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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/13692-pdf.pdf b/13692-pdf.pdf Binary files differnew file mode 100644 index 0000000..7316a7f --- /dev/null +++ b/13692-pdf.pdf diff --git a/13692-t.zip b/13692-t.zip Binary files differnew file mode 100644 index 0000000..f1883bb --- /dev/null +++ b/13692-t.zip diff --git a/13692-t/13692-t.tex b/13692-t/13692-t.tex new file mode 100644 index 0000000..21e17c4 --- /dev/null +++ b/13692-t/13692-t.tex @@ -0,0 +1,5275 @@ +%%================================================ +%% this file = nickalls13692vers4.tex --> fixes the index +%% Project Gutenberg book No 13692 +%%================================================ +%% The gutenberg file 13692-t.tex has been modified by +%% RWD Nickalls (dick@nickalls.org) June 26, 2018 +%% in order to implement the Index and TOC correctly +%% ---search for `RWD Nickalls' to see the changes i have made +%% in this LaTeX file. +%% +%% This work was done on a Debian Linux platform using the 2017 TeXLive system. +%% For details of LaTeX see: www.tug.org. +%% +%% To use the hyperlinks use Acrobat Reader for best results +%% (because many ordinary pdf viewers fail to implement all the pdf tools) +%%============================================== +\documentclass[oneside]{book} + +\usepackage[latin1]{inputenc} +\usepackage[reqno]{amsmath} +\usepackage{amssymb,graphicx,yfonts} +\usepackage{makeidx} +\makeindex +\renewcommand{\chaptername}{Article} +\DeclareMathOperator{\am}{am} +\DeclareMathOperator{\amh}{amh} +\DeclareMathOperator{\cg}{cg} +\DeclareMathOperator{\cn}{cn} +\DeclareMathOperator{\csch}{csch} +\DeclareMathOperator{\dn}{dn} +\DeclareMathOperator{\gd}{gd} +\DeclareMathOperator{\limdot}{lim.} +\DeclareMathOperator{\moddot}{mod.} +\DeclareMathOperator{\sech}{sech} +\DeclareMathOperator{\sg}{sg} +\DeclareMathOperator{\sn}{sn} +\DeclareMathOperator{\tg}{tg} + + +%%%=======================begin{Nickalls}====================================== +\usepackage{comment} +\begin{comment} +%% +%% RWD Nickalls +%% EMAIL: dick@nickalls.org +%% July 5, 2018 +%% +Original gutenberg file modified by RWD Nickalls (dick@nickalls.org) July 2018 +to facilitate makeindex, hyperref links and better margin alignment, +and to enable working Index entry in TOC and in PDF bookmarks + +In order to see all changes by RWD Nickalls, then search the +.tex file for instances of RWD Nickalls + +%%============introduction======================== +%% These commands help make makeindex work correctly with hyperref. +%% Really one has to either (a) edit the .ind file (awkward to do on the fly) +%% or (b) edit the <theindex> environment to include the \columnseprule and the +%% \addcontentsline{toc}... line. Here I have fashioned a +%% new working <theindex> environment to be used with the .tex file. +%% +%%=============notes============================== + +%% The standard environment <theindex> is defined in the LaTeX book.cls. +%% This has been modified it by Springer (for their open source svmono.cls, which +%% is one of my favourite classes). +%% I have further modified it by (a) adding \columnseprule, and (b) deleting several +%% of the Springer commands so it works OK here in the gutenberg case (book 13692): +%% +%% NOTE that the \addcontentsline{toc}... needs to be /inside/ the <theindex> environment +%% to work properly, as here: +%% I don't fully understand how all these commands work, but they fix the gutenberg +%% index problems OK. +%% +%% to make the index work from the PDF bookmarks +%% (1) put \printindex command AFTER \backmatter +%% +%% to create the pdf file with hyperlinks and index using +%% the makeindex TeX utlilty: +%% (A) run pdflatex TWICE +%% (B) run the command: makeindex nickalls13692vers4.idx +%% (C) run pdflatex TWICE again. +%% +%% see the book: The LaTeX Companion (2004), 2nd ed: (makeindex = p 649) +%% see the TUG website (www.tug.org) for details of LaTeX implementations +%% eg TeXLive and the annual TeX Collection DVD for all platforms. +%% +%% +%% **Remaining problems**: +%% The hyperlinks from text (in the articles) to particular pages mostly fail +%% to locate the correct page --- this is because +%% sections \& subsections are not used in this book (ie these page links home in on the +%% first page of the associated ``chapter'' etc. since Chapters are the only systematic +%% reference points used in this LaTeX typesetting +%% NB I have fixed one series of pagerefs as an example, (see Line 2510) and have left +%% some notes there (search for \subsubsection{} ). +%% The failure of the links to the Tables in the ListofTables is due to +%% the same reason---I have now fixed these too. +%% I have also fixed some indexing errors as well. +%% +%% Please send any feedback to Dick Nickalls (dick@nickalls.org). +\end{comment} + + +%%============ +\makeatletter +%% RWD Nickalls (dick@nickalls.org) July 2018 +%% +%% The following commands were extracted (and modified) +%% from: svmono.cls - v 4.17, 2006 (Springer) +%% an open-source monograph LaTeX class freely available from +%% the Springer website. +%% +\def\indexname{Index}% +%% +\renewenvironment{theindex}{% + \columnseprule=0.4pt %% this makes the vertical line in the index + \columnsep 1cc + \@nobreaktrue + \begin{multicols}{2}[\chapter*{\indexname}]% + \markboth{\MakeUppercase\indexname}{\MakeUppercase\indexname}% + \addcontentsline{toc}{chapter}{\indexname}% + \flushbottom + \parindent\z@ + \rightskip\z@ \@plus 40\p@ + \parskip\z@ \@plus .3\p@\relax + \flushbottom + \let\item\@idxitem + \def\,{\relax\ifmmode\mskip\thinmuskip + \else\hskip0.2em\ignorespaces\fi}% + \normalfont} + {\end{multicols}\clearpage} +%% +%% +\def\idxquad{\hskip 10\p@}% space that divides entry from number +%% +\def\@idxitem{\par\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax} + +\def\subitem{\par\noindent\setbox0=\hbox{--\enspace}% second order + \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax}% indexentry + +\def\subsubitem{\par\noindent\setbox0=\hbox{--\,--\enspace}% third order + \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax}% indexentry + +\def\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} +%% +\makeatother +%=========== + +\usepackage{multicol} %% required by makeindex to make 2 cols +\usepackage{ifpdf} +\ifpdf + \usepackage[unicode]{hyperref} + \hypersetup{hyperindex, + hyperfigures=true, + hyperfootnotes, + colorlinks, + urlcolor=blue, + linkcolor=blue, + bookmarksopenlevel=0, %%was 1, 0, %% (parts=-1, book=0, chap=1, article=2) + bookmarksnumbered=true, + bookmarkstype=toc, + pdfpagelayout=SinglePage, + %--------------------- + pdftitle={Hyperbolic functions}, + pdfauthor={James McMahon (1906) }, + pdfsubject={mathematics}, + pdfkeywords={Project Gutenberg, book No=13692}, + pdfcreator=pdfLaTeX, + } + \usepackage[verbose]{microtype} %% for better margin alignment + \usepackage{cmap} %% makes the pdf searchable +\fi +%%==============end{Nickalls}==================================================== + + + + +\begin{document} + +\thispagestyle{empty} +\small +\begin{verbatim} + +The Project Gutenberg EBook of Hyperbolic Functions, by James McMahon + +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.org + + +Title: Hyperbolic Functions + +Author: James McMahon + +Release Date: October 10, 2004 [EBook #13692] + +Language: English + +Character set encoding: TeX + +*** START OF THIS PROJECT GUTENBERG EBOOK HYPERBOLIC FUNCTIONS *** + + + + +Produced by David Starner, Joshua Hutchinson, John Hagerson, +and the Project Gutenberg On-line Distributed Proofreading Team. + + + + + + +\end{verbatim} +\normalsize +\newpage + +\index{Equations!Differential|see{Differential Equation}} +\index{Function!anti-gudermanian|see{Anti-gu\-der\-man\-i\-an}} +\index{Function!anti-hyperbolic|see{Anti-hy\-per\-bo\-lic +functions}} +\index{Function!circular|see{Circular functions}} +\index{Function!elliptic|see{Elliptic functions}} +\index{Function!gudermanian|see{Gudermanian function}} +\index{Imaginary|see{Complex}} + +\frontmatter + +%%=========================================== +%% RWD Nickalls (dick@nickalls.org) +\ifpdf\pdfbookmark[0]{Title}{Title}\fi %% 1 +%%=========================================== + +\begin{center} +\noindent \Large MATHEMATICAL MONOGRAPHS. + +\bigskip \footnotesize{\textsc{edited by}} \\ +\normalsize \textsc{MANSFIELD MERRIMAN and ROBERT S. WOODWARD.} + +\bigskip\bigskip\huge + +%------------------------ +%% RWD Nickalls (July 2018) +%% use a backslash after the No.\ to force an inter word space here +%% ie not an end-of-sentence space here. +No.\ 4. +%----------------------- + +\bigskip +HYPERBOLIC FUNCTIONS. + +\bigskip\bigskip\footnotesize\textsc{by} \\ +\bigskip\large JAMES McMAHON, \\ +\footnotesize\textsc{Professor of Mathematics in Cornell +University.} + +\bigskip\bigskip\normalsize NEW YORK: \\ +\medskip JOHN WILEY \& SONS. \\ +\medskip \textsc{London: CHAPMAN \& HALL, Limited.} \\ +\medskip 1906. + +\bigskip\bigskip +\tiny \textsc{Copyright 1896} \\ +\textsc{by} \\ +\textsc{MANSFIELD MERRIMAN and ROBERT S. WOODWARD} \\ +\textsc{under the title} \\ +\textsc{HIGHER MATHEMATICS} \normalsize +\end{center} + +\bigskip\bigskip +\scriptsize \noindent \textsc{Transcriber's Note:} \emph{I did my +best to recreate the index.} \normalsize + +\newpage + +\fbox{\parbox{11cm}{ +\begin{center} +\textbf{MATHEMATICAL MONOGRAPHS.} \\ +\small\textsc{edited by}\normalsize \\ +\textbf{Mansfield Merriman and Robert S. Woodward.} \\ +\smallskip \footnotesize \textbf{Octavo. Cloth.} \\ +\end{center} +\begin{tabbing} +No. 99999\= \kill +\textbf{No. 1.}\>\textbf{History of Modern Mathematics.} \\ +\>By \textsc{David Eugene Smith.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 2.}\>\textbf{Synthetic Projective Geometry.} \\ +\>By \textsc{George Bruce Halsted.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 3.}\>\textbf{Determinants.} \\ +\>By \textsc{Laenas Gifford Weld.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 4.}\>\textbf{Hyperbolic Functions.} \\ +\>By \textsc{James McMahon.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 5.}\>\textbf{Harmonic Functions.} \\ +\>By \textsc{William E. Byerly.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 6.}\>\textbf{Grassmann's Space Analysis.} \\ +\>By \textsc{Edward W. Hyde.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 7.}\>\textbf{Probability and Theory of Errors.} \\ +\>By \textsc{Robert S. Woodward.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 8.}\>\textbf{Vector Analysis and Quaternions.} \\ +\>By \textsc{Alexander Macfarlane.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 9.}\>\textbf{Differential Equations.} \\ +\>By \textsc{William Woolsey Johnson.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 10.}\>\textbf{The Solution of Equations.} \\ +\>By \textsc{Mansfield Merriman.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 11.}\>\textbf{Functions of a Complex Variable.} \\ +\>By \textsc{Thomas S. Fiske.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 12.}\>\textbf{The Theory of Relativity.} \\ +\>By \textsc{Robert D. Carmichael.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 13.}\>\textbf{The Theory of Numbers.} \\ +\>By \textsc{Robert D. Carmichael.} \$1.00 \emph{net}. \\ +\smallskip +\textbf{No. 14.}\>\textbf{Algebraic Invariants.} \\ +\>By \textsc{Leonard E. Dickson.} \$1.25 \emph{net}. \\ +\end{tabbing} +\begin{center} +\smallskip \normalsize PUBLISHED BY \\ +\smallskip \textbf{JOHN WILEY \& SONS, Inc., NEW YORK. \\ +CHAPMAN \& HALL, Limited, LONDON.} +\end{center}}} + +\chapter{Editors' Preface.} + +The volume called Higher Mathematics, the first edition of which was +published in 1896, contained eleven chapters by eleven authors, each +chapter being independent of the others, but all supposing the +reader to have at least a mathematical training equivalent to that +given in classical and engineering colleges. The publication of that +volume is now discontinued and the chapters are issued in separate +form. In these reissues it will generally be found that the +monographs are enlarged by additional articles or appendices which +either amplify the former presentation or record recent advances. +This plan of publication has been arranged in order to meet the +demand of teachers and the convenience of classes, but it is also +thought that it may prove advantageous to readers in special lines +of mathematical literature. + +It is the intention of the publishers and editors to add other +monographs to the series from time to time, if the call for the same +seems to warrant it. Among the topics which are under consideration +are those of elliptic functions, the theory of numbers, the group +theory, the calculus of variations, and non-Euclidean geometry; +possibly also monographs on branches of astronomy, mechanics, and +mathematical physics may be included. It is the hope of the editors +that this form of publication may tend to promote mathematical study +and research over a wider field than that which the former volume +has occupied. + +\smallskip \footnotesize December, 1905. \normalsize + +\chapter{Author's Preface.} + +This compendium of hyperbolic trigonometry was first published as a +chapter in Merriman and Woodward's Higher Mathematics. There is +reason to believe that it supplies a need, being adapted to two or +three different types of readers. College students who have had +elementary courses in trigonometry, analytic geometry, and +differential and integral calculus, and who wish to know something +of the hyperbolic trigonometry on account of its important and +historic relations to each of those branches, will, it is hoped, +find these relations presented in a simple and comprehensive way in +the first half of the work. Readers who have some interest in +imaginaries are then introduced to the more general trigonometry of +the complex plane, where the circular and hyperbolic functions merge +into one class of transcendents, the singly periodic functions, +having either a real or a pure imaginary period. For those who also +wish to view the subject in some of its practical relations, +numerous applications have been selected so as to illustrate the +various parts of the theory, and to show its use to the physicist +and engineer, appropriate numerical tables being supplied for these +purposes. + +With all these things in mind, much thought has been given to the +mode of approaching the subject, and to the presentation of +fundamental notions, and it is hoped that some improvements are +discernible. For instance, it has been customary to define the +hyperbolic functions in relation to a sector of the rectangular +hyperbola, and to take the initial radius of the sector coincident +with the principal radius of the curve; in the present work, these +and similar restrictions are discarded in the interest of analogy +and generality, with a gain in symmetry and simplicity, and the +functions are defined as certain characteristic ratios belonging to +any sector of any hyperbola. Such definitions, in connection with +the fruitful notion of correspondence of points on conics, lead to +simple and general proofs of the addition-theorems, from which +easily follow the conversion-formulas, the derivatives, the +Maclaurin expansions, and the exponential expressions. The proofs +are so arranged as to apply equally to the circular functions, +regarded as the characteristic ratios belonging to any elliptic +sector. For those, however, who may wish to start with the +exponential expressions as the definitions of the hyperbolic +functions, the appropriate order of procedure is indicated on +page~\pageref{def hyper as exp}, and a direct mode of bringing such +exponential definitions into geometrical relation with the +hyperbolic sector is shown in the Appendix. + +\enlargethispage*{1000pt} +\smallskip \footnotesize December, 1905. \normalsize + + +%%=========================================== +%% RWD Nickalls (dick@nickalls.org) +\cleardoublepage +\ifpdf\pdfbookmark[0]{Contents}{Contents}\fi +%%=========================================== + +\tableofcontents +\listoftables + +%% ART. 1. CORRESPONDENCE OF POINTS ON CONICS ...Page 7 +%% 2. AREAS OF CORRESPONDING TRIANGLES ...9 +%% 3. AREAS OF CORRESPONDING SECTORS ...9 +%% 4. CHARACTERISTIC RATIOS OF SECTORIAL MEASURES ...10 +%% 5. RATIOS EXPRESSED AS TRIANGLE-MEASURES ...10 +%% 6. FUNCTIONAL RELATIONS FOR ELLIPSE ...11 +%% 7. FUNCTIONAL RELATIONS FOR HYPERBOLA ...11 +%% 8. RELATIONS BETWEEN HYPERBOLIC FUNCTIONS ...12 +%% 9. VARIATIONS OF THE HYPERBOLIC FUNCTIONS ...14 +%% 10. ANTI HYPERBOLIC FUNCTIONS ...16 +%% 11. FUNCTIONS OF SUMS AND DIFFERENCES ...16 +%% 12. CONVERSION FORMULAS ...18 +%% 13. LIMITING RATIOS ...19 +%% 14. DERIVATIVES OF HYPERBOLIC FUNCTIONS ...20 +%% 15. DERIVATIVES OF ANTI-HYPERBOLIC FUNCTIONS ...22 +%% 16. EXPANSION OF HYPERBOLIC FUNCTIONS ...23 +%% 17. EXPONENTIAL EXPRESSIONS ...24 +%% 18. EXPANSION OF ANTI-FUNCTIONS ...25 +%% 19. LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS ...27 +%% 20. THE GUDERMANIAN FUNCTION ...28 +%% 21. CIRCULAR FUNCTIONS OF GUDERMANIAN ...28 +%% 22. GUDERMANIAN ANGLE ...29 +%% 23. DERIVATIVES OF GUDERMANIAN AND INVERSE ...30 +%% 24. SERIES FOR GUDERMANIAN AND ITS INVERSE ...31 +%% 25. GRAPHS OF HYPERBOLIC FUNCTIONS ...32 +%% 26. ELEMENTARY INTEGRALS ...35 +%% 27. FUNCTIONS OF COMPLEX NUMBERS ..38 +%% 28. ADDITION THEOREMS FOR COMPLEXES ...40 +%% 29. FUNCTIONS OF PURE IMAGINARIES ...41 +%% 30. FUNCTIONS OF \emph{x + iy} IN THE FORM \emph{X + iY} ...43 +%% 31. THE CATENARY ...47 +%% 32. THE CATENARY OF UNIFORM STRENGTH ...49 +%% 33. THE ELASTIC CATENARY ...50 +%% 34. THE TRACTORY ...51 +%% 35. THE LOXODROME ...52 +%% 36 COMBINED FLEXURE AND TENSION ...53 +%% 37. ALTERNATING CURRENTS ...55 +%% 38. MISCELLANEOUS APPLICATIONS ...60 +%% 39. EXPLANATION OF TABLES ...62 +%% +%% TABLE I. HYPERBOLIC FUNCTIONS ...64 +%% II. VALUES OF \textsc{cosh}(\emph{x+iy}) +%% AND \textsc{sinh}(\emph{x+iy}) ...66 +%% III. VALUES OF gd\emph{u} AND $0^\circ$ ...70 +%% IV. VALUES OF gd\emph{u}, \textsc{log sinh} \emph{u}, +%% \textsc{log cosh} \emph{u} ...70 +%% +%% APPENDIX. HISTORICAL AND BIBLIOGRAPHICAL ...71 +%% EXPONENTIAL EXPRESSIONS AS DEFINITIONS ...72 +%% +%% INDEX ...73 + +\mainmatter +\chapter{Correspondence of Points on Conics.}% +\index{Corresponding points!on conics}% +\index{Geometrical treatment of hyperbolic functions|(}% +\index{Hyperbola|(} + +To prepare the way for a general treatment of the hyperbolic +functions a preliminary discussion is given on the relations, +between hyperbolic sectors. The method adopted is such as to apply +at the same time to sectors of the ellipse, including the circle; +and the analogy of the hyperbolic and circular functions will be +obvious at every step, since the same set of equations can be read +in connection with either the hyperbola or the ellipse.\footnote{ +The hyperbolic functions are not so named on account of any analogy +with what are termed Elliptic Functions. ``The elliptic integrals, +and thence the elliptic functions, derive their name from the early +attempts of mathematicians at the rectification of the +ellipse.\,\ldots To a certain extent this is a disadvantage; \ldots\ +because we employ the name hyperbolic function to denote $\cosh u, +\sinh u$, etc., by analogy with which the elliptic functions would +be merely the circular functions $\cos \phi, \sin \phi$, +etc.\,\ldots'' (Greenhill, Elliptic Functions, p.\ +175.)\label{fnp7}}\index{Greenhill's!Elliptic Functions} It is +convenient to begin with the theory of correspondence of points on +two central conics of like species, i.e. +either both ellipses or both hyperbolas.% +\index{Circular functions}\index{Elliptic!functions}% +\index{Elliptic!integrals}\index{Elliptic!sectors} + +\begin{center} +\includegraphics[width=80mm]{fig01.png} +\end{center} + +To obtain a definition of corresponding points, let $O_1A_1, O_1B_1$ +be conjugate radii of a central conic, and $O_2A_2, O_2B_2$ +conjugate radii of any other central conic of the same species; let +$P_1, P_2$ be two points on the curves; and let their coordinates +referred to the respective pairs of conjugate directions be $(x_1, +y_1), (x_2, y_2)$; then, by analytic geometry, +\begin{equation} +\frac{x_1^2}{a_1^2} \pm \frac{y_1^2}{b_1^2} = 1,\qquad +\frac{x_2^2}{a_2^2} \pm \frac{y_2^2}{b_2^2} = 1. \tag{1} +\end{equation} +Now if the points $P_1, P_2$ be so situated that +\begin{equation} +\frac{x_1}{a_1} = \frac{x_2}{a_2},\qquad +\frac{y_1}{b_1} = \frac{y_2}{b_2}, \tag{2} +\end{equation} +the equalities referring to sign as well as magnitude, then $P_1, +P_2$ are called corresponding points in the two systems. If $Q_1, +Q_2$ be another pair of correspondents, then the sector and triangle +$P_1O_1Q_1$ are said to correspond respectively with the sector and +triangle $P_2O_2Q_2$. These definitions will apply also when the +conies coincide, the points $P_1, P_2$ being then referred to any +two pairs of conjugate diameters of the same conic. + +In discussing the relations between corresponding areas it is +convenient to adopt the following use of the word ``measure'': The +measure of any area connected with a given central conic is the +ratio which it bears to the constant area of the triangle formed by +two conjugate diameters of the same conic.