+since the electromotive intensity is approximately proportional
+to the projection of the resonator on the plane through the wire
+and the base line of the vibrator.
+
+Let the electromotive intensity in the air gap due to the
+waves coming from the vibrator independently of the wire be,
+when the plane of the resonator is at right angles to the wire,
+\[
+B \cos \frac{2\pi}{\lambda'} \bigl(V't - (z - \alpha)\bigr),
+\]
+where $\lambda'$~is the wave length and $V'$~the velocity of the air waves;
+then, if the plane of the resonator is turned until it makes an
+angle~$\phi$ with the plane through the wire and the base line, the
+electromotive intensity resolved parallel to the air gap is
+equal to
+\[
+B \sin\phi \cos \frac{2\pi}{\lambda'} \bigl(V't - (z - \alpha)\bigr).
+\]
+Thus, considering both the air waves and those along the wire,
+the electromotive intensity when the resonator is in this position
+is equal to
+\[
+A \cos\phi \cos \frac{2\pi}{\lambda} (Vt - z) +
+B \sin\phi \cos \frac{2\pi}{\lambda'} \bigl(V't - (z - \alpha)\bigr),
+\]
+which may, since $V/\lambda$~is equal to~$V'/\lambda'$, be written as
+\[
+R \cos \left\{\frac{2\pi}{\lambda} V(t + \epsilon) \right\},
+\]
+%% -----File: 471.png---Folio 457-------
+where
+\begin{multline*}
+R^2 = A^2 \cos^2 \phi + B^2 \sin^2 \phi \\
+  + 2AB \cos\phi \sin\phi
+    \cos \left\{\left( \frac{2\pi}{\lambda} - \frac{2\pi}{\lambda'} \right) z + \frac{2\pi}{\lambda'} \alpha \right\}.
+\end{multline*}
+
+Now $R$~is the maximum electromotive intensity acting on the
+air gap, and will be measured by the brightness of the spark.
+We see from the preceding expression that if $\lambda = \lambda'$, that is, if the
+velocity of the waves along the wire is the same as that of the
+air waves which are not affected by the wire, the last term in
+the expression for~$R^2$ will cease to be a periodic function of~$z$,
+so that in this case there will be no periodic change in the
+effect produced by a given rotation as we move the resonator
+along the wire. When however $\lambda$~is not equal to~$\lambda'$, the effect on
+the spark length of a given rotation of the resonator will vary
+harmonically along the wire. Since in Hertz's experiments the
+sparks were about equally long in the two extreme positions,
+$\phi = 0$ and $\phi = \pi/2$, we may in discussing these experiments put
+$A = B$, and therefore
+\[
+R^2 = A^2 \left( 1 + 2 \cos\phi \sin\phi
+  \cos \left\{\left( \frac{2\pi}{\lambda} - \frac{2\pi}{\lambda'} \right) z + \frac{2\pi}{\lambda'} \alpha \right\} \right);
+\]
+thus, if the resonator is rotated so that $\phi$~changes from $+\beta$ to~$-\beta$,
+$R^2$~is diminished by
+\[
+2A^2 \sin 2\beta
+  \cos \left\{\left( \frac{2\pi}{\lambda} - \frac{2\pi}{\lambda'} \right) z + \frac{2\pi}{\lambda'} \alpha \right\}.
+\]
+Thus when
+\[
+\left( \frac{2\pi}{\lambda} - \frac{2\pi}{\lambda'} \right) z + \frac{2\pi}{\lambda'} \alpha
+  = (2n + 1) \frac{\pi}{2},
+\]
+that is, at places separated by the intervals
+\[
+\tfrac{1}{2} \bigg/ \left\{\frac{1}{\lambda} - \frac{1}{\lambda'} \right\}
+\]
+along the wire the rotation of the resonator will produce no
+effect upon the sparks, while on one side of one of these positions
+it will increase, on the other side diminish the brightness
+of the sparks. If $\lambda'$~were very large compared with~$\lambda$, that is,
+if the velocity of the waves travelling freely through the air
+were very much greater than that of those travelling along the
+wire, the distance between the places where rotation produces
+no effect would be~$\frac{1}{2} \lambda$, which is the distance between the nodes
+observed in the experiments described in \artref{380}{Art.~380}. Hertz, however,
+%% -----File: 472.png---Folio 458-------
+came to the conclusion that the places where rotation produced
+no effect were separated by a much greater interval than
+the nodes. These he had determined to be about $2.8$~metres apart,
+whereas the places where rotation produced no effect seemed to
+be separated by about $7.5$~metres. Assuming these numbers we
+have
+\begin{gather*}
+\lambda = 5.6, \\
+  \tfrac{1}{2} \bigg{/} \left(\frac{1}{\lambda} - \frac{1}{\lambda'}\right) = 7.5;
+\end{gather*}
+hence $\lambda' = 8.94$. Thus from these experiments the velocity of
+the free air waves would appear to be greater than those along
+the wire in the proportion of $8.94$~to~$5.6$ or $1.6$~to~$1$; or the
+velocity of the air waves is about half as large again as that of
+the wire waves.
+
+We have, however, in the preceding investigations made
+several assumptions which it would be difficult to realise in
+practice; we have assumed, for example, that in the neighbourhood
+of the resonator the front of the air waves was at right
+angles to the wire. Since the resonator was close to the axis of
+the vibrator this assumption would be justifiable if there had
+been no reflection of the air waves from the walls or floors of
+the room. Since the thickness of the walls was small compared
+with the wave length it is not likely, unless they were very
+damp, that there would be much reflection from them; the case
+of the floor is however very different, and it is difficult to see
+how reflection from it could have been entirely avoided. Reflection
+from the floor would however introduce waves, the
+normals to whose fronts would make a finite angle with the
+wire. The electromotive intensity in the spark gap due to such
+waves would no longer be represented by a term of the form
+\[
+\cos \bigl(2\pi(V't - z)/\lambda'\bigr),
+\]
+but by one of the form
+\[
+\cos \bigl(2\pi(V't - z \cos\theta)/\lambda'\bigr),
+\]
+where $\theta$~is the angle between the normal to the wave front and
+the wire. Thus in the preceding investigation we must, for such
+waves, replace~$\lambda'$ by~$\lambda'\sec\theta$, and their apparent wave length
+along the wire would be $\lambda'\sec\theta$ and not~$\lambda'$, so that the reflection
+would have the effect of increasing the apparent wave
+length of the air waves. The result then of Hertz's experiments
+that the wave length of the air waves, measured parallel to the
+%% -----File: 473.png---Folio 459-------
+wire, was greater than that of the wire waves, may perhaps be
+explained by the reflection of the waves from the floor of the
+room, without supposing that the velocity of the free air waves
+is different from that of those guided by the wire.
+
+\includegraphicsmid{fig125}{Fig.~125.}
+
+\index{Sarasin and De la Rive, reflection of electromagnetic waves along wires@\subdashtwo electromagnetic waves along wires}%
+\index{Wires, Sarasin's and de la Rive's experiments on@\subdashone Sarasin's and de la Rive's experiments on}%
+\Article{385} The experiments of Sarasin and De~la~Rive (\textit{Archives des
+Sciences Physiques et Naturelles Genève}, 1890, t.~xxiii, p.~113) on
+the distance between the nodes (1)~along a wire, (2)~when produced
+by interference between direct air waves and waves reflected
+from a large metallic plate, seem to prove conclusively
+that the velocity of the waves guided by a wire is the same as
+that of free air waves. The experiments on the air waves have
+already been described in \artref{339}{Art.~339}; those on the wire waves
+were made in a slightly different way from Hertz's experiments.
+
+\includegraphicsmid{fig126}{Fig.~126.}
+
+The method used by Sarasin and De~la~Rive is indicated in
+\figureref{fig125}{Fig.~125}. Two metallic plates placed in front of the plates of
+the vibrator have parallel wires $F$,~$F$ soldered to them, the wires
+being of equal length and insulated. The plane of the resonator
+is at right angles to the wires, and the air gap is at the highest
+point, so that the air gap is parallel to the shortest distance
+between the wires. The resonator is mounted on a wagon by
+means of which it can be moved to and fro along the wires,
+while a scale on the bench along which the wagon slides enables
+the position of the latter to be determined. The resonator with
+its mounting is shown in \figureref{fig126}{Fig.~126}. Sarasin and De~la~Rive
+found that as long as the same resonator was used the distance
+between the nodes as determined by this apparatus was the
+%% -----File: 474.png---Folio 460-------
+same as when the nodes were produced by the interference of
+direct air waves and those reflected from a metallic plate. The
+relative distances are given in the table in \artref{340}{Art.~340}, where `$\lambda$~for
+wire' indicates twice the distance between the nodes measured
+along the wire. They found with the wires, as later on they
+found for the air waves, that the distance between the nodes
+depended entirely upon the size of the resonator and not upon
+that of the vibrator; in fact the distance between the nodes
+was directly proportional to the diameter of the resonator;
+while it did not seem to depend to any appreciable extent
+upon the size of the vibrator. These peculiarities can be explained
+in the same way as the corresponding ones for the air
+waves, see \artref{341}{Art.~341}.
+
+When the extremities of the wires remote from the vibrator
+are attached to large metallic plates, instead of being free, the
+electromotive intensity parallel to the plates at the ends must
+vanish; hence, whenever a bundle of positive Faraday tubes
+from the vibrator arrives at a plate an equal number of negative
+tubes must start from the plate and travel towards the
+vibrator, while, when the end of the wire is free, the tubes starting
+from the end of the wire in response to those coming from
+%% -----File: 475.png---Folio 461-------
+the vibrator are of the same sign as those arriving. Thus, when
+the end is free, the current vanishes and the electromotive intensity
+is a maximum, while when the end is attached to a large
+plate the electromotive intensity vanishes and the current is a
+maximum. Since the sparks in the resonator, when used as in
+Sarasin and De~la~Rive's experiments, are due to the tubes
+falling directly on the air gap, the sparks will be brightest when
+the electromotive intensity is a maximum, and will vanish when
+it vanishes; thus the loops when the ends are free will coincide
+with the nodes when the wires are attached to large plates.
+This was found by Sarasin and De~la~Rive to be the case.
+
+A similar point arises in connection with the experiments
+with wires to that which was mentioned in \artref{342}{Art.~342} in connection
+with the experiments on the air waves. The distance
+between the nodes, which is half the wave length of the vibration
+of the resonator, is, as is seen from the table in \artref{340}{Art.~340},
+very approximately four times the diameter; if the resonator
+were a straight wire the half wave length would be equal to the
+length of the wire, and we should expect that bending the wire
+into a circle would tend to shorten the period, we should therefore
+have expected the distance between the nodes to have been
+a little less than the circumference of the resonator. Sarasin
+and De~la~Rive's experiments show however that it was $80$~per
+cent.\ greater than this: it is remarkable however that the distance
+of the first node from the end of the wire, which is a loop,
+was always equal to half the circumference of the resonator,
+which is the value it would have had if the wave length of the
+vibration emitted by the resonator had been equal to twice its
+circumference.
+
+\Article{386} The experiments of Sarasin and De~la~Rive show that
+when vibrators of the kind shown in \figureref{fig113}{Fig.~113} are used, the
+oscillations which are detected by a circular resonator are those
+in the resonator rather than the vibrator.
+
+\index{Arons@Arons, electromagnetic waves}%
+\index{Paalzow, electromagnetic waves}%
+\index{Ritter, electromagnetic waves}%
+\index{Rubens, zelectromagnetic waves@\subdashone electromagnetic waves}%
+Rubens, Paalzow, Ritter, and Arons (\textit{Wied.\ Ann.}\ 37, p.~529,
+1889; 40, p.~55, 1890; 42, pp.~154, 581, 1891) have used another
+method of measuring wave lengths, which though it certainly
+requires great care and labour, yet when used in a particular
+way would seem to give very accurate results. The method
+depends upon the change which takes place in the resistance of
+a wire when it is heated by the passage of a current through
+%% -----File: 476.png---Folio 462-------
+it. Rubens finds that the rapidly alternating currents induced
+by the vibrator can produce heat sufficient to increase the resistance
+of a fine wire by an amount which can be made to
+cause a considerable deflection in a delicate galvanometer.
+
+\includegraphicsmid{fig127}{Fig.~127.}
+
+\Article{387} Rubens' apparatus, which is really a bolometer, is arranged
+as follows. Rapidly alternating currents pass through a very
+fine iron wire~$L$. This wire forms one of the arms of a Wheatstone's
+Bridge provided with a battery and a galvanometer.
+When the rapidly alternating currents do not pass through~$L$
+this bridge is balanced, and there is no deflection of the galvanometer.
+When however a rapidly alternating discharge passes
+through the fine wire it heats it and so alters its resistance, and
+as the Bridge is no longer balanced the galvanometer is deflected.
+This arrangement is so sensitive that it is not necessary
+to place~$L$ in series with the wires connected with the
+plates of the vibrating system. Rubens found if a wire in
+series with~$L$ encircled, without touching, one of the wires
+$EJ$, $DH$ in the experiment figured in \figureref{fig127}{Fig.~127} (Rubens, \textit{Wied.\
+Ann.}\ 42, p.~154, 1871), the deflection of the galvanometer was
+large enough to be
+easily measured. The apparatus was so
+delicate that a rise in temperature of $1/10,000$ of a degree
+in the wire produced a deflection of a millimetre on the galvanometer
+scale. In one of his experiments the wire joined
+%% -----File: 477.png---Folio 463-------
+in series with~$L$ was bent round two pieces of glass tubing
+through which the wires~$EJ$, $DH$ passed, the plane of the turns
+round the glass tube being at right angles to the wires. In this
+case each turn of the wire and the wire it surrounds acted like a
+little Leyden jar, and the electricity which flowed through the
+wire~$L$ and disturbed the balance in the Bridge was due to the
+charging and discharging of these jars.
+
+\includegraphicsmid{fig128}{Fig.~128.}
+
+The pieces of glass tube were attached to a frame work, see
+\figureref{fig128}{Fig.~128}, which was moved along the wire, and the deflection of
+the galvanometer observed as it moved along the wire. The
+relation between the galvanometer deflection and the position of
+the tubes is shown in \figureref{fig129}{Fig.~129}, where the ordinates represent the
+deflection of the galvanometer and the abscissae, the distance
+of the turns in the bolometer circuit from the point~$F$ in the
+wire. The curve shows very clearly the harmonic character of
+the disturbance along the wire.
+
+\Article{388} The results however of experiments of this kind were
+not very accordant, and in the majority of his experiments
+Rubens used another method which had previously been used
+by Lecher, who instead of a bolometer employed the brightness of
+\index{Lecher, xon electromagnetic waves@\subdashone on electromagnetic waves}%
+the discharge through an exhausted tube as a measure of the
+intensity of the waves.
+%% -----File: 478.png---Folio 464-------
+
+In these experiments the turns~$l$, $m$ (\figureref{fig128}{Fig.~128}) in the bolometer
+circuit were kept at the ends~$J$ and~$H$ of the main wire (\figureref{fig127}{Fig.~127}),
+while a metallic wire forming a bridge between the two parallel
+wires was moved along from one end of the wires to the other.
+The deflection of the bolometer depended on the position of the
+bridge, in the manner represented in \figureref{fig130}{Fig.~130}, where the ordinates
+represent the deflection of the galvanometer, the abscissae the
+position of the bridge.
+
+\includegraphicsmid{fig129}{Fig.~129.}
+
+Rubens found that the positions of the bridge, in which the
+deflection of the galvanometer was a maximum, were independent
+of the length of the wire connecting the plates of the vibrator to
+the balls between which the sparks passed, and therefore of
+the period of vibration of the vibrator. This result shows that
+the vibrations in the wires which are detected by the bolometer
+cannot be `forced' by the vibrator; for though, if this were the
+case, the deflection of the bolometer would vary with the position
+of the bridge, the places where the bridge produced a maximum
+deflection would depend upon the period of the vibrator. We
+can see this in the following way, if the bridge was at a place
+where the electromotive intensity at right angles to the wire
+%% -----File: 479.png---Folio 465-------
+vanished---which, if there were no capacity at the ends~$J$, $H$,
+would be an odd number of quarter wave lengths from these
+ends---the introduction of the bridge would, since no current
+would flow through it, produce no diminution in the electromotive
+intensity at the ends~$J$, $H$; in other positions of the
+bridge some of the current, which in its absence would go to the
+ends, would be diverted by the bridge, so that the electromotive
+intensity at the ends would be weakened. Thus, when the
+deflection of the bolometer was a maximum, the distances of
+the bridge from the ends~$J$, $H$ would be an odd multiple of a
+quarter of the wave length of the vibration travelling along the
+wire; thus, if these vibrations were `forced' by the vibrator,
+the positions of the bridge which give a maximum deflection in
+the bolometer would depend upon the period of the vibrator.
+Rubens' experiments show that this was not the case.
+
+\includegraphicsmid{fig130}{Fig.~130.}
+
+We may therefore, as the result of these experiments, assume
+that the effect of the sparks in the vibrator is to give an electrical
+impulse to the wires and start the `free' vibrations proper to
+them. The capacity of the plates at the ends of the wire makes
+the investigation of the free periods troublesome; we may however
+avail ourselves of the results of some experiments of Lecher's
+\index{Lecher, xon electromagnetic waves@\subdashone on electromagnetic waves}%
+(\textit{Wied.\ Ann.}\ 41, p.~850, 1890), who found that the addition of
+capacity to the ends might be represented by supposing the wires
+prolonged to an extent depending upon this additional capacity.
+
+\Article{389} Let $AB$, $CD$, \figureref{fig131}{Fig.~131}, be the original wires, $A\alpha$, $B\beta$, $C\gamma$,
+$D\delta$~the amount by which they have to be prolonged to represent
+the capacity at the ends, we shall call the wires~$\alpha\beta$, $\gamma\delta$ the
+`equivalent' wires. Let~$PQ$ represent the position of the
+bridge.
+
+\includegraphicsmid{fig131}{Fig.~131.}
+
+The electrical disturbance produced by the coil may start
+several systems of currents in the wires~$\alpha\beta$, $\gamma\delta$. Then there may
+be a system of longitudinal currents along~$\alpha\beta$, $\gamma\delta$ determined by
+the condition that the currents must vanish at~$\alpha$, $\beta$, and at~$\gamma$, $\delta$.
+Another system might flow round $\alpha PQ\gamma$, their wave length being
+determined by the condition that the currents along the wire
+must vanish at~$\alpha$ and~$\gamma$, and that by symmetry the electrification
+%% -----File: 480.png---Folio 466-------
+at these points must be equal and opposite. A third system of
+currents might flow round $\beta PQ \delta$, the flow vanishing at $\beta$~and~$\delta$.
+If the bridge~$PQ$ were near the ends $\alpha$,~$\gamma$, we might expect,
+\textit{a~priori}, that the current in the circuit $\alpha PQ \gamma$ would be the most
+intense. Since the currents induced in the wires by the coil
+would tend to distribute themselves so that their self-induction
+should be as small as possible they would therefore tend to take
+the shortest course, i.e.~that round the circuit~$\alpha PQ \gamma$: these currents
+would induce currents round the circuit~$\beta PQ \delta$. Lecher's
+experiments (\textit{Wied.\ Ann.}~41, p.~850, 1890) show that the currents
+circulating round $\alpha PQ \gamma$, $\beta PQ \delta$ are much more efficacious in
+producing the electrical disturbance at the ends than the
+longitudinal ones along $\alpha \beta$,~$\gamma \delta$. As a test of the magnitude of
+the disturbance at the ends, Lecher used an exhausted tube
+containing nitrogen and a little turpentine vapour; this was
+placed across the wires at the ends, and the brilliancy of the
+luminosity in the tube served as an indication of the magnitude
+of the electromotive intensity across~$\beta \delta$. In one of his
+experiments Lecher used a bridge formed of two wires, $PQ$,~$P'Q'$
+in parallel, and moved this about until the luminosity in the
+tube was a maximum; he then cut the wires $\alpha \beta$,~$\gamma \delta$ between
+$PQ$~and~$P'Q'$, so that the two circuits $\alpha PQ \gamma$, $\beta P'Q' \delta$ were no
+longer in metallic connection. Lecher found that this division
+of the circuit produced very little diminution in the brilliancy
+of the luminosity in the tube, though the longitudinal flow
+of the currents from $\alpha$ to~$\beta$ and from $\gamma$ to~$\delta$ must have been
+almost entirely destroyed by it. Lecher also found that the
+position of the bridge in which the luminosity of the tube was a
+maximum depended upon the length of the bridge; if the bridge
+were lengthened it had to be pushed towards, and if shortened
+away from the coil, to maintain the luminosity of the tube at
+its maximum value. He also found that, as might be expected,
+if the bridge were very short the tube at the end remained dark
+wherever the bridge was placed, while if the bridge were very
+long the tube was always bright whatever the position of the
+bridge. These experiments show that it is the currents round
+the circuits $\alpha PQ \gamma$, $\beta PQ \delta$ which chiefly cause the luminosity
+in the tube. Since the currents in the circuit $\beta PQ \delta$ are induced
+by those in the circuit~$\alpha PQ \gamma$, they will be greatest when
+the time of the electrical vibration of the system $\alpha PQ \gamma$ is
+%% -----File: 481.png---Folio 467-------
+the same as that of~$\beta PQ \delta$. The periods of vibration of these
+circuits are determined by the conditions that the current must
+vanish at their extremities and that these must be in opposite
+electrical conditions; these conditions entail that the wave
+lengths must be odd submultiples of the lengths of the circuit.
+If the two circuits are in unison the wave lengths must be the
+same, hence the ratio of the lengths of the two circuits must
+be of the form $(2n - 1)/(2m - 1)$, where $n$~and~$m$ are integers.
+
+This conclusion is verified in a remarkable way by Rubens'
+experiments with the bolometer. The relation between the
+deflections of the bolometer (the ordinates) and the distances
+of the bridge from~$G$ in \figureref{fig127}{Fig.~127} (the abscissae) is represented
+in \figureref{fig130}{Fig.~130}. The length of the bridge in these experiments
+was $14$~cm., that of the curved piece of the wire~$EG$ was $83$~cm.,
+and that of the straight portion~$GJ$ was $570$~cm. The lengths
+$A \alpha$,~$B \beta$  which had to be added to the wires to represent the
+effects of the capacity at the ends were assumed to be $55$~cm.\ for
+the end of the wire next the coil, and $60$~cm.\ for the end next
+the bolometer. These two lengths were chosen so as best to fit
+in with the observations, and were thus really determined by
+the measurements given in the following table; in spite of this,
+so many maxima were observed that the observations furnish
+satisfactory evidence of the truth of the theory just described.
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{4pt}
+\settowidth{\TmpLen}{$2m-1$.}
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+\parbox[c]{\TmpLen}{\bigskip\centering$m$.} &
+\parbox[c]{\TmpLen}{\bigskip\centering$n$.} &
+\parbox[c]{\TmpLen}{\bigskip\centering$2m-1$.} &
+\parbox[c]{\TmpLen}{\bigskip\centering$2n-1$.} &
+\multicolumn{2}{c|}{\settowidth{\TmpLen}{\itshape maximum deflection}%
+\parbox[c]{\TmpLen}{\centering\itshape\tablespaceup Distance of point of\\ maximum deflection\\ from~$G$.\tablespacedown}} &
+\settowidth{\TmpLen}{Corresponding point}%
+\parbox[t]{\TmpLen}{\centering\itshape Corresponding point\\ in \figureref{fig130}{Fig.~130}.}\\ \cline{5-6}
+    &     &     &     & \tablespaceup\textit{Calculated.} &  \tablespaceup\textit{Observed.} & \\
+\hline
+\tablespaceup$2$ & $1$ & $3$ & $1$ & $\Z51$ & $\Z50$ & $A$\\
+$4$ & $2$ & $7$ & $3$ & $\Z89$ & $\Z86$ & $B$\\
+$3$ & $2$ & $5$ & $3$ & $148$  & $143$  & $C$\\
+$4$ & $3$ & $7$ & $5$ & $181$  & $182$  & $D$\\
+$2$ & $2$ & $3$ & $3$ & $246$  & $245$  & $E$\\
+$3$ & $4$ & $5$ & $7$ & $311$  & $305$  & $F$\\
+$2$ & $3$ & $3$ & $5$ & $343$  & $334$  & $G$\\
+$2$ & $4$ & $3$ & $7$ & $402$  & $386$  & $H$\\
+$1$ & $2$ & $1$ & $3$ & $441$  & $443$  & $J$\\
+$1$ & $3$ & $1$ & $5$ & $506$  & $503$  & $K$\\
+$1$ & $4$ & $1$ & $7$ & $529$  & $523$  & $L$\tablespacedown\\
+\hline
+\end{tabular}
+\end{center}
+%% -----File: 482.png---Folio 468-------
+
+\bigskip
+
+\Subsection{Specific Inductive Capacity of Dielectrics in rapidly
+alternating Electric Fields.}
+\index{Specific inductive capacity}%
+\index{Capacity xspecific inductive@\subdashone specific inductive|indexetseq}%
+\index{Inductive capacity, specific}%
+
+\Article{390} Methods analogous to those we have just described
+have been applied to determine the specific inductive capacities
+of dielectrics when transmitting electrical waves a few metres
+long.
+
+One of the most striking results of Maxwell's \textit{Electromagnetic
+Theory of Light} is the connection which it entails between the
+specific inductive capacity and the refractive index of a transparent
+body. On this theory the refractive index for infinitely
+long waves is (Maxwell's \textit{Electricity and Magnetism}, vol.~ii, Art.~786)
+equal to the square root of the specific inductive capacity
+of the dielectric under a steady electric field.
+
+\Article{391} Some determinations of~$K$, the specific inductive capacity
+of various dielectrics in slowly varying fields, are given in the
+following table, which also contains the value of~$\mu^2$, the square of
+the refractive index for such dielectrics as are transparent. The
+letter following the value of~$\mu^2$ denotes the \DPtypo{Frauenhofer}{Fraunhofer} line for
+which the refractive index is measured; when $\infty$~is affixed to
+the value of~$\mu^2$ the number denotes the square of the refractive
+index for infinitely long waves deduced from Cauchy's formula.
+
+When $\mu$ is given by the observer of the specific inductive
+capacity this value has been used, in other cases $\mu$~has been taken
+from Landolt's and Börnstein's `Physicalisch-Chemische Tabellen.'
+
+\renewcommand\thefootnote{\arabic{footnote}}
+{
+\tabletextsize
+\setlength{\tabcolsep}{4pt}
+\begin{longtable}{|l|c|l|c||l|}
+\hline
+\multicolumn{1}{|c|}{Substance.}  &
+Observer.  &
+\multicolumn{1}{c|}{$K$.} &
+\settowidth{\TmpLen}{Tempera-}%
+\parbox[c]{\TmpLen}{\tablespaceup\centering Tempera-\\ture.\tablespacedown} &
+\multicolumn{1}{c|}{$\mu^2$.}\\
+\hline
+\endhead
+\hline
+\endfoot
+\tablespaceup Glass, very light flint\mdotfill & Hopkinson\tabfootmark[0]  &
+    $\Z6.57$ & \dots & $2.375~D$\\
+\PadTo{\text{Glass,}}{\Ditto} light flint\mdotfill & \Ditto &
+    $\Z6.85$ & \dots & $2.478~D$\\
+\PadTo{\text{Glass,}}{\Ditto} dense flint\mdotfill & \Ditto &
+    $\Z7.4$  & \dots & $2.631~D$\\
+\PadTo{\text{Glass,}}{\Ditto} extra dense flint\mdotfill & \Ditto &
+    $10.1$ & \dots & $2.924~D$\\
+\PadTo{\text{Glass,}}{\Ditto} hard crown\mdotfill  & \Ditto &
+    $\Z6.96$ &       & \\
+\PadTo{\text{Glass,}}{\Ditto} plate\mdotfill       & \Ditto &
+    $\Z8.45$ &       & \\
+Paraffin\mdotfill                                    & \Ditto &
+    $\Z2.29$ & \dots & $2.022~\infty$\\
+Sulphur, along greatest axis\mdotfill    & Boltzmann\tabfootmark  &
+    $\Z4.73$ & \dots & $4.89~B$\\
+\PadTo{\text{Sulphur, }}{\Ditto}\PadTo{\text{ along}}{\Ditto} mean axis\mdotfill & \Ditto &
+    $\Z3.970$ & \dots & $4.154~B$\\
+\PadTo{\text{Sulphur, }}{\Ditto}\PadTo{\text{ along}}{\Ditto} least axis\mdotfill& \Ditto &
+    $\Z3.811$ & \dots & $3.748~B$\\
+\PadTo{\text{Sulphur, }}{\Ditto} non-crystalline\mdotfill  & \Ditto &
+    $\Z3.84$ &       & \\
+Calcite, perpendicular to axis\mdotfill & Romich \& Nowak\tabfootmark &
+    $\Z7.7$  & \dots & $2.734~A$\\
+\PadTo{\text{Calcite, }}{\Ditto} along axis\mdotfill & \PadTo{\text{Romich}}{\Ditto}\ \PadTo{\text{Nowak}}{\Ditto} &
+    $\Z7.5$ & \dots & $2.197~A$\\
+%% -----File: 483.png---Folio 469-------
+\index{Arons and Cohn, specific inductive capacity of water}%
+\index{Cohn and Arons, specific inductive capacity}%
+\index{Curie, specific inductive capacity}%
+\index{Klemencic@Klemen\v{c}i\v{c}, specific inductive capacity}%
+\index{Negreano, specific inductive capacity}%
+\index{Nowak and Romich, specific inductive capacity}%
+\index{Romich and Nowak, specific inductive capacity}%
+\index{Rosa, specific inductive capacity}%
+Fluor Spar\mdotfill & \PadTo{\text{Romich}}{\Ditto}\ \PadTo{\text{Nowak}}{\Ditto} &
+    $\Z6.7$  &  \ldots & $2.050\ B$ \\
+Mica\mdotfill       & Klemen\v{c}i\v{c}\tabfootmark &
+    $\Z6.64$ &  \ldots & $2.526\ D$ \\
+Ebonite\mdotfill    & Boltzmann\tabfootmark[-2] &
+    $\Z3.15$ &         &        \\
+Resin\mdotfill      &  \Ditto         &
+    $\Z2.55$ &         &        \\
+Quartz along optic axis\mdotfill & Curie\tabfootmark[3] &
+    $\Z4.55$ &  \ldots & $2.41\ D$  \\
+\PadTo{\text{Quartz}}{\Ditto} perpendicular to axis\mdotfill &  \Ditto         &
+    $\Z4.49$ &  \ldots & $2.38\ D$  \\
+Tourmaline along axis\mdotfill &  \Ditto         &
+    $\Z6.05$ &  \ldots & $2.63\ D$  \\
+\settowidth{\TmpLen}{Tourmaline perpendicular to}
+\parbox[b]{\TmpLen}{\PadTo{\text{Tourmaline}}{\Ditto}perpendicular to\\
+                    \phantom{Tourmaline pe} axis\mdotfill} &  \Ditto         &
+    $\Z7.10$ &  \ldots & $2.70\ D$  \\
+Beryl along axis\mdotfill &  \Ditto         &
+    $\Z6.24$ &  \ldots & $2.48\ D$  \\
+\PadTo{\text{Beryl}}{\Ditto} perpendicular to axis\mdotfill &  \Ditto         &
+    $\Z7.58$ &  \ldots & $2.50\ D$  \\
+Topaz\mdotfill      &  \Ditto         &
+    $\Z6.56$ &  \ldots & $2.61\ D$  \\
+Gypsum\mdotfill     &  \Ditto         &
+    $\Z6.33$ &  \ldots & $2.32\ D$  \\
+Alum\mdotfill       &  \Ditto         &
+    $\Z6.4$  &  \ldots & $2.2\  D$  \\
+Rock Salt\mdotfill  &  \Ditto         &
+    $\Z5.85$ &  \ldots & $2.36\ D$\tablespacedown  \\
+\hline
+\tablespaceup Petroleum Spirit\mdotfill &  Hopkinson\tabfootmark[-4] &
+    $\Z1.92$ &  \ldots & $1.922\ \infty$ \\
+Petroleum Oil, Field's\mdotfill &  \Ditto         &
+    $\Z2.07$ &  \ldots & $2.075\ \infty$ \\
+\PadTo{\text{Petroleum}}{\Ditto} \PadTo{\text{Oil,}}{\Ditto} Common\mdotfill &  \Ditto         &
+    $\Z2.10$ &  \ldots & $2.078\ \infty$ \\
+Ozokerite\mdotfill  &  \Ditto         &
+    $\Z2.13$ &  \ldots & $2.086\ \infty$ \\
+Turpentine, commercial\mdotfill
+                   &  \Ditto         &
+    $\Z2.23$ &  \ldots & $2.128\ \infty$ \\
+Castor Oil\mdotfill &  \Ditto         &
+    $\Z4.78$ &  \ldots & $2.153\ \infty$ \\
+Sperm Oil\mdotfill  &  \Ditto         &
+    $\Z3.02$ &  \ldots & $2.135\ \infty$ \\
+Olive Oil\mdotfill  &  \Ditto         &
+    $\Z3.16$ &  \ldots & $2.131\ \infty$ \\
+Neat's-foot Oil\mdotfill
+                   &  \Ditto         &
+    $\Z3.07$ &  \ldots & $2.125\ \infty$ \\
+Benzene \ce{C6H6}\mdotfill & Hopkinson\tabfootmark[5] &
+    $\Z2.38$ &  \ldots & $2.2614\ D$ \\
+\PadTo{\text{Benzene}}{\Ditto}\PadTo{\ce{C6H6}}{\Ditto} & Negreano\tabfootmark &
+    $\Z2.2988$ & $25$  & $2.2434\ D$ \\
+\PadTo{\text{Benzene}}{\Ditto}\PadTo{\ce{C6H6}}{\Ditto} &  \Ditto         &
+    $\Z2.2921$ & $14$  & $2.2686\ D$ \\
+Toluene \ce{C7H8}\mdotfill &  \Ditto         &
+    $\Z2.242$  & $27$  & $2.224\  D$ \\
+\PadTo{\text{Toluene}}{\Ditto}\PadTo{\ce{C7H8}}{\Ditto} &  \Ditto         &
+    $\Z2.3013$ & $14$  & $2.245\  D$ \tablespacedown\\
+\tablespaceup Toluene\mdotfill    & Hopkinson\tabfootmark[-1] &
+    $\Z2.42$ &  \ldots & $2.2470\ D$ \\
+Xylene \ce{C8H10}\mdotfill &  \Ditto         &
+    $\Z2.39$ &  \ldots & $2.2238\ D$ \\
+\PadTo{\text{Xylene}}{\Ditto}\PadTo{\ce{C8H10}}{\Ditto} &  Negreano\tabfootmark &
+    $\Z2.2679$ & $27$  & $2.219\ D$ \\
+Metaxylene \ce{C8H10}\mdotfill &  \Ditto         &
+    $\Z2.3781$ & $12$  & $2.243\ D$ \\
+Pseudocumene \ce{C9H12}\mdotfill &  \Ditto         &
+    $\Z2.4310$ & $14$  & $2.201\ D$ \\
+Cymene \ce{C10H14}\mdotfill &  \Ditto         &
+    $\Z2.4706$ & $19$  & $2.201\ D$ \\
+\PadTo{\text{Cymene}}{\Ditto}\PadTo{\ce{C10H14}}{\Ditto} & Hopkinson\tabfootmark[-1] &
+    $\Z2.25$ &  \ldots & $2.2254\ D$ \\
+Terebenthine \ce{C10H16}\mdotfill & Negreano\tabfootmark &
+    $\Z2.2618$ & $20$  & $2.168\ D$ \\
+Carbon bisulphide\mdotfill &  Hopkinson\tabfootmark[-1] &
+    $\Z2.67$ &  \ldots & $2.673\ D$ \\
+                   &                 &
+           &         &  (at $10°$) \\
+Ether\mdotfill      &  \Ditto         &
+    $\Z4.75$ &  \ldots & $1.8055\ \infty$ \\
+Amylene\mdotfill    &  \Ditto         &
+    $\Z2.05$ &  \ldots & $1.9044\ D$ \\
+Distilled Water\mdotfill &  Cohn and Arons\tabfootmark[2] &
+    $76.$  & $15$\rlap{°?}  & $1.779\ D$ \\
+\PadTo{\text{Distilled}}{\Ditto}\PadTo{\text{Water}}{\Ditto}  &  Rosa\tabfootmark &
+    $75.7$ & $25\rlap{°}$   &     \\
+Ethyl alcohol ($98$\%)\mdotfill  & Cohn and Arons\tabfootmark[-1] &
+    $26.5$ & \ldots & $1.831\ \infty$ \\
+Amyl alcohol\mdotfill            & \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $15.$  & \ldots & $1.951\ \infty$ \\
+\settowidth{\TmpLen}{Mixture of Xylene and Ethyl}%
+\parbox[b]{\TmpLen}{\tabhang Mixture of Xylene and Ethyl\\ alcohol containing $x$~parts \\ of alcohol in unit volume} &  &
+           &        & \\
+          \qquad$x =\Z.00$  & \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $\Z2.36$&& \\
+\qquad$\phantom{x} = \Z.09$ & \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $\Z3.08$&& \\
+\qquad$\phantom{x} = \Z.17$ & \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $\Z3.98$&& \\
+\qquad$\phantom{x} = \Z.30$ & \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $\Z7.08$&& \\
+\qquad$\phantom{x} = \Z.40$ & \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $\Z9.53$&& \\
+\qquad$\phantom{x} = \Z.50$ & \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $13.0\Z$&& \\
+\qquad$\phantom{x} = 1.\Z\Z$& \PadTo{\text{Cohn and Arons}}{\Ditto\qquad\qquad\Ditto} &
+    $26.5\Z$&&\tablespacedown \\
+\end{longtable}
+}
+
+  \addtocounter{footnote}{-7}%
+\footnotetext{Hopkinson, \textit{Phil.\ Trans.}\ 1878, Part~I, p.~17, and \textit{Phil.\ Trans.}\ 1881, Part~II, p.~355.%
+\index{Hopkinson, specific inductive capacity}}
+  \addtocounter{footnote}{1}%
+\footnotetext{Boltzmann, \textit{Wien.\ Berichte}~70, 2nd~abth.\ p.~342, 1874.\index{Boltzmann, specific inductive capacity}}
+  \addtocounter{footnote}{1}%
+\footnotetext{Romich and Nowak, \textit{Wien.\ Berichte}~70, 2nd~abth.\ p.~380, 1874.}
+  \addtocounter{footnote}{1}%
+\footnotetext{Klemen\v{c}i\v{c}, \textit{Wien.\ Berichte} 96, 2nd~abth.\ p.~807, 1887.}
+  \addtocounter{footnote}{1}%
+\footnotetext{Curie, \textit{Annales de Chimie et de Physique}, 6, 17, p.~385, 1889.}
+  \addtocounter{footnote}{1}%
+\footnotetext{Hopkinson, \textit{Proc.\ Roy.\ Soc.}\ 43, p.~161, 1887.}
+  \addtocounter{footnote}{1}%
+\footnotetext{Negreano, \textit{Compt.\ rend.}\ 104, p.~425, 1887.}
+  \addtocounter{footnote}{1}%
+\footnotetext{Cohn and Arons, \textit{Wied.\ Ann.}\ 33, p.~13, 1888.}
+  \addtocounter{footnote}{1}%
+\footnotetext{Rosa, \textit{Phil.\ Mag.}\ [5], 31, p.~188, 1891.}
+%% -----File: 484.png---Folio 470-------
+
+The values of $K$ for the following gases at the pressure of
+$760$~mm.\ of mercury are expressed in terms of that for a
+vacuum. In deducing them it has been assumed that for air
+at different pressures the changes in~$K$ are proportional to the
+changes in the pressure.
+\begin{center}
+\tabletextsize
+\begin{tabular}{|l|c|l|c||l|}
+\hline
+\multicolumn{1}{|c|}{Gas.}  &   Observer. & \multicolumn{1}{c|}{$K$.} &
+\multicolumn{1}{c||}{\settowidth{\TmpLen}{Tempera-}%
+\parbox[c]{\TmpLen}{\centering\medskip Tempera-\\ture.\medskip}} & \multicolumn{1}{c|}{$\mu^2$.}\\
+\hline
+\tablespaceup Air\mdotfill  & Boltzmann\tabfootmark[-8]     & $1.000590$ & $\Z\Z0°$   & $1.000588\ D$ \\
+\PadTo{\text{Air}}{\Ditto}
+                       & Klemen\v{c}i\v{c}\tabfootmark     & $1.000586$ & $\Z\Z0°$   &               \\
+Hydrogen\mdotfill       & Boltzmann\tabfootmark[-1]         & $1.000264$ & $\Z\Z0°$   & $1.000278\ D$ \\
+\PadTo{\text{Hydrogen}}{\Ditto}
+                       & Klemen\v{c}i\v{c}\tabfootmark     & $1.000264$ & $\Z\Z0°$   &               \\
+Carbonic acid\mdotfill  & Boltzmann\tabfootmark[-1]         & $1.000946$ & $\Z\Z0°$   & $1.000908\ D$ \\
+\PadTo{\text{Carbonic}}{\Ditto} \PadTo{\text{Acid}}{\Ditto}
+                       & Klemen\v{c}i\v{c}\tabfootmark     & $1.000984$ & $\Z\Z0°$   &               \\
+Carbonic oxide\mdotfill & Boltzmann\tabfootmark[-1]         & $1.00069$  & $\Z\Z0°$   & $1.00067\ D$  \\
+\PadTo{\text{Carbonic}}{\Ditto} \PadTo{\text{Oxide}}{\Ditto}
+                       & Klemen\v{c}i\v{c}\tabfootmark     & $1.000694$ & $\Z\Z0°$   &               \\
+Nitrous oxide\mdotfill  & Boltzmann\tabfootmark[-1]         & $1.000994$ & $\Z\Z0°$   & $1.001032\ D$ \\
+\PadTo{\text{Nitric}}{\Ditto} \PadTo{\text{Oxide}}{\Ditto}
+                       & Klemen\v{c}i\v{c}\tabfootmark     & $1.001158$ & $\Z\Z0°$   &               \\
+Olefiant gas\mdotfill   & Boltzmann\tabfootmark[-1]         & $1.001312$ & $\Z\Z0°$   & $1.001356\ D$ \\
+Marsh gas\mdotfill      & Boltzmann\tabfootmark[0]          & $1.000944$ & $\Z\Z0°$   & $1.000886$    \\
+Methyl alcohol\mdotfill & Lebedew\tabfootmark[2]            & $1.0057$   & $100°$ &               \\
+Ethyl alcohol\mdotfill  & \Ditto                            &            &        & $1.001745\ D$ \\
+                       &                                   & $1.0065$   & $100°$ & \multicolumn{1}{c|}{(at~$0°$)} \\
+Methyl formate\mdotfill & \Ditto                            & $1.0069$   & $100°$ &               \\
+Ethyl formate\mdotfill  & \Ditto                            & $1.0083$   & $100°$ &  \\
+Methyl acetate\mdotfill & \Ditto                            & $1.0073$   & $100°$ &  \\
+Ethyl ether\mdotfill    & \Ditto                            & $1.0045$   & $100°$ &  \\
+\PadTo{\text{Ethyl}}{\Ditto} \PadTo{\text{Ether}}{\Ditto} & Klemen\v{c}i\v{c}\tabfootmark[-1]
+                                                           & $1.0074$   & $\Z\Z0°$   & $1.003048\ D$ \\
+Carbon bisulphide\mdotfill & \Ditto                         & $1.0029$   & $\Z\Z0°$   & $1.00296\ D$  \\
+Toluene\mdotfill & Lebedew\tabfootmark                      & $1.0043$   & $126°$ &   \\
+Benzene\mdotfill & \Ditto                                   & $1.0027$   & $100°$ &  \tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+  \addtocounter{footnote}{-2}%
+\footnotetext{Boltzmann, \textit{Pogg.\ Ann.}~155, p.~403, 1875.}%
+  \addtocounter{footnote}{1}%
+\footnotetext{Klemen\v{c}i\v{c}, \textit{Wien.\ Berichte}~91, 2nd~abth.\ p.~712, 1885.}%
+  \addtocounter{footnote}{1}%
+\footnotetext{Lebedew, \textit{Wied.\ Ann.}~44, p.~288, 1891.\index{Lebedew, specific inductive capacity}}%
+%% -----File: 485.png---Folio 471-------
+
+Ayrton and Perry (\textit{Practical Electricity}, p.~310) found that
+\index{Ayrton and Perry, specific inductive capacity of a `vacuum'}%
+the specific inductive capacity of a vacuum in which they
+estimated the pressure to be $.001$~mm.\ was about~$.994$. This
+would make $K$~for air referred to this vacuum as the unit about
+$1.006$, while $\mu^2$~from a vacuum to air is about $1.000588$, there is
+thus a serious discrepancy between these values.
+
+\Article{392} We see from the above table that for some substances,
+such as sulphur, paraffin, liquid hydrocarbons, and the permanent
+gases, the relation $K = \mu^2$ is very approximately fulfilled; while
+for most other substances the divergence between $K$~and~$\mu^2$ is considerable.
+When, however, we remember (1)~that even when $\mu$~is
+estimated for infinitely long waves this is done by Cauchy's
+formula, and that the values so deduced would be completely
+invalidated if there were any anomalous dispersion below the
+visible rays, (2)~that Maxwell's equations do not profess to contain
+any terms which would account for dispersion, the marvel is
+not that there should be substances for which the relation $K = \mu^2$
+does not hold, but that there should be any for which it does.
+To give the theory a fair trial we ought to measure the specific
+inductive capacity for electrical waves whose wave length is the
+same as the luminous waves we use to determine the refractive
+index.
+
+\Article{393} Though we are as yet unable to construct an electrical
+\index{Inductive capacity, specific in rapidly varying fields@\subdashtwo specific in rapidly varying fields}%
+system which emits electrical waves whose lengths approach
+those of the luminous rays, it is still interesting to measure the
+values of the specific inductive capacity for the shortest electrical
+waves we can produce.
+
+We can do this by a method used by von~Bezold (\textit{Pogg.\ Ann.}\
+140, p.~541, 1870) twenty years ago to prove that the velocity
+\index{Bezold@v.~Bezold, velocity of electromagnetic waves}%
+with which an electric pulse travels along a wire is independent
+of the material of wire, it was also used by Hertz in his experiments
+on electric waves.
+
+\includegraphicsouter{fig132}{Fig.~132.}
+
+This method is as follows. Let $ABCD$~be a rectangle of wires
+with an air space at~$EF$ in the middle of~$CD$; this rectangle is
+connected to one of the poles of an induction coil by a wire
+attached to a point~$K$ in~$AB$, then if $K$~is at the middle of~$AB$
+the pulse coming along the wire from the induction coil will
+divide at~$K$ and will travel to $E$~and~$F$, reaching these points
+simultaneously; thus $E$~and~$F$ will be in similar electric states
+and there will be no tendency to spark across the air gap~$EF$.
+%% -----File: 486.png---Folio 472-------
+If now we move~$K$ to a position which is not symmetrical with
+respect to $E$~and~$F$, then, when a pulse travels along the rectangle,
+it will reach one of these points before the other; their
+electric states will therefore be different
+and there will be a tendency to spark.
+
+Suppose that with $K$~at the middle
+point of~$AB$, we insert~$BC$ into a dielectric
+through which electromagnetic
+disturbances travel more slowly than
+they do through air, then the pulse
+which goes round~$AD$ will arrive at~$E$
+before the pulse which goes round~$BC$
+arrives at~$F$; thus $E$~and~$F$ will not be
+in the same electrical state and sparks
+will therefore pass across the air space.
+To get rid of the sparks we must either
+move~$K$ towards~$B$ or else keep~$K$ fixed
+and, as the waves travel more slowly
+through the dielectric than through air,
+lengthen the side~$AD$ of the figure. If
+we do this until the sparks disappear we may conclude that
+$E$~and~$F$ are in similar electric states, and therefore that the
+time taken by the pulse to travel round one arm of the circuit
+is the same as that round the other. By seeing how much the
+length of the one arm exceeds that of the other we can compare
+the velocity of electromagnetic action through the dielectric
+in which $BC$~is immersed with that through air.
+
+\Article{394} I have used (\textit{Phil.\ Mag.}\ [5], 30, p.~129, 1890) this method
+to determine the velocity of propagation of electromagnetic action
+through paraffin and sulphur. This was done by leading one of
+the wires, say~$BC$, through a long metal tube filled with either
+paraffin or sulphur, the wire being insulated from the tube
+which was connected to earth. By measuring the length of wire
+it was necessary to insert in~$AD$ to stop the sparks, I found that
+the velocities with which electromagnetic action travels through
+sulphur and paraffin are respectively $1/1.7$ and $1/1.35$~of the
+velocity through air. The corresponding values of the specific
+inductive capacities would be about $2.9$~and~$1.8$.
+
+\includegraphicsmid{fig133}{Fig.~133.}
+
+\Article{395} Rubens and Arons (\textit{Wied.\ Ann.}\ 42, p.~581; 44, p.~206),
+\index{Arons and Rubens, velocity of electromagnetic waves}%
+\index{Rubensx and Arons, velocity of electromagnetic waves}%
+while employing a method based on the same principles, have
+%% -----File: 487.png---Folio 473-------
+made it very much more sensitive by using a bolometer instead
+of observing the sparks and by using two quadrilaterals
+instead of one. The arrangement they used is represented in
+\figureref{fig133}{Fig.~133} (\textit{Wied.\ Ann.}\ 42, p.~584).
+
+The poles $P$~and~$Q$ of an induction coil are connected to
+the balls of a spark gap~$S$, to each of these balls a metal plate,
+$40$~cm.\ square, was attached by vertical brass rods $15$~cm.\
+long.
+
+Two small tin plates $x$,~$y$, $8$~cm.\ square, were placed at a distance
+of between $3$~and~$4$~cm.\ from the large plates. Then wires
+connected to these plates made sliding contacts at $u$~and~$v$ with the
+wire rectangles $ABCD$, $EFGH$ $.230$~cm.\ by $35$~cm. One of these rectangles
+was placed vertically over the other, the distance between
+them being $8$~cm. The points $u$,~$v$ were connected with each other
+by a vertical wooden rod, ending in a pointer which moved over
+a millimetre scale. The direct action of the coil on the rectangles
+was screened off by interposing a wire grating through which the
+%% -----File: 488.png---Folio 474-------
+wires $u$~$x$, $v$~$y$ were led. The wires $CD$,~$GH$ were cut in the
+middle and the free ends were attached to small metal plates
+$5.5$~cm.\ square; metal pieces attached to these plates went between
+the plates of the little condensers~$J$, $K$,~$L$,~$M$, the plates of these
+condensers were attached cross-wise to each other as in the figure.
+The two wires connecting the plates were attached to a bolometer
+circuit similar to that described in \artref{387}{Art.~387}. By means of a
+sliding coil attached to the bolometer circuit, Arons and Rubens
+investigated the electrical condition of the circuits $uADJ$,
+$uBCK$,~\&c., and found that approximately there was a node in
+the middle and a loop at each end; these circuits then may be
+regarded as executing electrical vibrations whose wave lengths
+are twice the lengths of the circuits. If the times of vibrations of
+the circuits on the left of~$u$,~$v$ are the same as those on the right,
+the plates $J$~and~$K$ will be in similar electrical states, as will also
+$L$~and~$M$, and there will be no deflection of the galvanometer in
+the bolometer circuit. When the wires are surrounded by air
+this will be when $u$,~$v$ are at the middle points of $AB$,~$EF$. In
+practice Arons and Rubens found that the deflection of the
+galvanometer never actually vanished, but attained a very
+decided minimum when~$u$,~$v$ were in the middle, and that the
+effect produced by sliding $u$,~$v$ through $1$~cm.\ could easily be
+detected.
+
+To determine the velocity of propagation of electromagnetic
+action through different dielectrics, one of the short sides of the
+rectangles was made so that the wires passed through a zinc box,
+$18$~cm.\ long, $13$~cm.\ broad, and $14$~cm.\ high; the wires were carefully
+insulated from the box; the wires outside the box were
+straight, but the part inside was sometimes straight and sometimes
+zigzag. This box could be filled with the dielectric under
+observation, and the velocity of propagation of the electromagnetic
+action through the dielectric was deduced from the
+alteration made in the null position (i.e.~the position in which
+the deflection of the galvanometer in the bolometer circuit was a
+minimum) of~$uv$ by filling the box with the dielectric.
+
+Let $p_1$~and~$p_2$ be the readings of the pointer attached to~$uv$
+when a straight wire of length~$D_g$ and a zigzag of length~$D_k$ are
+respectively inserted in the box, the box in this case being
+empty. Then since in each case the lengths of the circuits on the
+right and left of~$uv$ must be the same, the difference in the
+%% -----File: 489.png---Folio 475-------
+lengths of the circuits on the left, when the straight wire and
+the zigzag respectively are inserted, must be equal to the difference
+in the lengths of the circuits on the right. The length of
+the circuit on the left when the zigzag is in exceeds that when
+the straight wire is in by
+\[
+(D_k - p_2) - (D_g - p_1),
+\]
+while the difference in the length of the circuits on the right is
+\begin{DPgather*}
+p_2 - p_1; \\
+\lintertext{hence}
+D_k - D_g - (p_2 - p_1) = p_2 - p_1, \\
+\lintertext{or}
+D_k - D_g = 2(p_2 - P_1).
+\end{DPgather*}
+
+When the wires are surrounded by the dielectric, Arons and
+Rubens regard them as equivalent to wires in air, whose lengths
+are $n D_g$~and~$n D_k$, where $n$~is the ratio of the velocity of transmission
+of electromagnetic action through air to that through the
+dielectric; for the time taken by a pulse to travel over a wire of
+length~$n D_g$ in air, is the same as that required for the pulse to
+travel over the length~$D_g$ in the dielectric. We shall return to
+this point after describing the results of these experiments. If
+$p_3$~and~$p_4$ are the readings for the null positions of~$uv$ when the
+box is filled with the dielectric, then we have, on Arons and
+Rubens' hypothesis,
+\[
+n(D_k - D_g) = 2(p_4 - p_3);
+\]
+or, eliminating $D_k - D_g$,
+\[
+n = \frac{p_4 - p_3} {p_2 - p_1} ;
+\]
+hence, if $p_1$,~$p_2$, $p_3$,~$p_4$ are determined, the value of~$n$ follows
+immediately.
+
+In this way Arons and Rubens found as the values of~$n$ for
+the following substances:---
+\begin{center}
+\settowidth{\TmpLen}{Castor Oil Castor Oil}
+\begin{tabular}{p{\TmpLen} @{} c @{\quad\qquad} c}
+                   & $n$.   &  $\sqrt{K}$. \\
+Castor Oil\mdotfill & $2.05$ &  $2.16$ \\
+Olive Oil\mdotfill  & $1.71$ &  $1.75$ \\
+Xylol\mdotfill      & $1.50$ &  $1.53$ \\
+Petroleum\mdotfill  & $1.40$ &  $1.44$
+\end{tabular}
+\end{center}
+The values of~$K$, the specific inductive capacity in a slowly
+varying field, were determined by Arons and Rubens for the
+same samples as they used in their bolometer experiments.
+
+\Article{396} The method used by Arons and Rubens to reduce their
+observations leads to values of the specific inductive capacity
+%% -----File: 490.png---Folio 476-------
+which are in accordance with those found by other methods.
+It is however very difficult to see, using any theory of the
+action of the divided rectangle that has been suggested, why the
+values of the specific inductive capacity should be accurately
+deduced from the observations by this method, except in the
+particular case when the wires outside the box are very short
+compared with the wave length of the electrical vibrations.
+
+Considering the case of the single divided rectangle, there
+seem to be three ways in which it might be supposed to act.
+We may suppose that a single electrical impulse comes to~$K$ (\figureref{fig132}{Fig.~132}),
+and there splits up into two equal parts, one travelling
+round~$AD$ to~$E$, the other round~$BC$ to~$F$. If these impulses
+arrived at $E$~and~$F$ simultaneously they would, if they were of
+equal intensity, cause the electric states of $E$~and~$F$ to be similar,
+so that there would be no tendency to spark across the gap~$EF$.
+Thus, if the pulses arrived at $E$~and~$F$ undiminished in intensity,
+the condition for there to be no spark would be that the time
+taken by a pulse to travel from~$K$ to~$E$ should be equal to that from~$K$
+to~$F$. This reasoning is not applicable however when the pulse
+in its way round one side of the circuit passes through regions in
+which its velocity is not the same as when passing through air,
+because in this case the pulse will be partly reflected as it passes
+from one medium to another, and will therefore proceed with
+diminished intensity. Thus, though this pulse may arrive at the
+air gap at the same time as the pulse which has travelled round
+the other side of the rectangle, it will not have the same intensity
+as that pulse; the electrical conditions of the knobs will therefore
+be different, and there will therefore be a tendency to spark.
+When the pulse has to travel through media of high specific
+inductive capacity the reflection must be very considerable, and
+the inequality in the pulses on the two sides of the air gap
+so great that we should not expect to get under any circumstances
+such a diminution in the intensity of the sparks as we
+know from experience actually takes place. We conclude therefore
+that this method of regarding the action of the rectangle is
+not tenable.
+
+\Article{397} Another method of regarding the action is to look on
+the rectangle as the seat of vibrations, whose period is determined
+by the electrical system with which it is connected. Thus
+we may regard the potential at~$K$ as expressed by $\phi_0 \cos pt$;
+%% -----File: 491.png---Folio 477-------
+then the condition that there should be no sparks is that the
+potentials at $E$~and~$F$ should be the same. We can deduce the
+expressions for the potentials at $E$~and~$F$ from that at~$K$ when~$E$
+and~$F$ are nodes or loops. Let us consider the case when the capacity
+of the knobs $E$~and~$F$ is so small that the current at $E$~and~$F$
+vanishes. Then we can easily show by the method of \artref{298}{Art.~298}
+that if there is no discontinuity in the current along the wire,
+and if the self-induction per unit length of the wire is the same
+at all points in~$KADE$, and if the portions $AK$,~$DF$ are in air
+while $AD$~is immersed in a dielectric in which the velocity of
+propagation of electromagnetic action is~$V'$, that through air
+being~$V$, then if the potential at~$K$ is $\phi_0 \cos pt$, that at~$F$ is
+equal to
+{\setlength{\multlinegap}{0pt}%
+\begin{multline*}
+\hfill\frac{\phi_0 \cos pt} {\Delta},\hfill\\
+\shoveleft{\text{where }
+\Delta = \cos\left(\frac{p}{V'} AD\right) \cos \frac{p}{V} (KA + DF)
+  - \sin\left(\frac{p}{V'} AD\right) ×} \\
+  \left\{\mu \sin\left(\frac{p}{V} DF\right) \cos\left(\frac{p}{V} KA\right)
+    + \frac{1}{\mu} \sin\left(\frac{p}{V} KA\right) \cos\left(\frac{p}{V} DF\right) \right\},
+\end{multline*}%
+}%
+and $\mu = V/V'$.
+
+The potential at~$E$ is
+\[
+\frac{\phi_0 \cos pt} {\cos \dfrac{p}{V} KE}
+\]
+if $KE$~represents the total length $KB + BC + CE$, the whole of
+which is supposed to be surrounded by air. Hence, if the
+potentials at $E$~and~$F$ are the same, we have
+\begin{multline*}
+\cos\left(\frac{p}{V'} AD\right) \cos \frac{p}{V} (KA + DF)
+  - \sin\left(\frac{p}{V'} AD\right) × \\
+  \left\{\mu \sin\left(\frac{p}{V} DF\right) \cos\left(\frac{p}{V} KA\right)
+    + \frac{1}{\mu} \sin\left(\frac{p}{V} KA\right) \cos\left(\frac{p}{V} DF\right) \right\} \\
+  = \cos \frac{p}{V} KE.
+\Tag{1}
+\end{multline*}
+
+To make the interpretation of this equation as simple as
+possible, suppose $KA = DF$, equation~(\eqnref{397}{1}) then becomes
+\begin{multline*}
+\cos\left(\frac{p}{V'} AD\right) \cos\left(\frac{2p}{V} KA\right)
+  - \left(\mu+\frac{1}{\mu}\right)
+  \tfrac{1}{2} \sin\left(\frac{p}{V'} AD\right) \sin\left(\frac{2p}{V} KA\right) \\
+  = \cos \left(\frac{p}{V} KE\right).
+\Tag{2}
+\end{multline*}
+%% -----File: 492.png---Folio 478-------
+
+Let us now consider one or two special cases of this equation.
+Let us suppose that~$AD$ is so small that $\left(\mu + \dfrac{1}{\mu}\right) \sin \left(\dfrac{p}{V'} AD\right)$ is a
+small quantity, then equation~(\eqnref{397}{2}) may be written approximately
+\begin{DPgather*}
+\cos \left\{\frac{2p}{V} KA + \tfrac{1}{2}\left(\mu+\frac{1}{\mu}\right) \frac{p}{V'} AD \right\}
+  = \cos \left(\frac{p}{V} KE\right); \\
+\lintertext{hence}
+2KA + \tfrac{1}{2} (\mu^2 + 1)AD = KE, \\
+\lintertext{therefore}
+\frac{\delta KE}{\delta AD} = \tfrac{1}{2} (\mu^2 + 1),
+\end{DPgather*}
+so that in this case the process which Arons and Rubens applied
+to their measurements would give $(\mu^2+ 1)/2$ and not~$\mu$.
+
+If, on the other hand, $KA$~is so small that $\left(\mu + \dfrac{1}{\mu}\right) \sin \dfrac{2p}{V} KA$
+is small, equation~(\eqnref{397}{2}) may be written approximately
+{\allowdisplaybreaks
+\begin{DPgather*}
+\cos \left\{\frac{p}{V'} AD + \left(\mu + \frac{1}{\mu}\right) \frac{p}{V} KA \right\}
+  = \cos \left(\frac{p}{V} KE\right), \\
+\lintertext{or}
+\mu AD + \left(\mu + \frac{1}{\mu}\right) KA = KE, \\
+\lintertext{so that}
+\frac{\delta KE}{\delta AD} = \mu,
+\end{DPgather*}
+}
+and in this case Arons and Rubens' process gives the correct result.
+
+\Article{398} A third view of the action of the rectangle, which seems
+to be that taken by Arons and Rubens, is that the vibrations are
+not forced, but that each side of the rectangle executes its natural
+vibrations independently of the other. If the extremities are to
+keep in the same electrical states, then the times of vibration of
+the two sides must be equal.
+
+Arons and Rubens' measurements with the bolometer show
+that there is a loop at~$K$ and nodes at $E$~and~$F$.
+
+Now if $2 \pi/p$ is the time of vibration of a wire such as~$KADF$
+with a node at~$F$ and a loop at~$K$, surrounded by air along~$KA$,
+$DF$, and along~$AD$ by a medium through which electromagnetic
+action travels with the velocity~$V'$, then we can show by a
+process similar to that in \artref{298}{Art.~298} that $p$~is given by the equation
+\begin{multline*}
+\frac{1}{\mu} \cot \left(\frac{p}{V'} AD\right)
+  - \frac{1}{\mu} \cot \left(\frac{p}{V'} AD\right)
+                  \cot \left(\frac{p}{V} KA\right)
+                  \cot \left(\frac{p}{V} DF\right) \\
+  + \frac{1}{\mu^2}\cot\left(\frac{p}{V} KA\right)
+                  \cot \left(\frac{p}{V} DF\right) = 0.
+\Tag{3}
+\end{multline*}
+%% -----File: 493.png---Folio 479-------
+
+Let us take the case when $KA = DF$, then this equation
+becomes
+\begin{align*}
+\cot \left(\frac{p}{V'} AD\right)
+  &= \left(\mu + \frac{1}{\mu}\right)
+    \frac{\cot\left(\dfrac{p}{V} KA\right) }
+         {\cot^2\left(\dfrac{p}{V} KA\right) - 1 } \\
+  &= \tfrac{1}{2} \left(\mu + \frac{1}{\mu}\right) \tan\left(\frac{2p}{V} KA\right),
+\end{align*}
+\begin{DPgather*}
+\lintertext{or}
+\cot \left(\frac{2p}{V} KA\right)
+  = \tfrac{1}{2} \left(\mu + \frac{1}{\mu}\right) \tan \frac{p}{V'} AD.
+\Tag{4}
+\end{DPgather*}
+
+Let us consider the special case when $p \centerdot AD/V'$ is small, the
+solution of~(\eqnref{398}{4}) is then
+{\allowdisplaybreaks
+\begin{DPgather*}
+\frac{2p}{V} KA = \frac{\pi}{2}
+  - \tfrac{1}{2} \left(\mu + \frac{1}{\mu}\right) \frac{p}{V'} AD, \\
+\lintertext{or}
+\frac{p}{V} \left\{2KA + \frac{\mu^2 + 1}{2} AD \right\} = \frac{\pi}{2}.
+\end{DPgather*}
+}
+
+If $p'$ is the time of vibration of $KBCE$ with a loop at~$K$ and
+a node at~$E$, this wire being entirely surrounded by air, then
+\[
+\frac{p'}{V} (KE) = \frac{\pi}{2};
+\]
+hence if $p' = p$,
+\begin{DPgather*}
+2KA + \frac{\mu^2 + 1}{2} AD = KE, \\
+\lintertext{so that}
+\frac{\delta KE}{\delta AD} = \frac{\mu^2 + 1}{2}.
+\end{DPgather*}
+
+Arons and Rubens when reducing their observations took the
+ratio $\delta KE / \delta AD$ to be always equal to~$\mu$. The above investigation
+shows that this is not the case when $pAD/V'$ is small. We
+might show that $\delta KE / \delta AD$ is equal to~$\mu$ when $KA/AD$ is
+small.
+
+The results given on the third view of the electrical vibrations
+of the compound wire seem parallel to those which
+hold for vibrating strings and bars. Thus if we have three
+strings of different materials stretched in series between two
+points, the time of longitudinal vibration of this system is not
+proportional to the sum of the times a pulse would take to travel
+over the strings separately (see Routh's \textit{Advanced Rigid Dynamics},
+\index{Routh@Routh's \textit{Rigid Dynamics}}%
+p.~397), but is given by an equation somewhat resembling~(\eqnref{398}{3}).
+%% -----File: 494.png---Folio 480-------
+
+\Article{399} The discrepancy between the results of the preceding
+theory of the action of the divided rectangle and the method
+employed by Arons and Rubens to reduce their observations, may
+perhaps explain to some extent the difference between the values
+of the specific inductive capacity of glass in rapidly alternating
+electric fields obtained by these observers and those obtained by
+\index{Blondlot, velocity of electromagnetic waves@\subdashone velocity of electromagnetic waves}%
+M.~Blondlot and myself for the same quantity.
+
+Arons and Rubens (\textit{Wied.\ Ann.}\ 44, p.~206, 1891) determined
+the ratio of the velocity of electromagnetic action through air to
+that through glass by filling with glass blocks a box through which
+the wires on one side of their rectangle passed. Employing the
+same method of reduction as for liquid dielectrics, they found~$\mu$ (the
+ratio of the velocities) to be~$2.33$, whence $K = \mu^{2}$ is~$5.43$; while the
+value of~$K$ for the same glass, in slowly varying fields, was~$5.37$,
+which is practically identical with the preceding value. If, however,
+we adopted the method of reduction indicated by the preceding
+theory we should get a considerably smaller value of~$K$.
+In order to see what kind of diminution we might expect, let
+us suppose that the circuit through the glass is so short that
+the relation expressed by~(\eqnref{398}{4}) holds. This gives the same value
+for $(K+1)/2$ as Arons and Rubens get for~$\mu$; hence we find
+$K = 3.66$, a value considerably less than under steady fields.
+
+\includegraphicsouter{fig134}{Fig.~134.}
+
+\Article{400} Arons and Rubens checked their method by finding by
+means of it the specific inductive capacity of paraffin. This
+substance happens to be one for which either method of reduction
+leads to very much the same result. For example, for fluid
+paraffin their method of reduction gave $\mu = \sqrt{K} = 1.47$, $K = 2.16$;
+if we suppose that we ought to have $(K + 1)/2$ instead of~$\mu$ we
+get $K = 1.94$, while the value in slowly varying fields is~$1.98$;
+so that the result for this substance is not decisive between the
+methods of reduction.
+
+Both M.~Blondlot and myself found that the specific inductive
+capacity of glass was smaller under rapidly changing fields than
+in steady ones. The following is the method used by M.~Blondlot
+(\textit{Comptes Rendus}, May~11, 1891, p.~1058; \textit{Phil.\ Mag.}\ [5], 32,
+p.~230, 1891). A large rectangular plate of copper~$AA'$, \figureref{fig134}{Fig.~134},
+is fixed vertically, and a second parallel and smaller plate~$BB'$
+forms a condenser with the first. This condenser discharges
+itself by means of the knobs $a$,~$b$; $a$~is connected with the gas
+pipes, $b$~with one pole of an induction-coil, the other pole of
+%% -----File: 495.png---Folio 481-------
+which is connected to the gas pipes. When the coil is working,
+electrical oscillations take place in the condenser, the period
+of which is of the order $1/25,000,000$ of a second. There is thus
+on the side of~$AA'$ a periodic electric field which has~$xx$ as the
+plane of symmetry. Two square plates, $CD$,~$C'D'$, are placed in
+this field parallel to~$AA'$ and symmetrical
+with respect to~$xx$; two
+wires terminating in~$EE'$ are
+soldered at~$DD'$ to the middle
+points of the sides of these plates.
+The wires are connected at~$EE'$ to
+two carbon points kept facing each
+other at a very small distance apart.
+
+When the coil is working no
+sparks are observed between~$E$
+and~$E'$, this is due to the symmetry
+of the apparatus. When,
+however, a glass plate is placed
+between~$AA'$ and~$CD$ sparks immediately
+pass between~$E$ and~$E'$;
+these are caused by the induction
+received by~$CD$ differing from
+that received by~$C'D'$. By interposing
+between~$AA'$ and~$C'D'$ a
+sheet of sulphur of suitable thickness
+the sparks can be made to
+disappear again. We can thus
+find the relative thicknesses of
+plates of glass and sulphur which
+produce the same effect on the electromagnetic waves passing
+through them, and we can therefore compare the specific inductive
+capacity of glass and sulphur under similar electrical conditions.
+M.~Blondlot found the specific inductive capacity of the sulphur
+he employed by Curie's method (\textit{Annales de Chimie et de Physique},
+[6], 17, p.~385, 1889), and assuming that its inductive
+capacity was the same in rapidly alternating fields as in steady
+ones, he found the specific inductive capacity of the glass to be~$2.84$,
+which is considerably less than its value in steady fields.
+
+\Article{401} I had previously (\textit{Proc.\ Roy.\ Soc.}\ 46, p.~292) arrived at
+the same conclusion by measuring the lengths of the electrical
+%% -----File: 496.png---Folio 482-------
+waves emitted by a parallel plate condenser, (1)~when the
+plates were separated by air, (2)~when they were separated by
+glass. The period of vibration of the condenser depends upon
+its capacity, and this again upon the dielectric between the
+plates, so that the determination of the periods gives us the
+means of determining the specific inductive capacity of the glass.
+The parallel plate condenser loses its energy by radiation slowly,
+and will thus force the vibration of its own period upon any
+electrical system under its influence. It differs in this respect
+from the condenser in \figureref{fig113}{Fig.~113}, which radiates its energy away
+so rapidly that its action on neighbouring electrical conductors
+approximates to an impulse which starts the free vibrations of
+such systems.
+
+The wave lengths in those observations were determined by
+observations on sparks. This is not comparable in delicacy
+with the bolometric method of Arons and Rubens; the method
+was however sufficiently sensitive to show a considerable falling
+off in the specific inductive capacity of the glass, for which
+I obtained the value~$2.7$, almost coincident with that obtained
+by M.~Blondlot. Sulphur and ebonite on the other hand, when
+tested in the same way, showed no appreciable change in their
+specific inductive capacity.
+
+
+\Subsection{The Effects produced by a Magnetic Field on Light.}
+\index{Light, yeffect of magnetic field on@\subdashone effect of magnetic field on|indexetseq}%
+\index{Magnetic zzfield@\subdashone field, effect of on light|indexetseq}%
+
+\Article{402} The connection between optical and electromagnetic
+\index{Faraday, rotation of plane of polarization of light@\subdashone rotation of plane of polarization of light}%
+\index{Rotation of plane of polarization of light}%
+phenomena is illustrated by the effects produced by a magnetic
+field on light passing through it. Faraday was the first to discover
+the action of magnetism on light; he found (\textit{Experimental
+Researches}, vol.~3, p.~1) that when plane polarized light passes
+through certain substances, such as bisulphide of carbon or
+heavy glass, placed in a magnetic field where the lines of force
+are parallel to the direction of propagation of the light, the
+plane of polarization is twisted round the direction of the
+magnetic force. The laws of this phenomenon are described in
+Maxwell's \textit{Electricity and Magnetism}, Chapter~XXI\@.
+
+\Article{403} Subsequent investigations have shown that a magnetic
+field produces other effects upon light, which, though they
+probably have their origin in the same cause as that which produces
+the rotation of the plane of polarization in the magnetic
+field, manifest themselves in a different way.
+%% -----File: 497.png---Folio 483-------
+
+\index{Magnets, reflection of light from@\subdashone reflection of light from|indexetseq}%
+\index{Reflection of light from a magnet@\subdashtwo light from a magnet}%
+\index{Kerr, reflection of light from the pole of a magnet}%
+\index{Righi, reflection from a magnet@\subdashone reflection from a magnet}%
+\index{Kundt, reflection of light from a magnet@\subdashone reflection of light from a magnet}%
+\index{Du Bois, reflection of light from a magnet}%
+\index{Sissingh, reflection of light from a magnet}%
+Thus Kerr (\textit{Phil.\ Mag.}\ [5], 3, p.~321, 1877), whose experiments
+have been verified and extended by Righi (\textit{Annales de Chimie et
+de Physique}, [6], 4, p.~433, 1885; 9, p.~65, 1886; 10, p.~200,
+1887), Kundt (\textit{Wied.\ Ann.}\ 23, p.~228, 1884), Du~Bois (\textit{Wied.\
+Ann.}\ 39, p.~25, 1890), and Sissingh (\textit{Wied.\ Ann.}\ 42, p.~115, 1891)
+found that when plane polarized light is incident on the pole of
+an electromagnet, polished so as to act like a mirror, the plane
+of polarization of the reflected light is not the same when the
+magnet is `on' as when it is `off.'
+
+The simplest case is when the incident plane polarized light
+falls normally on the pole of an electromagnet. In this case,
+when the magnet is not excited, the reflected ray is plane
+polarized, and can be completely stopped by an analyser placed
+in a suitable position. If the analyser is kept in this position
+and the electromagnet excited, the field, as seen through the
+analyser, is no longer quite dark, but becomes so, or very nearly
+so, when the analyser is turned through a small angle, showing
+that the plane of polarization has been twisted through a small
+angle by reflection from the magnetized iron. Righi~(l.c.)\ has
+shown that the reflected light is not quite plane polarized, but
+that it is elliptically polarized, the axes of the ellipse being of
+very unequal magnitude. These axes are not respectively in and
+at right angles to the plane of incidence. If we regard for
+a moment the reflected elliptically \DPtypo{polarised}{polarized} light as approximately
+plane polarized, the plane of polarization being that through
+the major axis of the ellipse, the direction of rotation of the plane
+of polarization depends upon whether the pole from which the
+light is reflected is a north or south pole. Kerr found that the
+direction of rotation was opposite to that of the currents exciting
+the pole from which the light was reflected.
+
+The rotation produced is small. Kerr, who used a small
+electromagnet, had to concentrate the lines of magnetic force in
+the neighbourhood of the mirror by placing near to this a large
+mass of soft iron, before he could get any appreciable effects.
+\index{Gordon, reflection of light from a magnet}%
+By the use of more powerful magnets Gordon and Righi have
+succeeded in getting a difference of about half a degree between
+the positions of the analyser for maximum darkness with the
+magnetizing current flowing first in one direction and then in the
+opposite.
+
+A piece of gold-leaf placed over the pole entirely stops the
+%% -----File: 498.png---Folio 484-------
+magnetic rotation, thus proving that the rotation of the plane
+of polarization is not produced in the air.
+
+Hall (\textit{Phil.\ Mag.}\ [5], 12, p.~157, 1881) found that the rotation
+takes place when the light is reflected from nickel or cobalt,
+instead of from iron, and is in the same direction as for iron.
+
+Righi~(l.c.)\ showed that the amount of rotation depends on the
+nature of the light; the longer the wave length the greater (at
+least within the limits of the luminous spectrum) the rotation.
+
+
+\Subsection{Oblique Incidence on a Magnetic Pole.}
+
+\Article{404} When the light is incident obliquely and not normally
+on the polished pole of an electromagnet it is necessary, in
+order to be able to measure the rotation, that the incident light
+should be polarized either in or at right angles to the plane of
+incidence, since it is only in these two cases that plane polarized
+light remains plane polarized after reflection from a metallic
+surface, even though this is not in a magnetic field. When light
+polarized in either of these planes is incident on the polished pole
+of an electromagnet, the light, when the magnet is on, is elliptically
+polarized after reflection, and the major and minor axes
+of the ellipse are not respectively in and at right angles to the
+plane of incidence. The ellipticity of the reflected light is very
+small. If we regard the light as consisting of two plane polarized
+waves of unequal amplitudes and complementary phases, then
+the rotation from the plane of polarization of the incident wave
+to that of the plane in which the amplitude of the reflected wave
+is greatest is in the direction opposite to that of the currents
+which circulate round the poles of the electromagnet.
+
+According to Righi the amount of this rotation when the
+incident light is polarized in a plane perpendicular to that of
+incidence reaches a maximum when the angle of incidence is
+between~$44°$ and~$68°$; while when the light is polarized in the
+plane of incidence the rotation steadily decreases as the angle of
+incidence is increased. The rotation when the light is polarized
+in the plane of incidence is always less than when it is polarized
+at right angles to that plane, except when the incidence is normal,
+when of course the two rotations are equal.
+
+These results of Righi's differ in some respects from those of
+some preceding investigations by Kundt, who, when the light
+%% -----File: 499.png---Folio 485-------
+was polarized at right angles to the plane of incidence, obtained
+a reversal of the sign of the rotation of the plane of polarization
+near grazing incidence.
+
+
+\Subsection{Reflection from Tangentially Magnetized Iron.}
+
+\Article{405} In the preceding experiments the lines of magnetic force
+were at right angles to the reflecting surface; somewhat similar
+effects are however produced when the mirror is magnetized
+tangentially. In this case Kerr (\textit{Phil.\ Mag.}\ [5], 5, p.~161, 1878)
+found:---
+
+1. That when the plane of incidence is perpendicular to the
+lines of magnetic force no change is produced by the magnetization
+on the reflected light.
+
+2. No change is produced at normal incidence.
+
+3. When the incidence is oblique, the lines of magnetic force
+being in the plane of incidence, the reflected light is elliptically
+polarized after reflection from the magnetized surface, and the
+axes of the ellipse are not in and at right angles to the plane of
+incidence. When the light is polarized in the plane of incidence,
+the rotation of the plane of polarization (that is the rotation from
+the original plane to the plane through the major axis of the
+ellipse) is for all angles of incidence in the opposite direction to
+that of currents which would produce a magnetic field of the
+same sign as the magnet. When the light is polarized at right
+angles to the plane of incidence, the rotation is in the same
+direction as these currents when the angle of incidence is between
+$0°$~and~$75°$ according to Kerr, between $0°$~and~$80°$ according to
+Kundt, and between $0°$~and~$78°\,54'$ according to Righi. When the
+incidence is more oblique than this, the rotation of the plane of
+polarization is in the opposite direction to the electric currents
+which would produce a magnetic field of the same sign.
+
+\Article{406} Kerr's experiments were confined to the case of light
+reflected from metallic surfaces. Kundt (\textit{Phil.\ Mag.}\ [5], 18,
+p.~308, 1884) has made a most interesting series of observations
+of the effect of thin plates of the magnetic metals iron, nickel
+and cobalt, on the plane of polarization of light passing through
+these plates in a strong magnetic field where the lines of force are
+at right angles to the surface of the plates.
+
+Kundt found that in these circumstances the magnetic metals
+possess to an extraordinary degree the power of rotating the
+%% -----File: 500.png---Folio 486-------
+plane of polarization of the light. The rotation due to an iron
+plate is for the mean rays of the spectrum more than $30,000$
+times that of a glass plate of the same thickness in the same
+magnetic field, and nearly $1,500$ times the natural rotation
+(i.e.~the rotation independent of magnetic force) due to a plate
+of quartz of the same thickness. The rotation of the plane
+of polarization is with all three substances in the direction
+of the currents which would produce a magnetic field of the
+same sign as the one producing the rotation. The rotation
+under similar circumstances is nearly the same for iron and
+cobalt, while for nickel it is decidedly weaker. The rotation is
+greater for the red rays than for the blue.
+
+\Article{407} The phenomena discovered by Kerr show that when the
+rapidly alternating currents which accompany light waves are
+flowing through iron, nickel, or cobalt in a magnetic field,
+electromotive intensities are produced which are at right angles
+both to the current and the magnetic force. Let us take, for
+example, the simple case when light is incident normally on the
+pole of an electromagnet. Let us suppose that the incident
+light is polarized in the plane of~$zx$, where $z=0$ is the equation to
+the reflecting surface, so that in the incident wave the electromotive
+intensities and the currents are at right angles to this
+plane; Kerr found, however, that the reflected wave had a component
+polarized in the plane of~$yz$; thus after reflection there
+are electromotive intensities and currents parallel to~$x$, that is at
+right angles to both the direction of the external magnetic field
+which is parallel to~$z$ and to the intensities in the incident wave
+which are parallel to~$y$.
+
+
+\Subsection{The Hall Effect.}
+\index{Hall@`Hall effect'}%
+
+\Article{408} In the \textit{Philosophical Magazine} for November, 1880, Hall
+published an account of some experiments, which show that when
+a steady current is flowing in a steady magnetic field electromotive
+intensities are developed which are at right angles both to
+the magnetic force and to the current, and are proportional to the
+product of the intensity of the current, the magnetic force and the
+sine of the angle between the directions of these quantities.
+\includegraphicsouter{fig135}{Fig.~135.}
+The nature of the experiments by which this effect was demonstrated
+was as follows: A thin film of metal was deposited on a glass
+plate; this plate was placed over the pole of an electromagnet
+%% -----File: 501.png---Folio 487-------
+and a steady current sent through the film from two electrodes.
+The distribution of the current was indicated by finding two
+places in the film which were at the same potential; this was
+done by finding two points such that if they were placed in
+electrical connection with
+the terminals of a delicate
+galvanometer~($G$) they produced
+no current through it
+when the electromagnet was
+`off.' If now the current
+was sent through an electromagnet
+a deflection of the
+galvanometer~($G$) was produced,
+and this continued as
+long as the electromagnet was `on,' showing that the distribution
+of current in the film was altered by the magnetic field. The
+method used by Hall to measure this effect is described in the
+following extract taken from one of his papers on this subject
+(\textit{Phil.\ Mag.}\ [5], 19, p.~419, 1885). `In most cases, when possible,
+the metal was used in the form of a thin strip about $1.1$~centim.\
+wide and about $3$~centim.\ long between the two pieces of brass
+$B$,~$B$ (\figureref{fig135}{Fig.~135}), which, soldered to the ends of the strip, served
+as electrodes for the entrance
+and escape of the main current.
+To the arms $a$,~$a$, about
+$2$~millim.\ wide and perhaps $7$~millim.\
+long, were soldered the
+wires $w$,~$w$, which led to a
+Thomson galvanometer. The
+notches $c$,~$c$ show how adjustment
+was secured. The strip
+thus prepared was fastened to
+a plate of glass by means of a
+cement of beeswax and rosin,
+all the parts shown in the
+figure being imbedded in and
+covered by this cement, which was so hard and stiff as to be
+quite brittle at the ordinary temperature of the air.
+
+\includegraphicsouter{fig136}{Fig.~136.}
+
+`The plate of glass bearing the strip of metal so embedded was,
+when about to be tested, placed with $B$,~$B$ vertical in the narrow
+%% -----File: 502.png---Folio 488-------
+part of a tank whose horizontal section is shown in \figureref{fig136}{Fig.~136}.
+This tank,~$TT$, containing the plate of glass with the metal strip
+was placed between the poles~$PP$ of the electromagnet. The
+tank was filled with water which was sometimes at rest and
+sometimes flowing. By this means the temperature of the strip
+of metal was under tolerable control, and the inconvenience
+from thermoelectric effects at $a$~and~$a$ considerably lessened. The
+diameter of the plane circular ends of the pole pieces~$PP$ were
+about $3.7$~centim.'
+
+By means of experiments of this kind Hall arrived at the conclusion
+that if $\alpha$,~$\beta$,~$\gamma$; $u$,~$v$,~$w$ denote respectively the components
+of the magnetic force and the intensity of the current,
+electromotive intensities are set up whose components parallel
+to the axes of $x$,~$y$,~$z$ are respectively
+\[
+C(\beta  w - \gamma v), \quad
+C(\gamma u - \alpha w), \quad
+C(\alpha v - \beta u).
+\]
+The values of~$C$ in electromagnetic units for some metals at
+$20°$\,C, as determined by Hall (\textit{Phil.\ Mag.}\ [5], 19, p.~419, 1885),
+are given in the following table (l.c.\ p.~436):---
+\begin{center}
+\begin{tabular}{p{3in}}
+\quad\small Metal.\hfill $C × 10^{15}.\Z$\\
+Copper\mdotfill$- 520$ \\
+Zinc\mdotfill$+ 820$ \\
+Iron\mdotfill$+ 7850$ \\
+Steel, soft\mdotfill$+ 12060$ \\
+\PadTo{\text{Steel,}}{\Ditto} tempered\mdotfill$+ 33000$ \\
+Cobalt\mdotfill$+ 2460$ \\
+Nickel\mdotfill$- 14740$ \\
+Bismuth\mdotfill$- 8580000$ \\
+Antimony\mdotfill$+ 114000$ \\
+Gold\mdotfill$- 660$
+\end{tabular}
+\end{center}
+
+With regard to the magnetic metals, it is not certain that the
+quantity primarily involved in the Hall effect is the magnetic
+force rather than the magnetic induction, or the intensity of
+magnetization. Hall's experiments with nickel seem to point
+to its being the last of these three, as he found, using strong
+magnetic fields, that the effect ceased to be proportional to
+the external magnetic field, and fell off in a way similar to
+that in which the magnetization falls off when the field is increased.
+We must remember, if we use Hall's value of~$C$ for
+iron and the other magnetic metals, to use in the expression for
+%% -----File: 503.png---Folio 489-------
+the electromotive intensities the magnetic induction instead of
+the magnetic force. For in Hall's experiments the magnetic
+force measured was the normal magnetic force outside the iron.
+Since the plate was very thin the normal magnetic force outside
+the iron would be large compared with that inside; the normal
+magnetic induction inside would however be equal to the normal
+magnetic force outside, so that Hall in this case measured the
+relation between the electromotive intensity produced and the
+magnetic induction producing it.
+
+Hall has thus established for steady currents the existence of
+an effect of the same nature as that which Kerr's experiments
+proved (assuming the electromagnetic theory of light) to exist
+for the rapidly alternating currents which constitute light. Here
+however the resemblance ends; the values of the coefficient~$C$
+deduced by Hall from his experiments on steady currents do not
+apply to rapidly alternating light currents. Thus Hall found
+that for steady currents the sign of~$C$ was positive for iron,
+negative for nickel; the magneto-optical properties of these
+bodies are however quite similar. Again, both Hall and Righi
+found that the~$C$ for bismuth was enormously larger than that
+for iron or nickel. Righi, however, was unable to find any
+traces of magneto-optical effects in bismuth.
+
+The optical experiments previously described show that there
+is an electromotive intensity at right angles both to the magnetic
+force and to the electromotive intensity; they do not however
+show without further investigation on what function of the
+electromotive intensity the magnitude of the transverse intensity
+depends. Thus, for example, the complete current in the metal is
+the sum of the polarization and conduction currents. Thus, if
+the electromotive intensity is~$X$, the total current~$u$ is given by
+the equation
+\[
+u = \left(\frac{K'}{4\pi}\, \frac{d}{dt} + \frac{1}{\sigma}\right)X,
+\]
+or if the effects are periodic and proportional to~$\epsilon^{\iota pt}$,
+\[
+u = \left(\frac{K'}{4\pi}\, \iota p + \frac{1}{\sigma}\right)X,
+\]
+where $K'$~is the specific inductive capacity of the metal and~$\sigma$
+its specific resistance.
+
+We do not know from the experiments, without further discussion,
+%% -----File: 504.png---Folio 490-------
+whether the transverse electromotive intensity is proportional
+to~$u$, the total current, or only to $K'\iota pX/4\pi$, the polarization
+part of it, or to~$X/\sigma$, the conduction current.
+
+We shall assume that the components of the transverse electromotive
+intensity are given by the expressions
+\begin{gather*}
+k(bw - cv), \\
+k(cu - aw), \\
+k(av - bu);
+\end{gather*}
+where $a$,~$b$,~$c$ are the components of the magnetic induction,
+$u$,~$v$,~$w$ those of the total current.
+
+This form, if $k$~is a real constant, makes the transverse intensity
+proportional to the total current; the form is however sufficiently
+general analytically to cover the cases where the transverse
+intensity is proportional to the polarization current alone or to
+the conduction one. Thus, if we put
+\[
+k = \left(\frac{K'\iota p/4\pi}{K'\iota p/4\pi + 1/\sigma}\right)k',
+\]
+where $k'$~is a real constant, the transverse intensity will be proportional
+to the displacement current; while if we put
+\[
+k = \frac{k''}{K'\iota p/4\pi + 1/\sigma},
+\]
+where $k''$~is a real constant, the transverse intensity will be proportional
+to the conduction current. We shall now proceed to investigate
+which, if any, of these hypotheses will explain the
+results observed by Kerr.
+
+\Article{409} Let $P$,~$Q$,~$R$ be the components of the electromotive
+intensity in a conductor, $P'$,~$Q'$,~$R'$ the parts of these which arise
+from electromagnetic induction, $a$,~$b$,~$c$ the components of the
+magnetic induction, $\alpha$,~$\beta$,~$\gamma$ those of the magnetic force, $u$,~$v$,~$w$ the
+components of the current. $K'$,~$\mu'$,~$\sigma$ are respectively the specific
+inductive capacity, the magnetic permeability, and the specific
+resistance of the metal.
+
+Then we have in the metal
+\begin{align*}
+P & = P' + k(bw - cv), \\
+Q & = Q' + k(cu - aw), \\
+R & = R' + k(av - bu),
+\end{align*}
+where $k$~is a coefficient which bears the same relation to rapidly
+alternating currents as~$C$ (\artref{408}{Art.~408}) does to steady currents. If the
+%% -----File: 505.png---Folio 491-------
+external field is very strong, we may without appreciable error
+substitute for $a$,~$b$,~$c$, in the terms multiplied by~$k$, $a_0$,~$b_0$,~$c_0$, the
+components of the external field. We shall suppose that this
+field is uniform, so that $a_0$,~$b_0$,~$c_0$ are independent of~$x$,~$y$,~$z$.
+
+By equation~(\eqnref{256}{2}) of \artref{256}{Art.~256}
+\begin{align*}
+\frac{da}{dt} & = \frac{dQ'}{dz} - \frac{dR'}{dy} \\
+  & = \frac{dQ}{dz} - \frac{dR}{dy}
+    - k\left(a_0\, \frac{d}{dx} + b_0\, \frac{d}{dy} + c_0\, \frac{d}{dz}\right) u,
+\Tag{1}
+\end{align*}
+since $\dfrac{du}{dx} + \dfrac{dv}{dy} + \dfrac{dw}{dz} = 0$ on Maxwell's hypothesis that all the
+currents are closed. Now since~$u$ is the component of the total
+current parallel to~$x$, it is equal to the sum of the components of
+the polarization and conduction currents in that direction. The
+polarization current is equal to
+\[
+\frac{K'}{4\pi}\, \frac{dP}{dt},
+\]
+the conduction current to~$P/\sigma$, hence
+\[
+4\pi u = K'\, \frac{dP}{dt} + \frac{4\pi}{\sigma}\, P.
+\]
+We shall confine our attention to periodic currents and suppose
+that the variables are proportional to~$\epsilon^{\iota pt}$; in this case the preceding
+equation becomes
+\begin{DPalign*}
+4\pi u & = (K'\iota p + 4\pi /\sigma) P, \\
+\lintertext{but}
+4\pi u & = \frac{d\gamma}{dy} - \frac{d\beta}{dz};
+\end{DPalign*}
+hence we have
+\begin{DPalign*}
+(K'\iota p + 4\pi/\sigma) P & = \frac{d\gamma}{dy} - \frac{d\beta}{dz}; \\
+\lintertext{similarly}
+(K'\iota p + 4\pi/\sigma) Q & = \frac{d\alpha}{dz} - \frac{d\gamma}{dx}, \\
+(K'\iota p + 4\pi/\sigma) R & = \frac{d\beta}{dx} - \frac{d\alpha}{dy};
+\end{DPalign*}
+and therefore since
+\begin{gather*}
+\frac{d\alpha}{dx} + \frac{d\beta}{dy} + \frac{d\gamma}{dz} = 0, \\
+(K'\iota p + 4\pi/\sigma) \left(\frac{dQ}{dz} - \frac{dR}{dy}\right)
+  = \frac{d^2\alpha}{dx^2} + \frac{d^2\alpha}{dy^2} + \frac{d^2\alpha}{dz^2};
+\end{gather*}
+%% -----File: 506.png---Folio 492-------
+and hence equation~(\eqnref{409}{1}) becomes
+\begin{multline*}
+(K' \iota p + 4 \pi / \sigma) \mu'\, \frac{d \alpha}{dt}
+  = \frac{d^{2} \alpha}{dx^{2}} + \frac{d^{2} \alpha}{dy^{2}} + \frac{d^{2} \alpha}{dz^{2}} \\
+  - \frac{k}{4 \pi} (K' \iota p + 4 \pi / \sigma)
+    \left( a_0\, \frac{d}{dx} + b_0\, \frac{d}{dy} + c_0\, \frac{d}{dz} \right)
+    \left( \frac{d \gamma}{dy} - \frac{d \beta}{dz} \right).
+\end{multline*}
+Similarly we have
+\[
+\left.\begin{aligned}
+&(K' \iota p + 4 \pi / \sigma) \mu'\, \frac{d \beta}{dt}
+  = \frac{d^{2} \beta}{dx^{2}} + \frac{d^{2} \beta}{dy^{2}} + \frac{d^{2} \beta}{dz^{2}} \\
+&\quad - \frac{k}{4 \pi} (K' \iota p + 4 \pi / \sigma)
+    \left( a_0\, \frac{d}{dx} + b_0\, \frac{d}{dy} + c_0\, \frac{d}{dz} \right)
+    \left( \frac{d \alpha}{dz} - \frac{d \gamma}{dx} \right), \\
+&(K' \iota p + 4 \pi / \sigma) \mu'\, \frac{d \gamma}{dt}
+  = \frac{d^{2} \gamma}{dx^{2}} + \frac{d^{2} \gamma}{dy^{2}} + \frac{d^{2} \gamma}{dz^{2}} \\
+& \quad - \frac{k}{4 \pi} (K' \iota p + 4 \pi / \sigma)
+    \left( a_0\, \frac{d}{dx} + b_0\, \frac{d}{dy} + c_0\, \frac{d}{dz} \right)
+    \left( \frac{d \beta}{dx} - \frac{d \alpha}{dy} \right).
+\end{aligned}\right\}
+\Tag{2}
+\]
+We are now in a position to discuss the reflection of waves of
+light from a plane metallic surface. Let us take the plane
+separating the metal from the air as the plane of~$xy$, the plane of
+incidence as the plane of~$xz$; the positive direction along~$z$ is
+from the metal to the air.
+
+Let us suppose that waves of magnetic force are incidence on
+the metal, these incident waves may be expressed by equations
+of the form\nbpagebreak[0]
+\begin{DPgather*}
+\alpha = A_0\, \epsilon^{\iota(lx + mz + pt)},\\
+\beta  = B_0\, \epsilon^{\iota(lx + mz + pt)}; \\
+\lintertext{and since}
+\frac{d \alpha}{dx} + \frac{d \beta}{dy} + \frac{d \gamma}{dz} = 0,\\
+\gamma = - \frac{l}{m} A_0\, \epsilon^{\iota(lx + mz + pt)}, \\
+\lintertext{where}
+l^{2} + m^{2} = \frac{p^{2}}{V^{2}},
+\end{DPgather*}
+and $A_0$~and~$B_0$ are constants.
+
+$V$~is the velocity of propagation of electromagnetic action
+through the air and so is equal to~$1/K$, where $K$~is the electromagnetic
+measure of the specific inductive capacity of the air,
+whose magnetic permeability is taken as unity. Waves will be
+reflected from the surface of the metal, and the amplitudes of
+these waves will be proportional to $\epsilon^{\iota(lx - mz + pt)}$ , so that $\alpha$,~$\beta$,~$\gamma$,
+the components of the total magnetic force in the air, will, since
+%% -----File: 507.png---Folio 493-------
+it is due to both the incident and reflected waves, be represented
+by the equations
+\begin{align*}
+\alpha & = A_0\, \epsilon^{\iota(lx + mz + pt)} + A\, \epsilon^{\iota(lx - mz + pt)}, \\
+\beta  & = B_0\, \epsilon^{\iota(lx + mz + pt)} + B\, \epsilon^{\iota(lx - mz + pt)}, \\
+\gamma & = - \frac{l}{m} A_0\, \epsilon^{\iota(lx + mz + pt)}
+  + \frac{l}{m} A\, \epsilon^{\iota(lx - mz + pt)},
+\end{align*}
+where $A$~and~$B$ are constants.
+
+We shall suppose that the metal is so thick that there is no
+reflection except from the face $z = 0$; in this case the waves in
+the metal will travel in the negative direction of~$z$.
+
+Thus in the metal we may put
+\begin{align*}
+\alpha & = A'\, \epsilon^{\iota(lx + m'z + pt)}, \\
+\beta  & = B'\, \epsilon^{\iota(lx + m'z + pt)}, \\
+\gamma & = - \frac{l}{m'} A'\, \epsilon^{\iota(lx + m'z + pt)},
+\end{align*}
+where if $m'$~is complex the real part must be positive in order
+that the equations should represent a wave travelling in the
+negative direction of~$z$; the imaginary part of~$m'$ must be
+negative, otherwise the amplitude of the wave of magnetic force
+would increase indefinitely as the wave travelled along.\nbpagebreak[1]
+
+Substituting these values of $\alpha$,~$\beta$,~$\gamma$ in equations~(\eqnref{409}{2}), we get
+\begin{multline*}
+A'(-p^{2} \mu' K' + 4 \pi \mu' \iota p / \sigma + l^{2} + m'^{2}) \\
+  = - \frac{k}{4 \pi} (K' \iota p + 4 \pi / \sigma) m' (l a_0 + m' c_0) B',
+\Tag{3}
+\end{multline*}
+\begin{multline*}
+B'(-p^{2} \mu' K' + 4 \pi \mu' \iota p / \sigma + l^{2} + m'^{2}) \\
+  = \frac{k}{4 \pi} (K'\iota p + 4 \pi / \sigma)
+    \frac{l^{2} + m'^{2}}{m'} (l a_0 + m' c_0) A'.
+\Tag{4}
+\end{multline*}
+Eliminating $A'$~and~$B'$ from these equations, we get
+\begin{multline*}
+-p^{2} \mu' K' + 4 \pi \mu' \iota p / \sigma + l^{2} + m'^{2} \\
+  = ± \frac{\iota k}{4 \pi} (K' \iota p + 4 \pi / \sigma)
+    (l^{2} + m'^{2})^{\frac{1}{2}} (l a_0 + m' c_0).
+\Tag{5}
+\end{multline*}
+
+There are only two values of~$m'$ which satisfy this equation and
+which have their real parts positive and their imaginary parts
+negative. We shall denote these two roots by~$m_1$,~$m_2$; $m_1$~being
+the root when the plus sign is taken in the ambiguity in sign
+in equation~(\eqnref{409}{4}), $m_2$~the root when the minus sign is taken.
+%% -----File: 508.png---Folio 494-------
+
+We have from equation~(\eqnref{409}{3}), if $A_1$~and~$B_1$ are the values of~$A'$
+and~$B'$ corresponding to the root~$m_1$,
+\begin{DPgather*}
+A_1 \iota (l^{2} + m_1^{2})^{\frac{1}{2}} = -B_1 m_1; \\
+\lintertext{or if}
+l^{2} + m_1^{2} = \omega_1 ^{2}, \\
+A_1 \iota \omega_1 = -B_1 m_1.
+\end{DPgather*}
+If $A_2$~and~$B_2$ are the values of $A'$~and~$B'$ corresponding to the
+root~$m_2$, we have
+\begin{DPgather*}
+A_2 \iota \omega_2 = B_2 m_2, \\
+\lintertext{where}
+l^{2} + m_2^{2} = \omega_2^{2}.
+\end{DPgather*}
+Thus in the metal we have
+\begin{align*}
+\alpha & = A_1\, \epsilon^{\iota(lx + m_1 z + pt)}
+         + A_2\, \epsilon^{\iota(lx + m_2 z + pt)}, \\
+\beta  & = - \frac{\iota \omega_1}{m_1} A_1\, \epsilon^{\iota(lx + m_1 z + pt)}
+           + \frac{\iota \omega_2}{m_2} A_2\, \epsilon^{\iota(lx + m_2 z + pt)}, \\
+\gamma & = - \frac{l}{m_1} A_1\, \epsilon^{\iota(lx + m_1 z + pt)}
+           - \frac{l}{m_2} A_2\, \epsilon^{\iota(lx + m_2 z + pt)}.
+\end{align*}
+We thus see that the original plane wave is in the metal split
+up into two plane waves travelling with the velocities $p / \omega_1$,
+$p / \omega_2$ respectively. We also see from the equations for $\alpha$,~$\beta$,~$\gamma$
+that the waves are two circularly polarized ones travelling with
+different velocities. Starting from this result Prof.\ G.~F. Fitzgerald
+\index{Fitzgerald, rotation of plane of polarization of light@\subdashone rotation of plane of polarization of light}%
+(\textit{Phil.\ Trans.}\ 1880, p.~691) has calculated the rotation of the
+plane of polarization produced by reflection from the surface of a
+\emph{transparent} medium which under the action of magnetic force
+splits up a plane wave into two circularly polarized ones; some
+of the results which he has arrived at are not in accordance
+with the results of Kerr's and Righi's experiments on the reflection
+from metallic surfaces placed in a magnetic field, proving
+that in them we must take into account the opacity of the
+medium if we wish to completely explain the results of these
+experiments.
+
+\Article{410} In order to determine the reflected and transmitted waves
+we must introduce the boundary conditions. We assume (1)~that
+$\alpha$~and~$\beta$, the tangential components of the magnetic force,
+are continuous; (2)~that the normal magnetic induction is continuous;
+and (3)~that the part of the tangential electromotive
+intensity which is due to magnetic induction is continuous. It
+should be noticed that condition~(3) makes the total tangential
+%% -----File: 509.png---Folio 495-------
+electromotive intensity discontinuous, for the total electromotive
+intensity is made up of two parts, one due to electromagnetic
+induction, the other due to the causes which produce the Hall
+effect; it is only the first of these parts which we assume to be
+continuous.
+
+If $P$~is the component parallel to~$x$ of the total electromotive
+intensity, $P'$~the part of it due to electromagnetic induction,
+then
+\[
+P = P' + k(b_0 w - c_0 v);
+\]
+\begin{DPalign*}
+\lintertext{but}
+P &=   \frac{1}{K'\iota p + 4\pi/\sigma} \Bigl(\frac{d\gamma}{dy} - \frac{d\beta}{dz}\Bigr) \\
+  &= - \frac{1}{K'\iota p + 4\pi/\sigma}\, \frac{d\beta}{dz},
+\end{DPalign*}
+since in the present case $\gamma$~does not depend upon~$y$.
+
+Hence, substituting the values of $w$~and~$v$ in terms of the
+magnetic force, the condition that $P'$~is continuous is equivalent
+to that of
+\[
+- \frac{1}{K'\iota p + 4\pi/\sigma}\, \frac{d\beta}{dz}
+  - \frac{k}{4\pi} \left( b_0\, \frac{d\beta}{dx} - c_0\, \Bigl( \frac{d\alpha}{dz} - \frac{d\gamma}{dx} \Bigr) \right)
+\]
+being continuous.
+
+We shall suppose that in the air $k = 0$.
+
+The condition that $\alpha$~is continuous gives
+\[
+A_0 + A = A_1 + A_2;
+\Tag{6}
+\]
+the condition that $\beta$~is continuous gives
+\[
+B_0 + B = - \frac{\iota\omega_1}{m_1} A_1 + \frac{\iota\omega_2}{m_2} A_2;
+\Tag{7}
+\]
+the condition that the normal magnetic induction is continuous
+gives
+\begin{align*}
+-\frac{l}{m} (A_0 - A) &= -\mu' \left( \frac{l}{m_1} A_1 + \frac{l}{m_2} A_2 \right), \\
+\intertext{or dividing by~$l$,}
+-\frac{1}{m} (A_0 - A) &= -\mu' \left( \frac{1}{m_1} A_1 + \frac{1}{m_2} A_2 \right).
+\Tag{8}
+\end{align*}
+
+We can easily prove independently that this equation is true
+when $l = 0$, though in that case it cannot be legitimately deduced
+from the preceding equation.
+%% -----File: 510.png---Folio 496-------
+
+The condition that $P'$~is continuous gives, since $k = 0$ and $\sigma = \infty$
+for air,
+\begin{multline*}
+- \frac{m}{K\iota p} (B_0 - B)
+  = \frac{\iota(\omega_1 A_1 - \omega_2 A_2) }{K'\iota p + 4\pi/\sigma } \\
+  - \frac{k}{4\pi} \left\{
+    b_0 l \left(- \frac{\iota\omega_1}{m_1} A_1 + \frac{\iota\omega_2}{m_2} A_2\right)
+  - c_0 \left(\frac{\omega_1^2}{m_1} A_1 + \frac{\omega_2^2}{m_2} A_2\right) \right\}.
+\Tag{9}
+\end{multline*}
+
+The equations (\eqnref{410}{6}),~(\eqnref{410}{7}),~(\eqnref{410}{8}), and~(\eqnref{410}{9}) are sufficient to determine
+the four quantities~$A$, $A_1$, $A_2$, $B$, and thus to determine the
+amplitudes and phases of the reflected and transmitted waves.
+
+\Article{411} We shall now proceed to apply these equations to the case
+of reflection from a tangentially magnetized reflecting surface, as
+the peculiar reversal of the direction of rotation of the plane
+of polarization which (\artref{405}{Art.~405}) Kerr found to take place when
+the angle of incidence passes through~$75°$ seems to indicate that
+this case is the one which is best fitted to distinguish between
+rival hypotheses.
+
+Since in this case the magnetic force is tangential $c_0 = 0$; hence,
+referring to equation~(\eqnref{409}{5}), we see that there will only be one
+value of~$m'$ if $la_0$~vanishes, i.e.~if $l = 0$, in which case the incidence
+is normal, or if $a_0 = 0$, in which case the magnetic force is at
+right angles to the plane of incidence; hence, since there is only
+one value of~$m'$, there will not be any rotation of the plane of
+polarization in either of these cases; this agrees with Kerr's
+experiments (see \artref{405}{Art.~405}).
+
+Let us suppose that the light is polarized perpendicularly to
+the plane of incidence and that the mirror is magnetized in that
+plane. In the incident wave the magnetic force is at right
+angles to the plane of incidence, so that the~$A_0$ of equations (\eqnref{410}{6}),~(\eqnref{410}{7}),~(\eqnref{410}{8}),
+and~(\eqnref{410}{9}) vanishes. Putting
+\[
+A_0 = 0, \quad b_0 = 0, \quad c_0 = 0,
+\]
+we get from these equations
+\begin{gather*}
+A = A_1 + A_2, \\
+B_0 + B = -\iota \left( \frac{\omega_1}{m_1} A_1 - \frac{\omega_2}{m_2} A_2\right), \\
+A = -\mu' m \left(\frac{1}{m_1} A_1 + \frac{1}{m_2} A_2\right) \\
+- \frac{m}{K\iota p} (B_0 - B)
+  = \frac{\iota(\omega_1 A_1 - \omega_2 A_2)} {K'\iota p + 4\pi/\sigma}.
+\end{gather*}
+%% -----File: 511.png---Folio 497-------
+
+Since $(K'\iota p + 4\pi/\sigma) / K\iota p
+  = \dfrac{1}{\mu'} R^2 \epsilon^{2\iota\alpha}$, see \artref{353}{Art.~353}, the last
+equation may be written
+\[
+-(B_0 - B) = \frac{\iota\mu'} {R^2 \epsilon^{2\iota\alpha} m} (\omega_1 A_1 - \omega_2 A_2).
+\]
+
+The rotation observed is small, we shall therefore neglect the
+squares and higher powers of $(m_1 - m_2)$; doing this we find from
+the preceding equations that
+\[
+\frac{A}{B}
+  = \frac{\iota\mu'm \left(\dfrac{1}{m_1} - \dfrac{1}{m_2}\right) }
+         {\omega\left( \dfrac{\mu'}{R^2 \epsilon^{2\iota\alpha} m} - \dfrac{1}{M}\right)
+                 \left(1 + \mu' \dfrac{m}{M}\right) },
+\Tag{10}
+\]
+where $M$~is the value of~$m_1$ or~$m_2$, when $k = 0$, and $\omega^2 = l^2 + M^2$.
+
+From equation~(\eqnref{409}{5}) we have, when $c_0 = 0$,
+\begin{align*}
+-p^2\mu'K' + 4\pi\mu'\iota p/\sigma + l^2 + m_1^2 & =   \frac{\iota k}{4\pi} (K'\iota p + 4\pi/\sigma)(l^2 + m_1^2)^{\frac{1}{2}} la_0, \\
+-p^2\mu'K' + 4\pi\mu'\iota p/\sigma + l^2 + m_2^2 & = - \frac{\iota k}{4\pi} (K'\iota p + 4\pi/\sigma)(l^2 + m_2^2)^{\frac{1}{2}} la_0.
+\end{align*}
+
+Hence, when $m_1 - m_2$ is small, we have approximately
+\begin{align*}
+(m_1 - m_2)M &= \frac{\iota k}{4\pi} (K'\iota p + 4\pi/\sigma)\omega la_0 \\
+             &= \frac{\iota k}{4\pi\mu'} R^2 \epsilon^{2\iota\alpha} \iota p V_0^{-2} \omega la_0,
+\end{align*}
+where $V_0$~denotes the velocity of propagation of electromagnetic
+action through air. Substituting this value of $m_1 - m_2$ in
+equation~(\eqnref{411}{10}) we get
+\[
+\frac{A}{B} = \frac{\iota kp}{4\pi}\,
+  \frac{R^2 \epsilon^{2\iota\alpha} lmV_0^{-2} a_0\omega }
+       {M \left( \dfrac{\mu'M}{R^2 \epsilon^{2\iota\alpha} m} - 1 \right) (M + \mu'm) }.
+\Tag{11}
+\]
+
+\begin{DPgather*}
+\lintertext{\indent If}
+\frac{A}{B} = \theta + \iota\phi,
+\end{DPgather*}
+where $\theta$~and~$\phi$ are real quantities, then if the reflected light
+polarized perpendicularly to the plane of incidence is represented
+by
+\[
+\beta = \cos(pt + lx - mz),
+\]
+the reflected light polarized in the plane of incidence will be
+represented by
+\[
+\alpha = \theta \cos(pt + lx - mz) - \phi \sin(pt + lx - mz);
+\]
+%% -----File: 512.png---Folio 498-------
+thus, unless $\phi$~vanishes, the reflected light will be elliptically
+polarized. If however $\theta$~and~$\phi$ are small, then the angle between
+the major axis of the ellipse for the reflected light and that of
+the incident light (regarding this which is plane polarized as the
+limit of elliptically polarized light when the minor axis of the
+ellipse vanishes) will be approximately~$\theta$. Hence if the analysing
+prism is set so as to extinguish the light reflected from
+the mirror when it is not magnetized, the field after magnetization
+will be darkest when the analyser is turned through an
+angle~$\theta$, though even in this case it will not be absolutely dark.
+We proceed now to find~$\theta$ from equation~(\eqnref{411}{11}).
+
+We have by \artref{353}{Art.~\DPtypo{(353)}{353}}
+\begin{DPalign*}
+l^{2} + M^{2} &= R^{2} \epsilon^{2 \iota \alpha} (l^{2} + m^{2}),\\
+\lintertext{or}
+M^{2} &= (R^{2} \epsilon^{2 \iota \alpha} - 1) l^{2}
+        + R^{2} \epsilon^{2 \iota \alpha} m^{2}.
+\end{DPalign*}
+
+Now for metals the modulus of $R^{2} \epsilon^{2 \iota \alpha}$ is large, the table in
+\artref{355}{Art.~355} showing that for steel it is about~$17$; hence we have
+approximately
+\begin{align*}
+M^{2} &= R^{2} \epsilon^{2 \iota \alpha} (l^{2} + m^{2}) \\
+      &= R^{2} \epsilon^{2 \iota \alpha} \frac{p^2}{V_0^{2}}.
+\end{align*}
+
+We shall put $\mu' = 1$ in the denominator of the right-hand side
+of equation~(\eqnref{411}{11}), since there is no evidence that iron and steel
+retain their magnetic properties for magnetic forces alternating
+as rapidly as those in the light waves. Making this substitution
+and putting $m = (p / V_0) \cos i$, where $i$~is the angle of incidence,
+we find
+\begin{gather*}
+\left( \frac{M}{R^{2} \epsilon^{2 \iota \alpha} m} - 1 \right) (M + m)
+  = \frac{p}{V_0} \left( \frac{1}{\cos i} - R \epsilon^{\iota \alpha} + \frac{1}{R \epsilon^{\iota \alpha}} - \cos i \right) \\
+  = \frac{p}{V_0 \cos i} (1 - \cos^{2} i - R \epsilon^{\iota \alpha} \cos i)
+\end{gather*}
+approximately, since the modulus of $R \epsilon^{\iota \alpha}$ is large.
+
+Hence we see
+\[
+\frac{A}{B} = \frac{\iota k}{4 \pi}\,
+  \frac{p a_0 V_0^{-2} R \epsilon^{\iota \alpha} \sin i \cos^{2} i}
+       {\sin^{2} i - R\epsilon^{\iota \alpha} \cos i},
+\]
+so that if $k$~is real,
+\[
+\theta = - \frac{k}{4 \pi}\,
+  \frac{p a_0 V_0^{-2} \sin^{3} i \cos^{2}{i} R \sin \alpha}
+       {\sin^{4} i - 2 \sin^{2} i \cos{i} R \cos \alpha + R^{2} \cos^{2} i}.
+\]
+%% -----File: 513.png---Folio 499-------
+
+This does not change sign for any value of~$i$ between~$0$ and~$\pi/2$;
+this result is therefore inconsistent with Kerr's and Kundt's
+experiments, and we may conclude that the hypothesis on which
+it is founded---that the transverse intensity is proportional to
+the \emph{total} current---is erroneous.
+
+As Kerr's and Kundt's experiments were made with magnetic
+metals it seems desirable to consider the results of supposing
+these metals to retain their magnetic properties. When $\mu'$~is not
+put equal to unity, $\theta$~is proportional to
+\[
+\cos^2 i \sin i \sin \alpha
+  \left(\mu' \sin^2 i + \frac{2\mu'^2}{R} - \cos \alpha \cos i\right);
+\]
+this does not change sign for any value of~$i$ between~$0$ and~$\pi/2$,
+so that the preceding hypothesis cannot be made to agree with
+the facts by supposing the metals to retain their magnetic
+properties.
+
+\Article{412} Let us now consider the consequence of supposing that
+the transverse electromotive intensity is proportional not to the
+total current but to the polarization current; we can do this by
+putting
+\[
+k = \frac{K'\iota p/4\pi}{K'\iota p/4\pi + 1/\sigma}\, k',
+\]
+where $k'$~is a real quantity.
+
+This equation may be written
+\[
+k = \frac{K'V_0^2}{R^2 \epsilon^{2\iota\alpha}}\, k'.
+\]
+
+Substituting this value of~$k$ in equation~(\eqnref{411}{11}) we find
+\[
+\frac{A}{B} = \frac{\iota k'K'pa_0}{4\pi R\epsilon^{\iota\alpha}}\,
+  \frac{\sin i \cos^2 i }{\sin^2 i - R\epsilon^{\iota\alpha} \cos i}.
+\]
+
+If we write this in the form
+\[
+\frac{A}{B} = \theta' + \iota\phi',
+\]
+where $\theta'$~and~$\phi'$ are real, we find
+\[
+\theta' = \frac{k'K'pa_0}{4\pi R}\,
+  \frac{\sin i \cos^2 i (\sin \alpha \sin^2 i - R \sin 2 \alpha \cos i)}
+       {\sin^4 i - 2R\sin^2 i \cos i \cos \alpha + R^2 \cos^2 i}.
+\Tag{12}
+\]
+
+The angle through which the analyser has to be twisted in
+order to produce the greatest darkness is, as we have seen, equal
+to~$\theta'$ the real part of~$A/B$. Equation~(\eqnref{412}{12}) shows that this
+%% -----File: 514.png---Folio 500-------
+changes sign when $i$~passes through the value given by the
+equation
+\begin{DPgather*}
+\sin \alpha \sin^2 i - R \sin 2\alpha \cos i = 0, \\
+\lintertext{or}
+\sin^2 i = 2R \cos \alpha \cos i;
+\end{DPgather*}
+with the notation of the table in \artref{355}{Art.~355} this is
+\[
+\sin^2 i = 2n \cos i.
+\]
+
+If $\mu'$~is not equal to unity the corresponding equation may
+easily be shown to be
+\[
+\mu' \sin^2 i = 2n \cos i.
+\]
+
+From the table in \artref{355}{Art.~355} we see that for steel $n = 2.41$, the
+corresponding value of~$i$ when $\mu' = 1$ is about~$78°$, which agrees
+well with the results of Kerr's experiments. Hence we see that
+the consequences of the hypotheses, that the transverse electromotive
+intensity is proportional to the polarization current, and
+that $\mu' = 1$, agree with the results of experiments.
+
+We shall now consider the consequences of supposing that the
+transverse electromotive intensity is proportional to the conduction
+current. We can do this by putting
+\[
+k = \frac{k''}{K'\iota p/4\pi + 1/\sigma},
+\]
+where $k''$~is a constant real quantity.
+
+This equation may be written
+\[
+k = \frac{4\pi k''V_0^2}{\iota pR^2 \epsilon^{2\iota\alpha}}.
+\]
+
+Substituting this value of~$k$ in equation~(\eqnref{411}{11}) we find
+\[
+\frac{A}{B} = \frac{k'' \sin i \cos^2 i}{R\, \epsilon^{\iota\alpha} (\sin^2 i - R\, \epsilon^{\iota\alpha} \cos i)},
+\]
+the real part of which is
+\[
+\frac{k'' \sin i \cos^2 i}{R}\,
+  \frac{(\cos \alpha \sin^2 i - R \cos 2\alpha \cos i)}
+       {\sin^4 i - 2\sin^2 i \cos i R \cos\alpha + R^2 \cos^2 i}.
+\]
+
+This is the angle through which the analyser must be twisted
+in order to quench the reflected light as much as possible. The
+rotation of the analyser will change sign when $i$~passes through
+the value given by the equation
+\begin{DPalign*}
+\cos\alpha \sin^2 i &= R \cos 2\alpha \cos i, \\
+\lintertext{or}
+R \cos\alpha \sin^2 i &= R^2 \cos 2\alpha \cos i.
+\end{DPalign*}
+%% -----File: 515.png---Folio 501-------
+With the notation of \artref{355}{Art.~355} this may be written
+\[
+n \sin^{2}\iota = n^{2}(1-k^{2}) \cos \iota.
+\]
+From the table in \artref{355}{Art.~355} we see that $1-k^{2}$ is negative, hence,
+since $n$~is positive there is no real value of~$i$ less than~$\pi / 2$ which
+satisfies this equation, so that if this hypothesis were correct
+there would be no reversal of the direction of rotation of the
+analyser.
+
+Hence of the three hypotheses, (1)~that the transverse electromotive
+intensity concerned in these magnetic optical effects
+is proportional to the total current, (2)~that it is proportional
+to the polarization current, (3)~that it is proportional to the
+conduction current, we see that (1)~and~(3) are inconsistent
+with Kerr's experiments on the reflection from tangentially
+magnetized mirrors, while (2)~is completely in accordance with
+them.
+
+\Article{413} The transverse electromotive intensity indicated by
+hypothesis~(2) is of a totally different character from that
+discovered by Hall. In Hall's experiments the electromotive
+intensities, and therefore the currents through the metallic plates,
+were constant; when however this is the case the `polarization'
+current vanishes. Thus in Hall's experiments there could have
+been no electromotive intensity of the kind assumed in hypothesis~(2);
+there is therefore no reason to expect that the order
+of the metals with respect to Kerr's effect should be the same as
+that with respect to Hall's.
+
+It is worth noting that reflection from a transparent body
+placed in a magnetic field can be deduced from the preceding
+equations by putting $\alpha = 0$, since this makes the refractive index
+real. In this case we see, by equation~(\eqnref{412}{12}), that the real part of~$A / B$
+vanishes, so that the reflected light is elliptically polarized,
+with the major axis of the ellipse in the plane of incidence;
+any small rotation of the analyser would therefore in this case
+increase the brightness of the field.
+
+\Article{414} We now proceed to consider the case of reflection from
+a normally magnetized mirror. We shall confine ourselves to
+the case of normal incidence.
+
+If the incident light is plane polarized we may (using the
+notation of \artref{409}{Art.~409}) put $B_0=0$; we have also $l=0$, $\omega_1 = m_1$,
+$\omega_{2} = m_{2}$, and since the mirror is magnetized normally, $a_0 = 0$,
+%% -----File: 516.png---Folio 502-------
+$b_0 = 0$. Making these substitutions, equations (\eqnref{410}{6}),~(\eqnref{410}{7}),~(\eqnref{410}{8}); and~(\eqnref{410}{9})
+of \artref{410}{Art.~410} become, putting $\mu'= 1$,
+\begin{gather*}
+A_0 + A = A_1 + A_2, \Tag{13}\\
+B = -\iota(A_1 - A_2), \Tag{14}\\
+A_0 - A = m \left( \frac{A_1}{m_1} + \frac{A_2}{m_2} \right), \Tag{15}\\
+\frac{m}{K \iota p}\, B
+  = \frac{\iota(m_1 A_1 - m_2 A_2)}{K' \iota p + 4 \pi / \sigma}
+  + \frac{k}{4 \pi}\, c_0 (m_1 A_1 + m_2 A_2), \Tag{16}
+\end{gather*}
+where $K$~is the specific inductive capacity of air. The last equation
+by means of~(\eqnref{409}{5}) reduces to
+{\allowdisplaybreaks
+\begin{DPgather*}
+\frac{m}{K \iota p}\, B = p \left( \frac{A_1}{m_1} - \frac{A_2}{m_2} \right), \\
+\lintertext{or since}
+K p^{2} = m^{2},\\*
+B = \iota m \left( \frac{A_1}{m_1} - \frac{A_2}{m_2} \right).
+\Tag{17}
+\end{DPgather*}
+}
+
+Solving these equations we find
+\[
+\frac{B}{A} = - \frac{\iota m (m_1 - m_2)}{m_1 m_2 - m^{2}}.
+\]
+Now $m_1 - m_2$ is small, and we may therefore, if we neglect
+the squares of small quantities, in the denominator of the
+expression for~$B/A$, put~$M$ for either $m_1$~or~$m_2$, where $M$~is the
+value of these quantities when the magnetic field vanishes.
+
+We have by equation~(\eqnref{409}{5})
+\begin{align*}
+-p^{2} \mu' K' + 4 \pi \mu' \iota p / \sigma + m_1^{2} & =   \frac{\iota k}{4 \pi} (K' \iota p + 4 \pi / \sigma) m_1^{2} c_0,\\
+-p^{2} \mu' K' + 4 \pi \mu' \iota p / \sigma + m_2^{2} & = - \frac{\iota k}{4 \pi} (K' \iota p + 4 \pi / \sigma) m_2^{2} c_0;
+\end{align*}
+\begin{DPgather*}
+\lintertext{hence}
+m_1 - m_2 = \frac{\iota k}{4 \pi} (K' \iota p + 4 \pi / \sigma) M c_0
+\end{DPgather*}
+approximately.
+
+Since the transverse electromotive intensity is proportional to
+the polarization current we have
+\[
+k = \frac{K' \iota p / 4 \pi}{K'\iota p / 4 \pi + 1 / \sigma}\, k',
+\]
+where $k'$~is a real quantity. Substituting this value of~$k$ in the
+expression for $m_1 - m_2$ , we get
+\[
+m_1 - m_2 = - \frac{p K' k' M c_0}{4 \pi};
+\]
+%% -----File: 517.png---Folio 503-------
+but $M = R \epsilon^{\iota \alpha} m$, so that
+\[
+\frac{B}{A} = \frac{\iota p K' k' R \epsilon^{\iota \alpha} c_0}{4 \pi (R^{2} \epsilon^{2\iota \alpha} - 1)},
+\]
+or, since the modulus of $R^{2} \epsilon^{2 \iota \alpha}$ is large compared with unity,
+\begin{align*}
+\frac{B}{A} &= \frac{\iota p K' k' \epsilon^{-\iota\alpha}}{4\pi R} c_0 \\
+&= \frac{p K' k'}{4 \pi R}\, c_0 (\sin \alpha + \iota \cos \alpha)
+\end{align*}
+approximately.
+
+Hence, if the magnetic force in the reflected wave, which is
+polarized in the same plane as the incident wave, is represented by
+\[
+\cos(pt + mz),
+\]
+the magnetic force in the reflected wave polarized in the plane
+at right angles to this will be represented by
+\[
+\frac{p K' k'}{4 \pi R}\, c_0 \sin \alpha \cos(pt+mz) -
+\frac{p K' k'}{4 \pi R}\, c_0 \cos \alpha \sin(pt+mz).