% +\index{Areas}\index{Measure!defined} + +\index{Measure!of sector|(}For example, the measure of the sector +$A_1O_1P_1$ is the ratio +\begin{equation*} +\frac{\text{sector }A_1O_1P_1}{\text{triangle }A_1O_1B_1} +\end{equation*} +and is to be regarded as positive or negative according as +$A_1O_1P_1$ and $A_1O_1B_1$ are at the same or opposite sides of +their common initial line.\index{Hyperbola|)} + +\chapter{Areas of Corresponding Triangles.}% +\index{Areas} + +The areas of corresponding triangles have equal measures. For, let +the coordinates of $P_1, Q_1$ be $(x_1, y_1), (x'_1, y'_1)$, and let +those of their correspondents $P_2, Q_2$ be $(x_2, y_2), (x'_2, +y'_2)$; let the triangles $P_1O_1Q_1, P_2O_2Q_2$ be $T_1, T_2$, and +let the measuring triangles $A_1O_1B_1, A_2O_2B_2$ be $K_1, K_2$, +and their angles $\omega_1, \omega_1$; then, by analytic geometry, +taking account of both magnitude and direction of angles, areas, and +lines, +\begin{gather*} +\begin{aligned} +\frac{T_1}{K_1} &= + \frac{\frac{1}{2}(x_1y'_1 - x'_1y_1)\sin\omega_1} + {\frac{1}{2} a_1b_1\sin\omega_1} = + \frac{x_1}{a_1} \cdot \frac{y'_1}{b_1} - + \frac{x'_1}{a_1} \cdot \frac{y_1}{b_1}; \\ +\frac{T_2}{K_2} &= + \frac{\frac{1}{2}(x_2y'_2 - x'_2y_2)\sin\omega_2} + {\frac{1}{2} a_2b_2\sin\omega_2} = + \frac{x_2}{a_2} \cdot \frac{y'_2}{b_2} - + \frac{x'_2}{a_2} \cdot \frac{y_2}{b_2}. +\end{aligned} \\ +\intertext{Therefore, by (2),} +\frac{T_1}{K_1} = \frac{T_2}{K_2}. \tag{3} +\end{gather*}\index{Corresponding points!on sectors and triangles} + +\chapter{Areas of Corresponding Sectors.}\index{Sectors of conics} + +The areas of corresponding sectors have equal measures. For conceive +the sectors $S_1, S_2$ divided up into infinitesimal corresponding +sectors; then the respective infinitesimal corresponding triangles +have equal measures (Art.~2); but the given sectors are the limits +of the sums of these infinitesimal triangles, hence +\begin{equation*} +\frac{S_1}{K_1} = \frac{S_2}{K_2}. \tag{4} +\end{equation*} + +In particular, the sectors $A_1O_1P_1, A_2O_2P_2$ have equal +measures; for the initial points $A_1, A_2$ are corresponding +points. + +It may be proved conversely by an obvious reductio ad absurdum that +if the initial points of two equal-measured sectors correspond, then +their terminal points correspond. + +Thus if any radii $O_1A_1, O_2A_2$ be the initial lines of two +equal-measured sectors whose terminal radii are $O_1P_1, O_2P_2$, +then $P_1, P_2$ are corresponding points referred respectively to +the pairs of conjugate directions $O_1A_1, O_1B_1$, and $O_2A_2, +O_2A_B$; that is, +\begin{equation*} +\frac{x_1}{a_1} = \frac{x_2}{a_2},\quad +\frac{y_1}{b_1} = \frac{y_2}{b_2}. +\end{equation*} + +\small \begin{enumerate} +\item[Prob.~1.] Prove that the sector $P_1O_1Q_1$, is bisected by the +line joining $O_1$, to the mid-point of $P_1Q_1$. (Refer the points +$P_1, Q_1$, respectively, to the median as common axis of $x$, and +to the two opposite conjugate directions as axis of $y$, and show +that $P_1, Q_1$ are then corresponding points.) + +\item[Prob.~2.] Prove that the measure of a circular sector is equal +to the radian measure of its angle. + +\item[Prob.~3.] Find the measure of an elliptic quadrant, and of the +sector included by conjugate radii. +\end{enumerate} \normalsize + +\chapter{Charactersitic Ratios of Sectorial Measures.} + +Let $A_1O_1P_1 = S_1$, be any sector of a central conic; draw +$P_1M_1$ ordinate to $O_1A_1$, i.e.\ parallel to the tangent at +$A_1$; let $O_1M_1 = x_1, M_1P_1 = y_1, O_1A_1 = a_1$, and the +conjugate radius $O_1B_1 = b_1$; then the ratios $\dfrac{x_1}{a_1}, +\dfrac{y_1}{b_1}$ are called the characteristic ratios of the given +sectorial measure $\dfrac{S_1}{K_1}$. These ratios are constant both +in magnitude and sign for all sectors of the same measure and +species wherever these may be situated (Art.~3). Hence there exists +a functional relation between the sectorial measure and each of its +characteristic ratios.\label{sectoral-measures}\index{Characteristic +ratios}\index{Ratios!characteristic} + +\chapter{Ratios Expressed as Triangle-measures.} + +The triangle of a sector and its complementary triangle are measured +by the two characteristic ratios. For, let the triangle $A_1O_1P_1$ +and its complementary triangle $P_1O_1B_1$ be denoted by $T_1, +T'_1$; then +\begin{equation} +\left. +\begin{aligned} +\frac{T_1}{K_1} &=\frac{\frac{1}{2} a_1y_1 \sin\omega_1} + {\frac{1}{2} a_1b_1 \sin\omega_1} =\frac{y_1}{b_1}, +\\ +\frac{T'_1}{K_1}&=\frac{\frac{1}{2} b_1x_1 \sin\omega_1} + {\frac{1}{2} a_1b_1 \sin\omega_1} =\frac{x_1}{a_1}. +\end{aligned} +\right\} \tag{5} +\end{equation}\index{Complementary triangles}\index{Geometrical +treatment of hyperbolic functions|)} + +\chapter{Functional Relations for Ellipse.} + +\begin{center} +\includegraphics[width=60mm]{fig02.png} +\end{center} + +The functional relations that exist between the sectorial measure +and each of its characteristic ratios are the same for all elliptic, +including circular, sectors (Art.~4). Let $P_1, P_2$ be +corresponding points on an ellipse and a circle, referred to the +conjugate directions $O_1A_1, O_1B_1$ and $O_2A_2, O_2B_2$, the +latter pair being at right angles; let the angle $A_2O_2P_2 = +\theta$ in radian measure; then +\begin{gather*} +\frac{S_2}{K_2} = \frac{\frac{1}{2} a_2^2\theta}{\frac{1}{2} a_2^2} + = \theta. \tag{6} \\ +\therefore \frac{x_2}{a_2} = \cos \frac{S_2}{K_2}, \quad + \frac{y_2}{b_2} = \sin \frac{S_2}{K_2}; \qquad [ a_2 = b_2 \\ +\intertext{hence, in the ellipse, by Art.~3,} +\frac{x_1}{a_1} = \cos \frac{S_1}{K_1},\quad \frac{y_1}{b_1} = + \sin \frac{S_1}{K_1}. \tag{7} +\end{gather*}\index{Circular functions} + +\small \begin{enumerate} +\item[Prob.~4.] Given $x_1 = \tfrac{1}{2} a_1$; find the measure +of the elliptic sector $A_1O_1P_1$. Also find its area when $a_1 = +4, b_1 = 3, \omega = 60^\circ$. + +\item[Prob.~5.] Find the characteristic ratios of an elliptic +sector whose measure is $\frac{1}{4}\pi$. + +\item[Prob.~6.] Write down the relation between an elliptic +sector and its triangle. (See Art.~5.)\index{Measure!of sector|)} +\end{enumerate} \normalsize + +\chapter{Functional Relations for Hyperbola.}% +\index{Function!hyperbolic, defined}% +\index{Hyperbolic functions!defined}% +\index{Hyperbolic functions!relations among}% +\index{Relations among functions} + +The functional relations between a sectorial measure and its +characteristic ratios in the case of the hyperbola may be written in +the form +\begin{equation*} +\frac{x_1}{a_1} = \cosh \frac{S_1}{K_1},\quad +\frac{y_1}{b_1} = \sinh \frac{S_1}{K_1}, +\end{equation*} +and these express that the ratio of the two lines on the left is a +certain definite function of the ratio of the two areas on the +right. These functions are called by analogy the hyperbolic cosine +and the hyperbolic sine. Thus, writing $u$ for $\dfrac{S_1}{K_1}$ +the two equations +\begin{equation*} +\frac{x_1}{a_1} = \cosh u,\quad \frac{y_1}{b_1} = \sinh u \tag{8} +\end{equation*} +serve to define the hyperbolic cosine and sine of a given sectorial +measure $u$; and the hyperbolic tangent, cotangent, secant, and +cosecant are then defined as follows: +\begin{equation} +\left. +\begin{aligned} +\tanh u = \frac{\sinh u}{\cosh u}, &\quad +\coth u = \frac{\cosh u}{\sinh u},\\ +\sech u = \frac{ 1 }{\cosh u}, &\quad +\csch u = \frac{ 1 }{\sinh u}. +\end{aligned} +\right\} \tag{9} +\end{equation} + +The names of these functions may be read ``h-cosine,'' ''h-sine,'' +``h-tangent,'' etc., or ``hyper-cosine,'' etc. + +\chapter{Relations Among Hyperbolic Functions.} + +Among the six functions there are five independent relations, so +that when the numerical value of one of the functions is given, the +values of the other five can be found. Four of these relations +consist of the four defining equations (9). The fifth is derived +from the equation of the hyperbola +\begin{gather*} +\frac{x_1^2}{a_1^2} - \frac{y_1^2}{b_1^2} = 1,\\ +\intertext{giving} +\cosh^2 u - \sinh^2 u = 1. \tag{10} +\end{gather*} + +By a combination of some of these equations other subsidiary +relations may be obtained; thus, dividing (10) successively by +$\cosh^2 u, \sinh^2 u$, and applying (9), give +\begin{equation} +\left. +\begin{aligned} +1 - \tanh^2 u &= \sech^2 u, \\ +\coth^2 u - 1 &= \csch^2 u. +\end{aligned} +\right\} \tag{11} +\end{equation} + +Equations (9), (10), (11) will readily serve to express the value of +any function in terms of any other. For example, when $\tanh u$ is +given, +\begin{gather*} +\coth u = \frac{1}{\tanh u}, \quad \sech u = \sqrt{1 - \tanh^2 u}, \\ +\cosh u = \frac{ 1 }{\sqrt{1-\tanh^2 u}}, \quad +\sinh u = \frac{\tanh u}{\sqrt{1-\tanh^2 u}}, \\ +\csch u = \frac{\sqrt{1-\tanh^2 u}}{\tanh u}. +\end{gather*} + +The ambiguity in the sign of the square root may usually be removed +by the following considerations:% +\index{Ambiguity of value}\index{Multiple values} The functions +$\cosh u, \sech u$ are always positive, because the primary +characteristic ratio $\dfrac{x_1}{a_1}$ is positive, since the +initial line $O_1A_1$ and the abscissa $O_1M_1$ are similarly +directed from $O_1$ on whichever branch of the hyperbola $P_1$ maybe +situated; but the functions $\sinh u, \tanh u, \coth u, \csch u$, +involve the other characteristic ratio $\dfrac{y_1}{b_1}$, which is +positive or negative according as $y_1$ and $b_1$ have the same or +opposite signs, i.e., as the measure $u$ is positive or negative; +hence these four functions are either all positive or all negative. +Thus when any one of the functions $\sinh u, \tanh u, \csch u, \coth +u$, is given in magnitude and sign, there is no ambiguity in the +value of any of the six hyperbolic functions; but when either $\cosh +u$ or $\sech u$ is given, there is ambiguity as to whether the other +four functions shall be all positive or all negative. + +\begin{center} +\includegraphics[width=50mm]{fig03.png} +\end{center} + +The hyperbolic tangent may be expressed as the ratio of two lines. +For draw the tangent line $AC = t$; then +\begin{align*} +\tanh u &= \frac{y}{b} : \frac{x}{a} = \frac{a}b \cdot \frac{y}x \\ + &= \frac{a}{b} \cdot \frac{t}{a} = \frac{t}{b}. \tag{12} +\end{align*} + +The hyperbolic tangent is the measure of the triangle $OAC$. For +\begin{gather} +\frac{OAC}{OAB} = \frac{at}{ab} = \frac{t}{b} = \tanh u. \tag{13} +\end{gather} + +Thus the sector $AOP$, and the triangles $AOP, POB, AOC$, are +proportional to $u, \sinh u, \cosh u, \tanh u$ (eqs.\ 5, 13); hence +\begin{equation} +\sinh u > u > \tanh u. \tag{14} +\end{equation} + +\small \begin{enumerate} +\item[Prob.~7.] Express all the hyperbolic functions in terms of +$\sinh{u}$. Given $\cosh{u} = 2$, find the values of the other +functions. + +\item[Prob.~8.] Prove from eqs.\ 10, 11, that +$\cosh{u} > \sinh{u}, \cosh{u} > 1, \tanh{u} < 1, \sech{u} < 1$. + +\item[Prob.~9.] In the figure of Art.~1, let $OA = 2, OB = 1, +AOB = 60^{\circ}$, and area of sector $AOP = 3$; find the sectorial +measure, and the two characteristic ratios, in the elliptic sector, +and also in the hyperbolic sector; and find the area of the triangle +$AOP$. (Use tables of cos, sin, cosh, sinh.)% +\index{Areas} + +\item[Prob.~10.] Show that $\coth{u}, \sech{u}, \csch{u}$ may each +be expressed as the ratio of two lines, as follows: Let the tangent +at $P$ make on the conjugate axes $OA, OB$, intercepts $OS = m, OT = +n$; let the tangent at $B$, to the conjugate hyperbola, meet $OP$ in +$R$, making $BR = l$; then +\begin{equation*} +\coth{u} = \frac{l}{a},\quad \sech{u} = \frac{m}{a},\quad + \csch{u} = \frac{n}{b}. +\end{equation*} + +\item[Prob.~11.] The measure of segment $AMP$ is $\sinh{u}\cosh{u} - +u$. Modify this for the ellipse. Modify also eqs.\ 10--14, and +probs.\ 8, 10. +\end{enumerate} \normalsize + +\chapter{Variations of the Hyperbolic Functions.}% +\index{Variation of hyperbolic functions} + +\begin{center} +\includegraphics[width=40mm]{fig04.png}\label{ch9fig} +\end{center} + +Since the values of the hyperbolic functions depend only on the +sectorial measure, it is convenient, in tracing their variations, to +consider only sectors of one half of a rectangular hyperbola, whose +conjugate radii are equal, and to take the principal axis $OA$ as +the common initial line of all the sectors. The sectorial measure +$u$ assumes every value from $-\infty$, through $0$, to $+\infty$, +as the terminal point $P$ comes in from infinity on the lower +branch, and passes to infinity on the upper branch; that is, as the +terminal line $OP$ swings from the lower asymptotic position $y = +-x$, to the upper one, $y = x$. It is here assumed, but is proved in +Art.~17, that the sector $AOP$ becomes infinite as $P$ passes to +infinity. + +Since the functions $\cosh{u}, \sinh{u}, \tanh{u}$, for any position +of $OP$, are equal to the ratios of $x, y, t$, to the principal +radius $a$, it is evident from the figure that +\begin{equation} +\cosh 0 = 1,\quad \sinh 0 = 0,\quad \tanh 0 = 0, \tag{15} +\end{equation} +and that as $u$ increases towards positive infinity, $\cosh u, \sinh +u$ are positive and become infinite, but $\tanh u$ approaches unity +as a limit; thus +\begin{equation} +\cosh \infty = \infty,\quad +\sinh \infty = \infty,\quad +\tanh \infty = 1. \tag{16} +\end{equation} + +Again, as $u$ changes from zero towards the negative side, $\cosh u$ +is positive and increases from unity to infinity, but $\sinh u$ is +negative and increases numerically from zero to a negative infinite, +and $\tanh u$ is also negative and increases numerically from zero +to negative unity; hence +\begin{equation} +\cosh (-\infty) = \infty,\quad +\sinh (-\infty) = -\infty,\quad +\tanh (-\infty) = -1. \tag{17} +\end{equation} + +For intermediate values of $u$ the numerical values of these +functions can be found from the formulas of Arts.\ 16, 17, and are +tabulated at the end of this chapter. A general idea of their manner +of variation can be obtained from the curves in Art.~25, in which +the sectorial measure $u$ is represented by the abscissa, and the +values of the functions $\cosh u$, $\sinh u$, etc., are represented +by the ordinate. + +The relations between the functions of $-u$ and of $u$ are evident +from the definitions, as indicated above, and in Art.~8. Thus +\begin{equation} +\left. +\begin{aligned} +\cosh (-u) &= +\cosh u, &\quad \sinh (-u) &= -\sinh u, \\ +\sech (-u) &= +\sech u, &\quad \csch (-u) &= -\csch u, \\ +\tanh (-u) &= -\tanh u, &\quad \coth (-u) &= -\coth u. \\ +\end{aligned} +\right\} \tag{18} +\end{equation} + +\small \begin{enumerate} +\item[Prob.~12.] Trace the changes in $\sech u, \coth u, \csch u$, +as $u$ passes from $-\infty$ to $+\infty$. Show that $\sinh u, \cosh +u$ are infinites of the same order when $u$ is infinite. (It will +appear in Art.~17 that $\sinh u, \cosh u$ are infinites of an order +infinitely higher than the order of $u$.) + +\item[Prob.~13.] Applying eq.\ (12) to figure, +page~\pageref{ch9fig}, prove $\tanh u_1 = \tan AOP$. +\end{enumerate} \normalsize + +\chapter{Anti-hyperbolic Functions.}% +\index{Anti-hyperbolic functions} + +The equations $\dfrac{x}{a} = \cosh u, \dfrac{y}{b} = \sinh u, +\dfrac{t}{b} = \tanh u$, etc., may also be expressed by the inverse +notation $u = \cosh^{-1}\dfrac{x}{a}, u = \sinh^{-1}\dfrac{y}{b}, u += \tanh^{-1}\dfrac{t}{b}$, etc., which may be read: ``$u$ is the +sectorial measure whose hyperbolic cosine is the ratio $x$ to $a$,'' +etc.; or ``$u$ is the anti-h-cosine of $\dfrac{x}{a}$,'' etc. + +Since there are two values of $u$, with opposite signs, that +correspond to a given value of $\cosh u$, it follows that if $u$ be +determined from the equation $\cosh u = m$, where $m$ is a given +number greater than unity, $u$ is a two-valued function of $m$. The +symbol $\cosh^{-1} m$ will be used to denote the positive value of +$u$ that satisfies the equation $\cosh u = m$. Similarly the symbol +$\sech^{-1} m$ in will stand for the positive value of $u$ that +satisfies the equation $\sech u = m$. The signs of the other +functions $\sinh^{-1}m, \tanh^{-1}m, \coth^{-1}m, \csch^{-1}m$, are +the same as the sign of $m$. Hence all of the anti-hyperbolic +functions of real numbers are one-valued.% +\index{Ambiguity of value}\index{Multiple values} + +\small \begin{enumerate} +\item[Prob.~14.] Prove the following relations: +\begin{equation*} +\cosh^{-1}m = \sinh^{-1}\sqrt{m^2-1},\quad +\sinh^{-1}m = \pm \cosh^{-1}\sqrt{m^2+1}, +\end{equation*} +the upper or lower sign being used according as $m$ is positive or +negative. Modify these relations for $\sin^{-1}, \cos^{-1}$. + +\item[Prob.~15.] In figure, Art.~1, let $OA = 2, OB = 1, AOB = +60^\circ$; find the area of the hyperbolic sector $AOP$, and of the +segment $AMP$, if the abscissa of $P$ is 3. (Find $\cosh^{-1}$ from +the tables for $\cosh$.) +\end{enumerate} \normalsize + +\chapter{Functions of Sums and Differences.}% +\index{Addition-theorems}\index{Functions!of sum and difference}% +\index{Geometrical treatment of hyperbolic functions} + +(a) To prove the difference-formulas +\begin{equation} +\left. +\begin{aligned} +\sinh(u-v) &= \sinh u \cosh v - \cosh u \sinh v, \\ +\cosh(u-v) &= \cosh u \cosh v - \sinh u \sinh v. +\end{aligned} +\right\} \tag{19} +\end{equation}\index{Difference formula}% +\index{Hyperbolic funcitons!addition-theorems for} + +\begin{center} +\includegraphics[width=80mm]{fig05.png} +\end{center} + +Let $OA$ be any radius of a hyperbola, and let the sectors $AOP, +AOQ$ have the measures $u, v$; then $u-v$ is the measure of the +sector $QOP$. Let $OB, OQ'$ be the radii conjugate to $OA, OQ$; and +let the coördinates of $P, Q, Q'$ be $(x_1, y_1)$, $(x, y)$, $(x', +y')$ with reference to the axes $OA, OB$; then +\begin{align*} +\sinh (u-v) &= + \sinh\frac{\text{sector }QOP}{K} = + \frac{\text{triangle }QOP}{K}\quad \tag*{[Art.~5.} \\ +&= \frac{\frac{1}{2}(xy_1 - x_1y)\sin\omega} + {\frac{1}{2}a_1b_1\sin\omega} + = \frac{y_1}{b_1}\cdot\frac{x}{a_1} - + \frac{y}{b_1}\cdot\frac{x_1}{a_1} \notag \\ +&= \sinh u \cosh v - \cosh u \sinh v; \notag +\end{align*} +\begin{align*} +\cosh(u-v) &= \cosh\frac{\text{sector }QOP}{K} = + \frac{\text{triangle }POQ'}{K}\tag*{[Art.~5.} \\ +&= \frac{\frac{1}{2}(x_1y' - y_1x')\sin\omega} + {\frac{1}{2} a_1b_1\sin\omega} += \frac{y'}{b_1}\cdot\frac{x_1}{a_1} + - \frac{y}{b_1}\cdot\frac{x'}{a_1}; \\ +\intertext{but} +\frac{y'}{b_1} &= \frac{x}{a_1}, \qquad + \frac{x'}{a_1} = \frac{y}{b_1}, \tag{20} +\end{align*} +since $Q, Q'$ are extremities of conjugate radii; hence +\begin{equation*} +\cosh(u-v) = \cosh u \cosh v - \sinh u \sinh v. +\end{equation*} + +In the figures $u$ is positive and $v$ is positive or negative. +Other figures may be drawn with $u$ negative, and the language in +the text will apply to all. In the case of elliptic sectors, similar +figures may be drawn, and the same language will apply, except that +the second equation of (20) will be $\dfrac{x'}{a_1} = +\dfrac{-y}{b_1}$; therefore +\begin{align*} +\sin(u-v) &= \sin u \cos v - \cos u \sin v,\\ +\cos(u-v) &= \cos u \cos v + \sin u \sin v. +\end{align*}\index{Circular functions} + +(b) To prove the sum-formulas +\begin{equation} +\left. +\begin{aligned} +\sinh(u + v) &= \sinh u \cosh v + \cosh u \sinh v,\\ +\cosh(u + v) &= \cosh u \cosh v + \sinh u \sinh v. +\end{aligned} +\right\} \tag{21} +\end{equation} + +These equations follow from (19) by changing $v$ into $-v$, and then +for $\sinh (-v)$, $\cosh (-v)$, writing $-\sinh v$, $\cosh v$ (Art.\ +9, eqs.\ (18)). + +\medskip (c) To prove that +\begin{equation} +\tanh (u \pm v) = \frac{\tanh u \pm \tanh v} + {1 \pm \tanh u \tanh v}. \tag{22} +\end{equation} + +Writing $\tanh (u \pm v) = \dfrac{\sinh(u \pm v)}{\cosh(u \pm v)}$, +expanding and dividing numerator and denominator by $\cosh u \cosh +v$, eq.\ (22) is obtained. + +\small \begin{enumerate} + +\item[Prob.~16.] Given $\cosh u = 2, \cosh v = 3$, find +$\cosh(u + v)$. + +\item[Prob.~17.] Prove the following identities: +\begin{enumerate} +\item $\sinh 2u = 2 \sinh u \cosh u$. +\item $\cosh 2u = \cosh^2 u + \sinh^2 u = 1 + 2 \sinh^2 u + = 2 \cosh^2 u - 1$. +\item $1 + \cosh u = 2 \cosh^2 \frac{1}{2}u, + \cosh u - 1 = 2 \sinh^2 \frac{1}{2}u$. +\item $\tanh \frac{1}{2}u = \dfrac{\sinh u}{1 + \cosh u} + = \dfrac{\cosh u - 1}{\sinh u} + = \left( \dfrac{\cosh u - 1}{\cosh u + 1}\right) + ^{\frac{1}{2}}$. +\item $\sinh 2u = \dfrac{2\tanh u}{1 - \tanh^2 u},\ + \cosh 2u = \dfrac{1 + \tanh^2 u}{1 - \tanh^2 u}$. +\item $\sinh 3u = 3 \sinh u + 4 \sinh^3 u,\ + \cosh 3u = 4 \cosh^3u - 3 \cosh u$. +\item $\cosh u + \sinh u = \dfrac{1 + \tanh\frac{1}{2}u} + {1 - \tanh\frac{1}{2}u}$. +\item $(\cosh u + \sinh u)(\cosh v + \sinh v) + = \cosh (u+v) + \sinh (u+v)$. +\item Generalize (h); and show also what it becomes when +$u = v = \ldots$ +\item $\sinh^2 x \cos^2 y + \cosh^2 x \sin^2 y + = \sinh^2 x + \sin^2 y$. +\item $\cosh^{-1}m \pm \cosh^{-1}n = + \cosh^{-1}\left[ mn \pm\sqrt{(m^2-1)(n^2-1)} \right]$. +\item $\sinh^{-1}m \pm \sinh^{-1}n = + \sinh^{-1}\left[m\sqrt{1+n^2}\pm n\sqrt{1+m^2}\right]$. +\end{enumerate} + +\item[Prob.