+\]
+
+Thus in the expression for the light polarized in this plane one
+term represents a component in the same phase as the constituent
+in the original plane, while the phase of the component represented
+by the other term differs from this by quarter of a wave
+length. The resultant reflected light will thus be slightly
+elliptically polarized. As in \artref{411}{Art.~\DPtypo{(411)}{411}} however, we may show
+that the field can be darkened by twisting the analyser through
+a small angle from the position in which it completely quenched
+the light when the mirror was not magnetized. The angle for
+which the darkening is as great as possible is equal to the real
+term in the expression for~$B/A$, i.e.~to
+\[
+\frac{p K' k'}{4 \pi R}\, c_0 \sin \alpha.
+\]
+
+Thus though the reflected light cannot be completely quenched
+by rotating the analyser, its intensity can be very considerably
+reduced; this agrees with the results of Righi's experiments, see
+\artref{403}{Art.~403}.
+
+We can deduce from this case that of reflection from a transparent
+substance by putting $\alpha = 0$, as this assumption makes the
+refractive index wholly real; in this case the reflected light is elliptically
+polarized, but as the axes of the ellipse are respectively
+in and at right angles to the plane of the original polarization
+%% -----File: 518.png---Folio 504-------
+any small rotation of the analyser will increase the brightness
+of the field.
+
+We can solve by similar means the case of oblique reflection
+from a normally magnetized mirror; the results agree with Kerr's
+experiments; want of space compels us however to pass on to
+apply the same principles to the case where light, as in Kundt's
+experiments \artref{406}{Art.~406}, passes through thin metallic films placed
+in a magnetic field.
+
+
+\Subsection{On the Effect produced by a thin Magnetized Plate on Light
+passing through it.}
+\index{Films, transmission of light through when in magnetic field@\subdashone transmission of light through when in magnetic field}%
+\index{Light, zaction of magnet on light through thin films@\subdashone action of magnet on light through thin films}%
+\index{Magnets, zaction of light passing through thin films@\subdashone action of light passing through thin films}%
+\index{Rotation of plane of polarization xby a thin film@\subdashtwo of polarization by a thin film}%
+
+\Article{415} We shall assume that the plate is bounded by the planes
+$z = 0$, $z = -h$, the incident light falling normally on the plane
+$z = 0$. The external magnetic field is supposed to be parallel to
+the axis of~$z$.
+
+Let the incident light be plane polarized, the magnetic force in
+it being parallel to the axis of~$x$. The reflected light will consist
+of two portions, one polarized in the same plane as the incident
+light, the other polarized in the plane at right angles to this: the
+magnetic force in the latter part of the light will therefore be
+parallel to the axis of~$y$.
+
+If $\alpha$,~$\beta$ are the components of the magnetic force parallel to the
+axes of $x$~and~$y$ respectively, then in the region for which $z$~is
+positive we have
+\begin{align*}
+\alpha & = A_0 \epsilon^{\iota (mz+pt)} + A \epsilon^{\iota (-mz+pt)},\\
+\beta & = B \epsilon^{\iota (-mz+pt)},
+\end{align*}
+where $A_0 \epsilon^{\iota(mz+pt)}$ represents the magnetic force in the incident
+wave and $A$~and~$B$ are constants.
+
+In the plate we have
+\[
+\alpha
+  = A_1 \epsilon^{\iota (m_1 z + pt)} + A_1' \epsilon^{\iota (-m_1 z + pt)}
+  + A_2 \epsilon^{\iota (m_2 z + pt)} + A_2' \epsilon^{\iota (-m_2 z + pt)},
+\]
+and therefore, as in \artref{409}{Art.~409}, as $l=0$,
+\[
+\beta
+  = -\iota A_1 \epsilon^{\iota(m_1 z + pt)} - \iota A_1' \epsilon^{\iota (-m_1 z + pt)}
+  +  \iota A_2 \epsilon^{\iota(m_2 z + pt)} + \iota A_2' \epsilon^{\iota (-m_2 z + pt)},
+\]
+where $m_1$,~$m_2$ are the roots of equation~(\eqnref{409}{5}) and $A_1$,~$A_1'$, $A_2$,~$A_2'$
+are constants.
+
+After the light has passed through the plate, the components
+of the magnetic force will be given by equations of the form
+\begin{align*}
+\alpha & = C \epsilon^{\iota (mz+pt)},\\
+\beta  & = D \epsilon^{\iota (mz+pt)}.
+\end{align*}
+%% -----File: 519.png---Folio 505-------
+
+The four boundary conditions at the surface $z = 0$ give, if $\mu' = 1$,
+\begin{align*}
+&\left.\begin{aligned}
+A_0 + A &= A_1 + A_1' + A_2 + A_2', \\
+A_0 - A &= m\left(\frac{A_1}{m_1} - \frac{A_1'}{m_1} + \frac{A_2}{m_2} - \frac{A_2'}{m_2}\right);
+\end{aligned} \right\} \Tag{18} \\
+&\left.\begin{aligned}
+B &= -\iota(A_1 + A_1' - A_2 - A_2'), \\
+B &= \iota m \left(\frac{A_1}{m_1} - \frac{A_1'}{m_1} - \frac{A_2}{m_2} + \frac{A_2'}{m_2}\right).
+\end{aligned} \right\} \Tag{19}
+\end{align*}
+
+The boundary conditions when $z = -h$ give, writing $\theta$~and~$\phi$
+for $-\iota m_1 h$, $-\iota m_2 h$ respectively,
+\begin{align*}
+&\left.\begin{aligned}
+C\epsilon^{-\iota mh}
+  &= A_1 \epsilon^\theta + A_1' \epsilon^{-\theta}
+   + A_2 \epsilon^\phi   + A_2' \epsilon^{-\phi}, \\
+C\epsilon^{-\iota mh}
+  &= m \left(\frac{A_1 \epsilon^\theta}{m_1} - \frac{A_1' \epsilon^{-\theta}}{m_1}
+           + \frac{A_2 \epsilon^\phi  }{m_2} - \frac{A_2' \epsilon^{-\phi  }}{m_2} \right);
+\end{aligned} \right\} \Tag{20} \\
+&\left.\begin{aligned}
+D\epsilon^{-\iota mh}
+  &= -\iota (A_1 \epsilon^\theta + A_1' \epsilon^{-\theta}
+           - A_2 \epsilon^\phi   - A_2' \epsilon^{-\phi}), \\
+D\epsilon^{-\iota mh}
+  &= \iota m \left(\frac{A_1 \epsilon^\theta}{m_1} - \frac{A_1' \epsilon^{-\theta}}{m_1}
+                 - \frac{A_2 \epsilon^\phi  }{m_2} + \frac{A_2' \epsilon^{-\phi}}{m_2} \right).
+\end{aligned} \right\} \Tag{21}
+\end{align*}
+
+From equations (\eqnref{415}{19}),~(\eqnref{415}{20}), and~(\eqnref{415}{21}) we get
+\begin{align*}
+  &A_1 \epsilon^\theta \bigl(1 - \frac{m}{m_1}\bigr) + A_1' \epsilon^{-\theta} \bigl(1 + \frac{m}{m_1}\bigr)
+ + A_2 \epsilon^\phi   \bigl(1 - \frac{m}{m_2}\bigr) + A_2' \epsilon^{-\phi}   \bigl(1 + \frac{m}{m_2}\bigr) = 0,\\
+  &A_1 \epsilon^\theta \bigl(1 + \frac{m}{m_1}\bigr) + A_1' \epsilon^{-\theta} \bigl(1 - \frac{m}{m_1}\bigr)
+ - A_2 \epsilon^\phi   \bigl(1 + \frac{m}{m_2}\bigr) - A_2' \epsilon^{-\phi}   \bigl(1 - \frac{m}{m_2}\bigr) = 0,\\
+  &A_1 \bigl(1 + \frac{m}{m_1}\bigr) + A_1' \bigl(1 - \frac{m}{m_1}\bigr)
+ - A_2 \bigl(1 + \frac{m}{m_2}\bigr) - A_2' \bigl(1 - \frac{m}{m_2}\bigr) = 0,
+\end{align*}
+
+The solution of these equations may be expressed in the form
+{\footnotesize
+\begin{align*}
+\Delta &= \frac{A_1  \epsilon^{\theta}  }
+      {\epsilon^\phi      \Bigl(1 - \dfrac{m}{m_2}\Bigr) \Bigl(1 + \dfrac{m^2}{m_1m_2}\Bigr) -
+       \epsilon^{-\phi}   \Bigl(1 + \dfrac{m}{m_2}\Bigr) \Bigl(1 - \dfrac{m^2}{m_1m_2}\Bigr) +
+       2\epsilon^{\theta} \Bigl(1 - \dfrac{m}{m_1}\Bigr) \dfrac{m}{m_2}  },                       \\
+       &= \frac{-A_1' \epsilon^{-\theta} }{\epsilon^\phi    \Bigl(1 - \dfrac{m}{m_2}\Bigr) \Bigl(1 - \dfrac{m^2}{m_1m_2}\Bigr) - \epsilon^{-\phi}   \Bigl(1 + \dfrac{m}{m_2}\Bigr) \Bigl(1 + \dfrac{m^2}{m_1m_2}\Bigr) + 2\epsilon^{-\theta} \Bigl(1 + \dfrac{m}{m_1}\Bigr) \dfrac{m}{m_2}  },                       \\
+       &= \frac{A_2   \epsilon^{\phi   } }{\epsilon^\theta  \Bigl(1 - \dfrac{m}{m_1}\Bigr) \Bigl(1 + \dfrac{m^2}{m_1m_2}\Bigr) - \epsilon^{-\theta} \Bigl(1 + \dfrac{m}{m_1}\Bigr) \Bigl(1 - \dfrac{m^2}{m_1m_2}\Bigr) + 2\epsilon^{\phi}    \Bigl(1 - \dfrac{m}{m_2}\Bigr) \dfrac{m}{m_1}  },                       \\
+       &= \frac{-A_2' \epsilon^{-\phi  } }{\epsilon^\theta  \Bigl(1 - \dfrac{m}{m_1}\Bigr) \Bigl(1 - \dfrac{m^2}{m_1m_2}\Bigr) - \epsilon^{-\theta} \Bigl(1 + \dfrac{m}{m_1}\Bigr) \Bigl(1 + \dfrac{m^2}{m_1m_2}\Bigr) + 2\epsilon^{-\phi}   \Bigl(1 + \dfrac{m}{m_2}\Bigr) \dfrac{m}{m_1}  }.
+\end{align*}
+}
+%% -----File: 520.png---Folio 506-------
+
+Now by equations (\eqnref{415}{20})~and~(\eqnref{415}{21}) we have
+\[
+\frac{D}{C}
+  = - \frac{\iota ( A_1 \epsilon^{\theta} + A_1' \epsilon^{-\theta}
+                  - A_2 \epsilon^{\phi}   - A_2' \epsilon^{-\phi} ) }
+           {A_1 \epsilon^{\theta} + A_1' \epsilon^{-\theta}
+          + A_2 \epsilon^{\phi}   + A_2' \epsilon^{-\phi}}.
+\]
+
+Substituting the ratios of $A_1$,~$A_1'$, $A_2$,~$A_2'$ just found, we get
+{\footnotesize
+\begin{gather*}
+\text{\normalsize$-\dfrac{D}{C} =$} \\[6pt]
+  \frac{\iota \left\{\dfrac{1}{m_2} \Bigl( 1 + \dfrac{m^{2}}{m_1^{2}} \Bigr) (\epsilon^{\theta} - \epsilon^{-\theta})
+                    - \dfrac{1}{m_1} \Bigl( 1 + \dfrac{m^{2}}{m_2^{2}} \Bigr) (\epsilon^{\phi}   - \epsilon^{-\phi})
+        + \dfrac{2m}{m_1 m_2} (\epsilon^{\phi} + \epsilon^{-\phi} - \epsilon^{\theta} - \epsilon^{-\theta}) \right\} }
+       {              \dfrac{1}{m_2} \Bigl( 1 - \dfrac{m^{2}}{m_1^{2}} \Bigr) (\epsilon^{\theta} - \epsilon^{-\theta})
+                    + \dfrac{1}{m_1} \Bigl( 1 - \dfrac{m^{2}}{m_2^{2}} \Bigr) (\epsilon^{\phi}   - \epsilon^{-\phi})}
+\end{gather*}
+}
+
+We notice that the numerator vanishes when $m_1 = m_2$, in
+which case $\theta = \phi$: it therefore contains the factor $m_1 - m_2$; hence,
+if we neglect the squares and higher powers of $(m_1 - m_2)$, we may
+in the denominator put $m_1 = m_2 = M$ and $\phi = \theta$.
+
+If the thickness of the film is so small that $\theta$~and~$\phi$ are small
+quantities, then neglecting powers of~$h$ higher than the second,
+we find
+\[
+\frac{D}{C} = \tfrac{1}{2}\, \frac{m_2^{2} - m_1^{2}}{M^{2} - m^{2}} (\iota - m h).
+\]
+
+\nbpagebreak[1]
+Substituting the value of $m_2^{2} - m_1^{2}$ from equation~(\eqnref{409}{5}), and
+putting $M = R \epsilon^{\iota \alpha} m$, we see that
+\[
+\frac{D}{C}
+  = \frac{p K' k' c_0}{4 \pi}\,
+    \frac{(\iota - m h)}{1 - \dfrac{\epsilon^{- 2 \iota \alpha}}{R^{2}}}.
+\]
+
+Since $R^{2}$~is large for metals we may, as a first approximation,
+put
+\[
+\frac{D}{C} = \frac{p K' k' c_0}{4 \pi} (\iota - m h).
+\]
+
+The angle through which the plane of polarization is twisted
+is equal to the real part of~$D / C$, and is therefore equal to
+\[
+- p K' k' c_0 mh / 4 \pi;
+\]
+it is thus to our order of approximation independent of the
+opacity of the plate. We see from \artref{414}{Art.~414} that when light is
+incident normally on a magnetized mirror the rotation of the
+plane of polarization of the reflected light is proportional to~$\sin\alpha$,
+and thus depends primarily on the opacity of the mirror, vanishing
+when the mirror is transparent.
+
+The imaginary part of~$D / C$ remains finite though $h$~is made
+%% -----File: 521.png---Folio 507-------
+indefinitely small, we therefore infer that the transmitted light is
+elliptically polarized, and that the ratio of the axes of the ellipse
+is approximately independent of the thickness of the plate.
+
+Let us now consider the light reflected from the plate. We
+have by equations (\eqnref{415}{18})~and~(\eqnref{415}{19})
+\[
+\frac{B}{2A}
+  = - \frac{\iota(A_1 + A_1' - A_2 - A_2')}
+           {A_1 \left( 1 - \dfrac{m}{m_1} \right) + A_1' \left( 1 + \dfrac{m}{m_1} \right)
+          + A_2 \left( 1 - \dfrac{m}{m_2} \right) + A_2' \left( 1 + \dfrac{m}{m_2} \right) }.
+\]
+Substituting the values of $A_1$,~$A_1'$, $A_2$,~$A_2'$ previously given,
+we find, neglecting squares and higher powers of $m_1 - m_2$,
+{\footnotesize
+\begin{gather*}
+\text{\normalsize$-\dfrac{B}{2A} =$} \\[6pt]
+\iota \frac{\left\{(\epsilon^{\theta - \phi} - \epsilon^{-(\theta - \phi)}) \dfrac{2 m}{M} \Bigl( 1 + \dfrac{m^2}{M^2} \Bigr)
+           + \Bigl( \epsilon^{-(\theta + \phi)} - \epsilon^{\theta + \phi} \Bigr) m \Bigl( \dfrac{1}{m_1} - \dfrac{1}{m_2} \Bigr) \Bigl( 1 - \dfrac{m^2}{M^2} \Bigr) \right\} }
+           {2 \Bigl( 1 - \dfrac{m^2}{M^2} \Bigr)^{2} \{2 - ( \epsilon^{2 \theta} + \epsilon^{-2 \theta} ) \} }.
+\end{gather*}
+}
+If the plate is so thin that $\theta$~and~$\phi$ are small, we have approximately
+\begin{align*}
+\frac{B}{A}
+  &= \frac{(m_1 - m_2) \dfrac{m}{M} \left\{1 + \dfrac{m^2}{M^2} + \left( 1 - \dfrac{m^2}{M^2} \right) \right\} }
+          {\left( 1 - \dfrac{m^2}{M^2} \right)^2 M^2 h } \\
+  &= (m_1 - m_2)\, \frac{2m}{M^3 h \left( 1 - \dfrac{m^2}{M^2} \right)^2} \\
+  &= \frac{2(m_1 - m_2)m}{M^3 h},
+\end{align*}
+since $m/M$~is small for metals.
+
+Substituting the value of $m_1 - m_2$ from equation~(\eqnref{409}{5}) we get,
+putting $M = R \epsilon^{\iota \alpha} m$,
+\[
+\frac{B}{A} = - \frac{pK'k'c_0}{2\pi}\, \frac{1}{m R^{2} \epsilon^{2 \iota \alpha} h};
+\]
+the rotation of the plane of polarization is equal to the real part
+of~$B / A$, and hence to
+\[
+- \frac{K' k' c_0}{2 \pi}\, \frac{p}{m h}\, \frac{\cos 2 \alpha}{R^{2}}.
+\]
+
+Since this is proportional to~$1 / h$ we see that the rotation
+increases as the thickness of the plate diminishes. The explanation
+%% -----File: 522.png---Folio 508-------
+of this is that while the intensities of the two components
+reflected light, viz.~the component polarized in the same plane as
+the incident wave and the component polarized in the plane at
+right angles to this, both diminish as the thickness of the plate
+diminishes; the first component diminishes much more rapidly
+than the second; thus the ratio of the second component to the
+first and therefore the angle of rotation of the plane of polarization
+increases as the thickness of the plate diminishes.
+
+\Article{416} The effect of a magnetic field in producing rotation of
+the plane of polarization thus seems to afford strong evidence of
+the existence of a transverse electromotive intensity in a conductor
+placed in a magnetic field, this intensity being quite
+distinct from that discovered by Hall, inasmuch as the former
+is proportional to the rate of variation of the electromotive intensity,
+whereas the Hall effect is proportional to the electromotive
+intensity itself. We shall now endeavour to form some estimate
+of the magnitude of this transverse intensity revealed to us by
+optical phenomena.
+
+Kundt (\textit{Wied.\ Ann.}~23, p.~238, 1884) found from his experiments
+\index{Kundt, transmission of light through thin films@\subdashone transmission of light through thin films}%
+that if~$\phi$, the rotation of the plane of polarization
+produced by the passage of light of wave length~$\lambda$ through a
+magnetized plate of thickness~$h$, is given by an equation of the
+form
+\[
+\phi = \frac{\pi h}{\lambda} (n - n'),
+\]
+then $\phi = 1°.48'$ when
+\begin{DPgather*}
+\lambda = 5.8 × 10^{-5}, \quad \text{and} \quad h = 5.5 × 10^{-6}, \\
+\lintertext{thus}
+n - n' = .1.
+\end{DPgather*}
+
+But we have seen that the rotation in this case is equal to
+\[
+\frac{pK'k'c_0}{2\pi}\, \frac{\pi h}{\lambda};
+\]
+hence, comparing this with Kundt's result, we find
+\[
+\frac{pK'k'c_0}{2\pi} = .1,
+\]
+but if $\lambda = 5.8 × 10^{-5}$, $p = 2\pi × 3 × 10^{10} × 10^5 /5.8 = 3.2 × 10^{15}$.
+
+Substituting these values, we find
+\[
+\frac{K'k'}{2\pi}\, c_0 = 3.1 × 10^{-17}.
+\]
+%% -----File: 523.png---Folio 509-------
+
+Now if $f$~is the electric polarization parallel to~$x$, the transverse
+electromotive intensity is equal to
+\begin{align*}
+k'c_0 \frac{df}{dt}
+  &= k' \iota pc_0 f \\
+  &= k' \frac{K'}{4\pi} \iota pc_0 X,
+\end{align*}
+where $X$~is the electromotive intensity parallel to~$x$. Hence
+$k'K'pc_0 /4\pi$ the ratio of the magnitude of the transverse intensity
+to that producing the current; this ratio is for iron therefore
+equal to
+\[
+1.6 × 10^{-17} p
+\]
+for magnetic fields of the strength used by Kundt. The factor
+multiplying~$p$ is so small as to make it probable that the effects
+of this transverse force are insensible except when the electromotive
+intensity is changing with a rapidity comparable with
+the rate of change in light waves, in other words, that it is
+only in optical phenomena that this transverse electromotive
+intensity produces any measurable effect.
+%% -----File: 524.png---Folio 510-------
+
+\Chapter{Chapter VI.}{The Distribution of Rapidly Alternating Currents.}
+\index{Alternating currents, distribution among a net-work of conductors}%
+\index{Rayleigh, Lord, distribution of alternating currents@\subdashtwo distribution of alternating currents|indexetseq}%
+
+\Article{417} \Firstsc{Problems} concerning alternating currents have become
+in recent years of much greater importance than they were at
+the time when Maxwell's Treatise was published; this is due to
+the extensive use of such currents for electric lighting, and to the
+important part which the much more rapidly oscillating currents
+produced by the discharge of Leyden jars now play in electrical
+researches. It is therefore desirable to consider more
+fully than is done in the \textit{Electricity and Magnetism} the application
+of Maxwell's principles to such currents. In doing this
+we shall follow the methods used by Lord Rayleigh in his papers
+on `The Reaction upon the Driving-Point of a System executing
+Forced Harmonic Oscillations of various Periods, with Applications
+to Electricity,' \textit{Phil.\ Mag.}~[5], 21, p.~369, 1886, and on
+`The Sensitiveness of the Bridge Method in its Application to
+Periodic Electric Currents,' \textit{Proc.\ Roy.\ Soc.}~49, p.~203, 1891.
+
+\Article{418} When the currents are steady their distribution among a
+net-work of conductors is determined by the condition that the
+rate of heat production must be a minimum, see Maxwell's
+\textit{Electricity and Magnetism}, vol.~i.\ p.~408. Thus, if $F$~is the
+Dissipation Function (\textit{Electricity and Magnetism}, vol.~i.\ p.~408),
+$\dot{x}_1$,~$\dot{x}_2$, $\dot{x}_3 \ldots$ the currents flowing through the circuits, these
+variables being chosen so that they are sufficient but not more
+than sufficient to determine the currents flowing through each
+branch of the net-work, then $\dot{x}_1$,~$\dot{x}_2$,~\&c.\ are determined by the
+equations
+\[
+\frac{dF}{d\dot{x}_1} = \frac{dF}{d\dot{x}_2} = \frac{dF}{d\dot{x}_3} = \ldots = 0.
+\]
+%% -----File: 525.png---Folio 511-------
+When, however, the currents are variable these equations are no
+longer true; we have instead of them the equations
+\begin{gather*}
+\frac{d}{dt}\, \frac{dT}{d\dot{x}_1} + \frac{dF}{d\dot{x}_1} - \frac{dV}{dx_1} = 0,\\
+.\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .
+\end{gather*}
+where $T$~is the Kinetic Energy due to the Self and Mutual induction
+of the circuits, $F$~as before is the Dissipation Function,
+and $V$~is the Potential Energy arising from the charges that
+may be in any condensers in the system.
+
+If the currents are periodic and proportional to~$\epsilon^{\iota pt}$, the preceding
+equation may be written as
+\[
+\iota p\, \frac{dT}{d\dot{x}_1} + \frac{dF}{d\dot{x}_1} - \frac{dV}{dx_1} = 0
+\]
+and thus when $p$~increases indefinitely the preceding equation
+approximates to
+\[
+\frac{dT}{d\dot{x}_1} = 0;
+\]
+we have similarly
+\[
+\frac{dT}{d\dot{x}_2} = \frac{dT}{d\dot{x}_3} = \ldots = 0.
+\]
+Thus in this case the distribution of currents is independent
+of the resistances, and is determined by the condition that
+the Kinetic Energy and not the Dissipation Function is a
+minimum.
+
+\index{Kinetic energy, xa minimum for rapidly alternating currents@\subdashtwo a minimum for rapidly alternating currents}%
+\Article{419} We have already considered several instances of this
+effect. Thus, when a rapidly alternating current travels along a
+wire, the currents fly to the outside of the wire, since by doing
+this the mean distance between the parts of the current is a
+maximum and the Kinetic Energy therefore a minimum. Again,
+when two currents in opposite directions flow through two
+parallel plates the currents congregate on the adjacent surfaces
+of the plates, since by so doing the average distance between
+the opposite currents, and therefore the Kinetic Energy, is a
+minimum.
+
+\includegraphicsmid[b]{fig137}{Fig.~137.}
+
+\includegraphicsmid[t]{fig138}{Fig.~138.}
+
+Mr.\ G.~F.~C. Searle has devised an experiment which shows
+\index{Searle, experiment on alternating currents}%
+this tendency of the currents in a very striking way. $AB$,~\figureref{fig123}{Fig.~123},
+is an exhausted tube through which the periodic currents
+produced by the discharge of a Leyden jar are sent. When none
+of the wires leading from the jar to the tube passes parallel to it
+%% -----File: 526.png---Folio 512-------
+in its neighbourhood, the glow produced by the currents fills the
+tube uniformly. When however one of the leads is bent, as in
+\figureref{fig137}{Fig.~137}, so as to pass near the tube in such a way that the current
+through the lead is in the opposite direction to that through the
+tube, the glow no longer fills the tube but concentrates itself on
+the side of the tube next the wire, thus getting as near as possible
+to the current in the opposite direction through the wire. When
+however the wire is bent, as in \figureref{fig138}{Fig.~138}, so that the current
+through the lead is in the same direction as that through the
+tube, the glow flies to the part of the tube most remote from the
+wire.
+
+\includegraphicsmid{fig139}{Fig.~139.}
+
+\Article{420} We shall now proceed to consider the distribution of
+\index{Alternating currents, distribution of between two circuits in parallel@\subdashtwo distribution of between two circuits in parallel}%
+alternating currents among various systems of conductors. The
+first case we shall consider is the distribution of an alternating
+current between two conductors $ACB$,~$ADB$ in parallel. Let the
+resistance and self-induction in the arm~$ACB$ be respectively
+$R$,~$L$, the corresponding quantities in the arm~$ADB$ being denoted
+by $S$,~$N$, and let~$M$ be the coefficient of mutual induction between
+the circuits $ACB$,~$ADB$. We shall suppose that the rate of
+alternation of the current is not so rapid as to produce any
+%% -----File: 527.png---Folio 513-------
+appreciable variation in the intensity of the current from one
+end of~$ACB$ or~$ADB$ to the other, in other words, that the wave
+length corresponding to the rate of alternation of the current
+is large compared with the length~$ACB$ or~$ADB$; the case
+when this wave length is comparable with the length of the
+circuit is considered separately in \artref{298}{Art.~298}. Let the current
+flowing in along~$OA$ and out along~$BP$ be denoted by~$\dot{x}$; we
+shall assume that $\dot{x}$~varies as~$\epsilon^{\iota pt}$. Let the current in~$ACB$ be~$\dot{y}$,
+that in~$ADB$ will be $\dot{x} - \dot{y}$. Then~$T$, the Kinetic Energy in
+the branch $ACDB$ of the circuit, is expressed by the equation
+\[
+T = \tfrac{1}{2} \{L \dot{y}^{2} + 2 M ( \dot{x} - \dot{y} ) \dot{y} + N ( \dot{x} - \dot{y} )^{2} \}.
+\]
+The dissipation function~$F$ is given by
+\[
+F = \tfrac{1}{2} \{R \dot{y}^{2} + S ( \dot{x} - \dot{y} )^{2} \},
+\]
+and we have
+\begin{gather*}
+\frac{d}{dt}\, \frac{dT}{d \dot{y}} + \frac{dF}{d \dot{y}} = 0, \\
+\intertext{or}
+( L + N - 2 M ) \frac{d \dot{y}}{dt} + ( R + S ) \dot{y} - ( N - M ) \frac{d \dot{x}}{dt} - S \dot{x} = 0.
+\end{gather*}
+
+Let $\dot{x} = \epsilon^{\iota pt}$, then from this equation we have
+\[
+\dot{y} = \frac{(N - M) \iota p + S}{( L + N - 2M ) \iota p + ( R + S ) }\, \epsilon^{\iota pt},
+\]
+or, taking the real part of this, corresponding to the current $\cos pt$
+along $OA$, we find
+{\footnotesize
+\setlength{\multlinegap}{0pt}
+\eqnlabel{\eqnart.1}
+\begin{multline*}
+\dot{y} =\\
+  \frac{\{S(R + S) + (L + N - 2M)(N - M)p^{2} \} \cos pt - p \{R(N - M) - S(L - M)\} \sin pt}
+          {(L + N - 2M)^{2} p^{2} + (R + S)^{2}},\nbtag{1}
+\end{multline*}
+\eqnlabel{\eqnart.2}
+\begin{multline*}
+\dot{x} - \dot{y} =\\
+  \frac{\{R(S + R) + (L + N - 2M)(L - M)p^{2} \} \cos pt + p \{R(N - M) - S(L - M) \} \sin pt}
+          {(L + N - 2M)^{2} p^{2} + (R + S)^{2}}.\nbtag{2}
+\end{multline*}
+}
+These expressions may be written in the forms
+\begin{gather*}
+\dot{y}
+  = \Bigl\{\frac{S^{2} + (N - M)^{2} p^{2}}{(L + N - 2M)^{2} p^{2} + (R + S)^{2}} \Bigr\}^{\frac{1}{2}} \cos(pt + \epsilon)  = A \cos(pt+\epsilon), \text{ say},\\
+\dot{x} - \dot{y}
+  = \left\{\frac{R^{2} + (L - M)^{2} p^{2}}{(L + N - 2M)^{2} p^{2} + (R + S)^{2}} \right\}^{\frac{1}{2}} \cos(pt + \epsilon') = B \cos(pt+\epsilon'),
+\end{gather*}
+\begin{DPalign*}
+\lintertext{where}
+\tan \epsilon  & =   \frac{p \{R(N - M) - S(L - M) \}}{S(R + S) + (L + N - 2M)(N - M)p^{2}}, \\
+\lintertext{and}
+\tan \epsilon' & = - \frac{p \{R(N - M) - S(L - M) \}}{R(R + S) + (L + N - 2M)(L - M)p^{2}}.
+\end{DPalign*}
+%% -----File: 528.png---Folio 514-------
+
+The maximum currents through $ACB$,~$ADB$ are proportional
+to $A$~and $B$, and we see from the preceding equations that
+\[
+\frac{A}{\{S^{2} + (N - M)^{2} p^{2} \}^{\frac{1}{2}}} =
+\frac{B}{\{R^{2} + (L - M)^{2} p^{2} \}^{\frac{1}{2}}}.
+\]
+When $p$~is very large, this equation becomes
+\[
+\frac{A}{N - M} = \frac{B}{L - M},
+\]
+so that in this case the distribution of the currents is governed
+entirely by the induction in the circuits, and not at all by their
+resistances. Referring to equations (\eqnref{420}{1})~and~(\eqnref{420}{2}) we see that when
+$p$~is infinite
+\begin{align*}
+\dot{y} &= \frac{N - M}{L + N - 2M} \cos pt, \Tag{3} \\
+\dot{x} - \dot{y} &= \frac{L-M}{L + N - 2M} \cos pt. \Tag{4}
+\end{align*}
+
+An inspection of these equations leads to the interesting result
+that when the alternations are very rapid the maximum current
+in one or both of the branches may be greater than that in the
+leads. Consider the case when the two circuits $ACB$,~$ADB$ are
+wound close together. Suppose, for example, that they are parts
+of a circular coil, and that there are $m$~turns in the circuit~$ACB$,
+and $n$~turns in~$ADB$, then if the coils are close together we may
+put
+\[
+L = Km^{2},\quad M = Knm,\quad N = Kn^{2},
+\]
+where $K$~is a constant.
+
+Substituting these values for $L$,~$M$,~$N$ in equations (\eqnref{420}{3})~and~(\eqnref{420}{4})
+we find
+\begin{align*}
+\dot{y}
+  &= \frac{n^{2} - nm}{(n - m)^{2}} \cos pt
+   = \frac{n}{n - m} \cos pt, \Tag{5} \\
+\dot{x} - \dot{y}
+  &= \frac{m^{2} - nm}{(n - m)^{2}}
+   = - \frac{m}{n - m} \cos pt. \Tag{6}
+\end{align*}
+
+Thus the currents are of opposite signs in the two coils, the
+current in the coil with the smallest number of turns flows in
+the same direction as the current in the leads. When $n - m$ is
+very small both currents become large, being now much greater
+than the current in the leads whose maximum value was taken
+%% -----File: 529.png---Folio 515-------
+as unity; thus by introducing an alternating current of small
+intensity into a divided circuit, we can produce in the arms of
+this circuit currents of very much greater intensity. The reason
+of this becomes clear when we consider the energy in the loop,
+when the rate of alternation is exceedingly rapid. The effects
+of the inertia of the system become all important, and the distribution
+of currents is that which would result if we considered
+merely the Kinetic Energy of the system. In this case, in
+accordance with dynamical principles, the actual solution is that
+which makes the Kinetic Energy as small as possible consistent
+with the condition that the algebraical sum of the currents in
+$ACB$,~$ADB$ shall be equal to~$\dot{x}$.
+
+Thus, as the Kinetic Energy is to be as small as possible, and
+this energy is in the field around the loop and proportional at each
+place to the square of the magnetic force, the currents will
+distribute themselves in the wires so as to neutralize as much as
+possible each other's magnetic effect. Thus if the wires are wound
+close together the currents will flow in opposite directions, the
+branch having the smallest number of turns having the largest
+current, so as to be on equal terms as far as magnetic force is
+concerned with the branch with the larger number of turns. In
+fact we see from equations (\eqnref{420}{5})~and~(\eqnref{420}{6}) that the current in each
+branch is inversely proportional to the number of turns. If the
+two branches are exactly equal in all respects the current in
+each will be in the same direction, but this distribution will be
+unstable, the slightest difference of the coefficients of induction in
+the two branches being sufficient to make the current in the
+branch of least inductance flow in the direction of that in the
+leads, and the current in the other branch in the opposite direction,
+the intensity in either branch at the same time increasing
+largely.
+
+When the currents are distributed in accordance with equations
+(\eqnref{420}{3})~and (\eqnref{420}{4}), the Kinetic Energy in the loop is
+\[
+\tfrac{1}{2}\, \frac{LN-M^2}{L+N-2M}\, p^2 \cos^2 pt.
+\]
+
+We notice that $(LN - M^2)/(L + N - 2M)$ is always less than
+$L$~or~$N$. $L + N - 2M$~is always positive, since it is proportional
+to the Kinetic Energy in the loop when the currents are equal
+and opposite.
+%% -----File: 530.png---Folio 516-------
+
+We see from equations (\eqnref{420}{1})~and~(\eqnref{420}{2}) that when
+\begin{gather*}
+R(N - M) = S(L - M), \\
+\dot{y} = \frac{S}{R+S} \cos pt, \\
+\dot{x} - \dot{y} = \frac{R}{R+S} \cos pt.
+\end{gather*}
+So that in this case the distribution of alternating currents of
+any frequency is the same as when the currents are steady.
+
+\Article{421} We shall now consider the self-induction and resistance
+\index{Induction, self, for two wires in parallel@\subdashtwo for two wires in parallel}%
+\index{Self-induction, expression for, of two wires in parallel@\subdashtwo of two wires in parallel}%
+\index{Multiple arc, xself-induction of wires in@\subdashtwo self-induction of wires in}%
+of the two wires in parallel. Let $L_0$ and~$r$ be respectively the
+self-induction and resistance of the leads, and suppose that
+there is no mutual induction between the leads and the branches
+$ACB$,~$ADB$.
+
+Then we have
+\begin{multline*}
+(L_0 + N)\frac{d\dot{x}}{dt} - (N - M)\frac{d\dot{y}}{dt} + (r + S)\dot{x} - S\dot{y} \\
+= \text{external electromotive force tending to increase~$x$}.
+\end{multline*}
+
+Substituting in this expression the value of~$\dot{y}$ in terms of~$\dot{x}$
+previously obtained in \artref{420}{Art.~420}, we find
+\begin{multline*}
+(L_0 + N) \frac{d\dot{x}}{dt}
+  - \frac{{(N-M)\iota p + S}^2}{(L+N-2M)\iota p+R+S} \dot{x} + (r+S)\dot{x} \\
+  = \text{external electromotive force tending to increase~$x$}.
+\end{multline*}
+
+Remembering that $\iota p\dot{x} = d\dot{x}/dt$, we see that the left-hand side
+of this equation may be written
+\begin{multline*}
+\left\{L_0 + \frac{NR^2 + LS^2 + 2MRS + p^2(LN-M^2)(L+N-2M)}
+                  {(R+S)^2 + p^2(L+N-2M)^2}\right\} \frac{d\dot{x}}{dt} \\
+  + \left\{r + \frac{RS(R+S) + p^2{R(N-M)^2 + S(L-M)^2}}
+                    {(R+S)^2 + p^2(L+N-2M)^2}\right\} \dot{x}.
+\end{multline*}
+
+From the form of this equation we see that the self-induction
+of the two wires in parallel is
+\[
+\frac{NR^2 + LS^2 + 2MRS + p^2(LN-M^2)(L+N-2M)}
+     {(R+S)^2 + p^2(L+N-2M)^2},
+\]
+which may be written as
+\begin{multline*}
+\frac{NR^2 + LS^2 + 2MRS}{(R+S)^2} \\
+- \frac{p^2(L+N-2M)}{(R+S)^2+p^2(L+N-2M)^2}\, \{R(N-M)-S(L-M)\}^2.
+\end{multline*}
+%% -----File: 531.png---Folio 517-------
+
+The impedance of the loop is
+\[
+\frac{RS(R+S) + p^2\{R(N-M)^2 + S(L-M)^2\}}
+     {(R+S)^2 + p^2(L+N-2M)^2},
+\]
+which is equal to
+\[
+\frac{RS}{R+S}
+  + \frac{p^2\{R(N-M) - S(L-M)\}^2}
+         {(R+S)\{(R+S)^2 + p^2(L+N-2M)^2\}}.
+\]
+
+We see from the expression for the self-induction of the loop
+that it is greatest when $p = 0$, when its value is
+\[
+\frac{NR^2 + 2MRS + LS^2}{(R+S)^2},
+\]
+and least when $p$~is infinite when it is equal to
+\begin{DPgather*}
+\frac{LN-M^2}{L+N-2M}.\\
+\lintertext{\indent If}
+R(N - M) = S(L - M),
+\end{DPgather*}
+the self-induction of the loop is independent of the period.
+
+From the expression for the impedance of the loop we see that
+it is least when $p = 0$ when its value is
+\[
+\frac{RS}{R+S}
+\]
+and greatest when $p$~is infinite when it is equal to
+\begin{DPgather*}
+\frac{R(N-M)^2 + S(L-M)^2}{(L+N-2M)^2}; \\
+\lintertext{and if}
+R(N - M) = S(L - M),
+\end{DPgather*}
+the impedance is independent of the period. Thus in this case
+the self-induction and the impedance are unaltered, whatever the
+frequency of the currents. In all other cases the self-induction
+diminishes and the impedance increases as the frequency of the
+currents increases.
+
+\Article{422} We shall now proceed to investigate the general case
+\index{Alternating currents, expression for self-induction of systems of wires@\subdashtwo expression for self-induction of systems of wires}%
+\index{Alternating currents, expression for `impedance' of systems of wire@\subdashtwo expression for `impedance' of systems of wire}%
+\index{Impedance, for two wires in parallel@\subdashone for two wires in parallel}%
+\index{Multiple arc, ximpedance of wires in@\subdashtwo impedance of wires in}%
+when there are any number of wires in parallel. Let~$\dot{x}_0$ be the
+current in the leads, $\dot{x}_1, \dot{x}_2, \ldots \dot{x}_n$ the currents in the $n$~wires in
+parallel; we shall assume, as before, that there is no induction
+between these wires and the leads. Let $a_{rr}$~be the self-induction
+and $r_r$~the resistance of the wire through which the current is~$\dot{x}_r$,
+$a_{rs}$~the coefficient of mutual induction between this wire and the
+wire through which the current is~$\dot{x}_s$. Let $a_0$~be the self-induction,
+$r_0$~the resistance of the leads, $E_0$~the electromotive
+force in the external circuit; we shall suppose that this varies as~$\epsilon^{\iota pt}$.
+%% -----File: 532.png---Folio 518-------
+The current through the leads and those through the
+wires in parallel are connected by the relation
+\[
+\dot{x}_0 - (\dot{x}_1 + \dot{x}_2 + \ldots \dot{x}_n) = 0;
+\]
+we shall denote this by
+\[
+\phi = 0.
+\]
+
+Then $T$~being the Kinetic Energy, $F$~the Dissipation function,
+and $\lambda$~an arbitrary multiplier, the equations determining the
+currents are of the form
+\begin{multline*}
+\frac{d}{dt}\, \frac{dT}{d \dot{x}_s} + \frac{dF}{d \dot{x}_s} + \lambda \frac{d\phi}{d \dot{x}_s}\\
+  = \text{external electromotive force tending to increase~$\dot{x}_s$}.
+\end{multline*}
+
+From these equations we get
+\begin{gather*}
+(a_0 \iota p + r_0) \dot{x}_0 + \lambda = E_0,
+\Tag{7} \\
+\left.
+\begin{aligned}
+(a_{11} \iota p + r_1) \dot{x}_1 + a_{12} \iota p \dot{x}_2 + \ldots -\lambda = 0,&\\
+a_{12} \iota p \dot{x}_1 + (a_{22} \iota p + r_2) \dot{x}_2 + \ldots -\lambda = 0,&\\
+.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\;\;&\\%[TN:manual spacing of dots]
+a_{1n} \iota p \dot{x}_1 + \PadTo[l]{(a_{22} \iota p + r_2) \dot{x}_2 + \ldots}{a_{2n} \iota p \dot{x}_2 + \ldots} -\lambda = 0.&
+\end{aligned}
+\right\}
+\Tag{8}
+\end{gather*}
+
+Solving equations~(\eqnref{422}{8}) we find
+\begin{multline*}
+\frac{\dot{x}_1}{A_{11} + A_{12} + \ldots A_{1n}}
+  = \frac{\dot{x}_2}{A_{12} + A_{22} + \ldots A_{2n}} \\
+  = \frac{\dot{x}_n}{A_{1n} + A_{2n} + \ldots A_{nn}}
+  = \frac{\lambda}{\Delta}, \Tag{9}
+\end{multline*}
+where
+\[
+\Delta=
+\begin{vmatrix}
+a_{11} \iota p + r_1, &a_{12} \iota p\hfill, &\ldots &a_{1n} \iota p\\
+a_{12} \iota p\hfill, &a_{22} \iota p + r_2, &\ldots  &a_{2n} \iota p\\
+\hdotsfor[6]{4}\\
+a_{1n} \iota p\hfill, &a_{2n} \iota p\hfill, &\ldots  &a_{nn} \iota p + r_n
+\end{vmatrix},
+\]
+and $A_{pq}$~denotes the minor of~$\Delta$ corresponding to the constituent
+$a_{pq} \iota p$.
+\begin{DPgather*}
+\lintertext{\indent Since}
+\dot{x}_0 = \dot{x}_1 + \dot{x}_2 + \ldots,
+\end{DPgather*}
+we have from the above equations
+\[
+\frac{\dot{x}_0}{A_{11} + A_{22} + \ldots A_{nn} + 2A_{12} + 2A_{13} + 2A_{23} + \ldots}
+  = \frac{\lambda}{\Delta}.
+\]
+
+Substituting this value of~$\lambda$ in equation~(\eqnref{422}{7}) we find
+\[
+\left( a_0 \iota p + r_0 + \frac{\Delta}{S} \right) \dot{x}_0 = E_0,
+\Tag{10}
+\]
+where $S$~is written for
+\[
+A_{11} + A_{22} + \ldots A_{nn} + 2A_{12} + 2A_{13} + 2A_{23} + \ldots .
+\]
+%% -----File: 533.png---Folio 519-------
+
+The self-induction and impedance of the leads can be deduced
+from (\eqnref{422}{10}); the expressions for them are however in general very
+complicated, but they take comparatively simple forms when
+$\iota p$~is either very large or very small.
+
+When $\iota p$~is very large,
+\[
+\frac{\Delta}{S} = \iota p \frac{D}{S'}
+  + \frac{r_1 ( A'_{11} + A'_{12} + \ldots A'_{1n} )^{2}
+        + r_2 ( A'_{12} + A'_{22} + \ldots A'_{2n} )^{2} + \ldots}{S'^{2}},
+\]
+where
+\[
+D =
+\begin{vmatrix}
+a_{11}, & a_{12}, & a_{1n} \\
+a_{12}, & a_{22}, & a_{2n} \\
+\hdotsfor[6]{3} \\
+a_{1n}, & a_{2n}, & a_{nn}
+\end{vmatrix},
+\]
+and $A'_{pq}$~is the minor of~$D$ corresponding to the constituent~$a_{pq}$,
+while
+\[
+S' = A'_{11} + A'_{22} + \ldots A'_{nn} + 2A'_{12} + 2A'_{13} + 2A'_{23} + \ldots.
+\]
+
+Thus the self-induction of the wires in parallel is in this case
+\[
+\frac{D}{S'},
+\]
+while the impedance is
+\[
+\{r_1 ( A'_{11} + A'_{12} + \ldots A'_{1n})^{2}
+ + r_2 ( A'_{12} + A'_{22} + \ldots A'_{2n})^{2} + \ldots \} / S'^{2}.
+\]
+
+When $\iota p$~is very small,
+\[
+\frac{\Delta}{S} = \iota p\,
+  \frac{\left( \dfrac{a_{11}}{r_1^2} + \dfrac{a_{22}}{r_2^2} + \ldots
+              + \dfrac{2a_{12}}{r_1r_2} + \dfrac{2a_{13}}{r_1r_3} + \ldots \right)}
+       {\left( \dfrac{1}{r_1} + \dfrac{1}{r_2} + \ldots \dfrac{1}{r_n} \right)^{2}}
+  + \frac{1}{\dfrac{1}{r_1} + \dfrac{1}{r_2} + \ldots \dfrac{1}{r_n}}.
+\]
+
+So that in this case the self-induction of the wires in
+parallel is
+\[
+\frac{\dfrac{a_{11}}{r_1^2} + \dfrac{a_{22}}{r_2^2} + \ldots
+     + \dfrac{2a_{12}}{r_1 r_2} + \dfrac{2a_{13}}{r_1 r_3} + \ldots}
+     {\left( \dfrac{1}{r_1} + \dfrac{1}{r_2} + \ldots \dfrac{1}{r_n} \right)^{2}}
+\]
+and the resistance is
+\[
+\frac{1}{\dfrac{1}{r_1} + \dfrac{1}{r_2} + \ldots \dfrac{1}{r_n}}.
+\]
+
+When there is no induction between the wires in parallel,
+%% -----File: 534.png---Folio 520-------
+$a_{12}$,~$a_{13}$, $a_{23}$,~\&c.\ all vanish; hence, when $\iota p$~is infinite, the
+self-induction is
+\[
+\frac{1}{\dfrac{1}{a_{11}} + \dfrac{1}{a_{22}} + \ldots \dfrac{1}{a_{nn}}},
+\]
+and the impedance
+\[
+\frac{\dfrac{r_1}{a_{11}^{2}} + \dfrac{r_2}{a_{22}^{2}} + \ldots}
+     {\left( \dfrac{1}{a_{11}} + \dfrac{1}{a_{22}} + \ldots\right)^{2}}.
+\]
+
+\Article{423} We shall now consider the case of any number of
+\index{Impedance, for a network of wire@\subdashone for a network of wire}%
+\index{Induction, self, for a network of wires@\subdashtwo for a network of wires}%
+\index{Self-induction, expression for, of a net-work of wires@\subdashtwo of a net-work of wires}%
+circuits; the investigation will apply whether the circuits are
+arranged so as to form separate circuits or whether some or all
+of them are metallically connected so as to form a net-work
+of conductors.
+
+Let $\dot{x}_1, \dot{x}_2, \ldots \dot{x}_n$ be the variables required to fix the distribution
+of currents through the circuits; let~$T$, the Kinetic Energy due
+to these currents, be expressed by the equation
+\[
+T = \tfrac{1}{2} \{a_{11} \dot{x_1^2} + a_{22} \dot{x_2^2} + \ldots
+  + 2a_{12} \dot{x_1} \dot{x_2} + \ldots \},
+\]
+while the Dissipation Function~$F$ is given by
+\[
+F = \tfrac{1}{2} \{r_{11} \dot{x_1^2} + r_{22} \dot{x_2^2} + \ldots
+  + 2r_{12} \dot{x_1} \dot{x_2} + \ldots\}.
+\]
+
+Let us suppose that there are no external forces of types
+$\dot{x}_2$,~$\dot{x}_3$,~\&c., and that~$X_1$, the external force of type~$x_1$, is proportional
+to~$\epsilon^{\iota pt}$.
+
+The equations giving the currents are
+\begin{align*}
+(a_{11} \iota p + r_{11}) \dot{x_1} + (a_{12} \iota p + r_{12}) \dot{x_2} + \ldots &{} = X,\\
+(a_{12} \iota p + r_{12}) \dot{x_1} + (a_{22} \iota p + r_{22}) \dot{x_2} + \ldots &{} = 0,\\
+(a_{13} \iota p + r_{13}) \dot{x_1} + (a_{23} \iota p + r_{23}) \dot{x_2} + \ldots &{} = 0,\\
+.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.&\quad.%[TN:manual spacing of dots]
+\end{align*}
+
+From the last~$(n-1)$ of these equations we have
+\[
+\frac{\dot{x}_{1}}{B_{11}}
+  = \frac{\dot{x}_{2}}{B_{12}}
+  = \frac{\dot{x}_{3}}{B_{13}} = \ldots,
+\Tag{11}
+\]
+where $B_{pq}$~denotes the minor of the determinant
+\[
+\begin{vmatrix}
+a_{11} \iota p + r_{11}, & a_{12} \iota p + r_{12}, \ldots \\
+a_{12} \iota p + r_{12}, & a_{22} \iota p + r_{22}, \ldots \\
+\hdotsfor[6]{2}
+\end{vmatrix}
+\]
+corresponding to the constituent $a_{pq} \iota p + r_{pq}$; we shall denote the
+determinant by~$\Delta$.
+%% -----File: 535.png---Folio 521-------
+
+Substituting the values of~$\dot{x}_2, \dot{x}_3, \ldots$ in the first equation, we
+have
+\[
+(a_{11} \iota p + r_{11})\dot{x}_1
+  + \frac{1}{B_{11}} \{(a_{12} \iota p + r_{12})B_{12}
+                     + (a_{13} \iota p + r_{13})B_{13} + \dots \} \dot{x}_1 = X_1,
+\]
+which may be written
+\[
+\frac{\Delta}{B_{11}} \dot{x}_1 = X_1.
+\Tag{12}
+\]
+
+If $\Delta / B_{11}$~be written in the form $L\iota p + R$, where $L$~and~$R$ are
+real quantities, then $L$~is the effective self-induction of the circuit
+and $R$~the impedance.
+
+By equation~(\eqnref{423}{11}) we have
+\[
+\frac{\Delta}{B_{12}} \dot{x}_2 = X_1.
+\]
+
+If an electromotive force~$X_2$ of the same period as~$X_1$ acted
+on the second circuit, then the current~$\dot{x}_1$ induced in the first
+circuit would be given by
+\[
+\frac{\Delta}{B_{12}} \dot{x}_1 = X_2.
+\]
+
+Comparing these results we get Lord Rayleigh's theorem, that
+when a periodic electromotive force~$F$ acts on a circuit~$A$ the
+current induced in another circuit~$B$ is the same in amplitude
+and phase as the current induced in~$A$ when an electromotive
+force equal in amplitude and phase to~$F$ acts on the circuit~$B$.
+
+When there are only two circuits in the field,
+\[
+\frac{\Delta}{B_{11}} = a_{11} \iota p + r_{11}
+  - \frac{(a_{12} \iota p + r_{12})^2}{a_{22} \iota p + r_{22}};
+\]
+if the circuits are not in metallic connection $r_{12}=0$, and we have
+\[
+\frac{\Delta}{B_{11}}
+  = \left(a_{11} - \frac{p^2 a_{22} a^2_{12}}{a^2_{22} p^2 +r^2_{22}}\right) \iota p
+  + r_{11}       + \frac{p^2 r_{22} a^2_{12}}{a^2_{22} p^2 +r^2_{22}}.
+\]
+
+Thus the presence of the second circuit diminishes the self-induction
+of the first by
+\[
+\frac{p^2 a_{22} a^2_{12}}{a^2_{22} p^2 + r^2_{22}},
+\]
+while it increases the impedance by
+\[
+\frac{p^2 r_{22} a^2_{12}}{a^2_{22} p^2 + r^2_{22}}.
+\]
+%% -----File: 536.png---Folio 522-------
+
+These results were given by Maxwell in his paper `A Dynamical
+Theory of the Electromagnetic Field' (\textit{Phil.\ Trans.}\ 155, p.~459,
+1865). We see from these expressions that the diminution in the
+self-induction and the increase in the impedance increase continuously
+as the frequency of the electromotive force increases.
+
+\Article{424} Lord Rayleigh has shown that this result is true whatever
+may be the number of circuits. We have by~(\eqnref{423}{12})
+\[
+\frac{\Delta}{B_{11}} \dot{x}_1 = X_1.
+\]
+
+Now while keeping $\dot{x}_1$ the same we can choose $\dot{x}_2$,~$\dot{x}_3$,~\&c., so
+that the two quadratic expressions
+\begin{align*}
+& a_{22} \dot{x}^2_2 + a_{33} \dot{x}_3^2 + \ldots 2a_{23} \dot{x}_2 \dot{x}_3 + \ldots, \\
+& r_{22} \dot{x}^2_2 + r_{33} \dot{x}_3^2 + \ldots 2r_{23} \dot{x}_2 \dot{x}_3 + \ldots,
+\end{align*}
+i.e.~the expressions got by putting $\dot{x}_1 = 0$ in~$2T$ and~$2F$ respectively,
+reduce to the sums of squares of $\dot{x}_2$,~$\dot{x}_3$,~\&c.; when $\dot{x}_2$,~$\dot{x}_3$,~\&c.\
+are chosen in this way,
+\[
+a_{23} = a_{24} = a_{pq} = 0,
+\]
+when $p$~is not equal to~$q$ and both are greater than unity.
+
+In this case
+\begin{align*}
+\Delta &=
+\begin{vmatrix}
+a_{11} \iota p + r_{11}, &a_{12} \iota p + r_{12}, &a_{13} \iota p + r_{13}, &\ldots &a_{1n} \iota p + r_{1n}\\
+a_{12} \iota p + r_{12}, &a_{22} \iota p + r_{22}, &0\hfill,                 &\ldots &0 \\
+a_{13} \iota p + r_{13}, &0\hfill,                 &a_{33} \iota p + r_{33}, &\ldots &0 \\
+\hdotsfor[6]{5}\\
+a_{1n} \iota p + r_{1n}, &0\hfill,                 &0\hfill,                 &\ldots &a_{nn} \iota p + r_{nn}
+\end{vmatrix} \\
+  &= (a_{11} \iota p +r_{11})
+     (a_{22} \iota p +r_{22}) \ldots
+     (a_{nn} \iota p +r_{nn}) × {}\\
+ &\quad\begin{aligned}
+  \biggl\{1 - \frac{(a_{12} \iota p + r_{12})^2}{(a_{11} \iota p +r_{11}) (a_{22}\iota p +r_{22})}
+    &- \frac{(a_{13} \iota p + r_{13})^2}{(a_{11} \iota p +r_{11}) (a_{33}\iota p +r_{33})} \\
+  \llap{${}- \ldots$} &- \frac{(a_{1n} \iota p + r_{1n})^2}{(a_{11} \iota p + r_{11})(a_{nn}\iota p + r_{nn})} \biggr\},
+\end{aligned} \\
+B_{11} &= (a_{22} \iota p + r_{22})^2 \ldots (a_{nn} \iota p + r_{nn}).
+\end{align*}
+
+Hence
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}
+\frac{\Delta}{B_{11}}
+  = a_{11} \iota p + r_{11}
+    - \frac{(a_{12} \iota p + r_{12})^2}{a_{22} \iota p + r_{22}}
+    - \frac{(a_{13} \iota p + r_{13})^2}{a_{33} \iota p + r_{33}} - \ldots
+    - \frac{(a_{1n} \iota p + r_{1n})^2}{a_{nn} \iota p + r_{nn}} \\
+  = \Bigl\{a_{11}
+    + \tsum \bigl( \dfrac{a_{nn} r_{1n}^2 - 2 a_{1n} r_{1n} r_{nn}}{r_{nn}^2} \bigr)
+    - \tsum \bigl( \dfrac{a_{nn} p^2 (a_{1n} r_{nn} - a_{nn} r_{1n})^2}{r^2_{nn} (a^2_{nn} p^2 +r^2_{nn})} \bigr) \Bigr\} \iota p\\
+  + r_{11} - \tsum \dfrac{r^2_{1n}}{r_{nn}}
+    + \tsum \bigl( \dfrac{p^2 (a_{1n} r_{nn} - a_{nn} r_{1n})^2}{r_{nn}(a^2_{nn} p^2 + r^2_{nn})} \bigr).
+\end{multline*}
+}
+%% -----File: 537.png---Folio 523-------
+
+The coefficient of~$\iota p$ in the first line is the coefficient of self-induction
+of the first circuit,---we see that it is diminished by
+any increase in~$p$; the second line is the impedance, and we see
+that this is increased by any increase in~$p$.
+
+\Article{425} We shall now return to the general case. The reduction
+of~$\Delta/B_{11}$ to the form $L\iota p + R$ without any limitation as to the
+value of~$p$ would usually lead to very complicated expressions;
+we can, however, obtain without difficulty the values of $L$~and~$R$,
+(1)~when $p$~is very large, (2)~when it is very small.
+
+When $\iota p$~is very large we see that
+\begin{DPgather*}
+L = \frac{D}{A_{11}},\\
+\lintertext{where}
+D = \begin{vmatrix}
+  a_{11}, & a_{12} & \ldots & a_{1n} \\
+  a_{12}, & a_{22} & \ldots & a_{2n} \\
+  \hdotsfor{4}[6] \\
+  a_{1n}, & a_{12} & \ldots & a_{nn} \end{vmatrix},
+\end{DPgather*}
+and $A_{11}$~is the minor of~$D$ corresponding to the constituent~$a_{11}$.
+If $A_{pq}$~denotes the minor of~$D$ corresponding to the constituent~$a_{pq}$,
+then we have by~(\eqnref{423}{11})
+\[
+\frac{\dot{x}_1}{A_{11}} =
+\frac{\dot{x}_2}{A_{12}} = \ldots =
+\frac{\dot{x}_n}{A_{1n}}.
+\Tag{13}
+\]
+
+Substituting these values of $\dot{x}_2$,~$\dot{x}_3$,~\&c.\ in terms of~$\dot{x}_1$, in the
+Dissipation Function, we find that
+\[
+R = \frac{1}{A_{11}^2}
+  \{r_{11} A_{11}^2 + r_{22} A_{12}^2 + \dots r_{nn} A_{1n}^2
+ + 2r_{12} A_{11} A_{12} + 2r_{pq} A_{1p} A_{1q} + \ldots \};
+\]
+we might of course have deduced this value directly from that
+of~$\Delta/B_{11}$.
+
+When $\iota p$~is very small, we see by putting $\iota p = 0$ in~$\Delta/B_{11}$ that
+\begin{DPgather*}
+R = \frac{C}{R_{11}},\\
+\lintertext{where}
+C = \begin{vmatrix}
+  r_{11}, & r_{12} & \ldots & r_{1n} \\
+  r_{12}, & r_{22} & \ldots & r_{2n} \\
+  \hdotsfor{4}[6] \\%[TN:Dots added}
+  r_{1n}, & r_{2n} & \ldots & r_{nn} \end{vmatrix},
+\end{DPgather*}
+and $R_{11}$~is the minor of~$C$ corresponding to the constituent~$r_{11}$;
+if $R_{pq}$~denotes the minor of~$C$ corresponding to the constituent~$r_{pq}$,
+then we have by~(\eqnref{423}{11})
+\[
+\frac{\dot{x}_1}{R_{11}} =
+\frac{\dot{x}_2}{R_{12}} = \ldots =
+\frac{\dot{x}_n}{R_{1n}}.
+\]
+%% -----File: 538.png---Folio 524-------
+
+Substituting these values of $\dot{x}_1, \dot{x}_2, \dot{x}_3, \ldots$ in the expression for
+the Kinetic Energy, we see that
+\[
+L = \frac{1}{R_{11}^2}
+  \{a_{11} R_{11}^2 + a_{22} R_{12}^2 + \ldots 2a_{pq} R_{1p} R_{1q} + \ldots \}.
+\]
+
+\Article{426} Suppose we have a series of circuits arranged so that
+each circuit acts by induction only on the two adjacent ones;
+this is expressed by the condition that $a_{12}$~is finite but that $a_{1p}$~vanishes
+when $p > 2$; again, $a_{12}$,~$a_{23}$ are finite, but $a_{2p}$~vanishes if
+$p$~differs from~$2$ by more than unity. Substituting these values
+of~$a_{1p}, a_{2p}, a_{3p} \ldots$, we easily find
+\begin{align*}
+A_{12} &= -a_{12}\, \frac{dA_{11}} {da_{22}}, \\
+A_{13} &= a_{12} a_{23}\, \frac{d^2 A_{11}} {da_{22}\, da_{33}}, \\
+A_{14} &= -a_{12} a_{23} a_{34}\, \frac{d^3 A_{11}} {da_{22}\, da_{33}\, da_{44}}, \\
+.\quad&\quad.\qquad.\qquad.\qquad.\qquad.\qquad.\\
+A_{1n} &= (-1)^{n-1} a_{12} a_{23} a_{34} \ldots a_{n-1 n}.
+\end{align*}
+
+Now $T$, the Kinetic Energy, is always positive, but the condition
+for this is (Maxwell's \textit{Electricity and Magnetism}, vol.~i.\ p.~111) that
+\[
+D,\quad A_{11},\quad \frac{d A_{11}} {da_{22}},\quad
+\frac{d^2 A_{11}} {da_{22}\, da_{33}}\ \ldots
+\]
+should all be positive; hence we see if we take $a_{12}, a_{23} \ldots$,~\&c.\ all
+positive, $A_{11}$,~$A_{12}$,~$A_{13}$ will be alternately plus and minus, but
+when the frequency of the electromotive force is very great, $\dot{x}_1, \dot{x}_2, \ldots$
+are by~(\eqnref{425}{13}) respectively proportional to $A_{11}, A_{12} \ldots$; hence we see
+that in this case the adjacent currents are flowing in opposite
+directions: a result given by Lord Rayleigh. Another way of
+stating this result is to say that the direction of the currents is
+such that all the terms involving the product of two currents
+in the expression for the Kinetic Energy of the system of currents
+are negative, and in this form we recognise it as a consequence
+of the principle that the distribution of the currents must be
+such as to make the Kinetic Energy a minimum.
+
+\Article{427} We shall now apply these results to the case when the
+circuits are a series of $m$~co-axial right circular solenoids of equal
+length, which act inductively on each other but which are not
+%% -----File: 539.png---Folio 525-------
+in metallic connection. We shall suppose that $a$~is the radius
+of the first solenoid, $b$~that of the second, $c$~that of the third, and
+so on, $a$,~$b$,~$c$ being in ascending order of magnitude; and that
+$n_1, n_2, n_3 \ldots$ are the numbers of turns of wire per unit length
+of the first, second, and third circuits. Then if $l$~is the length of
+the solenoids, we have
+\begin{align*}
+a_{11} &= 4\pi^2 n_1^2 la^2, & a_{22} &= 4\pi^2 n_2^2 lb^2, & a_{33} &= 4\pi^2 n_3^2 lc^2, \\
+a_{12} &= 4\pi^2 n_1 n_2 la^2, & a_{23} &= 4\pi^2 n_2 n_3 lb^2, & a_{34} &= 4\pi^2 n_3 n_4 lc^2, \\
+a_{13} &= 4\pi^2 n_1 n_3 la^2, & a_{24} &= 4\pi^2 n_2 n_4 lb^2, & . \quad &. \quad . \quad . \quad . \quad . \\
+. \quad &. \quad . \quad . \quad . \quad . & . \quad &. \quad . \quad . \quad . \quad .
+\end{align*}
+
+Hence
+\begin{gather*}
+D = \begin{vmatrix}
+a_{11}, & a_{12}, & a_{13} & \ldots \\
+a_{12}, & a_{22}, & a_{23} & \ldots \\
+a_{13}, & a_{23}, & a_{33} & \ldots \\
+\hdotsfor[6]{4} \\
+\end{vmatrix} \\[0.5ex]
+= (4\pi^2 l)^m n_1^2 n_2^2 n_3^2 \ldots a^2 (b^2 - a^2)(c^2 - b^2)(d^2 - c^2) \ldots,
+\end{gather*}
+\begin{DPalign*}
+\lintertext{and}
+A_{11} &= \frac{dD} {da_{11}} \\
+       &= (4\pi^2 l)^{m-1} n_2^2 n_3^2 \ldots b^2 (c^2 - b^2)(d^2 - c^2) \ldots.
+\end{DPalign*}
+
+Now the coefficient of self-induction of the first circuit for
+very rapidly alternating current is
+\[
+\frac{D} {A_{11}}.
+\]
+Substituting the preceding expressions for $D$~and~$A_{11}$ we find that
+the self-induction equals
+\[
+4\pi^2 l n_1^2 a^2 \left(1 - \frac{a^2} {b^2} \right).
+\]
+
+Thus the only one of the circuits which affects the self-induction
+of the first is the one immediately adjacent to it. We
+can at once see the reason for this if we notice that
+\begin{DPgather*}
+\frac{a_{12}} {a_{22}} = \frac{a_{13}} {a_{23}} = \frac{a_{14}} {a_{24}} = \dots , \\
+\lintertext{and therefore}
+A_{13} = A_{14} = A_{15} = \ldots = 0.
+\end{DPgather*}
+
+Now when the rate of alternation is very rapid, $\dot{x}_3, \dot{x}_4, \dot{x}_5 \ldots$,
+the currents in the third, fourth, and fifth circuits,~\&c.\ are by
+equation~(\eqnref{425}{13}) \artref{425}{Art.~\DPtypo{(425)}{425}} proportional to~$A_{13}, A_{14}, A_{15} \ldots$; hence
+we see that in this case these currents all vanish, in other words
+%% -----File: 540.png---Folio 526-------
+the second solenoid forms a perfect electric screen, and screens
+off all induction from the solenoids outside it.
+
+\Article{428} Let us consider the case of three solenoids each of
+length~$l$ when the frequency is not infinitely rapid; we shall
+suppose that the primary coil is inside and has a radius~$a$,
+number of turns per unit length~$n_1$, resistance~$r$; next to this
+is the secondary, radius~$b$, turns per unit length~$n_2$, resistance~$s$;
+and outside this is the tertiary, radius~$c$, turns per unit length
+$n_3$, resistance~$t$. Since the circuits are not in metallic connection
+$r_{12} = r_{13} = r_{23} = 0$. If $X_1$, the electromotive force acting on the
+primary, is proportional to~$\epsilon^{\iota pt}$, then we have by equations (\eqnref{423}{11})~and~(\eqnref{423}{12})
+\[
+x_3 = - \frac{n_1 n_3 a^2 s\iota p} {n_1^2 n_2^2 n_3^2 (4\pi^2 l)^2} \,
+      \frac{X_1} {
+\begin{vmatrix}
+a^2 \iota p + \dfrac{r} {4\pi^2 ln_1^2}\hfill, & a^2 \iota p\hfill, & a^2 \iota p \\
+a^2 \iota p\hfill, & b^2 \iota p + \dfrac{s} {4\pi^2 ln_2^2}\hfill, & b^2 \iota p \\
+a^2 \iota p\hfill, & b^2 \iota p\hfill, & c^2 \iota p + \dfrac{t} {4\pi^2 ln_3^2}
+\end{vmatrix} }~.
+\]
+
+We see from this expression that as long as the radius and
+length of the secondary remain the same, the effect produced by
+it on the current in the tertiary circuit depends on the ratio~$s/n_2^2$,
+since $s$~and~$n_2$ only enter into the expression for~$\dot{x}_3$ as
+constituents of the factor~$s/n_2^2$. Thus all secondaries of radius~$b$
+and length~$l$ will produce the same effect if $s/n_2^2$~remains
+constant.
+
+\sloppy
+We can apply this result to compare resistances in the
+following way: take two similar systems $A$~and~$B$ each consisting
+of three co-axial solenoids, the primaries of $A$~and~$B$ being
+exactly equal, as are also the two tertiaries, while the two
+secondaries are of the same size but differ as to the materials of
+which they are made. Let us use $A$~and~$B$ as a Hughes' Induction
+\index{Balance, Induction}%
+\index{Hughes, induction balance@\subdashone induction balance}%
+\index{Induction balance}%
+Balance, putting the two primaries in series and connecting
+the tertiaries so that the currents generated in them by their
+respective primaries tend to circulate in opposite directions;
+then if, by altering if necessary the resistance in one of the
+secondaries, we make the resultant current in the combined
+tertiaries vanish, we know that $s/n_2^2$ is the same for $A$~and~$B$.
+Suppose that the secondary in~$B$ is a thin tube of thickness~$\tau$
+%% -----File: 541.png---Folio 527-------
+and specific resistance~$\sigma$, then considering the tube as a solenoid
+wound with wire of square section~$\alpha$ packed close together,
+we see that for the tube
+\[
+s = 2\pi bln_2\, \frac{\sigma}{\alpha} = 2\pi bln_2^2\, \frac{\sigma}{\tau}.
+\]
+
+\fussy
+Now $s/n_2^2$ for the tube is equal to $s/n_2^2$ for the secondary
+of~$A$, which may be an ordinary solenoid. We thus have
+\[
+\frac{s}{n_2^2} = 2\pi bl\sigma/\tau,
+\]
+a relation by which we can deduce~$\sigma$.
+
+In order that this method should be sensitive the interposition
+of the secondary ought to produce a considerable effect on the
+currents induced in the tertiary. If the resistance of the
+secondary is large this will not happen unless the frequency of
+the electromotive force is very great; for ordinary metals a
+frequency of about a thousand is sufficient, but this would be
+useless if the specific resistance of the tube were comparable with
+that of electrolytes.
+
+On the other hand, if the frequency is infinite, there will not
+be any current in the tertiaries whatever the resistance of the
+secondaries may be.
+
+
+\Subsection{Wheatstone's Bridge with Self-Induction in the Arms.}
+\index{Wheatstone's Bridge with alternating current}%
+
+\includegraphicsouter{fig140}{Fig.~140.}
+
+\Article{429} The preceding investigation can be applied to find the
+effect of self-induction in the arms of a Wheatstone's Bridge.
+Let $ABCO$ represent the bridge, let an electromotive
+force~$X$ proportional to~$\epsilon^{\iota pt}$ act in
+the arm~$CB$. Let $x$~be the current in~$CB$,
+$y$~that in~$BA$, $z$~that in~$AO$, then the currents
+along~$BO$, $AC$,~$OC$ are respectively $x-y$,
+$y-z$, and~$x-y+z$.
+
+Let the self-induction in~$CB$, $BA$, $AC$, $AO$,
+$BO$, $CO$ be respectively $A$,~$C$,~$B$, $L$,~$M$,~$N$,
+while the resistance in these arms are respectively
+$a$,~$c$,~$b$, $\alpha$,~$\beta$,~$\gamma$. We suppose, moreover, that there is
+no mutual induction between the various arms of the Bridge.
+Then the Kinetic Energy~$T$ of the system of currents is expressed
+by the equation
+\[
+2T = Ax^2 + Cy^2 + B(y - z)^2 + Lz^2 + M(x - y)^2 + N(x - y + z)^2.
+\]
+%% -----File: 542.png---Folio 528-------
+
+The Dissipation Function~$F$ is given by the expression
+\[
+2F = ax^2 + cy^2 + b(y - z)^2 + \alpha z^2 + \beta(x - y)^2 + \gamma(x - y + z)^2.
+\]
+
+Comparing this with our previous notation, we must put
+\begin{align*}
+a_{11} & = A + M + N, & a_{12} & = -(M + N), \\
+a_{22} & = B + C + M + N, & a_{13} & = N, \\
+a_{33} & = B + L + N, & a_{23} & = -(N+B); \\
+r_{11} & = \alpha + \beta + \gamma, & r_{12} & = -(\beta + \gamma), \\
+r_{22} & = b + c + \beta + \gamma, & r_{13} & = \gamma, \\
+r_{33} & = b + \alpha + \gamma, & r_{23} & = -(\gamma + b).
+\end{align*}
+
+Now by equations (\eqnref{423}{11})~and~(\eqnref{423}{12})
+\[
+z = \frac{B_{13}}{\Delta}\, X,
+\]
+where $B_{13}$~is the minor of~$\Delta$ corresponding to the constituent
+$a_{13} \iota p + r_{13}$, i.e.\
+\[
+B_{13} = (a_{12} \iota p + r_{12})(a_{23} \iota p + r_{23})
+       - (a_{22} \iota p + r_{22})(a_{13} \iota p + r_{13}).
+\]
+
+Substituting the preceding values for the $a$'s and the~$r$'s, we
+find
+\[
+B_{13} = -p^2 (MB - NC) + \iota p(Mb + B\beta - Nc - C\gamma) + b\beta - c\gamma.
+\]
+
+Now if $z$~vanishes $B_{13}$ must vanish; hence if the Bridge is
+balanced for all values of~$p$ we must have
+\begin{gather*}
+MB - NC = 0, \\
+Mb + B\beta - Nc - C\gamma = 0, \\
+b\beta - c\gamma = 0;
+\end{gather*}
+while if the Bridge is only balanced for a particular value of~$p$,
+we have
+\begin{gather*}
+b\beta - c\gamma = p^2(MB - NC), \\
+p(Mb + B\beta - Nc - C\gamma) = 0.
+\end{gather*}
+
+When the frequency is very great the most important term in
+the expression for~$B_{13}$ is $-p^2 (MB - NC)$, so that the most important
+condition to be fulfilled when the Bridge is balanced is
+\[
+MB - NC = 0;
+\]
+thus for high frequencies the Bridge tests the self-induction
+rather than the resistances of its arms.
+%% -----File: 543.png---Folio 529-------
+
+
+\Subsection{Combination of Self-Induction and Capacity.}
+\index{Capacity zelectrostatic neutralizes self@\subdashone electrostatic neutralizes self-induction}%
+\index{Induction, self, xand capacity@\subdashtwo and capacity}%
+\index{Self-induction, expression for, xand capacity@\subdashtwo and capacity}%
+
+\Article{430} We have supposed in the preceding investigations that
+the circuits were closed and devoid of capacity; very interesting
+results, however, occur when some or all of the circuits are cut
+and their free ends connected to condensers of suitable capacity.
+We can by properly adjusting the capacity inserted in a circuit
+in relation to the frequency of the electromotive force and the
+self-induction of the circuit, make the circuit behave under the
+action of an electromotive force of given frequency as if it possessed
+no apparent self-induction.
+
+The explanation of this will, perhaps, be clear if we consider
+the behaviour of a simple mechanical system under the action of
+a periodic force. The system we shall take is that of the
+rectilinear motion of a mass attached to a spring and resisted by
+a frictional force proportional to its velocity.
+
+Suppose that an external periodic force~$X$ acts on the system,
+then at any instant $X$~must be in equilibrium with the resultant
+of (1)~minus the rate of change of momentum of the system,
+(2)~the force due to the compression or extension of the spring,
+(3)~the resistance. If the frequency of~$X$ is very great, then for a
+given momentum (1)~will be very large, so that unless it is
+counterbalanced by~(2) a finite force of infinite frequency would
+produce an infinitely small momentum. Let us, however, suppose
+that the frequency of the force is the same as that of the
+free vibrations of the system when the friction is zero. When
+the mass vibrates with this frequency (1)~and~(2) will balance
+each other, thus all the external force has to do is to balance
+the resistance. The system will thus behave like one without
+either mass or stiffness resisted by a frictional force.
+
+In the corresponding electrical system, self-induction corresponds
+to mass, the reciprocal of the capacity to the stiffness of
+the spring, and the electric resistance to the frictional resistance.
+If now we choose the capacity so that the period of the electrical
+vibrations, calculated on the supposition that the resistance of
+the circuit vanishes, is the same as that of the external electromotive
+force, the system will behave as if it had neither self-induction
+nor capacity but only resistance. Hence, if $L$~is the
+self-induction of a circuit whose ends are connected to the plates
+of a condenser whose capacity in electromagnetic measure is~$C$,
+%% -----File: 544.png---Folio 530-------
+the system will behave as if it had no self-induction under an
+electromotive force whose frequency is $p / 2 \pi$ if $LCp^2 = 1$.
+
+\includegraphicsmid{fig141}{Fig.~141.}
+
+\Article{431} We shall now consider the case represented in the figure,
+where we have two circuits in parallel, one of the circuits being
+cut and its ends connected to the plates of a condenser. Let $\Lambda$~be
+the self-induction of the leads, $r$~their resistance; $L$,~$N$ the
+coefficients of self-induction of~$ACB$ and the condenser circuit
+respectively, $M$~the coefficient of mutual induction between these
+circuits. Let $R$,~$S$ be the resistances respectively of~$ACB$ and the
+condenser circuit, $C$~the capacity of the condenser. Let $\dot{x}$~be the
+current in the leads, $\dot{y}$~that in the condenser circuit, then that in
+the circuit~$ACB$ will be $\dot{x} - \dot{y}$. Let $X$, the electromotive force in
+the leads, be proportional to~$\epsilon^{\iota pt}$. If there is no mutual induction
+between the leads and the wires in parallel, the equations
+giving $\dot{x}$,~$\dot{y}$ are
+\begin{gather*}
+(\Lambda + L)\, \frac{d \dot{x}}{dt} - (L - M)\, \frac{d \dot{y}}{dt} + (r + R)\dot{x} - R \dot{y} = X,\\
+(L + N - 2M)\,  \frac{d \dot{y}}{dt} - (L - M)\, \frac{d \dot{x}}{dt} + (S + R)\dot{y} - R \dot{x} + \frac{y}{C} = 0.
+\end{gather*}
+
+Substituting the value of~$\dot{y}$ in terms of~$\dot{x}$ and remembering
+that $d / dt = \iota p$, we get
+\begin{multline*}
+\left\{\Lambda + L
+  + \frac{\xi \{R^2 - (L - M)^2 p^2 \} - 2R(R + S)(L - M)}
+         {p^2 \xi^2 + (R + S)^2} \right\} \iota p \dot{x}\\
+  + \left\{r + R
+  - \frac{(R + S) \{R^2 - (L - M)^2 p^2 \} + 2p^2 \xi R(L - M)}
+         {p^2 \xi^2 + (R + S)^2} \right\} \dot{x} = X,
+\Tag{14}
+\end{multline*}
+\begin{DPgather*}
+\lintertext{where}
+\xi = (L + N - 2M) - \frac{1}{Cp^2}.
+\end{DPgather*}
+
+From the form of this equation we see that the self-induction
+of the two circuits in parallel is
+\[
+L + \frac{\xi \{R^2 - (L - M)^2 p^2 \} - 2R(R + S)(L - M)}
+         {p^2 \xi^2 + (R + S)^2},
+\]
+%% -----File: 545.png---Folio 531-------
+this will vanish if
+\begin{multline*}
+Lp^2 \xi^2 + \xi \{R^2 - (L - M)^2 p^2\}\\
+  + (R + S)\{L(R + S) - 2R(L - M)\} = 0.
+\Tag{15}
+\end{multline*}
+If the roots of this quadratic are real, then it is possible to choose~$C$
+so that the self-induction of the loop vanishes. An important
+special case is when $S = 0$, $M = 0$, when the quadratic reduces to
+\begin{DPgather*}
+Lp^2 \xi^2 + \xi(R^2 - L^2 p^2) - LR^2 = 0; \\
+\lintertext{thus}
+\xi = -\frac{R^2}{Lp^2} \text{ or } L;
+\end{DPgather*}
+the first root gives
+\[
+\frac{1}{C} = (L + N)p^2 + \frac{R^2}{L},
+\]
+the second
+\[
+\frac{1}{C} = Np^2;
+\]
+this last value of~$1/C$ makes $\dot{x} = \dot{y}$, so that none of the current
+goes through~$ACB$.
+
+When $\xi$ satisfies~(\eqnref{431}{15}) the self-induction of the loop vanishes.
+If in that equation we substitute $L + \Lambda$ for~$L$ and $M + \Lambda$ for~$M$,
+the values of~$\xi$ which satisfy the new equation will make the
+self-induction of the whole circuit vanish.
+
+\Article{432} We shall next consider the case of an induction coil or
+transformer, the primary of which is cut and its free ends connected
+to the plates of a condenser whose capacity is~$C$. Let
+$L$,~$N$ be the self-induction of the primary and secondary respectively,
+$M$~the coefficient of mutual induction between the two, $R$~the
+resistance of the primary, $S$~that of the secondary, $\dot{x}$,~$\dot{y}$ the
+currents in the primary and secondary respectively; then if $X$~is
+the electromotive force acting on the primary, we have
+\begin{gather*}
+L\, \frac{d\dot{x}}{dt} + M\, \frac{d\dot{y}}{dt} + R\dot{x} + \frac{x}{C} = X,\\
+M\, \frac{d\dot{x}}{dt} + N\, \frac{d\dot{y}}{dt} + S\dot{y} = 0.
+\end{gather*}
+
+Hence if $X$~varies as~$\epsilon^{\iota pt}$, we find
+\begin{DPgather*}
+\dot{y} = \frac{-M\iota pX }{-p^2(\xi N-M^2) + RS + \iota p(RN+S\xi)}, \\
+\lintertext{where}
+\xi = L - \frac{1}{Cp^2}.
+\end{DPgather*}
+%% -----File: 546.png---Folio 532-------
+
+The amplitude of~$\dot{y}$ for a given amplitude of~$X$ is proportional
+to
+\[
+\frac{XMp}{\bigl\{\bigl(RS-p^2(\xi N-M^2)\bigr)^2 + (RN+S\xi)^2p^2 \bigr\}^{\frac{1}{2}}}.
+\]
+
+This vanishes when $p = 0$, because in this case the current in
+the primary is steady; it also vanishes in general when $p$~is
+infinite, because in consequence of the self-induction of the
+primary only an indefinitely small current passes through it in
+this case. If however
+\begin{DPgather*}
+\xi N = M^2, \\
+\lintertext{or}
+\frac{1}{Cp^2} = L - \frac{M^2}{N},
+\end{DPgather*}
+then the amplitude of the current in the secondary is finite when
+$p$~is infinite, and is equal to
+\[
+\frac{MNX}{RN^2 + SM^2};
+\]
+thus when the frequency of the electromotive force is very high
+the amplitude of the current in the secondary may be increased
+enormously by cutting the primary circuit and connecting its
+ends to a condenser of suitable capacity.
+
+\Article{432*} We can apply a method similar to that of \artref{424}{Art.~424} to
+\index{Time of vibration of adjacent electrical systems@\subdashtwo vibration of adjacent electrical systems}%
+determine the effect of placing a vibrating electrical system near
+a number of other such systems.
+
+We shall suppose that the systems are not in electrical connection,
+and neglect the resistances of the circuits. Let $T$~be the
+Kinetic, $V$~the Potential Energy of the system of currents; let~$\dot{x}_1$
+denote the current in the first circuit, and let $\dot{x}_2, \dot{x}_3, \ldots$, the currents
+in the other circuits, be so chosen that when $x_1$~is put equal
+to zero the expressions for $T$~and~$V$ reduce to the sums of
+squares of $\dot{x}_2, \dot{x}_3, \ldots$; $x_2, x_3, \ldots$ respectively.
+
+Let $T$~be given by the same expression as in \artref{424}{Article~424},
+while
+\[
+V = \tfrac{1}{2} \left\{\frac{x_1^2}{c_1} + \frac{x_2^2}{c_2} + \frac{{x_3}^2}{c_3} + \ldots \right\}.
+\]
+
+Then the equations of the type
+\[
+\frac{d}{dt}\, \frac{dT}{d \dot{x}} + \frac{dV}{dx} = 0
+\]
+%% -----File: 547.png---Folio 533-------
+give, if all the variables are proportional to~$\epsilon^\iota pt$,
+\[
+\begin{aligned}
+\left(-a_{11} p^2 + \frac{1}{c_1}\right)x_1 - a_{12} p^2 x_2 - a_{13} p^2 x_3 - \ldots &= 0 \\
+-a_{12} p^2 x_1 + \left(-a_{22} p^2+ \frac{1}{c_2} \right)x_2 &=0 \\
+-a_{13} p^2 x_1 + \left(-a_{33} p^2+ \frac{1}{c_3} \right)x_3 &=0. \\
+.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad &
+\end{aligned}
+\]
+
+Hence substituting for $x_2$,~$x_3$ in terms of~$x_1$ we get
+\[
+-a_{11} p^2 + \frac{1}{c_1}
+  = \frac{a_{12}{^2} p^4}{\dfrac{1}{c_2} - a_{22} p^2}
+  + \frac{a_{13}{^2} p^4}{\dfrac{1}{c_3} - a_{33} p^2} + \ldots.
+\]
+
+Let us suppose that the period of the first system is only
+slightly changed, so that we may in the right-hand side of this
+equation write~$p_1$ for~$p$, where~$p_1$ is the value of~$p$ when the first
+vibrator is alone in the field.
+
+Let $p_2, p_3, \ldots$ be the values of~$p$ for the other vibrators when
+the first one is absent, then
+\begin{align*}
+\frac{1}{c^2} & = p_2{^2} a_{22} \\
+\frac{1}{c^3} & = p_3{^2} a_{33}.
+\end{align*}
+
+Thus if $\delta p_1{^2}$~denotes the increase in~$p_1{^2}$ due to the presence of
+the other vibrators, we have
+\[
+-a_{11}\, \delta p_1{^2}
+  = p_1{^4} \left\{\frac{a_{12}{^2}}{a_{22}(p_2{^2} - p_1{^2})}
+                  + \frac{a_{13}{^2}}{a_{33}(p_3{^2} - p_1{^2})} + \ldots \right\}.
+\]
+
+Thus we see that if~$p_2$ is greater than~$p_1$ the effect of the
+proximity of the circuit whose period is~$p_2$ is to diminish~$p_1$,
+while if~$p_2$ is less than~$p_1$ the proximity of this circuit increases~$p_1$.
+Similar remarks apply to the other circuits. Thus the first
+system, if its free period is slower than that of the second, is
+made to vibrate still more slowly by the presence of the latter;
+while if its free period is faster than that of the second the
+presence of the latter makes it vibrate still more quickly. In
+other words, the effect of putting two vibrators near together is
+to make the difference between their periods greater than it is
+when the vibrators are free from each other's influence; the
+quicker period is accelerated, the slower one retarded.
+%% -----File: 548.png---Folio 534-------
+
+\Chapter{Chapter VII.}{Electromotive Intensity in Moving Bodies.}
+
+\Article{433} \Firstsc{The} equations~$(B)$ given in Art.~598 of Maxwell's \textit{Electricity
+and Magnetism}, for the components of the electromotive intensity
+in a moving body involve a quantity~$\Psi$, whose physical meaning
+it is desirable to consider more fully. The investigation by
+which the equations themselves are deduced tells us nothing
+about~$\Psi$; it is introduced after the investigation is finished, so
+as to make the expressions for the electromotive intensity as
+general as it is possible for them to be and yet be consistent
+with Faraday's Law of the induction of currents in a variable
+magnetic field.
+
+Let $u$,~$v$,~$w$ denote the components of the velocity of the
+medium; $a$,~$b$,~$c$ the components of the magnetic induction;
+$F$,~$G$,~$H$ those of the vector potential; $X$,~$Y$,~$Z$ those of the
+electromotive intensity.
+
+In the course of Maxwell's investigation of the values of
+$X$,~$Y$,~$Z$ due to induction, the terms
+\begin{gather*}
+-\frac{d}{dx} (Fu + Gv + Hw), \quad
+-\frac{d}{dy} (Fu + Gv + Hw), \\
+-\frac{d}{dz} (Fu + Gv + Hw)
+\end{gather*}
+respectively in the final expressions for $X$,~$Y$,~$Z$ are included
+under the $\Psi$~terms. We shall find it clearer to keep these
+terms separate and write the expressions for $X$,~$Y$,~$Z$ as
+\[
+\left. \begin{aligned}
+X& = cv - bw - \frac{dF}{dt} - \frac{d}{dx} (Fu + Gv + Hw) - \frac{d\phi}{dx}, \\
+Y& = aw - cu - \frac{dG}{dt} - \frac{d}{dy} (Fu + Gv + Hw) - \frac{d\phi}{dy}, \\
+Z& = bu - av - \frac{dH}{dt} - \frac{d}{dz} (Fu + Gv + Hw) - \frac{d\phi}{dz}.
+\end{aligned} \right\} \Tag{1}
+\]
+%% -----File: 549.png---Folio 535-------
+
+For Faraday's law to hold, the line integral of the electromotive
+intensity taken round any closed curve must be independent
+of~$\phi$, hence $\phi$~must be a continuous function.
+
+When there is no free electricity
+\[
+\frac{dX}{dx} + \frac{dY}{dy} + \frac{dZ}{dz} = 0.
+\]
+
+Substituting the values of $X$,~$Y$,~$Z$ just given, we find, using
+\[
+\frac{dF}{dx} + \frac{dG}{dy} + \frac{dH}{dz} = 0,
+\]
+\begin{multline*}
+F\nabla^2 u + G\nabla^2 v + H\nabla^2 w
+  + 2 \left(\frac{dF}{dx}\, \frac{du}{dx}
+          + \frac{dG}{dy}\, \frac{dv}{dy}
+          + \frac{dH}{dz}\, \frac{dw}{dz}\right) \\
+  + \left(\frac{dH}{dy} + \frac{dG}{dz}\right) \left(\frac{dw}{dy} + \frac{dv}{dz}\right)
+  + \left(\frac{dF}{dz} + \frac{dH}{dx}\right) \left(\frac{du}{dz} + \frac{dw}{dx}\right)\\
+  + \left(\frac{dG}{dx} + \frac{dF}{dy}\right) \left(\frac{dv}{dx} + \frac{du}{dy}\right)
+  = -\nabla^2 \phi.
+\end{multline*}
+
+If the medium is moving like a rigid body, then
+\begin{align*}
+u &= p + \omega_2 z - \omega_3 y,\\
+v &= q + \omega_3 x - \omega_1 z,\\
+w &= r + \omega_1 y - \omega_2 x;
+\end{align*}
+where $p$,~$q$,~$r$ are the components of the velocity of the origin and
+$\omega_1$,~$\omega_2$,~$\omega_3$ the rotations about the axes of $x$,~$y$,~$z$ respectively.
+
+Substituting these values we see that whenever the system moves
+as a rigid body
+\[
+\nabla^2 \phi = 0.
+\]
+
+\Article{434} In order to see the meaning of~$\phi$ we shall take the case
+\index{Rotating sphere in a symmetrical magnetic field@\subdashone sphere in a symmetrical magnetic field|indexetseq}%
+\index{Sphere, rotating in a symmetrical magnetic field@\subdashone rotating in a symmetrical magnetic field|indexetseq}%
+of a solid sphere rotating with uniform angular velocity~$\omega$ about
+the axis of~$z$ in a uniform magnetic field where the magnetic
+induction is parallel to the axis~$z$ and is equal to~$c$. We may
+suppose that the magnetic induction is produced by a large
+cylindrical solenoid with the axis of~$z$ for its axis; in this case
+\[
+F = -\tfrac{1}{2} cy, \quad
+G =  \tfrac{1}{2} cx, \quad
+H = 0.
+\]
+
+In the rotating sphere
+\[
+u = -\omega y, \quad
+v =  \omega x, \quad
+w = 0.
+\]
+
+If the system is in a steady state, $dF/dt$, $dG/dt$, $dH/dt$ all
+vanish.
+%% -----File: 550.png---Folio 536-------
+
+Thus in the sphere
+\begin{alignat*}{2}
+X &= c \omega x - \tfrac{1}{2} \frac{d}{dx} \{c\omega(x^2 + y^2) \} &{}- \frac{d\phi}{dx},\\
+Y &= c \omega y - \tfrac{1}{2} \frac{d}{dy} \{c\omega(x^2 + y^2) \} &{}- \frac{d\phi}{dy},\\
+Z &=  &{}-\frac{d\phi}{dz};
+\end{alignat*}
+these equations reduce to
+\begin{align*}
+X &= -\frac{d\phi}{dx}, \\
+Y &= -\frac{d\phi}{dy}, \\
+Z &= -\frac{d\phi}{dz},
+\end{align*}
+and we have also $\nabla^2 \phi = 0$.
+
+In the space outside the sphere the medium does not move as a
+rigid body. The process by which the equations~(\eqnref{433}{1}) were obtained
+could not without further investigation be held to justify us in
+applying them to cases where the velocity is discontinuous, for in
+the investigation, see Maxwell, Art.~598, it is assumed that the
+variations $\delta x$,~$\delta y$,~$\delta z$ are continuous, and that these are proportional
+to the components of the velocity. To avoid any
+discontinuity in the velocity at the surface of the sphere we
+shall suppose that the medium in contact with the sphere moves
+at the same rate as the sphere, but that as we recede from the
+surface of the sphere the velocity diminishes in the same way
+as it does in a viscous fluid surrounding a rotating sphere. Thus
+we shall suppose that the rotating sphere whose radius is~$\smallbold{a}$ is
+surrounded by a fixed sphere whose radius is~$\smallbold{b}$, and that between
+the spheres the components of the velocity are given by the
+expressions
+\[
+u = -(A\, \frac{d}{dy}\, \frac{1}{r} + By),\quad
+v =  (A\, \frac{d}{dx}\, \frac{1}{r} + Bx),\quad
+w = 0,
+\]
+where $r$~is the distance from the centre of the rotating sphere.
+
+When $r = \smallbold{b}$,\quad $u = 0$,\quad $v = 0$, hence
+\[
+-\frac{A}{\smallbold{b}^3} + B = 0;
+\]
+%% -----File: 551.png---Folio 537-------
+when $r = \smallbold{a}$, $u = -\omega y$, $v = \omega x$, hence
+\[
+-\frac{A}{\smallbold{a}^3} + B = \omega,
+\]
+hence
+\[
+A = -\frac{\omega \smallbold{a}^3 \smallbold{b}^3}{\smallbold{b}^3 - \smallbold{a}^3}.
+\]
+
+Substituting these values of $u$,~$v$ in equation~(\eqnref{433}{1}), we find that
+when $\smallbold{a} < r < \smallbold{b}$,
+\begin{align*}
+X & = \tfrac{1}{2} cA(x^2 + y^2)\,\frac{d}{dx}\, \frac{1}{r^3} - \frac{d\phi}{dx}, \\
+Y & = \tfrac{1}{2} cA(x^2 + y^2)\,\frac{d}{dy}\, \frac{1}{r^3} - \frac{d\phi}{dy}, \\
+Z & = \tfrac{1}{2} cA(x^2 + y^2)\,\frac{d}{dz}\, \frac{1}{r^3} - \frac{d\phi}{dz};
+\end{align*}
+\begin{DPgather*}
+\lintertext{hence, since}
+\frac{dX}{dx} + \frac{dY}{dy} + \frac{dZ}{dz} = 0, \\
+\lintertext{we have}
+\nabla^2 \phi = 0.
+\end{DPgather*}
+
+Again, when $r > \smallbold{b}$ the medium is at rest, here we have
+\begin{align*}
+X & = -\frac{d\phi}{dx},\\
+Y & = -\frac{d\phi}{dy},\\
+Z & = -\frac{d\phi}{dz},
+\end{align*}
+and $\nabla^2 \phi = 0$.
+
+The boundary conditions satisfied by~$\phi$ and its differential
+coefficients will depend upon whether the sphere is a conductor
+or an insulator. We shall first consider the case when it is an
+insulated conductor. In this case, when the system is in a
+steady state, the radial currents in the sphere must vanish, otherwise
+the electrical condition of the surface of the sphere could
+not be constant.
+
+Thus at any point on the surface of the sphere
+\[
+xX + yY + zZ = 0,
+\]
+this is equivalent to
+\[
+\frac{d\phi_1}{dr} = 0,
+\]
+where $\phi_1$~is the value of~$\phi$ inside the rotating sphere; hence we
+have
+\[
+\phi_1 = K,
+\]
+where $K$~is a constant.
+%% -----File: 552.png---Folio 538-------
+
+If $\phi_{2}$,~$\phi_{3}$ are the values of~$\phi$ in the region between the fixed
+and moving spheres, and in the fixed sphere respectively, then
+we may put
+\begin{align*}
+\phi_2 &= L + \frac{M}{r} + NQ_{2} \left( \frac{r^{2}}{\smallbold{a}^{2}} - \frac{\smallbold{a}^{3}}{r^{3}} \right), \\
+\phi_{3} &= \frac{PQ_{2}}{r^{3}},
+\end{align*}
+where $L$,~$M$, $N$,~$P$ are constants, and $Q_{2}$~is the second zonal
+harmonic with~$z$ for its axis.
+
+The continuity of~$\phi$ gives
+\begin{align*}
+K &= L + \frac{M}{\smallbold{a}},\quad 0 = L + \frac{M}{\smallbold{b}}, \\
+P &= \frac{N(\smallbold{b}^{5} - \smallbold{a}^{5} )}{\smallbold{a}^{2}}.
+\end{align*}
+
+If $K_{1}$~is the specific inductive capacity of the medium between
+the two spheres, $K_{2}$~that of the medium beyond the outer sphere;
+then, since the normal electric polarization must be continuous
+when $r = \smallbold{b}$, we have
+\[
+3 K_{2}P \frac{Q_{2}}{\smallbold{b}^{4}}
+  = K_{1} \left\{cA \frac{(Q_{2}-1)}{\smallbold{b}^{2}}
+    + \frac{M}{\smallbold{b}^{2}}
+    - NQ_{2} \left( \frac{2 \smallbold{b}}{\smallbold{a}^{2}} + \frac{3 \smallbold{a}^{3}}{\smallbold{b}^{4}} \right) \right\}.
+\]
+
+Solving these equations we find
+\begin{DPgather*}
+\left.
+\begin{aligned}
+P &= \frac{cAK_{1} \smallbold{b}^{2}(\smallbold{b}^{5} - \smallbold{a}^{5})}
+          {3K_{2}( \smallbold{b}^{5} - \smallbold{a}^{5}) + K_{1} (2 \smallbold{b}^{5} + 3 \smallbold{a}^{5})},\\
+N &= \frac{cAK_{1} \smallbold{a}^{2} \smallbold{b}^{2}}
+          {3K_{2}( \smallbold{b}^{5} - \smallbold{a}^{5}) + K_{1} (2 \smallbold{b}^{5} + 3 \smallbold{a}^{5})},\\
+M &= cA,\quad
+L = -cA/\smallbold{b},\quad
+K= cA(\smallbold{b} - \smallbold{a})/ \smallbold{a} \smallbold{b},
+\end{aligned}
+\right\}
+\Tag{2} \\
+\lintertext{where}
+A = -\omega \smallbold{a}^{3} \smallbold{b}^{3} / ( \smallbold{b}^{3} - \smallbold{a}^{3}).
+\end{DPgather*}
+
+The surface density of the electricity on the moving sphere is
+\[
+\frac{K_{1} cA}{4 \pi \smallbold{a}^{2}}
+  \left\{\frac{K_1(2 \smallbold{b}^{5}+3 \smallbold{a}^{5} - 5 \smallbold{a}^{3} \smallbold{b}^{2}) + 3K_{2}( \smallbold{b}^{5} - \smallbold{a}^{5})}
+               {3K_{2}( \smallbold{b}^{5} - \smallbold{a}^{5}) + K_{1}(2 \smallbold{b}^{5} + 3 \smallbold{a}^{5})} \right\} Q_2.
+\]
+
+The preceding formulæ are general; we shall now consider
+some particular cases.
+
+\Article{435} The first we shall consider is when $\smallbold{b} - \smallbold{a} = \delta$ is small
+compared with either~$\smallbold{b}$ or~$\smallbold{a}$. In this case we have approximately,
+when $K_{2}$~is not infinite,
+\begin{gather*}
+P = -\tfrac{1}{3} c \omega \smallbold{a}^{5},\quad
+N = -\tfrac{1}{15}\, \frac{c \omega \smallbold{a}^{3}}{\delta}, \quad
+M = -\tfrac{1}{3}\, \frac{c \omega \smallbold{a}^{4}}{\delta},\\
+%
+L = \tfrac{1}{3}\, \frac{c \omega \smallbold{a}^{3}}{\delta},\quad
+K = -\tfrac{1}{3} c \omega \smallbold{a}^{2}.
+\end{gather*}
+%% -----File: 553.png---Folio 539-------
+
+Thus in the outer fixed sphere the components of the electromotive
+intensity are equal to the differential coefficients with
+respect to $x$,~$y$,~$z$ of the function
+\[
+\tfrac{1}{3} c\omega \smallbold{a}^5 \frac{Q_2}{r^3}.
+\]
+
+Thus the radial electromotive intensity close to the surface of
+the rotating sphere is
+\[
+-c\omega \smallbold{a} Q_2,
+\]
+while the tangential intensity is
+\[
+-c\omega \smallbold{a} \sin\theta \cos\theta.
+\]
+
+These results show that the effects produced by rotating
+uncharged spheres in a strong magnetic field ought to be quite
+large enough to be measurable. Thus if the sphere is rotating so
+fast that a point on its equator moves with the velocity $3 × 10^3$,
+which is about $100$~feet per second, and if $c = 10^3$, then the
+maximum radial intensity is about $1/33$~of a volt per centimetre,
+and the maximum tangential intensity about $1/2$~of this: these
+are quite measurable quantities, and if it were necessary to increase
+the effect both $c$~and~$\omega$ might be made considerably greater
+than the values we have assumed.
+
+The surface density of the electricity on the rotating sphere
+when $(\smallbold{b}-\smallbold{a})/\smallbold{a}$ is small is
+\[
+-\frac{1}{4\pi}\, K_2 c\omega \smallbold{a} Q_2.
+\]
+
+\Article{436} If the outer fixed sphere is a conductor, the electromotive
+intensity must vanish when $r > \smallbold{b}$, hence $P = 0$, so that $N = 0$,
+while $M$,~$L$,~$K$ have the same values as before. In this case the
+surface density of the electricity on the surface of the rotating
+sphere is
+\[
+\frac{K_1}{4\pi \smallbold{a}^2}\, cAQ_2,
+\]
+and when $\smallbold{b} - \smallbold{a}$ is small, this is equal to
+\[
+-\frac{K_1}{12\pi\delta}\, c\omega \smallbold{a}^2 Q_2.
+\]
+
+Since this expression is proportional to~$1/\delta$, the surface density
+can be increased to any extent by diminishing the distance
+between the rotating and fixed surfaces.
+
+In the general case, when $\smallbold{b}-\smallbold{a}$ is not necessarily small, the
+%% -----File: 554.png---Folio 540-------
+surface density of the electricity on the rotating sphere is
+\[
+\frac{K_{1}}{4 \pi \smallbold{a}^{2}}\, cAQ_{2},
+\]
+the surface density on the fixed sphere is
+\[
+- \frac{K_1}{4 \pi \smallbold{b}^{2}}\, cAQ_{2}.
+\]
+The electrostatic potential due to this distribution of electricity
+at a distance~$r$ from the centre of the rotating sphere is, when
+$r > \smallbold{b}$,
+\[
+- \frac{cA}{5} ( \smallbold{b}^{2} - \smallbold{a}^{2} )\, \frac{Q_{2}}{r^{3}},
+\]
+while when $r < \smallbold{a}$ it is
+\[
+- \frac{cA}{5} \left( \frac{1}{\smallbold{b}^{3}} - \frac{1}{\smallbold{a}^{3}} \right) r^{2} Q_{2}.
+\]
+
+The values of~$\phi$ in these regions are respectively zero and a
+constant. Hence this example is sufficient to show us that $\phi$~is
+not equal to the electrostatic potential due to the free electricity
+on the surface of the conductors.
+
+\Article{437} We may (though there does not seem to be any advantage
+gained by so doing) regard~$\phi$ as the sum of two parts,
+one of which,~$\phi_{e}$, is the electrostatic potential due to the distribution
+of free electricity over the surfaces separating the different
+media; the other,~$\phi_{m}$, being regarded as peculiarly due to electromagnetic
+induction.
+
+Let us consider the case of a body moving in any manner,
+then we must have, since there is no volume distribution of
+electricity,
+\[
+\nabla^{2} \phi_{e} = 0.
+\]
+
+If $\sigma$~is the surface density of the electricity over any surface of
+separation at a point where the direction cosines of the outward
+drawn normal are $l$,~$m$,~$n$, then if $K$~is the specific inductive
+capacity
+\[
+4 \pi \sigma = \bigl[K(lX + mY + nZ)\bigr]_{1}^{2},
+\]
+where the expression on the right-hand side of this equation
+denotes the excess of the value of $K(lX + mY + nZ)$ in the
+outer medium over its value in the inner. But if $\phi_{e}$~is the
+electrostatic potential, then
+\[
+4 \pi \sigma
+  = - \left[ K \left( l\, \frac{d \phi_{e}}{dx} + m\, \frac{d \phi_{e}}{dy} + n\, \frac{d \phi_{e}}{dz} \right) \right]_{1}^{2}.
+\]
+%% -----File: 555.png---Folio 541-------
+
+From these conditions we see from equations~(\eqnref{433}{1}) that
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}
+\nabla^2 (\phi_m + Fu + Gv + Hw) = \frac{d}{dx} (cv - bw)
+  + \frac{d}{dy} (aw - cu)
+  + \frac{d}{dz} (bu - av),\\
+\shoveleft{\text{and\quad}
+\left[ K ( l\, \frac{d}{dx} + m\, \frac{d}{dy} + n\, \frac{d}{dz} )
+         ( \phi_m + Fu + Gv + Hw) \right]_{1}^{2}} \\
+= \bigl[ K \{l(cv-bw) + m(aw-cu) + n(bu-av) \} \bigr]_{1}^{2}.\quad
+\end{multline*}
+}
+From these equations $\phi_m$~is uniquely determined, for we see that
+$\phi_m + Fu + Gv + Hw$ is the potential due to a distribution of
+electricity whose volume density is
+\[
+- \frac{1}{4 \pi} \left\{\frac{d}{dx} (cv - bw) + \frac{d}{dy} (aw - cu) + \frac{d}{dz} (bu - av) \right\},
+\]
+together with a distribution whose surface density is
+\[
+- \frac{1}{4 \pi} \bigl[ K \{l(cv-bw) + m(aw-cu) + n(bu-av) \} \bigr]_{1}^{2}.
+\]
+
+Having thus determined~$\phi_m$ and deducing~$\phi$ by the process
+exemplified in the preceding examples we can determine~$\phi_e$.
+
+\Article{438} The question as to whether or not the equations~(\eqnref{433}{1}) are
+true for moving insulators as well as for moving conductors,
+$u$,~$v$,~$w$ being the components of the velocity of the insulator, is
+a very important one. The truth of these equations for conductors
+has been firmly established by experiment, but we have,
+so far as I am aware, no experimental verification of them for
+insulators. The following considerations suggest, I think, that
+some further evidence is required before we can feel assured of
+the validity of the application of these equations to insulators.
+We may regard a steady magnetic field as one in which Faraday
+tubes are moving about according to definite laws, the positive
+tubes moving in one direction, the negative ones in the opposite,
+the tubes being arranged so that as many positive as negative
+tubes pass through any area. When a conductor is moved
+about in such a magnetic field it disturbs the motion of the
+tubes, so that at some parts of the field the positive tubes no
+longer balance the negative and an electromotive intensity is
+produced in such regions. To assume the truth of equations~(\eqnref{433}{1}),
+whatever the nature of the moving body may be, is, from this
+point of view, to assume that the effect on these tubes is the
+same whether the moving body be a conductor or an insulator of
+%% -----File: 556.png---Folio 542-------
+large or small specific inductive capacity. Now it is quite conceivable
+that though a conductor, or a dielectric with a considerable
+inductive capacity, might when in motion produce a considerable
+disturbance of the Faraday tubes in the ether in and
+around it, yet little or no effect might be produced by the
+motion of a substance of small specific inductive capacity such
+as a gas, and thus it might be expected that the electromotive
+intensity due to the motion of a conductor in a magnetic field
+would be much greater than that due to the motion of a gas
+moving with the same speed.
+
+\Article{439} As one of the most obvious methods of determining
+whether or not equations~(\eqnref{433}{1}) are true for dielectrics is to investigate
+the effect of rotating an insulating sphere in a magnetic
+field: we give the solution of the case similar to the one discussed
+in \artref{434}{Art.~434}, with the exception that the metallic rotating sphere
+of that article is replaced by an insulating one, specific inductive
+capacity~$K_0$, of the same radius. Using the notation of that
+article, we easily find that in this case
+\begin{multline*}
+P \left\{\frac{3K_2}{\smallbold{b}^4}
+  + \frac{2 \smallbold{b} ( 3K_1 + 2K_0 ) K_1 - 6K_1 ( K_1 - K_0 ) \smallbold{a}^5 / \smallbold{b}^4}
+         {2(K_1-K_0) \smallbold{a}^5 + (3K_1 + 2K_0) \smallbold{b}^5} \right\} \\
+  = cK_1 A \left\{\frac{1}{\smallbold{b}^2}
+    - \frac{5 \smallbold{a}^2 \smallbold{b} K_1}
+           {2(K_1 - K_0) \smallbold{a}^5 + (3K_1 + 2K_0)\smallbold{b}^5} \right\}.
+\end{multline*}
+When $\smallbold{b} - \smallbold{a}$ is small, this becomes
+\[
+P = - \tfrac{1}{3}\, \frac{2K_0}{3K_2 + 2K_0}\, c \omega \smallbold{a}^5.
+\]
+So that in this case the components of electromotive intensities
+in the region at rest are equal to the differential coefficients
+with respect to $x$,~$y$,~$z$ of the function
+\[
+\tfrac{1}{3}\, \frac{2K_0}{3K_2 + 2K_0}\, \frac{c \omega \smallbold{a}^5}{r^3}\, Q_2,
+\]
+and thus, by \artref{435}{Art.~435}, bear to the intensities produced by the
+rotating conductor the ratio of~$2K_0$ to $3K_2 + 2K_0$.
+
+Thus, if equations~(\eqnref{433}{1}) are true for insulators, a rotating sphere
+made of an insulating material ought to produce an electric field
+comparable with that due to a rotating metallic sphere of the
+same size.
+
+\sloppy
+The greatest difficulty in experimenting with the insulating
+sphere would be that it would probably get electrified by
+friction, but unless this completely overpowered the effect due
+%% -----File: 557.png---Folio 543-------
+to the rotation we ought to be able to distinguish between the
+two effects, since the rotational one is reversed when the direction
+of rotation is reversed as well as when the magnetic field is
+reversed.
+
+\fussy
+In deducing equations~(\eqnref{434}{2}) of \artref{434}{Art.~434}, we assumed that equations~(\eqnref{433}{1})
+held in the medium between the fixed and moving
+surfaces, the general equations will therefore only be true on
+this assumption. In the special case, however, when the layer of
+this medium is indefinitely thin, the results will be the same
+whether this medium is an insulator or conductor, so that the
+results in this special case would not throw any light on whether
+equations~(\eqnref{433}{1}) do or do not hold for a moving dielectric.
+
+
+\Subsection{Propagation of Light through a Moving Dielectric.}
+
+\Article{440} We might expect that some light would be thrown on
+the electromotive intensity developed in a dielectric moving in a
+magnetic field by the consideration of the effect which the motion
+of the dielectric would have on the velocity of light passing
+through it. We shall therefore investigate the laws of propagation
+of light through a dielectric moving uniformly with the
+velocity components $u$,~$v$,~$w$.
+
+In this case, since we have only to deal with insulators, all
+the currents in the field are polarization currents due to alterations
+in the intensity of the polarization. When the dielectric
+is moving we are confronted with a question which we have not
+had to consider previously, and that is whether the equivalent
+current is to be taken as equal to the time rate of variation of
+the polarization at a point fixed in space or at a point fixed in
+the dielectric and moving with it; i.e.~if $f$~is the dielectric
+polarization parallel to~$x$, is the current parallel to~$x$
+\begin{DPgather*}
+\frac{df}{dt}, \\
+\lintertext{or}
+\frac{df}{dt} + u\, \frac{df}{dx} + v\, \frac{df}{dy} + w\, \frac{df}{dz}?
+\end{DPgather*}
+
+In the first case we should have, if $\alpha$,~$\beta$,~$\gamma$ are the components of
+the magnetic force,
+\[
+4 \pi\, \frac{df}{dt} = \frac{d \gamma}{dy} - \frac{d \beta}{dz};
+\Tag{3}
+\]
+in the second,
+\[
+4 \pi \left( \frac{df}{dt} + u\, \frac{df}{dx} + v\, \frac{df}{dy} + w\, \frac{df}{dz} \right)
+  = \frac{d \gamma}{dy} - \frac{d \beta}{dz}.
+\Tag{4}
+\]
+%% -----File: 558.png---Folio 544-------
+
+This point seems one which can only be settled by experiment.
+It seems desirable, however, to look at the question from
+as many points of view as possible; the equation connecting the
+current with the magnetic force is the expression of the fact that
+the line integral of the magnetic force round any closed curve is
+equal to $4 \pi$~times the rate of increase of the number of Faraday
+tubes passing through the curve. We saw in \chapref{Chapter I.}{Chapter~I.} that this
+was equivalent to saying that a Faraday tube when in motion
+gave rise to a magnetic force at right angles to itself, and to the
+direction in which it is moving and proportional to its velocity
+at right angles to itself.
+
+\index{Dielectric, electromotive forces in a moving}%
+\index{Equations for a moving dielectric}%
+\index{Moving dielectrics, electromotive intensity in}%
+\index{Propagation of light through moving dielectrics}%
+When the medium is moving, the question then arises whether
+this velocity to which the magnetic force is proportional is the
+velocity of the tube relative (1)~to a fixed point in the region
+under consideration, or (2)~relative to the moving dielectric, or
+(3)~relative to the ether in this region. If the first supposition
+is true we have equation~(\eqnref{440}{3}), if the second equation~(\eqnref{440}{4}), if the
+third an equation similar to~(\eqnref{440}{4}) with the components of the
+velocity of the ether written for $u$,~$v$,~$w$. I am not aware of any
+experiments which would enable us to decide absolutely which,
+if any, of the assumptions (1),~(2),~(3) is correct; \textit{a~priori} (3)~appears
+the most probable.
+
+If $X$,~$Y$,~$Z$ are the components of the electromotive intensity;
+$a$,~$b$,~$c$ those of magnetic induction; $f$,~$g$,~$h$ those of electric
+polarization, and $F$,~$G$,~$H$ those of the vector potential, then we
+have
+\[
+\left.
+\begin{aligned}
+X & = \frac{4 \pi}{K}\, f = cv - bw - \frac{dF}{dt} - \frac{d \psi}{dx},\\
+Y & = \frac{4 \pi}{K}\, g = aw - cu - \frac{dG}{dt} - \frac{d \psi}{dy},\\
+Z & = \frac{4 \pi}{K}\, h = bu - av - \frac{dH}{dt} - \frac{d \psi}{dz}.
+\end{aligned}
+\right\}
+\Tag{5}
+\]
+
+Then, since the dielectric is moving uniformly, we have
+\begin{align*}
+\frac{4 \pi}{K} \left( \frac{df}{dy} - \frac{dg}{dx} \right)
+  &= u\, \frac{dc}{dx} + v\, \frac{dc}{dy} + w\, \frac{dc}{dz} + \frac{dc}{dt}
+\Tag{6}\\
+%
+  &= \left( \frac{d}{dt} + u\, \frac{d}{dx} + v\, \frac{d}{dy} + w\, \frac{d}{dz} \right) c.
+\end{align*}
+
+Now if equation~(\eqnref{440}{3}) is true
+\[
+\frac{df}{dt} = \frac{1}{4 \pi \mu} \left( \frac{dc}{dy} - \frac{db}{dz} \right),
+\]
+%% -----File: 559.png---Folio 545-------
+with similar equations for $dg/dt$,~$dh/dt$; hence from~(\eqnref{440}{6}) we
+have
+\[
+\frac{1}{K \mu}\, \nabla^2 c
+  = \frac{d}{dt} \left( \frac{d}{dt} + u\, \frac{d}{dx} + v\, \frac{d}{dy} + w\, \frac{d}{dz} \right) c.
+\Tag{7}
+\]
+
+If, on the other hand, equation~(\eqnref{440}{4}) is true, we get
+\[
+\frac{1}{K \mu}\, \nabla^2 c
+  = \left( \frac{d}{dt} + u\, \frac{d}{dx} + v\, \frac{d}{dy} + w\, \frac{d}{dz} \right)^{2} c,
+\Tag{8}
+\]
+with similar equations for $a$~and~$b$.
+
+\index{Dielectric, velocity of light through a moving@\subdashone velocity of light through a moving}%
+\index{Light, zvelocity of through moving dielectric@\subdashone velocity of through moving dielectric}%
+\index{Moving dielectrics, velocity of light through@\subdashtwo velocity of light through}%
+\index{Velocity of ylight through moving dielectrics@\subdashtwo light through moving dielectrics}%
+Let us apply these equations to a wave of plane polarized
+light travelling along the axis of~$x$, the dielectric moving with
+velocity~$u$ in that direction. In this case equation~(\eqnref{440}{7}) becomes
+\[
+\frac{1}{K \mu}\, \frac{d^2 c}{dx^2} = \frac{d^2 c}{dt^2} + u\, \frac{d^2 c}{dx\,dt}.
+\Tag{9}
+\]
+
+Let $c = \cos(pt - mx)$; then if $V$~is the velocity of light through
+the dielectric when at rest, equation~(\eqnref{440}{9}) gives
+\begin{DPgather*}
+V^2 m^2 = p^2 - upm, \\
+\lintertext{or}
+\frac{p^2}{m^2} - \frac{up}{m} = V^2.
+\end{DPgather*}
+
+Since $u$~is small compared with~$V$, we have approximately
+\[
+\frac{p}{m} = \tfrac{1}{2} u + V.
+\]
+
+Thus the velocity of light through the moving dielectric is
+increased by half the velocity of the dielectric.
+
+If we take equation~(\eqnref{440}{8}), then
+\[
+V^2\, \frac{d^2 c}{dx^2} = \left( \frac{d}{dt} + u\, \frac{d}{dx} \right)^{2} c,
+\]
+or putting as before,
+\begin{DPgather*}
+c = \cos(pt - mx),\\
+V^2 m^2 = (p - mu)^2, \\
+\lintertext{hence}
+\frac{p}{m} = V + u;
+\end{DPgather*}
+so that in this case the velocity of the light is increased by that
+of the dielectric.
+
+If we suppose that the condition~(\eqnref{440}{3}) is the true one, viz., that
+\[
+4 \pi \mu \left( \frac{df}{dt} + u_0\, \frac{df}{dx} + v_0\, \frac{df}{dy} + w_0\, \frac{df}{dz} \right)
+  = \frac{dc}{dy} - \frac{db}{dz},
+\]
+where $u_0$,~$v_0$,~$w_0$ are the components of the velocity of the ether,
+%% -----File: 560.png---Folio 546-------
+then, when equations~(\eqnref{433}{1}) are supposed to hold, the relation
+between $p$~and~$m$ for the plane polarized wave is easily found to be
+\[
+V^2 m^2 = (p - mu)(p - mu_0),
+\]
+or if $u$ and~$u_0$ are small compared with~$V$,
+\[
+\frac{p}{m} = V + \tfrac{1}{2} (u + u_0),
+\]
+so that in this case the velocity of the light is increased by the
+mean of the velocities of the dielectric and the ether.
+
+Fizeau's result that the increase in the velocity of light passing
+through a current of air is a very small fraction of the velocity
+of the air, shows that all of the preceding suppositions are
+incorrect.
+
+Thus, if we retain the Electromagnetic Theory of Light, we
+must admit that equations~(\eqnref{433}{1}) do not represent the electromotive
+intensities in a dielectric in motion if $u$,~$v$,~$w$ are the velocities
+of the \emph{dielectric itself}.
+
+If we suppose that in these equations $u$,~$v$,~$w$ ought to refer to
+the velocity of the \emph{ether and not of the dielectric}, then the preceding
+work shows that if supposition~(1) is true, the velocity
+of light passing through moving ether is increased by one half
+the velocity of the ether, while if supposition~(3) is true it is
+increased by the velocity of the ether.
+
+As we could not suppose that the motion of the dielectric
+makes the ether move faster than itself, the discovery of a case
+in which the velocity of light was increased by more than half
+the velocity of the dielectric would be sufficient to disprove
+supposition~(1).
+
+
+\Subsection{Currents induced in a Rotating conducting Sphere.}
+\index{Himstedt, xcurrents induced in rotating sphere@\subdashone currents induced in rotating sphere}%
+\index{Rotating sphere in an unsymmetrical field@\subdashtwo in an unsymmetrical field|indexetseq}%
+\index{Sphere, rotating in an unsymmetrical field@\subdashone rotating in an unsymmetrical field|indexetseq}%
+
+\Article{441} When the external magnetic field is not symmetrical
+about the axis of rotation electric currents will be produced in
+the sphere. These have been discussed by Himstedt (\textit{Wied.\ Ann.}\
+11, p.~812, 1880), and Larmor (\textit{Phil.\ Mag.}\ [5], 17, p.~1, 1884). We
+\index{Larmor, currents in a rotating sphere}%
+can find these currents by the methods given in \chapref{Chapter IV.}{Chapters IV}~and~\chapref{Chapter V.}{V}
+for dealing with spherical conductors.
+
+From equations~(\eqnref{433}{1}) we have, since
+\[
+\frac{da}{dz} + \frac{db}{dy} + \frac{dc}{dz} = 0,
+\]
+%% -----File: 561.png---Folio 547-------
+\begin{multline*}
+\frac{dX}{dy} - \frac{dY}{dx}
+  = u \frac{dc}{dx} + v \frac{dc}{dy} + w \frac{dc}{dz}
+    + c\left(\frac{du}{dx} + \frac{dv}{dy} + \frac{dw}{dz}\right) \\
+  - \left(a \frac{dw}{dx} + b \frac{dw}{dy} + c \frac{dw}{dz}\right),
+\Tag{10}
+\end{multline*}
+with similar equations for
+\[
+\frac{dZ}{dx} - \frac{dX}{dz}, \quad
+\frac{dY}{dz} - \frac{dZ}{dy}.
+\]
+
+If the sphere is rotating with angular velocity~$\omega$ about the
+axis of~$z$,
+\[
+u = -\omega y, \quad v = \omega x, \quad w = 0;
+\]
+so that equation~(\eqnref{441}{10}) becomes
+\[
+\frac{dX}{dy} - \frac{dY}{dx}
+  = \omega\left(x \frac{dc}{dy} - y \frac{dc}{dx}\right).
+\Tag{11}
+\]
+
+If $\sigma$~is the specific resistance of the sphere, $\mu$~its magnetic
+permeability, $\smallbold{p}$,~$\smallbold{q}$,~$\smallbold{r}$ the components of the current,
+\[
+\left.\begin{aligned}
+X & = \sigma \smallbold{p} = \frac{\sigma}{4\pi\mu} \left(\frac{dc}{dy} - \frac{db}{dz}\right), \\
+Y & = \sigma \smallbold{q} = \frac{\sigma}{4\pi\mu} \left(\frac{da}{dz} - \frac{dc}{dx}\right), \\
+Z & = \sigma \smallbold{r} = \frac{\sigma}{4\pi\mu} \left(\frac{db}{dx} - \frac{da}{dy}\right).
+\end{aligned}\right\}
+\Tag{12}
+\]
+
+If we substitute these values for $X$~and~$Y$, equation~(\eqnref{441}{11})
+becomes
+\begin{DPgather*}
+\lintertext{\raisebox{\baselineskip}{similarly}}
+\left.\begin{aligned}
+\frac{\sigma}{4\pi\mu} \nabla^2 c & = \omega\left(x \frac{dc}{dy} - y \frac{dc}{dx}\right), \\
+\frac{\sigma}{4\pi\mu} \nabla^2 b & = \omega\left(x \frac{db}{dy} - y \frac{db}{dx}\right) - \omega a,  \\
+\frac{\sigma}{4\pi\mu} \nabla^2 a & = \omega\left(x \frac{da}{dy} - y \frac{da}{dx}\right) + \omega b.
+\end{aligned}\right\}
+\Tag{13}
+\end{DPgather*}
+
+From these equations we find by the aid of~(\eqnref{441}{12})
+\begin{align*}
+\frac{\sigma}{4\pi\mu} \nabla^2 \smallbold{p} & = \omega\left(x \frac{d\smallbold{p}}{dy} - y \frac{d\smallbold{p}}{dx}\right) + \omega \smallbold{q}, \\
+\frac{\sigma}{4\pi\mu} \nabla^2 \smallbold{q} & = \omega\left(x \frac{d\smallbold{q}}{dy} - y \frac{d\smallbold{q}}{dx}\right) + \omega \smallbold{p}, \\
+\frac{\sigma}{4\pi\mu} \nabla^2 \smallbold{r} & = \omega\left(x \frac{d\smallbold{r}}{dy} - y \frac{d\smallbold{r}}{dx}\right).
+\end{align*}
+%% -----File: 562.png---Folio 548-------
+
+Hence
+\[
+\frac{\sigma}{4\pi\mu} \nabla^2 (x\smallbold{p} + y\smallbold{q} + z\smallbold{r})
+  = \omega\left(x \frac{d}{dy} - y \frac{d}{dx}\right)(x\smallbold{p} + y\smallbold{q} + z\smallbold{r}).