~18.] What modifications of signs are required in (21), +(22), in order to pass to circular functions? + +\item[Prob.~19.] Modify the identities of Prob. 17 for the same +purpose. +\end{enumerate} \normalsize + +\chapter{Conversion Formulas.}\index{Conversion-formulas} + +To prove that +\begin{equation} +\left. +\begin{aligned} +\cosh u_1 + \cosh u_2 &= 2\cosh\tfrac{1}{2}(u_1+u_2) + \cosh\tfrac{1}{2}(u_1-u_2), \\ +\cosh u_1 - \cosh u_2 &= 2\sinh\tfrac{1}{2}(u_1+u_2) + \sinh\tfrac{1}{2}(u_1-u_2), \\ +\sinh u_1 + \sinh u_2 &= 2\sinh\tfrac{1}{2}(u_1+u_2) + \cosh\tfrac{1}{2}(u_1-u_2), \\ +\sinh u_1 - \sinh u_2 &= 2\cosh\tfrac{1}{2}(u_1+u_2) + \sinh\tfrac{1}{2}(u_1-u_2). +\end{aligned} +\right\} \tag{23} +\end{equation} +From the addition formulas it follows that +\begin{align*} +\cosh (u+v) + \cosh (u-v) &= 2 \cosh u \cosh v, \\ +\cosh (u+v) - \cosh (u-v) &= 2 \sinh u \sinh v, \\ +\sinh (u+v) + \sinh (u-v) &= 2 \sinh u \cosh v, \\ +\sinh (u+v) - \sinh (u-v) &= 2 \cosh u \sinh v, +\end{align*} +and then by writing $u + v = u_1$, $u-v = u_2$, $u = \frac{1}{2}(u_1 ++ u_2)$, $v = \frac{1}{2}(u_1 - u_2)$, these equations take the form +required. + +\small \begin{enumerate} +\item[Prob.~20.] In passing to circular functions, show that the +only modification to be made in the conversion formulas is in the +algebraic sign of the right-hand member of the second formula. + +\item[Prob.~21.] Simplify $\dfrac{\cosh 2u + \cosh 4v}{\sinh 2u + \sinh +4v}$, $\dfrac{\cosh 2u + \cosh 4v}{\cosh 2u - \cosh 4v}$. + +\item[Prob.~22.] Prove $\sinh^2 x - \sinh^2 y = \sinh (x+y) \sinh +(x-y)$. + +\item[Prob.~23.] Simplify $\cosh^2 x \cosh^2 y \pm \sinh^2 x \sinh^2 +y$. + +\item[Prob.~24.] Simplify $\cosh^2 x \cos^2 y + \sinh^2 x \sin^2 y$. +\end{enumerate} \normalsize + +\chapter{Limiting Ratios.}\index{Limiting ratios}% +\index{Ratios!limiting} + +To find the limit, as $u$ approaches zero, of +\begin{equation*} +\frac{\sinh u}{u}, \frac{\tanh u}{u}, +\end{equation*} which are then indeterminate in form. + +By eq.\ (14), $\sinh u > u > \tanh u$; and if $\sinh u$ and $\tanh +u$ be successively divided by each term of these inequalities, it +follows that +\begin{gather*} +1 < \frac{\sinh u}{u} < \cosh u ,\\ +\sech u < \frac{\tanh u}{u} <1, +\end{gather*} +but when $u \doteq 0$, $\cosh u \doteq 1$, $\sech u \doteq 1$, hence +\begin{equation} +\lim_{u\doteq 0} \frac{\sinh u}{u} = 1, + \lim_{u\doteq 0} \frac{\tanh u}{u} = 1. \tag{24} +\end{equation} + +\chapter{Derivatives of Hyperbolic Functions.}% +\index{Derived functions}\index{Hyperbolic functions!derivatives of}% +\index{Hyperbolic functions!variation of} + +To prove that +\begin{equation} +\left. +\begin{aligned} +(\textit{a}) && \frac{d(\sinh u)}{du} &= \cosh u, \\ +(\textit{b}) && \frac{d(\cosh u)}{du} &= \sinh u, \\ +(\textit{c}) && \frac{d(\tanh u)}{du} &= \sech^2 u, \\ +(\textit{d}) && \frac{d(\sech u)}{du} &= -\sech u\; \tanh u, \\ +(\textit{e}) && \frac{d(\coth u)}{du} &= -\csch^2 u, \\ +(\textit{f}) && \frac{d(\csch u)}{du} &= -\csch u\; \coth u, \\ +\end{aligned} \right\} \tag{25} +\end{equation} + +\begin{enumerate} +\item[(a)] Let +\begin{align*} +y &= \sinh u, \\ +\Delta y &= \sinh \left( {u + \Delta u} \right) - \sinh u \\ + &= 2\cosh \frac{1}{2}\left( {2u + \Delta u} \right) + \sinh \frac{1}{2}\Delta u, \\ +\frac{\Delta y}{\Delta u} &= + \cosh \left( {u + \frac{1}{2}\Delta u} \right) + \frac{\sinh \frac{1}{2}\Delta u}{\frac{1}{2}\Delta u}. \\ +\intertext{Take the limit of both sides, as $\Delta u \doteq 0$, and +put} +\limdot &\frac{\Delta y}{\Delta u} = \frac{dy}{du} = + \frac{d\left( {\sinh u} \right)}{du}, \\ +\limdot &\cosh \left( {u + \frac{1}{2}\Delta u} \right) = \cosh u, \\ +\limdot &\frac{\sinh \frac{1}{2}\Delta u} + {\frac{1}{2}\Delta u} = 1; \tag{see Art. 13} \\ +\intertext{then } +&\frac{d\left( {\sinh u} \right)}{du} = \cosh u. \\ +\end{align*} + +\item[(b)] Similar to (a). + +\item[(c)] \begin{align*} + \frac{d\left( {\tanh u} \right)}{du} &= \frac{d}{du} \cdot + \frac{\sinh u}{\cosh u} \\ + &= \frac{\cosh ^2 u - \sinh ^2 u}{\cosh ^2 u} = + \frac{1}{\cosh ^2 u} = \sech^{2} u. +\end{align*} + +\item[(d)] Similar to (c). + +\item[(e)] \begin{equation*} +\frac{d(\sech u)}{du} = \frac{d}{du} \cdot \frac{1}{\cosh u} + = -\frac{\sinh u}{\cosh^2 u} = -\sech u \tanh u. +\end{equation*} + +\item[(f)] Similar to (e). +\end{enumerate} + +It thus appears that the functions $\sinh u, \cosh u$ reproduce +themselves in two differentiations; and, similarly, that the +circular functions $\sin u, \cos u$ produce their opposites in two +differentiations. In this connection it may be noted that the +frequent appearance of the hyperbolic (and circular) functions in +the solution of physical problems is chiefly due to the fact that +they answer the question: What function has its second derivative +equal to a positive (or negative) constant multiple of the function +itself? (See Probs.\ 28--30.) An answer such as $y = \cosh mx$ is +not, however, to be understood as asserting that $mx$ is an actual +sectorial measure and $y$ its characteristic ratio; but only that +the relation between the numbers $mx$ and $y$ is the same as the +known relation between the measure of a hyperbolic sector and its +characteristic ratio; and that the numerical value of $y$ could be +found from a table of hyperbolic cosines. + +\small \begin{enumerate} +\item[Prob.~25.] Show that for circular functions the only +modifications required are in the algebraic signs of (b), (d). + +\item[Prob.~26.] Show from their derivatives which of the +hyperbolic and circular functions diminish as $u$ increases. + +\item[Prob.~27.] Find the derivative of $\tanh u$ independently +of the derivatives of $\sinh u$, $\cosh u$. + +\item[Prob.~28.] Eliminate the constants by differentiation from +the equation +\begin{equation*} +y = A \cosh mx + B \sinh mx, +\end{equation*} and prove that $\dfrac{d^2y}{dx^2} = m^2y.$ + +\item[Prob.~29.] Eliminate the constants from the equation +\begin{equation*} +y = A \cos mx + B \sin mx, +\end{equation*} +and prove that $\dfrac{d^2y}{dx^2} = -m^2y.$% +\index{Circular functions}\index{Elimination of constants} + +\item[Prob.~30.] Write down the most general solutions of the +differential equations +\begin{equation*} +\frac{d^2y}{dx^2} = m^2y, \quad +\frac{d^2y}{dx^2} = -m^2y, \quad +\frac{d^4y}{dx^4} = m^4y. +\end{equation*}\index{Differential equation} +\end{enumerate} \normalsize + +\chapter{Derivatives of Anti-hyperbolic Functions.}% +\index{Anti-hyperbolic functions}\index{Derived functions} + +\begin{equation} +\left. +\begin{aligned} +(\textit{a}) && \frac{d(\sinh^{-1} x)}{dx} &= + \frac{1}{\sqrt{x^2+1}}, \\ +(\textit{b}) && \frac{d(\cosh^{-1} x)}{dx} &= + \frac{1}{\sqrt{x^2-1}}, \\ +(\textit{c}) && \frac{d(\tanh^{-1} x)}{dx} &= + \left. \frac{1}{1-x^2} \right]_{x<1}, \\ +(\textit{d}) && \frac{d(\coth^{-1} x)}{dx} &= + \left. \frac{1}{1-x^2} \right]_{x>1}, \\ +(\textit{e}) && \frac{d(\sech^{-1} x)}{dx} &= + -\frac{1}{x\sqrt{1-x^2}}, \\ +(\textit{f}) && \frac{d(\csch^{-1} x)}{dx} &= + -\frac{1}{x\sqrt{x^2+1}}, \\ +\end{aligned} +\right\} \tag{26} +\end{equation} + +\begin{enumerate} +\item[(a)] Let $u = \sinh^{-1} x$, then $x = \sinh u$, $dx = +\cosh u\,du = \sqrt{1 + \sinh^2 u} = \sqrt{1 + x^2} du$, $du = +\dfrac{dx}{\sqrt{1 + x^2}}$. + +\item[(b)] Similar to (a). + +\item[(c)] Let $u = \tanh^{-1} x$, then $x = \tanh u$, $dx = +\sech^2 u\,du = (1 - \tanh^2 u)du = (1 - x^2)du$, $du = \dfrac{dx}{1 +- x^2}$. + +\item[(d)] Similar to (c). + +\item[(e)] +\begin{equation*} +\frac{d(\sech^{-1} x)}{dx} + = \frac{d}{dx}\left( \cosh^{-1} \frac{1}{x} \right) + = \frac{\frac{-1}{x^2}}{\left( \frac{1}{x^2} - 1 \right)^{\frac{1}{2}}} + = \frac{-1}{x\sqrt{1-x^2}}. +\end{equation*} + +\item[(f)] Similar to (e). +\end{enumerate} + +\small \begin{enumerate} +\item[Prob. 31.] Prove +\begin{align*} + \frac{d(\sin^{-1} x)}{dx} &= \frac{1}{\sqrt{1 - x^2}}, +& \frac{d(\cos^{-1} x)}{dx} &= -\frac{1}{\sqrt{1 - x^2}}, \\ + \frac{d(\tan^{-1} x)}{dx} &= \frac{1}{1 + x^2}, +& \frac{d(\cot^{-1} x)}{dx} &= -\frac{1}{1 + x^2} . +\end{align*} + +\item[Prob.~32.] Prove +\begin{align*} +d\sinh^{-1}\frac{x}a &= \frac{dx}{\sqrt{x^2+a^2}}, & +d\cosh^{-1}\frac{x}a &= \frac{dx}{\sqrt{x^2-a^2}}, \\ +d\tanh^{-1}\frac{x}a &= \left.\frac{adx}{a^2-x^2}\right]_{x<a}, & +d\coth^{-1}\frac{x}a &= -\left.\frac{adx}{x^2-a^2}\right]_{x>a}. +\end{align*} + +\item[Prob.~33.] Find $d(\sech^{-1} x)$ independently of $\cosh^{-1} +x$. + +\item[Prob.~34.] When $\tanh^{-1} x$ is real, prove that $\coth^{-1} x$ +is imaginary, and conversely; except when $x = 1$. + +\item[Prob.~35.] Evaluate $\dfrac{\sinh^{-1}x}{\log x}$, +$\dfrac{\cosh^{-1}x}{\log x}$ when $x = \infty$. +\end{enumerate} \normalsize + +\chapter{Expansion of Hyperbolic Functions.}% +\index{Hyperbolic functions!expansions of}\index{Limiting ratios}% +\index{Series} + +For this purpose take Maclaurin's Theorem, +\begin{gather*} +f(u) = f(0) + uf'(0) + \frac{1}{2!}u^2f''(0) + + \frac{1}{3!}u^3f'''(0) + \ldots, \\ +\intertext{and put} +f(u) = \sinh u,\quad f'(u) = \cosh u,\quad + f''(u) = \sinh u, \ldots, \\ +\intertext{then} +f(0) = \sinh 0 = 0,\quad f'(0) = \cosh 0 = 1, \ldots; \\ +\intertext{hence} +\sinh u = u + \frac{1}{3!}u^3 + \frac{1}{5!}u^5 + \ldots; \tag{27} +\intertext{and similarly, or by differentiation,} +\cosh u = 1 + \frac{1}{2!}u^2 + \frac{1}{4!}u^4 + \ldots. \tag{28} +\end{gather*}\index{Expansion in series} + +By means of these series the numerical values of $\sinh u, \cosh u$, +can be computed and tabulated for successive values of the +independent variable $u$. They are convergent for all values of $u$, +because the ratio of the $n$th term to the preceding is in the first +case $\dfrac{u^2}{(2n-1)(2n-2)}$, and in the second case +$\dfrac{u^2}{(2n-2)(2n-3)}$, both of which ratios can be made less +than unity by taking $n$ large enough, no matter what value $u$ has. +Lagrange's remainder shows equivalence of function and series.% +\index{Convergence} + +From these series the following can be obtained by division: +\begin{equation} +\left. +\begin{aligned} + \tanh u &= u - \frac{1}{3} u^3 + \frac{2}{ 15} u^5 + + \frac{17}{ 315} u^7 + \ldots, \\ + \sech u &= 1 - \frac{1}{2} u^2 + \frac{5}{ 24} u^4 - + \frac{61}{ 720} u^6 + \ldots, \\ +u\coth u &= 1 + \frac{1}{3} u^2 - \frac{1}{ 45} u^4 + + \frac{ 2}{ 945} u^6 - \ldots, \\ +u\csch u &= 1 - \frac{1}{6} u^2 + \frac{7}{360} u^4 - + \frac{31}{15120} u^6 + \ldots. +\end{aligned} +\right\} \tag{29} +\end{equation} + +These four developments are seldom used, as there is no observable +law in the coefficients, and as the functions $\tanh u, \sech u, +\coth u, \csch u$, can be found directly from the previously +computed values of $\cosh u, \sinh u$. + +\small \begin{enumerate} +\item[Prob. 36.] Show that these six developments can be adapted to +the circular functions by changing the alternate signs. +\end{enumerate} \normalsize + +\chapter{Exponential Expressions.}\index{Exponential expressions}% +\index{Hyperbolic functions!exponential functions for} + +Adding and subtracting (27), (28) give the identities +\begin{gather*} +\begin{aligned} +\cosh u + \sinh u &= 1 + u + \frac{1}{2!} u^2 + \frac{1}{3!} u^3 + + \frac{1}{4!} u^4 + \ldots = e^u, \\ +\cosh u - \sinh u &= 1 - u + \frac{1}{2!} u^2 - \frac{1}{3!} u^3 + + \frac{1}{4!} u^4 - \ldots = e^{-u}, +\end{aligned} +\intertext{hence} +\left. +\begin{aligned} +\cosh u &= \tfrac{1}{2}(e^u + e^{-u}), & +\sinh u &= \tfrac{1}{2}(e^u - e^{-u}), \\ +\tanh u &= \frac{e^u - e^{-u}}{e^u + e^{-u}}, & \sech u &= +\frac{2}{e^u + e^{-u}},\quad\text{etc.} +\end{aligned} +\right\} \tag{30} +\end{gather*} + +The analogous exponential expressions for $\sin u, \cos u$ are +\begin{equation*} +\cos u = \frac{1}{2} (e^{ui} + e^{-ui}),\quad +\sin u = \frac{1}{2i}(e^{u} - e^{-ui}),\quad (i=\sqrt{-1}) +\end{equation*} +where the symbol $e^{ui}$ stands for the result of substituting $ui$ +for $x$ in the exponential development +\begin{equation*} +e^x = 1 + x + \frac{1}{2!} x^2 + \frac{1}{3!} x^3 + \ldots +\end{equation*}\index{Circular functions} + +This will be more fully explained in treating of complex numbers, +Arts.~28, 29. + +\subsubsection{} %% RWD Nickalls a hook for the \label below +\small \begin{enumerate} +\item[Prob.~37.] Show that the properties of the hyperbolic functions +could be placed on a purely algebraic basis by starting with +equations (30) as their definitions; for example, verify the +identities:\label{def hyper as exp} +\begin{gather*} +\sinh (-u) = -\sinh u,\quad \cosh (-u) = \cosh u,\\ +\cosh^2 u - \sinh^2 u = 1,\quad + \sinh (u+v) = \sinh u \cosh v + \cosh u \sinh v,\\ +\frac{d^2(\cosh mu)}{du^2} = m^2 \cosh mu,\quad + \frac{d^2(\sinh mu)}{du^2} = m^2 \sinh mu. +\end{gather*} + +\item[Prob.~38.] Prove $(\cosh u + \sinh u)^n = \cosh nu + \sinh nu$. + +\item[Prob.~39.] Assuming from Art.~14 that $\cosh u$, $\sinh u$ +satisfy the differential equation $\dfrac{d^2y}{du^2} = y$, whose +general solution may be written $y = Ae^n + Be^{-n}$, where $A$, $B$ +are arbitrary constants; show how to determine $A$, $B$ in order to +derive the expressions for $\cosh u$, $\sinh u$, respectively. [Use +eq.\ (15).]\index{Differential equation} + +\item[Prob.~40.] Show how to construct a table of exponential functions +from a table of hyperbolic sines and cosines, and \emph{vice versa}. + +\item[Prob.~41.] Prove $u = \log_e (\cosh u + \sinh u)$. + +\item[Prob.~42.] Show that the area of any hyperbolic sector is +infinite when its terminal line is one of the asymptotes. + +\item[Prob.~43.] From the relation $2 \cosh u = e^n + e^{-n}$ prove +\begin{equation*} +2^{n-1}(\cosh u)^n = \cosh nu + n \cosh (n-2)u + + \tfrac{1}{2} n(n-1)\cosh (n-4)u + \ldots, +\end{equation*} +and examine the last term when $n$ is odd or even. Find also the +corresponding expression for $2^{n-1} (\sinh u)^n$. +\end{enumerate}\index{Exponential expressions} \normalsize + +\chapter{Expansion of Anti-functions.}% +\index{Anti-hyperbolic functions} + +Since +\begin{gather*} +\begin{aligned} +\frac{d(\sinh^{-1} x)}{dx} &= \frac{1}{\sqrt{1+x^2}} + = (1+x^2)^{-\frac{1}{2}} \\ +&= 1 - \frac{1}{2}\cdot x^2 + \frac{1}{2}\cdot\frac{3}{4}\cdot x^4 + - \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot x^6 + + \ldots,\\ +\end{aligned} +\intertext{hence, by integration,} +\sinh^{-1} x = x - \frac{1}{2}\cdot \frac{x^3}{3} + + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{x^5}{5} - + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\frac{x^7}{7} + + \ldots, \tag{31} +\end{gather*} +the integration-constant being zero, since $\sinh^{-1} x$ vanishes +with $x$. This series is convergent, and can be used in computation, +only when $x < 1$. Another series, convergent when $x > 1$, is +obtained by writing the above derivative in the form +\begin{align*} +\frac{d(\sinh^{-1} x)}{dx} &= (x^2+1)^{-\frac{1}{2}} = + \frac{1}{x} \left(1 + \frac{1}{x^2}\right)^{-\frac{1}{2}} \\ +&= \frac{1}{x} \left[ 1 - \frac{1}{2}\cdot\frac{1}{x^2} + + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{1}{x^4} - + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot \frac{1}{x^6} + + \dotsb \right], \\ +\therefore\; \sinh^{-1} &= C + \log x + + \frac{1}{2}\cdot\frac{1}{2x^2} - + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{1}{4x^4} + + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\frac{1}{6x^6} + - \dotsb, \tag{32} +\end{align*} +where $C$ is the integration-constant, which will be shown in +Art.~19 to be equal to $\log_e 2$.\index{Convergence}% +\index{Expansion in series} + +A development of similar form is obtained for $\cosh^{-1} x$; for +\begin{gather*} +\begin{aligned} +\frac{d(\cosh^{-1} x)}{dx} &= (x^2-1)^{-\frac{1}{2}} = + \frac{1}{x}\left(1-\frac{1}{x^2}\right)^{-\frac{1}{2}} \\ +&= \frac{1}{x} \left[ 1 + \frac{1}{2}\cdot\frac{1}{x^2} + + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{1}{x^4} + + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\frac{1}{x^6} + + \dotsb \right], +\end{aligned} \\ +\intertext{hence} +\cosh^{-1} x = C + \log x - \frac{1}{2}\cdot \frac{1}{2x^2} - + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{1}{4x^4} - + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot \frac{1}{6x^6} + - \dotsb, \tag{33} +\end{gather*} +in which $C$ is again equal to $\log_e 2$ [Art.~19, Prob.~46]. In +order that the function $\cosh^{-1} x$ may be real, $x$ must not be +less than unity; but when $x$ exceeds unity, this series is +convergent, hence it is always available for computation. + +Again +\begin{align*} +\frac{d(\tanh^{-1} x)}{dx} &= \frac{1}{1-x^2} + = 1 + x^2 + x^4 + x^6 + \dotsb, \\ +\intertext{and hence} +\tanh^{-1} x &= x + \frac{1}{3} x^3 + + \frac{1}{5} x^5 + \frac{1}{7} x^7 + \dotsb, \tag{34} +\end{align*} + +From (32), (33), (34) are derived: +\begin{align*} +\sech^{-1} x &= \cosh^{-1} \frac{1}{x} \\ +&= C - \log x - \frac{x^2}{2 \cdot 2} - + \frac{1 \cdot 3 \cdot x^4}{2 \cdot 4\cdot 4} + - \frac{1 \cdot 3\cdot 5 \cdot x^6} + {2 \cdot 4\cdot 6\cdot 6 } - \dotsb; \tag{35} \\ +\csch^{-1} x &= \sinh^{-1} \frac{1}{x} = + \frac{1}{x} - \frac{1}{2}\cdot\frac{1}{3x^3} + + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{1}{5x^5} - + \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\frac{1}{7x^7} + + \dotsb, \\ +&= C - \log x + \frac{x^2}{2 \cdot 2} - + \frac{1 \cdot 3 \cdot x^4} {2 \cdot 4 \cdot 4} + + \frac{1 \cdot 3 \cdot 5 \cdot x^6} + {2 \cdot 4 \cdot 6 \cdot 6} - \dotsb; \tag{36} \\ +\coth^{-1} x &= \tanh^{-1} \frac{1}{x} = + \frac{1}{x} + \frac{1}{3x^3} + \frac{1}{5x^5} + + \frac{1}{7x^7} + \dotsb. \tag{37} +\end{align*} + +\small \begin{enumerate} +\item[Prob.~44.] Show that the series for $\tanh^{-1} x, \coth^{-1} +x, \sech^{-1} x$, are always available for computation. + +\item[Prob.~45.] Show that one or other of the two developments of the +inverse hyperbolic cosecant is available. +\end{enumerate} \normalsize + +\chapter{Logarithmic Expression of Anti-Functions.}% +\index{Logarithmic!expressions} + +Let +\begin{align*} +x &= \cosh u, \\ +\intertext{then} +\sqrt{x^2 - 1} &= \sinh u; \\ +\intertext{therefore} +x + \sqrt{x^2 - 1} &= \cosh u + \sinh u = e^u, \\ +\intertext{and} +u = \cosh^{-1} x &= \log{\left(x + \sqrt{x^2 - 1}\right)}. \tag{38} \\ +\intertext{Similarly,} +\sinh^{-1} x &= \log{\left(x + \sqrt{x^2 + 1}\right)}. \tag{39} \\ +\intertext{Also} +\sech^{-1} x &= \cosh^{-1}\frac{1}{x} = + \log{\frac{1 + \sqrt{1 - x^2}}{x}}, \tag{40} \\ +\csch^{-1} x &= \sinh^{-1}\frac{1}{x} = + \log{\frac{1 + \sqrt{1 + x^2}}{x}}. \tag{41} \\ +\intertext{Again, let} +x &= \tanh u = \frac{e^u - e^{-u}}{e^u + e^{-u}}, \\ +\intertext{therefore} +\frac{1 + x}{1 - x} &= \frac{e^u}{e^{-u}} = e^{2u}, \\ +2u &= \log{\frac{1 + x}{1 - x}}, \quad + \tanh^{-1} = \tfrac{1}{2}\log{\frac{1 + x}{1 - x}}; \tag{42} \\ +\intertext{and} +\coth^{-1} x &= \tanh^{-1} \frac{1}{x} = + \tfrac{1}{2}\log{\frac{x + 1}{x - 1}}. \tag{43} +\end{align*} + +\small \begin{enumerate} +\item[Prob.~46.] Show from (38), (39), that, when $x \doteq \infty$, +\begin{equation*} +\sinh^{-1} x - \log x \doteq \log 2,\qquad +\cosh^{-1} x - \log x \doteq \log 2, +\end{equation*} +and hence show that the integration-constants in (32), (33) are each +equal to $\log 2$. + +\item[Prob.~47.] Derive from (42) the series for $\tanh^{-1}x$ given in +(34). + +\item[Prob.~48.] Prove the identities: +\begin{align*} +& \log x = 2\tanh^{-1} \frac{x - 1}{x + 1} += \tanh^{-1} \frac{x^2 - 1}{x^2 + 1} += \sinh^{-1} \tfrac{1}{2} (x - x^{-1}) += \cosh^{-1} \tfrac{1}{2} (x + x^{-1}); \\ +& \log\sec x = 2\tanh^{-1} \tfrac{1}{2} x;\ + \log\csc x = 2\tanh^{-1} + \tan^2\left(\frac{1}{4}\pi + \frac{1}{2}x\right); \\ +& \log\tan x = -\tanh^{-1} \cos 2x += -\sinh^{-1} \cot 2x = \cosh^{-1} \csc 2x. +\end{align*} +\end{enumerate} \normalsize + +\chapter{The Gudermanian Function.} +\label{gudermanian}% +\index{Gudermanian!function} + +The correspondence of sectors of the same species was discussed in +Arts.~1--4. It is now convenient to treat of the correspondence that +may exist between sectors of different species. + +Two points $P_1, P_2$, on any hyperbola and ellipse, are said to +correspond with reference to two pairs of conjugates $O_1 A_1, O_1 +B_1$, and $O_2 A_2, O_2 B_2$, respectively, when +\begin{equation} +\frac{x_1}{a_1} = \frac{a_2}{x_2} ,\tag{44} +\end{equation} +and when $y_1, y_2$ have the same sign. The sectors $A_1 O_1 P_1, +A_2 O_2 P_2$ are then also said to correspond. Thus corresponding +sectors of central conics of different species are of the same sign +and have their primary characteristic ratios reciprocal. Hence there +is a fixed functional relation between their respective measures. +The elliptic sectorial measure is called the gudermanian of the +corresponding hyperbolic sectorial measure, and the latter the +anti-gudermanian of the former. This relation is expressed by +\begin{gather*} +\frac{S_2}{K_2} = \gd \frac{S_1}{K_1} \\ + \text{or } v = \gd u, \text{ and } u = \gd^{-1} v. \tag{45} +\end{gather*}% +\index{Anti-gudermanian}\index{Corresponding points!on conics}% +\index{Corresponding points!on sectors and triangles}% +\index{Sectors of conics} + +\chapter{Circular Functions of Gudermanian.