+\Tag{14}
+\]
+
+\begin{DPgather*}
+\lintertext{\indent Let}
+x\smallbold{p} + y\smallbold{q} + z\smallbold{r} = F(r)Y_n^s\, \epsilon^{\iota s\phi},
+\end{DPgather*}
+where $r$,~$\theta$,~$\phi$ are the polar coordinates of a point, $\theta$~being
+measured from the axis of~$z$. $Y_n^s\, \epsilon^{\iota s\phi}$ is a surface harmonic of
+degree~$n$. Substituting this value in~(\eqnref{441}{14}), we find
+\[
+\frac{d^2F}{dr^2} + \frac{2}{r}\, \frac{dF}{dr}
+  - \left(\frac{n(n + 1)}{r^2} + \frac{4\pi\iota\mu s\omega}{\sigma}\right)F = 0.
+\]
+
+The solution of this is, \artref{308}{Art.~308},
+\begin{DPgather*}
+F(r) = S_n (kr), \\
+\lintertext{where}
+k^2 = -4\pi\mu\iota s\omega/\sigma. \\
+\lintertext{\indent Thus}
+x\smallbold{p} + y\smallbold{q} + z\smallbold{r} = AS_n (kr)Y_n^s\, \epsilon^{\iota s\phi},
+\end{DPgather*}
+where $A$~is a constant.
+
+Now $x\smallbold{p} + y\smallbold{q} + z\smallbold{r}$ is proportional to the current along the
+radius, and this vanishes at the surface of the sphere where
+$r = \smallbold{a}$; hence we have $AS_n (k\smallbold{a}) = 0$, but since the roots of
+$S_n (x) = 0$ are real, and $k$~is partly imaginary, $S_n (k\smallbold{a})$ cannot
+vanish, thus $A$~must vanish. In other words, the radial currents
+must vanish throughout the sphere; the currents thus flow along
+the surfaces of spheres concentric with the rotating one.
+\begin{DPgather*}
+\lintertext{\indent Since}
+x\smallbold{p} + y\smallbold{q} + z\smallbold{r} = 0, \\
+\intertext{we may by \artref{370}{Art.~370} put}
+\left.\begin{aligned}
+\smallbold{p} = f_n (kr) \left(y \frac{d}{dz} - z \frac{d}{dy}\right)\omega_n, \\
+\smallbold{q} = f_n (kr) \left(z \frac{d}{dx} - x \frac{d}{dz}\right)\omega_n, \\
+\smallbold{r} = f_n (kr) \left(x \frac{d}{dy} - y \frac{d}{dx}\right)\omega_n;
+\end{aligned}\right\}
+\Tag{15} \\
+\lintertext{where}
+f_n (kr) = \frac{S_n (kr)}{(kr)^n}, \\
+k^2 = -4\pi\mu\iota \DPtypo{S}{s}\omega/\sigma,
+\end{DPgather*}
+and $\omega_n$~is a solid spherical harmonic of degree~$n$.
+
+By \artref{372}{Art.~372}, $\alpha$,~$\beta$,~$\gamma$, the components of magnetic force, will be
+given by
+\[
+\alpha = \frac{4\pi}{(2n+1)k^2}
+  \left\{(n+1)f_{n-1} (kr)\, \frac{d\omega_n}{dx}
+    - nk^2 r^{2n+3} f_{n+1} (kr)\, \frac{d}{dx} \left(\frac{\omega_n}{r^{2n+1}}\right)\right\},
+\Tag{16}
+\]
+with similar expressions for $\beta$~and~$\gamma$.
+%% -----File: 563.png---Folio 549-------
+
+Now the magnetic force may be regarded as made up of two
+parts, one due to the currents induced in the sphere, the other to
+the external magnetic field; the latter part will be derived from
+a potential. Let~$\Omega_n$ be the value of this potential in the sphere;
+we may regard~$\Omega_n$ as a solid spherical harmonic of degree~$n$,
+since the most general expression for the potential is the sum
+of terms of this type. If $\alpha_1$,~$\beta_1$,~$\gamma_1$ are the components of the
+magnetic force due to the currents, $\alpha_0$,~$\beta_0$,~$\gamma_0$ those due to the
+magnetic field, then
+\[
+a = a_1 + a_0 = a_1 - \frac{d}{dx} \Omega_n.
+\]
+
+Hence in the sphere
+\begin{multline*}
+a_1 = \frac{d\Omega_n}{dx}
+  + \frac{4\pi}{(2n+1)k^2} \left\{(n+1)f_{n-1} (kr)\, \frac{d\omega_n}{dx} \right. \\
+    \left. -nk^2 r^{2n+3} f_{n+1} (kr)\, \frac{d}{dx} \left(\frac{\omega_n}{r^{2n+1}}\right)\right\},
+\Tag{17}
+\end{multline*}
+with similar expressions for $\beta_1$~and~$\gamma_1$.
+
+Outside the sphere the magnetic force due to the currents will
+(neglecting the displacement currents in the dielectric) be derivable
+from a potential which satisfies Laplace's equation; hence
+outside the sphere we may put, if $\omega_n'$~represents a solid harmonic,
+\[
+a_1 = -\smallbold{a}^{2n+1}\, \frac{d}{dx}\, \frac{\omega_n'}{r^{2n+1}},
+\]
+with similar expressions for $\beta_1$~and~$\gamma_1$, where $\smallbold{a}$~is the radius of
+the sphere. The magnetic force tangential to the sphere due
+to these currents is continuous, as is also the normal magnetic
+induction; hence, $\mu$~being the magnetic permeability of the
+sphere, we have
+\begin{gather*}
+\Omega_n + \frac{4\pi}{(2n+1)k^2} \left\{(n+1) f_{n-1} (k\smallbold{a}) \omega_n - nk^2 \smallbold{a}^2 f_{n+1} (k\smallbold{a}) \omega_n \right\} \\
+  = - \omega_n', \\
+\mu (n\Omega_n) + \frac{\mu n(n+1) 4\pi}{(2n+1)k^2} \left\{f_{n-1} (k\smallbold{a}) \omega_n + k^2 \smallbold{a}^2 f_{n+1} (k\smallbold{a}) \omega_n \right\} \\
+  = (n+1) \omega_n'.
+\end{gather*}
+
+Solving these equations, we find at the surface of the sphere
+\begin{align*}
+4\pi\omega_n
+  & = -\frac{(2n+1)(\mu n + n + 1) k^2 \Omega_n}
+            {(n+1)\{(\mu n + n + 1) f_{n-1} (k\smallbold{a}) + n(\mu-1) k^2 \smallbold{a}^2 f_{n+1} (k\smallbold{a})\}},
+\Tag{18} \\
+%
+\omega_n'
+  & = -\frac{n(2n+1)\mu k^2 \smallbold{a}^2 f_{n+1} (k\smallbold{a})\Omega_n}
+            {(n+1)\{(\mu n + n + 1) f_{n-1} (k\smallbold{a}) + n(\mu-1) k^2 \smallbold{a}^2 f_{n+1} (k\smallbold{a})\}}.
+\Tag{19}
+\end{align*}
+%% -----File: 564.png---Folio 550-------
+
+If we substitute these values of $\omega_n$,~$\omega_n'$ in equations (\eqnref{441}{15})~and~(\eqnref{441}{17}),
+we get the currents induced in the sphere and the magnetic
+force produced by those currents.
+
+\Article{442} We shall consider in detail the case when $n = 1$, i.e.~when
+the sphere is rotating in a uniform magnetic field. Let
+the magnetic potential of the external field be equal to the real
+part of
+\[
+Cr \cos \theta + Br \sin \theta\, \epsilon^{\iota \phi},
+\]
+where $C$~is the force parallel to~$z$ and $B$~that parallel to~$x$.
+
+Then in the sphere
+\[
+\Omega_1 = \frac{3}{\mu+ 2} (Cr \cos \theta + Br \sin \theta\, \epsilon^{\iota \phi}).
+\]
+
+We shall first consider the case when $kr$~is very small, so
+that approximately by \artref{309}{Art.~309}
+\[
+f_0 (kr) = 1 - \tfrac{1}{6} k^2 r^2, \quad
+f_1 (kr) = -\tfrac{1}{3}, \quad
+f_2 (kr) = \tfrac{1}{15}.
+\]
+
+Substituting these values in (\eqnref{441}{18})~and~(\eqnref{441}{19}) and retaining only
+the lowest powers of~$k$, we find
+\begin{gather*}
+4\pi\omega_1
+  = - \tfrac{9}{2}\, \frac{k^2}{\mu+2} \left(1+ \frac{\mu+4}{10(\mu+2)} k^2 \smallbold{a}^2\right)
+    Br \sin \theta\, \epsilon^{\iota \phi}, \\
+\omega_1' = - \frac{3k^2 \smallbold{a}^2}{10(\mu+2)^2}
+    Br \sin \theta\, \epsilon^{\iota \phi}.
+\end{gather*}
+
+The term $Cr \cos \theta$ in~$\Omega$ does not give rise to any terms in
+$\omega_n$,~$\omega_n'$ since~$s$ and therefore~$k$ vanishes for this term. Substituting
+these values we get by equations~(\eqnref{441}{15})
+\[
+\left.
+\begin{aligned}
+\smallbold{p} &= -\tfrac{3}{2}\, \frac{\mu \omega}{(\mu + 2)\sigma} zB, \\
+\smallbold{q} &= 0, \\
+\smallbold{r} &=  \tfrac{3}{2}\, \frac{\mu \omega}{(\mu + 2)\sigma} xB.
+\end{aligned} \right\}
+\Tag{20}
+\]
+
+Thus the currents flow in parallel circles, having for their
+common axis the line through the centre of the sphere which
+is at right angles both to the axis of rotation and to the direction
+of magnetic force in the external field. The intensity of the
+current at any point is proportional to the distance of the point
+from this axis.
+
+The components of the magnetic induction in the sphere are
+given by the equations
+%% -----File: 565.png---Folio 551-------
+\[
+\left.
+\begin{aligned}
+a &= - \frac{3 \mu B}{\mu + 2} \left( 1 + \frac{2 \pi \mu \omega} {5 \sigma} xy \right), \\
+b &=   \frac{3 \pi \mu^2 \omega B}{(\mu + 2) \sigma} (\tfrac{4}{5} r^2 - \tfrac{2}{5} y^2 - \tfrac{2}{5}\, \frac{\mu + 4}{\mu + 2} \smallbold{a}^2), \\
+c &= - \frac{3 \mu}{\mu + 2} \left( C + \frac{2 \pi \mu \omega}{5 \sigma} Byz \right).
+\end{aligned} \right\}
+\Tag{21}
+\]
+
+Thus the magnetic force due to currents consists of a radial
+force proportional to~$yr$, together with a force parallel to~$y$
+proportional to $2 r^2 - (\mu+4) \smallbold{a}^2 /(\mu+2)$.
+
+Outside the sphere the total magnetic potential is
+\[
+(Cz + Bx) \left( 1 - \frac{(\mu - 1)}{\mu + 2}\, \frac{\smallbold{a}^3}{r^3} \right)
+  - \frac{6 \pi B}{5 (\mu + 2)^2}\, \frac{\mu^2 \omega \smallbold{a}^5}{\sigma}\, \frac{y}{r^3}.
+\]
+
+Thus the magnetic effect of the currents at a point outside
+the sphere is the same as that of a small magnet at the centre,
+with its axis at right angles to the axis of rotation and the
+external magnetic field, and whose moment is
+\[
+\frac{6 \pi B}{5(\mu + 2)^2}\, \frac{\mu^2 \omega \smallbold{a}^5}{\sigma}.
+\]
+
+\Article{443} Let us now consider the case when $k \smallbold{a}$~is large, since,
+when $s = 1$
+\begin{DPalign*}
+k^2 &= - \frac{4 \pi \mu \omega \iota}{\sigma},\\
+\lintertext{we have}
+k &= \sqrt{2} K\, \epsilon^{-\frac{\iota \pi}{4}}, \\
+\lintertext{where}
+K^2 &= \frac{2 \pi \mu \omega}{\sigma},
+\end{DPalign*}
+thus the real part of~$\iota k \smallbold{a}$ is positive and large; hence we have
+approximately
+\begin{align*}
+f_0(k \smallbold{a}) &=  \frac{\epsilon^{\iota k \smallbold{a}}}{2 \iota k \smallbold{a}}, \\
+f_1(k \smallbold{a}) &=  \frac{\epsilon^{\iota k \smallbold{a}}}{2k^2 \smallbold{a}^2}, \\
+f_2(k \smallbold{a}) &= -\frac{\epsilon^{\iota k \smallbold{a}}}{2 \iota k^3 \smallbold{a}^3}.
+\end{align*}
+
+Hence we find
+\begin{gather*}
+4 \pi \omega_1
+  = - 3 \iota k^3 \smallbold{a}\, \epsilon^{-\iota k \smallbold{a}}\, Br \sin \theta\, \epsilon^{\iota \phi},\\
+\omega_1' = \tfrac{3}{2}\, \frac{\mu}{\mu + 2} Br \sin \theta\, \epsilon^{\iota \phi},
+\end{gather*}
+%% -----File: 566.png---Folio 552-------
+so that by~(\eqnref{441}{15})
+{\footnotesize
+\[
+\left.
+\begin{aligned}
+\smallbold{p} &= - \frac{3 \sqrt{2} K}{8 \pi}\, \frac{z \smallbold{a}}{r^2}\, B \epsilon^{-K(\smallbold{a}-r)} \cos \left\{K(\smallbold{a}-r) + \frac{\pi}{4} \right\},\\
+\smallbold{q} &= - \frac{3 \sqrt{2} K}{8 \pi}\, \frac{z \smallbold{a}}{r^2}\, B \epsilon^{-K(\smallbold{a}-r)} \sin \left\{K(\smallbold{a}-r) + \frac{\pi}{4} \right\},\\
+\smallbold{r} &=   \frac{3 \sqrt{2} K}{8 \pi}\, \frac{\smallbold{a}}{r^2}\, B \epsilon^{-K(\smallbold{a}-r)}
+  \left[ x \cos \left\{K(\smallbold{a}-r) + \frac{\pi}{4} \right\}
+       + y \sin \left\{K(\smallbold{a}-r) + \frac{\pi}{4} \right\} \right].
+\end{aligned}
+\right\}
+\eqnlabel{\eqnart.22}\tag*{\text{\normalsize(22)}}
+\]
+}
+
+The total components of the magnetic induction inside the
+sphere are given by
+{\footnotesize
+\[
+\left.
+\begin{aligned}
+a = - \mu B \frac{\smallbold{a}}{r}\, \epsilon^{-K(\smallbold{a}-r)} \cos K (\smallbold{a}-r)
+  &- \tfrac{1}{2} \mu B \epsilon^{-K (\smallbold{a}-r)} \smallbold{a}r^2 \cos K (\smallbold{a}-r)\, \frac{d}{dx}\, \frac{x}{r^3} \\
+  &- \tfrac{1}{2} \mu B \epsilon^{-K (\smallbold{a}-r)} \smallbold{a}r^2 \sin K (\smallbold{a}-r)\, \frac{d}{dx}\, \frac{y}{r^3},\\
+%
+b = - \mu B \frac{\smallbold{a}}{r}\, \epsilon^{-K(\smallbold{a}-r)} \sin K (\smallbold{a}-r)
+  &- \tfrac{1}{2} \mu B \epsilon^{-K (\smallbold{a}-r)} \smallbold{a}r^2 \cos K (\smallbold{a}-r)\, \frac{d}{dy}\, \frac{x}{r^3} \\
+  &- \tfrac{1}{2} \mu B \epsilon^{-K (\smallbold{a}-r)} \smallbold{a}r^2 \sin K (\smallbold{a}-r)\, \frac{d}{dy}\, \frac{y}{r^3},\\
+%
+c = - \frac{3 \mu C}{\mu + 2} - \tfrac{1}{2} \mu B\, \epsilon^{-K(\smallbold{a}-r)} \smallbold{a}r^2 &\cos K (\smallbold{a}-r)\, \frac{d}{dz}\, \frac{x}{r^3} \\
+  &- \tfrac{1}{2} \mu B \epsilon^{-K (\smallbold{a}-r)} \smallbold{a}r^2 \sin K (\smallbold{a}-r)\, \frac{d}{dz}\, \frac{y}{r^3},
+\end{aligned}
+\right\}
+\eqnlabel{\eqnart.23}\tag*{\text{\normalsize(23)}}
+\]
+}
+while the magnetic potential outside due to the currents in
+the sphere is
+\[
+\frac{3}{2}\, \frac{\mu B}{\mu+2}\, \smallbold{a}^3\, \frac{x}{r^3}.
+\Tag{24}
+\]
+
+If we compare these results with those we obtained when~$k \smallbold{a}$,
+was small, we see that they differ in the same way as the
+distribution of rapidly varying currents in a conductor differs
+from that of steady or slowly varying ones. When $k \smallbold{a}$~is small
+the currents spread through the whole of the sphere, while
+when $k \smallbold{a}$~is large they are, as equations~(\eqnref{443}{22}) show, confined
+to a thin shell. The currents flow along the surfaces of spheres
+concentric with the rotating one, and the intensity of the
+currents diminishes in Geometrical Progression as the distance
+from the surface of the sphere increases in Arithmetical Progression.
+
+The magnetic field due to these currents annuls in the interior
+of the sphere, as equation~(\eqnref{443}{23}) shows, that part of the external
+%% -----File: 567.png---Folio 553-------
+magnetic field which is not symmetrical about the axis of
+rotation. Thus the rotating sphere screens its interior from all
+but symmetrical distributions of magnetic force if $\{4 \pi \mu \omega / \sigma \}^{\frac{1}{2}} \smallbold{a}$ is
+large.
+
+A very interesting case of the rotating sphere is that of the
+\index{Earth's magnetism}%
+earth; in this case
+\[
+\smallbold{a} = 6.37 × 10^8,\quad \omega = 2 \pi / (24 × 60 × 60),
+\]
+so that approximately
+\[
+\{4 \pi \mu \omega / \sigma \}^{\frac{1}{2}} \smallbold{a} = 2 × 10^7 \sigma^{-\frac{1}{2}}.
+\]
+
+Thus if $\sigma$~is comparable with~$10^8$, which is of the order of the
+specific resistance of electrolytes, $k \smallbold{a}$~will be about $2000$, and this
+will be large enough to keep the earth a few miles below its
+surface practically free from the effects of an external unsymmetrical
+magnetic field.
+
+Again, we have seen, \artref{84}{Art.~84}, that rarefied gases have considerable
+conductivity for discharges travelling along closed
+curves inside them. For gases in the normal state this conductivity
+only manifests itself under large electromotive intensities,
+but when the gas is in the state similar to that
+produced by the passage of a previous discharge, it has considerable
+conductivity even for small electromotive intensities. We
+see from the preceding results that if there were a belt of gas in
+this condition in the upper regions of the earth's atmosphere,
+and if the part of the solar system traversed by the earth were
+a magnetic field, this gas would screen off from the earth all
+magnetic effects which were not symmetrical about the axis of
+rotation. Thus the magnetic field at the earth's surface would,
+on this hypothesis, resemble that which actually exists in being
+roughly symmetrical about the earth's axis. The thickness of
+a shell required to reduce the magnetic field to $1 / \epsilon$~of its value at
+the outer surface of the shell is $\{4 \pi \omega / \sigma \}^{-\frac{1}{2}}$, or if $\sigma = 10^8$, about
+two miles. The result mentioned in Art.~470 of Maxwell's
+\textit{Electricity and Magnetism}, that by far the greater part of the
+mean value of the magnetic elements arises from some cause
+inside the earth, shows, however, that we cannot assign the
+earth's permanent magnetic field to this cause.
+
+\Article{444} The total magnetic potential outside the sphere is, when
+$k \smallbold{a}$~is large, by equation~(\eqnref{443}{24}),
+%% -----File: 568.png---Folio 554-------
+\begin{multline*}
+Cz \left( 1 - \frac{\mu - 1}{\mu + 2}\, \frac{\smallbold{a}^3}{r^3} \right)
+  + B \left( x-\frac{\mu-1}{\mu+2}\, \frac{\smallbold{a}^3}{r^3} x
+    + \tfrac{3}{2}\, \frac{\mu}{\mu + 2}\, \frac{\smallbold{a}^3}{r^3} x \right) \\
+= Cz \left( 1 - \frac{\mu - 1}{\mu + 2}\, \frac{\smallbold{a}^3}{r^3} \right)
+  + Bx \left( 1 + \tfrac{1}{2}\, \frac{\smallbold{a}^3}{r^3} \right).
+\end{multline*}
+
+Thus the effect of the rotating sphere on the part of the
+external magnetic field which is unsymmetrical about the axis
+of rotation, i.e.~upon the term $Br \sin \theta\, \epsilon^{\iota \phi}$, is exactly the same
+as if this sphere were replaced by a sphere of diamagnetic
+substance for which $\mu = 0$; in other words, the rotating sphere
+behaves like a diamagnetic body. Thus we could make a
+model which would exhibit the properties of a feebly diamagnetic
+body in a steady field, by having a large number of rotating
+conductors arranged so that the distance between their centres
+was large compared with their linear dimensions.
+
+
+\Subsection{Couples and Forces on the Rotating Sphere.}
+
+\Article{445} We shall now proceed to investigate the couples and
+forces on the sphere caused by the action of the magnetic field
+on the currents induced in the sphere.
+
+If $X$,~$Y$,~$Z$ are the components of the mechanical force per
+unit volume, then (Maxwell's \textit{Electricity and Magnetism}, vol.~ii.
+Art.~603, equations~\textit{C})
+\begin{align*}
+X & = c \smallbold{q} - b \smallbold{r},\\
+Y & = a \smallbold{r} - c \smallbold{p},\\
+Z & = b \smallbold{p} - a \smallbold{q}.
+\end{align*}
+
+The couple on the sphere round the axis of~$z$ is
+\[
+\iiint (Yx - Xy)\,dx\,dy\,dz,
+\]
+the integration extending throughout the sphere.
+
+Substituting the preceding values for $Y$~and~$X$, we see that
+this may be written
+\[
+\iiint ( \smallbold{r} (ax + by + cz) - c ( \smallbold{p}x + \smallbold{q}y + \smallbold{r}z))\,dx\,dy\,dz.
+\]
+
+But since the radial current vanishes,
+\[
+\smallbold{p}x + \smallbold{q}y + \smallbold{r}z = 0;
+\]
+thus the couple round~$z$ reduces to
+\[
+\iiint \smallbold{r}Rr\,dx\,dy\,dz,
+\]
+where $R$~is the magnetic induction along the radius.
+%% -----File: 569.png---Folio 555-------
+
+Similarly the couple round~$x$ is equal to
+\index{Couple on a sphere rotating in a magnetic field}%
+\[
+\iiint \smallbold{p}Rr\,dx\,dy\,dz,
+\]
+while that round~$y$ is
+\[
+\iiint \smallbold{q}Rr\,dx\,dy\,dz.
+\]
+
+From equation~(\eqnref{441}{16}) we see that
+\[
+Rr = \frac{4\pi\mu}{(2n+1)k^2}\, n(n+1)\{f_{n-1} (kr)+k^2 r^2 f_{n+1} (kr)\}\omega_n.
+\]
+
+Now by~(\eqnref{370}{4}), \artref{370}{Art.~370},
+\[
+f_{n-1} (kr)+k^2 r^2 f_{n+1} (kr) = -(2n+1)f_n (kr),
+\]
+so that
+\[
+Rr = - \frac{4\pi\mu}{k^2}\, n(n+1)f_n (kr)\omega_n,
+\Tag{25}
+\]
+or by~(\eqnref{441}{15})
+\[
+Rr = - \frac{\sigma}{\omega s^2}\, n(n+1)\smallbold{r}.
+\]
+
+Thus the couple around~$z$ is
+\[
+- \frac{\sigma}{\omega s^2}\, n\centerdot(n+1) \iiint \smallbold{r}^2\,dx\,dy\,dz.
+\]
+
+When $\omega$~is small we find, by substituting the value of~$r$ given
+in equation~(\eqnref{442}{20}), that when the sphere is rotating in a uniform
+magnetic field the couple tending to stop it is
+\[
+\frac{6\mu^2}{5(\mu+2)^2}\, B^2\, \frac{\omega}{\sigma}\, \pi \smallbold{a}^5.
+\]
+
+\nbpagebreak[1]\Article{446} We see by equation~(\eqnref{445}{25}) that the normal component of
+the magnetic force is proportional to $f_n (kr)$, while by~(\eqnref{441}{16}) the
+other components contain terms proportional to $f_{n-1} (kr)$, but
+when $k\smallbold{a}$~is very large we have approximately
+\begin{gather*}
+f_{n-1} (k\smallbold{a}) = \tfrac{1}{2}\, \iota^{n-2}\, \frac{\epsilon^{\iota k\smallbold{a}}}{(k\smallbold{a})^n}, \\
+f_n     (k\smallbold{a}) = \tfrac{1}{2}\, \iota^{n-1}\, \frac{\epsilon^{\iota k\smallbold{a}}}{(k\smallbold{a})^{n+1}}.
+\end{gather*}
+
+Thus when $k\smallbold{a}$~is very large $f_n (k\smallbold{a})$, and near the surface of the
+sphere $f_n (k\smallbold{r})$, is very small compared with $f_{n-1} (k\smallbold{a})$, so that by~(\eqnref{445}{25})
+the magnetic force along the normal to the sphere vanishes in
+comparison with the tangential force, in other words the magnetic
+force is tangential to the surface.
+
+This result can be shown to be true, whatever the shape of
+the body, provided it is rotating with very great velocity. If
+%% -----File: 570.png---Folio 556-------
+we consider the part of the magnetic field which is not symmetrical
+about the axis of rotation we have the following
+results:---
+
+Since the magnetic potential outside the rotating bodies is
+determined by the conditions (1)~that it should have at an infinite
+distance from these bodies the same value as for the undisturbed
+external field, and (2)~that the magnetic force at right angles to
+these bodies should vanish over their surface, we see that the
+magnetic force at any point will be the same as the velocity of
+an incompressible fluid moving irrotationally and surrounding
+these bodies supposed at rest, the velocity potential at an infinite
+distance from these being equal to the magnetic potential in the
+undisturbed magnetic field.
+
+\Article{447} If we substitute the value of~$R$, given by equation~(\eqnref{445}{25}), in
+the expression for the couple round~$z$, we find that if we neglect
+powers of~$1/k\smallbold{a}$ the couple vanishes. Thus the couple vanishes
+when $\omega = 0$ and when $\omega = \infty$, there must therefore be some
+intermediate value of~$\omega$ for which the couple is a maximum.
+
+Let us now consider the forces on the sphere. The force
+parallel to~$x$ is equal to
+\begin{gather*}
+\iiint (c\smallbold{q}-b\smallbold{r})\,dx\,dy\,dz \\
+\begin{aligned}
+  &= \frac{1}{4\pi} \iiint
+     \left\{c\left(\frac{d\alpha}{dz} - \frac{d\gamma}{dx}\right)
+          - b\left(\frac{d\beta}{dx}  - \frac{d\alpha}{dy}\right)\right\} dx\,dy\,dz \\
+  &= \frac{1}{4\pi} \iint
+     \left\{\alpha(la + mb + nc) - \tfrac{1}{2} l(a\alpha + b\beta + c\gamma)\right\}\,dS,
+\end{aligned}
+\end{gather*}
+where $dS$~is an element of the surface, and $l$,~$m$,~$n$ the direction
+cosines of the outward drawn normal. The forces parallel to
+$y$~and~$z$ are given by similar expressions. We see that the force
+is equivalent to a tension parallel to the magnetic force inside
+the sphere and equal to
+\[
+\frac{1}{4\pi}(\alpha^2 + \beta^2 + \gamma^2 )^{\frac{1}{2}} R
+\]
+per unit of surface, $R$~being the magnetic induction along the
+outward normal; and to a normal pressure equal to
+\[
+\frac{1}{8\pi}(a\alpha + b\beta + c\gamma).
+\]
+
+When the sphere is rotating so rapidly that $k\smallbold{a}$~is very large
+%% -----File: 571.png---Folio 557-------
+$R$~vanishes, and the force on the rotating sphere is that due
+to a pressure
+\[
+\frac{\mu}{8\pi}(\alpha^2 + \beta^2 + \gamma^2);
+\]
+this pressure will tend to make the sphere move from the strong
+to the weak places of the field. We see, therefore, that not only
+does the rotating sphere disturb the magnetic field in the same
+way as a diamagnetic body, but that it tends to move as such
+a body would move, i.e.~from the strong to the weak parts of
+the field.
+
+\Article{448} If instead of a rotating sphere in a steady magnetic
+field we have a fixed sphere in a variable field, varying as~$\epsilon^{\iota pt}$,
+the preceding results will apply if instead of putting
+$k^2 = -4\pi\mu\omega\iota s/\sigma$ we put $k^2 = -4\pi\mu\iota p/\sigma$, and neglect the polarization
+currents in the dielectric. We can prove this at once by
+seeing that the equations for $a$,~$b$,~$c$ in the two cases become
+identical if we make this change.
+
+The results we have already obtained in this chapter, when
+applied to the case of alternating currents, show that in a
+variable field when $k\smallbold{a}$~is large the currents and magnetic force
+will be confined to a thin layer near the surface, and that a conductor
+will act like a diamagnetic body both in the way it
+disturbs the field and the way it tends to move under the
+influence of that field. The movement of currents from the
+strong to the weak parts of the field has been demonstrated in
+\index{Electromagnetic vrepulsion@\subdashone repulsion}%
+\index{Fleming, electromagnetic repulsion@\subdashone electromagnetic repulsion}%
+\index{Repulsion, electromagnetic}%
+\index{Thompson, Elihu, electromagnetic repulsion}%
+\index{Walker, electromagnetic repulsion}%
+some very striking experiments made by Professor Elihu Thomson,
+\textit{Electrical World}, 1887, p.~258 (see also Professor J.~A.
+Fleming on `Electromagnetic Repulsion,' \textit{Electrician}, 1891, pp.~567
+and~601, and Mr.\ G.~T. Walker, \textit{Phil.\ Trans}.\ A. p.~279, 1892).
+The correspondence of the magnetic force to the velocity of an
+incompressible fluid, flowing round the conductors, is more complete
+in this case than in that of the rotating sphere, inasmuch
+as we have not to except any part of the magnetic potential,
+whereas in the case of the rotating sphere we have to except
+that part of the magnetic potential which is symmetrical about
+the axis of rotation.
+%% -----File: 572.png---Folio 558-------
+%% -----File: 573.png---Folio 559-------
+
+\Chapter{Appendix.}{}
+
+\artlabel{App}\Firstsc{In} \artref{201}{Art.~201} of the text there is a description of Perrot's experiments
+on the electrolysis of steam. As these experiments throw a great deal of
+light on the way in which electrical discharges pass through gases I have,
+while this work has been passing through the press, made a series of
+experiments on the same subject.
+
+The apparatus I used was the same in principle as Perrot's. I made
+some changes, however, in order to avoid some inconveniences to which
+it seemed to me Perrot's form was liable. One source of doubt in
+Perrot's experiments arose from the proximity of the tubes surrounding
+the electrodes to the surface of the water, and their liability to get
+damp in consequence. These tubes were narrow, and if they got damp
+the sparks instead of passing directly through the steam might conceivably
+have passed from one platinum electrode to the film of moisture
+on the adjacent tube, then through the steam to the film of moisture on
+the other tube and thence to the other electrode. If anything of this
+kind happened it might be urged that since the discharge passed
+through water in its passage from one terminal to the other, some of
+the gases collected in the tubes~$gg$ (\figureref{fig84}{Fig.~84}) might have been due to
+the decomposition of the water and not to that of the steam.
+
+To overcome this objection I (1)~removed the terminals to a very
+much greater distance from the surface of the water and placed them in
+a region surrounded by a ring-burner by means of which the steam was
+heated to a temperature of $140°$\,C to $150°$\,C. \hspace{0.5em}(2)~I got rid of the
+narrow tubes surrounding the electrodes altogether by making the tubes
+through which the steam escaped partly of metal and using the metallic
+part of these tubes as the electrodes.
+
+Instead of following Perrot's plan of removing the mixed gases from
+the collecting tubes~$ee$ (\figureref{fig84}{Fig.~84}) and then exploding them in a separate
+vessel, I collected the gases on their escape from the discharge tubes in
+%% -----File: 574.png---Folio 560-------
+graduated eudiometers provided with platinum terminals, by means of
+which the mixed gases were exploded \textit{in~situ} at short intervals during
+the course of the experiments.
+
+\Subsection{Description of Apparatus.}
+
+This apparatus is represented in \figureref{fig142}{Fig.~142}. \smallsanscap{H} is a glass bulb $1.5$~to
+$2$~litres in volume containing the water which supplies the steam; a
+%% -----File: 575.png---Folio 561-------
+glass tube about $.75$~cm.\ in diameter and $35$~cm.\ in length is joined on to
+this bulb, and the top of this tube is fused on to the discharge tube~\smallsanscap{CD};
+this tube is blown out into a bulb in the region where the sparks pass,
+so that when long sparks are used they may not fly to the walls of the
+tube. This part of the tube is encircled by the ring-burner~\smallsanscap{K} by means
+of which the steam can be superheated.
+
+\includegraphicsmid{fig142}{Fig.~142.}
+
+The electrodes between which the sparks pass are shown in detail in
+\figureref{fig143}{Fig.~143}; \smallsanscap{A},~\smallsanscap{B}~are metal tubes, these must be made of a metal which does
+not oxidise. In the following experiment \smallsanscap{A},~\smallsanscap{B}~are either brass tubes
+thickly plated with gold, or tubes made by winding thick platinum wire
+into a coil. These tubes are placed in pieces of glass tubing to hold
+them in position. These tubes stop short of the places \smallsanscap{F},~\smallsanscap{G} where the
+delivery tubes join the discharge tube. The discharge tube is closed at
+the ends by the glass tubes \smallsanscap{P}~and~\smallsanscap{Q}, and wires connected to the electrodes
+\smallsanscap{A}~and~\smallsanscap{B} are fused through these tubes.
+
+\includegraphicsmid{fig143}{Fig.~143.}
+
+The delivery tubes which terminate in fine openings were fused on to
+the discharge tube at \smallsanscap{F}~and~\smallsanscap{G}.
+
+To get rid of the air which is in the apparatus or which is absorbed
+by the water, the apparatus is filled so full of water at the beginning of
+the experiment that when the water is heated it expands sufficiently
+to fill the discharge tube and overflow through the delivery tubes. The
+water is boiled vigorously for $6$~or $7$~hours with the ends of the delivery
+tubes open to the atmosphere. The eudiometer tubes filled with
+mercury are then placed over the ends of the delivery tubes, so that if
+any air is mixed with the steam it will be collected in these tubes.
+The sparking is not commenced until after the steam has run into the
+delivery tubes for about an hour without carrying with it a quantity of
+air large enough to be detected.
+
+The sparks are produced by a large induction coil giving sparks
+about $5$~cm.\ long when the current from five large storage cells is sent
+through the primary. When a condenser of about $6$~or $7$~micro-farads
+capacity is added to that supplied with the instrument a current was
+produced which, when the distance between the electrodes \smallsanscap{A}~and~\smallsanscap{B} in
+%% -----File: 576.png---Folio 562-------
+the discharge tube is not more than about $4$~mm., will liberate about
+$4$~c.c.\ of hydrogen per hour in a water voltameter placed in series with
+the discharge tube.
+
+
+\Subsection{Method of making the Experiments.}
+
+When it had been ascertained that all the air had been expelled from
+the vessel and from the water, and that the rates of flow of the gases
+through the delivery tubes were approximately equal, the eudiometer
+tubes filled with mercury were placed over the ends of the delivery
+tubes, a water voltameter was placed in series with the steam tube, and
+the coil set in action.
+
+The steam which went up the eudiometer tubes condensed into hot
+water which soon displaced the mercury; the mixture of oxygen and
+hydrogen produced by the spark went up the eudiometer tubes and
+was collected over this hot water and exploded at short intervals of time
+by the sparks from a Wimshurst machine. The gases did not disappear
+entirely when the sparks passed; a small fraction of the volume remained
+over after each explosion, and the volume which remained was
+greater in one tube than the other. The residual gas which had the
+greatest volume was found on analysis to be hydrogen, the other was
+oxygen. When a sufficient quantity of the residual gases had been
+collected they were analysed. The result of the analysis was that when
+the sparks were not too long the residual gas in one tube was pure
+hydrogen, that in the other pure oxygen; if any other gases were present
+their volume was too small to be detected by my analyses. When the
+sparks were very long there was always some other gas (nitrogen?)\
+present, sometimes in considerable quantities.
+
+
+\Subsection{Results of the Experiments.}
+
+The results obtained by the preceding method varied greatly in their
+character with the length of the spark, I shall therefore consider them
+under the heads---`short sparks,' `medium sparks,' and `long sparks.'
+
+The lengths at which a spark changes from `short' to `medium' and
+then again to `long' depend on the intensity of the current passing
+through the steam, and therefore upon the size of the induction coil and
+the battery power used to drive it. The limits of `short,' `medium,'
+and `long' sparks given below must therefore be understood to have
+reference to the particular coil and current used in these experiments.
+With a larger coil and current these limits would expand, with a smaller
+one they would contract.
+%% -----File: 577.png---Folio 563-------
+
+\Subsection{Short Sparks.}
+\index{Arc discharge, connection between chemical change and quantity of electricity passing@\subdashtwo connection between chemical change and quantity of electricity passing}%
+
+These sparks were from $1.5$~mm.~to $4$~mm.\ long. The appearance of
+the spark showed all the characteristics of an arc discharge, it was a
+thickish column with ill-defined edges and was blown out by a wind to
+a broad flame-like appearance. For these arcs the following laws were
+found to hold:---
+
+1. That within the limits of error of the experiments the volumes of
+the excesses of hydrogen in one tube and of oxygen in the other which
+remain after the explosion of the mixed gases are respectively equal to
+the volumes of the hydrogen and oxygen liberated in the water voltameter
+placed in series with the steam tube.
+
+2. The excess of hydrogen appears in the tube which is in connection
+with the \emph{positive} electrode, the excess of oxygen in the tube which is in
+connection with the \emph{negative} electrode.
+
+It thus appears that with these short sparks or arcs the hydrogen
+appears at the \emph{positive} electrode instead of as in ordinary electrolysis at
+the \emph{negative}.
+
+The following table contains the results of some measurements of the
+relation between the excesses of hydrogen and oxygen in the eudiometer
+tubes attached to the steam tube and the quantity of hydrogen liberated
+in a water voltameter placed in series with the discharge tube. The
+ordinary vibrating break supplied with induction coils was used unless
+the contrary is specified:---
+\begin{center}
+\tabletextsize
+\setlength{\tabcolsep}{2pt}
+\settowidth{\TmpPadLen}{Platinum}
+\begin{tabular}{|*{6}{c|}}
+\hline
+\settowidth{\TmpLen}{Spark length}%
+\parbox[c]{\TmpLen}{\medskip\centering Spark length\\in milli-\\metres.\medskip} &
+\settowidth{\TmpLen}{Metal used for}%
+\parbox[c]{\TmpLen}{\centering Metal used for\\electrodes.} &
+\settowidth{\TmpLen}{Excess of $H$ in}%
+\parbox[c]{\TmpLen}{\centering Excess of $H$ in\\tube next $+$\\electrode.} &
+\settowidth{\TmpLen}{Excess of $O$ in}%
+\parbox[c]{\TmpLen}{\centering Excess of $O$ in\\tube next $-$\\electrode.} &
+\settowidth{\TmpLen}{$H$ liberated}%
+\parbox[c]{\TmpLen}{\centering $H$ liberated\\in water\\voltameter.} &
+\settowidth{\TmpLen}{Duration of}%
+\parbox[c]{\TmpLen}{\centering Duration of\\experiment\\in minutes.} \\
+\hline
+\tablespaceup$1.5$ & \parbox[c]{\TmpPadLen}{Gold} & $3.25$\rlap{ c.c.}& $1.5\Z$\rlap{c.c.} & $3.2\Z$\rlap{c.c.}     & $40$  \\
+$1.5$              & Platinum                      & $2.8\Z$           & $1.6\Z$      & $3\phantom{.00}$ & $30$  \\
+$1.5$              & \parbox[c]{\TmpPadLen}{Gold} & $1.7\Z$           & $\Z.8\Z$     & $1.8\Z$          & $20$  \\
+$2\phantom{.0}$    & \parbox[c]{\TmpPadLen}{Gold} & $2\phantom{.00}$  & $1.08$       & $1.95$           & $30$  \\
+$2\phantom{.0}$    & \parbox[c]{\TmpPadLen}{Gold} & $3.25$            & $1.75$       & $3.2\Z$          & $60$  \\
+$2\phantom{.0}$    & Platinum                      & $1.8\Z$           & Tube broke   & $2\phantom{.00}$ & Not noted  \\
+$2\phantom{.0}$    & Platinum                      & $3\phantom{.00}$  & $1.5\Z$      & $3\phantom{.00}$ & $60$  \\
+$2\phantom{.0}$    & \parbox[c]{\TmpPadLen}{Gold} & $2.5\Z$           & $1.5\Z$      & $3\phantom{.00}$ & $60$  \\
+$3\phantom{.0}$    & \parbox[c]{\TmpPadLen}{Gold} & $1.8\Z$           & Not noted    & $1.8\Z$          & Not noted  \\
+$3\rlap{\tabfootmark}\phantom{.0}$%
+                   & \parbox[c]{\TmpPadLen}{Gold} & $\Z.7\Z$          & $\Z.4\Z$     & $\Z.8\Z$         & $90$  \\
+$3\rlap{\tabfootmark}\phantom{.0}$%
+                   & \parbox[c]{\TmpPadLen}{Gold} & $1.6\Z$           & Not noted    & $1.75$           & Not noted  \\
+$4\phantom{.0}$    & \parbox[c]{\TmpPadLen}{Gold} & $\Z.9\Z$          & $\Z.37$      & $\Z.7\Z$         & $20$  \\
+$4\phantom{.0}$    & \parbox[c]{\TmpPadLen}{Gold} & $2.75$            & $1.25$       & $2.7\Z$          & $60$  \\
+$4\rlap{\tabfootmark[0]}\phantom{.0}$%
+                   & \parbox[c]{\TmpPadLen}{Gold} & $1.0\Z$           & Not noted    & $1.25$           & Not noted  \\
+$4\phantom{.0}$    & \parbox[c]{\TmpPadLen}{Gold} & $2.5\Z$           & $1.25$       & $2.3\Z$          & $45$\tablespacedown \\
+\hline
+\end{tabular}
+\end{center}
+
+\addtocounter{footnote}{-1}%
+\footnotetext[\value{footnote}]{In this experiment a slow mercury break, making about four breaks a second, was used.}
+\addtocounter{footnote}{1}%
+\footnotetext[\value{footnote}]{In these experiments Leyden jars were attached to the electrodes.}
+%% -----File: 578.png---Folio 564-------
+
+The results tabulated above show that the excesses of hydrogen and
+oxygen from the steam are approximately equal to the quantities of
+hydrogen and oxygen liberated in the water voltameter.
+
+
+\Subsection{Medium Sparks.}
+
+When the spark length is greater than $4$~mm.\ the first of the preceding
+results ceases to hold. The second, that the hydrogen comes off
+at the positive electrode, remains true until the sparks are about $11$~mm.\
+long, but the hydrogen from the steam, instead of being equal to that
+from the voltameter, is, when the increase in the spark length is not too
+large, considerably greater.
+
+\noindent\begin{minipage}{\textwidth} %[TN: minipage to keep sentence with table]
+The following are a few instances of this:---
+\begin{center}
+\tabletextsize
+\begin{tabular}{c *{2}{>{\qquad}{c}}}
+Spark length &
+\settowidth{\TmpLen}{Hydrogen from}%
+\parbox[c]{\TmpLen}{\centering Hydrogen from\\steam in c.c.\medskip} &
+\settowidth{\TmpLen}{voltameter in c.c.}%
+\parbox[c]{\TmpLen}{\centering Hydrogen from\\voltameter in c.c.} \\
+$5$ mm. & $1.8\Z$          & $1.2$           \\
+$5$ mm. & $3.75$           & $3\phantom{.0}$ \\
+$5$ mm. & $4.4\Z$          & $2.1$           \\
+$6$ mm. & $4\phantom{.00}$ & $1.6$           \\
+$7$ mm. & $4.25$           & $3\phantom{.0}$  \\
+$7$ mm. & $3.75$           & $2\phantom{.0}$  \\
+$8$ mm. & $3.75$           & $2.6$
+\end{tabular}
+\medskip
+\end{center}
+\end{minipage}
+
+The increase in the ratio of the hydrogen from the steam to that from
+the voltameter does not continue when the spark length is still further
+increased. When the spark length exceeds $8$~mm.\ this ratio begins to
+fall off very rapidly as the spark length increases, and we soon reach a
+critical spark length at which it seems almost a matter of chance whether
+the hydrogen from the steam appears at the positive or the negative
+electrode.
+
+
+\Subsection{Long Sparks.}
+
+When the spark length is increased beyond the critical value, the
+excess of hydrogen instead of appearing at the positive electrode as with
+shorter sparks changes over to the negative, the excess of oxygen at the
+same time going over from the negative to the positive electrode. Thus
+the gases, when the spark length is greater than its critical value, appear
+at the same terminals in the steam tube as when liberated from an
+ordinary electrolyte, instead of at the opposite ones as they do when the
+sparks are shorter.
+
+The critical length depends very largely upon the current sent through
+the steam; the smaller the current the shorter this length. It also
+depends upon a number of small differences, some of which are not easily
+specified, and it will sometimes change suddenly without any apparent
+reason. I have found, however, that this capriciousness disappears if
+%% -----File: 579.png---Folio 565-------
+Leyden jars are attached to the terminals of the steam tube or if an
+air-break is placed in series with that tube.
+
+It will be seen that the results when the spark length is greater than
+the critical length agree with those obtained by Perrot (\artref{201}{Art.~201}) and
+Ludeking (\artref{210}{Art.~210}), as both these observers found that the hydrogen
+appeared at the negative, the oxygen at the positive electrode. Ludeking
+worked with long sparks, so that his results are quite in accordance
+with mine. In Perrot's experiments the spark length was $6$~mm.
+I have never been able to reduce the critical length quite so low as this,
+though I diminished the current to the magnitude of that used by
+Perrot; I have, however, got it as low as $8$~mm., and it is probable that
+the critical length may not be governed entirely by the current.
+
+\nblabel{pag:565}I was not able to detect any decided change in the appearance of the
+spark as the spark length passed through the critical value. My
+observations on the connection between the appearance of the discharge
+and the electrode at which the hydrogen appears may be expressed by
+the statement that when the discharge is plainly an arc the hydrogen
+appears at the positive electrode, and that when the hydrogen appears
+at the negative electrode the discharge shows all the characteristics of
+a spark. It however looks much more like a spark than an arc long
+before the spark length reaches the critical value.
+
+With regard to the ratio of the quantities of hydrogen liberated from
+the steam tube and from the water voltameter, I found that when the
+spark length was a few millimetres greater than the critical length the
+amount of hydrogen from the steam was the same as that from the voltameter.
+The following table contains a few measurements on this
+point:---
+\begin{center}
+\tabletextsize
+\begin{tabular}{c *{2}{>{\qquad}{c}}}
+Spark length. &
+\settowidth{\TmpLen}{Hydrogen from}%
+\parbox[c]{\TmpLen}{\medskip\centering Hydrogen from\\steam in c.c.\medskip} &
+\settowidth{\TmpLen}{voltameter in c.c.}%
+\parbox[c]{\TmpLen}{\centering Hydrogen from\\voltameter in c.c.} \\
+$10$ mm. & $.7\Z$ & $\Z.8$ \\
+$12$ mm.\rlap{\tabfootmark[-1]}
+         & $.75$  & $\Z.9$ \\
+$14$ mm. & $.8\Z$ & $1.1$
+\end{tabular}
+\footnotetext[\value{footnote}]{In this experiment there was an air break $9$~mm.~long in series with the steam
+tube.}
+\end{center}
+
+When the sparks were longer than $14$~mm.\ the amount of hydrogen
+from the steam was no longer equal to that from the voltameter. The
+results became irregular, and there was a further reversal of the electrode
+at which the hydrogen appeared when the spark length exceeded
+$22$~mm. In this case, however, the current was so small that it took
+several hours to liberate $1$~c.c.\ of hydrogen in the voltameter. With these
+very long sparks the proportion between the hydrogen from the steam
+and that from the voltameter was too irregular to allow of any conclusions
+being drawn.
+%% -----File: 580.png---Folio 566-------
+
+\index{Arc discharge, electrification in@\subdashtwo electrification in}%
+\index{Electrification xin arc discharge@\subdashone in arc discharge}%
+We see from the preceding results that in the electrolysis of steam, as
+in that of water, there is a very close connection between the amounts of
+hydrogen and oxygen liberated at the electrodes and the quantity of
+electricity which has passed through the steam, and that this relation for
+certain lengths of arc is the same for steam as for water. There is,
+however, this remarkable difference between the electrolysis of steam and
+that of water, that whereas in water the hydrogen always comes off at the
+negative, the oxygen at the positive electrode, in steam the hydrogen and
+oxygen come off sometimes at one terminal, sometimes at the other,
+according to the nature of the spark.
+
+\includegraphicsmid{fig144}{Fig.~144.}
+
+The results obtained when the discharge passed as an arc, i.e.~that the
+oxygen appears at the negative electrode, the hydrogen at the positive, is
+what would happen if the oxygen in the arc had a positive charge, the
+hydrogen a negative one. With the view of seeing if I could obtain any
+other evidence of this peculiarity I tried the following experiments, the
+arrangement of which is represented in \figureref{fig144}{Fig.~144}.
+
+An arc discharge between the platinum terminals \smallsanscap{A},~\smallsanscap{B} was produced
+by a large transformer, which transformed up in the ratio of $400$~to~$1$;
+a current of about $40$~Ampères making $80$~alternations per second was
+%% -----File: 581.png---Folio 567-------
+sent through the primary. A current of the gas under examination
+entered the discharge tube through a glass tube~\smallsanscap{C} and blew the gas in
+the neighbourhood of the arc against the platinum electrode~\smallsanscap{E}, which was
+connected to one quadrant of an electrometer, the other quadrant of
+which was connected to earth. To screen~\smallsanscap{E} from external electrical influences
+it was enclosed in a platinum tube~\smallsanscap{D}, which was closed in by fine
+platinum wire gauze, which though it screened~\smallsanscap{E} from external electrostatic
+action, yet allowed the gases in the neighbourhood of the arc to pass
+through it. This tube was connected to earth. The electrode~\smallsanscap{E} after
+passing out of this tube was attached to one end of gutta-percha covered
+wire wound round with tin-foil connected to earth.
+
+The experiments were of the following kind. The quadrants of the
+electrometer were charged up by a battery, the connection with the
+battery was then broken and the rate of leak observed. When the arc
+was not passing the insulation was practically perfect. As soon, however,
+as the arc was started, and for as long as it continued, the insulation of
+the gas in many cases completely gave way. There are, however, many
+remarkable exceptions to this which we proceed to consider.
+
+
+\Subsection{Oxygen.}
+
+We shall begin by considering the case when a well-developed arc
+passed through the oxygen.
+
+If the electrode~\smallsanscap{E} was charged negatively, it lost its charge very
+rapidly; it did not however remain uncharged, but acquired a positive
+charge, this charge increasing until~\smallsanscap{E} acquired a potential~$V$; $V$~depended
+greatly upon the size of the arc and the proximity to it of the
+electrode~\smallsanscap{E}, in many of my experiments it was as large as $10$~or $12$~volts.
+
+When \smallsanscap{E}~was charged positively to a high potential the electricity
+leaked from it until the potential fell to~$V$; after reaching this potential
+the leak stopped and the gas seemed to insulate as well as when no discharge
+passed through it. If the potential to which~\smallsanscap{E} was initially raised
+was less than~$V$ (a particular case being when it was without charge to
+begin with) the positive charge increased until the potential of~\smallsanscap{E} was
+equal to~$V$, after which it remained constant. Thus we see (1)~that an
+electrode immersed in the oxygen of the arc can insulate a small positive
+charge perfectly, while it very rapidly loses a negative one; (2)~that an
+uncharged electrode immersed in this gas acquires a positive charge.
+
+When the distance between the electrodes \smallsanscap{A},~\smallsanscap{B} was increased until the
+discharge passed as a spark then the electrode~\smallsanscap{E} leaked slowly, whether
+charged positively or negatively. The rate of leak in this case was however
+%% -----File: 582.png---Folio 568-------
+exceedingly small compared to that which existed when the discharge
+passed as an arc.
+
+
+\Subsection{Hydrogen.}
+
+When similar experiments were tried in hydrogen the results were
+quite different. When the \emph{arc} discharge passed through the hydrogen
+the electrode~\smallsanscap{E} always leaked when it was positively electrified, and it
+did not merely lose its charge but acquired a negative one, its potential
+falling to~$-U$, where $U$~is a quantity which depended upon the size of
+the arc and on its proximity to the electrode~\smallsanscap{E}. In my experiments $5$~to
+$6$~volts was a common value of~$U$.
+
+When the electrode~\smallsanscap{E} was initially uncharged it acquired a negative
+charge, the potential falling to~$-U$; when it was initially charged negatively,
+it leaked if its initial negative potential was greater than~$U$ until
+its potential fell to this value, when no further leak occurred. When the
+initial negative potential of~\smallsanscap{E} was less than~$U$ the negative charge
+increased until the potential had fallen to~$-U$.
+
+It is more difficult to get a good arc in hydrogen than in oxygen,
+so that the experiments with the former gas are a little more troublesome
+than those with the latter. When short arcs are used the electrode~\smallsanscap{E}
+must be placed close to the arc.
+
+The following experiment was made to see if the charging up of the
+electrode was due to an electrification developed by the contact of the gas
+in the arc with the electrode, or whether this gas behaved as if it had a
+charge of electricity independent of its contact with the metal of the
+electrode. If the electrification were due to the contact of the gas with
+the electrode it would disappear if the electrode were covered with a non-conducting
+layer; if however the gas in the arc behaved as if it were
+charged with electricity, then even though the electrode were covered
+with a non-conducting layer the electrostatic induction due to the charge
+in the gas ought to produce a deflection of the electrometer in the same
+direction as if the electrode were uncovered. To test this point the
+electrode~\smallsanscap{E} was coated with glass, with mica, with ebonite, and with
+sulphur; in all these cases the needle of the electrometer was deflected as
+long as the arc was passing, and the deflection corresponded to a positive
+charge on the gas when the arc passed through oxygen and to a negative
+one when it passed through hydrogen; this deflection disappeared almost
+entirely as soon as the arc stopped.
+
+In another experiment tried with the same object the arc was surrounded
+by a large glass tube coated inside and out with a thin layer of
+sulphur so as to prevent conduction over the surface. A ring of tin-foil
+was placed outside the tube so as to surround the place where the arc
+%% -----File: 583.png---Folio 569-------
+passed, and this ring was connected with one of the quadrants of an
+electrometer. As a further precaution against the creeping of the electricity
+over the surface of the tube two thin rings of tin-foil connected to
+the earth were placed round the ends of the tube. When the arc passed
+through oxygen the quadrants of the electrometer connected with the
+ring of tin-foil were positively electrified by induction, when the arc
+passed through hydrogen they were negatively charged.
+
+These experiments show that the oxygen in the arc behaves as if it
+had a \emph{positive} charge of electricity, while the hydrogen in the arc
+behaves as if it had a \emph{negative} charge.
+
+In all the above experiments the electrodes were so large that they
+were not heated sufficiently by the discharge to become luminous.
+
+\nblabel{add:2}Elster and Geitel found (\artref{43}{Art.~43}) that a metal plate placed near a red-hot
+\index{Elster and Geitel, electrification produced by glowing bodies}%
+platinum wire became positively electrified if the wire and the plate
+were surrounded by oxygen, and negatively electrified if they were surrounded
+by hydrogen. If we suppose that the effect of the hot wire is to
+put the gas around it in a condition resembling the gas in the arc, Elster
+and Geitel's results would be explained by the preceding experiments, for
+these have shown that when this gas is oxygen it is positively electrified,
+while when it is hydrogen it is negatively electrified.
+
+These experiments suggest the following explanation of the results of
+the investigation on the electrolysis of steam. We have seen (\artref{212}{Art.~212})
+that when an electric discharge passes through a gas the properties of
+the gas in the neighbourhood of the line of discharge are modified, and
+(\artref{84}{Art.~84}) that this modified gas possesses very considerable conductivity.
+When the discharge stops, this modified gas goes back to its original condition.
+If now the discharges through the gas follow one another so
+rapidly that the modified gas produced by one discharge has not time to
+revert to its original condition before the next discharge passes, the successive
+discharges will pass through the modified gas. If, on the other
+hand, the gas has time to return to its original condition before the next
+discharge passes, each discharge will have to make its way through the
+unmodified gas.
+
+We regard the arc discharge as corresponding to the first of the
+preceding cases when the discharge passes through the modified gas, the
+spark discharge as corresponding to the second case when the discharge
+passes through the gas in its unmodified condition.
+
+From this point of view the explanation of the results observed in the
+electrolysis of steam are very simple. The modified gas produced by the
+passage of the discharge through the steam consists of a mixture of
+hydrogen and oxygen, these gases being in the same condition as when
+the arc discharge passes through hydrogen and oxygen respectively, when,
+%% -----File: 584.png---Folio 570-------
+as we have seen, the hydrogen behaves as if it had a negative charge, the
+oxygen as if it had a positive one. Thus in the case of the \emph{arc} in steam
+the oxygen, since it behaves as if it had a positive charge, will move in
+the direction of the current and appear at the \emph{negative} electrode; the
+hydrogen will move in the opposite direction and appear at the \emph{positive}
+electrode.
+
+The equality which we found to exist between the quantities of hydrogen
+and oxygen from the electrolysis of the steam and those liberated from
+the electrolysis of water by the same current, shows that the charges on
+the atoms of the modified oxygen and hydrogen are the same in amount
+but opposite in sign to the charges we ascribe to them in ordinary
+electrolytes.
+
+In the case of the long sparks when the discharge goes through the
+steam itself, since the molecule of steam consists of two positively
+charged hydrogen atoms and one negatively charged oxygen atom, when
+this splits up in the electric field the hydrogen atoms will go towards the
+negative, the oxygen atom towards the positive electrode, as in ordinary
+electrolysis. The experiments described on page~\pageref{pag:565} show that with these
+long sparks the hydrogen appears at the negative, the oxygen at the
+positive electrode.
+%% -----File: 585.png---Folio 571-------
+
+%Include the index here
+\printindex
+
+\begin{center}
+\vspace{2ex}
+THE END.
+\end{center}
+%% -----File: 586.png---Folio 572-------
+%% -----File: 587.png---Folio 573-------
+%% -----File: 588.png---Folio 574-------
+%% -----File: 589.png---Folio 575-------
+%% -----File: 580.png---Folio 576-------
+%% -----File: 591.png---Folio 577-------
+%% -----File: 592.png---Folio 578-------
+%% -----File: 593.png---Folio 579-------
+
+\pagebreak
+\fancyhf{}
+\renewcommand{\headrulewidth}{0.5pt}
+\thispagestyle{empty}
+\fancyhead[C]{\itshape Select Works Published by the Clarendon Press}
+
+\begin{center}
+{\huge Select Works}\\[2ex]
+{PUBLISHED BY THE CLARENDON PRESS}\\[2ex]
+\rule{1in}{0.5pt}
+\end{center}
+
+\sloppy
+\selectedwork{ALDIS}{A Text-Book of Algebra: {\normalfont with} Answers to the Examples}
+By \selectedauthor{W.~S. Aldis, M.A\@.} Crown 8vo, \selectedprice{7}{6}
+
+\selectedwork{BAYNES}{Lessons on Thermodynamics} By \selectedauthor{R.~E. Baynes},
+M.A\@. Crown 8vo, \selectedprice{7}{6}
+
+\selectedwork{CHAMBERS}{A Handbook of Descriptive Astronomy} By
+\selectedauthor{G.~F. Chambers}, F.R.A.S\@. \textit{Fourth Edition.}
+
+\selectedvol{Vol.~I}  The Sun, Planets, and Comets. 8vo, \selectedprice{21}{}
+
+\selectedvol{Vol.~II} Instruments and Practical Astronomy. 8vo, \selectedprice{21}{}
+
+\selectedvol{Vol.~III} The Starry Heavens. 8vo, \selectedprice{14}{}
+
+\selectedwork{CLARKE}{Geodesy} By \selectedauthor[Col.]{A.~R. Clarke}, C.B., R.E\@. \selectedprice{12}{6}
+
+\selectedwork{CREMONA}{Elements of Projective Geometry} By \selectedauthor{Luigi
+Cremona}. Translated by \selectedauthor{C.~Leudesdorf}, M.A\@. Demy 8vo, \selectedprice{12}{6}
+
+\selectedwork{}{Graphical Statics}  Two Treatises on the Graphical
+Calculus and Reciprocal Figures in Graphical Statics. By  \selectedauthor{Luigi
+Cremona}. Translated by \selectedauthor{T.~Hudson Beare}. Demy 8vo, \selectedprice{8}{6}
+
+\selectedwork{DONKIN}{Acoustics} By \selectedauthor{W.~F. Donkin}, M.A., F.R.S\@.
+\textit{Second Edition.} Crown 8vo, \selectedprice{7}{6}
+
+\selectedwork{EMTAGE}{An Introduction to the Mathematical Theory of Electricity and Magnetism}
+By \selectedauthor{W.~T.~A. Emtage}, M.A\@. \selectedprice{7}{6}
+
+\selectedwork{JOHNSTON}{A Text-Book of Analytical Geometry} By
+\selectedauthor{W.~J. Johnston}, M.A\@. Crown 8vo. \hfill[\textit{Immediately.}
+
+\selectedwork{MAXWELL}{A Treatise on Electricity and Magnetism}
+By \selectedauthor{J.~Clerk Maxwell}, M.A\@. \textit{Third Edition.} 2~vols. 8vo, \selectedprice[1]{12}{}
+
+\selectedwork{}{An Elementary Treatise on Electricity} Edited by
+\selectedauthor{William Garnett}, M.A\@. 8vo, \selectedprice{7}{6}
+
+\selectedwork{MINCHIN}{A Treatise on Statics, with Applications to Physics}
+By \selectedauthor{G.~M. Minchin}, M.A\@. \textit{Fourth Edition.}
+
+\selectedvol{Vol.~I} Equilibrium of Coplanar Forces. 8vo, \selectedprice{10}{6}
+
+\selectedvol{Vol.~II} Non-Coplanar Forces. 8vo, \selectedprice{16}{}
+
+\selectedwork{}{Uniplanar Kinematics of Solids and Fluids} Crown
+8vo, \selectedprice{7}{6}
+
+\selectedwork{}{Hydrostatics and Elementary Hydrokinetics} Crown
+8vo, \selectedprice{10}{6}
+%% -----File: 594.png---Folio 580-------
+
+\selectedwork{PRICE}{Treatise on Infinitesimal Calculus} By \selectedauthor{Bartholomew
+Price}, D.D., F.R.S\@.
+
+\selectedvol{Vol.~I}  Differential Calculus. \textit{Second Edition.} 8vo, \selectedprice{14}{6}
+
+\selectedvol{Vol.~II} Integral Calculus, Calculus of Variations, and Differential
+    Equations. \textit{Second Edition.} 8vo, \selectedprice{18}{}
+
+\selectedvol{Vol.~III} Statics, including Attractions; Dynamics of a Material Particle.
+    \textit{Second Edition.} 8vo, \selectedprice{16}{}
+
+\selectedvol{Vol.~IV} Dynamics of Material Systems. \textit{Second Edition.} 8vo, \selectedprice{18}{}
+
+\selectedwork{RUSSELL}{An Elementary Treatise on Pure Geometry, \normalfont with numerous Examples}
+By \selectedauthor{J.~Wellesley Russell}, M.A\@. Crown 8vo, \selectedprice{10}{6}
+
+\selectedwork{SELBY}{Elementary Mechanics of Solids and Fluids} By
+\selectedauthor{A.~L. Selby}, M.A\@. Crown 8vo, \selectedprice{7}{6}
+
+\selectedwork{SMYTH}{A Cycle of Celestial Objects} Observed, Reduced,
+and Discussed by \selectedauthor[Admiral]{W.~H. Smyth}, R.N\@. Revised, condensed, and
+greatly enlarged by \selectedauthor{G.~F. Chambers}, F.R.A.S\@. 8vo, \selectedprice{12}{}
+
+\selectedwork{STEWART}{An Elementary Treatise on Heat} With numerous
+Woodcuts and Diagrams. By \selectedauthor{Balfour Stewart}, LL.D., F.R.S\@. \textit{Fifth
+Edition.} Extra fcap.\ 8vo, \selectedprice{7}{6}
+
+\selectedwork{VAN 'T HOFF}{Chemistry in Space} Translated and Edited
+by \selectedauthor{J.~E. Marsh}, B.A\@. Crown 8vo, \selectedprice{4}{6}
+
+\selectedwork{WALKER}{The Theory of a Physical Balance} By \selectedauthor{James
+Walker}, M.A\@. 8vo, stiff cover, \selectedprice{3}{6}
+
+\selectedwork{WATSON and BURBURY}{}
+
+\selectedpart{I}{A Treatise on the Application of Generalised Coordinates
+    to the Kinetics of a Material System} By \selectedauthor{H.~W.
+    Watson}, D.Sc., and \selectedauthor{S.~H. Burbury}, M.A\@. 8vo, \selectedprice{6}{}
+
+\selectedpart{II}{The Mathematical Theory of Electricity and Magnetism}
+
+\selectedvol{Vol.~I}  Electrostatics. 8vo, \selectedprice{10}{6}
+
+\selectedvol{Vol.~II} Magnetism and Electrodynamics. 8vo, \selectedprice{10}{6}
+\fussy
+
+\begin{center}
+\rule{1in}{0.5pt}\\[2ex]
+
+$\mathfrak{Oxford:}$\\[1ex]
+
+AT THE CLARENDON PRESS.\\[1ex]
+
+\small LONDON: HENRY FROWDE,\\[1ex]
+
+\footnotesize OXFORD UNIVERSITY PRESS WAREHOUSE, AMEN CORNER, E.C.
+\end{center}
+
+\clearpage
+
+\pdfbookmark[0]{Project Gutenberg License}{Project Gutenberg License}
+
+\renewcommand{\headrulewidth}{0pt}
+\fancyhead[C]{\footnotesize LICENSING}
+\pagenumbering{Alph}
+
+\begin{verbatim}
+End of the Project Gutenberg EBook of Notes on Recent Researches in
+Electricity and Magnetism, by J. J. Thomson
+
+*** END OF THIS PROJECT GUTENBERG EBOOK RECENT RESEARCHES--ELECTRICITY, MAGNETISM ***
+
+***** This file should be named 36525-t.tex or 36525-t.zip *****
+This and all associated files of various formats will be found in:
+        http://www.gutenberg.org/3/6/5/2/36525/
+
+Produced by Robert Cicconetti, Nigel Blower and the Online
+Distributed Proofreading Team at http://www.pgdp.net (This
+file was produced from images generously made available
+by Cornell University Digital Collections)
+
+
+Updated editions will replace the previous one--the old editions
+will be renamed.
+
+Creating the works from public domain print editions means that no
+one owns a United States copyright in these works, so the Foundation
+(and you!) can copy and distribute it in the United States without
+permission and without paying copyright royalties.  Special rules,
+set forth in the General Terms of Use part of this license, apply to
+copying and distributing Project Gutenberg-tm electronic works to
+protect the PROJECT GUTENBERG-tm concept and trademark.  Project
+Gutenberg is a registered trademark, and may not be used if you
+charge for the eBooks, unless you receive specific permission.  If you
+do not charge anything for copies of this eBook, complying with the
+rules is very easy.  You may use this eBook for nearly any purpose
+such as creation of derivative works, reports, performances and
+research.  They may be modified and printed and given away--you may do
+practically ANYTHING with public domain eBooks.  Redistribution is
+subject to the trademark license, especially commercial
+redistribution.
+
+
+
+*** START: FULL LICENSE ***
+
+THE FULL PROJECT GUTENBERG LICENSE
+PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
+
+To protect the Project Gutenberg-tm mission of promoting the free
+distribution of electronic works, by using or distributing this work
+(or any other work associated in any way with the phrase "Project
+Gutenberg"), you agree to comply with all the terms of the Full Project
+Gutenberg-tm License (available with this file or online at
+http://gutenberg.net/license).
+
+
+Section 1.  General Terms of Use and Redistributing Project Gutenberg-tm
+electronic works
+
+1.A.  By reading or using any part of this Project Gutenberg-tm
+electronic work, you indicate that you have read, understand, agree to
+and accept all the terms of this license and intellectual property
+(trademark/copyright) agreement.  If you do not agree to abide by all
+the terms of this agreement, you must cease using and return or destroy
+all copies of Project Gutenberg-tm electronic works in your possession.
+If you paid a fee for obtaining a copy of or access to a Project
+Gutenberg-tm electronic work and you do not agree to be bound by the
+terms of this agreement, you may obtain a refund from the person or
+entity to whom you paid the fee as set forth in paragraph 1.E.8.
+
+1.B.  "Project Gutenberg" is a registered trademark.  It may only be
+used on or associated in any way with an electronic work by people who
+agree to be bound by the terms of this agreement.  There are a few
+things that you can do with most Project Gutenberg-tm electronic works
+even without complying with the full terms of this agreement.  See
+paragraph 1.C below.  There are a lot of things you can do with Project
+Gutenberg-tm electronic works if you follow the terms of this agreement
+and help preserve free future access to Project Gutenberg-tm electronic
+works.  See paragraph 1.E below.
+
+1.C.  The Project Gutenberg Literary Archive Foundation ("the Foundation"
+or PGLAF), owns a compilation copyright in the collection of Project
+Gutenberg-tm electronic works.  Nearly all the individual works in the
+collection are in the public domain in the United States.  If an
+individual work is in the public domain in the United States and you are
+located in the United States, we do not claim a right to prevent you from
+copying, distributing, performing, displaying or creating derivative
+works based on the work as long as all references to Project Gutenberg
+are removed.  Of course, we hope that you will support the Project
+Gutenberg-tm mission of promoting free access to electronic works by
+freely sharing Project Gutenberg-tm works in compliance with the terms of
+this agreement for keeping the Project Gutenberg-tm name associated with
+the work.  You can easily comply with the terms of this agreement by
+keeping this work in the same format with its attached full Project
+Gutenberg-tm License when you share it without charge with others.
+
+1.D.  The copyright laws of the place where you are located also govern
+what you can do with this work.  Copyright laws in most countries are in
+a constant state of change.  If you are outside the United States, check
+the laws of your country in addition to the terms of this agreement
+before downloading, copying, displaying, performing, distributing or
+creating derivative works based on this work or any other Project
+Gutenberg-tm work.  The Foundation makes no representations concerning
+the copyright status of any work in any country outside the United
+States.
+
+1.E.  Unless you have removed all references to Project Gutenberg:
+
+1.E.1.  The following sentence, with active links to, or other immediate
+access to, the full Project Gutenberg-tm License must appear prominently
+whenever any copy of a Project Gutenberg-tm work (any work on which the
+phrase "Project Gutenberg" appears, or with which the phrase "Project
+Gutenberg" is associated) is accessed, displayed, performed, viewed,
+copied or distributed:
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.net
+
+1.E.2.  If an individual Project Gutenberg-tm electronic work is derived
+from the public domain (does not contain a notice indicating that it is
+posted with permission of the copyright holder), the work can be copied
+and distributed to anyone in the United States without paying any fees
+or charges.  If you are redistributing or providing access to a work
+with the phrase "Project Gutenberg" associated with or appearing on the
+work, you must comply either with the requirements of paragraphs 1.E.1
+through 1.E.7 or obtain permission for the use of the work and the
+Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or
+1.E.9.
+
+1.E.3.  If an individual Project Gutenberg-tm electronic work is posted
+with the permission of the copyright holder, your use and distribution
+must comply with both paragraphs 1.E.1 through 1.E.7 and any additional
+terms imposed by the copyright holder.  Additional terms will be linked
+to the Project Gutenberg-tm License for all works posted with the
+permission of the copyright holder found at the beginning of this work.
+
+1.E.4.  Do not unlink or detach or remove the full Project Gutenberg-tm
+License terms from this work, or any files containing a part of this
+work or any other work associated with Project Gutenberg-tm.
+
+1.E.5.  Do not copy, display, perform, distribute or redistribute this
+electronic work, or any part of this electronic work, without
+prominently displaying the sentence set forth in paragraph 1.E.1 with
+active links or immediate access to the full terms of the Project
+Gutenberg-tm License.
+
+1.E.6.  You may convert to and distribute this work in any binary,
+compressed, marked up, nonproprietary or proprietary form, including any
+word processing or hypertext form.  However, if you provide access to or
+distribute copies of a Project Gutenberg-tm work in a format other than
+"Plain Vanilla ASCII" or other format used in the official version
+posted on the official Project Gutenberg-tm web site (www.gutenberg.net),
+you must, at no additional cost, fee or expense to the user, provide a
+copy, a means of exporting a copy, or a means of obtaining a copy upon
+request, of the work in its original "Plain Vanilla ASCII" or other
+form.  Any alternate format must include the full Project Gutenberg-tm
+License as specified in paragraph 1.E.1.
+
+1.E.7.  Do not charge a fee for access to, viewing, displaying,
+performing, copying or distributing any Project Gutenberg-tm works
+unless you comply with paragraph 1.E.8 or 1.E.9.
+
+1.E.8.  You may charge a reasonable fee for copies of or providing
+access to or distributing Project Gutenberg-tm electronic works provided
+that
+
+- You pay a royalty fee of 20% of the gross profits you derive from
+     the use of Project Gutenberg-tm works calculated using the method
+     you already use to calculate your applicable taxes.  The fee is
+     owed to the owner of the Project Gutenberg-tm trademark, but he
+     has agreed to donate royalties under this paragraph to the
+     Project Gutenberg Literary Archive Foundation.  Royalty payments
+     must be paid within 60 days following each date on which you
+     prepare (or are legally required to prepare) your periodic tax
+     returns.  Royalty payments should be clearly marked as such and
+     sent to the Project Gutenberg Literary Archive Foundation at the
+     address specified in Section 4, "Information about donations to
+     the Project Gutenberg Literary Archive Foundation."
+
+- You provide a full refund of any money paid by a user who notifies
+     you in writing (or by e-mail) within 30 days of receipt that s/he
+     does not agree to the terms of the full Project Gutenberg-tm
+     License.  You must require such a user to return or
+     destroy all copies of the works possessed in a physical medium
+     and discontinue all use of and all access to other copies of
+     Project Gutenberg-tm works.
+
+- You provide, in accordance with paragraph 1.F.3, a full refund of any
+     money paid for a work or a replacement copy, if a defect in the
+     electronic work is discovered and reported to you within 90 days
+     of receipt of the work.
+
+- You comply with all other terms of this agreement for free
+     distribution of Project Gutenberg-tm works.
+
+1.E.9.  If you wish to charge a fee or distribute a Project Gutenberg-tm
+electronic work or group of works on different terms than are set
+forth in this agreement, you must obtain permission in writing from
+both the Project Gutenberg Literary Archive Foundation and Michael
+Hart, the owner of the Project Gutenberg-tm trademark.  Contact the
+Foundation as set forth in Section 3 below.
+
+1.F.
+
+1.F.1.  Project Gutenberg volunteers and employees expend considerable
+effort to identify, do copyright research on, transcribe and proofread
+public domain works in creating the Project Gutenberg-tm
+collection.  Despite these efforts, Project Gutenberg-tm electronic
+works, and the medium on which they may be stored, may contain
+"Defects," such as, but not limited to, incomplete, inaccurate or
+corrupt data, transcription errors, a copyright or other intellectual
+property infringement, a defective or damaged disk or other medium, a
+computer virus, or computer codes that damage or cannot be read by
+your equipment.
+
+1.F.2.  LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
+of Replacement or Refund" described in paragraph 1.F.3, the Project
+Gutenberg Literary Archive Foundation, the owner of the Project
+Gutenberg-tm trademark, and any other party distributing a Project
+Gutenberg-tm electronic work under this agreement, disclaim all
+liability to you for damages, costs and expenses, including legal
+fees.  YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
+LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
+PROVIDED IN PARAGRAPH 1.F.3.  YOU AGREE THAT THE FOUNDATION, THE
+TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
+LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
+INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
+DAMAGE.
+
+1.F.3.  LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
+defect in this electronic work within 90 days of receiving it, you can
+receive a refund of the money (if any) you paid for it by sending a
+written explanation to the person you received the work from.  If you
+received the work on a physical medium, you must return the medium with
+your written explanation.  The person or entity that provided you with
+the defective work may elect to provide a replacement copy in lieu of a
+refund.  If you received the work electronically, the person or entity
+providing it to you may choose to give you a second opportunity to
+receive the work electronically in lieu of a refund.  If the second copy
+is also defective, you may demand a refund in writing without further
+opportunities to fix the problem.
+
+1.F.4.  Except for the limited right of replacement or refund set forth
+in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER
+WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO
+WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE.
+
+1.F.5.  Some states do not allow disclaimers of certain implied
+warranties or the exclusion or limitation of certain types of damages.
+If any disclaimer or limitation set forth in this agreement violates the
+law of the state applicable to this agreement, the agreement shall be
+interpreted to make the maximum disclaimer or limitation permitted by
+the applicable state law.  The invalidity or unenforceability of any
+provision of this agreement shall not void the remaining provisions.
+
+1.F.6.  INDEMNITY - You agree to indemnify and hold the Foundation, the
+trademark owner, any agent or employee of the Foundation, anyone
+providing copies of Project Gutenberg-tm electronic works in accordance
+with this agreement, and any volunteers associated with the production,
+promotion and distribution of Project Gutenberg-tm electronic works,
+harmless from all liability, costs and expenses, including legal fees,
+that arise directly or indirectly from any of the following which you do
+or cause to occur: (a) distribution of this or any Project Gutenberg-tm
+work, (b) alteration, modification, or additions or deletions to any
+Project Gutenberg-tm work, and (c) any Defect you cause.
+
+
+Section  2.  Information about the Mission of Project Gutenberg-tm
+
+Project Gutenberg-tm is synonymous with the free distribution of
+electronic works in formats readable by the widest variety of computers
+including obsolete, old, middle-aged and new computers.  It exists
+because of the efforts of hundreds of volunteers and donations from
+people in all walks of life.
+
+Volunteers and financial support to provide volunteers with the
+assistance they need are critical to reaching Project Gutenberg-tm's
+goals and ensuring that the Project Gutenberg-tm collection will
+remain freely available for generations to come.  In 2001, the Project
+Gutenberg Literary Archive Foundation was created to provide a secure
+and permanent future for Project Gutenberg-tm and future generations.
+To learn more about the Project Gutenberg Literary Archive Foundation
+and how your efforts and donations can help, see Sections 3 and 4
+and the Foundation web page at http://www.pglaf.org.
+
+
+Section 3.  Information about the Project Gutenberg Literary Archive
+Foundation
+
+The Project Gutenberg Literary Archive Foundation is a non profit
+501(c)(3) educational corporation organized under the laws of the
+state of Mississippi and granted tax exempt status by the Internal
+Revenue Service.  The Foundation's EIN or federal tax identification
+number is 64-6221541.  Its 501(c)(3) letter is posted at
+http://pglaf.org/fundraising.  Contributions to the Project Gutenberg
+Literary Archive Foundation are tax deductible to the full extent
+permitted by U.S. federal laws and your state's laws.
+
+The Foundation's principal office is located at 4557 Melan Dr. S.
+Fairbanks, AK, 99712., but its volunteers and employees are scattered
+throughout numerous locations.  Its business office is located at
+809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email
+business@pglaf.org.  Email contact links and up to date contact
+information can be found at the Foundation's web site and official
+page at http://pglaf.org
+
+For additional contact information:
+     Dr. Gregory B. Newby
+     Chief Executive and Director
+     gbnewby@pglaf.org
+
+
+Section 4.  Information about Donations to the Project Gutenberg
+Literary Archive Foundation
+
+Project Gutenberg-tm depends upon and cannot survive without wide
+spread public support and donations to carry out its mission of
+increasing the number of public domain and licensed works that can be
+freely distributed in machine readable form accessible by the widest
+array of equipment including outdated equipment.  Many small donations
+($1 to $5,000) are particularly important to maintaining tax exempt
+status with the IRS.
+
+The Foundation is committed to complying with the laws regulating
+charities and charitable donations in all 50 states of the United
+States.  Compliance requirements are not uniform and it takes a
+considerable effort, much paperwork and many fees to meet and keep up
+with these requirements.  We do not solicit donations in locations
+where we have not received written confirmation of compliance.  To
+SEND DONATIONS or determine the status of compliance for any
+particular state visit http://pglaf.org
+
+While we cannot and do not solicit contributions from states where we
+have not met the solicitation requirements, we know of no prohibition
+against accepting unsolicited donations from donors in such states who
+approach us with offers to donate.
+
+International donations are gratefully accepted, but we cannot make
+any statements concerning tax treatment of donations received from
+outside the United States.  U.S. laws alone swamp our small staff.
+
+Please check the Project Gutenberg Web pages for current donation
+methods and addresses.  Donations are accepted in a number of other
+ways including including checks, online payments and credit card
+donations.  To donate, please visit: http://pglaf.org/donate
+
+
+Section 5.  General Information About Project Gutenberg-tm electronic
+works.
+
+Professor Michael S. Hart is the originator of the Project Gutenberg-tm
+concept of a library of electronic works that could be freely shared
+with anyone.  For thirty years, he produced and distributed Project
+Gutenberg-tm eBooks with only a loose network of volunteer support.
+
+
+Project Gutenberg-tm eBooks are often created from several printed
+editions, all of which are confirmed as Public Domain in the U.S.
+unless a copyright notice is included.  Thus, we do not necessarily
+keep eBooks in compliance with any particular paper edition.
+
+
+Most people start at our Web site which has the main PG search facility:
+
+     http://www.gutenberg.net
+
+This Web site includes information about Project Gutenberg-tm,
+including how to make donations to the Project Gutenberg Literary
+Archive Foundation, how to help produce our new eBooks, and how to
+subscribe to our email newsletter to hear about new eBooks.
+\end{verbatim}
+
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of the Project Gutenberg EBook of Notes on Recent Researches in     %
+% Electricity and Magnetism, by J. J. Thomson                             %
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK RECENT RESEARCHES--ELECTRICITY, MAGNETISM ***
+%                                                                         %
+% ***** This file should be named 36525-t.tex or 36525-t.zip *****        %
+% This and all associated files of various formats will be found in:      %
+%         http://www.gutenberg.org/3/6/5/2/36525/                         %
+%                                                                         %
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+
+\end{document}
+
+###
+@MathEnvironments = (
+  ['\\begin{DPgather*}','\\end{DPgather*}','<MATH DPgather>'],
+  ['\\begin{DPalign*}','\\end{DPalign*}','<MATH DPalign>']
+  );
+@ControlwordReplace = (
+  ['\\ToCHead','Art. Page'],
+  ['\\ToCHeadPage','Art. Page']
+  );
+@ControlwordArguments = (
+  ['\\Article', 1,1,'',".]"],
+  ['\\Chapter', 1,1,"\n\n\n\n\n","\n\n", 1,1,'',"\n"],
+  ['\\DPtypo', 1,0,'','',1,1,'',''],
+  ['\\Section', 1,1,"\n\n","\n"],
+  ['\\Subsection', 1,1,"\n\n","\n"],
+  ['\\ToCLine', 0,0,'','', 1,1,'','. ', 1,1,'',''],
+  ['\\ToCLinR', 0,0,'','', 1,1,'','-', 1,1,'','. ', 1,1,'',''],
+  ['\\TocChapter', 1,1,'',' ', 1,1,'',''],
+  ['\\addtocounter', 1,0,'','', 1,0,'',''],
+  ['\\nblabel', 1,0,'',''],
+  ['\\artlabel', 1,0,'',''],
+  ['\\chaplabel', 1,0,'',''],
+  ['\\figurelabel', 1,0,'',''],
+  ['\\eqnlabel', 1,0,'',''],
+  ['\\chapref', 1,0,'','', 1,1,'',''],
+  ['\\figureref', 1,0,'','', 1,1,'',''],
+  ['\\artref', 1,0,'','', 1,1,'',''],
+  ['\\eqnref', 1,0,'','', 1,1,'',''],
+  ['\\fancypagestyle', 1,0,'','', 1,0,'',''],
+  ['\\includegraphicsmid', 0,0,'','', 1,0,'','', 1,1,'[Illustration: ',']'],
+  ['\\includegraphicsouter', 0,0,'','', 1,0,'','', 1,1,'[Illustration: ',']'],
+  ['\\includegraphicssideways', 0,0,'','', 1,0,'','', 1,1,'[Illustration: ',']'],
+  ['\\includegraphicstwo', 0,0,'','', 1,0,'','', 1,1,'[Illustration: ',']\n', 1,0,'','', 1,1,'[Illustration: ',']'],
+  ['\\selectedwork', 1,1,'','. ', 1,1,'','. '],
+  ['\\selectedpart', 1,1,'','. ', 1,1,'','. '],
+  ['\\selectedauthor', 0,1,'',' ', 1,1,'',''],
+  ['\\selectedprice', 0,0,'','', 1,1,'','s. ', 1,1,'','d.'],
+  ['\\selectedvol', 1,1,'','. '],
+  ['\\nbvspace', 0,0,"\n",'']
+  );
+###
+This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2010.5.6)  27 JUN 2011 16:35
+entering extended mode
+ %&-line parsing enabled.
+**36525-t.tex
+(./36525-t.tex
+LaTeX2e <2005/12/01>
+Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
+yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov
+ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
+ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
+nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
+eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia
+n, swedish, ukenglish, pinyin, loaded.
+\@indexfile=\write3
+\openout3 = `36525-t.idx'.
+
+Writing index file 36525-t.idx
+(/usr/share/texmf-texlive/tex/latex/base/book.cls
+Document Class: book 2005/09/16 v1.4f Standard LaTeX document class
+(/usr/share/texmf-texlive/tex/latex/base/bk12.clo
+File: bk12.clo 2005/09/16 v1.4f Standard LaTeX file (size option)
+)
+\c@part=\count79
+\c@chapter=\count80
+\c@section=\count81
+\c@subsection=\count82
+\c@subsubsection=\count83
+\c@paragraph=\count84
+\c@subparagraph=\count85
+\c@figure=\count86
+\c@table=\count87
+\abovecaptionskip=\skip41
+\belowcaptionskip=\skip42
+\bibindent=\dimen102
+) (/usr/share/texmf-texlive/tex/latex/geometry/geometry.sty
+Package: geometry 2002/07/08 v3.2 Page Geometry
+(/usr/share/texmf-texlive/tex/latex/graphics/keyval.sty
+Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
+\KV@toks@=\toks14
+)
+\Gm@cnth=\count88
+\Gm@cntv=\count89
+\c@Gm@tempcnt=\count90
+\Gm@bindingoffset=\dimen103
+\Gm@wd@mp=\dimen104
+\Gm@odd@mp=\dimen105
+\Gm@even@mp=\dimen106
+\Gm@dimlist=\toks15
+(/usr/share/texmf-texlive/tex/xelatex/xetexconfig/geometry.cfg))
+
+LaTeX Warning: You have requested, on input line 118, version
+               `2008/11/13' of package geometry,
+               but only version
+               `2002/07/08 v3.2 Page Geometry'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
+Package: inputenc 2006/05/05 v1.1b Input encoding file
+\inpenc@prehook=\toks16
+\inpenc@posthook=\toks17
+(/usr/share/texmf-texlive/tex/latex/base/latin1.def
+File: latin1.def 2006/05/05 v1.1b Input encoding file
+)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
+Package: amsmath 2000/07/18 v2.13 AMS math features
+\@mathmargin=\skip43
+For additional information on amsmath, use the `?' option.
+(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
+Package: amstext 2000/06/29 v2.01
+(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty
+File: amsgen.sty 1999/11/30 v2.0
+\@emptytoks=\toks18
+\ex@=\dimen107
+)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty
+Package: amsbsy 1999/11/29 v1.2d
+\pmbraise@=\dimen108
+) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty
+Package: amsopn 1999/12/14 v2.01 operator names
+)
+\inf@bad=\count91
+LaTeX Info: Redefining \frac on input line 211.
+\uproot@=\count92
+\leftroot@=\count93
+LaTeX Info: Redefining \overline on input line 307.
+\classnum@=\count94
+\DOTSCASE@=\count95
+LaTeX Info: Redefining \ldots on input line 379.
+LaTeX Info: Redefining \dots on input line 382.
+LaTeX Info: Redefining \cdots on input line 467.
+\Mathstrutbox@=\box26
+\strutbox@=\box27
+\big@size=\dimen109
+LaTeX Font Info:    Redeclaring font encoding OML on input line 567.
+LaTeX Font Info:    Redeclaring font encoding OMS on input line 568.
+\macc@depth=\count96
+\c@MaxMatrixCols=\count97
+\dotsspace@=\muskip10
+\c@parentequation=\count98
+\dspbrk@lvl=\count99
+\tag@help=\toks19
+\row@=\count100
+\column@=\count101
+\maxfields@=\count102
+\andhelp@=\toks20
+\eqnshift@=\dimen110
+\alignsep@=\dimen111
+\tagshift@=\dimen112
+\tagwidth@=\dimen113
+\totwidth@=\dimen114
+\lineht@=\dimen115
+\@envbody=\toks21
+\multlinegap=\skip44
+\multlinetaggap=\skip45
+\mathdisplay@stack=\toks22
+LaTeX Info: Redefining \[ on input line 2666.
+LaTeX Info: Redefining \] on input line 2667.
+) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
+Package: amssymb 2002/01/22 v2.2d
+(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty
+Package: amsfonts 2001/10/25 v2.2f
+\symAMSa=\mathgroup4
+\symAMSb=\mathgroup5
+LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'
+(Font)                  U/euf/m/n --> U/euf/b/n on input line 132.
+)) (/usr/share/texmf-texlive/tex/latex/fancyhdr/fancyhdr.sty
+\fancy@headwidth=\skip46
+\f@ncyO@elh=\skip47
+\f@ncyO@erh=\skip48
+\f@ncyO@olh=\skip49
+\f@ncyO@orh=\skip50
+\f@ncyO@elf=\skip51
+\f@ncyO@erf=\skip52
+\f@ncyO@olf=\skip53
+\f@ncyO@orf=\skip54
+) (/usr/share/texmf-texlive/tex/latex/fancyhdr/extramarks.sty
+\@temptokenb=\toks23
+) (/usr/share/texmf-texlive/tex/latex/tools/verbatim.sty
+Package: verbatim 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements
+\every@verbatim=\toks24
+\verbatim@line=\toks25
+\verbatim@in@stream=\read1
+) (/usr/share/texmf-texlive/tex/latex/graphics/graphicx.sty
+Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
+(/usr/share/texmf-texlive/tex/latex/graphics/graphics.sty
+Package: graphics 2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
+(/usr/share/texmf-texlive/tex/latex/graphics/trig.sty
+Package: trig 1999/03/16 v1.09 sin cos tan (DPC)
+) (/etc/texmf/tex/latex/config/graphics.cfg
+File: graphics.cfg 2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
+)
+Package graphics Info: Driver file: pdftex.def on input line 90.
+(/usr/share/texmf-texlive/tex/latex/pdftex-def/pdftex.def
+File: pdftex.def 2007/01/08 v0.04d Graphics/color for pdfTeX
+\Gread@gobject=\count103
+))
+\Gin@req@height=\dimen116
+\Gin@req@width=\dimen117
+) (/usr/share/texmf-texlive/tex/latex/wrapfig/wrapfig.sty
+\wrapoverhang=\dimen118
+\WF@size=\dimen119
+\c@WF@wrappedlines=\count104
+\WF@box=\box28
+\WF@everypar=\toks26
+Package: wrapfig 2003/01/31  v 3.6
+) (/usr/share/texmf-texlive/tex/latex/rotating/rotating.sty
+Package: rotating 1997/09/26, v2.13 Rotation package
+(/usr/share/texmf-texlive/tex/latex/base/ifthen.sty
+Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
+)
+\c@r@tfl@t=\count105
+\rot@float@box=\box29
+) (/usr/share/texmf-texlive/tex/latex/caption/caption.sty
+Package: caption 2007/01/07 v3.0k Customising captions (AR)
+(/usr/share/texmf-texlive/tex/latex/caption/caption3.sty
+Package: caption3 2007/01/07 v3.0k caption3 kernel (AR)
+\captionmargin=\dimen120
+\captionmarginx=\dimen121
+\captionwidth=\dimen122
+\captionindent=\dimen123
+\captionparindent=\dimen124
+\captionhangindent=\dimen125
+)
+Package caption Info: rotating package v2.0 (or newer) detected on input line 3
+82.
+)
+
+LaTeX Warning: You have requested, on input line 129, version
+               `2008/08/24' of package caption,
+               but only version
+               `2007/01/07 v3.0k Customising captions (AR)'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/multirow/multirow.sty
+\bigstrutjot=\dimen126
+) (/usr/share/texmf-texlive/tex/latex/base/makeidx.sty
+Package: makeidx 2000/03/29 v1.0m Standard LaTeX package
+) (/usr/share/texmf-texlive/tex/latex/tools/multicol.sty
+Package: multicol 2006/05/18 v1.6g multicolumn formatting (FMi)
+\c@tracingmulticols=\count106
+\mult@box=\box30
+\multicol@leftmargin=\dimen127
+\c@unbalance=\count107
+\c@collectmore=\count108
+\doublecol@number=\count109
+\multicoltolerance=\count110
+\multicolpretolerance=\count111
+\full@width=\dimen128
+\page@free=\dimen129
+\premulticols=\dimen130
+\postmulticols=\dimen131
+\multicolsep=\skip55
+\multicolbaselineskip=\skip56
+\partial@page=\box31
+\last@line=\box32
+\mult@rightbox=\box33
+\mult@grightbox=\box34
+\mult@gfirstbox=\box35
+\mult@firstbox=\box36
+\@tempa=\box37
+\@tempa=\box38
+\@tempa=\box39
+\@tempa=\box40
+\@tempa=\box41
+\@tempa=\box42
+\@tempa=\box43
+\@tempa=\box44
+\@tempa=\box45
+\@tempa=\box46
+\@tempa=\box47
+\@tempa=\box48
+\@tempa=\box49
+\@tempa=\box50
+\@tempa=\box51
+\@tempa=\box52
+\@tempa=\box53
+\c@columnbadness=\count112
+\c@finalcolumnbadness=\count113
+\last@try=\dimen132
+\multicolovershoot=\dimen133
+\multicolundershoot=\dimen134
+\mult@nat@firstbox=\box54
+\colbreak@box=\box55
+) (/usr/share/texmf-texlive/tex/latex/tools/calc.sty
+Package: calc 2005/08/06 v4.2 Infix arithmetic (KKT,FJ)
+\calc@Acount=\count114
+\calc@Bcount=\count115
+\calc@Adimen=\dimen135
+\calc@Bdimen=\dimen136
+\calc@Askip=\skip57
+\calc@Bskip=\skip58
+LaTeX Info: Redefining \setlength on input line 75.
+LaTeX Info: Redefining \addtolength on input line 76.
+\calc@Ccount=\count116
+\calc@Cskip=\skip59
+) (/usr/share/texmf-texlive/tex/latex/tools/array.sty
+Package: array 2005/08/23 v2.4b Tabular extension package (FMi)
+\col@sep=\dimen137
+\extrarowheight=\dimen138
+\NC@list=\toks27
+\extratabsurround=\skip60
+\backup@length=\skip61
+) (/usr/share/texmf-texlive/tex/latex/mhchem/mhchem.sty
+Package: mhchem 2006/12/17 v3.05 for typesetting chemical formulae
+(/usr/share/texmf-texlive/tex/latex/oberdiek/twoopt.sty
+Package: twoopt 2006/02/20 v1.4 Definitions with two optional arguments (HO)
+)
+\tok@mhchem@ce@ii=\toks28
+\mhchem@arrowminlength=\skip62
+\mhchem@arrows@box=\box56
+\mhchem@arrowlength@pgf=\skip63
+\mhchem@arrowminlength@pgf=\skip64
+\mhchem@bondwidth=\skip65
+\mhchem@bondheight=\skip66
+\mhchem@smallbondwidth@tmpA=\skip67
+\mhchem@smallbondwidth@tmpB=\skip68
+\mhchem@smallbondwidth=\skip69
+\mhchem@cf@element=\toks29
+\mhchem@cf@number=\toks30
+\mhchem@cf@sup=\toks31
+\tok@mhchem@cf@i=\toks32
+\c@mhchem@cf@length=\count117
+\mhchem@cf@replaceminus@tok=\toks33
+\mhchem@option@minussidebearingleft=\skip70
+\mhchem@option@minussidebearingright=\skip71
+)
+
+LaTeX Warning: You have requested, on input line 135, version
+               `2007/05/19' of package mhchem,
+               but only version
+               `2006/12/17 v3.05 for typesetting chemical formulae'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/tools/indentfirst.sty
+Package: indentfirst 1995/11/23 v1.03 Indent first paragraph (DPC)
+) (/usr/share/texmf-texlive/tex/latex/tools/longtable.sty
+Package: longtable 2004/02/01 v4.11 Multi-page Table package (DPC)
+\LTleft=\skip72
+\LTright=\skip73
+\LTpre=\skip74
+\LTpost=\skip75
+\LTchunksize=\count118
+\LTcapwidth=\dimen139
+\LT@head=\box57
+\LT@firsthead=\box58
+\LT@foot=\box59
+\LT@lastfoot=\box60
+\LT@cols=\count119
+\LT@rows=\count120
+\c@LT@tables=\count121
+\c@LT@chunks=\count122
+\LT@p@ftn=\toks34
+) (/usr/share/texmf-texlive/tex/latex/footmisc/footmisc.sty
+Package: footmisc 2005/03/17 v5.3d a miscellany of footnote facilities
+\FN@temptoken=\toks35
+\footnotemargin=\dimen140
+\c@pp@next@reset=\count123
+\c@@fnserial=\count124
+Package footmisc Info: Declaring symbol style bringhurst on input line 817.
+Package footmisc Info: Declaring symbol style chicago on input line 818.
+Package footmisc Info: Declaring symbol style wiley on input line 819.
+Package footmisc Info: Declaring symbol style lamport-robust on input line 823.
+
+Package footmisc Info: Declaring symbol style lamport* on input line 831.
+Package footmisc Info: Declaring symbol style lamport*-robust on input line 840
+.
+)
+
+LaTeX Warning: You have requested, on input line 139, version
+               `2009/09/15' of package footmisc,
+               but only version
+               `2005/03/17 v5.3d a miscellany of footnote facilities'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/hyperref/hyperref.sty
+Package: hyperref 2007/02/07 v6.75r Hypertext links for LaTeX
+\@linkdim=\dimen141
+\Hy@linkcounter=\count125
+\Hy@pagecounter=\count126
+(/usr/share/texmf-texlive/tex/latex/hyperref/pd1enc.def
+File: pd1enc.def 2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
+) (/etc/texmf/tex/latex/config/hyperref.cfg
+File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
+) (/usr/share/texmf-texlive/tex/latex/oberdiek/kvoptions.sty
+Package: kvoptions 2006/08/22 v2.4 Connects package keyval with LaTeX options (
+HO)
+)
+Package hyperref Info: Option `hyperfootnotes' set `false' on input line 2238.
+Package hyperref Info: Option `pdfdisplaydoctitle' set `true' on input line 223
+8.
+Package hyperref Info: Option `pdfpagelabels' set `true' on input line 2238.
+Package hyperref Info: Option `bookmarksopen' set `true' on input line 2238.
+Package hyperref Info: Option `colorlinks' set `false' on input line 2238.
+Package hyperref Info: Hyper figures OFF on input line 2288.
+Package hyperref Info: Link nesting OFF on input line 2293.
+Package hyperref Info: Hyper index ON on input line 2296.
+Package hyperref Info: Plain pages OFF on input line 2303.
+Package hyperref Info: Backreferencing OFF on input line 2308.
+Implicit mode ON; LaTeX internals redefined
+Package hyperref Info: Bookmarks ON on input line 2444.
+(/usr/share/texmf-texlive/tex/latex/ltxmisc/url.sty
+\Urlmuskip=\muskip11
+Package: url 2005/06/27  ver 3.2  Verb mode for urls, etc.
+)
+LaTeX Info: Redefining \url on input line 2599.
+\Fld@menulength=\count127
+\Field@Width=\dimen142
+\Fld@charsize=\dimen143
+\Choice@toks=\toks36
+\Field@toks=\toks37
+Package hyperref Info: Hyper figures OFF on input line 3102.
+Package hyperref Info: Link nesting OFF on input line 3107.
+Package hyperref Info: Hyper index ON on input line 3110.
+Package hyperref Info: backreferencing OFF on input line 3117.
+Package hyperref Info: Link coloring OFF on input line 3122.
+\Hy@abspage=\count128
+\c@Item=\count129
+)
+*hyperref using driver hpdftex*
+(/usr/share/texmf-texlive/tex/latex/hyperref/hpdftex.def
+File: hpdftex.def 2007/02/07 v6.75r Hyperref driver for pdfTeX
+\Fld@listcount=\count130
+)
+
+LaTeX Warning: You have requested, on input line 155, version
+               `2008/11/18' of package hyperref,
+               but only version
+               `2007/02/07 v6.75r Hypertext links for LaTeX'
+               is available.
+
+\TmpLen=\skip76
+\TmpPadLen=\skip77
+\HangPadLen=\skip78
+\HangLen=\skip79
+\ToCLen=\skip80
+\ToCLenA=\skip81
+\ToCLenB=\skip82
+\DP@lign@no=\count131
+\DP@lignb@dy=\toks38
+(./36525-t.aux)
+\openout1 = `36525-t.aux'.
+
+LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 567.
+LaTeX Font Info:    ... okay on input line 567.
+LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 567.
+LaTeX Font Info:    ... okay on input line 567.
+LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 567.
+LaTeX Font Info:    ... okay on input line 567.
+LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 567.
+LaTeX Font Info:    ... okay on input line 567.
+LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 567.
+LaTeX Font Info:    ... okay on input line 567.
+LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 567.
+LaTeX Font Info:    ... okay on input line 567.
+LaTeX Font Info:    Checking defaults for PD1/pdf/m/n on input line 567.
+LaTeX Font Info:    ... okay on input line 567.
+-------------------- Geometry parameters
+paper: user defined
+landscape: --
+twocolumn: --
+twoside: --
+asymmetric: --
+h-parts: 9.03374pt, 379.41751pt, 9.03374pt
+v-parts: 37.9413pt, 540.2187pt, 18.06749pt
+hmarginratio: --
+vmarginratio: --
+lines: --
+heightrounded: --
+bindingoffset: 0.0pt
+truedimen: --
+includehead: --
+includefoot: --
+includemp: --
+driver: pdftex
+-------------------- Page layout dimensions and switches
+\paperwidth  397.48499pt
+\paperheight 596.2275pt
+\textwidth  379.41751pt
+\textheight 540.2187pt
+\oddsidemargin  -63.23625pt
+\evensidemargin -63.23625pt
+\topmargin  -63.23622pt
+\headheight 14.5pt
+\headsep    14.45377pt
+\footskip   30.0pt
+\marginparwidth 47.0pt
+\marginparsep   7.0pt
+\columnsep  10.0pt
+\skip\footins  10.8pt plus 4.0pt minus 2.0pt
+\hoffset 0.0pt
+\voffset 0.0pt
+\mag 1000
+
+(1in=72.27pt, 1cm=28.45pt)
+-----------------------
+(/usr/share/texmf/tex/context/base/supp-pdf.tex
+[Loading MPS to PDF converter (version 2006.09.02).]
+\scratchcounter=\count132
+\scratchdimen=\dimen144
+\scratchbox=\box61
+\nofMPsegments=\count133
+\nofMParguments=\count134
+\everyMPshowfont=\toks39
+\MPscratchCnt=\count135
+\MPscratchDim=\dimen145
+\MPnumerator=\count136
+\everyMPtoPDFconversion=\toks40
+) (/usr/share/texmf-texlive/tex/latex/ragged2e/ragged2e.sty
+Package: ragged2e 2003/03/25 v2.04 ragged2e Package (MS)
+(/usr/share/texmf-texlive/tex/latex/everysel/everysel.sty
+Package: everysel 1999/06/08 v1.03 EverySelectfont Package (MS)
+LaTeX Info: Redefining \selectfont on input line 125.
+)
+\CenteringLeftskip=\skip83
+\RaggedLeftLeftskip=\skip84
+\RaggedRightLeftskip=\skip85
+\CenteringRightskip=\skip86
+\RaggedLeftRightskip=\skip87
+\RaggedRightRightskip=\skip88
+\CenteringParfillskip=\skip89
+\RaggedLeftParfillskip=\skip90
+\RaggedRightParfillskip=\skip91
+\JustifyingParfillskip=\skip92
+\CenteringParindent=\skip93
+\RaggedLeftParindent=\skip94
+\RaggedRightParindent=\skip95
+\JustifyingParindent=\skip96
+)
+Package caption Info: hyperref package v6.74m (or newer) detected on input line
+ 567.
+Package caption Info: longtable package v3.15 (or newer) detected on input line
+ 567.
+Package hyperref Info: Link coloring OFF on input line 567.
+(/usr/share/texmf-texlive/tex/latex/hyperref/nameref.sty
+Package: nameref 2006/12/27 v2.28 Cross-referencing by name of section
+(/usr/share/texmf-texlive/tex/latex/oberdiek/refcount.sty
+Package: refcount 2006/02/20 v3.0 Data extraction from references (HO)
+)
+\c@section@level=\count137
+)
+LaTeX Info: Redefining \ref on input line 567.
+LaTeX Info: Redefining \pageref on input line 567.
+(./36525-t.out) (./36525-t.out)
+\@outlinefile=\write4
+\openout4 = `36525-t.out'.
+
+[1
+
+{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
+LaTeX Font Info:    Try loading font information for U+msa on input line 640.
+(/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd
+File: umsa.fd 2002/01/19 v2.2g AMS font definitions
+)
+LaTeX Font Info:    Try loading font information for U+msb on input line 640.
+(/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd
+File: umsb.fd 2002/01/19 v2.2g AMS font definitions
+)
+LaTeX Font Info:    Try loading font information for U+euf on input line 640.
+(/usr/share/texmf-texlive/tex/latex/amsfonts/ueuf.fd
+File: ueuf.fd 2002/01/19 v2.2g AMS font definitions
+) [1
+
+] [2
+
+] [3
+
+] [4
+
+
+] [5] [6] [7] [8
+
+] [9] [10] [11] [12] [13] [14] [15] [16] [17
+
+]
+LaTeX Font Info:    Try loading font information for OMS+cmr on input line 1209
+.
+(/usr/share/texmf-texlive/tex/latex/base/omscmr.fd
+File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions
+)
+LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <12> not available
+(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 1209.
+
+[1
+
+
+] [2] [3] [4] [5] [6] [7] [8] [9] [10] <./images/fig01.png, id=1125, 236.17836p
+t x 229.24043pt>
+File: ./images/fig01.png Graphic file (type png)
+<use ./images/fig01.png>
+\CF@figure=\count138
+[11] [12 <./images/fig01.png (PNG copy)>] [13] [14] [15] [16] [17] [18] [19] [2
+0] [21] [22] [23] [24] [25] [26] [27] [28] <./images/fig02.png, id=1254, 250.92
+143pt x 300.06503pt>
+File: ./images/fig02.png Graphic file (type png)
+<use ./images/fig02.png> [29] [30 <./images/fig02.png (PNG copy)>] <./images/fi
+g03.png, id=1268, 155.52504pt x 223.16975pt>
+File: ./images/fig03.png Graphic file (type png)
+<use ./images/fig03.png> [31 <./images/fig03.png (PNG copy)>] [32] <./images/fi
+g04.png, id=1285, 183.85487pt x 241.09271pt>
+File: ./images/fig04.png Graphic file (type png)
+<use ./images/fig04.png> [33 <./images/fig04.png (PNG copy)>] <./images/fig05.p
+ng, id=1296, 232.7094pt x 78.34068pt>
+File: ./images/fig05.png Graphic file (type png)
+<use ./images/fig05.png> [34 <./images/fig05.png (PNG copy)>] [35] [36] <./imag
+es/fig06.png, id=1318, 196.5744pt x 121.4136pt>
+File: ./images/fig06.png Graphic file (type png)
+<use ./images/fig06.png> [37 <./images/fig06.png (PNG copy)>] <./images/fig07.p
+ng, id=1328, 182.01334pt x 160.06467pt>
+File: ./images/fig07.png Graphic file (type png)
+<use ./images/fig07.png> <./images/fig08.png, id=1329, 182.01334pt x 146.68134p
+t>
+File: ./images/fig08.png Graphic file (type png)
+<use ./images/fig08.png> <./images/fig09.png, id=1330, 182.01334pt x 149.89333p
+t>
+File: ./images/fig09.png Graphic file (type png)
+<use ./images/fig09.png> <./images/fig10.png, id=1331, 182.01334pt x 146.146pt>
+File: ./images/fig10.png Graphic file (type png)
+<use ./images/fig10.png> <./images/fig11.png, id=1332, 182.01334pt x 124.73267p
+t>
+File: ./images/fig11.png Graphic file (type png)
+<use ./images/fig11.png> <./images/fig12.png, id=1333, 182.01334pt x 128.21233p
+t>
+File: ./images/fig12.png Graphic file (type png)
+<use ./images/fig12.png> [38 <./images/fig07.png (PNG copy)> <./images/fig08.pn
+g (PNG copy)> <./images/fig09.png (PNG copy)> <./images/fig10.png (PNG copy)>] 
+[39 <./images/fig11.png (PNG copy)> <./images/fig12.png (PNG copy)>] <./images/
+fig13.png, id=1361, 379.56204pt x 223.74792pt>
+File: ./images/fig13.png Graphic file (type png)
+<use ./images/fig13.png> [40 <./images/fig13.png (PNG copy)>] [41] [42] <./imag
+es/fig14.png, id=1382, 328.39488pt x 54.05795pt>
+File: ./images/fig14.png Graphic file (type png)
+<use ./images/fig14.png> <./images/fig15.png, id=1383, 330.9966pt x 56.65968pt>
+File: ./images/fig15.png Graphic file (type png)
+<use ./images/fig15.png> <./images/fig16.png, id=1387, 330.9966pt x 59.83955pt>
+File: ./images/fig16.png Graphic file (type png)
+<use ./images/fig16.png> [43] [44 <./images/fig14.png (PNG copy)> <./images/fig
+15.png (PNG copy)> <./images/fig16.png (PNG copy)>] <./images/fig17.png, id=140
+7, 280.98576pt x 202.93416pt>
+File: ./images/fig17.png Graphic file (type png)
+<use ./images/fig17.png> [45 <./images/fig17.png (PNG copy)>] [46] <./images/fi
+g18.png, id=1418, 245.718pt x 358.4592pt>
+File: ./images/fig18.png Graphic file (type png)
+<use ./images/fig18.png> [47] [48 <./images/fig18.png (PNG copy)>] [49] [50] [5
+1] [52
+
+] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [6
+8] [69] [70] <./images/fig19.png, id=1538, 492.8814pt x 310.761pt>
+File: ./images/fig19.png Graphic file (type png)
+<use ./images/fig19.png>
+File: ./images/fig19.png Graphic file (type png)
+<use ./images/fig19.png> [71 <./images/fig19.png (PNG copy)>] [72] <./images/fi
+g20.png, id=1552, 242.8272pt x 198.30888pt>
+File: ./images/fig20.png Graphic file (type png)
+<use ./images/fig20.png> [73 <./images/fig20.png (PNG copy)>] [74] [75] [76] <.
+/images/fig21.png, id=1579, 335.91096pt x 213.34103pt>
+File: ./images/fig21.png Graphic file (type png)
+<use ./images/fig21.png> [77 <./images/fig21.png (PNG copy)>] <./images/fig22.p
+ng, id=1587, 246.29616pt x 244.85075pt>
+File: ./images/fig22.png Graphic file (type png)
+<use ./images/fig22.png> [78 <./images/fig22.png (PNG copy)>] [79] [80] [81] <.
+/images/fig23.png, id=1612, 139.62564pt x 140.78197pt>
+File: ./images/fig23.png Graphic file (type png)
+<use ./images/fig23.png> [82 <./images/fig23.png (PNG copy)>] [83] <./images/fi
+g24.png, id=1627, 304.40125pt x 243.98352pt>
+File: ./images/fig24.png Graphic file (type png)
+<use ./images/fig24.png> [84 <./images/fig24.png (PNG copy)>] <./images/fig25.p
+ng, id=1636, 241.3818pt x 280.98576pt>
+File: ./images/fig25.png Graphic file (type png)
+<use ./images/fig25.png> [85 <./images/fig25.png (PNG copy)>] <./images/fig26.p
+ng, id=1645, 214.78644pt x 246.87431pt>
+File: ./images/fig26.png Graphic file (type png)
+<use ./images/fig26.png> <./images/fig27.png, id=1655, 272.31335pt x 240.51456p
+t>
+File: ./images/fig27.png Graphic file (type png)
+<use ./images/fig27.png>
+Underfull \hbox (badness 4647) in paragraph at lines 5118--5156
+\OT1/cmr/m/n/12 in-ter-est-ing ex-per-i-ments has
+ []
+
+
+Underfull \hbox (badness 2626) in paragraph at lines 5118--5156
+\OT1/cmr/m/n/12 these curves in the neigh-
+ []
+
+
+Underfull \hbox (badness 1137) in paragraph at lines 5118--5156
+\OT1/cmr/m/n/12 cu-racy, as it was pro-duced
+ []
+
+
+Underfull \hbox (badness 1448) in paragraph at lines 5118--5156
+\OT1/cmr/m/n/12 could very eas-ily be de-ter-
+ []
+
+[86 <./images/fig26.png (PNG copy)>] [87 <./images/fig27.png (PNG copy)>] <./im
+ages/fig28.png, id=1670, 265.37544pt x 262.19556pt>
+File: ./images/fig28.png Graphic file (type png)
+<use ./images/fig28.png> [88] [89 <./images/fig28.png (PNG copy)>] [90] [91] [9
+2] <./images/fig29.png, id=1705, 300.93228pt x 171.71352pt>
+File: ./images/fig29.png Graphic file (type png)
+<use ./images/fig29.png> [93 <./images/fig29.png (PNG copy)>] [94] [95] [96] <.
+/images/fig30.png, id=1729, 185.0112pt x 183.5658pt>
+File: ./images/fig30.png Graphic file (type png)
+<use ./images/fig30.png> [97] <./images/fig31.png, id=1736, 147.14172pt x 146.8
+5265pt>
+File: ./images/fig31.png Graphic file (type png)
+<use ./images/fig31.png> [98 <./images/fig30.png (PNG copy)> <./images/fig31.pn
+g (PNG copy)>] [99] <./images/fig32.png, id=1751, 97.41995pt x 173.73708pt>
+File: ./images/fig32.png Graphic file (type png)
+<use ./images/fig32.png> [100 <./images/fig32.png (PNG copy)>] <./images/fig33.
+png, id=1760, 99.7326pt x 100.02168pt>
+File: ./images/fig33.png Graphic file (type png)
+<use ./images/fig33.png> [101 <./images/fig33.png (PNG copy)>] [102] <./images/
+fig34.png, id=1778, 210.45024pt x 147.14172pt>
+File: ./images/fig34.png Graphic file (type png)
+<use ./images/fig34.png>
+Underfull \hbox (badness 6157) in paragraph at lines 5932--5950
+\OT1/cmr/m/n/12 can con-tinue in the same
+ []
+
+
+Underfull \hbox (badness 1502) in paragraph at lines 5932--5950
+\OT1/cmr/m/n/12 medium all the way it can
+ []
+
+[103 <./images/fig34.png (PNG copy)>] [104] <./images/fig35.png, id=1794, 259.0
+1569pt x 163.04112pt>
+File: ./images/fig35.png Graphic file (type png)
+<use ./images/fig35.png> [105 <./images/fig35.png (PNG copy)>] <./images/fig36.
+png, id=1803, 50.589pt x 272.31335pt>
+File: ./images/fig36.png Graphic file (type png)
+<use ./images/fig36.png> [106 <./images/fig36.png (PNG copy)>] <./images/fig37.
+png, id=1812, 262.48463pt x 58.39417pt>
+File: ./images/fig37.png Graphic file (type png)
+<use ./images/fig37.png> <./images/fig38.png, id=1813, 263.93004pt x 81.52055pt
+>
+File: ./images/fig38.png Graphic file (type png)
+<use ./images/fig38.png> [107 <./images/fig37.png (PNG copy)> <./images/fig38.p
+ng (PNG copy)>] [108] <./images/fig39.png, id=1831, 274.91508pt x 74.00449pt>
+File: ./images/fig39.png Graphic file (type png)
+<use ./images/fig39.png> [109] <./images/fig40.png, id=1840, 333.02016pt x 503.
+57736pt>
+File: ./images/fig40.png Graphic file (type png)
+<use ./images/fig40.png> [110 <./images/fig39.png (PNG copy)>] [111 <./images/f
+ig40.png (PNG copy)>] [112] <./images/fig41.png, id=1861, 206.6922pt x 411.0717
+6pt>
+File: ./images/fig41.png Graphic file (type png)
+<use ./images/fig41.png>
+Underfull \hbox (badness 1210) in paragraph at lines 6349--6372
+\OT1/cmr/m/n/12 po-si-tion of the pos-i-tive elec-
+ []
+
+
+Underfull \hbox (badness 2478) in paragraph at lines 6349--6372
+\OT1/cmr/m/n/12 trode, nor do they in-crease
+ []
+
+[113 <./images/fig41.png (PNG copy)>] <./images/fig42.png, id=1869, 347.18507pt
+ x 360.77184pt>
+File: ./images/fig42.png Graphic file (type png)
+<use ./images/fig42.png> [114] [115 <./images/fig42.png (PNG copy)>] [116] [117
+] <./images/fig43.png, id=1894, 214.20828pt x 132.68771pt>
+File: ./images/fig43.png Graphic file (type png)
+<use ./images/fig43.png> [118 <./images/fig43.png (PNG copy)>] <./images/fig44.
+png, id=1905, 206.11404pt x 132.68771pt>
+File: ./images/fig44.png Graphic file (type png)
+<use ./images/fig44.png> [119 <./images/fig44.png (PNG copy)>] <./images/fig45.
+png, id=1914, 268.8444pt x 54.34705pt>
+File: ./images/fig45.png Graphic file (type png)
+<use ./images/fig45.png> [120 <./images/fig45.png (PNG copy)>] <./images/fig46.
+png, id=1924, 169.97903pt x 41.04936pt>
+File: ./images/fig46.png Graphic file (type png)
+<use ./images/fig46.png> <./images/fig47.png, id=1927, 372.62411pt x 201.48875p
+t>
+File: ./images/fig47.png Graphic file (type png)
+<use ./images/fig47.png> [121 <./images/fig46.png (PNG copy)>] <./images/fig48.
+png, id=1937, 238.20192pt x 100.59984pt>
+File: ./images/fig48.png Graphic file (type png)
+<use ./images/fig48.png> <./images/fig49.png, id=1938, 271.15704pt x 216.81pt>
+File: ./images/fig49.png Graphic file (type png)
+<use ./images/fig49.png> [122 <./images/fig47.png (PNG copy)>] [123 <./images/f
+ig48.png (PNG copy)> <./images/fig49.png (PNG copy)>] [124] <./images/fig50.png
+, id=1963, 132.10956pt x 222.30252pt>
+File: ./images/fig50.png Graphic file (type png)
+<use ./images/fig50.png> [125 <./images/fig50.png (PNG copy)>] [126] <./images/
+fig51.png, id=1978, 379.56204pt x 120.54636pt>
+File: ./images/fig51.png Graphic file (type png)
+<use ./images/fig51.png> [127 <./images/fig51.png (PNG copy)>] [128] [129] <./i
+mages/fig52.png, id=1999, 131.5314pt x 39.60396pt>
+File: ./images/fig52.png Graphic file (type png)
+<use ./images/fig52.png> <./images/fig53.png, id=2000, 220.85712pt x 120.25728p
+t>
+File: ./images/fig53.png Graphic file (type png)
+<use ./images/fig53.png> [130 <./images/fig52.png (PNG copy)> <./images/fig53.p
+ng (PNG copy)>] <./images/fig54.png, id=2010, 140.49287pt x 167.08824pt>
+File: ./images/fig54.png Graphic file (type png)
+<use ./images/fig54.png> <./images/fig55.png, id=2011, 153.50148pt x 167.08824p
+t>
+File: ./images/fig55.png Graphic file (type png)
+<use ./images/fig55.png> <./images/fig56.png, id=2016, 302.95584pt x 116.78831p
+t>
+File: ./images/fig56.png Graphic file (type png)
+<use ./images/fig56.png> <./images/fig57.png, id=2021, 270.2898pt x 115.632pt>
+File: ./images/fig57.png Graphic file (type png)
+<use ./images/fig57.png> <./images/fig58.png, id=2022, 298.90872pt x 80.36424pt
+>
+File: ./images/fig58.png Graphic file (type png)
+<use ./images/fig58.png> <./images/fig59.png, id=2023, 304.9794pt x 116.21016pt
+>
+File: ./images/fig59.png Graphic file (type png)
+<use ./images/fig59.png> <./images/fig60.png, id=2024, 261.6174pt x 124.01532pt
+>
+File: ./images/fig60.png Graphic file (type png)
+<use ./images/fig60.png> [131 <./images/fig54.png (PNG copy)> <./images/fig55.p
+ng (PNG copy)>] [132 <./images/fig56.png (PNG copy)> <./images/fig57.png (PNG c
+opy)> <./images/fig58.png (PNG copy)>] [133 <./images/fig59.png (PNG copy)> <./
+images/fig60.png (PNG copy)>] [134] [135] <./images/fig61.png, id=2079, 140.781
+97pt x 272.31335pt>
+File: ./images/fig61.png Graphic file (type png)
+<use ./images/fig61.png> [136 <./images/fig61.png (PNG copy)>] [137] <./images/
+fig62.png, id=2094, 133.55496pt x 385.05457pt>
+File: ./images/fig62.png Graphic file (type png)
+<use ./images/fig62.png> [138 <./images/fig62.png (PNG copy)>] [139] [140] [141
+]
+Underfull \hbox (badness 1189) in paragraph at lines 7593--7593
+[]\OT1/cmr/m/n/10 Intensity of cur-rent in mil-
+ []
+
+
+Underfull \hbox (badness 1810) in paragraph at lines 7598--7598
+\OT1/cmr/m/n/10 the charg-ing of the con-
+ []
+
+[142] <./images/fig63.png, id=2125, 305.84663pt x 75.44987pt>
+File: ./images/fig63.png Graphic file (type png)
+<use ./images/fig63.png> [143 <./images/fig63.png (PNG copy)>] [144] <./images/
+fig64.png, id=2139, 377.2494pt x 196.28532pt>
+File: ./images/fig64.png Graphic file (type png)
+<use ./images/fig64.png> [145 <./images/fig64.png (PNG copy)>] [146]
+Underfull \hbox (badness 1546) in paragraph at lines 7803--7803
+[][]\OT1/cmr/m/n/10 Number fix-ing the
+ []
+
+
+Underfull \hbox (badness 5189) in paragraph at lines 7813--7813
+\OT1/cmr/m/n/10 meter from the
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7813--7813
+\OT1/cmr/m/n/10 charge in the
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7813--7813
+\OT1/cmr/m/n/10 con-denser due
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7813--7813
+\OT1/cmr/m/n/10 to the po-ten-
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7813--7813
+\OT1/cmr/m/n/10 tial dif-fer-ence
+ []
+
+[147] [148] [149]
+Underfull \hbox (badness 10000) in paragraph at lines 7970--7970
+[]\OT1/cmr/m/n/10 Intensity of cur-rent in
+ []
+
+
+Underfull \hbox (badness 1137) in paragraph at lines 7973--7973
+\OT1/cmr/m/n/10 ten-tial dif-fer-ence be-tween
+ []
+
+[150] <./images/fig65.png, id=2181, 32.95512pt x 129.21877pt>
+File: ./images/fig65.png Graphic file (type png)
+<use ./images/fig65.png> [151 <./images/fig65.png (PNG copy)>] <./images/fig66.
+png, id=2193, 154.07964pt x 290.5254pt>
+File: ./images/fig66.png Graphic file (type png)
+<use ./images/fig66.png> [152 <./images/fig66.png (PNG copy)>] [153] [154] [155
+] [156] <./images/fig67.png, id=2226, 126.32796pt x 219.7008pt>
+File: ./images/fig67.png Graphic file (type png)
+<use ./images/fig67.png> [157 <./images/fig67.png (PNG copy)>] <./images/fig68.
+png, id=2237, 354.99023pt x 81.80965pt>
+File: ./images/fig68.png Graphic file (type png)
+<use ./images/fig68.png> <./images/fig69.png, id=2239, 293.4162pt x 451.25388pt
+>
+File: ./images/fig69.png Graphic file (type png)
+<use ./images/fig69.png> [158 <./images/fig68.png (PNG copy)>] [159 <./images/f
+ig69.png (PNG copy)>] <./images/fig70.png, id=2255, 228.66228pt x 149.16528pt>
+File: ./images/fig70.png Graphic file (type png)
+<use ./images/fig70.png> [160 <./images/fig70.png (PNG copy)>] [161] <./images/
+fig71.png, id=2269, 172.86984pt x 285.90012pt>
+File: ./images/fig71.png Graphic file (type png)
+<use ./images/fig71.png> [162 <./images/fig71.png (PNG copy)>] [163] [164] [165
+] <./images/fig72.png, id=2299, 227.21687pt x 66.4884pt>
+File: ./images/fig72.png Graphic file (type png)
+<use ./images/fig72.png> [166] <./images/fig73.png, id=2306, 300.93228pt x 74.0
+0449pt>
+File: ./images/fig73.png Graphic file (type png)
+<use ./images/fig73.png> <./images/fig74.png, id=2307, 96.26364pt x 100.59984pt
+>
+File: ./images/fig74.png Graphic file (type png)
+<use ./images/fig74.png> <./images/fig75.png, id=2308, 219.7008pt x 100.59984pt
+>
+File: ./images/fig75.png Graphic file (type png)
+<use ./images/fig75.png> [167 <./images/fig72.png (PNG copy)> <./images/fig73.p
+ng (PNG copy)>] [168 <./images/fig74.png (PNG copy)> <./images/fig75.png (PNG c
+opy)>] <./images/fig76.png, id=2328, 209.00484pt x 51.45624pt>
+File: ./images/fig76.png Graphic file (type png)
+<use ./images/fig76.png> <./images/fig77.png, id=2330, 214.20828pt x 213.9192pt
+>
+File: ./images/fig77.png Graphic file (type png)
+<use ./images/fig77.png> <./images/fig78.png, id=2331, 136.15668pt x 130.37508p
+t>
+File: ./images/fig78.png Graphic file (type png)
+<use ./images/fig78.png> [169 <./images/fig76.png (PNG copy)> <./images/fig78.p
+ng (PNG copy)>] <./images/fig79.png, id=2344, 253.52316pt x 156.1032pt>
+File: ./images/fig79.png Graphic file (type png)
+<use ./images/fig79.png> [170 <./images/fig77.png (PNG copy)>] <./images/fig80.