}% +\index{Circular functions!of gudermanian}% +\index{Hyperbolic functions!relations to circular functions} + +The six hyperbolic functions of $u$ are expressible in terms of the +six circular functions of its gudermanian; for since +\begin{equation} +\frac{x_1}{a_1} = \cosh u, \quad \frac{x_2}{a_2} = \cosh v, + \tag{see Arts.\ 6, 7} +\end{equation} +in which $u, v$ are the measures of corresponding hyperbolic and +elliptic sectors, hence +\begin{equation} +\left. +\begin{aligned} + \cosh u &= \sec v, \qquad [\text{eq.\ (44)}] \\ + \sinh u &= \sqrt{\sec^2 v - 1} = \tan v, \\ + \tanh u &= \frac{\tan v}{\sec v} = \sin v, \\ + \coth u &= \csc v, \\ + \sech u &= \cos v, \\ + \csch u &= \cot v. \\ +\end{aligned} +\right\}\tag{46} +\end{equation} + +The gudermanian is sometimes useful in computation; for instance, if +$\sinh u$ be given, $v$ can be found from a table of natural +tangents, and the other circular functions of $v$ will give the +remaining hyperbolic functions of $u$. Other uses of this function +are given in Arts.\ 22--26, 32--36.\index{Relations among functions} + +\small \begin{enumerate} +\item[Prob.~49.] Prove that +\begin{align*} +\gd u &= \sec^{-1} (\cosh u) = \tan^{-1} (\sinh u) \\ + &= \cos^{-1} (\sech u) = \sin^{-1} (\tanh u). +\end{align*}% +\index{Anti-hyperbolic functions}\index{Circular functions} + +\item[Prob.~50.] Prove +\begin{align*} +\gd^{-1} v &= \cosh^{-1} (\sec v) = \sinh^{-1} (\tan v) \\ + &= \sech^{-1} (\cos v) = \tanh^{-1} (\sin v). +\end{align*} + +\item[Prob.~51.] Prove +\begin{align*} +\gd 0 &= 0, \; + \gd \infty = \tfrac{1}{2}\pi,\ + \gd (-\infty) = -\tfrac{1}{2}\pi, \\ +\gd^{-1} 0 &= 0, \; + \gd^{-1}\left( \tfrac{1}{2}\pi\right) = \infty, \; + \gd^{-1}\left(-\tfrac{1}{2}\pi\right) = -\infty. +\end{align*} + +\item[Prob.~52.] Show that $\gd u$ and $\gd^{-1} v$ are odd +functions of $u, v$. + +\item[Prob.~53.] From the first identity in 4, Prob.~17, derive the +relation $\tanh \frac{1}{2} u = \tan \frac{1}{2} v$. + +\item[Prob.~54.] Prove $\tanh^{-1} (\tan u) = \tfrac{1}{2} \gd 2u,$ +and $\tan^{-1} (\tanh x) = \tfrac{1}{2} \gd^{-1} 2x.$ +\end{enumerate} \normalsize + +\chapter{Gudermanian Angle}\index{Gudermanian!angle}% +\index{Hyperbolic functions!relations to gudermanian} + +If a circle be used instead of the ellipse of Art.~20, the +gudermanian of the hyperbolic sectorial measure will be equal to the +radian measure of the angle of the corresponding circular sector +(see eq.~(6), and Art.~3, Prob.~2). This angle will be called the +gudermanian angle; but the gudermanian function $v$, as above +defined, is merely a number, or ratio; and this number is equal to +the radian measure of the gudermanian angle $\theta$, which is +itself usually tabulated in degree measure; thus +\begin{equation} + \theta = \frac{180^\circ v}{\pi} \tag{47} +\end{equation} + +\small \begin{enumerate} +\item[Prob.~55.] Show that the gudermanian angle of $u$ may be +constructed as follows: + +\begin{center} +\includegraphics[width=35mm]{fig06.png} +\end{center} + +Take the principal radius $OA$ of an equilateral hyperbola, as the +initial line, and $OP$ as the terminal line, of the sector whose +measure is $u$; from $M$, the foot of the ordinate of $P$, draw $MT$ +tangent to the circle whose diameter is the transverse axis; then +$AOT$ is the angle required.% +\footnote{This angle was called by Gudermann the longitude of $u$, +and denoted by $lu$. His inverse symbol was $\textgoth{L}$; thus $u += \textgoth{L}(lu)$. (Crelle's Journal, vol.~6, 1830.) Lambert, who +introduced the angle $\theta$, named it the transcendent angle. +(Hist.\ de l'acad.\ roy.\ de Berlin, 1761). Hoüel (Nouvelles +Annales, vol.~3, 1864) called it the hyperbolic amplitude of $u$, +and wrote it $\amh{u}$, in analogy with the amplitude of an elliptic +function, as shown in Prob.~62. Cayley (Elliptic Functions, 1876) +made the usage uniform by attaching to the angle the name of the +mathematician who had used it extensively in tabulation and in the +theory of elliptic functions of modulus unity.}% +\index{Cayley's Elliptic Functions}% +\index{Construction!for gudermanian}\index{Gudermann's notation}% +\index{Hoüel's notation, etc.}\index{Hyperbola}% +\index{Lambert's!notation}\index{Modulus} + +\item[Prob.~56.] Show that the angle $\theta$ never exceeds +$90^{\circ}$. + +\item[Prob.~57.] The bisector of angle $AOT$ bisects the sector $AOP$ +(see Prob.~13, Art.~9, and Prob.~53, Art.~21), and the line $AP$. +(See Prob.~1, Art.~3.) + +\item[Prob.~58.] This bisector is parallel to $TP$, and the points +$T$, $P$ are in line with the point diametrically opposite to $A$. + +\item[Prob.~59.] The tangent at $P$ passes through the foot of the +ordinate of $T$, and intersects $TM$ on the tangent at $A$. + +\item[Prob.~60.] The angle $APM$ is half the gudermanian angle. +\end{enumerate} \normalsize + +\chapter{Derivatives of Gudermanian and Inverse.}% +\index{Derived functions} + +Let +\begin{align*} + v & = \gd{u}, \quad u = \gd^{-1}{v}, \\ +\intertext{then} + \sec{v} & = \cosh{u}, \\ + \sec{v}\tan{v} \,dv & = \sinh{u} \,du, \\ + \sec{v} \,dv & = du, \\ +\intertext{therefore} + d(\gd^{-1}{v}) & = \sec{v} \,dv. \tag{48} \\ +\intertext{\qquad Again,} + dv & = \cos{v} \,du = \sech{u} \,du, \\ +\intertext{therefore} + d(\gd{u}) & = \sech{u} \,du. \tag{49} +\end{align*}% +\index{Anti-gudermanian} + +\small \begin{enumerate} +\item[Prob.~61.] Differentiate: +\begin{align*} + y & = \sinh{u} - \gd{u}, +& y & = \sin{v} + \gd^{-1}{v}, \\ + y & = \tanh{u}\sech{u} + \gd{u}, +& y & = \tan{v}\sec{v} + \gd^{-1}{v}. +\end{align*} + +\item[Prob.~62.] Writing the ``elliptic integral of the first +kind'' in the form +\begin{equation*} +u = \int_0^\phi \frac{d\phi}{\sqrt{1 - \kappa^2\sin^2\phi}}, +\end{equation*} +$\kappa$ being called the modulus, and $\phi$ the amplitude; that is, +\begin{equation*} +\phi = \am u, (\moddot \kappa), +\end{equation*} +show that, in the special case when $\kappa = 1$, +\begin{align*} +u &= \gd^{-1} \phi, & \am u &= \gd u, & \sin \am u &= \tanh u, \\ +\cos \am u &= \sech u, & \tan \am u &= \sinh u; +\end{align*} +and that thus the elliptic functions $\sin \am u$, etc., degenerate +into the hyperbolic functions, when the modulus is unity.% +\footnote{The relation $\gd u = \am u, (\moddot 1)$, led Hoüel to +name the function $\gd u$, the hyperbolic amplitude of $u$, and to +write it $\amh u$ (see note, Art.~22). In this connection Cayley +expressed the functions $\tanh u$, $\sech u$, $\sinh u$ in the form +$\sin \gd u$, $\cos \gd u$, $\tan \gd u$, and wrote them $\sg u$, +$\cg u$, $\tg u$, to correspond with the abbreviations $\sn u$, $\cn +u$, $\dn u$ for $\sin \am u$, $\cos \am u$, $\tan \am u$. Thus +$\tanh u = \sg u = \sn u, (\moddot 1)$; etc.\index{Modulus} + +\indent It is well to note that neither the elliptic nor the +hyperbolic functions received their names on account of the relation +existing between them in a special case. (See foot-note, +p.~\ref{fnp7})}% +\index{Amplitude!hyperbolic}\index{Cayley's Elliptic Functions}% +\index{Elliptic functions}\index{Elliptic integrals}% +\index{Elliptic sectors}\index{Hoüel's notation, etc.} +\end{enumerate} \normalsize + +\chapter{Series for Gudermanian and its Inverse.}% +\index{Gudermanian!function}\index{Series} + +Substitute for $\sech u, \sec v$ in (49), (48) their expansions, +Art.~16, and integrate, then +\begin{align*} +\gd u &= u - \frac{1}{6} u^3 + \frac{1}{24} u^5 - \frac{61}{5040} +u^7 + \dotsb \tag{50} \\ +\gd^{-1} v &= v + \frac{1}{6} v^3 + \frac{1}{24} v^5 - +\frac{61}{5040} v^7 + \dotsb \tag{51} +\end{align*} +No constants of integration appear, since $\gd u$ vanishes with $u$, +and $\gd^{-1} v$ with $v$. These series are seldom used in +computation, as $\gd u$ is best found and tabulated by means of +tables of natural tangents and hyperbolic sines, from the equation +\begin{equation*} +\gd u = \tan^{-1}(\sinh u), +\end{equation*} +and a table of the direct function can be used to furnish the +numerical values of the inverse function; or the latter can be +obtained from the equation, +\begin{equation*} +\gd^{-1} v = \sinh^{-1}(\tan v) = \cosh^{-1}(\sec v). +\end{equation*}\index{Expansion in series} + +To obtain a logarithmic expression for $\gd^{-1} v$, let +\begin{align*} +\gd^{-1} v &= u,\quad v = \gd u, \\ +\intertext{therefore} +\sec v &= \cosh u,\quad \tan v = \sinh u, \\ +\sec v + \tan v &= \cosh u + \sinh u = e^u, \\ +e^u = \frac{1 + \sin v}{\cos v} &= + \frac{1-\cos(\frac{1}{2}\pi + v)}{\sin(\frac{1}{2}\pi + v)} = + \tan\left(\frac{1}{4}\pi + \frac{1}{2}v\right), \\ +u = \gd^{-1} v &= + \log_e\tan\left(\frac{1}{4}\pi + \frac{1}{2}v\right). \tag{52} +\end{align*} + +\small \begin{enumerate} +\item[Prob.~63.] Evaluate $\left.\dfrac{\gd u - +u}{u^3}\right]_{u\doteq 0}$, $\left.\dfrac{\gd^{-1} v - +v}{v^3}\right]_{v\doteq 0}$.\index{Limiting ratios} + +\item[Prob.~64.] Prove that $\gd u - \sin u$ is an infinitesimal of the +fifth order, when $u \doteq 0$.\index{Logarithmic!expressions} + +\item[Prob.~65.] Prove the relations $\frac{1}{4}\pi + \frac{1}{2} v +\tan^{-1}e^u$, $\frac{1}{4}\pi - \frac{1}{2} v = \tan^{-1}e^{-u}$. +\end{enumerate} \normalsize + +\chapter{Graphs of Hyperbolic Functions.}% +\index{Construction!of graphs}\index{Graphs}% +\index{Hyperbolic functions!graphs of} + +\begin{figure*}[p] +\begin{center} +\includegraphics[width=60mm]{fig07.png} \\ +\includegraphics[width=50mm]{fig08.png} +\includegraphics[width=50mm]{fig09.png} \\ +\includegraphics[width=100mm]{fig10.png} +\end{center} +\end{figure*} + +Drawing two rectangular axes, and laying down a series of points +whose abscissas represent, on any convenient scale, successive +values of the sectorial measure, and whose ordinates represent, +preferably on the same scale, the corresponding values of the +function to be plotted, the locus traced out by this series of +points will be a graphical representation of the variation of the +function as the sectorial measure varies. The equations of the +curves in the ordinary cartesian notation are: + +\medskip \begin{center} +\begin{tabular}{l l l} +\multicolumn{1}{c}{Fig.} & + \multicolumn{1}{c}{Full Lines.} + & \multicolumn{1}{c}{Dotted Lines.} \\ +A & $y = \cosh x,$ & $y = \sech x;$ \\ +B & $y = \sinh x,$ & $y = \csch x;$ \\ +C & $y = \tanh x,$ & $y = \coth x;$ \\ +D & $y = \gd x.$ & +\end{tabular} +\end{center} + +Here $x$ is written for the sectorial measure $u$, and $y$ for the +numerical value of $\cosh u$, etc. It is thus to be noted that the +variables $x$, $y$ are numbers, or ratios, and that the equation $y += \cosh x$ merely expresses that the relation between the numbers +$x$ and $y$ is taken to be the same as the relation between a +sectorial measure and its characteristic ratio. The numerical values +of $\cosh u, \sinh u, \tanh u$ are given in the tables at the end of +this chapter for values of $u$ between $0$ and $4$. For greater +values they may be computed from the developments of Art.~16. + +The curves exhibit graphically the relations: +\begin{gather*} +\sech u = \frac{1}{\cosh u}, \quad + \csch u = \frac{1}{\sinh u}, \quad + \coth u = \frac{1}{\tanh u}; \\ +\cosh u \nless 1, \quad \sech u \ngtr 1, \quad + \tanh u \ngtr 1, \quad \gd u < \tfrac{1}{2}\pi, + \text{ etc.}; \\ +\sinh(-u) = -\sinh u, \quad \cosh(-u) = \cosh u, \\ +\tanh(-u) = -\tanh u, \quad \gd(-u) = -\gd u, \text{ etc.}; \\ +\cosh 0 = 1, \quad \sinh 0 = 0, \quad + \tanh 0 = 0, \quad \csch(0) = \infty, \text{ etc.}; \\ +\cosh(\pm\infty) = \infty, \quad + \sinh(\pm\infty) = \pm\infty, \quad + \tanh(\pm\infty) = \pm 1, \text{ etc.} +\end{gather*} + +The slope of the curve $y = \sinh x$ is given by the equation +$\dfrac{dy}{dx} = \cosh x$, showing that it is always positive, and +that the curve becomes more nearly vertical as $x$ becomes infinite. +Its direction of curvature is obtained from $\dfrac{d^2y}{dx^2} = +\sinh x$, proving that the curve is concave downward when $x$ is +negative, and upward when $x$ is positive. The point of inflexion is +at the origin, and the inflexional tangent bisects the angle between +the axes. + +The direction of curvature of the locus $y = \sech x$ is given by +$\dfrac{d^2y}{dx^2}=$ $\sech x (2 \tanh^2 x - 1)$, and thus the +curve is concave downwards or upwards according as $2 \tanh^2 x - 1$ +is negative or positive. The inflexions occur at the points $x = \pm +\tanh^{-1} .707, = \pm .881$, $y =.707$; and the slopes of the +inflexional tangents are $\mp\frac{1}{2}$. + +The curve $y = \csch x$ is asymptotic to both axes, but approaches +the axis of $x$ more rapidly than it approaches the axis of $y$, for +when $x = 3$, $y$ is only $.1$, but it is not till $y = 10$ that $x$ +is so small as $.1$. The curves $y = \csch x$, $y = \sinh x$ cross +at the points $x = \pm .881$, $y = \pm 1$. + +\small \begin{enumerate} +\item[Prob.~66.] Find the direction of curvature, the inflexional +tangent, and the asymptotes of the curves $y = \gd x$, $y = \tanh +x$.\index{Gudermanian!function} + +\item[Prob.~67.] Show that there is no inflexion-point on the curves +$y = \cosh x$, $y = \coth x$. + +\item[Prob.~68.] Show that any line $y = mx + n$ meets the curve $y = +\tanh x$ in either three real points or one. Hence prove that the +equation $\tanh x = mx + n$ has either three real roots or one. From +the figure give an approximate solution of the equation $\tanh x = x +- 1$. + +\item[Prob.~69.] Solve the equations: $\cosh x = x + 2$; $\sinh x = +\frac{3}{2} x$; $\gd x = x - \frac{1}{2}\pi$. + +\item[Prob.~70.] Show which of the graphs represent even functions, +and which of them represent odd ones. +\end{enumerate} \normalsize + +\chapter{Elementary Integrals.}% +\index{Anti-hyperbolic functions}\index{Circular functions}% +\index{Equations!Numerical}% +\index{Hyperbolic functions!integrals involving}\index{Integrals} + +The following useful indefinite integrals follow from Arts. 14, 15, +23: + +\newcommand{\dint}{\displaystyle\int} +\medskip\begin{tabular}{rll} +& \multicolumn{1}{c}{Hyperbolic.} & \multicolumn{1}{c}{Circular.} \\ +1. & $\dint \sinh u\: du = \cosh u,$ + & $\dint \sin u\: du = -\cos u,$ \\ +2. & $\dint \cosh u\: du = \sinh u,$ + & $\dint \cos u\: du = \sin u,$ \\ +3. & $\dint \tanh u\: du = \log \cosh u,$ + & $\dint \tan u\: du = -\log \cos u,$ \\ +4. & $\dint \coth u\: du = \log \sinh u,$ + & $\dint \cot u\: du = \log \sin u,$ \\ +5. & $\dint \csch u\: du = \log \tanh \frac{u}{2},$ + & $\dint \csc u\: du = \log \tan\dfrac{u}2,$ \\ + & $\qquad = -\sinh^{-1}(\csch u),$ + & $\qquad = -\cosh^{-1}(\csch u),$ \\ +6. & $\dint \sech u\: du = \gd u,$ + & $\dint \sec u\: du = \gd^{-1} u,$ \\ +7. & $\dint \frac{dx}{\sqrt{x^2+a^2}} = + \sinh^{-1}\frac{x}{a},$\footnotemark + & $\dint \frac{dx}{\sqrt{a^2-x^2}} = + \sin^{-1}\frac{x}{a},$ \\ +8. & $\dint \frac{dx}{\sqrt{x^2-a^2}} = + \cosh^{-1}\frac{x}{a},$ + & $\dint \frac{-dx}{\sqrt{a^2-x^2}} = + \cos^{-1}\frac{x}{a},$ \\ +9. & $\dint \left.\frac{dx}{a^2-x^2}\right]_{x<a} = + \frac{1}{a}\tanh^{-1}\frac{x}{a},$ + & $\dint \frac{dx}{a^2+x^2} = \frac{1}{a}\tan^{-1}\frac{x}{a},$ +\\ +10. & $\dint \left.\frac{-dx}{x^2-a^2}\right]_{x>a} = + \frac{1}{a}\coth^{-1} \frac{x}{a},$ + & $\dint \frac{-dx}{a^2+x^2} = + \frac{1}{a}\cot^{-1}\frac{x}{a},$\\ +11. & $\dint \frac{-dx}{x\sqrt{a^2-x^2}} = + \frac{1}{a}\sech^{-1}\frac{x}a,$ + & $\dint \frac{dx}{x\sqrt{x^2-a^2}} = + \frac{1}{a}\sec^{-1}\frac{x}{a},$ \\ +12. & $\dint \frac{-dx}{x\sqrt{a^2+x^2}} = + \frac{1}{a}\csch^{-1} \frac{x}{a},$ + & $\dint \frac{-dx}{x\sqrt{x^2-a^2}} = + \frac{1}{a}\csc^{-1}\frac{x}{a}.$ +\end{tabular} +\footnotetext{Forms 7--12 are preferable to the respective +logarithmic expressions (Art.~19), on account of the close analogy +with the circular forms, and also because they involve functions +that are directly tabulated. This advantage appears more clearly in +13--20.} + +From these fundamental integrals the following may be +derived: +\begin{equation*} +\begin{aligned} +13.\quad \int\frac{dx}{\sqrt{ax^2+2bx+c}} &= +\frac{1}{\sqrt{a}}\sinh^{-1} \frac{ax+b}{\sqrt{ac-b^2}},\ +a \text{ positive, } ac > b^2;\\ +&= \frac{1}{\sqrt{a}}\cosh^{-1} \frac{ax+b}{\sqrt{b^2-ac}},\ +a \text{ positive, } ac < b^2;\\ +&= \frac{1}{\sqrt{-a}}\cos^{-1} \frac{ax+b}{\sqrt{b^2-ac}},\ a +\text{ negative}. +\end{aligned} +\end{equation*} + +\begin{equation*} +\begin{aligned} +14.\quad \int\frac{dx}{ax^2+2bx+c} &= + \frac{1}{\sqrt{ac-b^2}}\tan^{-1}\frac{ax+b}{\sqrt{ac-b^2}},\ + ac > b^2; \\ +&= \frac{-1}{\sqrt{b^2-ac}}\tanh^{-1} \frac{ax+b}{\sqrt{b^2-ac}},\ + ac < b^2, ax+b < \sqrt{b^2-ac};\\ +&= \frac{-1}{\sqrt{b^2-ac}}\coth^{-1} \frac{ax+b}{\sqrt{b^2-ac}},\ + ac < b^2, ax+b > \sqrt{b^2-ac}; +\end{aligned} +\end{equation*} + +Thus, +\begin{align*} +\int_4^5 \frac{dx}{x^2-4x+3} +&= \left.-\coth^{-1}(x-2)\right]_4^5 = \coth^{-1}2 -\coth^{-1}3 \\ +&= \tanh^{-1}(.5) - \tanh^{-1}(.3333) = .5494 - .3466 += .2028.\footnotemark \\ +\int_2^{2.5} \frac{dx}{x^2-4x+3} +&= \left.-\tanh^{-1}(x-2)\right]_2^{2.5} = \tanh^{-1}0 - +\tanh^{-1}(0.5) =-.5494. +\end{align*} +\footnotetext{For $\tanh^{-1}(.5)$ interpolate between $\tanh (.54) += .4930$, $\tanh (.56) =.5080$ (see tables, pp.~\pageref{Table1p1}, +\pageref{Table1p2}); and similarly for $\tanh^{-1}(.3333)$.}% +\index{Interpolation} + +(By interpreting these two integrals as areas, show graphically +that the first is positive, and the second negative.)% +\index{Areas} + +\begin{align*} +15.\quad \int \frac{dx}{(a-x)\sqrt{x-b}} &= + \frac{2}{\sqrt{a-b}}\tanh^{-1}\sqrt{\frac{x-b}{a-b}}, \\ +&\text{or } \frac{-2}{\sqrt{b-a}}\tan^{-1}\sqrt{\frac{x-b}{b-a}}, \\ +&\text{or } \frac{2}{\sqrt{a-b}}\coth^{-1}\sqrt{\frac{x-b}{a-b}}; +\end{align*} +the real form to be taken. (Put $x - b = z^2$, and apply 9, 10.) + +\begin{align*} +16.\quad \int\frac{dx}{(a-x)\sqrt{b-x}} &= + \frac{2}{\sqrt{b-a}}\tanh^{-1}\sqrt{\frac{b-x}{b-a}}, \\ +&\text{or }\frac{2}{\sqrt{b-a}}\coth^{-1}\sqrt{\frac{b-x}{b-a}}, \\ +&\text{or } \frac{-2}{\sqrt{a-b}}\tan^{-1}\sqrt{\frac{b-x}{a-b}}; +\end{align*} +the real form to be taken. + +\begin{equation*} +17.\quad \int(x^2-a^2)^{\frac{1}{2}}dx = + \frac{1}{2}x(x^2-a^2)^{\frac{1}{2}} - + \frac{1}{2}a^2\cosh^{-1}\frac{x}{a}. +\end{equation*} + +By means of a reduction-formula this integral is easily made to +depend on 8. It may also be obtained by transforming the expression +into hyperbolic functions by the assumption $x = a\cosh u$, when the +integral takes the form +\begin{align*} +a^2\int\sinh^2u\,du = \frac{a^2}{2}\int(\cosh 2u-1)du +&= \frac{1}{4}a^2(\sinh 2u-2u) \\ +&= \frac{1}{2}a^2(\sinh u\cosh u-u), +\end{align*} +which gives 17 on replacing $a\cosh u$ by $x$, and $a\sinh u$ by +$(x^2-a^2)^{\frac{1}{2}}$. The geometrical interpretation of the +result is evident, as it expresses that the area of a +rectangular-hyperbolic segment $AMP$ is the difference between a +triangle $OMP$ and a sector $OAP$.% +\index{Areas}\index{Geometrical interpretation}\index{Hyperbola}% +\index{Reduction formula} + +\begin{align*} +18.\quad \int(a^2-x^2)^{\frac{1}{2}}dx &= + \frac{1}{2}x(a^2-x^2)^{\frac{1}{2}} + + \frac{1}{2}a^2\sin^{-1}\frac{x}{a}.\\ +19.\quad \int(x^2+a^2)^{\frac{1}{2}} dx &= + \frac{1}{2}x(x^2-a^2)^{\frac{1}{2}} + + \frac{1}{2}a^2\sinh^{-1}\frac{x}{a}.\\ +20.\quad \int\sec^3\phi\,d\phi &= + \int(1+\tan^2\phi)^{\frac{1}{2}}d\tan\phi\\ +&= \frac{1}{2}\tan\phi(1+\tan^2\phi)^{\frac{1}{2}} + +\frac{1}{2}\sinh^{-1}(\tan\phi) \\ +&= \frac{1}{2}\sec\phi\tan\phi + \frac{1}{2}\gd^{-1}\phi. \\ +21.\quad \int\sech^3u\,du &= + \frac{1}{2}\sech u\tanh u + \frac{1}{2}\gd u. +\end{align*} + +\small \begin{enumerate} +\item[Prob.~71.] What is the geometrical interpretation of 18, 19? + +\item[Prob.~72.] Show that $\int(ax^2+2bx+c)^{\frac{1}{2}}dx$ reduces +to 17, 18, 19, respectively: when $a$ is positive, with $ac < b^2$; +when $a$ is negative; and when $a$ is positive, with $ac > b^2$. + +\item[Prob.~73.] Prove +\begin{align*} +\int\sinh u\tanh u\,du &= \sinh u - \gd u,\\ +\int\cosh u\coth u\,du &= \cosh u + \log\tanh\frac{u}{2}. +\end{align*} + +\item[Prob.~74.] Integrate $(x^2+2x+5)^{-\frac{1}{2}}dx$, +$(x^2+2x+5)^{-1}dx$, $(x^2+2x+5)^{\frac{1}{2}}dx$. + +\item[Prob.~75.] In the parabola $y^2 = 4px$, if $s$ be the length of +arc measured from the vertex, and $\phi$ the angle which the tangent +line makes with the vertical tangent, prove that the intrinsic +equation of the curve is $\dfrac{ds}{d\phi} = 2p\sec^3\phi$, $s = +p\sec\phi\tan\phi + p\gd^{-1}\phi$.\index{Intrinsic equation}% +\index{Parabola} + +\item[Prob.~76.] The polar equation of a parabola being +$r = a\sec^2\theta$, referred to its focus as pole, express $s$ in +terms of $\theta$. + +\item[Prob.~77.] Find the intrinsic equation of the curve +$\dfrac{y}{a} = \cosh \dfrac{x}{a}$, and of the curve $\dfrac{y}{a} += \log\sec\dfrac{x}{a}$. + +\item[Prob.~78.] Investigate a formula of reduction for +$\dint\cosh^nx\,dx$; also integrate by parts \\ +$\cosh^{-1}x\,dx$, $\tanh^{-1}x\,dx$, $(\sinh^{-1}x)^2dx$; and show +that the ordinary methods of reduction for $\dint\cos^mx\sin^nx\,dx$ +can be applied to $\dint\cosh^mx\sinh^nx\,dx$.