+png, id=2354, 281.56392pt x 84.41136pt>
+File: ./images/fig80.png Graphic file (type png)
+<use ./images/fig80.png> [171 <./images/fig79.png (PNG copy)> <./images/fig80.p
+ng (PNG copy)>] <./images/fig81.png, id=2363, 345.16151pt x 211.31747pt>
+File: ./images/fig81.png Graphic file (type png)
+<use ./images/fig81.png> [172] [173 <./images/fig81.png (PNG copy)>] <./images/
+fig82.png, id=2378, 191.37096pt x 193.10544pt>
+File: ./images/fig82.png Graphic file (type png)
+<use ./images/fig82.png> [174 <./images/fig82.png (PNG copy)>] [175] <./images/
+fig83.png, id=2395, 308.15929pt x 162.75204pt>
+File: ./images/fig83.png Graphic file (type png)
+<use ./images/fig83.png> [176 <./images/fig83.png (PNG copy)>] [177] [178] <./i
+mages/fig84.png, id=2416, 261.6174pt x 154.6578pt>
+File: ./images/fig84.png Graphic file (type png)
+<use ./images/fig84.png> [179 <./images/fig84.png (PNG copy)>] [180] [181] <./i
+mages/fig85.png, id=2434, 192.52728pt x 145.11816pt>
+File: ./images/fig85.png Graphic file (type png)
+<use ./images/fig85.png> [182] [183 <./images/fig85.png (PNG copy)>] [184] <./i
+mages/fig86.png, id=2453, 275.49324pt x 59.2614pt>
+File: ./images/fig86.png Graphic file (type png)
+<use ./images/fig86.png> [185 <./images/fig86.png (PNG copy)>] <./images/fig87.
+png, id=2464, 171.71352pt x 176.3388pt>
+File: ./images/fig87.png Graphic file (type png)
+<use ./images/fig87.png> [186 <./images/fig87.png (PNG copy)>] [187] [188]
+Underfull \hbox (badness 1264) in paragraph at lines 9690--9695
+[]\OT1/cmr/m/n/12 Then, again, we have the very in-ter-est-ing re-sult dis-cov-
+ered by
+ []
+
+[189] [190] [191] <./images/fig88.png, id=2505, 293.12712pt x 23.1264pt>
+File: ./images/fig88.png Graphic file (type png)
+<use ./images/fig88.png> [192 <./images/fig88.png (PNG copy)>] [193] [194] [195
+] [196] [197] [198] [199] [200] [201] [202] [203] [204
+
+] <./images/fig89.png, id=2603, 351.2322pt x 31.7988pt>
+File: ./images/fig89.png Graphic file (type png)
+<use ./images/fig89.png> [205 <./images/fig89.png (PNG copy)>] [206] [207] <./i
+mages/fig90.png, id=2629, 354.123pt x 54.9252pt>
+File: ./images/fig90.png Graphic file (type png)
+<use ./images/fig90.png> <./images/fig91.png, id=2630, 354.123pt x 54.05795pt>
+File: ./images/fig91.png Graphic file (type png)
+<use ./images/fig91.png> [208 <./images/fig90.png (PNG copy)> <./images/fig91.p
+ng (PNG copy)>] [209] [210] [211] <./images/fig92.png, id=2675, 354.123pt x 58.
+97232pt>
+File: ./images/fig92.png Graphic file (type png)
+<use ./images/fig92.png> <./images/fig93.png, id=2676, 341.40347pt x 45.38556pt
+>
+File: ./images/fig93.png Graphic file (type png)
+<use ./images/fig93.png> [212 <./images/fig92.png (PNG copy)> <./images/fig93.p
+ng (PNG copy)>] [213] <./images/fig94.png, id=2706, 327.23856pt x 51.74532pt>
+File: ./images/fig94.png Graphic file (type png)
+<use ./images/fig94.png> [214 <./images/fig94.png (PNG copy)>]
+Underfull \hbox (badness 10000) in paragraph at lines 10901--10904
+
+ []
+
+[215] [216] [217] <./images/fig95.png, id=2742, 156.97044pt x 106.38144pt>
+File: ./images/fig95.png Graphic file (type png)
+<use ./images/fig95.png> [218 <./images/fig95.png (PNG copy)>] [219] <./images/
+fig96.png, id=2765, 269.13348pt x 130.37508pt>
+File: ./images/fig96.png Graphic file (type png)
+<use ./images/fig96.png> [220] [221 <./images/fig96.png (PNG copy)>] <./images/
+fig97.png, id=2787, 306.13573pt x 48.27637pt>
+File: ./images/fig97.png Graphic file (type png)
+<use ./images/fig97.png> <./images/fig98.png, id=2788, 298.61964pt x 19.65744pt
+>
+File: ./images/fig98.png Graphic file (type png)
+<use ./images/fig98.png> [222] [223 <./images/fig97.png (PNG copy)> <./images/f
+ig98.png (PNG copy)>] [224] [225] [226] [227] <./images/fig99.png, id=2845, 267
+.68808pt x 112.7412pt>
+File: ./images/fig99.png Graphic file (type png)
+<use ./images/fig99.png> [228 <./images/fig99.png (PNG copy)>] [229] [230] [231
+] [232] <./images/fig100.png, id=2886, 109.56133pt x 15.8994pt>
+File: ./images/fig100.png Graphic file (type png)
+<use ./images/fig100.png> [233 <./images/fig100.png (PNG copy)>] <./images/fig1
+01.png, id=2895, 130.95325pt x 44.22923pt>
+File: ./images/fig101.png Graphic file (type png)
+<use ./images/fig101.png> [234 <./images/fig101.png (PNG copy)>] [235] [236] [2
+37] <./images/fig102.png, id=2928, 183.5658pt x 46.2528pt>
+File: ./images/fig102.png Graphic file (type png)
+<use ./images/fig102.png> [238] [239 <./images/fig102.png (PNG copy)>] <./image
+s/fig103.png, id=2939, 225.19331pt x 59.2614pt>
+File: ./images/fig103.png Graphic file (type png)
+<use ./images/fig103.png> [240 <./images/fig103.png (PNG copy)>] <./images/fig1
+04.png, id=2948, 123.43716pt x 107.53777pt>
+File: ./images/fig104.png Graphic file (type png)
+<use ./images/fig104.png> [241] <./images/fig105.png, id=2956, 250.92143pt x 21
+.39192pt>
+File: ./images/fig105.png Graphic file (type png)
+<use ./images/fig105.png> [242 <./images/fig104.png (PNG copy)> <./images/fig10
+5.png (PNG copy)>] [243] <./images/fig106.png, id=2974, 207.27036pt x 36.135pt>
+File: ./images/fig106.png Graphic file (type png)
+<use ./images/fig106.png> [244 <./images/fig106.png (PNG copy)>] <./images/fig1
+07.png, id=2985, 113.03027pt x 101.178pt>
+File: ./images/fig107.png Graphic file (type png)
+<use ./images/fig107.png> [245 <./images/fig107.png (PNG copy)>] [246] [247
+
+] [248] [249] [250] [251] [252] [253] [254] [255] [256] [257] [258] [259] [260]
+[261] [262] [263]
+Underfull \hbox (badness 3439) in paragraph at lines 13102--13105
+[]\OT1/cmr/m/n/12 Condition (2) gives, writ-ing $\OML/cmm/m/it/12 J[]\OT1/cmr/m
+/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )$ for $\OML/cmm/m/it/12 dJ[]\OT1/cmr/
+m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 =dx$\OT1/cmr/m/n/12
+ , and $\OML/cmm/m/it/12 K[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 
+)$
+ []
+
+[264] [265] [266] [267] [268] [269] [270] [271] [272] [273] [274] [275] [276] [
+277] [278] [279] [280] <./images/fig108.png, id=3331, 307.8702pt x 187.03476pt>
+File: ./images/fig108.png Graphic file (type png)
+<use ./images/fig108.png> [281 <./images/fig108.png (PNG copy)>] [282] [283] [2
+84] [285] [286] [287] [288] [289]
+Underfull \hbox (badness 1968) in paragraph at lines 14419--14425
+[]\OT1/cmr/m/n/12 273[][].] We shall now pass on to the case when $\OML/cmm/m/i
+t/12 n\OT1/cmr/bx/n/12 a$ \OT1/cmr/m/n/12 is large
+ []
+
+[290] [291] [292] [293] [294] [295] [296] [297] [298] [299] [300] [301] [302] [
+303] [304] [305] [306] [307] [308] [309] [310] [311] [312] [313] [314] [315] [3
+16] [317] [318] [319] [320] [321] [322] [323] [324] [325] [326] [327] [328] [32
+9] [330] [331] [332] [333] [334] [335] [336] [337] <./images/fig109.png, id=389
+1, 277.5168pt x 63.88667pt>
+File: ./images/fig109.png Graphic file (type png)
+<use ./images/fig109.png> [338 <./images/fig109.png (PNG copy)>] [339] [340] [3
+41] [342] [343] [344] [345] [346] [347] <./images/fig110.png, id=3989, 218.8335
+6pt x 213.63013pt>
+File: ./images/fig110.png Graphic file (type png)
+<use ./images/fig110.png> <./images/fig111.png, id=3991, 203.8014pt x 161.59572
+pt>
+File: ./images/fig111.png Graphic file (type png)
+<use ./images/fig111.png> [348 <./images/fig110.png (PNG copy)>] [349 <./images
+/fig111.png (PNG copy)>] [350] [351] [352] [353] [354] [355] [356] [357] [358] 
+[359] [360] [361] [362] [363] [364] [365] [366] [367] [368] [369] [370] [371] [
+372] [373] [374] [375] [376] [377] [378] <./images/fig112.png, id=4240, 234.732
+96pt x 234.73296pt>
+File: ./images/fig112.png Graphic file (type png)
+<use ./images/fig112.png> [379 <./images/fig112.png (PNG copy)>] [380] [381] [3
+82] [383] [384] [385] [386] [387] <./images/fig113.png, id=4406, 177.49512pt x 
+51.16716pt>
+File: ./images/fig113.png Graphic file (type png)
+<use ./images/fig113.png> [388
+
+ <./images/fig113.png (PNG copy)>] [389] [390] <./images/fig114.png, id=4424, 1
+00.59984pt x 97.70905pt>
+File: ./images/fig114.png Graphic file (type png)
+<use ./images/fig114.png> [391 <./images/fig114.png (PNG copy)>] <./images/fig1
+15.png, id=4436, 278.09496pt x 182.69856pt>
+File: ./images/fig115.png Graphic file (type png)
+<use ./images/fig115.png> [392] [393 <./images/fig115.png (PNG copy)>] <./image
+s/fig116.png, id=4452, 339.37991pt x 371.17873pt>
+File: ./images/fig116.png Graphic file (type png)
+<use ./images/fig116.png> <./images/fig117.png, id=4453, 281.56392pt x 112.7412
+pt>
+File: ./images/fig117.png Graphic file (type png)
+<use ./images/fig117.png> [394] [395 <./images/fig116.png (PNG copy)>] [396 <./
+images/fig117.png (PNG copy)>] <./images/fig118.png, id=4474, 119.39005pt x 129
+.21877pt>
+File: ./images/fig118.png Graphic file (type png)
+<use ./images/fig118.png> <./images/fig119.png, id=4478, 226.06056pt x 106.3814
+4pt>
+File: ./images/fig119.png Graphic file (type png)
+<use ./images/fig119.png> [397 <./images/fig118.png (PNG copy)>] [398 <./images
+/fig119.png (PNG copy)>] [399] [400] [401] [402] [403] <./images/fig120.png, id
+=4521, 256.41396pt x 121.70268pt>
+File: ./images/fig120.png Graphic file (type png)
+<use ./images/fig120.png> [404 <./images/fig120.png (PNG copy)>] [405] [406] [4
+07] [408] [409] [410] [411] [412] [413] [414] [415] [416] [417] [418] [419] [42
+0] [421] [422] [423] [424] [425] [426] [427] [428] [429] [430] [431] <./images/
+fig121.png, id=4735, 283.00932pt x 207.55943pt>
+File: ./images/fig121.png Graphic file (type png)
+<use ./images/fig121.png> [432 <./images/fig121.png (PNG copy)>] [433] [434] [4
+35] [436] <./images/fig122.png, id=4764, 324.63684pt x 188.19109pt>
+File: ./images/fig122.png Graphic file (type png)
+<use ./images/fig122.png> [437 <./images/fig122.png (PNG copy)>] [438] [439] [4
+40] [441] [442] [443] [444] [445] [446] [447] [448] [449] [450] [451] [452] [45
+3] [454] <./images/fig123.png, id=4917, 282.14207pt x 140.49287pt>
+File: ./images/fig123.png Graphic file (type png)
+<use ./images/fig123.png> [455 <./images/fig123.png (PNG copy)>] [456] <./image
+s/fig124.png, id=4935, 184.43304pt x 120.25728pt>
+File: ./images/fig124.png Graphic file (type png)
+<use ./images/fig124.png> [457 <./images/fig124.png (PNG copy)>] [458] [459] [4
+60] [461] <./images/fig125.png, id=4962, 255.25764pt x 193.97269pt>
+File: ./images/fig125.png Graphic file (type png)
+<use ./images/fig125.png> <./images/fig126.png, id=4964, 232.13124pt x 269.1334
+8pt>
+File: ./images/fig126.png Graphic file (type png)
+<use ./images/fig126.png> [462] [463 <./images/fig125.png (PNG copy)>] [464 <./
+images/fig126.png (PNG copy)>] <./images/fig127.png, id=4989, 302.95584pt x 87.
+01308pt>
+File: ./images/fig127.png Graphic file (type png)
+<use ./images/fig127.png> [465 <./images/fig127.png (PNG copy)>] <./images/fig1
+28.png, id=4999, 137.313pt x 132.39864pt>
+File: ./images/fig128.png Graphic file (type png)
+<use ./images/fig128.png> [466 <./images/fig128.png (PNG copy)>] <./images/fig1
+29.png, id=5012, 256.99213pt x 251.21053pt>
+File: ./images/fig129.png Graphic file (type png)
+<use ./images/fig129.png> [467 <./images/fig129.png (PNG copy)>] <./images/fig1
+30.png, id=5020, 370.60056pt x 282.72025pt>
+File: ./images/fig130.png Graphic file (type png)
+<use ./images/fig130.png> <./images/fig131.png, id=5022, 245.718pt x 56.3706pt>
+File: ./images/fig131.png Graphic file (type png)
+<use ./images/fig131.png> [468] [469 <./images/fig130.png (PNG copy)> <./images
+/fig131.png (PNG copy)>] [470] [471] [472] [473] [474] <./images/fig132.png, id
+=5062, 113.89752pt x 208.1376pt>
+File: ./images/fig132.png Graphic file (type png)
+<use ./images/fig132.png> [475 <./images/fig132.png (PNG copy)>] <./images/fig1
+33.png, id=5070, 343.42703pt x 309.60468pt>
+File: ./images/fig133.png Graphic file (type png)
+<use ./images/fig133.png> [476] [477 <./images/fig133.png (PNG copy)>] [478] [4
+79] [480] [481] [482] [483] <./images/fig134.png, id=5127, 145.69632pt x 324.05
+869pt>
+File: ./images/fig134.png Graphic file (type png)
+<use ./images/fig134.png>
+Underfull \hbox (badness 1077) in paragraph at lines 23995--24018
+\OT1/cmr/m/n/12 used by M. Blond-lot (\OT1/cmr/m/it/12 Comptes Ren-dus\OT1/cmr/
+m/n/12 ,
+ []
+
+[484 <./images/fig134.png (PNG copy)>] [485] [486] [487] [488] [489] <./images/
+fig135.png, id=5162, 179.2296pt x 104.35788pt>
+File: ./images/fig135.png Graphic file (type png)
+<use ./images/fig135.png> [490 <./images/fig135.png (PNG copy)>] <./images/fig1
+36.png, id=5171, 169.40088pt x 170.5572pt>
+File: ./images/fig136.png Graphic file (type png)
+<use ./images/fig136.png> [491 <./images/fig136.png (PNG copy)>] [492] [493] [4
+94] [495] [496] [497] [498] [499] [500] [501] [502] [503] [504] [505] [506] [50
+7] [508] [509] [510] [511] [512] [513] [514] [515
+
+] <./images/fig137.png, id=5378, 174.02615pt x 38.73672pt>
+File: ./images/fig137.png Graphic file (type png)
+<use ./images/fig137.png> <./images/fig138.png, id=5379, 164.19743pt x 74.29356
+pt>
+File: ./images/fig138.png Graphic file (type png)
+<use ./images/fig138.png> [516] <./images/fig139.png, id=5387, 246.58524pt x 61
+.57404pt>
+File: ./images/fig139.png Graphic file (type png)
+<use ./images/fig139.png> [517 <./images/fig137.png (PNG copy)>] [518 <./images
+/fig138.png (PNG copy)> <./images/fig139.png (PNG copy)>] [519] [520] [521] [52
+2] [523] [524] [525] [526] [527] [528] [529] [530] [531] [532] <./images/fig140
+.png, id=5510, 84.41136pt x 93.951pt>
+File: ./images/fig140.png Graphic file (type png)
+<use ./images/fig140.png> [533 <./images/fig140.png (PNG copy)>] [534] <./image
+s/fig141.png, id=5524, 240.51456pt x 60.12865pt>
+File: ./images/fig141.png Graphic file (type png)
+<use ./images/fig141.png> [535] [536 <./images/fig141.png (PNG copy)>] [537] [5
+38] [539] [540] [541
+
+] [542] [543] [544] [545] [546] [547] [548] [549] [550] [551] [552] [553] [554]
+[555] [556] [557] [558] [559] [560] [561] [562] [563] [564] [565] [566] [567
+
+] <./images/fig142.png, id=5788, 379.28366pt x 454.23033pt>
+File: ./images/fig142.png Graphic file (type png)
+<use ./images/fig142.png> <./images/fig143.png, id=5790, 359.32645pt x 78.62976
+pt>
+File: ./images/fig143.png Graphic file (type png)
+<use ./images/fig143.png> [568] [569 <./images/fig142.png (PNG copy)>] [570 <./
+images/fig143.png (PNG copy)>] [571] [572] [573] [574] <./images/fig144.png, id
+=5833, 315.38628pt x 301.51044pt>
+File: ./images/fig144.png Graphic file (type png)
+<use ./images/fig144.png> [575 <./images/fig144.png (PNG copy)>] [576] [577] [5
+78] (./36525-t.ind [579] [580
+
+]
+Underfull \hbox (badness 7740) in paragraph at lines 98--100
+[]\OT1/cmr/m/n/10.95 --- elec-tro-static neu-tral-izes self-
+ []
+
+[581] [582] [583] [584] [585] [586] [587] [588] [589] [590] [591] [592]) [593] 
+[594] [595] [1
+
+] [2] [3] [4] [5] [6] [7] (./36525-t.aux)
+
+ *File List*
+    book.cls    2005/09/16 v1.4f Standard LaTeX document class
+    bk12.clo    2005/09/16 v1.4f Standard LaTeX file (size option)
+geometry.sty    2002/07/08 v3.2 Page Geometry
+  keyval.sty    1999/03/16 v1.13 key=value parser (DPC)
+geometry.cfg
+inputenc.sty    2006/05/05 v1.1b Input encoding file
+  latin1.def    2006/05/05 v1.1b Input encoding file
+ amsmath.sty    2000/07/18 v2.13 AMS math features
+ amstext.sty    2000/06/29 v2.01
+  amsgen.sty    1999/11/30 v2.0
+  amsbsy.sty    1999/11/29 v1.2d
+  amsopn.sty    1999/12/14 v2.01 operator names
+ amssymb.sty    2002/01/22 v2.2d
+amsfonts.sty    2001/10/25 v2.2f
+fancyhdr.sty    
+extramarks.sty    
+verbatim.sty    2003/08/22 v1.5q LaTeX2e package for verbatim enhancements
+graphicx.sty    1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
+graphics.sty    2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
+    trig.sty    1999/03/16 v1.09 sin cos tan (DPC)
+graphics.cfg    2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
+  pdftex.def    2007/01/08 v0.04d Graphics/color for pdfTeX
+ wrapfig.sty    2003/01/31  v 3.6
+rotating.sty    1997/09/26, v2.13 Rotation package
+  ifthen.sty    2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
+ caption.sty    2007/01/07 v3.0k Customising captions (AR)
+caption3.sty    2007/01/07 v3.0k caption3 kernel (AR)
+multirow.sty    
+ makeidx.sty    2000/03/29 v1.0m Standard LaTeX package
+multicol.sty    2006/05/18 v1.6g multicolumn formatting (FMi)
+    calc.sty    2005/08/06 v4.2 Infix arithmetic (KKT,FJ)
+   array.sty    2005/08/23 v2.4b Tabular extension package (FMi)
+  mhchem.sty    2006/12/17 v3.05 for typesetting chemical formulae
+  twoopt.sty    2006/02/20 v1.4 Definitions with two optional arguments (HO)
+indentfirst.sty    1995/11/23 v1.03 Indent first paragraph (DPC)
+longtable.sty    2004/02/01 v4.11 Multi-page Table package (DPC)
+footmisc.sty    2005/03/17 v5.3d a miscellany of footnote facilities
+hyperref.sty    2007/02/07 v6.75r Hypertext links for LaTeX
+  pd1enc.def    2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
+hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
+kvoptions.sty    2006/08/22 v2.4 Connects package keyval with LaTeX options (HO
+)
+     url.sty    2005/06/27  ver 3.2  Verb mode for urls, etc.
+ hpdftex.def    2007/02/07 v6.75r Hyperref driver for pdfTeX
+supp-pdf.tex
+ragged2e.sty    2003/03/25 v2.04 ragged2e Package (MS)
+everysel.sty    1999/06/08 v1.03 EverySelectfont Package (MS)
+ nameref.sty    2006/12/27 v2.28 Cross-referencing by name of section
+refcount.sty    2006/02/20 v3.0 Data extraction from references (HO)
+ 36525-t.out
+ 36525-t.out
+    umsa.fd    2002/01/19 v2.2g AMS font definitions
+    umsb.fd    2002/01/19 v2.2g AMS font definitions
+    ueuf.fd    2002/01/19 v2.2g AMS font definitions
+  omscmr.fd    1999/05/25 v2.5h Standard LaTeX font definitions
+./images/fig01.png
+./images/fig02.png
+./images/fig03.png
+./images/fig04.png
+./images/fig05.png
+./images/fig06.png
+./images/fig07.png
+./images/fig08.png
+./images/fig09.png
+./images/fig10.png
+./images/fig11.png
+./images/fig12.png
+./images/fig13.png
+./images/fig14.png
+./images/fig15.png
+./images/fig16.png
+./images/fig17.png
+./images/fig18.png
+./images/fig19.png
+./images/fig19.png
+./images/fig20.png
+./images/fig21.png
+./images/fig22.png
+./images/fig23.png
+./images/fig24.png
+./images/fig25.png
+./images/fig26.png
+./images/fig27.png
+./images/fig28.png
+./images/fig29.png
+./images/fig30.png
+./images/fig31.png
+./images/fig32.png
+./images/fig33.png
+./images/fig34.png
+./images/fig35.png
+./images/fig36.png
+./images/fig37.png
+./images/fig38.png
+./images/fig39.png
+./images/fig40.png
+./images/fig41.png
+./images/fig42.png
+./images/fig43.png
+./images/fig44.png
+./images/fig45.png
+./images/fig46.png
+./images/fig47.png
+./images/fig48.png
+./images/fig49.png
+./images/fig50.png
+./images/fig51.png
+./images/fig52.png
+./images/fig53.png
+./images/fig54.png
+./images/fig55.png
+./images/fig56.png
+./images/fig57.png
+./images/fig58.png
+./images/fig59.png
+./images/fig60.png
+./images/fig61.png
+./images/fig62.png
+./images/fig63.png
+./images/fig64.png
+./images/fig65.png
+./images/fig66.png
+./images/fig67.png
+./images/fig68.png
+./images/fig69.png
+./images/fig70.png
+./images/fig71.png
+./images/fig72.png
+./images/fig73.png
+./images/fig74.png
+./images/fig75.png
+./images/fig76.png
+./images/fig77.png
+./images/fig78.png
+./images/fig79.png
+./images/fig80.png
+./images/fig81.png
+./images/fig82.png
+./images/fig83.png
+./images/fig84.png
+./images/fig85.png
+./images/fig86.png
+./images/fig87.png
+./images/fig88.png
+./images/fig89.png
+./images/fig90.png
+./images/fig91.png
+./images/fig92.png
+./images/fig93.png
+./images/fig94.png
+./images/fig95.png
+./images/fig96.png
+./images/fig97.png
+./images/fig98.png
+./images/fig99.png
+./images/fig100.png
+./images/fig101.png
+./images/fig102.png
+./images/fig103.png
+./images/fig104.png
+./images/fig105.png
+./images/fig106.png
+./images/fig107.png
+./images/fig108.png
+./images/fig109.png
+./images/fig110.png
+./images/fig111.png
+./images/fig112.png
+./images/fig113.png
+./images/fig114.png
+./images/fig115.png
+./images/fig116.png
+./images/fig117.png
+./images/fig118.png
+./images/fig119.png
+./images/fig120.png
+./images/fig121.png
+./images/fig122.png
+./images/fig123.png
+./images/fig124.png
+./images/fig125.png
+./images/fig126.png
+./images/fig127.png
+./images/fig128.png
+./images/fig129.png
+./images/fig130.png
+./images/fig131.png
+./images/fig132.png
+./images/fig133.png
+./images/fig134.png
+./images/fig135.png
+./images/fig136.png
+./images/fig137.png
+./images/fig138.png
+./images/fig139.png
+./images/fig140.png
+./images/fig141.png
+./images/fig142.png
+./images/fig143.png
+./images/fig144.png
+ 36525-t.ind
+ ***********
+
+ ) 
+Here is how much of TeX's memory you used:
+ 9564 strings out of 94074
+ 127205 string characters out of 1165154
+ 246539 words of memory out of 1500000
+ 10503 multiletter control sequences out of 10000+50000
+ 22253 words of font info for 85 fonts, out of 1200000 for 2000
+ 647 hyphenation exceptions out of 8191
+ 34i,21n,44p,1169b,594s stack positions out of 5000i,500n,6000p,200000b,5000s
+</usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmbx10.pfb></usr/share/texmf
+-texlive/fonts/type1/bluesky/cm/cmbx12.pfb></usr/share/texmf-texlive/fonts/type
+1/bluesky/cm/cmbx6.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmbx7.p
+fb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmbx8.pfb></usr/share/texmf
+-texlive/fonts/type1/bluesky/cm/cmbxti10.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmcsc10.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cme
+x10.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmmi10.pfb></usr/share
+/texmf-texlive/fonts/type1/bluesky/cm/cmmi12.pfb></usr/share/texmf-texlive/font
+s/type1/bluesky/cm/cmmi6.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/c
+mmi7.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmmi8.pfb></usr/share
+/texmf-texlive/fonts/type1/bluesky/cm/cmr10.pfb></usr/share/texmf-texlive/fonts
+/type1/bluesky/cm/cmr12.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cm
+r17.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmr6.pfb></usr/share/t
+exmf-texlive/fonts/type1/bluesky/cm/cmr7.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmr8.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmss12
+.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmss8.pfb></usr/share/tex
+mf-texlive/fonts/type1/bluesky/cm/cmsy10.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmsy6.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmsy7
+.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmsy8.pfb></usr/share/tex
+mf-texlive/fonts/type1/bluesky/cm/cmti10.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmti12.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmtt
+10.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmtt12.pfb></usr/share/
+texmf-texlive/fonts/type1/bluesky/cm/cmtt8.pfb></usr/share/texmf-texlive/fonts/
+type1/bluesky/ams/eufm10.pfb></usr/share/texmf-texlive/fonts/type1/public/cmex/
+fmex8.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/ams/msam10.pfb>
+Output written on 36525-t.pdf (620 pages, 3968199 bytes).
+PDF statistics:
+ 7277 PDF objects out of 7423 (max. 8388607)
+ 2067 named destinations out of 2073 (max. 131072)
+ 849 words of extra memory for PDF output out of 10000 (max. 10000000)
+
+This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2010.5.6)  28 JUN 2011 21:23
+entering extended mode
+ %&-line parsing enabled.
+**36525-t.tex
+(./36525-t.tex
+LaTeX2e <2005/12/01>
+Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
+yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov
+ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
+ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
+nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
+eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia
+n, swedish, ukenglish, pinyin, loaded.
+\@indexfile=\write3
+\openout3 = `36525-t.idx'.
+
+Writing index file 36525-t.idx
+(/usr/share/texmf-texlive/tex/latex/base/book.cls
+Document Class: book 2005/09/16 v1.4f Standard LaTeX document class
+(/usr/share/texmf-texlive/tex/latex/base/bk12.clo
+File: bk12.clo 2005/09/16 v1.4f Standard LaTeX file (size option)
+)
+\c@part=\count79
+\c@chapter=\count80
+\c@section=\count81
+\c@subsection=\count82
+\c@subsubsection=\count83
+\c@paragraph=\count84
+\c@subparagraph=\count85
+\c@figure=\count86
+\c@table=\count87
+\abovecaptionskip=\skip41
+\belowcaptionskip=\skip42
+\bibindent=\dimen102
+) (/usr/share/texmf-texlive/tex/latex/geometry/geometry.sty
+Package: geometry 2002/07/08 v3.2 Page Geometry
+(/usr/share/texmf-texlive/tex/latex/graphics/keyval.sty
+Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
+\KV@toks@=\toks14
+)
+\Gm@cnth=\count88
+\Gm@cntv=\count89
+\c@Gm@tempcnt=\count90
+\Gm@bindingoffset=\dimen103
+\Gm@wd@mp=\dimen104
+\Gm@odd@mp=\dimen105
+\Gm@even@mp=\dimen106
+\Gm@dimlist=\toks15
+(/usr/share/texmf-texlive/tex/xelatex/xetexconfig/geometry.cfg))
+
+LaTeX Warning: You have requested, on input line 118, version
+               `2008/11/13' of package geometry,
+               but only version
+               `2002/07/08 v3.2 Page Geometry'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
+Package: inputenc 2006/05/05 v1.1b Input encoding file
+\inpenc@prehook=\toks16
+\inpenc@posthook=\toks17
+(/usr/share/texmf-texlive/tex/latex/base/latin1.def
+File: latin1.def 2006/05/05 v1.1b Input encoding file
+)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
+Package: amsmath 2000/07/18 v2.13 AMS math features
+\@mathmargin=\skip43
+For additional information on amsmath, use the `?' option.
+(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
+Package: amstext 2000/06/29 v2.01
+(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty
+File: amsgen.sty 1999/11/30 v2.0
+\@emptytoks=\toks18
+\ex@=\dimen107
+)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty
+Package: amsbsy 1999/11/29 v1.2d
+\pmbraise@=\dimen108
+) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty
+Package: amsopn 1999/12/14 v2.01 operator names
+)
+\inf@bad=\count91
+LaTeX Info: Redefining \frac on input line 211.
+\uproot@=\count92
+\leftroot@=\count93
+LaTeX Info: Redefining \overline on input line 307.
+\classnum@=\count94
+\DOTSCASE@=\count95
+LaTeX Info: Redefining \ldots on input line 379.
+LaTeX Info: Redefining \dots on input line 382.
+LaTeX Info: Redefining \cdots on input line 467.
+\Mathstrutbox@=\box26
+\strutbox@=\box27
+\big@size=\dimen109
+LaTeX Font Info:    Redeclaring font encoding OML on input line 567.
+LaTeX Font Info:    Redeclaring font encoding OMS on input line 568.
+\macc@depth=\count96
+\c@MaxMatrixCols=\count97
+\dotsspace@=\muskip10
+\c@parentequation=\count98
+\dspbrk@lvl=\count99
+\tag@help=\toks19
+\row@=\count100
+\column@=\count101
+\maxfields@=\count102
+\andhelp@=\toks20
+\eqnshift@=\dimen110
+\alignsep@=\dimen111
+\tagshift@=\dimen112
+\tagwidth@=\dimen113
+\totwidth@=\dimen114
+\lineht@=\dimen115
+\@envbody=\toks21
+\multlinegap=\skip44
+\multlinetaggap=\skip45
+\mathdisplay@stack=\toks22
+LaTeX Info: Redefining \[ on input line 2666.
+LaTeX Info: Redefining \] on input line 2667.
+) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
+Package: amssymb 2002/01/22 v2.2d
+(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty
+Package: amsfonts 2001/10/25 v2.2f
+\symAMSa=\mathgroup4
+\symAMSb=\mathgroup5
+LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'
+(Font)                  U/euf/m/n --> U/euf/b/n on input line 132.
+)) (/usr/share/texmf-texlive/tex/latex/fancyhdr/fancyhdr.sty
+\fancy@headwidth=\skip46
+\f@ncyO@elh=\skip47
+\f@ncyO@erh=\skip48
+\f@ncyO@olh=\skip49
+\f@ncyO@orh=\skip50
+\f@ncyO@elf=\skip51
+\f@ncyO@erf=\skip52
+\f@ncyO@olf=\skip53
+\f@ncyO@orf=\skip54
+) (/usr/share/texmf-texlive/tex/latex/fancyhdr/extramarks.sty
+\@temptokenb=\toks23
+) (/usr/share/texmf-texlive/tex/latex/tools/verbatim.sty
+Package: verbatim 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements
+\every@verbatim=\toks24
+\verbatim@line=\toks25
+\verbatim@in@stream=\read1
+) (/usr/share/texmf-texlive/tex/latex/graphics/graphicx.sty
+Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
+(/usr/share/texmf-texlive/tex/latex/graphics/graphics.sty
+Package: graphics 2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
+(/usr/share/texmf-texlive/tex/latex/graphics/trig.sty
+Package: trig 1999/03/16 v1.09 sin cos tan (DPC)
+) (/etc/texmf/tex/latex/config/graphics.cfg
+File: graphics.cfg 2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
+)
+Package graphics Info: Driver file: pdftex.def on input line 90.
+(/usr/share/texmf-texlive/tex/latex/pdftex-def/pdftex.def
+File: pdftex.def 2007/01/08 v0.04d Graphics/color for pdfTeX
+\Gread@gobject=\count103
+))
+\Gin@req@height=\dimen116
+\Gin@req@width=\dimen117
+) (/usr/share/texmf-texlive/tex/latex/wrapfig/wrapfig.sty
+\wrapoverhang=\dimen118
+\WF@size=\dimen119
+\c@WF@wrappedlines=\count104
+\WF@box=\box28
+\WF@everypar=\toks26
+Package: wrapfig 2003/01/31  v 3.6
+) (/usr/share/texmf-texlive/tex/latex/rotating/rotating.sty
+Package: rotating 1997/09/26, v2.13 Rotation package
+(/usr/share/texmf-texlive/tex/latex/base/ifthen.sty
+Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
+)
+\c@r@tfl@t=\count105
+\rot@float@box=\box29
+) (/usr/share/texmf-texlive/tex/latex/caption/caption.sty
+Package: caption 2007/01/07 v3.0k Customising captions (AR)
+(/usr/share/texmf-texlive/tex/latex/caption/caption3.sty
+Package: caption3 2007/01/07 v3.0k caption3 kernel (AR)
+\captionmargin=\dimen120
+\captionmarginx=\dimen121
+\captionwidth=\dimen122
+\captionindent=\dimen123
+\captionparindent=\dimen124
+\captionhangindent=\dimen125
+)
+Package caption Info: rotating package v2.0 (or newer) detected on input line 3
+82.
+)
+
+LaTeX Warning: You have requested, on input line 129, version
+               `2008/08/24' of package caption,
+               but only version
+               `2007/01/07 v3.0k Customising captions (AR)'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/multirow/multirow.sty
+\bigstrutjot=\dimen126
+) (/usr/share/texmf-texlive/tex/latex/base/makeidx.sty
+Package: makeidx 2000/03/29 v1.0m Standard LaTeX package
+) (/usr/share/texmf-texlive/tex/latex/tools/multicol.sty
+Package: multicol 2006/05/18 v1.6g multicolumn formatting (FMi)
+\c@tracingmulticols=\count106
+\mult@box=\box30
+\multicol@leftmargin=\dimen127
+\c@unbalance=\count107
+\c@collectmore=\count108
+\doublecol@number=\count109
+\multicoltolerance=\count110
+\multicolpretolerance=\count111
+\full@width=\dimen128
+\page@free=\dimen129
+\premulticols=\dimen130
+\postmulticols=\dimen131
+\multicolsep=\skip55
+\multicolbaselineskip=\skip56
+\partial@page=\box31
+\last@line=\box32
+\mult@rightbox=\box33
+\mult@grightbox=\box34
+\mult@gfirstbox=\box35
+\mult@firstbox=\box36
+\@tempa=\box37
+\@tempa=\box38
+\@tempa=\box39
+\@tempa=\box40
+\@tempa=\box41
+\@tempa=\box42
+\@tempa=\box43
+\@tempa=\box44
+\@tempa=\box45
+\@tempa=\box46
+\@tempa=\box47
+\@tempa=\box48
+\@tempa=\box49
+\@tempa=\box50
+\@tempa=\box51
+\@tempa=\box52
+\@tempa=\box53
+\c@columnbadness=\count112
+\c@finalcolumnbadness=\count113
+\last@try=\dimen132
+\multicolovershoot=\dimen133
+\multicolundershoot=\dimen134
+\mult@nat@firstbox=\box54
+\colbreak@box=\box55
+) (/usr/share/texmf-texlive/tex/latex/tools/calc.sty
+Package: calc 2005/08/06 v4.2 Infix arithmetic (KKT,FJ)
+\calc@Acount=\count114
+\calc@Bcount=\count115
+\calc@Adimen=\dimen135
+\calc@Bdimen=\dimen136
+\calc@Askip=\skip57
+\calc@Bskip=\skip58
+LaTeX Info: Redefining \setlength on input line 75.
+LaTeX Info: Redefining \addtolength on input line 76.
+\calc@Ccount=\count116
+\calc@Cskip=\skip59
+) (/usr/share/texmf-texlive/tex/latex/tools/array.sty
+Package: array 2005/08/23 v2.4b Tabular extension package (FMi)
+\col@sep=\dimen137
+\extrarowheight=\dimen138
+\NC@list=\toks27
+\extratabsurround=\skip60
+\backup@length=\skip61
+) (/usr/share/texmf-texlive/tex/latex/mhchem/mhchem.sty
+Package: mhchem 2006/12/17 v3.05 for typesetting chemical formulae
+(/usr/share/texmf-texlive/tex/latex/oberdiek/twoopt.sty
+Package: twoopt 2006/02/20 v1.4 Definitions with two optional arguments (HO)
+)
+\tok@mhchem@ce@ii=\toks28
+\mhchem@arrowminlength=\skip62
+\mhchem@arrows@box=\box56
+\mhchem@arrowlength@pgf=\skip63
+\mhchem@arrowminlength@pgf=\skip64
+\mhchem@bondwidth=\skip65
+\mhchem@bondheight=\skip66
+\mhchem@smallbondwidth@tmpA=\skip67
+\mhchem@smallbondwidth@tmpB=\skip68
+\mhchem@smallbondwidth=\skip69
+\mhchem@cf@element=\toks29
+\mhchem@cf@number=\toks30
+\mhchem@cf@sup=\toks31
+\tok@mhchem@cf@i=\toks32
+\c@mhchem@cf@length=\count117
+\mhchem@cf@replaceminus@tok=\toks33
+\mhchem@option@minussidebearingleft=\skip70
+\mhchem@option@minussidebearingright=\skip71
+)
+
+LaTeX Warning: You have requested, on input line 135, version
+               `2007/05/19' of package mhchem,
+               but only version
+               `2006/12/17 v3.05 for typesetting chemical formulae'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/tools/indentfirst.sty
+Package: indentfirst 1995/11/23 v1.03 Indent first paragraph (DPC)
+) (/usr/share/texmf-texlive/tex/latex/tools/longtable.sty
+Package: longtable 2004/02/01 v4.11 Multi-page Table package (DPC)
+\LTleft=\skip72
+\LTright=\skip73
+\LTpre=\skip74
+\LTpost=\skip75
+\LTchunksize=\count118
+\LTcapwidth=\dimen139
+\LT@head=\box57
+\LT@firsthead=\box58
+\LT@foot=\box59
+\LT@lastfoot=\box60
+\LT@cols=\count119
+\LT@rows=\count120
+\c@LT@tables=\count121
+\c@LT@chunks=\count122
+\LT@p@ftn=\toks34
+) (/usr/share/texmf-texlive/tex/latex/footmisc/footmisc.sty
+Package: footmisc 2005/03/17 v5.3d a miscellany of footnote facilities
+\FN@temptoken=\toks35
+\footnotemargin=\dimen140
+\c@pp@next@reset=\count123
+\c@@fnserial=\count124
+Package footmisc Info: Declaring symbol style bringhurst on input line 817.
+Package footmisc Info: Declaring symbol style chicago on input line 818.
+Package footmisc Info: Declaring symbol style wiley on input line 819.
+Package footmisc Info: Declaring symbol style lamport-robust on input line 823.
+
+Package footmisc Info: Declaring symbol style lamport* on input line 831.
+Package footmisc Info: Declaring symbol style lamport*-robust on input line 840
+.
+)
+
+LaTeX Warning: You have requested, on input line 139, version
+               `2009/09/15' of package footmisc,
+               but only version
+               `2005/03/17 v5.3d a miscellany of footnote facilities'
+               is available.
+
+(/usr/share/texmf-texlive/tex/latex/hyperref/hyperref.sty
+Package: hyperref 2007/02/07 v6.75r Hypertext links for LaTeX
+\@linkdim=\dimen141
+\Hy@linkcounter=\count125
+\Hy@pagecounter=\count126
+(/usr/share/texmf-texlive/tex/latex/hyperref/pd1enc.def
+File: pd1enc.def 2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
+) (/etc/texmf/tex/latex/config/hyperref.cfg
+File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
+) (/usr/share/texmf-texlive/tex/latex/oberdiek/kvoptions.sty
+Package: kvoptions 2006/08/22 v2.4 Connects package keyval with LaTeX options (
+HO)
+)
+Package hyperref Info: Option `hyperfootnotes' set `false' on input line 2238.
+Package hyperref Info: Option `pdfdisplaydoctitle' set `true' on input line 223
+8.
+Package hyperref Info: Option `pdfpagelabels' set `true' on input line 2238.
+Package hyperref Info: Option `bookmarksopen' set `true' on input line 2238.
+Package hyperref Info: Option `colorlinks' set `false' on input line 2238.
+Package hyperref Info: Hyper figures OFF on input line 2288.
+Package hyperref Info: Link nesting OFF on input line 2293.
+Package hyperref Info: Hyper index ON on input line 2296.
+Package hyperref Info: Plain pages OFF on input line 2303.
+Package hyperref Info: Backreferencing OFF on input line 2308.
+Implicit mode ON; LaTeX internals redefined
+Package hyperref Info: Bookmarks ON on input line 2444.
+(/usr/share/texmf-texlive/tex/latex/ltxmisc/url.sty
+\Urlmuskip=\muskip11
+Package: url 2005/06/27  ver 3.2  Verb mode for urls, etc.
+)
+LaTeX Info: Redefining \url on input line 2599.
+\Fld@menulength=\count127
+\Field@Width=\dimen142
+\Fld@charsize=\dimen143
+\Choice@toks=\toks36
+\Field@toks=\toks37
+Package hyperref Info: Hyper figures OFF on input line 3102.
+Package hyperref Info: Link nesting OFF on input line 3107.
+Package hyperref Info: Hyper index ON on input line 3110.
+Package hyperref Info: backreferencing OFF on input line 3117.
+Package hyperref Info: Link coloring OFF on input line 3122.
+\Hy@abspage=\count128
+\c@Item=\count129
+)
+*hyperref using driver hpdftex*
+(/usr/share/texmf-texlive/tex/latex/hyperref/hpdftex.def
+File: hpdftex.def 2007/02/07 v6.75r Hyperref driver for pdfTeX
+\Fld@listcount=\count130
+)
+
+LaTeX Warning: You have requested, on input line 155, version
+               `2008/11/18' of package hyperref,
+               but only version
+               `2007/02/07 v6.75r Hypertext links for LaTeX'
+               is available.
+
+\TmpLen=\skip76
+\TmpPadLen=\skip77
+\HangPadLen=\skip78
+\HangLen=\skip79
+\ToCLen=\skip80
+\ToCLenA=\skip81
+\ToCLenB=\skip82
+\DP@lign@no=\count131
+\DP@lignb@dy=\toks38
+(./36525-t.aux)
+\openout1 = `36525-t.aux'.
+
+LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 568.
+LaTeX Font Info:    ... okay on input line 568.
+LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 568.
+LaTeX Font Info:    ... okay on input line 568.
+LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 568.
+LaTeX Font Info:    ... okay on input line 568.
+LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 568.
+LaTeX Font Info:    ... okay on input line 568.
+LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 568.
+LaTeX Font Info:    ... okay on input line 568.
+LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 568.
+LaTeX Font Info:    ... okay on input line 568.
+LaTeX Font Info:    Checking defaults for PD1/pdf/m/n on input line 568.
+LaTeX Font Info:    ... okay on input line 568.
+-------------------- Geometry parameters
+paper: user defined
+landscape: --
+twocolumn: --
+twoside: --
+asymmetric: --
+h-parts: 9.03374pt, 379.41751pt, 9.03374pt
+v-parts: 37.9413pt, 540.2187pt, 18.06749pt
+hmarginratio: --
+vmarginratio: --
+lines: --
+heightrounded: --
+bindingoffset: 0.0pt
+truedimen: --
+includehead: --
+includefoot: --
+includemp: --
+driver: pdftex
+-------------------- Page layout dimensions and switches
+\paperwidth  397.48499pt
+\paperheight 596.2275pt
+\textwidth  379.41751pt
+\textheight 540.2187pt
+\oddsidemargin  -63.23625pt
+\evensidemargin -63.23625pt
+\topmargin  -63.23622pt
+\headheight 14.5pt
+\headsep    14.45377pt
+\footskip   30.0pt
+\marginparwidth 47.0pt
+\marginparsep   7.0pt
+\columnsep  10.0pt
+\skip\footins  10.8pt plus 4.0pt minus 2.0pt
+\hoffset 0.0pt
+\voffset 0.0pt
+\mag 1000
+
+(1in=72.27pt, 1cm=28.45pt)
+-----------------------
+(/usr/share/texmf/tex/context/base/supp-pdf.tex
+[Loading MPS to PDF converter (version 2006.09.02).]
+\scratchcounter=\count132
+\scratchdimen=\dimen144
+\scratchbox=\box61
+\nofMPsegments=\count133
+\nofMParguments=\count134
+\everyMPshowfont=\toks39
+\MPscratchCnt=\count135
+\MPscratchDim=\dimen145
+\MPnumerator=\count136
+\everyMPtoPDFconversion=\toks40
+) (/usr/share/texmf-texlive/tex/latex/ragged2e/ragged2e.sty
+Package: ragged2e 2003/03/25 v2.04 ragged2e Package (MS)
+(/usr/share/texmf-texlive/tex/latex/everysel/everysel.sty
+Package: everysel 1999/06/08 v1.03 EverySelectfont Package (MS)
+LaTeX Info: Redefining \selectfont on input line 125.
+)
+\CenteringLeftskip=\skip83
+\RaggedLeftLeftskip=\skip84
+\RaggedRightLeftskip=\skip85
+\CenteringRightskip=\skip86
+\RaggedLeftRightskip=\skip87
+\RaggedRightRightskip=\skip88
+\CenteringParfillskip=\skip89
+\RaggedLeftParfillskip=\skip90
+\RaggedRightParfillskip=\skip91
+\JustifyingParfillskip=\skip92
+\CenteringParindent=\skip93
+\RaggedLeftParindent=\skip94
+\RaggedRightParindent=\skip95
+\JustifyingParindent=\skip96
+)
+Package caption Info: hyperref package v6.74m (or newer) detected on input line
+ 568.
+Package caption Info: longtable package v3.15 (or newer) detected on input line
+ 568.
+Package hyperref Info: Link coloring OFF on input line 568.
+(/usr/share/texmf-texlive/tex/latex/hyperref/nameref.sty
+Package: nameref 2006/12/27 v2.28 Cross-referencing by name of section
+(/usr/share/texmf-texlive/tex/latex/oberdiek/refcount.sty
+Package: refcount 2006/02/20 v3.0 Data extraction from references (HO)
+)
+\c@section@level=\count137
+)
+LaTeX Info: Redefining \ref on input line 568.
+LaTeX Info: Redefining \pageref on input line 568.
+(./36525-t.out) (./36525-t.out)
+\@outlinefile=\write4
+\openout4 = `36525-t.out'.
+
+[1
+
+{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
+LaTeX Font Info:    Try loading font information for U+msa on input line 641.
+(/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd
+File: umsa.fd 2002/01/19 v2.2g AMS font definitions
+)
+LaTeX Font Info:    Try loading font information for U+msb on input line 641.
+(/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd
+File: umsb.fd 2002/01/19 v2.2g AMS font definitions
+)
+LaTeX Font Info:    Try loading font information for U+euf on input line 641.
+(/usr/share/texmf-texlive/tex/latex/amsfonts/ueuf.fd
+File: ueuf.fd 2002/01/19 v2.2g AMS font definitions
+) [1
+
+] [2
+
+] [3
+
+] [4
+
+
+] [5] [6] [7] [8
+
+] [9] [10] [11] [12] [13] [14] [15] [16] [17
+
+]
+LaTeX Font Info:    Try loading font information for OMS+cmr on input line 1210
+.
+(/usr/share/texmf-texlive/tex/latex/base/omscmr.fd
+File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions
+)
+LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <12> not available
+(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 1210.
+
+[1
+
+
+] [2] [3] [4] [5] [6] [7] [8] [9] [10] <./images/fig01.png, id=1126, 236.17836p
+t x 229.24043pt>
+File: ./images/fig01.png Graphic file (type png)
+<use ./images/fig01.png>
+\CF@figure=\count138
+[11] [12 <./images/fig01.png (PNG copy)>] [13] [14] [15] [16] [17] [18] [19] [2
+0] [21] [22] [23] [24] [25] [26] [27] [28] <./images/fig02.png, id=1255, 250.92
+143pt x 300.06503pt>
+File: ./images/fig02.png Graphic file (type png)
+<use ./images/fig02.png> [29] [30 <./images/fig02.png (PNG copy)>] <./images/fi
+g03.png, id=1269, 155.52504pt x 223.16975pt>
+File: ./images/fig03.png Graphic file (type png)
+<use ./images/fig03.png> [31 <./images/fig03.png (PNG copy)>] [32] <./images/fi
+g04.png, id=1286, 183.85487pt x 241.09271pt>
+File: ./images/fig04.png Graphic file (type png)
+<use ./images/fig04.png> [33 <./images/fig04.png (PNG copy)>] <./images/fig05.p
+ng, id=1297, 232.7094pt x 78.34068pt>
+File: ./images/fig05.png Graphic file (type png)
+<use ./images/fig05.png> [34 <./images/fig05.png (PNG copy)>] [35] [36] <./imag
+es/fig06.png, id=1319, 196.5744pt x 121.4136pt>
+File: ./images/fig06.png Graphic file (type png)
+<use ./images/fig06.png> [37 <./images/fig06.png (PNG copy)>] <./images/fig07.p
+ng, id=1329, 182.01334pt x 160.06467pt>
+File: ./images/fig07.png Graphic file (type png)
+<use ./images/fig07.png> <./images/fig08.png, id=1330, 182.01334pt x 146.68134p
+t>
+File: ./images/fig08.png Graphic file (type png)
+<use ./images/fig08.png> <./images/fig09.png, id=1331, 182.01334pt x 149.89333p
+t>
+File: ./images/fig09.png Graphic file (type png)
+<use ./images/fig09.png> <./images/fig10.png, id=1332, 182.01334pt x 146.146pt>
+File: ./images/fig10.png Graphic file (type png)
+<use ./images/fig10.png> <./images/fig11.png, id=1333, 182.01334pt x 124.73267p
+t>
+File: ./images/fig11.png Graphic file (type png)
+<use ./images/fig11.png> <./images/fig12.png, id=1334, 182.01334pt x 128.21233p
+t>
+File: ./images/fig12.png Graphic file (type png)
+<use ./images/fig12.png> [38 <./images/fig07.png (PNG copy)> <./images/fig08.pn
+g (PNG copy)> <./images/fig09.png (PNG copy)> <./images/fig10.png (PNG copy)>] 
+[39 <./images/fig11.png (PNG copy)> <./images/fig12.png (PNG copy)>] <./images/
+fig13.png, id=1362, 379.56204pt x 223.74792pt>
+File: ./images/fig13.png Graphic file (type png)
+<use ./images/fig13.png> [40 <./images/fig13.png (PNG copy)>] [41] [42] <./imag
+es/fig14.png, id=1383, 328.39488pt x 54.05795pt>
+File: ./images/fig14.png Graphic file (type png)
+<use ./images/fig14.png> <./images/fig15.png, id=1384, 330.9966pt x 56.65968pt>
+File: ./images/fig15.png Graphic file (type png)
+<use ./images/fig15.png> <./images/fig16.png, id=1388, 330.9966pt x 59.83955pt>
+File: ./images/fig16.png Graphic file (type png)
+<use ./images/fig16.png> [43] [44 <./images/fig14.png (PNG copy)> <./images/fig
+15.png (PNG copy)> <./images/fig16.png (PNG copy)>] <./images/fig17.png, id=140
+8, 280.98576pt x 202.93416pt>
+File: ./images/fig17.png Graphic file (type png)
+<use ./images/fig17.png> [45 <./images/fig17.png (PNG copy)>] [46] <./images/fi
+g18.png, id=1419, 245.718pt x 358.4592pt>
+File: ./images/fig18.png Graphic file (type png)
+<use ./images/fig18.png> [47] [48 <./images/fig18.png (PNG copy)>] [49] [50] [5
+1] [52
+
+] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [6
+8] [69] [70] <./images/fig19.png, id=1539, 492.8814pt x 310.761pt>
+File: ./images/fig19.png Graphic file (type png)
+<use ./images/fig19.png>
+File: ./images/fig19.png Graphic file (type png)
+<use ./images/fig19.png> [71 <./images/fig19.png (PNG copy)>] [72] <./images/fi
+g20.png, id=1553, 242.8272pt x 198.30888pt>
+File: ./images/fig20.png Graphic file (type png)
+<use ./images/fig20.png> [73 <./images/fig20.png (PNG copy)>] [74] [75] [76] <.
+/images/fig21.png, id=1580, 335.91096pt x 213.34103pt>
+File: ./images/fig21.png Graphic file (type png)
+<use ./images/fig21.png> [77 <./images/fig21.png (PNG copy)>] <./images/fig22.p
+ng, id=1588, 246.29616pt x 244.85075pt>
+File: ./images/fig22.png Graphic file (type png)
+<use ./images/fig22.png> [78 <./images/fig22.png (PNG copy)>] [79] [80] <./imag
+es/fig23.png, id=1609, 139.62564pt x 140.78197pt>
+File: ./images/fig23.png Graphic file (type png)
+<use ./images/fig23.png> [81 <./images/fig23.png (PNG copy)>] [82] <./images/fi
+g24.png, id=1624, 304.40125pt x 243.98352pt>
+File: ./images/fig24.png Graphic file (type png)
+<use ./images/fig24.png> [83 <./images/fig24.png (PNG copy)>] <./images/fig25.p
+ng, id=1633, 241.3818pt x 280.98576pt>
+File: ./images/fig25.png Graphic file (type png)
+<use ./images/fig25.png> [84 <./images/fig25.png (PNG copy)>] <./images/fig26.p
+ng, id=1642, 214.78644pt x 246.87431pt>
+File: ./images/fig26.png Graphic file (type png)
+<use ./images/fig26.png> <./images/fig27.png, id=1652, 272.31335pt x 240.51456p
+t>
+File: ./images/fig27.png Graphic file (type png)
+<use ./images/fig27.png>
+Underfull \hbox (badness 4647) in paragraph at lines 5119--5157
+\OT1/cmr/m/n/12 in-ter-est-ing ex-per-i-ments has
+ []
+
+
+Underfull \hbox (badness 2626) in paragraph at lines 5119--5157
+\OT1/cmr/m/n/12 these curves in the neigh-
+ []
+
+
+Underfull \hbox (badness 1137) in paragraph at lines 5119--5157
+\OT1/cmr/m/n/12 cu-racy, as it was pro-duced
+ []
+
+
+Underfull \hbox (badness 1448) in paragraph at lines 5119--5157
+\OT1/cmr/m/n/12 could very eas-ily be de-ter-
+ []
+
+[85 <./images/fig26.png (PNG copy)>] [86 <./images/fig27.png (PNG copy)>] <./im
+ages/fig28.png, id=1667, 265.37544pt x 262.19556pt>
+File: ./images/fig28.png Graphic file (type png)
+<use ./images/fig28.png> [87] [88 <./images/fig28.png (PNG copy)>] [89] [90] [9
+1] <./images/fig29.png, id=1702, 300.93228pt x 171.71352pt>
+File: ./images/fig29.png Graphic file (type png)
+<use ./images/fig29.png> [92 <./images/fig29.png (PNG copy)>] [93] [94] [95] <.
+/images/fig30.png, id=1726, 185.0112pt x 183.5658pt>
+File: ./images/fig30.png Graphic file (type png)
+<use ./images/fig30.png> [96] <./images/fig31.png, id=1732, 147.14172pt x 146.8
+5265pt>
+File: ./images/fig31.png Graphic file (type png)
+<use ./images/fig31.png> [97 <./images/fig30.png (PNG copy)> <./images/fig31.pn
+g (PNG copy)>] [98] <./images/fig32.png, id=1748, 97.41995pt x 173.73708pt>
+File: ./images/fig32.png Graphic file (type png)
+<use ./images/fig32.png> [99 <./images/fig32.png (PNG copy)>] <./images/fig33.p
+ng, id=1757, 99.7326pt x 100.02168pt>
+File: ./images/fig33.png Graphic file (type png)
+<use ./images/fig33.png> [100 <./images/fig33.png (PNG copy)>] [101] <./images/
+fig34.png, id=1775, 210.45024pt x 147.14172pt>
+File: ./images/fig34.png Graphic file (type png)
+<use ./images/fig34.png>
+Underfull \hbox (badness 6157) in paragraph at lines 5933--5951
+\OT1/cmr/m/n/12 can con-tinue in the same
+ []
+
+
+Underfull \hbox (badness 1502) in paragraph at lines 5933--5951
+\OT1/cmr/m/n/12 medium all the way it can
+ []
+
+[102 <./images/fig34.png (PNG copy)>] [103] <./images/fig35.png, id=1791, 259.0
+1569pt x 163.04112pt>
+File: ./images/fig35.png Graphic file (type png)
+<use ./images/fig35.png> [104 <./images/fig35.png (PNG copy)>] <./images/fig36.
+png, id=1800, 50.589pt x 272.31335pt>
+File: ./images/fig36.png Graphic file (type png)
+<use ./images/fig36.png> [105 <./images/fig36.png (PNG copy)>] <./images/fig37.
+png, id=1809, 262.48463pt x 58.39417pt>
+File: ./images/fig37.png Graphic file (type png)
+<use ./images/fig37.png> <./images/fig38.png, id=1810, 263.93004pt x 81.52055pt
+>
+File: ./images/fig38.png Graphic file (type png)
+<use ./images/fig38.png> [106 <./images/fig37.png (PNG copy)> <./images/fig38.p
+ng (PNG copy)>] [107] <./images/fig39.png, id=1828, 274.91508pt x 74.00449pt>
+File: ./images/fig39.png Graphic file (type png)
+<use ./images/fig39.png> [108 <./images/fig39.png (PNG copy)>] <./images/fig40.
+png, id=1837, 333.02016pt x 503.57736pt>
+File: ./images/fig40.png Graphic file (type png)
+<use ./images/fig40.png> [109] [110 <./images/fig40.png (PNG copy)>] [111] <./i
+mages/fig41.png, id=1859, 206.6922pt x 411.07176pt>
+File: ./images/fig41.png Graphic file (type png)
+<use ./images/fig41.png>
+Underfull \hbox (badness 1210) in paragraph at lines 6350--6373
+\OT1/cmr/m/n/12 po-si-tion of the pos-i-tive elec-
+ []
+
+
+Underfull \hbox (badness 2478) in paragraph at lines 6350--6373
+\OT1/cmr/m/n/12 trode, nor do they in-crease
+ []
+
+[112 <./images/fig41.png (PNG copy)>] <./images/fig42.png, id=1866, 347.18507pt
+ x 360.77184pt>
+File: ./images/fig42.png Graphic file (type png)
+<use ./images/fig42.png> [113] [114 <./images/fig42.png (PNG copy)>] [115] [116
+] <./images/fig43.png, id=1891, 214.20828pt x 132.68771pt>
+File: ./images/fig43.png Graphic file (type png)
+<use ./images/fig43.png> [117 <./images/fig43.png (PNG copy)>] <./images/fig44.
+png, id=1902, 206.11404pt x 132.68771pt>
+File: ./images/fig44.png Graphic file (type png)
+<use ./images/fig44.png> [118 <./images/fig44.png (PNG copy)>] <./images/fig45.
+png, id=1912, 268.8444pt x 54.34705pt>
+File: ./images/fig45.png Graphic file (type png)
+<use ./images/fig45.png> [119 <./images/fig45.png (PNG copy)>] <./images/fig46.