\index{Reduction +formula} +\end{enumerate}\normalsize + +\chapter{Functions of Complex Numbers.}% +\index{Complex numbers|(}\index{Function!of complex numbers}% +\index{Hyperbolic functions!of complex numbers|(}% +\index{Numbers, complex|(} + +As vector quantities are of frequent occurrence in Mathematical +Physics; and as the numerical measure of a vector in terms of a +standard vector is a complex number of the form $x+iy$, in which $x, +y$ are real, and $i$ stands for $\sqrt{-1}$; it becomes necessary in +treating of any class of functional operations to consider the +meaning of these operations when performed on such generalized +numbers.\footnote{% +The use of vectors in electrical theory is shown in Bedell and +Crehore's Alternating Currents, Chaps, XIV--XX (first published in +1892). The advantage of introducing the complex measures of such +vectors into the differential equations is shown by Steinmetz, Proc. +Elec. Congress, 1893; while the additional convenience of expressing +the solution in hyperbolic functions of these complex numbers is +exemplified by Kennelly, Proc. American Institute Electrical +Engineers, April 1895. (See below, Art.~37.)}% +\index{Alternating currents}% +\index{Bedel and Crehore's alternating currents}% +\index{Kennelly on alternating currents}% +\index{Steinmetz on alternating currents}\index{Vectors} The +geometrical definitions of $\cosh u$, $\sinh u$, given in Art.~7, +being then no longer applicable, it is necessary to assign to each +of the symbols $\cosh(x+iy)$, $\sinh(x+iy)$, a suitable algebraic +meaning, which should be consistent with the known algebraic values +of $\cosh x$, $\sinh x$, and include these values as a particular +case when $y = 0$. The meanings assigned should also, if possible, +be such as to permit the addition-formulas of Art.~11 to be made +general, with all the consequences that flow from them. + +Such definitions are furnished by the algebraic developments in +Art.~16, which are convergent for all values of $u$, real or +complex. Thus the definitions of $\cosh(x+iy)$, $\sinh(x+iy)$ are to +be +\begin{equation*} + \left. + \begin{aligned} + \cosh(x+iy) &= 1 + \frac{1}{2!}(x+iy)^2 + \frac{1}{4!}(x+iy)^4 + + \dots,\\ + \sinh(x+iy) & = (x+iy) + \frac{1}{3!}(x+iy)^3 + \dots + \end{aligned} + \right\}\tag{52} +\end{equation*} + +From these series the numerical values of $\cosh(x+iy)$, +$\sinh(x+iy)$ could be computed to any degree of approximation, when +$x$ and $y$ are given. In general the results will come out in the +complex form\footnote{% +It is to be borne in mind that the symbols cosh, sinh, here stand +for algebraic operators which convert one number into another; or +which, in the language of vector-analysis, change one vector into +another, by stretching and turning.} +\begin{align*} + \cosh(x+iy) &= a+ib,\\ + \sinh(x+iy) &= c+id. +\end{align*} +The other functions are defined as in Art.~7, eq.~(9). + +\small \begin{enumerate} +\item[Prob.~79.] Prove from these definitions that, whatever $u$ may +be, +\begin{align*} + \cosh(-u) &= \cosh u, & + \sinh(-u) &= -\sinh u, \\ + \frac{d}{du} \cosh u &= \sinh u, & + \frac{d}{du} \sinh u &= \cosh u,\\ + \frac{d^2}{du^2} \cosh mu &= m^2\cosh mu, & + \frac{d^2}{du^2} \sinh mu &= m^2\sinh mu.\footnotemark +\end{align*} +\footnotetext{The generalized hyperbolic functions usually present +themselves in Mathematical Physics as the solution of the +differential equation $\dfrac{d^2\phi}{du^2} = m^2\phi$, where +$\phi$, $m$, $u$ are complex numbers, the measures of vector +quantities. (See Art.~37.)}\index{Operators, generalized}% +\index{Permanence of equivalence} +\end{enumerate}\index{Circular functions}\normalsize + +\chapter{Addition-Theorems for Complexes.}% +\index{Addition-theorems} + +The addition-theorems for $\cosh(u+v)$, etc., where $u$, $v$ are +complex numbers, may be derived as follows. First take $u$, $v$ as +real numbers, then, by Art.~11, +\begin{align*} +\cosh(u+v) &= \cosh u \cosh v + \sinh u\ \sinh v; \\ +\intertext{hence} +1 + \frac{1}{2!}(u+v)^2+ \dots + &= \left(1+\frac{1}{2!}u^2+\dots\right) + \left(1+\frac{1}{2!}v^2+\dots\right) \\ + &\quad + \left(u+\frac{1}{3!}u^3+\dots\right) + \left(v+\frac{1}{3!}v^3+\dots\right) +\end{align*} + +This equation is true when $u$, $v$ are any real numbers. It must, +then, be an algebraic identity. For, compare the terms of the $r$th +degree in the letters $u, v$ on each side. Those on the left are +$\dfrac{1}{r!}(u+v)^r$; and those on the right, when collected, form +an $r$th-degree function which is numerically equal to the former +for more than $r$ values of $u$ when $v$ is constant, and for more +than $r$ values of $v$ when $u$ is constant. Hence the terms of the +$r$th degree on each side are algebraically identical functions of +$u$ and $v$.\footnote{% +``If two $r$th-degree functions of a single variable be equal for +more than $r$ values of the variable, then they are equal for all +values of the variable, and are algebraically identical.''} +Similarly for the terms of any other degree. Thus the equation above +written is an algebraic identity, and is true for all values of $u$, +$v$, whether real or complex. Then writing for each side its symbol, +it follows that +\begin{align*} + \cosh(u+v) &= \cosh u \cosh v + \sinh u \sinh v;\tag{53} \\ +\intertext{and by changing $v$ into $-v$,} + \cosh(u-v) &= \cosh u \cosh v - \sinh u \sinh v.\tag{54} +\end{align*} + +In a similar manner is found +\begin{equation} + \sinh(u\pm v) = \sinh u \cosh v \pm \cosh u \sinh v.\tag{55} +\end{equation} + +In particular, for a complex argument, +\begin{equation} + \left. + \begin{aligned} + \cosh(x\pm iy) &= \cosh x \cosh iy \pm \sinh x \sinh iy,\\ + \sinh(x\pm iy) &= \sinh x \cosh iy \pm \cosh x \sinh iy. + \end{aligned} + \right\}\tag{56} +\end{equation} + +\small \begin{enumerate} +\item[Prob.~79.] Show, by a similar process of generalization,% +\footnote{This method of generalization is sometimes called the +principle of the ``permanence of equivalence of forms.'' It is not, +however, strictly speaking, a ``principle,'' but a method; for, the +validity of the generalization has to be demonstrated, for any +particular form, by means of the principle of the algebraic identity +of polynomials enunciated in the preceding foot-note. (See Annals of +Mathematics, Vol.~6, p.~81.)} that if $\sin u$, $\cos u$, $\exp u$% +\footnote{The symbol $\exp u$ stands for ``exponential function of +u,'' which is identical with $e^u$ when $u$ is real.} be defined by +their developments in powers of $u$, then, whatever $u$ may be, +\begin{align*} + \sin (u+v) &= \sin u \cos v + \cos u \sin v, \\ + \cos (u+v) &= \cos u \cos v - \sin u \sin v, \\ + \exp (u+v) &= \exp u \exp v. +\end{align*}\index{Generalization} + +\item[Prob. 80.] Prove that the following are identities: +\begin{align*} + \cosh^2 u - \sinh^2 u &= 1, \\ + \cosh u + \sinh u &= \exp u, \\ + \cosh u - \sinh u &= \exp (-u), \\ + \cosh u &= \tfrac{1}{2} [\exp u + \exp (-u)], \\ + \sinh u &= \tfrac{1}{2} [\exp u - \exp (-u)]. +\end{align*} +\end{enumerate} \normalsize + +\chapter{Functions of Pure Imaginaries.}% +\index{Functions!of pure imaginaries}\index{Pure imaginary} + +In the defining identities% +\index{Algebraic identity} +\begin{align*} + \cosh u &= 1 + \frac{1}{2!} u^2 + \frac{1}{4!} u^4 + \dotsb, \\ + \sinh u &= 1 + \frac{1}{3!} u^3 + \frac{1}{5!} u^5 + \dotsb, +\end{align*} +put for $u$ the pure imaginary $iy$, then +\begin{align*} +\cosh iy &= 1 - \frac{1}{2!} y^2 + \frac{1}{4!} y^4 - \dotsb + = \cos y, \tag{57} \\ +\sinh iy &= iy + \frac{1}{3!} (iy)^3 + \frac{1}{5!} (iy)^5 + + \dotsb \\ +&= i\left[ y - \frac{1}{3!} y^3 + \frac{1}{5!} y^5 - \dotsb \right] += i\sin y, \tag{58} \\ +\intertext{and, by division,} +\tanh iy &= i\tan y. \tag{59} +\end{align*} + +These formulas serve to interchange hyperbolic and circular +functions. The hyperbolic cosine of a pure imaginary is real, and +the hyperbolic sine and tangent are pure imaginaries. + +The following table exhibits the variation of $\sinh u$, $\cosh u$. +$\tanh u$, $\exp u$, as $u$ takes a succession of pure imaginary +values. +%%\index{Tables} %% RWD Nickalls moved \index{Tables} into table footnote below + +\begin{minipage}{10cm}{ +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline \rule[-5pt]{0pt}{16pt} + $u$ & $\sinh u$ & $\cosh u$ & $\tanh u$ & $\exp u$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $0$ & $0$ & $1$ & $0$ & $1$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $\frac{1}{4}i\pi$ & $.7i$ & $.7\footnotemark$ & $i$ & $.7(1+i)$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $\frac{1}{2}i\pi$ & $i$ & $0$ & $\infty i$ & $i$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $\frac{3}{4}i\pi$ & $.7i$ & $-.7$ & $-i$ & $.7(1-i)$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $i\pi$ & $0$ & $-1$ & $0$ & $-1$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $\frac{5}{4}i\pi$ & $-.7i$ & $-.7$ & $i$ & $-.7(1+i)$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $\frac{3}{2}i\pi$ & $-i$ & $0$ & $\infty i$ & $-i$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $\frac{7}{4}i\pi$ & $-.7i$ & $.7$ & $-i$ & $-.7(1-i)$ \\ +\hline \rule[-5pt]{0pt}{16pt} + $2i\pi$ & $0$ & $1$ & $0$ & $1$ \\ +\hline +\end{tabular} +\footnotetext{In this table $.7$ is written for +$\frac{1}{2}\sqrt{2}, = .707\dotsc$.\index{Tables}} +%% RWD Nickalls: I have moved the above \index{} command to here% +\end{center} +}\end{minipage} + + + +\small \begin{enumerate} +\item[Prob.~81.] Prove the following identities: +\begin{align*} +\cos y = \cosh iy \phantom{\frac{1}{i}} + &= \frac{1}{2} \left[\exp iy + \exp (-iy)\right], \\ +\sin y = \frac{1}{i} \sinh iy + &= \frac{1}{2i} \left[\exp iy - \exp (-iy)\right], \\ +\cos y + i\sin y &= \cosh iy + \sinh iy = \exp iy, \\ +\cos y - i\sin y &= \cosh iy - \sinh iy = \exp\, (-iy), \\ +\cos iy &= \cosh y, \quad \sin iy = i\sinh y. +\end{align*}\index{Circular functions}% +\index{Hyperbolic functions!relations to circular functions}% +\index{Interchange of hyperbolic and circular functions}% +\index{Relations among functions} + +\item[Prob.~82.] Equating the respective real and imaginary parts +on each side of the equation $\cos ny + i\sin ny = (\cos y + i\sin +y)^n$, express $\cos ny$ in powers of $\cos y$, $\sin y$; and hence +derive the corresponding expression for $\cosh ny$.\index{Circular +functions!of complex numbers} + +\item[Prob.~83.] Show that, in the identities (57) and (58), +$y$ may be replaced by a general complex, and hence that +\begin{align*} +\sinh (x \pm iy) &= \pm i \sin (y \mp ix),\\ +\cosh (x \pm iy) &= \cos (y \mp ix),\\ +\sin (x \pm iy) &= \pm i \sinh (y \mp ix),\\ +\cos (x \pm iy) &= \cosh (y \mp ix). +\end{align*} + +\item[Prob.~84.] From the product-series for $\sin x$ derive +that for $\sinh x$: +\begin{align*} +\sin x &= x\left(1 - \frac{x^2}{ \pi^2}\right) + \left(1 - \frac{x^2}{2^2\pi^2}\right) + \left(1 - \frac{x^2}{3^2\pi^2}\right) \ldots,\\ +\sinh x &= x\left(1 + \frac{x^2}{ \pi^2}\right) + \left(1 + \frac{x^2}{2^2\pi^2}\right) + \left(1 + \frac{x^2}{3^2\pi^2}\right) \ldots. +\end{align*}\index{Product-series} +\end{enumerate} \normalsize + + +\chapter{Functions of $x+iy$ in the Form $X+iY$.} + +By the addition-formulas, +\begin{gather*} +\begin{aligned} +\cosh (x + iy) &= \cosh x \cosh iy + \sinh x \sinh iy,\\ +\sinh (x + iy) &= \sinh x \cosh iy + \cosh x \sinh iy, +\end{aligned} +\intertext{but} +\cosh iy = \cos y,\quad \sinh iy = i \sin y, \\ +\intertext{hence} +\left. +\begin{aligned} +\cosh (x + iy) &= \cosh x \cos y + i \sinh x \sin y,\\ +\sinh (x + iy) &= \sinh x \cos y + i \cosh x \sin y. +\end{aligned} +\right\}\tag{60} +\end{gather*} + +Thus if $\cosh (x + iy) = a+ib$, $\sinh (x + iy) = c + id$, then +\begin{equation} +\left. +\begin{aligned} +a &= \cosh x \cos y, &\quad b &= \sinh x \sin y,\\ +c &= \sinh x \cos y, &\quad d &= \cosh x \sin y. +\end{aligned} +\right\}\tag{61} +\end{equation} + +From these expressions the complex tables at the end of this chapter +have been computed. + +Writing $\cosh z = Z$, where $z = x + iy$, $Z = X + iY$; let the +complex numbers $z, Z$ be represented on Argand diagrams,% +\index{Argand diagram}\index{Construction!of charts} in the usual +way, by the points whose coordinates are $(x, y)$, $(X, Y)$; and let +the point $z$ move parallel to the $y$-axis, on a given line $x = +m$, then the point $Z$ will describe an ellipse whose equation, +obtained by eliminating $y$ between the equations $X= \cosh m \cos +y$, $Y= \sinh m \sin y$, is +\begin{equation*} +\frac{X^2}{(\cosh m)^2} + \frac{Y^2}{(\sinh m)^2} = 1, +\end{equation*} +and which, as the parameter $m$ varies, represents a series of +confocal ellipses, the distance between whose foci is unity. +Similarly, if the point $z$ move parallel to the $x$-axis, on a +given line $y=n$, the point $Z$ will describe an hyperbola whose +equation, obtained by eliminating the variable $x$ from the +equations. $X = \cosh x \cos n$, $Y = \sinh x \sin n$, is +\begin{equation*} +\frac{X^2}{(\cos n)^2} - \frac{Y^2}{(\sin n)^2} = 1, +\end{equation*} +and which, as the parameter $n$ varies, represents a series of +hyperbolas confocal with the former series of +ellipses.\index{Ellipses, chart of confocal}\index{Hyperbola} + +These two systems of curves, when accurately drawn at close +intervals on the $Z$ plane, constitute a chart of the hyperbolic +cosine; and the numerical value of $\cosh (m + in)$ can be read off +at the intersection of the ellipse whose parameter is $m$ with the +hyperbola whose parameter is $n$.\footnote{% +Such a chart is given by Kennelly, Proc.~A.~I.~E.~E., April 1895, +and is used by him to obtain the numerical values of $\cosh (x + +iy)$, $\sinh (x+iy)$, which present themselves as the measures of +certain vector quantities in the theory of alternating currents. +(See Art.~37.) The chart is constructed for values of $x$ and of $y$ +between 0 and 1.2; but it is available for all values of $y$, on +account of the periodicity of the functions.}% +\index{Alternating currents}\index{Chart!of hyperbolic functions}% +\index{Kennelly's chart} A similar chart can be drawn for $\sinh +(x+iy)$, as indicated in Prob.~85. + +\medskip + +\begin{comment} +RWD Nickalls : I have inserted the following `empty' subsubsection{} command, +---which is effectively invisible. LaTeX requires some form of `section' command +as an anchor in order to make the pageref links work correctly. Otherwise, the +links will just go to the page of the current chapter (which is one page earlier +in this particular case). +Note that this particular \label{} is used (pointed to) by three other locations +using the \pageref{period-hyp-funct} command. These 3 links all work correctly now. +\end{comment} + +\subsubsection*{}% +Periodicity of Hyperbolic Functions. \label{period-hyp-funct}---The +functions $\sinh u$ and $\cosh u$ have the pure imaginary period +$2i\pi$. For +\begin{align*} +\sinh(u+2i\pi) &= \sinh u \cos 2\pi + i \cosh u \sin 2\pi = \sinh u,\\ +\cosh(u+2i\pi) &= \cosh u \cos 2\pi + i \sinh u \sin 2\pi = \cosh u. +\end{align*}\index{Function!periodic}\index{Periodicity} + +The functions $\sinh u$ and $\cosh u$ each change sign when the +argument $u$ is increased by the half period $i\pi$. For +\begin{align*} +\sinh (u+i\pi) &= \sinh u \cos \pi + i \cosh u \sin \pi = -\sinh u,\\ +\cosh (u+i\pi) &= \cosh u \cos \pi + i \sinh u \sin \pi = -\cosh u. +\end{align*} + +The function $\tanh u$ has the period $i\pi$. For, it follows from +the last two identities, by dividing member by member, that +\begin{equation*} +\tanh (u+i\pi) = \tanh u. +\end{equation*} + +By a similar use of the addition formulas it is shown that +\begin{equation*} +\sinh (u + \frac{1}{2} i\pi) = i \cosh u,\quad +\cosh (u + \frac{1}{2} i\pi) = i \sinh u. +\end{equation*} + +By means of these periodic, half-periodic, and quarter-periodic +relations, the hyperbolic functions of $x + iy$ are easily +expressible in terms of functions of $x+iy'$, in which $y'$ is less +than $\frac{1}{2} \pi$. + +The hyperbolic functions are classed in the modern function-theory +of a complex variable as functions that are singly periodic with a +pure imaginary period, just as the circular functions are singly +periodic with a real period, and the elliptic functions are doubly +periodic with both a real and a pure imaginary period. + +\medskip Multiple Values of Inverse Hyperbolic Functions.---It follows +from the periodicity of the direct functions that the inverse +functions $\sinh^{-1} m$ and $\cosh^{-1} m$ have each an indefinite +number of values arranged in a series at intervals of $2i\pi$. That +particular value of $\sinh^{-1} m$ which has the coefficient of $i$ +not greater than $\frac{1}{2}\pi$ nor less than $-\frac{1}{2}\pi$ is +called the principal value of $\sinh^{-1} m$; and that particular +value of $\cosh^{-1} m$ which has the coefficient of $i$ not greater +than $\pi$ nor less than zero is called the principal value of +$\cosh^{-1} m$. When it is necessary to distinguish between the +general value and the principal value the symbol of the former will +be capitalized; thus +\begin{gather*} +\text{Sinh}^{-1} m = \sinh^{-1} m + 2ir\pi,\quad +\text{Cosh}^{-1} m = \cosh^{-1} m + 2ir\pi,\\ +\text{Tanh}^{-1} m = \tanh^{-1} m + ir\pi, +\end{gather*} +in which $r$ is any integer, positive or negative.% +\index{Ambiguity of value}\index{Anti-hyperbolic functions}% +\index{Multiple values} + +\medskip Complex Roots of Cubic Equations.---It is well known that when +the roots of a cubic equation are all real they are expressible in +terms of circular functions. Analogous hyperbolic expressions are +easily found when two of the roots are complex. Let the cubic, with +second term removed, be written +\begin{equation*} +x^3 \pm 3bx = 2c. +\end{equation*} + +Consider first the case in which $b$ has the positive sign. Let +$x = r \sinh u$, substitute, and divide by $r^3$, then +\begin{equation*} +\sinh^3 u + \frac{3b}{r^2} \sinh u = \frac{2c}{r^3}. +\end{equation*} + +Comparison with the formula $\sinh^3 u + \frac{3}{4} \sinh u = +\frac{1}{4} \sinh 3u$ gives +\begin{gather*} +\frac{3b}{r^2} = \frac{3}{4},\quad +\frac{2c}{r^3} = \frac{\sinh 3u}{4},\\ +\intertext{whence} +r = 2b^{\frac{1}{2}},\quad +\sinh 3u = \frac{c}{b^{\frac{3}{2}}},\quad +u = \frac{1}{3} \sinh^{-1} \frac{c}{b^{\frac{3}{2}}}; \\ +\intertext{therefore} +x = 2b^{\frac{1}{2}} + \sinh \left(\frac{1}{3}\sinh^{-1}\frac{c}{b^{\frac{3}{2}}} + \right), +\end{gather*} +in which the sign of $b^{\frac{1}{2}}$ is to be taken the same as +the sign of $c$. Now let the principal value of +$\sinh^{-1}\dfrac{c}{b^{\frac{3}{2}}}$, found from the tables, be +$n$; then two of the imaginary values are $n\pm 2i\pi$, hence the +three values of $x$ are $2b^{\frac{1}{2}} \sinh\dfrac{n}3$ and +$2b^{\frac{1}{2}} \sinh\left(\dfrac{n}{3} \pm \dfrac{2i\pi}{3} +\right)$. The last two reduce to $-b^{\frac{1}{2}} +\sinh\left(\dfrac{n}{3} \pm i\sqrt{3}\cosh\dfrac{n}{3} \right)$. + +Next, let the coefficient of $x$ be negative and equal to $-3b$. It +may then be shown similarly that the substitution $x = r \sin +\theta$ leads to the three solutions +\begin{equation*} +-2b^{\frac{1}{2}} \sin\frac{n}{3},\quad +b^{\frac{1}{2}} \left(\sin\frac{n}{3} \pm + \sqrt{3}\cos\frac{n}{3}\right),\quad +\text{ where } n = \sin^{-1}\frac{c}{b^{\frac{3}{2}}}. +\end{equation*} +These roots are all real when $c \ngtr b^{\frac{3}{2}}$. If $c +> b^{\frac{3}{2}}$, the substitution $x = r\cosh u$ leads to the +solution +\begin{equation*} +x = 2b^{\frac{1}{2}} \cosh\left(\frac{1}{3} + \cosh^{-1}\frac{c}{b^{\frac{3}{2}}} \right), +\end{equation*} +which gives the three roots +\begin{equation*} +2b^{\frac{1}{2}} \cosh\frac{n}{3},\quad +-b^{\frac{1}{2}} \left( \cosh\frac{n}{3} + \pm i\sqrt{3}\sinh\frac{n}3 \right),\quad +\text{ wherein } n = \cosh^{-1}\frac{c}{b^{\frac{3}{2}}}, +\end{equation*} +in which the sign of $b^{\frac{1}{2}}$ is to be taken the same as +the sign of $c$. + +\index{Hyperbolic functions!applictions of|(} +\small \begin{enumerate} +\item[Prob.~85.] Show that the chart of $\cosh (x + iy)$ can be adapted +to $\sinh (x + iy)$, by turning through a right angle; also to $\sin +(x + iy)$. + +\item[Prob.~86.] Prove the identity +\begin{equation*} +\tanh (x + iy) = \frac{\sinh 2x + i \sin 2y}{\cosh 2x + \cos 2y}. +\end{equation*} + +\item[Prob.~87.] If $\cosh (x + iy) = a + ib$, be written in the +``modulus and amplitude'' form as $r(\cos\theta + i\sin \theta), = r +\exp i\theta$, then +\begin{align*} +r^2 = a^2 + b^2 &= \cosh^2 x = \sin^2 y = \cos^2 y - \sinh^2 x,\\ +\tan \theta = \frac{b}{a} &= \tanh x \tan y. +\end{align*}% +\index{Amplitude!of complex number}\index{Modulus} + +\index{Applications|(} +\item[Prob.~88.] Find the modulus and amplitude of $\sinh (x + iy)$. + +\item[Prob.~89.] Show that the period of $\exp \dfrac{2\pi u}{a}$ is $ia$. + +\item[Prob.