+png, id=1921, 169.97903pt x 41.04936pt>
+File: ./images/fig46.png Graphic file (type png)
+<use ./images/fig46.png> <./images/fig47.png, id=1924, 372.62411pt x 201.48875p
+t>
+File: ./images/fig47.png Graphic file (type png)
+<use ./images/fig47.png> [120 <./images/fig46.png (PNG copy)> <./images/fig47.p
+ng (PNG copy)>] <./images/fig48.png, id=1934, 238.20192pt x 100.59984pt>
+File: ./images/fig48.png Graphic file (type png)
+<use ./images/fig48.png> <./images/fig49.png, id=1935, 271.15704pt x 216.81pt>
+File: ./images/fig49.png Graphic file (type png)
+<use ./images/fig49.png> [121] [122 <./images/fig48.png (PNG copy)> <./images/f
+ig49.png (PNG copy)>] [123] <./images/fig50.png, id=1960, 132.10956pt x 222.302
+52pt>
+File: ./images/fig50.png Graphic file (type png)
+<use ./images/fig50.png> [124 <./images/fig50.png (PNG copy)>] [125] <./images/
+fig51.png, id=1975, 379.56204pt x 120.54636pt>
+File: ./images/fig51.png Graphic file (type png)
+<use ./images/fig51.png> [126 <./images/fig51.png (PNG copy)>] [127] [128] <./i
+mages/fig52.png, id=1996, 131.5314pt x 39.60396pt>
+File: ./images/fig52.png Graphic file (type png)
+<use ./images/fig52.png> <./images/fig53.png, id=1997, 220.85712pt x 120.25728p
+t>
+File: ./images/fig53.png Graphic file (type png)
+<use ./images/fig53.png> <./images/fig54.png, id=1999, 140.49287pt x 167.08824p
+t>
+File: ./images/fig54.png Graphic file (type png)
+<use ./images/fig54.png> <./images/fig55.png, id=2000, 153.50148pt x 167.08824p
+t>
+File: ./images/fig55.png Graphic file (type png)
+<use ./images/fig55.png> <./images/fig56.png, id=2005, 302.95584pt x 116.78831p
+t>
+File: ./images/fig56.png Graphic file (type png)
+<use ./images/fig56.png> [129 <./images/fig52.png (PNG copy)> <./images/fig53.p
+ng (PNG copy)>] <./images/fig57.png, id=2018, 270.2898pt x 115.632pt>
+File: ./images/fig57.png Graphic file (type png)
+<use ./images/fig57.png> <./images/fig58.png, id=2019, 298.90872pt x 80.36424pt
+>
+File: ./images/fig58.png Graphic file (type png)
+<use ./images/fig58.png> <./images/fig59.png, id=2020, 304.9794pt x 116.21016pt
+>
+File: ./images/fig59.png Graphic file (type png)
+<use ./images/fig59.png> <./images/fig60.png, id=2021, 261.6174pt x 124.01532pt
+>
+File: ./images/fig60.png Graphic file (type png)
+<use ./images/fig60.png> [130 <./images/fig54.png (PNG copy)> <./images/fig55.p
+ng (PNG copy)> <./images/fig56.png (PNG copy)>] [131 <./images/fig57.png (PNG c
+opy)> <./images/fig58.png (PNG copy)>] [132 <./images/fig59.png (PNG copy)> <./
+images/fig60.png (PNG copy)>] [133] [134] <./images/fig61.png, id=2077, 140.781
+97pt x 272.31335pt>
+File: ./images/fig61.png Graphic file (type png)
+<use ./images/fig61.png> [135 <./images/fig61.png (PNG copy)>] <./images/fig62.
+png, id=2086, 133.55496pt x 385.05457pt>
+File: ./images/fig62.png Graphic file (type png)
+<use ./images/fig62.png> [136] [137 <./images/fig62.png (PNG copy)>] [138] [139
+] [140]
+Underfull \hbox (badness 1189) in paragraph at lines 7594--7594
+[]\OT1/cmr/m/n/10 Intensity of cur-rent in mil-
+ []
+
+
+Underfull \hbox (badness 1810) in paragraph at lines 7599--7599
+\OT1/cmr/m/n/10 the charg-ing of the con-
+ []
+
+<./images/fig63.png, id=2117, 305.84663pt x 75.44987pt>
+File: ./images/fig63.png Graphic file (type png)
+<use ./images/fig63.png> [141] [142 <./images/fig63.png (PNG copy)>] <./images/
+fig64.png, id=2132, 377.2494pt x 196.28532pt>
+File: ./images/fig64.png Graphic file (type png)
+<use ./images/fig64.png> [143] [144 <./images/fig64.png (PNG copy)>]
+Underfull \hbox (badness 1546) in paragraph at lines 7804--7804
+[][]\OT1/cmr/m/n/10 Number fix-ing the
+ []
+
+
+Underfull \hbox (badness 5189) in paragraph at lines 7814--7814
+\OT1/cmr/m/n/10 meter from the
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7814--7814
+\OT1/cmr/m/n/10 charge in the
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7814--7814
+\OT1/cmr/m/n/10 con-denser due
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7814--7814
+\OT1/cmr/m/n/10 to the po-ten-
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 7814--7814
+\OT1/cmr/m/n/10 tial dif-fer-ence
+ []
+
+[145] [146] [147] [148]
+Underfull \hbox (badness 10000) in paragraph at lines 7971--7971
+[]\OT1/cmr/m/n/10 Intensity of cur-rent in
+ []
+
+
+Underfull \hbox (badness 1137) in paragraph at lines 7974--7974
+\OT1/cmr/m/n/10 ten-tial dif-fer-ence be-tween
+ []
+
+<./images/fig65.png, id=2174, 32.95512pt x 129.21877pt>
+File: ./images/fig65.png Graphic file (type png)
+<use ./images/fig65.png> [149] [150 <./images/fig65.png (PNG copy)>] <./images/
+fig66.png, id=2190, 154.07964pt x 290.5254pt>
+File: ./images/fig66.png Graphic file (type png)
+<use ./images/fig66.png> [151 <./images/fig66.png (PNG copy)>] [152] [153] [154
+] [155] <./images/fig67.png, id=2224, 126.32796pt x 219.7008pt>
+File: ./images/fig67.png Graphic file (type png)
+<use ./images/fig67.png> [156 <./images/fig67.png (PNG copy)>] <./images/fig68.
+png, id=2233, 354.99023pt x 81.80965pt>
+File: ./images/fig68.png Graphic file (type png)
+<use ./images/fig68.png> <./images/fig69.png, id=2235, 293.4162pt x 451.25388pt
+>
+File: ./images/fig69.png Graphic file (type png)
+<use ./images/fig69.png> [157 <./images/fig68.png (PNG copy)>] [158 <./images/f
+ig69.png (PNG copy)>] <./images/fig70.png, id=2253, 228.66228pt x 149.16528pt>
+File: ./images/fig70.png Graphic file (type png)
+<use ./images/fig70.png> [159 <./images/fig70.png (PNG copy)>] [160] <./images/
+fig71.png, id=2266, 172.86984pt x 285.90012pt>
+File: ./images/fig71.png Graphic file (type png)
+<use ./images/fig71.png> [161 <./images/fig71.png (PNG copy)>] [162] [163] [164
+] <./images/fig72.png, id=2296, 227.21687pt x 66.4884pt>
+File: ./images/fig72.png Graphic file (type png)
+<use ./images/fig72.png> <./images/fig73.png, id=2298, 300.93228pt x 74.00449pt
+>
+File: ./images/fig73.png Graphic file (type png)
+<use ./images/fig73.png> [165 <./images/fig72.png (PNG copy)> <./images/fig73.p
+ng (PNG copy)>] <./images/fig74.png, id=2308, 96.26364pt x 100.59984pt>
+File: ./images/fig74.png Graphic file (type png)
+<use ./images/fig74.png> <./images/fig75.png, id=2309, 219.7008pt x 100.59984pt
+>
+File: ./images/fig75.png Graphic file (type png)
+<use ./images/fig75.png> [166 <./images/fig74.png (PNG copy)> <./images/fig75.p
+ng (PNG copy)>] <./images/fig76.png, id=2321, 209.00484pt x 51.45624pt>
+File: ./images/fig76.png Graphic file (type png)
+<use ./images/fig76.png> <./images/fig77.png, id=2323, 214.20828pt x 213.9192pt
+>
+File: ./images/fig77.png Graphic file (type png)
+<use ./images/fig77.png> [167 <./images/fig76.png (PNG copy)>] <./images/fig78.
+png, id=2330, 136.15668pt x 130.37508pt>
+File: ./images/fig78.png Graphic file (type png)
+<use ./images/fig78.png> [168 <./images/fig77.png (PNG copy)> <./images/fig78.p
+ng (PNG copy)>] <./images/fig79.png, id=2342, 253.52316pt x 156.1032pt>
+File: ./images/fig79.png Graphic file (type png)
+<use ./images/fig79.png> <./images/fig80.png, id=2344, 281.56392pt x 84.41136pt
+>
+File: ./images/fig80.png Graphic file (type png)
+<use ./images/fig80.png> [169 <./images/fig79.png (PNG copy)>] [170 <./images/f
+ig80.png (PNG copy)>] <./images/fig81.png, id=2361, 345.16151pt x 211.31747pt>
+File: ./images/fig81.png Graphic file (type png)
+<use ./images/fig81.png> [171 <./images/fig81.png (PNG copy)>] <./images/fig82.
+png, id=2370, 191.37096pt x 193.10544pt>
+File: ./images/fig82.png Graphic file (type png)
+<use ./images/fig82.png> [172 <./images/fig82.png (PNG copy)>] [173] <./images/
+fig83.png, id=2387, 308.15929pt x 162.75204pt>
+File: ./images/fig83.png Graphic file (type png)
+<use ./images/fig83.png> [174 <./images/fig83.png (PNG copy)>] [175] [176] <./i
+mages/fig84.png, id=2408, 261.6174pt x 154.6578pt>
+File: ./images/fig84.png Graphic file (type png)
+<use ./images/fig84.png> [177 <./images/fig84.png (PNG copy)>] [178] [179] [180
+] <./images/fig85.png, id=2430, 192.52728pt x 145.11816pt>
+File: ./images/fig85.png Graphic file (type png)
+<use ./images/fig85.png> [181 <./images/fig85.png (PNG copy)>] [182] <./images/
+fig86.png, id=2446, 275.49324pt x 59.2614pt>
+File: ./images/fig86.png Graphic file (type png)
+<use ./images/fig86.png> [183 <./images/fig86.png (PNG copy)>] <./images/fig87.
+png, id=2457, 171.71352pt x 176.3388pt>
+File: ./images/fig87.png Graphic file (type png)
+<use ./images/fig87.png> [184 <./images/fig87.png (PNG copy)>] [185] [186]
+Underfull \hbox (badness 1264) in paragraph at lines 9691--9696
+[]\OT1/cmr/m/n/12 Then, again, we have the very in-ter-est-ing re-sult dis-cov-
+ered by
+ []
+
+[187] [188] [189] <./images/fig88.png, id=2498, 293.12712pt x 23.1264pt>
+File: ./images/fig88.png Graphic file (type png)
+<use ./images/fig88.png> [190 <./images/fig88.png (PNG copy)>] [191] [192] [193
+] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203
+
+] <./images/fig89.png, id=2600, 351.2322pt x 31.7988pt>
+File: ./images/fig89.png Graphic file (type png)
+<use ./images/fig89.png> [204 <./images/fig89.png (PNG copy)>] [205] [206] <./i
+mages/fig90.png, id=2626, 354.123pt x 54.9252pt>
+File: ./images/fig90.png Graphic file (type png)
+<use ./images/fig90.png> <./images/fig91.png, id=2627, 354.123pt x 54.05795pt>
+File: ./images/fig91.png Graphic file (type png)
+<use ./images/fig91.png> [207 <./images/fig90.png (PNG copy)> <./images/fig91.p
+ng (PNG copy)>] [208] [209] [210] <./images/fig92.png, id=2671, 354.123pt x 58.
+97232pt>
+File: ./images/fig92.png Graphic file (type png)
+<use ./images/fig92.png> <./images/fig93.png, id=2672, 341.40347pt x 45.38556pt
+>
+File: ./images/fig93.png Graphic file (type png)
+<use ./images/fig93.png> [211 <./images/fig92.png (PNG copy)> <./images/fig93.p
+ng (PNG copy)>] [212] <./images/fig94.png, id=2703, 327.23856pt x 51.74532pt>
+File: ./images/fig94.png Graphic file (type png)
+<use ./images/fig94.png> [213 <./images/fig94.png (PNG copy)>]
+Underfull \hbox (badness 10000) in paragraph at lines 10902--10905
+
+ []
+
+[214] [215] [216] <./images/fig95.png, id=2738, 156.97044pt x 106.38144pt>
+File: ./images/fig95.png Graphic file (type png)
+<use ./images/fig95.png> [217 <./images/fig95.png (PNG copy)>] [218] <./images/
+fig96.png, id=2762, 269.13348pt x 130.37508pt>
+File: ./images/fig96.png Graphic file (type png)
+<use ./images/fig96.png> [219] [220 <./images/fig96.png (PNG copy)>] <./images/
+fig97.png, id=2784, 306.13573pt x 48.27637pt>
+File: ./images/fig97.png Graphic file (type png)
+<use ./images/fig97.png> <./images/fig98.png, id=2785, 298.61964pt x 19.65744pt
+>
+File: ./images/fig98.png Graphic file (type png)
+<use ./images/fig98.png> [221] [222 <./images/fig97.png (PNG copy)> <./images/f
+ig98.png (PNG copy)>] [223] [224] [225] <./images/fig99.png, id=2838, 267.68808
+pt x 112.7412pt>
+File: ./images/fig99.png Graphic file (type png)
+<use ./images/fig99.png> [226] [227 <./images/fig99.png (PNG copy)>] [228] [229
+] [230] <./images/fig100.png, id=2875, 109.56133pt x 15.8994pt>
+File: ./images/fig100.png Graphic file (type png)
+<use ./images/fig100.png> [231] [232 <./images/fig100.png (PNG copy)>] <./image
+s/fig101.png, id=2892, 130.95325pt x 44.22923pt>
+File: ./images/fig101.png Graphic file (type png)
+<use ./images/fig101.png> [233 <./images/fig101.png (PNG copy)>] [234] [235] [2
+36] <./images/fig102.png, id=2925, 183.5658pt x 46.2528pt>
+File: ./images/fig102.png Graphic file (type png)
+<use ./images/fig102.png> [237] [238 <./images/fig102.png (PNG copy)>] <./image
+s/fig103.png, id=2936, 225.19331pt x 59.2614pt>
+File: ./images/fig103.png Graphic file (type png)
+<use ./images/fig103.png> [239 <./images/fig103.png (PNG copy)>] <./images/fig1
+04.png, id=2945, 123.43716pt x 107.53777pt>
+File: ./images/fig104.png Graphic file (type png)
+<use ./images/fig104.png> [240] <./images/fig105.png, id=2952, 250.92143pt x 21
+.39192pt>
+File: ./images/fig105.png Graphic file (type png)
+<use ./images/fig105.png> [241 <./images/fig104.png (PNG copy)> <./images/fig10
+5.png (PNG copy)>] [242] <./images/fig106.png, id=2971, 207.27036pt x 36.135pt>
+File: ./images/fig106.png Graphic file (type png)
+<use ./images/fig106.png> [243 <./images/fig106.png (PNG copy)>] <./images/fig1
+07.png, id=2982, 113.03027pt x 101.178pt>
+File: ./images/fig107.png Graphic file (type png)
+<use ./images/fig107.png> [244 <./images/fig107.png (PNG copy)>] [245] [246
+
+] [247] [248] [249] [250] [251] [252] [253] [254] [255] [256] [257] [258] [259]
+[260] [261] [262]
+Underfull \hbox (badness 3439) in paragraph at lines 13103--13106
+[]\OT1/cmr/m/n/12 Condition (2) gives, writ-ing $\OML/cmm/m/it/12 J[]\OT1/cmr/m
+/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )$ for $\OML/cmm/m/it/12 dJ[]\OT1/cmr/
+m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 =dx$\OT1/cmr/m/n/12
+ , and $\OML/cmm/m/it/12 K[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 
+)$
+ []
+
+[263] [264] [265] [266] [267] [268] [269] [270] [271] [272] [273] [274] [275] [
+276] [277] [278] [279] <./images/fig108.png, id=3328, 307.8702pt x 187.03476pt>
+File: ./images/fig108.png Graphic file (type png)
+<use ./images/fig108.png> [280 <./images/fig108.png (PNG copy)>] [281] [282] [2
+83] [284] [285] [286] [287] [288]
+Underfull \hbox (badness 1968) in paragraph at lines 14420--14426
+[]\OT1/cmr/m/n/12 273[][].] We shall now pass on to the case when $\OML/cmm/m/i
+t/12 n\OT1/cmr/bx/n/12 a$ \OT1/cmr/m/n/12 is large
+ []
+
+[289] [290] [291] [292] [293] [294] [295] [296] [297] [298] [299] [300] [301] [
+302] [303] [304] [305] [306] [307] [308] [309] [310] [311] [312] [313] [314] [3
+15] [316] [317] [318] [319] [320] [321] [322] [323] [324] [325] [326] [327] [32
+8] [329] [330] [331] [332] [333] [334] [335] [336] <./images/fig109.png, id=388
+7, 277.5168pt x 63.88667pt>
+File: ./images/fig109.png Graphic file (type png)
+<use ./images/fig109.png> [337 <./images/fig109.png (PNG copy)>] [338] [339] [3
+40] [341] [342] [343] [344] [345] [346] <./images/fig110.png, id=3986, 218.8335
+6pt x 213.63013pt>
+File: ./images/fig110.png Graphic file (type png)
+<use ./images/fig110.png> <./images/fig111.png, id=3988, 203.8014pt x 161.59572
+pt>
+File: ./images/fig111.png Graphic file (type png)
+<use ./images/fig111.png> [347 <./images/fig110.png (PNG copy)>] [348 <./images
+/fig111.png (PNG copy)>] [349] [350] [351] [352] [353] [354] [355] [356] [357] 
+[358] [359] [360] [361] [362] [363] [364] [365] [366] [367] [368] [369] [370] [
+371] [372] [373] [374] [375] [376] [377] <./images/fig112.png, id=4237, 234.732
+96pt x 234.73296pt>
+File: ./images/fig112.png Graphic file (type png)
+<use ./images/fig112.png> [378 <./images/fig112.png (PNG copy)>] [379] [380] [3
+81] [382] [383] [384] [385] [386] <./images/fig113.png, id=4403, 177.49512pt x 
+51.16716pt>
+File: ./images/fig113.png Graphic file (type png)
+<use ./images/fig113.png> [387
+
+ <./images/fig113.png (PNG copy)>] [388] <./images/fig114.png, id=4417, 100.599
+84pt x 97.70905pt>
+File: ./images/fig114.png Graphic file (type png)
+<use ./images/fig114.png> [389] [390 <./images/fig114.png (PNG copy)>] <./image
+s/fig115.png, id=4432, 278.09496pt x 182.69856pt>
+File: ./images/fig115.png Graphic file (type png)
+<use ./images/fig115.png> [391 <./images/fig115.png (PNG copy)>] [392] <./image
+s/fig116.png, id=4449, 339.37991pt x 371.17873pt>
+File: ./images/fig116.png Graphic file (type png)
+<use ./images/fig116.png> [393 <./images/fig116.png (PNG copy)>] <./images/fig1
+17.png, id=4456, 281.56392pt x 112.7412pt>
+File: ./images/fig117.png Graphic file (type png)
+<use ./images/fig117.png> [394 <./images/fig117.png (PNG copy)>] <./images/fig1
+18.png, id=4467, 119.39005pt x 129.21877pt>
+File: ./images/fig118.png Graphic file (type png)
+<use ./images/fig118.png> [395] <./images/fig119.png, id=4476, 226.06056pt x 10
+6.38144pt>
+File: ./images/fig119.png Graphic file (type png)
+<use ./images/fig119.png> [396 <./images/fig118.png (PNG copy)> <./images/fig11
+9.png (PNG copy)>] [397] [398] [399] [400] [401] <./images/fig120.png, id=4513,
+ 256.41396pt x 121.70268pt>
+File: ./images/fig120.png Graphic file (type png)
+<use ./images/fig120.png> [402 <./images/fig120.png (PNG copy)>] [403] [404] [4
+05] [406] [407] [408] [409] [410] [411] [412] [413] [414] [415] [416] [417] [41
+8] [419] [420] [421] [422] [423] [424] [425] [426] [427] [428] [429] <./images/
+fig121.png, id=4728, 283.00932pt x 207.55943pt>
+File: ./images/fig121.png Graphic file (type png)
+<use ./images/fig121.png> [430 <./images/fig121.png (PNG copy)>] [431] [432] [4
+33] [434] <./images/fig122.png, id=4757, 324.63684pt x 188.19109pt>
+File: ./images/fig122.png Graphic file (type png)
+<use ./images/fig122.png> [435 <./images/fig122.png (PNG copy)>] [436] [437] [4
+38] [439] [440] [441] [442] [443] [444] [445] [446] [447] [448] [449] [450] [45
+1] [452] <./images/fig123.png, id=4910, 282.14207pt x 140.49287pt>
+File: ./images/fig123.png Graphic file (type png)
+<use ./images/fig123.png> [453 <./images/fig123.png (PNG copy)>] [454] <./image
+s/fig124.png, id=4928, 184.43304pt x 120.25728pt>
+File: ./images/fig124.png Graphic file (type png)
+<use ./images/fig124.png> [455 <./images/fig124.png (PNG copy)>] [456] [457] [4
+58] [459] <./images/fig125.png, id=4955, 255.25764pt x 193.97269pt>
+File: ./images/fig125.png Graphic file (type png)
+<use ./images/fig125.png> [460 <./images/fig125.png (PNG copy)>] <./images/fig1
+26.png, id=4963, 232.13124pt x 269.13348pt>
+File: ./images/fig126.png Graphic file (type png)
+<use ./images/fig126.png> [461 <./images/fig126.png (PNG copy)>] [462] <./image
+s/fig127.png, id=4981, 302.95584pt x 87.01308pt>
+File: ./images/fig127.png Graphic file (type png)
+<use ./images/fig127.png> [463 <./images/fig127.png (PNG copy)>] <./images/fig1
+28.png, id=4992, 137.313pt x 132.39864pt>
+File: ./images/fig128.png Graphic file (type png)
+<use ./images/fig128.png> [464 <./images/fig128.png (PNG copy)>] <./images/fig1
+29.png, id=5006, 256.99213pt x 251.21053pt>
+File: ./images/fig129.png Graphic file (type png)
+<use ./images/fig129.png> [465 <./images/fig129.png (PNG copy)>] <./images/fig1
+30.png, id=5013, 370.60056pt x 282.72025pt>
+File: ./images/fig130.png Graphic file (type png)
+<use ./images/fig130.png> [466 <./images/fig130.png (PNG copy)>] <./images/fig1
+31.png, id=5020, 245.718pt x 56.3706pt>
+File: ./images/fig131.png Graphic file (type png)
+<use ./images/fig131.png> [467 <./images/fig131.png (PNG copy)>] [468] [469] [4
+70] [471] [472] <./images/fig132.png, id=5055, 113.89752pt x 208.1376pt>
+File: ./images/fig132.png Graphic file (type png)
+<use ./images/fig132.png> [473 <./images/fig132.png (PNG copy)>] <./images/fig1
+33.png, id=5062, 343.42703pt x 309.60468pt>
+File: ./images/fig133.png Graphic file (type png)
+<use ./images/fig133.png> [474] [475 <./images/fig133.png (PNG copy)>] [476] [4
+77] [478] [479] [480] [481] <./images/fig134.png, id=5121, 145.69632pt x 324.05
+869pt>
+File: ./images/fig134.png Graphic file (type png)
+<use ./images/fig134.png>
+Underfull \hbox (badness 1077) in paragraph at lines 23996--24019
+\OT1/cmr/m/n/12 used by M. Blond-lot (\OT1/cmr/m/it/12 Comptes Ren-dus\OT1/cmr/
+m/n/12 ,
+ []
+
+[482 <./images/fig134.png (PNG copy)>] [483] [484] [485] [486] <./images/fig135
+.png, id=5150, 179.2296pt x 104.35788pt>
+File: ./images/fig135.png Graphic file (type png)
+<use ./images/fig135.png> [487] <./images/fig136.png, id=5157, 169.40088pt x 17
+0.5572pt>
+File: ./images/fig136.png Graphic file (type png)
+<use ./images/fig136.png> [488 <./images/fig135.png (PNG copy)>] [489 <./images
+/fig136.png (PNG copy)>] [490] [491] [492] [493] [494] [495] [496] [497] [498] 
+[499] [500] [501] [502] [503] [504] [505] [506] [507] [508] [509] [510] [511] [
+512
+
+] <./images/fig137.png, id=5367, 174.02615pt x 38.73672pt>
+File: ./images/fig137.png Graphic file (type png)
+<use ./images/fig137.png> <./images/fig138.png, id=5368, 164.19743pt x 74.29356
+pt>
+File: ./images/fig138.png Graphic file (type png)
+<use ./images/fig138.png> [513 <./images/fig137.png (PNG copy)>] <./images/fig1
+39.png, id=5378, 246.58524pt x 61.57404pt>
+File: ./images/fig139.png Graphic file (type png)
+<use ./images/fig139.png> [514 <./images/fig138.png (PNG copy)> <./images/fig13
+9.png (PNG copy)>] [515] [516] [517] [518] [519] [520] [521] [522] [523] [524] 
+[525] [526] [527] [528] [529] <./images/fig140.png, id=5499, 84.41136pt x 93.95
+1pt>
+File: ./images/fig140.png Graphic file (type png)
+<use ./images/fig140.png> [530 <./images/fig140.png (PNG copy)>] [531] <./image
+s/fig141.png, id=5513, 240.51456pt x 60.12865pt>
+File: ./images/fig141.png Graphic file (type png)
+<use ./images/fig141.png> [532] [533 <./images/fig141.png (PNG copy)>] [534] [5
+35] [536] [537] [538
+
+] [539] [540] [541] [542] [543] [544] [545] [546] [547] [548] [549] [550] [551]
+[552] [553] [554] [555] [556] [557] [558] [559] [560] [561] [562] [563] [564
+
+] <./images/fig142.png, id=5776, 379.28366pt x 454.23033pt>
+File: ./images/fig142.png Graphic file (type png)
+<use ./images/fig142.png> <./images/fig143.png, id=5778, 359.32645pt x 78.62976
+pt>
+File: ./images/fig143.png Graphic file (type png)
+<use ./images/fig143.png> [565] [566 <./images/fig142.png (PNG copy)>] [567 <./
+images/fig143.png (PNG copy)>] [568] [569] [570] [571] <./images/fig144.png, id
+=5822, 315.38628pt x 301.51044pt>
+File: ./images/fig144.png Graphic file (type png)
+<use ./images/fig144.png> [572 <./images/fig144.png (PNG copy)>] [573] [574] [5
+75] (./36525-t.ind [576] [577
+
+]
+Underfull \hbox (badness 7740) in paragraph at lines 98--100
+[]\OT1/cmr/m/n/10.95 --- elec-tro-static neu-tral-izes self-
+ []
+
+[578] [579] [580] [581] [582] [583] [584] [585] [586] [587] [588] [589]) [590] 
+[591] [592] [1
+
+] [2] [3] [4] [5] [6] [7] (./36525-t.aux)
+
+ *File List*
+    book.cls    2005/09/16 v1.4f Standard LaTeX document class
+    bk12.clo    2005/09/16 v1.4f Standard LaTeX file (size option)
+geometry.sty    2002/07/08 v3.2 Page Geometry
+  keyval.sty    1999/03/16 v1.13 key=value parser (DPC)
+geometry.cfg
+inputenc.sty    2006/05/05 v1.1b Input encoding file
+  latin1.def    2006/05/05 v1.1b Input encoding file
+ amsmath.sty    2000/07/18 v2.13 AMS math features
+ amstext.sty    2000/06/29 v2.01
+  amsgen.sty    1999/11/30 v2.0
+  amsbsy.sty    1999/11/29 v1.2d
+  amsopn.sty    1999/12/14 v2.01 operator names
+ amssymb.sty    2002/01/22 v2.2d
+amsfonts.sty    2001/10/25 v2.2f
+fancyhdr.sty    
+extramarks.sty    
+verbatim.sty    2003/08/22 v1.5q LaTeX2e package for verbatim enhancements
+graphicx.sty    1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
+graphics.sty    2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
+    trig.sty    1999/03/16 v1.09 sin cos tan (DPC)
+graphics.cfg    2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
+  pdftex.def    2007/01/08 v0.04d Graphics/color for pdfTeX
+ wrapfig.sty    2003/01/31  v 3.6
+rotating.sty    1997/09/26, v2.13 Rotation package
+  ifthen.sty    2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
+ caption.sty    2007/01/07 v3.0k Customising captions (AR)
+caption3.sty    2007/01/07 v3.0k caption3 kernel (AR)
+multirow.sty    
+ makeidx.sty    2000/03/29 v1.0m Standard LaTeX package
+multicol.sty    2006/05/18 v1.6g multicolumn formatting (FMi)
+    calc.sty    2005/08/06 v4.2 Infix arithmetic (KKT,FJ)
+   array.sty    2005/08/23 v2.4b Tabular extension package (FMi)
+  mhchem.sty    2006/12/17 v3.05 for typesetting chemical formulae
+  twoopt.sty    2006/02/20 v1.4 Definitions with two optional arguments (HO)
+indentfirst.sty    1995/11/23 v1.03 Indent first paragraph (DPC)
+longtable.sty    2004/02/01 v4.11 Multi-page Table package (DPC)
+footmisc.sty    2005/03/17 v5.3d a miscellany of footnote facilities
+hyperref.sty    2007/02/07 v6.75r Hypertext links for LaTeX
+  pd1enc.def    2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
+hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
+kvoptions.sty    2006/08/22 v2.4 Connects package keyval with LaTeX options (HO
+)
+     url.sty    2005/06/27  ver 3.2  Verb mode for urls, etc.
+ hpdftex.def    2007/02/07 v6.75r Hyperref driver for pdfTeX
+supp-pdf.tex
+ragged2e.sty    2003/03/25 v2.04 ragged2e Package (MS)
+everysel.sty    1999/06/08 v1.03 EverySelectfont Package (MS)
+ nameref.sty    2006/12/27 v2.28 Cross-referencing by name of section
+refcount.sty    2006/02/20 v3.0 Data extraction from references (HO)
+ 36525-t.out
+ 36525-t.out
+    umsa.fd    2002/01/19 v2.2g AMS font definitions
+    umsb.fd    2002/01/19 v2.2g AMS font definitions
+    ueuf.fd    2002/01/19 v2.2g AMS font definitions
+  omscmr.fd    1999/05/25 v2.5h Standard LaTeX font definitions
+./images/fig01.png
+./images/fig02.png
+./images/fig03.png
+./images/fig04.png
+./images/fig05.png
+./images/fig06.png
+./images/fig07.png
+./images/fig08.png
+./images/fig09.png
+./images/fig10.png
+./images/fig11.png
+./images/fig12.png
+./images/fig13.png
+./images/fig14.png
+./images/fig15.png
+./images/fig16.png
+./images/fig17.png
+./images/fig18.png
+./images/fig19.png
+./images/fig19.png
+./images/fig20.png
+./images/fig21.png
+./images/fig22.png
+./images/fig23.png
+./images/fig24.png
+./images/fig25.png
+./images/fig26.png
+./images/fig27.png
+./images/fig28.png
+./images/fig29.png
+./images/fig30.png
+./images/fig31.png
+./images/fig32.png
+./images/fig33.png
+./images/fig34.png
+./images/fig35.png
+./images/fig36.png
+./images/fig37.png
+./images/fig38.png
+./images/fig39.png
+./images/fig40.png
+./images/fig41.png
+./images/fig42.png
+./images/fig43.png
+./images/fig44.png
+./images/fig45.png
+./images/fig46.png
+./images/fig47.png
+./images/fig48.png
+./images/fig49.png
+./images/fig50.png
+./images/fig51.png
+./images/fig52.png
+./images/fig53.png
+./images/fig54.png
+./images/fig55.png
+./images/fig56.png
+./images/fig57.png
+./images/fig58.png
+./images/fig59.png
+./images/fig60.png
+./images/fig61.png
+./images/fig62.png
+./images/fig63.png
+./images/fig64.png
+./images/fig65.png
+./images/fig66.png
+./images/fig67.png
+./images/fig68.png
+./images/fig69.png
+./images/fig70.png
+./images/fig71.png
+./images/fig72.png
+./images/fig73.png
+./images/fig74.png
+./images/fig75.png
+./images/fig76.png
+./images/fig77.png
+./images/fig78.png
+./images/fig79.png
+./images/fig80.png
+./images/fig81.png
+./images/fig82.png
+./images/fig83.png
+./images/fig84.png
+./images/fig85.png
+./images/fig86.png
+./images/fig87.png
+./images/fig88.png
+./images/fig89.png
+./images/fig90.png
+./images/fig91.png
+./images/fig92.png
+./images/fig93.png
+./images/fig94.png
+./images/fig95.png
+./images/fig96.png
+./images/fig97.png
+./images/fig98.png
+./images/fig99.png
+./images/fig100.png
+./images/fig101.png
+./images/fig102.png
+./images/fig103.png
+./images/fig104.png
+./images/fig105.png
+./images/fig106.png
+./images/fig107.png
+./images/fig108.png
+./images/fig109.png
+./images/fig110.png
+./images/fig111.png
+./images/fig112.png
+./images/fig113.png
+./images/fig114.png
+./images/fig115.png
+./images/fig116.png
+./images/fig117.png
+./images/fig118.png
+./images/fig119.png
+./images/fig120.png
+./images/fig121.png
+./images/fig122.png
+./images/fig123.png
+./images/fig124.png
+./images/fig125.png
+./images/fig126.png
+./images/fig127.png
+./images/fig128.png
+./images/fig129.png
+./images/fig130.png
+./images/fig131.png
+./images/fig132.png
+./images/fig133.png
+./images/fig134.png
+./images/fig135.png
+./images/fig136.png
+./images/fig137.png
+./images/fig138.png
+./images/fig139.png
+./images/fig140.png
+./images/fig141.png
+./images/fig142.png
+./images/fig143.png
+./images/fig144.png
+ 36525-t.ind
+ ***********
+
+ ) 
+Here is how much of TeX's memory you used:
+ 9561 strings out of 94074
+ 127181 string characters out of 1165154
+ 247535 words of memory out of 1500000
+ 10503 multiletter control sequences out of 10000+50000
+ 22253 words of font info for 85 fonts, out of 1200000 for 2000
+ 647 hyphenation exceptions out of 8191
+ 34i,21n,44p,1169b,594s stack positions out of 5000i,500n,6000p,200000b,5000s
+</usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmbx10.pfb></usr/share/texmf
+-texlive/fonts/type1/bluesky/cm/cmbx12.pfb></usr/share/texmf-texlive/fonts/type
+1/bluesky/cm/cmbx6.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmbx7.p
+fb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmbx8.pfb></usr/share/texmf
+-texlive/fonts/type1/bluesky/cm/cmbxti10.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmcsc10.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cme
+x10.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmmi10.pfb></usr/share
+/texmf-texlive/fonts/type1/bluesky/cm/cmmi12.pfb></usr/share/texmf-texlive/font
+s/type1/bluesky/cm/cmmi6.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/c
+mmi7.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmmi8.pfb></usr/share
+/texmf-texlive/fonts/type1/bluesky/cm/cmr10.pfb></usr/share/texmf-texlive/fonts
+/type1/bluesky/cm/cmr12.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cm
+r17.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmr6.pfb></usr/share/t
+exmf-texlive/fonts/type1/bluesky/cm/cmr7.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmr8.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmss12
+.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmss8.pfb></usr/share/tex
+mf-texlive/fonts/type1/bluesky/cm/cmsy10.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmsy6.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmsy7
+.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmsy8.pfb></usr/share/tex
+mf-texlive/fonts/type1/bluesky/cm/cmti10.pfb></usr/share/texmf-texlive/fonts/ty
+pe1/bluesky/cm/cmti12.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmtt
+10.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmtt12.pfb></usr/share/
+texmf-texlive/fonts/type1/bluesky/cm/cmtt8.pfb></usr/share/texmf-texlive/fonts/
+type1/bluesky/ams/eufm10.pfb></usr/share/texmf-texlive/fonts/type1/public/cmex/
+fmex8.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/ams/msam10.pfb>
+Output written on 36525-t.pdf (617 pages, 3964342 bytes).
+PDF statistics:
+ 7264 PDF objects out of 7423 (max. 8388607)
+ 2064 named destinations out of 2073 (max. 131072)
+ 849 words of extra memory for PDF output out of 10000 (max. 10000000)
+
diff --git a/36525-t/images/fig01.png b/36525-t/images/fig01.png
new file mode 100644
index 0000000..8e431ae
--- /dev/null
+++ b/36525-t/images/fig01.png
diff --git a/36525-t/images/fig02.png b/36525-t/images/fig02.png
new file mode 100644
index 0000000..34fd667
--- /dev/null
+++ b/36525-t/images/fig02.png
diff --git a/36525-t/images/fig03.png b/36525-t/images/fig03.png
new file mode 100644
index 0000000..2b71c0d
--- /dev/null
+++ b/36525-t/images/fig03.png
diff --git a/36525-t/images/fig04.png b/36525-t/images/fig04.png
new file mode 100644
index 0000000..39d1c16
--- /dev/null
+++ b/36525-t/images/fig04.png
diff --git a/36525-t/images/fig05.png b/36525-t/images/fig05.png
new file mode 100644
index 0000000..73b77af
--- /dev/null
+++ b/36525-t/images/fig05.png
diff --git a/36525-t/images/fig06.png b/36525-t/images/fig06.png
new file mode 100644
index 0000000..df90ef4
--- /dev/null
+++ b/36525-t/images/fig06.png
diff --git a/36525-t/images/fig07.png b/36525-t/images/fig07.png
new file mode 100644
index 0000000..842aee6
--- /dev/null
+++ b/36525-t/images/fig07.png
diff --git a/36525-t/images/fig08.png b/36525-t/images/fig08.png
new file mode 100644
index 0000000..0c20c64
--- /dev/null
+++ b/36525-t/images/fig08.png
diff --git a/36525-t/images/fig09.png b/36525-t/images/fig09.png
new file mode 100644
index 0000000..57fc2d3
--- /dev/null
+++ b/36525-t/images/fig09.png
diff --git a/36525-t/images/fig10.png b/36525-t/images/fig10.png
new file mode 100644
index 0000000..57750ee
--- /dev/null
+++ b/36525-t/images/fig10.png
diff --git a/36525-t/images/fig100.png b/36525-t/images/fig100.png
new file mode 100644
index 0000000..bc2b2c1
--- /dev/null
+++ b/36525-t/images/fig100.png
diff --git a/36525-t/images/fig101.png b/36525-t/images/fig101.png
new file mode 100644
index 0000000..325f041
--- /dev/null
+++ b/36525-t/images/fig101.png
diff --git a/36525-t/images/fig102.png b/36525-t/images/fig102.png
new file mode 100644
index 0000000..c3a6954
--- /dev/null
+++ b/36525-t/images/fig102.png
diff --git a/36525-t/images/fig103.png b/36525-t/images/fig103.png
new file mode 100644
index 0000000..e095900
--- /dev/null
+++ b/36525-t/images/fig103.png
diff --git a/36525-t/images/fig104.png b/36525-t/images/fig104.png
new file mode 100644
index 0000000..7142b0b
--- /dev/null
+++ b/36525-t/images/fig104.png
diff --git a/36525-t/images/fig105.png b/36525-t/images/fig105.png
new file mode 100644
index 0000000..dc1abb8
--- /dev/null
+++ b/36525-t/images/fig105.png
diff --git a/36525-t/images/fig106.png b/36525-t/images/fig106.png
new file mode 100644
index 0000000..da4c2da
--- /dev/null
+++ b/36525-t/images/fig106.png
diff --git a/36525-t/images/fig107.png b/36525-t/images/fig107.png
new file mode 100644
index 0000000..978a557
--- /dev/null
+++ b/36525-t/images/fig107.png
diff --git a/36525-t/images/fig108.png b/36525-t/images/fig108.png
new file mode 100644
index 0000000..2d1c6d4
--- /dev/null
+++ b/36525-t/images/fig108.png
diff --git a/36525-t/images/fig109.png b/36525-t/images/fig109.png
new file mode 100644
index 0000000..cdcf659
--- /dev/null
+++ b/36525-t/images/fig109.png
diff --git a/36525-t/images/fig11.png b/36525-t/images/fig11.png
new file mode 100644
index 0000000..4d8ce54
--- /dev/null
+++ b/36525-t/images/fig11.png
diff --git a/36525-t/images/fig110.png b/36525-t/images/fig110.png
new file mode 100644
index 0000000..533f85a
--- /dev/null
+++ b/36525-t/images/fig110.png
diff --git a/36525-t/images/fig111.png b/36525-t/images/fig111.png
new file mode 100644
index 0000000..54960ae
--- /dev/null
+++ b/36525-t/images/fig111.png
diff --git a/36525-t/images/fig112.png b/36525-t/images/fig112.png
new file mode 100644
index 0000000..9232426
--- /dev/null
+++ b/36525-t/images/fig112.png
diff --git a/36525-t/images/fig113.png b/36525-t/images/fig113.png
new file mode 100644
index 0000000..c12f118
--- /dev/null
+++ b/36525-t/images/fig113.png
diff --git a/36525-t/images/fig114.png b/36525-t/images/fig114.png
new file mode 100644
index 0000000..b6ccb9b
--- /dev/null
+++ b/36525-t/images/fig114.png
diff --git a/36525-t/images/fig115.png b/36525-t/images/fig115.png
new file mode 100644
index 0000000..d67a339
--- /dev/null
+++ b/36525-t/images/fig115.png
diff --git a/36525-t/images/fig116.png b/36525-t/images/fig116.png
new file mode 100644
index 0000000..d47227e
--- /dev/null
+++ b/36525-t/images/fig116.png
diff --git a/36525-t/images/fig117.png b/36525-t/images/fig117.png
new file mode 100644
index 0000000..e69f20c
--- /dev/null
+++ b/36525-t/images/fig117.png
diff --git a/36525-t/images/fig118.png b/36525-t/images/fig118.png
new file mode 100644
index 0000000..5963338
--- /dev/null
+++ b/36525-t/images/fig118.png
diff --git a/36525-t/images/fig119.png b/36525-t/images/fig119.png
new file mode 100644
index 0000000..46999b5
--- /dev/null
+++ b/36525-t/images/fig119.png
diff --git a/36525-t/images/fig12.png b/36525-t/images/fig12.png
new file mode 100644
index 0000000..58f8382
--- /dev/null
+++ b/36525-t/images/fig12.png
diff --git a/36525-t/images/fig120.png b/36525-t/images/fig120.png
new file mode 100644
index 0000000..f7c6c38
--- /dev/null
+++ b/36525-t/images/fig120.png
diff --git a/36525-t/images/fig121.png b/36525-t/images/fig121.png
new file mode 100644
index 0000000..78ba7fe
--- /dev/null
+++ b/36525-t/images/fig121.png
diff --git a/36525-t/images/fig122.png b/36525-t/images/fig122.png
new file mode 100644
index 0000000..b41a16f
--- /dev/null
+++ b/36525-t/images/fig122.png
diff --git a/36525-t/images/fig123.png b/36525-t/images/fig123.png
new file mode 100644
index 0000000..528915f
--- /dev/null
+++ b/36525-t/images/fig123.png
diff --git a/36525-t/images/fig124.png b/36525-t/images/fig124.png
new file mode 100644
index 0000000..ed77f02
--- /dev/null
+++ b/36525-t/images/fig124.png
diff --git a/36525-t/images/fig125.png b/36525-t/images/fig125.png
new file mode 100644
index 0000000..1be90d8
--- /dev/null
+++ b/36525-t/images/fig125.png
diff --git a/36525-t/images/fig126.png b/36525-t/images/fig126.png
new file mode 100644
index 0000000..4acb8a7
--- /dev/null
+++ b/36525-t/images/fig126.png
diff --git a/36525-t/images/fig127.png b/36525-t/images/fig127.png
new file mode 100644
index 0000000..98f7727
--- /dev/null
+++ b/36525-t/images/fig127.png
diff --git a/36525-t/images/fig128.png b/36525-t/images/fig128.png
new file mode 100644
index 0000000..4fcbbac
--- /dev/null
+++ b/36525-t/images/fig128.png
diff --git a/36525-t/images/fig129.png b/36525-t/images/fig129.png
new file mode 100644
index 0000000..5fb4cbe
--- /dev/null
+++ b/36525-t/images/fig129.png
diff --git a/36525-t/images/fig13.png b/36525-t/images/fig13.png
new file mode 100644
index 0000000..644ec58
--- /dev/null
+++ b/36525-t/images/fig13.png
diff --git a/36525-t/images/fig130.png b/36525-t/images/fig130.png
new file mode 100644
index 0000000..2c4c401
--- /dev/null
+++ b/36525-t/images/fig130.png
diff --git a/36525-t/images/fig131.png b/36525-t/images/fig131.png
new file mode 100644
index 0000000..2a66c16
--- /dev/null
+++ b/36525-t/images/fig131.png
diff --git a/36525-t/images/fig132.png b/36525-t/images/fig132.png
new file mode 100644
index 0000000..35b05fb
--- /dev/null
+++ b/36525-t/images/fig132.png
diff --git a/36525-t/images/fig133.png b/36525-t/images/fig133.png
new file mode 100644
index 0000000..4792683
--- /dev/null
+++ b/36525-t/images/fig133.png
diff --git a/36525-t/images/fig134.png b/36525-t/images/fig134.png
new file mode 100644
index 0000000..eaa1f87
--- /dev/null
+++ b/36525-t/images/fig134.png
diff --git a/36525-t/images/fig135.png b/36525-t/images/fig135.png
new file mode 100644
index 0000000..c70e705
--- /dev/null
+++ b/36525-t/images/fig135.png
diff --git a/36525-t/images/fig136.png b/36525-t/images/fig136.png
new file mode 100644
index 0000000..5e372d2
--- /dev/null
+++ b/36525-t/images/fig136.png
diff --git a/36525-t/images/fig137.png b/36525-t/images/fig137.png
new file mode 100644
index 0000000..b4ec456
--- /dev/null
+++ b/36525-t/images/fig137.png
diff --git a/36525-t/images/fig138.png b/36525-t/images/fig138.png
new file mode 100644
index 0000000..b9437da
--- /dev/null
+++ b/36525-t/images/fig138.png
diff --git a/36525-t/images/fig139.png b/36525-t/images/fig139.png
new file mode 100644
index 0000000..d5a2a51
--- /dev/null
+++ b/36525-t/images/fig139.png
diff --git a/36525-t/images/fig14.png b/36525-t/images/fig14.png
new file mode 100644
index 0000000..02685da
--- /dev/null
+++ b/36525-t/images/fig14.png
diff --git a/36525-t/images/fig140.png b/36525-t/images/fig140.png
new file mode 100644
index 0000000..b3f099c
--- /dev/null
+++ b/36525-t/images/fig140.png
diff --git a/36525-t/images/fig141.png b/36525-t/images/fig141.png
new file mode 100644
index 0000000..67a7729
--- /dev/null
+++ b/36525-t/images/fig141.png
diff --git a/36525-t/images/fig142.png b/36525-t/images/fig142.png
new file mode 100644
index 0000000..4880f47
--- /dev/null
+++ b/36525-t/images/fig142.png
diff --git a/36525-t/images/fig143.png b/36525-t/images/fig143.png
new file mode 100644
index 0000000..589fb72
--- /dev/null
+++ b/36525-t/images/fig143.png
diff --git a/36525-t/images/fig144.png b/36525-t/images/fig144.png
new file mode 100644
index 0000000..a672408
--- /dev/null
+++ b/36525-t/images/fig144.png
diff --git a/36525-t/images/fig15.png b/36525-t/images/fig15.png
new file mode 100644
index 0000000..0e8075f
--- /dev/null
+++ b/36525-t/images/fig15.png
diff --git a/36525-t/images/fig16.png b/36525-t/images/fig16.png
new file mode 100644
index 0000000..dc54754
--- /dev/null
+++ b/36525-t/images/fig16.png
diff --git a/36525-t/images/fig17.png b/36525-t/images/fig17.png
new file mode 100644
index 0000000..9698cda
--- /dev/null
+++ b/36525-t/images/fig17.png
diff --git a/36525-t/images/fig18.png b/36525-t/images/fig18.png
new file mode 100644
index 0000000..b72154d
--- /dev/null
+++ b/36525-t/images/fig18.png
diff --git a/36525-t/images/fig19.png b/36525-t/images/fig19.png
new file mode 100644
index 0000000..e547240
--- /dev/null
+++ b/36525-t/images/fig19.png
diff --git a/36525-t/images/fig20.png b/36525-t/images/fig20.png
new file mode 100644
index 0000000..4cabd24
--- /dev/null
+++ b/36525-t/images/fig20.png
diff --git a/36525-t/images/fig21.png b/36525-t/images/fig21.png
new file mode 100644
index 0000000..62f49da
--- /dev/null
+++ b/36525-t/images/fig21.png
diff --git a/36525-t/images/fig22.png b/36525-t/images/fig22.png
new file mode 100644
index 0000000..8edf8c3
--- /dev/null
+++ b/36525-t/images/fig22.png
diff --git a/36525-t/images/fig23.png b/36525-t/images/fig23.png
new file mode 100644
index 0000000..5f40794
--- /dev/null
+++ b/36525-t/images/fig23.png
diff --git a/36525-t/images/fig24.png b/36525-t/images/fig24.png
new file mode 100644
index 0000000..f8efcde
--- /dev/null
+++ b/36525-t/images/fig24.png
diff --git a/36525-t/images/fig25.png b/36525-t/images/fig25.png
new file mode 100644
index 0000000..9ba19cc
--- /dev/null
+++ b/36525-t/images/fig25.png
diff --git a/36525-t/images/fig26.png b/36525-t/images/fig26.png
new file mode 100644
index 0000000..dc2e93f
--- /dev/null
+++ b/36525-t/images/fig26.png
diff --git a/36525-t/images/fig27.png b/36525-t/images/fig27.png
new file mode 100644
index 0000000..9ac703c
--- /dev/null
+++ b/36525-t/images/fig27.png
diff --git a/36525-t/images/fig28.png b/36525-t/images/fig28.png
new file mode 100644
index 0000000..5e8edd1
--- /dev/null
+++ b/36525-t/images/fig28.png
diff --git a/36525-t/images/fig29.png b/36525-t/images/fig29.png
new file mode 100644
index 0000000..090dd65
--- /dev/null
+++ b/36525-t/images/fig29.png
diff --git a/36525-t/images/fig30.png b/36525-t/images/fig30.png
new file mode 100644
index 0000000..cb383de
--- /dev/null
+++ b/36525-t/images/fig30.png
diff --git a/36525-t/images/fig31.png b/36525-t/images/fig31.png
new file mode 100644
index 0000000..c48285e
--- /dev/null
+++ b/36525-t/images/fig31.png
diff --git a/36525-t/images/fig32.png b/36525-t/images/fig32.png
new file mode 100644
index 0000000..c028ad3
--- /dev/null
+++ b/36525-t/images/fig32.png
diff --git a/36525-t/images/fig33.png b/36525-t/images/fig33.png
new file mode 100644
index 0000000..3326e91
--- /dev/null
+++ b/36525-t/images/fig33.png
diff --git a/36525-t/images/fig34.png b/36525-t/images/fig34.png
new file mode 100644
index 0000000..f8607e0
--- /dev/null
+++ b/36525-t/images/fig34.png
diff --git a/36525-t/images/fig35.png b/36525-t/images/fig35.png
new file mode 100644
index 0000000..302be8e
--- /dev/null
+++ b/36525-t/images/fig35.png
diff --git a/36525-t/images/fig36.png b/36525-t/images/fig36.png
new file mode 100644
index 0000000..0b5ba5b
--- /dev/null
+++ b/36525-t/images/fig36.png
diff --git a/36525-t/images/fig37.png b/36525-t/images/fig37.png
new file mode 100644
index 0000000..075cddf
--- /dev/null
+++ b/36525-t/images/fig37.png
diff --git a/36525-t/images/fig38.png b/36525-t/images/fig38.png
new file mode 100644
index 0000000..3bd7c1a
--- /dev/null
+++ b/36525-t/images/fig38.png
diff --git a/36525-t/images/fig39.png b/36525-t/images/fig39.png
new file mode 100644
index 0000000..47024fa
--- /dev/null
+++ b/36525-t/images/fig39.png
diff --git a/36525-t/images/fig40.png b/36525-t/images/fig40.png
new file mode 100644
index 0000000..877ec55
--- /dev/null
+++ b/36525-t/images/fig40.png
diff --git a/36525-t/images/fig41.png b/36525-t/images/fig41.png
new file mode 100644
index 0000000..568b8bc
--- /dev/null
+++ b/36525-t/images/fig41.png
diff --git a/36525-t/images/fig42.png b/36525-t/images/fig42.png
new file mode 100644
index 0000000..6cc6a1e
--- /dev/null
+++ b/36525-t/images/fig42.png
diff --git a/36525-t/images/fig43.png b/36525-t/images/fig43.png
new file mode 100644
index 0000000..0914c9e
--- /dev/null
+++ b/36525-t/images/fig43.png
diff --git a/36525-t/images/fig44.png b/36525-t/images/fig44.png
new file mode 100644
index 0000000..a289031
--- /dev/null
+++ b/36525-t/images/fig44.png
diff --git a/36525-t/images/fig45.png b/36525-t/images/fig45.png
new file mode 100644
index 0000000..f98620b
--- /dev/null
+++ b/36525-t/images/fig45.png
diff --git a/36525-t/images/fig46.png b/36525-t/images/fig46.png
new file mode 100644
index 0000000..eaed2b6
--- /dev/null
+++ b/36525-t/images/fig46.png
diff --git a/36525-t/images/fig47.png b/36525-t/images/fig47.png
new file mode 100644
index 0000000..4a51309
--- /dev/null
+++ b/36525-t/images/fig47.png
diff --git a/36525-t/images/fig48.png b/36525-t/images/fig48.png
new file mode 100644
index 0000000..8df4481
--- /dev/null
+++ b/36525-t/images/fig48.png
diff --git a/36525-t/images/fig49.png b/36525-t/images/fig49.png
new file mode 100644
index 0000000..5c1b3b0
--- /dev/null
+++ b/36525-t/images/fig49.png
diff --git a/36525-t/images/fig50.png b/36525-t/images/fig50.png
new file mode 100644
index 0000000..10260b3
--- /dev/null
+++ b/36525-t/images/fig50.png
diff --git a/36525-t/images/fig51.png b/36525-t/images/fig51.png
new file mode 100644
index 0000000..1025267
--- /dev/null
+++ b/36525-t/images/fig51.png
diff --git a/36525-t/images/fig52.png b/36525-t/images/fig52.png
new file mode 100644
index 0000000..36e1ee5
--- /dev/null
+++ b/36525-t/images/fig52.png
diff --git a/36525-t/images/fig53.png b/36525-t/images/fig53.png
new file mode 100644
index 0000000..4a2ba90
--- /dev/null
+++ b/36525-t/images/fig53.png
diff --git a/36525-t/images/fig54.png b/36525-t/images/fig54.png
new file mode 100644
index 0000000..3677b93
--- /dev/null
+++ b/36525-t/images/fig54.png
diff --git a/36525-t/images/fig55.png b/36525-t/images/fig55.png
new file mode 100644
index 0000000..4825ca8
--- /dev/null
+++ b/36525-t/images/fig55.png
diff --git a/36525-t/images/fig56.png b/36525-t/images/fig56.png
new file mode 100644
index 0000000..60500d3
--- /dev/null
+++ b/36525-t/images/fig56.png
diff --git a/36525-t/images/fig57.png b/36525-t/images/fig57.png
new file mode 100644
index 0000000..504e454
--- /dev/null
+++ b/36525-t/images/fig57.png
diff --git a/36525-t/images/fig58.png b/36525-t/images/fig58.png
new file mode 100644
index 0000000..bfcfc5b
--- /dev/null
+++ b/36525-t/images/fig58.png
diff --git a/36525-t/images/fig59.png b/36525-t/images/fig59.png
new file mode 100644
index 0000000..68559cb
--- /dev/null
+++ b/36525-t/images/fig59.png
diff --git a/36525-t/images/fig60.png b/36525-t/images/fig60.png
new file mode 100644
index 0000000..a09eb64
--- /dev/null
+++ b/36525-t/images/fig60.png
diff --git a/36525-t/images/fig61.png b/36525-t/images/fig61.png
new file mode 100644
index 0000000..ba1c971
--- /dev/null
+++ b/36525-t/images/fig61.png
diff --git a/36525-t/images/fig62.png b/36525-t/images/fig62.png
new file mode 100644
index 0000000..5cb1a2a
--- /dev/null
+++ b/36525-t/images/fig62.png
diff --git a/36525-t/images/fig63.png b/36525-t/images/fig63.png
new file mode 100644
index 0000000..5ee4b0e
--- /dev/null
+++ b/36525-t/images/fig63.png
diff --git a/36525-t/images/fig64.png b/36525-t/images/fig64.png
new file mode 100644
index 0000000..5c4ab0a
--- /dev/null
+++ b/36525-t/images/fig64.png
diff --git a/36525-t/images/fig65.png b/36525-t/images/fig65.png
new file mode 100644
index 0000000..00e76b4
--- /dev/null
+++ b/36525-t/images/fig65.png
diff --git a/36525-t/images/fig66.png b/36525-t/images/fig66.png
new file mode 100644
index 0000000..c069181
--- /dev/null
+++ b/36525-t/images/fig66.png
diff --git a/36525-t/images/fig67.png b/36525-t/images/fig67.png
new file mode 100644
index 0000000..8f99258
--- /dev/null
+++ b/36525-t/images/fig67.png
diff --git a/36525-t/images/fig68.png b/36525-t/images/fig68.png
new file mode 100644
index 0000000..66930be
--- /dev/null
+++ b/36525-t/images/fig68.png
diff --git a/36525-t/images/fig69.png b/36525-t/images/fig69.png
new file mode 100644
index 0000000..864b7e6
--- /dev/null
+++ b/36525-t/images/fig69.png
diff --git a/36525-t/images/fig70.png b/36525-t/images/fig70.png
new file mode 100644
index 0000000..bae7baf
--- /dev/null
+++ b/36525-t/images/fig70.png
diff --git a/36525-t/images/fig71.png b/36525-t/images/fig71.png
new file mode 100644
index 0000000..3eb9c4f
--- /dev/null
+++ b/36525-t/images/fig71.png
diff --git a/36525-t/images/fig72.png b/36525-t/images/fig72.png
new file mode 100644
index 0000000..0c38244
--- /dev/null
+++ b/36525-t/images/fig72.png
diff --git a/36525-t/images/fig73.png b/36525-t/images/fig73.png
new file mode 100644
index 0000000..b5d0fd5
--- /dev/null
+++ b/36525-t/images/fig73.png
diff --git a/36525-t/images/fig74.png b/36525-t/images/fig74.png
new file mode 100644
index 0000000..c0e689d
--- /dev/null
+++ b/36525-t/images/fig74.png
diff --git a/36525-t/images/fig75.png b/36525-t/images/fig75.png
new file mode 100644
index 0000000..fbec2e1
--- /dev/null
+++ b/36525-t/images/fig75.png
diff --git a/36525-t/images/fig76.png b/36525-t/images/fig76.png
new file mode 100644
index 0000000..9fed3f4
--- /dev/null
+++ b/36525-t/images/fig76.png
diff --git a/36525-t/images/fig77.png b/36525-t/images/fig77.png
new file mode 100644
index 0000000..bdf8ee0
--- /dev/null
+++ b/36525-t/images/fig77.png
diff --git a/36525-t/images/fig78.png b/36525-t/images/fig78.png
new file mode 100644
index 0000000..14b649a
--- /dev/null
+++ b/36525-t/images/fig78.png
diff --git a/36525-t/images/fig79.png b/36525-t/images/fig79.png
new file mode 100644
index 0000000..f9e60f0
--- /dev/null
+++ b/36525-t/images/fig79.png
diff --git a/36525-t/images/fig80.png b/36525-t/images/fig80.png
new file mode 100644
index 0000000..767a725
--- /dev/null
+++ b/36525-t/images/fig80.png
diff --git a/36525-t/images/fig81.png b/36525-t/images/fig81.png
new file mode 100644
index 0000000..a28db36
--- /dev/null
+++ b/36525-t/images/fig81.png
diff --git a/36525-t/images/fig82.png b/36525-t/images/fig82.png
new file mode 100644
index 0000000..fb5514f
--- /dev/null
+++ b/36525-t/images/fig82.png
diff --git a/36525-t/images/fig83.png b/36525-t/images/fig83.png
new file mode 100644
index 0000000..8505783
--- /dev/null
+++ b/36525-t/images/fig83.png
diff --git a/36525-t/images/fig84.png b/36525-t/images/fig84.png
new file mode 100644
index 0000000..d0aa1f4
--- /dev/null
+++ b/36525-t/images/fig84.png
diff --git a/36525-t/images/fig85.png b/36525-t/images/fig85.png
new file mode 100644
index 0000000..090e8df
--- /dev/null
+++ b/36525-t/images/fig85.png
diff --git a/36525-t/images/fig86.png b/36525-t/images/fig86.png
new file mode 100644
index 0000000..2f88afa
--- /dev/null
+++ b/36525-t/images/fig86.png
diff --git a/36525-t/images/fig87.png b/36525-t/images/fig87.png
new file mode 100644
index 0000000..6ba9505
--- /dev/null
+++ b/36525-t/images/fig87.png
diff --git a/36525-t/images/fig88.png b/36525-t/images/fig88.png
new file mode 100644
index 0000000..680509e
--- /dev/null
+++ b/36525-t/images/fig88.png
diff --git a/36525-t/images/fig89.png b/36525-t/images/fig89.png
new file mode 100644
index 0000000..a0768bf
--- /dev/null
+++ b/36525-t/images/fig89.png
diff --git a/36525-t/images/fig90.png b/36525-t/images/fig90.png
new file mode 100644
index 0000000..c2bbc76
--- /dev/null
+++ b/36525-t/images/fig90.png
diff --git a/36525-t/images/fig91.png b/36525-t/images/fig91.png
new file mode 100644
index 0000000..2f5bce2
--- /dev/null
+++ b/36525-t/images/fig91.png
diff --git a/36525-t/images/fig92.png b/36525-t/images/fig92.png
new file mode 100644
index 0000000..3652427
--- /dev/null
+++ b/36525-t/images/fig92.png
diff --git a/36525-t/images/fig93.png b/36525-t/images/fig93.png
new file mode 100644
index 0000000..3ad5411
--- /dev/null
+++ b/36525-t/images/fig93.png
diff --git a/36525-t/images/fig94.png b/36525-t/images/fig94.png
new file mode 100644
index 0000000..b6d12d2
--- /dev/null
+++ b/36525-t/images/fig94.png
diff --git a/36525-t/images/fig95.png b/36525-t/images/fig95.png
new file mode 100644
index 0000000..9f83665
--- /dev/null
+++ b/36525-t/images/fig95.png
diff --git a/36525-t/images/fig96.png b/36525-t/images/fig96.png
new file mode 100644
index 0000000..936e80a
--- /dev/null
+++ b/36525-t/images/fig96.png
diff --git a/36525-t/images/fig97.png b/36525-t/images/fig97.png
new file mode 100644
index 0000000..250a28f
--- /dev/null
+++ b/36525-t/images/fig97.png
diff --git a/36525-t/images/fig98.png b/36525-t/images/fig98.png
new file mode 100644
index 0000000..dd3dbbc
--- /dev/null
+++ b/36525-t/images/fig98.png
diff --git a/36525-t/images/fig99.png b/36525-t/images/fig99.png
new file mode 100644
index 0000000..3e4349b
--- /dev/null
+++ b/36525-t/images/fig99.png
diff --git a/LICENSE.txt b/LICENSE.txt
new file mode 100644
index 0000000..6312041
--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,11 @@
+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
diff --git a/README.md b/README.md
new file mode 100644
index 0000000..6408137
--- /dev/null
+++ b/README.md
@@ -0,0 +1,2 @@
+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #36525 (https://www.gutenberg.org/ebooks/36525)