~90.] When $m$ is real and $> 1$, $\cos^{-1} m = i +\cosh^{-1} m$, $\sin^{-1} m = \frac{\pi}2 - i \cosh^{-1} m$. + +When $m$ is real and $< 1$, $\cosh^{-1} m = i \cos^{-1} m$. +\end{enumerate}\index{Complex numbers|)}% +\index{Hyperbolic functions!of complex numbers|)}% +\index{Numbers, complex|)} \normalsize + +\chapter{The Catenary}\index{Catenary}\index{Physical problems|(} + +A flexible inextensible string is suspended from two fixed points, +and takes up a position of equilibrium under the action of gravity. +It is required to find the equation of the curve in which it hangs. + +Let $w$ be the weight of unit length, and $s$ the length of arc $AP$ +measured from the lowest point $A$; then $ws$ is the weight of the +portion $AP$. This is balanced by the terminal tensions, $T$ acting +in the tangent line at $P$, and $H$ in the horizontal tangent. +Resolving horizontally and vertically gives +\begin{equation*} +T\cos\phi = H,\quad T\sin\phi = ws, +\end{equation*} +in which $\phi$ is the inclination of the tangent at $P$; hence +\begin{equation*} +\tan\phi = \frac{ws}{H} = \frac{s}{c}, +\end{equation*} +where $c$ is written for $\dfrac{H}{w}$, the length whose weight is +the constant horizontal tension; therefore +\begin{gather*} +\frac{dy}{dx}=\frac{s}{c},\quad +\frac{ds}{dx}=\sqrt{1+\frac{s^2}{c^2}},\quad +\frac{dx}{c}=\frac{ds}{\sqrt{s^2+c^2}}, \\ +\frac{x}{c}=\sinh^{-1}\frac{s}{c},\quad +\sinh\frac{x}{c}=\frac{s}{c}=\frac{dy}{dx},\quad +\frac{y}{c}=\cosh\frac{x}{c}, +\end{gather*} +which is the required equation of the catenary, referred to an axis +of $x$ drawn at a distance $c$ below $A$. + +The following trigonometric method illustrates the use of the +gudermanian: The ``intrinsic equation,'' $s = c\tan\phi$, gives $ds += c\sec^2\phi\, d\phi$; hence $dx = ds\cos\phi = c\sec\phi\, d\phi$; +$dy = ds\sin\phi = c\sec\phi\tan\phi\, d\phi$; thus $x = +c\gd^{-1}\phi, y = c\sec\phi$; whence $\frac{y}{c} = \sec\phi = +\sec\gd\frac{x}{c} = \cosh \frac{x}{c}$; and $\frac{s}{c} = \tan +\gd\frac{x}{c} = \sinh\frac{x}{c}$.% +\index{Anti-gudermanian}\index{Differential equation}% +\index{Gudermanian!function}\index{Intrinsic equation} + +\medskip Numerical Exercise.---A chain whose length is 30 feet is +suspended from two points 20 feet apart in the same horizontal; find +the parameter $c$, and the depth of the lowest point. + +The equation $\frac{s}{c} = \sinh\frac{x}{c}$ gives $\frac{15}{c} = +\sinh\frac{10}{c}$, which, by putting $\frac{10}{c} = z$, may be +written $1.5 z = \sinh z$. By examining the intersection of the +graphs of $y = \sinh z$, $y = 1.5 z$, it appears that the root of +this equation is $z = 1.6$, nearly. To find a closer approximation +to the root, write the equation in the form $f(z) = \sinh z - 1.5 z += 0$, then, by the tables, +\begin{align*} + f(1.60) &= 2.3756 - 2.4000 = -.0244, \\ + f(1.62) &= 2.4276 - 2.4300 = -.0024, \\ + f(1.64) &= 2.4806 - 2.4600 = +.0206; +\end{align*} +whence, by interpolation, it is found that $f(1.6221) = 0$, and $z = +1.6221$, $c = \frac{10}{z} = 6.1649$. The ordinate of either of the +fixed points is given by the equation +\begin{equation*} + \frac{y}{c} = \cosh\frac{x}{c} = \cosh\frac{10}{c} = + \cosh 1.6221 = 2.6306, +\end{equation*} +from tables; hence $y = 16.2174$, and required depth of the vertex $ += y - c = 10.0525$ feet.\footnote{See a similar problem in Chap.~1, +Art.~7.}\index{Interpolation} + +\small \begin{enumerate} +\item[Prob.~91.] In the above numerical problem, find the inclination +of the terminal tangent to the horizon.\index{Equations!Numerical} + +\item[Prob.~92.] If a perpendicular $MN$ be drawn from the foot of +the ordinate to the tangent at $P$, prove that $MN$ is equal to the +constant $c$, and that $NP$ is equal to the arc $AP$. Hence show +that the locus of $N$ is the involute of the catenary, and has the +property that the length of the tangent, from the point of contact +to the axis of $x$, is constant. (This is the characteristic +property of the tractory).\index{Involute!of catenary}% +\index{Tractory} + +\item[Prob.~93.] The tension $T$ at any point is equal to the weight +of a portion of the string whose length is equal to the ordinate $y$ +of that point. + +\item[Prob.~94.] An arch in the form of an inverted catenary\footnote{ +For the theory of this form of arch, see ``Arch'' in the +Encyclop\ae{}dia Britannica.} is $30$ feet wide and $10$ feet high; +show that the length of the arch can be obtained from the equations +$\cosh z - \frac{2}{3}z = i$, $2s = \dfrac{30}{z} \sinh z$.% +\index{Arch} +\end{enumerate} \normalsize + +\chapter{Catenary of Uniform Strength.}\index{Catenary!of uniform +strength} + +If the area of the normal section at any point be made proportional +to the tension at that point, there will then be a constant tension +per unit of area, and the tendency to break will be the same at all +points. To find the equation of the curve of equilibrium under +gravity, consider the equilibrium of an element $PP'$ whose length +is $ds$, and whose weight is $g\rho\omega\, ds$, where $\omega$ is +the section at $P$, and $\rho$ the uniform density. This weight is +balanced by the difference of the vertical components of the +tensions at $P$ and $P'$, hence +\begin{align*} +d(T\sin\phi) &= g\rho\omega\, ds,\\ +d(T\cos\phi) &= 0; +\end{align*} +therefore $T\cos\phi = H$, the tension at the lowest point, and $T = +H \sec \phi$. Again, if $\omega_0$ be the section at the lowest +point, then by hypothesis $\frac{\omega}{\omega_0} = \frac{T}{H} = +\sec \phi$, and the first equation becomes +\begin{gather*} +Hd(\sec\phi\sin\phi) = g\rho\omega_0\sec\phi\, ds, \\ +\intertext{or} +cd\tan\phi = \sec\phi\, ds, +\end{gather*} +where $c$ stands for the constant $\dfrac{H}{g\rho\omega_0}$, the +length of string (of section $\omega_0$) whose weight is equal to +the tension at the lowest point; hence, +\begin{equation*} +ds = c \sec\phi\, d\phi,\quad \frac{s}{c} = \gd^{-1}\phi, +\end{equation*} +the intrinsic equation of the catenary of uniform strength.% +\index{Intrinsic equation} + +Also +\begin{gather*} +dx = ds\cos\phi = c d\phi,\quad +dy = ds\sin\phi = c\tan\phi\, d\phi; \\ +\intertext{hence} +x = c\phi,\quad y = c \log\sec\phi, +\intertext{and thus the Cartesian equation is} +\frac{y}{c} = \log \sec\frac{x}{c}, +\end{gather*} +in which the axis of $x$ is the tangent at the lowest +point.\index{Differential equation} + +\small \begin{enumerate} +\item[Prob.~95.] Using the same data as in Art.~31, find the parameter +$c$ and the depth of the lowest point. (The equation $\dfrac{x}{c} = +\gd\dfrac{s}{c}$ gives $\dfrac{10}{c} = \gd\dfrac{15}{c}$, which, by +putting $\dfrac{15}{c} = z$, becomes $\gd{z}= \dfrac{2}{3}z$. From +the graph it is seen that $z$ is nearly $1.8$. If $f(z) = +\gd{z}-\dfrac{2}{3}z$, then, from the tables of the gudermanian at +the end of this chapter, +\begin{align*} +f(1.80) & = 1.2432 - 1.2000 = +.0432,\\ +f(1.90) & = 1.2739 - 1.2667 = +.0072,\\ +f(1.95) & = 1.2881 - 1.3000 = -.0119, +\end{align*} +whence, by interpolation, $z = 1.9189$ and $c = 7.8170$. Again, +$\dfrac{y}{c} = \log_e{\sec{\dfrac{x}{c}}}$; but $\dfrac{x}{c} - +\dfrac{10}{c} = 1.2793$; and $1.2793 \text{ radians } = +73^{\circ}\,17'\,55''$; hence $y = 7.8170 \times .41914 \times +2.3026 = 7.5443$, the required depth.)\index{Interpolation} + +\item[Prob.~96.] Find the inclination of the terminal tangent. + +\item[Prob.~97.] Show that the curve has two vertical asymptotes. + +\item[Prob.~98.] Prove that the law of the tension $T$, and of the +section $\omega$, at a distance $s$, measured from the lowest point +along the curve, is +\begin{equation*} +\frac{T}{H} = \frac{\omega}{\omega_0} = \cosh{\frac{c}{h}}; +\end{equation*} +and show that in the above numerical example the terminal section is +$3.48$ times the minimum section.\index{Equations!Numerical} + +\item[Prob.~99.] Prove that the radius of curvature is given by $\rho = +c \cosh{\dfrac{s}{f}}$. Also that the weight of the arc $s$ is given +by $W = H \sinh{\dfrac{s}{f}}$, in which $s$ is measured from the +vertex. +\end{enumerate} \normalsize + +\chapter{The Elastic Catenary.}% +\index{Catenary!Elastic}\index{Curvature} + +An elastic string of uniform section and density in its natural +state is suspended from two points. Find its equation of +equilibrium. + +Let the element $d\sigma$ stretch into $ds$; then, by Hooke's law, +$ds = d\sigma(1 + \lambda T)$, where $\lambda$ is the elastic +constant of the string; hence the weight of the stretched element +$ds = g\rho\omega\, d\sigma = \dfrac{g\rho\omega\, ds}{(1 + \lambda +T)}$. Accordingly, as before, +\begin{align*} +d(T\sin{\phi}) & = \frac{g\rho\omega\, ds}{(1 + \lambda T)},\\ +\intertext{and} +T\cos{\phi} & = H = g\rho\omega c,\\ +\intertext{hence} +cd(\tan{\phi}) & = \frac{ds}{(1 + \mu\sec{\phi})}, +\intertext{in which $\mu$ stands for $\lambda H$, the extension at +the lowest point; therefore} +ds &= c(\sec^2\phi + \mu\sec^3\phi)d\phi, \\ +\frac{s}{c} &= \tan\phi + \frac{1}{2}\mu(\sec\phi\tan\phi + + \gd^{-1}\phi), \tag*{[prob.~20, p.~37} +\end{align*} +which is the intrinsic equation of the curve, and reduces to that of +the common catenary when $\mu = 0$. The coordinates $x$, $y$ may be +expressed in terms of the single parameter $\phi$ by putting +\begin{align*} +dx &= ds\cos\phi = c(\sec\phi + \mu\sec^2\phi)d\phi, \\ +dy &= ds\sin\phi = c(\sec^2\phi + \mu\sec^3\phi)\sin\phi\, d\phi. \\ +\intertext{Whence} +\frac{x}{c} &= \gd^{-1}\phi + \mu\tan\phi,\quad +\frac{y}{c} = \sec\phi + \frac{1}{2}\mu\tan^2\phi. +\end{align*}\index{Intrinsic equation} + +These equations are more convenient than the result of eliminating +$\phi$, which is somewhat complicated. + +\chapter{The Tractory.}% +\index{Arch}\index{Tractory} + +[Note.\footnote{This curve is used in Schiele's anti-friction pivot +(Minchin's Statics, Vol.~I, p.~242); and in the theory of the skew +circular arch, the horizontal projection of the joints being a +tractory. (See ``Arch,'' Encyclopædia Britannica.) The equation +$\phi=\gd\frac{t}{c}$ furnishes a convenient method of plotting the +curve.}] + +To find the equation of the curve which possesses the property that +the length of the tangent from the point of contact to the axis of +$x$ is constant. + +\begin{center} +\includegraphics[width=40mm]{fig11.png} +\end{center} + +Let $PT$, $P'T'$ be two consecutive tangents such that $PT = P'T' = +c$, and let $OT = t$; draw $TS$ perpendicular to $P'T'$; then if +$PP' = ds$, it is evident that $ST'$ differs from $ds$ by an +infinitesimal of a higher order. Let $PT$ make an angle $\phi$ with +$OA$, the axis of $y$; then (to the first order of infinitesimals) +$PT d\phi = TS = TT'\cos\phi$; that is, +\begin{gather*} +c\,d\phi = \cos\phi\, dt,\quad t = c\,\gd^{-1}\phi, \\ +x = t-c\,\sin\phi = c(\gd^{-1}\phi-\sin\phi),\quad y = c\,\cos\phi. +\end{gather*}% +\index{Anti-gudermanian}\index{Differential equation} + +This is a convenient single-parameter form, which gives all values +of $x$, $y$ as $\phi$ increases from $0$ to $\frac{1}{2}\pi$. The +value of $s$, expressed in the same form, is found from the relation +\begin{equation*} +ds = ST' = dt\,\sin\phi = c\tan\,\phi\,d\phi,\quad + s = c\,\log_e\sec\phi. +\end{equation*} + +At the point $A$, $\phi=0$, $x=0$, $s=0$, $t=0$, $y=c$. The +Cartesian equation, obtained by eliminating $\phi$, is +\begin{equation*} +\frac{x}{c}= \gd^{-1}\left(\cos^{-1}\frac{y}{c}\right) - + \sin\left(\cos^{-1}\frac{y}{c}\right) = + \cosh^{-1}\frac{c}{y} - \sqrt{1-\frac{y^2}{c^2}}. +\end{equation*} + +If $u$ be put for $\dfrac{t}{c}$, and be taken as independent +variable, $\phi=\gd u$, $\dfrac{x}{c} = u - \tanh u$, $\dfrac{y}{c} += \sech u$, $\dfrac{s}{c} = \log\cosh u.$ + +\small \begin{enumerate} +\item[Prob.~100.] Given $t = 2c$, show that $\phi = 74^\circ\, 35'$, +$s = 1.3249c$, $y = .2658c$, $x = 1.0360c.$ At what point is $t = +c$? + +\item[Prob.~101.] Show that the evolute of the tractory is the +catenary. (See Prob.~92.)\index{Evolute of tractory} + +\item[Prob.~102.] Find the radius of curvature of the tractory in +terms of $\phi$; and derive the intrinsic equation of the involute.% +\index{Intrinsic equation}\index{Involute!of tractory} +\end{enumerate} \normalsize + +\chapter{The Loxodrome.}\index{Curvature}\index{Loxodrome} + +On the surface of a sphere a curve starts from the equator in a +given direction and cuts all the meridians at the same angle. To +find its equation in latitude-and-longitude coordinates: + +\begin{center} +\includegraphics[width=45mm]{fig12.png} +\end{center} + +Let the loxodrome cross two consecutive meridians $AM$, $AN$ in the +points $P$, $Q$; let $P\!R$ be a parallel of latitude; let $O\!M = +x$, $M\!P = y$, $M\!N' = dx$, $RQ = dy$, all in radian measure; and +let the angle $M\!O\!P = RPQ = \alpha$; then +\begin{equation*} +\tan\alpha = \frac{RQ}{P\!R}\text{, but } P\!R = M\!N\cos +M\!P,\footnotemark +\end{equation*} +\footnotetext{Jones, Trigonometry (Ithaca, 1890), p.~185.}% +\index{Jones' Trigonometry} hence $dx\,\tan\alpha = dy\,\sec y$, and +$x\tan\alpha = \gd^{-1}y$, there being no integration-constant since +$y$ vanishes with $x$; thus the required equation is +\begin{equation*} +y = \gd(x\,\tan\alpha). +\end{equation*}% +\index{Anti-gudermanian}\index{Differential equation} + +To find the length of the arc $OP$: Integrate the equation +\begin{equation*} +ds = dy\,\csc\alpha, \text{ whence } s = y\,\csc\alpha. +\end{equation*} + +To illustrate numerically, suppose a ship sails northeast, from a +point on the equator, until her difference of longitude is +$45^\circ$, find her latitude and distance: + +Here $\tan\alpha = 1$, and $y = \gd x = \gd\frac{1}{4}\pi = \gd +(.7854) = .7152$ radians; $s = y\sqrt{2} = 1.0114$ radii. The +latitude in degrees is $40.980$. + +If the ship set out from latitude $y_1$, the formula must be +modified as follows: Integrating the above differential equation +between the limits $(x_1, y_1)$ and $(x_2, y_2)$ gives +\begin{equation*} +(x_2 - x_1)\tan\alpha = \gd^{-1}y_2 - \gd^{-1}y_1; +\end{equation*} +hence $\gd^{-1}y_2 = \gd^{-1}y_1 + (x_2 - x_1)\tan\alpha$, from +which the final latitude can be found when the initial latitude and +the difference of longitude are given. The distance sailed is equal +to $(y_2 - y_1)\csc\alpha$ radii, a radius being $60 \times +\frac{180}{\pi}$ nautical miles.\index{Gudermanian!function} + +\medskip Mercator's Chart.---In this projection the meridians are +parallel straight lines, and the loxodrome becomes the straight line +$y' = x\tan\alpha$, hence the relations between the coordinates of +corresponding points on the plane and sphere are $x' = x$, $y' = +\gd^{-1}y$. Thus the latitude $y$ is magnified into $\gd^{-1}y$, +which is tabulated under the name of ``meridional part for latitude +$y$''; the values of $y$ and of $y'$ being given in minutes. A chart +constructed accurately from the tables can be used to furnish +graphical solutions of problems like the one proposed above.% +\index{Chart!Mercator's}\index{Mercator's Chart} + +\small \begin{enumerate} +\item[Prob.~103.] Find the distance on a rhumb line between the +points ($30^\circ$ N, $20^\circ$ E) and ($30^\circ$ S, $40^\circ$ +E).\index{Rhumb line} +\end{enumerate} \normalsize + +\chapter{Combined Flexure and Tension.}% +\index{Beams, flexure of}\index{Flexure of Beams} + +A beam that is built-in at one end carries a load $P$ at the other, +and is also subjected to a horizontal tensile force $Q$ applied at +the same point; to find the equation of the curve assumed by its +neutral surface: Let $x, y$ be any point of the elastic curve, +referred to the free end as origin, then the bending moment for this +point is $Qy - Px$. Hence, with the usual notation of the theory of +flexure,\footnote{Merriman, Mechanics of Materials (New York, 1895), +pp.~70--77, 267--269.} +\begin{gather*} +EI\frac{d^2y}{dx^2} = Qy - Px,\quad + \frac{d^2y}{dx^2} = n^2(y - mx), + \tag*{$\left[ m = \dfrac{P}{Q}\right.,\ n^2=\dfrac{Q}{EI}$.} \\ +\intertext{which, on putting $y - mx = u$, and $\dfrac{d^2y}{dx^2} = +\dfrac{d^2u}{dx^2}$, becomes} +\frac{d^2u}{dx^2} = n^2u, \\ +\intertext{whence} +u = A \cosh nx + B \sinh nx, \tag*{[probs.~28, 30} \\ +\intertext{that is,} +y = mx + A \cosh nx + B \sinh nx. +\end{gather*} + +The arbitrary constants $A$, $B$ are to be determined by the +terminal conditions.\index{Terminal conditions} At the free end $x = +0$, $y = 0$; hence $A$ must be zero, and +\begin{align*} +y &= mx + B \sinh nx, \\ +\frac{dy}{dx} &= m+nB \cosh nx; \\ +\intertext{but at the fixed end, $x = l$, and $\dfrac{dy}{dx} = 0$, +hence} +B &= -\frac{m}{n} \cosh nl, \\ +\intertext{and accordingly} +y &= mx - \frac{m \sinh nx}{n \cosh nl}. +\end{align*} + +To obtain the deflection of the loaded end, find the ordinate of the +fixed end by putting $x = l$, giving +\begin{equation*} +\text{deflection} = m(l - \frac{1}{n}\tanh nl), +\end{equation*}\index{Deflection of beams} + +\small \begin{enumerate} +\item[Prob.~104.] Compute the deflection of a cast-iron beam, +$2 \times 2$ inches section, and $6$ feet span, built-in at one end +and carrying a load of $100$ pounds at the other end, the beam being +subjected to a horizontal tension of $8000$ pounds. [In this case $I += \frac{4}{3}, E = 15 \times 10^6, Q = 8000, P = 100$; hence $n = +\frac{1}{50}, m = \frac{1}{80}$, deflection $= \frac{1}{80}(72 - 50 +\tanh 1.44) = \frac{1}{80}(72 - 44.69) = .341$ inches.] + +\item[Prob.~105.] If the load be uniformly distributed over the beam, +say $w$ per linear unit, prove that the differential equation is +\begin{equation*} + EI \frac{d^2 y}{dx^2} = Qy - \tfrac{1}{2}wx^2, \text{ or } + \frac{d^2 y}{dx^2} = n^2(y - mx^2), +\end{equation*} +and that the solution is $y = A \cosh nx + B \sinh nx + mx^2 + +\dfrac{2m}{n^2}$. Show also how to determine the arbitrary +constants. +\end{enumerate}\index{Distributed load} \normalsize + +\chapter{Alternating Currents.}% +\index{Alternating currents}% +\index{Complex numbers!Applications of|(}% +\index{Currents, alternating} + +[Note.\footnote{See references in foot-note Art.~27.}] + +In the general problem treated the cable or wire is regarded as +having resistance, distributed capacity, self-induction, and +leakage; although some of these may be zero in special cases.% +\index{Self-induction of conductor} The line will also be considered +to feed into a receiver circuit of any description; and the general +solution will include the particular cases in which the receiving +end is either grounded or insulated. The electromotive force may, +without loss of generality, be taken as a simple harmonic function +of the time, because any periodic +function can be expressed in a Fourier series of simple harmonics.% +\footnote{Chapter V, Art.~8.} The E.M.F.\ and the current, which may +differ in phase by any angle, will be supposed to have given values +at the terminals of the receiver circuit; and the problem then is to +determine the E.M.F.\ and current that must be kept up at the +generator terminals; and also to express the values of these +quantities at any intermediate point, distant $x$ from the receiving +end; the four line-constants being supposed known, viz.: +\begin{verse} + $R$ = resistance, in ohms per mile, \\ + $L$ = coefficient of self-induction, in henrys per mile, \\ + $C$ = capacity, in farads per mile, \\ + $G$ = coefficient of leakage, in mhos per mile.% +\footnote{Kennelly denotes these constants by $r$, $l$, $c$, $g$. +Steinmetz writes $s$ for $\omega L$, $\kappa$ for $\omega C$, +$\theta$ for $G$, and he uses $C$ for current.} +\end{verse}% +\index{Capacity of conductor}\index{Electromotive force}% +\index{Fourier series}\index{Phase angle}% +\index{Resistance of conductor} + +It is shown in standard works% +\footnote{Thomson and Tait, Natural Philosophy, Vol.~I. p.~40; +Rayleigh, Theory of Sound, Vol.~I. p.~20; Bedell and Crehore, +Alternating Currents, p.~214.}% +\index{Bedel and Crehore's alternating currents}% +\index{Rayleigh's Theory of Sound} that if any simple harmonic +function $a \sin(\omega t + \theta)$ be represented by a vector of +length $a$ and angle $\theta$, then two simple harmonics of the same +period $\dfrac{2\pi}{\omega}$, but having different values of the +phase-angle $\theta$, can be combined by adding their representative +vectors.\index{Vectors} Now the E.M.F. and the current at any point +of the circuit, distant $x$ from the receiving end, are of the form +\begin{equation} +e = e_1\sin{(\omega t + \theta)},\quad +i = i_1\sin{(\omega t + \theta')}, \tag{64} +\end{equation} +in which the maximum values $e_1$, $i_1$, and the phase-angles +$\theta$, $\theta'$, are all functions of $x$. These simple +harmonics will be represented by the vectors +$e_1\underline{/\theta}$, $i_1\underline{/\theta'}$; whose numerical +measures are the complexes $e_1(\cos\theta + j\sin\theta)$\footnote{ +In electrical theory the symbol $j$ is used, instead of $i$, for +$\sqrt{-1}$.}, $i_1(\cos{\theta'} + j\sin{\theta'})$, which will be +denoted by $\bar{e}$, $\bar{\imath}$. The relations between +$\bar{e}$ and $\bar{\imath}$ may be obtained from the ordinary +equations\footnote{Bedell and Crehore, Alternating Currents, p.~181. +The sign of $dx$ is changed, because $x$ is measured from the +receiving end. The coefficient of leakage, $G$, is usually taken +zero, but is here retained for generality and symmetry.} +\begin{equation} +\frac{di}{dx} = Ge + C\frac{de}{dt},\quad +\frac{de}{dx} = Ri + L\frac{di}{dt}; \tag{65} +\end{equation} +for, since $\dfrac{de}{dt} = \omega e_1\cos{(\omega t + \theta)} = +\omega e_1 \sin{(\omega t + \theta + \frac{1}{2}\pi)}$, then +$\dfrac{de}{dt}$ will be represented by the vector $\omega +e_1\underline{/\theta + \frac{1}{2}\pi}$; and $\dfrac{di}{dx}$ by +the sum of the two vectors $Ge_1\underline{/\theta}, C\omega +e_1\underline{/\theta + \frac{1}{2}\pi}$; whose numerical measures +are the complexes $G\bar{e}$, $j\omega C\bar{e}$; and similarly for +$\dfrac{de}{dx}$ in the second equation; thus the relations between +the complexes $\bar{e}$, $\bar{\imath}$ are +\begin{equation} +\frac{d\bar{\imath}}{dx} = (G + j\omega C)\bar{e},\quad +\frac{d\bar{e}}{dx} = (R + j\omega L)\bar{\imath}. + \tag*{(66)\footnotemark} +\end{equation} +\footnotetext{These relations have the advantage of not involving +the time. Steinmetz derives them from first principles without using +the variable $t$. For instance, he regards $R+j\omega L$ as a +generalized resistance-coefficient, which, when applied to $i$, +gives an E.M.F., part of which is in phase with $i$, and part in +quadrature with $i$. Kennelly calls $R + j\omega L$ the conductor +impedance; and $G + j\omega C$ the dielectric admittance; the +reciprocal of which is the dielectric impedance.}% +\index{Admittance of dielectric}\index{Impedance}% +\index{Kennelly's chart}\index{Operators, generalized}% +\index{Steinmetz on alternating currents} + +Differentiating and substituting give +\begin{equation} +\left. \begin{aligned} +\frac{d^2\bar{e}}{dx^2} &= + (R + j\omega L)(G + j\omega C)\bar{e}, \\ +\frac{d^2\bar{\imath}}{dx^2} &= + (R + j\omega L)(G + j\omega C)\bar{\imath}, +\end{aligned} +\right\} \tag{67} +\end{equation} +and thus $\bar{e}, \bar{\imath}$ are similar functions of $x$, to be +distinguished only by their terminal values. + +It is now convenient to define two constants $m$, $m_1$ by the +equations\footnote{The complex constants $m$, $m_1$ are written $z, +y$ by Kennelly; and the variable length $x$ is written $L_2$. +Steinmetz writes $v$ for $m$.}% +\index{Kennelly on alternating currents} +\begin{equation} +m^2 = (R+j\omega L)(G + j\omega C),\quad +m_1 = \frac{m}{(G + j\omega C)}; \tag{68} +\end{equation} +and the differential equations may then be written +\begin{equation} +\frac{d^2\bar{e}}{dx^2} = m^2\bar{e},\quad +\frac{d^2\bar{\imath}}{dx^2} = m^2\bar{\imath}, \tag{69} +\end{equation} +the solutions of which are\footnote{See Art.~14, Probs.~28--30; and +Art.~27, foot-note.} +\begin{equation*} +\bar{e} = A \cosh mx + B\ \sinh mx,\quad +\bar{\imath} = A' \cosh mx + B' \sinh mx, +\end{equation*} +wherein only two of the four constants are arbitrary; for +substituting in either of the equations (66), and equating +coefficients, give +\begin{gather*} +(G + j\omega C)A = mB',\quad (G + j\omega C)B = mA', \\ +\intertext{whence} +B' = \frac{A}{m_1},\quad A' = \frac{B}{m_1}. +\end{gather*}\index{Differential equation} + +Next let the assigned terminal values of $\bar{e}$, $\bar{\imath}$, +at the receiver, be denoted by $\bar{E}, \bar{I}$; then putting $x = +0$ gives $\bar{E} = A, \bar{I} = A'$, whence $B = m_1\bar{I}, B' = +\dfrac{\bar{E}}{m_1}$; and thus the general solution is +\begin{equation} +\left. \begin{aligned} +\bar{e} &= \bar{E}\cosh mx + m_1\bar{I}\sinh mx,\\ +\bar{\imath} &= \bar{I}\cosh mx + \frac{I}{m_1}\bar{E}\sinh mx. +\end{aligned} +\right\} \tag{70} +\end{equation} + +If desired, these expressions could be thrown into the ordinary +complex form $X + jY, X' + jY'$, by putting for the letters their +complex values, and applying the addition-theorems for the +hyperbolic sine and cosine. The quantities $X, Y, X', Y'$ would then +be expressed as functions of $x$; and the representative vectors of +$e, i$, would be $e_1\underline{/\theta}, i_1\underline{/\theta'}$, +where ${e_{1}}^{2} = X^2 + Y^2, i^2 = {X'}^2 + {Y'}^2, \tan{\theta} += \dfrac{Y}{X}, \tan{\theta'} = \dfrac{Y'}{X'}$.% +\index{Argand diagram} + +For purposes of numerical computation, however, the formulas ($70$) +are the most convenient, when either a chart,\footnote{Art.~30, +footnote.} or a table,\footnote{See Table II.} of $\cosh{u}$, +$\sinh{u}$, is available, for complex values of $u$.% +\index{Chart!of hyperbolic functions} + +\small \begin{enumerate} +\item[Prob.~106.\footnotemark]\footnotetext{The data for this example +are taken from Kennelly's article (l.~c., p.~38).}% +\index{Conductor resistance and impedance} Given the four +line-constants: $R$ = 2 ohms per mile, $L$ = 20 millihenrys per +mile, $C$ = $\frac{1}{2}$ microfarad per mile, $G$ = 0; and given +$\omega$, the angular velocity of E.M.F. to be 2000 radians per +second;\index{Electromotive force} then +\begin{align*} +\omega L &= 40 \text{ ohms, conductor reactance per mile};\\ +R + j\omega L &= 2 + 40j \text{ ohms, conductor impedance per mile};\\ +\omega C &= .001 \text{ mho, dielectric susceptance per mile};\\ +G + j\omega C &= .001j \text{ mho, dielectric admittance per mile};\\ +(G + j\omega C)^{-1} &= -1000j \text{ ohms, dielectric impedance per + mile};\\ +m^2 &= (R + j\omega L)(G + j\omega C) = .04 + .002j, \\ +&\qquad \text{which is the measure of } + .04005\underline{/177^{\circ}8'}; \\ +\intertext{therefore} +m &= \text{ measure of } .2001\underline{/88^{\circ}34'} + = .0050 + .2000j, \\ +&\qquad \text{an abstract coefficient per mile, of + dimensions } [\mathrm{length}]^{-1}, \\ +m_1 &= \dfrac{m}{(G + j\omega C)} = 200 - 5j \text{ ohms}. +\end{align*}\index{Reactance of conductor}% +\index{Suceptance of dielectric} + +\indent Next let the assigned terminal conditions at the receiver +be: $I = 0$ (line insulated)\index{Terminal conditions}; and $E = +1000$ volts, whose phase may be taken as the standard (or zero) +phase; then at any distance $x$, by (70), +\begin{align*} +\bar{e} &= E\cosh{mx}, & \bar{\imath} &= \frac{E}{m_1}\sinh{mx}, +\end{align*} +in which $mx$ is an abstract complex. + +\indent Suppose it is required to find the E.M.F. and current that +must be kept up at a generator $100$ miles away; then +\begin{gather*} +\bar{e} = 1000 \cosh(.5 + 20 j),\quad +\bar{\imath} = 200 (40 - j)^{-1} \sinh(.5 + 20j), \\ +\intertext{but, by Prob.~89,} +\begin{aligned} +\cosh(.5 + 20 j) &= \cosh(.5 + 20 j - 6\pi j) \\ + &= \cosh(.5 + 1.15 j) = .4600 + .4750j +\end{aligned} +\end{gather*} +obtained from Table II, by interpolation between $\cosh (.5 + +1.1j)$ and $\cosh (.5 + 1.2j)$; hence +\begin{equation*} +\bar{e} = 460 + 475j = e_1 (\cos \theta + j \sin\theta), +\end{equation*} +where $\log \tan \theta = \log 475 - \log 460 = .0139$, $\theta = +45^\circ\: 55'$, and $e_1 = 460 \sec \theta = 661.2$ volts, the +required E.M.F.\index{Interpolation} + +\smallskip \indent Similarly $\sinh (.5 + 20j) = +\sinh (.5 + 1.15j) = .2126 + 1.0280j$, and hence +\begin{align*} +\bar{\imath} = \frac{200}{1601}(40+j)(.2126+1.028j) + &= \frac{1}{1601}(1495+8266j) \\ + &= i_1(\cos\theta' + j\sin\theta'), +\end{align*} +where $\log \tan \theta' = 10.7427$, $\theta' = 79^\circ\, 45'$, +$i_1 = 1495 \sec \dfrac{\theta'}{1601} = 5.25$ amperes, the phase +and magnitude of required current.\index{Phase angle} + +\indent Next let it be required to find $e$ at $x = 8$; then +\begin{equation*} +\bar{e}= 1000 \cosh (.04 + 1.6j) = 1000j \sinh (.04 + .03j), +\end{equation*} +by subtracting $\frac{1}{2}\pi j$, and applying +page~\pageref{period-hyp-funct}. Interpolation +between $\sinh (0 + 0j)$ and $\sinh (0 +.1j)$ gives +\begin{align*} +\sinh ( 0 + .03j) &= .00000 + .02995j. \\ +\intertext{Similarly} +\sinh (.1 + .03j) &= .10004 + .03004j. \\ +\intertext{Interpolation between the last two gives} +\sinh (.04 + .03j) &= .04002 + .02999j. +\end{align*} +Hence $\bar{e} = j(40.02 + 29.99j) = -29.99 + 40.02j = e_1 +(\cos\theta + j\sin\theta)$, where $\log \tan \theta = .12530$, +$\theta = 126^\circ\, 51'$, $e_1 = -29.99 \sec 126^\circ\, 51' = +50.01$ volts.\index{Interpolation} + +\indent Again, let it be required to find $e$ at $x = 16$; here +\begin{gather*} +\bar{e} = 1000 \cosh (.08 + 3.2j) = -1000 \cosh (.08 + .06j), \\ +\intertext{but} +\cosh (0 + .06j) = .9970 + 0j,\ + \cosh (.1 + .06j) = 1.0020 + .006j; \\ +\intertext{hence} +\cosh (.08 +.06j) = 1.0010 + .0048j, \\ +\intertext{and} +\bar{e} = -1001 + 4.8j = e_1(\cos\theta + j\sin\theta), +\end{gather*} +where $\theta = 180^\circ\, 17'$, $e_1 = 1001$ volts. Thus at a +distance of about 16 miles the E.M.F. is the same as at the +receiver, but in opposite phase. Since $\bar{e}$ is proportional to +$\cosh (.005 + .2j)x$, the value of $x$ for which the phase is +exactly $180^\circ$ is $\frac{\pi}{.2} = 15.7$. Similarly the phase +of the E.M.F.\ at $x = 7.85$ is $90^\circ$. There is agreement in +phase at any two points whose distance apart is $31.4$ miles. + +\indent In conclusion take the more general terminal conditions in +which the line feeds into a receiver circuit, and suppose the +current is to be kept at $50$ amperes, in a phase $40^\circ$ in +advance of the electromotive force; then $\bar{I} 50(\cos 40^\circ + +\sin 40^\circ) = 38.30 + 32.14j$, and substituting the constants in +(70) gives +\begin{align*} +\bar{c} +&= 1000 \cosh (.005 + .2j)x + (7821 + 6236j) \sinh (.005 + .2j)x \\ +&= 460 + 475j - 4748 + 9366j = -4288 + 9841j + = e_1(\cos\theta + j\sin\theta), +\end{align*} +where $\theta = 113^\circ\: 33'$, $e_1 = 10730$ volts, the E.M.F.\ +at sending end. This is 17 times what was required when the other +end was insulated.\index{Terminal conditions} + +\item[Prob.~107.] If $L = 0$, $G = 0$, $I = 0$; then $m = (1 + j)n$, +$m_1 = (1 + j)n_1$ where $n^2 = \dfrac{\omega RC}{2}$, $n_1^2 = +\dfrac{R}{2\omega C}$; and the solution is +\begin{align*} + e_1 &= \frac{1}{\sqrt{2}} E\sqrt{\cosh 2nx + \cos 2nx}, + &\tan \theta &= \tan nx \tanh nx, +\\ + i_1 &= \frac{1}{2n_1} E\sqrt{\cosh 2nx - \cos 2nx}, + &\tan \theta' &= \tan nx \coth nx . +\end{align*} + +\item[Prob.~108.] If self-induction and capacity be zero, and the +receiving end be insulated, show that the graph of the electromotive +force is a catenary if $G \neq 0$, a line if $G = 0$. + +\item[Prob.~109.] Neglecting leakage and capacity, prove that the +solution of equations (66) is $\bar{\imath} = \bar{I}$, $\bar{c} = +\bar{E} + (R + j\omega L)\bar{I}x$. + +\item[Prob.~110.] If $x$ be measured from the sending end, show how +equations (65), (66) are to be modified; and prove that +\begin{equation*} + \bar{e} = \bar{E}_0 \cosh mx - m_1\bar{I}_0 \sinh mx,\ + \bar{\imath} = \bar{I}_0 \cosh mx - \frac{1}{m_1}\bar{E}_0 \sinh mx, +\end{equation*} +where $\bar{E}_0$, $\bar{I}_0$ refer to the sending end. +\end{enumerate}\index{Complex numbers!Applications of|)} \normalsize + + + +\chapter{Miscellaneous Applications.} + +\begin{enumerate} +\item[1.] The length of the arc of the logarithmic curve $y = a^x$ +is $s = M(\cosh u + \log\tanh\frac{1}{2} u)$, in which $M = +\dfrac{1}{\log a}$, $\sinh u = \dfrac{y}{M}$.% +\index{Curvature}\index{Logarithmic!curve} + +\item[2.] The length of arc of the spiral of Archimedes +$r = a\theta$ is $s = \frac{1}{4} a(\sinh 2u + 2u)$, where $\sinh u += \theta$.\index{Spiral of Archimedes} + +\item[3.] In the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ +the radius of curvature is \\ +$\rho = \dfrac{(a^2 \sinh^2 u + b^2 \cosh^2 u)^{\frac{3}{2}}}{ab}$; +in which $u$ is the measure of the sector $AOP$, i.e.\ $\cosh u = +\dfrac{x}{a}$, $\sinh u = \dfrac{y}{b}$.% +\index{Areas}\index{Hyperbola} + +\item[4.] In an oblate spheroid, the superficial area of the zone +between the equator and a parallel plane at a distance $y$ is +$S=\dfrac{\pi b^2(\sinh 2u+2u)}{2e}$, wherein $b$ is the axial +radius, $e$ eccentricity, $\sinh u = \dfrac{ey}{p}$, and $p$ +parameter of generating ellipse.\index{Spheroid, area of oblate} + +\item[5.] The length of the arc of the parabola $y^2 = 2px$, +measured from the vertex of the curve, is $l = \frac{1}{4}p(\sinh 2u ++ 2u)$, in which $\sinh u = \dfrac{y}{p} = \tan\phi$, where $\phi$ +is the inclination of the terminal tangent to the initial one.% +\index{Parabola} + +\item[6.] The centre of gravity of this arc is given by +\begin{equation*} +3l\bar{x} = p^2(\cosh^3u-1),\quad 64l\bar{y} = p^2(\sinh 4u-4u); +\end{equation*} +and the surface of a paraboloid of revolution is $S = 2\pi\bar{y}l$.% +\index{Center of gravity} + +\item[7.] The moment of inertia of the same arc about its terminal +ordinate is $I = \mu\left[xl(x - 2\bar{x}) + +\frac{1}{64}p^3N\right]$, where $\mu$ is the mass of unit length, +and +\begin{equation*} +N = u - \frac{1}{4}\sinh 2u + - \frac{1}{4}\sinh 4u + \frac{1}{12}\sinh 6u. +\end{equation*}\index{Moment of inertia} + +\item[8.] The centre of gravity of the arc of a catenary measured +from the lowest point is given by +\begin{equation*} +4l\bar{y} = c^2(\sinh 2u + 2u),\ + l\bar{x} = c^2(u\sinh u - \cosh u+1), +\end{equation*} +in which $u = \frac{x}{c}$; and the moment of inertia of this arc +about its terminal abscissa is +\begin{equation*} +I = \mu c^3\left(\frac{1}{12}\sinh 3u + \frac{3}{4}\sinh u - + u\cosh u\right). +\end{equation*} + +\item[9.] Applications to the vibrations of bars are given in Rayleigh, +Theory of Sound, Vol.~I, art.~170; to the torsion of prisms in Love, +Elasticity, pp.~166--74; to the flow of heat and electricity in +Byerly, Fourier Series, pp.~75--81; to wave motion in fluids in +Rayleigh, Vol.~I, Appendix, p.~477, and in Bassett, Hydrodynamics, +arts.~120, 384; to the theory of potential in Byerly p.~135, and in +Maxwell, Electricity, arts.~172--4\index{Maxwell's Electricity}; to +Non-Euclidian geometry and many other subjects in +Günther\index{Gunther's Die Lehre, etc.}, Hyperbelfunktionen, +Chaps.~V and VI. Several numerical examples are +worked out in Laisant, Essai sur les fonctions hyperboliques.% +\index{Bassett's Hydrodynamics}\index{Byerly's Fourier Series, etc.}% +\index{Fourier series}\index{Laisant's Essai, etc.}% +\index{Love's elasticity}\index{Potential theory}% +\index{Vibrations of bars} +\end{enumerate} +\index{Applications|)}\index{Hyperbolic functions!applictions of|)}% +\index{Physical problems|)} + + + +\chapter{Explanation of Tables.}\index{Complex numbers!Tables}% +\index{Tables} + +In Table I the numerical values of the hyperbolic functions $\sinh +u, \cosh u, \tanh u$ are tabulated for values of $u$ increasing from +0 to 4 at intervals of .02. When $u$ exceeds 4, Table IV may be +used. + +Table II gives hyperbolic functions of complex arguments, in which +\begin{equation*} +\cosh (x \pm iy) = a \pm ib,\ \sinh (x \pm iy) = c \pm id, +\end{equation*} +and the values of $a, b, c, d$ are tabulated for values of $x$ and +of $y$ ranging separately from 0 to 1.5 at intervals of .1. When +interpolation is necessary it may be performed in three +stages.\index{Interpolation} For example, to find $\cosh (.82 + +1.34i)$: First find $\cosh (.82 + 1.3i)$, by keeping $y$ at 1.3 and +interpolating between the entries under $x =.8$ and $x =.9$; next +find $\cosh (.82 + 1.4i)$, by keeping $y$ at 1.4 and interpolating +between the entries under $x =.8$ and $x =.9$, as before; then by +interpolation between $\cosh (.82 + 1.3i)$ and $\cosh (.82 + 1.4i)$ +find $\cosh(.82 + 1.34i)$, in which $x$ is kept at .82. The table is +available for all values of $y$, however great, by means of the +formulas on page~\pageref{period-hyp-funct}: +\begin{equation*} +\sinh (x + 2i\pi) = \sinh x,\ + \cosh (x + 2i\pi) = \cosh x, \text{ etc.} +\end{equation*} +It does not apply when $x$ is greater than 1.5, but this case seldom +occurs in practice. This table can also be used as a complex table +of circular functions, for +\begin{equation*} +\cos (y \pm ix) = a \mp ib,\ \sin (y \pm ix) = d \pm ic; +\end{equation*} +and, moreover, the exponential function is given by +\begin{equation*} +\exp (\pm x \pm iy) = a \pm c \pm i(b \pm d), +\end{equation*} +in which the signs of $c$ and $d$ are to be taken the same as the +sign of $x$, and the sign of $i$ on the right is to be the product +of the signs of $x$ and of $i$ on the left.\index{Periodicity} + +Table III gives the values of $v = \gd u$, and of the gudermanian +angle $\theta = \dfrac{180^\circ v}{\pi}$, as $u$ changes from 0 to +1 at intervals of .02, from 1 to 2 at intervals of .05, and from 2 +to 4 at intervals of .1. + +In Table IV are given, the values of $\gd u$, $\log \sinh u$, $\log +\cosh u$, as $u$ increases from 4 to 6 at intervals of .1, from 6 to +7 at intervals of .2, and from 7 to 9 at intervals of .5. + +In the rare cases in which more extensive tables are necessary, +reference may be made to the tables\footnote{% +Gudermann in Crelle's Journal, vols. 6--9, 1831--2 (published +separately under the title Theorie der hyperbolischen Functionen, +Berlin, 1833). Glaisher in Cambridge Phil.\ Trans., vol.\ 13, 1881. +Geipel and Kilgour's Electrical Handbook.}\index{Geipel and +Kilgour's Electrical Handbook}\index{Glaisher's exponential tables} +of Gudermann, Glaisher, and Geipel and Kilgour. In the first the +Gudermanian angle (written $k$) is taken as the independent +variable, and increases from 0 to 100 grades at intervals of .01, +the corresponding value of $u$ (written $Lk$) being tabulated. In +the usual case, in which the table is entered with the value of $u$, +it gives by interpolation the value of the gudermanian angle, whose +circular functions would then give the hyperbolic functions of +$u$.\index{Interpolation} When $u$ is large, this angle is so nearly +right that interpolation is not reliable. To remedy this +inconvenience Gudermann's second table gives directly $\log\sinh u$, +$\log\cosh u$, $\log\tanh u$, to nine figures, for values of $u$ +varying by .001 from 2 to 5, and by .01 from 5 to +12.\index{Gudermanian!function} + +Glaisher has tabulated the values of $e^x$ and $e^{-x}$, to nine +significant figures, as $x$ varies by .001 from 0 to .1, by .01 from +0 to 2, by .1 from 0 to 10, and by 1 from 0 to 500. From these the +values of $\cosh x$, $\sinh x$ are easily obtained. + +Geipel and Kilgour's handbook gives the values of $\cosh x$, $\sinh +x$, to seven figures, as $x$ varies by .01 from 0 to 4. + +There are also extensive tables by Forti, Gronau, Vassal, Callet, +and Hoüel; and there are four-place tables in Byerly's Fourier +Series, and in Wheeler's Trigonometry.% +\index{Byerly's Fourier Series, etc.}\index{Callet's Tables}% +\index{Forti's Tavoli e teoria}\index{Gronau's!Tafeln}% +\index{Vassal's Tables}\index{Wheeler's Trigonometry} + +In the following tables a dash over a final digit indicates that the +number has been increased. + + + +\newpage +\markright{TABLES} + +\subsubsection*{} %% RWD Nickalls : +%%% empty subsubsection{} command as a hook for \addcontentsline... and for the \label. +%%% place this /before/ the addtocontentsline.. and before the \label +\addcontentsline{lot}{table}{Table I.---Hyperbolic Functions} +\index{Hyperbolic functions!tables of|(} +\begin{center} +\textsc{Table I.---Hyperbolic Functions.} +\label{Table1p1} \\ +\medskip\scriptsize +\begin{tabular}{r|r|r|r||r|r|r|r} +\hline + +\multicolumn{1}{c|}{$u$} + &$ \sinh u. $&$ \cosh u. $&$ \tanh u. $ & +\multicolumn{1}{c|}{$u$} + &$ \sinh u. $&$ \cosh u. $&$ \tanh u. $ +\\ +\hline +&&&&&&& \\ +$ .00 $&$ .0000 $&$ 1.0000 $&$ .0000 $ & +$ 1.00 $&$ 1.1752 $&$ 1.543\bar{1} $&$ .7616 $ +\\ +$ 02 $&$ 0200 $&$ 1.0002 $&$ 0200 $ & +$ 1.02 $&$ 1.206\bar{3} $&$ 1.566\bar{9} $&$ 769\bar{9} $ +\\ +$ 04 $&$ 0400 $&$ 1.0008 $&$ 040\bar{0} $ & +$ 1.04 $&$ 1.237\bar{9} $&$ 1.5913 $&$ 777\bar{9} $ +\\ +$ 06 $&$ 0600 $&$ 1.0018 $&$ 0599 $ & +$ 1.06 $&$ 1.270\bar{0} $&$ 1.6164 $&$ 785\bar{7} $ +\\ +$ 08 $&$ 080\bar{1} $&$ 1.0032 $&$ 0798 $ & +$ 1.08 $&$ 1.3025 $&$ 1.6421 $&$ 793\bar{2} $ +\\ +&&&&&&& \\ +$ .10 $&$ .100\bar{2} $&$ 1.0050 $&$ .099\bar{7} $ & +$ 1.10 $&$ 1.3356 $&$ 1.6685 $&$ .8005 $ +\\ +$ 12 $&$ 120\bar{3} $&$ 1.0072 $&$ 1194 $ & +$ 1.12 $&$ 1.369\bar{3} $&$ 1.695\bar{6} $&$ 807\bar{6} $ +\\ +$ 14 $&$ 140\bar{5} $&$ 1.0098 $&$ 139\bar{1} $ & +$ 1.14 $&$ 1.403\bar{5} $&$ 1.7233 $&$ 8144 $ +\\ +$ 16 $&$ 160\bar{7} $&$ 1.0128 $&$ 1586 $ & +$ 1.16 $&$ 1.4382 $&$ 1.7517 $&$ 8210 $ +\\ +$ 18 $&$ 181\bar{0} $&$ 1.0162 $&$ 178\bar{1} $ & +$ 1.18 $&$ 1.4735 $&$ 1.7808 $&$ 827\bar{5} $ +\\ +&&&&&&& \\ +$ .20 $&$ .2013 $&$ 1.020\bar{1} $&$ .197\bar{4} $ & +$ 1.20 $&$ 1.509\bar{5} $&$ 1.810\bar{7} $&$ .833\bar{7} $ +\\ +$ 22 $&$ 221\bar{8} $&$ 1.024\bar{3} $&$ 2165 $ & +$ 1.22 $&$ 1.546\bar{0} $&$ 1.8412 $&$ 839\bar{7} $ +\\ +$ 24 $&$ 2423 $&$ 1.0289 $&$ 235\bar{5} $ & +$ 1.24 $&$ 1.5831 $&$ 1.872\bar{5} $&$ 845\bar{5} $ +\\ +$ 26 $&$ 2629 $&$ 1.034\bar{0} $&$ 254\bar{3} $ & +$ 1.26 $&$ 1.620\bar{9} $&$ 1.9045 $&$ 851\bar{1} $ +\\ +$ 28 $&$ 283\bar{7} $&$ 1.0395 $&$ 2729 $ & +$ 1.28 $&$ 1.6593 $&$ 1.9373 $&$ 856\bar{5} $ +\\ +&&&&&&& \\ +$ .30 $&$ .3045 $&$ 1.0453 $&$ .2913 $ & +$ 1.30 $&$ 1.6984 $&$ 1.9709 $&$ .8617 $ +\\ +$ 32 $&$ 325\bar{5} $&$ 1.0516 $&$ 3095 $ & +$ 1.32 $&$ 1.7381 $&$ 2.005\bar{3} $&$ 8668 $ +\\ +$ 34 $&$ 3466 $&$ 1.058\bar{4} $&$ 327\bar{5} $ & +$ 1.34 $&$ 1.778\bar{6} $&$ 2.0404 $&$ 871\bar{7} $ +\\ +$ 36 $&$ 3678 $&$ 1.0655 $&$ 3452 $ & +$ 1.36 $&$ 1.819\bar{8} $&$ 2.0764 $&$ 876\bar{4} $ +\\ +$ 38 $&$ 3892 $&$ 1.0731 $&$ 3627 $ & +$ 1.38 $&$ 1.861\bar{7} $&$ 2.1132 $&$ 881\bar{0} $ +\\ +&&&&&&& \\ +$ .40 $&$ .410\bar{8} $&$ 1.081\bar{1} $&$ .3799 $ & +$ 1.40 $&$ 1.9043 $&$ 2.150\bar{9} $&$ .8854 $ +\\ +$ 42 $&$ 432\bar{5} $&$ 1.0895 $&$ 3969 $ & +$ 1.42 $&$ 1.9477 $&$ 2.1894 $&$ 889\bar{6} $ +\\ +$ 44 $&$ 4543 $&$ 1.098\bar{4} $&$ 4136 $ & +$ 1.44 $&$ 1.991\bar{9} $&$ 2.2288 $&$ 893\bar{7} $ +\\ +$ 46 $&$ 476\bar{4} $&$ 1.107\bar{7} $&$ 430\bar{1} $ & +$ 1.46 $&$ 2.036\bar{9} $&$ 2.269\bar{1} $&$ 897\bar{7} $ +\\ +$ 48 $&$ 4986 $&$ 1.1174 $&$ 4462 $ & +$ 1.48 $&$ 2.082\bar{7} $&$ 2.310\bar{3} $&$ 901\bar{5} $ +\\ +&&&&&&& \\ +$ .50 $&$ .521\bar{1} $&$ 1.1276 $&$ .4621 $ & +$ 1.50 $&$ 2.129\bar{3} $&$ 2.3524 $&$ .9051 $ +\\ +$ 52 $&$ 543\bar{8} $&$ 1.138\bar{3} $&$ 4777 $ & +$ 1.52 $&$ 2.176\bar{8} $&$ 2.395\bar{5} $&$ 908\bar{7} $ +\\ +$ 54 $&$ 5666 $&$ 1.149\bar{4} $&$ 493\bar{0} $ & +$ 1.54 $&$ 2.2251 $&$ 2.439\bar{5} $&$ 9121 $ +\\ +$ 56 $&$ 5897 $&$ 1.1609 $&$ 508\bar{0} $ & +$ 1.56 $&$ 2.2743 $&$ 2.484\bar{5} $&$ 9154 $ +\\ +$ 58 $&$ 613\bar{1} $&$ 1.173\bar{0} $&$ 522\bar{7} $ & +$ 1.58 $&$ 2.324\bar{5} $&$ 2.530\bar{5} $&$ 9186 $ +\\ +&&&&&&& \\ +$ .60 $&$ .636\bar{7} $&$ 1.185\bar{5} $&$ .5370 $ & +$ 1.60 $&$ 2.375\bar{6} $&$ 2.577\bar{5} $&$ .921\bar{7} $ +\\ +$ 62 $&$ 660\bar{5} $&$ 1.1984 $&$ 5511 $ & +$ 1.62 $&$ 2.427\bar{6} $&$ 2.625\bar{5} $&$ 9246 $ +\\ +$ 64 $&$ 684\bar{6} $&$ 1.211\bar{9} $&$ 564\bar{9} $ & +$ 1.64 $&$ 2.480\bar{6} $&$ 2.674\bar{6} $&$ 927\bar{5} $ +\\ +$ 66 $&$ 709\bar{0} $&$ 1.2258 $&$ 578\bar{4} $ & +$ 1.66 $&$ 2.534\bar{6} $&$ 2.7247 $&$ 9302 $ +\\ +$ 68 $&$ 7336 $&$ 1.2402 $&$ 5915 $ & +$ 1.68 $&$ 2.5896 $&$ 2.776\bar{0} $&$ 932\bar{9} $ +\\ +&&&&&&& \\ +$ .70 $&$ .758\bar{6} $&$ 1.255\bar{2} $&$ .604\bar{4} $ & +$ 1.70 $&$ 2.6456 $&$ 2.8283 $&$ .9354 $ +\\ +$ 72 $&$ 7838 $&$ 1.270\bar{6} $&$ 6169 $ & +$ 1.72 $&$ 2.7027 $&$ 2.881\bar{8} $&$ 937\bar{9} $ +\\ +$ 74 $&$ 8094 $&$ 1.2865 $&$ 6291 $ & +$ 1.74 $&$ 2.7609 $&$ 2.9364 $&$ 9402 $ +\\ +$ 76 $&$ 8353 $&$ 1.303\bar{0} $&$ 641\bar{1} $ & +$ 1.76 $&$ 2.820\bar{2} $&$ 2.9922 $&$ 9425 $ +\\ +$ 78 $&$ 8615 $&$ 1.3199 $&$ 6527 $ & +$ 1.78 $&$ 2.8806 $&$ 3.0492 $&$ 944\bar{7} $ +\\ +&&&&&&& \\ +$ .80 $&$ .8881 $&$ 1.3374 $&$ .6640 $ & +$ 1.80 $&$ 2.942\bar{2} $&$ 3.107\bar{5} $&$ .9468 $ +\\ +$ 82 $&$ 9150 $&$ 1.355\bar{5} $&$ 675\bar{1} $ & +$ 1.82 $&$ 3.0049 $&$ 3.1669 $&$ 9488 $ +\\ +$ 84 $&$ 9423 $&$ 1.3740 $&$ 6858 $ & +$ 1.84 $&$ 3.068\bar{9} $&$ 3.227\bar{7} $&$ 950\bar{8} $ +\\ +$ 86 $&$ 970\bar{0} $&$ 1.393\bar{2} $&$ 696\bar{3} $ & +$ 1.86 $&$ 3.1340 $&$ 3.2897 $&$ 952\bar{7} $ +\\ +$ 88 $&$ 998\bar{1} $&$ 1.4128 $&$ 7064 $ & +$ 1.88 $&$ 3.200\bar{5} $&$ 3.3530 $&$ 954\bar{5} $ +\\ +&&&&&&& \\ +$ .90 $&$ 1.0265 $&$ 1.433\bar{1} $&$ .716\bar{3} $ & +$ 1.90 $&$ 3.268\bar{2} $&$ 3.4177 $&$ .9562 $ +\\ +$ 92 $&$ 1.0554 $&$ 1.4539 $&$ 725\bar{9} $ & +$ 1.92 $&$ 3.337\bar{2} $&$ 3.483\bar{8} $&$ 9579 $ +\\ +$ 94 $&$ 1.084\bar{7} $&$ 1.4753 $&$ 7352 $ & +$ 1.94 $&$ 3.4075 $&$ 3.5512 $&$ 9595 $ +\\ +$ 96 $&$ 1.1144 $&$ 1.497\bar{3} $&$ 744\bar{3} $ & +$ 1.96 $&$ 3.4792 $&$ 3.620\bar{1} $&$ 961\bar{1} $ +\\ +$ 98 $&$ 1.144\bar{6} $&$ 1.519\bar{9} $&$ 753\bar{1} $ & +$ 1.98 $&$ 3.5523 $&$ 3.6904 $&$ 962\bar{6} $ +\\ +&&&&&&& \\ +\hline +\end{tabular} \end{center} \normalsize + +\newpage + +\subsubsection*{} %% RWD Nickalls : empty subsubsection{} command as a hook for \label +\begin{center} +\textsc{Table I.---Hyperbolic Functions} (\emph{continued})% +\label{Table1p2} \\ +\medskip\scriptsize +\begin{tabular}{r|r|r|r||r|r|r|r} +\hline \multicolumn{1}{c|}{$u$} + &$ \sinh u. $&$ \cosh u. $&$ \tanh u. $ & +\multicolumn{1}{c|}{$u$} + &$ \sinh u. $&$ \cosh u. $&$ \tanh u. $ +\\ +\hline +&&&&&&& \\ +$ 2.00 $&$ 3.626\bar{9} $&$ 3.762\bar{2} $&$ .9640 $ & +$ 3.00 $&$ 10.017\bar{9} $&$ 10.067\bar{7} $&$ .99505 $ +\\ +$ 2.02 $&$ 3.7028 $&$ 3.835\bar{5} $&$ 9654 $ & +$ 3.02 $&$ 10.2212 $&$ 10.2700 $&$ 99524 $ +\\ +$ 2.04 $&$ 3.780\bar{3} $&$ 3.9103 $&$ 9667 $ & +$ 3.04 $&$ 10.4287 $&$ 10.4765 $&$ 99543 $ +\\ +$ 2.06 $&$ 3.859\bar{3} $&$ 3.9867 $&$ 9680 $ & +$ 3.06 $&$ 10.6403 $&$ 10.6872 $&$ 99561 $ +\\ +$ 2.08 $&$ 3.9398 $&$ 4.0647 $&$ 969\bar{3} $ & +$ 3.08 $&$ 10.8562 $&$ 10.902\bar{2} $&$ 99578 $ +\\ +&&&&&&& \\ +$ 2.10 $&$ 4.021\bar{9} $&$ 4.1443 $&$ .970\bar{5} $ & +$ 3.10 $&$ 11.076\bar{5} $&$ 11.1215 $&$ .99594 $ +\\ +$ 2.12 $&$ 4.1055 $&$ 4.225\bar{6} $&$ 971\bar{6} $ & +$ 3.12 $&$ 11.3011 $&$ 11.345\bar{3} $&$ 99610 $ +\\ +$ 2.14 $&$ 4.190\bar{9} $&$ 4.3085 $&$ 972\bar{7} $ & +$ 3.14 $&$ 11.530\bar{3} $&$ 11.573\bar{6} $&$ 99626 $ +\\ +$ 2.16 $&$ 4.2779 $&$ 4.3932 $&$ 9737 $ & +$ 3.16 $&$ 11.764\bar{1} $&$ 11.8065 $&$ 99640 $ +\\ +$ 2.18 $&$ 4.3666 $&$ 4.479\bar{7} $&$ 974\bar{8} $ & +$ 3.18 $&$ 12.002\bar{6} $&$ 12.044\bar{2} $&$ 99654 $ +\\ +&&&&&&& \\ +$ 2.20 $&$ 4.4571 $&$ 4.5679 $&$ .9757 $ & +$ 3.20 $&$ 12.245\bar{9} $&$ 12.2866 $&$ .99668 $ +\\ +$ 2.22 $&$ 4.549\bar{4} $&$ 4.658\bar{0} $&$ 976\bar{7} $ & +$ 3.22 $&$ 12.494\bar{1} $&$ 12.5340 $&$ 99681 $ +\\ +$ 2.24 $&$ 4.6434 $&$ 4.749\bar{9} $&$ 977\bar{6} $ & +$ 3.24 $&$ 12.747\bar{3} $&$ 12.7864 $&$ 99693 $ +\\ +$ 2.26 $&$ 4.739\bar{4} $&$ 4.8437 $&$ 978\bar{5} $ & +$ 3.26 $&$ 13.005\bar{6} $&$ 13.044\bar{0} $&$ 99705 $ +\\ +$ 2.28 $&$ 4.837\bar{2} $&$ 4.939\bar{5} $&$ 979\bar{3} $ & +$ 3.28 $&$ 13.269\bar{1} $&$ 13.3067 $&$ 99717 $ +\\ +&&&&&&& \\ +$ 2.30 $&$ 4.937\bar{0} $&$ 5.0372 $&$ .980\bar{1} $ & +$ 3.30 $&$ 13.537\bar{9} $&$ 13.574\bar{8} $&$ .99728 $ +\\ +$ 2.32 $&$ 5.0387 $&$ 5.137\bar{0} $&$ 980\bar{9} $ & +$ 3.32 $&$ 13.812\bar{1} $&$ 13.848\bar{3} $&$ 99738 $ +\\ +$ 2.34 $&$ 5.1425 $&$ 5.238\bar{8} $&$ 9816 $ & +$ 3.34 $&$ 14.0918 $&$ 14.127\bar{3} $&$ 99749 $ +\\ +$ 2.36 $&$ 5.248\bar{3} $&$ 5.342\bar{7} $&$ 9823 $ & +$ 3.36 $&$ 14.3772 $&$ 14.412\bar{0} $&$ 99758 $ +\\ +$ 2.38 $&$ 5.356\bar{2} $&$ 5.4487 $&$ 9830 $ & +$ 3.38 $&$ 14.668\bar{4} $&$ 14.7024 $&$ 99768 $ +\\ +&&&&&&& \\ +$ 2.40 $&$ 5.4662 $&$ 5.5569 $&$ .983\bar{7} $ & +$ 3.40 $&$ 14.965\bar{4} $&$ 14.9987 $&$ .99777 $ +\\ +$ 2.42 $&$ 5.5785 $&$ 5.667\bar{4} $&$ 9843 $ & +$ 3.42 $&$ 15.268\bar{4} $&$ 15.301\bar{1} $&$ 99786 $ +\\ +$ 2.44 $&$ 5.6929 $&$ 5.7801 $&$ 9849 $ & +$ 3.44 $&$ 15.5774 $&$ 15.6095 $&$ 99794 $ +\\ +$ 2.46 $&$ 5.809\bar{7} $&$ 5.8951 $&$ 9855 $ & +$ 3.46 $&$ 15.892\bar{8} $&$ 15.9242 $&$ 99802 $ +\\ +$ 2.48 $&$ 5.928\bar{8} $&$ 6.0125 $&$ 986\bar{1} $ & +$ 3.48 $&$ 16.2144 $&$ 16.245\bar{3} $&$ 99810 $ +\\ +&&&&&&& \\ +$ 2.50 $&$ 6.0502 $&$ 6.132\bar{3} $&$ .9866 $ & +$ 3.50 $&$ 16.5426 $&$ 16.5728 $&$ .99817 $ +\\ +$ 2.52 $&$ 6.174\bar{1} $&$ 6.2545 $&$ 9871 $ & +$ 3.52 $&$ 16.8774 $&$ 16.9070 $&$ 99824 $ +\\ +$ 2.54 $&$ 6.3004 $&$ 6.379\bar{3} $&$ 9876 $ & +$ 3.54 $&$ 17.219\bar{0} $&$ 17.248\bar{0} $&$ 99831 $ +\\ +$ 2.56 $&$ 6.429\bar{3} $&$ 6.506\bar{6} $&$ 9881 $ & +$ 3.56 $&$ 17.567\bar{4} $&$ 17.5958 $&$ 99831 $ +\\ +$ 2.58 $&$ 6.560\bar{7} $&$ 6.6364 $&$ 988\bar{6} $ & +$ 3.58 $&$ 17.9228 $&$ 17.9507 $&$ 99844 $ +\\ +&&&&&&& \\ +$ 2.60 $&$ 6.6947 $&$ 6.7690 $&$ .9890 $ & +$ 3.60 $&$ 18.2854 $&$ 18.312\bar{8} $&$ .99850 $ +\\ +$ 2.62 $&$ 6.831\bar{5} $&$ 6.904\bar{3} $&$ 989\bar{5} $ & +$ 3.62 $&$ 18.655\bar{4} $&$ 18.682\bar{2} $&$ 99856 $ +\\ +$ 2.64 $&$ 6.9709 $&$ 7.042\bar{3} $&$ 989\bar{9} $ & +$ 3.64 $&$ 19.032\bar{8} $&$ 19.0590 $&$ 99862 $ +\\ +$ 2.66 $&$ 7.113\bar{2} $&$ 7.183\bar{2} $&$ 990\bar{3} $ & +$ 3.66 $&$ 19.4178 $&$ 19.4435 $&$ 99867 $ +\\ +$ 2.68 $&$ 7.258\bar{3} $&$ 7.3268 $&$ 9906 $ & +$ 3.68 $&$ 19.810\bar{6} $&$ 19.8358 $&$ 99872 $ +\\ +&&&&&&& \\ +$ 2.70 $&$ 7.406\bar{3} $&$ 7.473\bar{5} $&$ .9910 $ & +$ 3.70 $&$ 20.211\bar{3} $&$ 20.2360 $&$ .99877 $ +\\ +$ 2.72 $&$ 7.5572 $&$ 7.623\bar{1} $&$ 991\bar{4} $ & +$ 3.72 $&$ 20.620\bar{1} $&$ 20.6443 $&$ 99882 $ +\\ +$ 2.74 $&$ 7.7112 $&$ 7.775\bar{8} $&$ 991\bar{7} $ & +$ 3.74 $&$ 21.0371 $&$ 21.060\bar{9} $&$ 99887 $ +\\ +$ 2.76 $&$ 7.868\bar{3} $&$ 7.931\bar{6} $&$ 9920 $ & +$ 3.76 $&$ 21.462\bar{6} $&$ 21.485\bar{9} $&$ 99891 $ +\\ +$ 2.78 $&$ 8.028\bar{5} $&$ 8.0905 $&$ 9923 $ & +$ 3.78 $&$ 21.8966 $&$ 21.9194 $&$ 9989\bar{6} $ +\\ +&&&&&&& \\ +$ 2.80 $&$ 8.1919 $&$ 8.2527 $&$ .9926 $ & +$ 3.80 $&$ 22.3394 $&$ 22.361\bar{8} $&$ .9990\bar{0} $ +\\ +$ 2.82 $&$ 8.3586 $&$ 8.4182 $&$ 9929 $ & +$ 3.82 $&$ 22.7911 $&$ 22.813\bar{1} $&$ 9990\bar{4} $ +\\ +$ 2.84 $&$ 8.528\bar{7} $&$ 8.587\bar{1} $&$ 993\bar{2} $ & +$ 3.84 $&$ 23.252\bar{0} $&$ 23.273\bar{5} $&$ 99907 $ +\\ +$ 2.86 $&$ 8.7021 $&$ 8.759\bar{4} $&$ 993\bar{5} $ & +$ 3.86 $&$ 23.7221 $&$ 23.7432 $&$ 99911 $ +\\ +$ 2.88 $&$ 8.879\bar{1} $&$ 8.9352 $&$ 9937 $ & +$ 3.88 $&$ 24.2018 $&$ 24.2224 $&$ 9991\bar{5} $ +\\ +&&&&&&& \\ +$ 2.90 $&$ 9.059\bar{6} $&$ 9.114\bar{6} $&$ .994\bar{0} $ & +$ 3.90 $&$ 24.6911 $&$ 24.7113 $&$ .99918 $ +\\ +$ 2.92 $&$ 9.243\bar{7} $&$ 9.2976 $&$ 994\bar{2} $ & +$ 3.92 $&$ 25.1903 $&$ 25.2101 $&$ 99921 $ +\\ +$ 2.94 $&$ 9.431\bar{5} $&$ 9.484\bar{4} $&$ 9944 $ & +$ 3.94 $&$ 25.699\bar{6} $&$ 25.7190 $&$ 99924 $ +\\ +$ 2.96 $&$ 9.623\bar{1} $&$ 9.674\bar{9} $&$ 994\bar{7} $ & +$ 3.96 $&$ 26.2191 $&$ 26.238\bar{2} $&$ 99927 $ +\\ +$ 2.98 $&$ 9.8185 $&$ 9.8693 $&$ 994\bar{9} $ & +$ 3.98 $&$ 26.749\bar{2} $&$ 26.767\bar{9} $&$ 99930 $ +\\ +&&&&&&& \\ +\hline +\end{tabular} \end{center} \normalsize + +\newpage + +\subsubsection*{} %% RWD Nickalls : empty subsubsection{} command as a hook for the addcontentsline... +\begin{center} +\addcontentsline{lot}{table}{Table II.---Values of $\cosh(x + iy)$ +and $\sinh (x + iy)$} +\textsc{Table II.---Values of $\cosh(x + iy)$ and $\sinh(x + iy)$.}% +\index{Complex numbers!Tables} +\\ +\medskip \footnotesize +\begin{tabular}{r| rc| cr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = 0 $} & \multicolumn{4}{|c}{$ x = .1 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&&& \\ + 0 &$ 1.0000 $&$ 0000 $&$ 0000 $&$ .0000 $ + &$ 1.0050 $&$ .00000 $&$ .1001\bar{7} $&$ .0000 $ +\\ + .1 &$ 0.9950 $& '' & '' &$ 0998 $ + &$ 1.000\bar{0} $&$ 01000 $&$ 09967 $&$ 1003 $ +\\ + .2 &$ 0.980\bar{1} $& '' & '' &$ 198\bar{7} $ + &$ 0.9850 $&$ 0199\bar{0}$&$ 09817 $&$ 199\bar{7} $ +\\ + .3 &$ 0.9553 $& '' & '' &$ 2955 $ + &$ 0.9601 $&$ 02960 $&$ 0957\bar{0} $&$ 297\bar{0} $ +\\ +&&&&&&&& \\ + .4 &$ .921\bar{1} $& '' & '' &$ .3894 $ + &$ .925\bar{7} $&$ .03901 $&$ .09226 $&$ .3914 $ +\\ + .5 &$ 8776 $& '' & '' &$ 4794 $ + &$ 882\bar{0} $&$ 04802 $&$ 0879\bar{1} $&$ 4818 $ +\\ + .6 &$ 8253 $& '' & '' &$ 5646 $ + &$ 829\bar{5} $&$ 05656 $&$ 08267 $&$ 567\bar{5} $ +\\ + .7 &$ 7648 $& '' & '' &$ 6442 $ + &$ 768\bar{7} $&$ 06453 $&$ 07661 $&$ 6474 $ +\\ +&&&&&&&& \\ + .8 &$ .6967 $& '' & '' &$ .717\bar{4} $ + &$ .700\bar{2} $&$ .0718\bar{6}$&$ .0697\bar{9} $&$ .7800 $ +\\ + .9 &$ 6216 $& '' & '' &$ 7833 $ + &$ 624\bar{7} $&$ 0784\bar{7}$&$ 0622\bar{7} $&$ 7872 $ +\\ + +1.0 &$ 5403 $& '' & '' &$ 841\bar{5} $ + &$ 5430 $&$ 08429 $&$ 05412 $&$ 845\bar{7} $ +\\ +1.1 &$ 4536 $& '' & '' &$ 8912 $ + &$ 455\bar{9} $&$ 08927 $&$ 04544 $&$ 895\bar{7} $ +\\ +&&&&&&&& \\ +1.2 &$ .362\bar{4} $& '' & '' &$ .9320 $ + &$ .364\bar{2} $&$ .09336 $&$ .0363\bar{0} $&$0.936\bar{7} $ +\\ +1.3 &$ 2675 $& '' & '' &$ 963\bar{6} $ + &$ 268\bar{8} $&$ 0965\bar{2}$&$ 0268\bar{0} $&$0.968\bar{4} $ +\\ +1.4 &$ 170\bar{0} $& '' & '' &$ 9854 $ + &$ 1708 $&$ 09871 $&$ 0170\bar{3} $&$0.990\bar{4} $ +\\ +1.5 &$ 0707 $& '' & '' &$ 997\bar{5} $ + &$ 0711 $&$ 0999\bar{2}$&$ 0070\bar{9} $&$1.002\bar{5} $ +\\ +&&&&&&&& \\ +$\tfrac{1}{2}\pi $ + &$ 0000 $& '' & '' &$1.0000 $ + &$ 0000 $&$ 1001\bar{7}$&$ 00000 $&$1.0050 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} \\ + +\bigskip +\begin{tabular}{r| rr| rr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = .2 $} & \multicolumn{4}{|c}{$ x = .3 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&&& \\ + 0 &$ 1.020\bar{1} $&$ .0000 $&$ .2013 $&$ .0000 $ + &$ 1.0453 $&$ .0000 $&$ .3045 $&$ .0000 $ +\\ + .1 &$ 1.015\bar{0} $&$ 0201 $&$ 2003 $&$ 1018 $ + &$ 1.040\bar{1} $&$ 0304 $&$ 303\bar{0} $&$ 1044 $ +\\ + .2 &$ 0.9997 $&$ 0400 $&$ 1973 $&$ 202\bar{7} $ + &$ 1.024\bar{5} $&$ 0605 $&$ 298\bar{5} $&$ 207\bar{7} $ +\\ + .3 &$ 0.9745 $&$ 0595 $&$ 1923 $&$ 3014 $ + &$ 9987 $&$ 090\bar{0} $&$ 2909 $&$ 3089 $ +\\ +&&&&&&&& \\ + .4 &$ .9395 $&$ .0784 $&$ .1854 $&$ .3972 $ + &$ .9628 $&$ .1186 $&$ .280\bar{5} $&$ .407\bar{1} $ +\\ + .5 &$ 895\bar{2} $&$ 0965 $&$ 176\bar{7} $&$ 4890 $ + &$ 917\bar{4} $&$ 146\bar{0} $&$ 267\bar{2} $&$ 501\bar{2} $ +\\ + .6 &$ 8419 $&$ 113\bar{7} $&$ 166\bar{2} $&$ 576\bar{0} $ + &$ 8687 $&$ 1719 $&$ 2513 $&$ 590\bar{3} $ +\\ + .7 &$ 780\bar{2} $&$ 1297 $&$ 154\bar{0} $&$ 6571 $ + &$ 7995 $&$ 196\bar{2} $&$ 2329 $&$ 6734 $ +\\ +&&&&&&&& \\ + .8 &$ .710\bar{7} $&$ .1444 $&$ .140\bar{3} $&$ .731\bar{8} $ + &$ .728\bar{3} $&$ .2184 $&$ .212\bar{2} $&$ .7498 $ +\\ + .9 &$ 634\bar{1} $&$ 1577 $&$ 125\bar{2} $&$ 7990 $ + &$ 649\bar{8} $&$ 2385 $&$ 189\bar{3} $&$ 8188 $ +\\ +1.0 &$ 5511 $&$ 1694 $&$ 108\bar{8} $&$ 858\bar{4} $ + &$ 5648 $&$ 2562 $&$ 1645 $&$ 8796 $ +\\ +1.1 &$ 4627 $&$ 179\bar{5} $&$ 0913 $&$ 909\bar{1} $ + &$ 474\bar{2} $&$ 2714 $&$ 1381 $&$ 9316 $ +\\ +&&&&&&&& \\ +1.2 &$ .3696 $&$ .187\bar{7} $&$ .073\bar{0} $&$0.9507 $ + &$ .378\bar{8} $&$ .2838 $&$ .1103 $&$0.974\bar{3} $ +\\ +1.3 &$ 272\bar{9} $&$ 1940 $&$ 053\bar{9} $&$0.982\bar{9} $ + &$ 2796 $&$ 2934 $&$ 081\bar{5} $&$1.0072 $ +\\ +1.4 &$ 173\bar{4} $&$ 1984 $&$ 0342 $&$1.0052 $ + &$ 177\bar{7} $&$ 3001 $&$ 051\bar{8} $&$1.0301 $ +\\ +1.5 &$ 072\bar{2} $&$ 2008 $&$ 0142 $&$1.0175 $ + &$ 0739 $&$ 303\bar{8} $&$ 0215 $&$1.042\bar{7} $ +\\ +&&&&&&&& \\ +$\tfrac{1}{2}\pi $ + &$ 0000 $&$ 2013 $&$ 0000 $&$1.020\bar{1} $ + &$ 0000 $&$ 3045 $&$ 0000 $&$1.0453 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} \normalsize + + +\newpage + +\subsubsection*{} %% RWD Nickalls : empty subsubsection{} command as a hook for the addcontentsline... +\textsc{Table II.---Values of $\cosh(x + iy)$ and $\sinh(x + iy)$.} +(\emph{continued}) \\ +\footnotesize +\bigskip +\begin{tabular}{r| rr| rr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = .4 $} & \multicolumn{4}{|c}{$ x = .5 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&& \\ + 0 &$ 1.081\bar{1} $&$ .0000 $&$ .410\bar{8} $&$ .0000 $ + &$ 1.1276 $&$ .0000 $&$ .521\bar{1} $&$ .0000 $ +\\ + .1 &$ 1.0756 $&$ 0410 $&$ 408\bar{7} $&$ 1079 $ + &$ 1.122\bar{0} $&$ 0520 $&$ 518\bar{5} $&$ 1126 $ +\\ + .2 &$ 1.0595 $&$ 0816 $&$ 402\bar{6} $&$ 214\bar{8}$ + &$ 1.1051 $&$ 1025 $&$ 5107 $&$ 2240 $ +\\ + .3 &$ 1.032\bar{8} $&$ 121\bar{4} $&$ 3924 $&$ 319\bar{5}$ + &$ 1.077\bar{3} $&$ 154\bar{0} $&$ 4978 $&$ 3332 $ +\\ +&&&&&&& \\ + .4 &$ .9957 $&$ .160\bar{0} $&$ .3783 $&$ .421\bar{0}$ + &$ 1.0386 $&$ .2029 $&$ .480\bar{0} $&$ .4391 $ +\\ + .5 &$ 9487 $&$ 1969 $&$ 360\bar{5} $&$ 518\bar{3}$ + &$ 0.989\bar{6} $&$ 2498 $&$ 4573 $&$ 5406 $ +\\ + .6 &$ 8922 $&$ 2319 $&$ 3390 $&$ 6104 $ + &$ 0.9306 $&$ 2942 $&$ 430\bar{1} $&$ 6367 $ +\\ + .7 &$ 8268 $&$ 2646 $&$ 314\bar{2} $&$ 6964 $ + &$ 0.8624 $&$ 335\bar{7} $&$ 398\bar{6} $&$ 7264 $ +\\ +&&&&&&& \\ + .8 &$ .753\bar{2} $&$ .2947 $&$ .286\bar{2} $&$ .7755 $ + &$ .7856 $&$ .3738 $&$ .363\bar{1} $&$0.8089 $ +\\ + .9 &$ 672\bar{0} $&$ 3218 $&$ 2553 $&$ 8468 $ + &$ 7009 $&$ 408\bar{2} $&$ 3239 $&$0.8833 $ +\\ +1.0 &$ 5841 $&$ 3456 $&$ 2219 $&$ 909\bar{7}$ + &$ 609\bar{3} $&$ 438\bar{5} $&$ 2815 $&$0.948\bar{9}$ +\\ +1.1 &$ 4904 $&$ 366\bar{1} $&$ 1863 $&$ 963\bar{5}$ + &$ 511\bar{5} $&$ 4644 $&$ 236\bar{4} $&$1.005\bar{0}$ +\\ +&&&&&&& \\ +1.2 &$ .3917 $&$ .328\bar{9} $&$ .1488 $&$1.0076 $ + &$ .4056 $&$ .485\bar{7} $&$ .1888 $&$1.051\bar{0}$ +\\ +1.3 &$ 289\bar{2} $&$ 395\bar{8} $&$ 109\bar{9} $&$1.041\bar{7}$ + &$ 3016 $&$ 5021 $&$ 139\bar{4} $&$1.0865 $ +\\ +1.4 &$ 183\bar{8} $&$ 404\bar{8} $&$ 0698 $&$1.0653 $ + &$ 191\bar{7} $&$ 5135 $&$ 088\bar{6} $&$1.1163 $ +\\ +1.5 &$ 076\bar{5} $&$ 4097 $&$ 029\bar{1} $&$1.078\bar{4}$ + &$ 079\bar{8} $&$ 519\bar{8} $&$ 036\bar{9} $&$1.124\bar{8}$ +\\ +&&&&&&& \\ +$ \tfrac{1}{2}\pi$ + &$ 0000 $&$ 410\bar{8} $&$ 0000 $&$1.081\bar{1}$ + &$ 0000 $&$ 521\bar{1} $&$ 0000 $&$1.1276 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} + +\bigskip +\begin{tabular}{r| rr| rr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = .6 $} & \multicolumn{4}{|c}{$ x = .7 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&& \\ + 0 &$ 1.185\bar{5} $&$ .0000 $&$ .636\bar{7} $&$ .0000 $ + &$ 1.2552 $&$ .0000 $&$ .758\bar{6} $&$ .0000 $ +\\ + .1 &$ 1.1795 $&$ 063\bar{6} $&$ 633\bar{5} $&$ 1183 $ + &$ 1.248\bar{9} $&$ 0757 $&$ 754\bar{8} $&$ 1253 $ +\\ + .2 &$ 1.161\bar{8} $&$ 126\bar{5} $&$ 624\bar{0} $&$ 2355 $ + &$ 1.2301 $&$ 1542 $&$ 743\bar{5} $&$ 249\bar{4}$ +\\ + .3 &$ 1.132\bar{5} $&$ 1881 $&$ 6082 $&$ 3503 $ + &$ 1.1991 $&$ 224\bar{2} $&$ 7247 $&$ 3709 $ +\\ +&&&&&&& \\ + .4 &$ 1.0918 $&$ .2479 $&$ .5864 $&$ .461\bar{7}$ + &$ 1.156\bar{1} $&$ .2954 $&$ .6987 $&$ .488\bar{8}$ + \\ + .5 &$ 1.0403 $&$ 3052 $&$ 5587 $&$ 5684 $ + &$ 1.1015 $&$ 363\bar{7} $&$ 6657 $&$ 601\bar{8}$ + \\ + .6 &$ 0.9784 $&$ 395\bar{5} $&$ 525\bar{5} $&$ 669\bar{4}$ + &$ 1.0359 $&$ 4253 $&$ 626\bar{1} $&$ 7087 $ + \\ + .7 &$ 0.906\bar{7} $&$ 4101 $&$ 4869 $&$ 763\bar{7}$ + &$ 0.960\bar{0} $&$ 488\bar{7} $&$ 580\bar{2} $&$ 8086 $ +\\ +&&&&&&& \\ + .8 &$ .8259 $&$ .4567 $&$ .443\bar{6} $&$0.8504 $ + &$ .874\bar{5} $&$ .544\bar{2} $&$ .5285 $&$0.9004 $ +\\ + .9 &$ 736\bar{9} $&$ 4987 $&$ 3957 $&$0.9286 $ + &$ 7802 $&$ 5942 $&$ 4715 $&$0.9832 $ +\\ +1.0 &$ 6405 $&$ 5357 $&$ 344\bar{0} $&$0.9975 $ + &$ 678\bar{2} $&$ 6383 $&$ 409\bar{9} $&$1.056\bar{2}$ +\\ +1.1 &$ 5377 $&$ 567\bar{4} $&$ 288\bar{8} $&$1.056\bar{5}$ + &$ 5693 $&$ 6760 $&$ 344\bar{1} $&$1.1186 $ +\\ +&&&&&&& \\ +1.2 &$ .429\bar{6} $&$ .593\bar{4} $&$ .230\bar{7} $&$1.104\bar{9}$ + &$ .4548 $&$ .7070 $&$ .274\bar{9} $&$1.169\bar{9}$ +\\ +1.3 &$ 3171 $&$ 613\bar{5} $&$ 1703 $&$1.1422 $ + &$ 335\bar{8} $&$ 7309 $&$ 2029 $&$1.2094 $ +\\ +1.4 &$ 201\bar{5} $&$ 627\bar{4} $&$ 1082 $&$1.1682 $ + &$ 2133 $&$ 7475 $&$ 1289 $&$1.2369 $ +\\ +1.5 &$ 083\bar{9} $&$ 635\bar{1} $&$ 0450 $&$1.182\bar{5}$ + &$ 088\bar{8} $&$ 756\bar{7} $&$ 053\bar{7} $&$1.2520 $ +\\ +&&&&&&& \\ +$ \tfrac{1}{2}\pi$ + &$ 0000 $&$ 636\bar{7} $&$ 0000 $&$1.185\bar{5}$ + &$ 0000 $&$ 7586 $&$ 0000 $&$1.2552 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} \normalsize + + +\newpage + +\subsubsection*{} %% RWD Nickalls : empty subsubsection{} command as a hook for the addcontentsline... +\textsc{Table II.---Values of $\cosh(x + iy)$ and $\sinh(x + iy)$.} +(\emph{continued}) \\ +\footnotesize +\bigskip +\begin{tabular}{r| rr| rr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = .8 $} & \multicolumn{4}{|c}{$ x = .9 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&& \\ + 0 &$ 1.3374 $&$ .0000 $&$ .8881 $&$ .0000 $ + &$ 1.433\bar{1}$&$ .0000 $&$ 1.0265 $&$ .0000 $ +\\ + .1 &$ 1.330\bar{8}$&$ 088\bar{7} $&$ 883\bar{7}$&$ 1335 $ + &$ 1.4259 $&$ 102\bar{5} $&$ 1.021\bar{4}$&$ 143\bar{1}$ +\\ + .2 &$ 1.3108 $&$ 1764 $&$ 8704 $&$ 2657 $ + &$ 1.4045 $&$ 2039 $&$ 1.006\bar{1}$&$ 2847 $ +\\ + .3 &$ 1.2776 $&$ 262\bar{5} $&$ 8484 $&$ 3952 $ + &$ 1.3691 $&$ 303\bar{4} $&$ 0.980\bar{7}$&$ 4235 $ +\\ +&&&&&&&& \\ + .4 &$ 1.231\bar{9}$&$ .3458 $&$ .8180 $&$ .5208 $ + &$ 1.320\bar{0}$&$ .3997 $&$ .945\bar{5}$&$ .558\bar{1}$ +\\ + .5 &$ 1.173\bar{7}$&$ 425\bar{8} $&$ 779\bar{4}$&$ 641\bar{2}$ + &$ 1.257\bar{7}$&$ 4921 $&$ 9008 $&$ 687\bar{1}$ +\\ + .6 &$ 1.1038 $&$ 501\bar{5} $&$ 733\bar{0}$&$ 755\bar{2}$ + &$ 1.182\bar{8}$&$ 5796 $&$ 8472 $&$ 809\bar{2}$ +\\ + .7 &$ 1.0229 $&$ 5721 $&$ 679\bar{3}$&$ 861\bar{6}$ + &$ 1.096\bar{1}$&$ 661\bar{3} $&$ 7851 $&$ 9232 $ +\\ +&&&&&&&& \\ + .8 &$ .931\bar{8}$&$ .637\bar{1} $&$ .618\bar{8}$&$0.9595 $ + &$ .9984 $&$ .736\bar{4} $&$ .715\bar{2}$&$1.0280 $ +\\ + .9 &$ 831\bar{4}$&$ 695\bar{7} $&$ 552\bar{1}$&$1.0476 $ + &$ 8908 $&$ 804\bar{1} $&$ 638\bar{1}$&$1.1226 $ +\\ +1.0 &$ 7226 $&$ 7472 $&$ 4798 $&$1.1254 $ + &$ 7743 $&$ 8638 $&$ 5546 $&$1.205\bar{9}$ +\\ +1.1 &$ 606\bar{7}$&$ 791\bar{5} $&$ 4028 $&$1.1919 $ + &$ 6500 $&$ 9148 $&$ 4656 $&$1.277\bar{2}$ +\\ +&&&&&&&& \\ +1.2 &$ .4846 $&$ .827\bar{8} $&$ .3218 $&$1.2465 $ + &$ .519\bar{3}$&$0.956\bar{8} $&$ .372\bar{0}$&$1.335\bar{7}$ +\\ +1.3 &$ 357\bar{8}$&$ 8557 $&$ 237\bar{6}$&$1.288\bar{7}$ + &$ 383\bar{4}$&$0.9891 $&$ 274\bar{6}$&$1.380\bar{9}$ +\\ +1.4 &$ 2273 $&$ 875\bar{2} $&$ 151\bar{0}$&$1.3180 $ + &$ 2436 $&$1.0124 $&$ 1745 $&$1.4122 $ +\\ +1.5 &$ 0946 $&$ 885\bar{9} $&$ 0628 $&$1.334\bar{1}$ + &$ 101\bar{4}$&$1.0239 $&$ 0726 $&$1.429\bar{5}$ +\\ +&&&&&&&& \\ +$\tfrac{1}{2} \pi$ + &$ 0000 $&$ .8881 $&$ 0000 $&$1.3374 $ + &$ 0000 $&$1.0265 $&$ 0000 $&$1.433\bar{1}$ +\\ +&&&&&&&& \\ +\hline +\end{tabular} + +\bigskip +\begin{tabular}{r| rr| rr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = 1.0 $} & \multicolumn{4}{|c}{$ x = 1.1 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&& \\ + 0 &$1.543\bar{1} $&$ .0000 $&$1.1752 $&$ .0000 $ + &$1.6685 $&$ .0000 $&$1.3356 $&$ .0000 $ +\\ + .1 &$1.535\bar{4} $&$ 1173 $&$1.1693 $&$ 154\bar{1}$ + &$1.660\bar{2} $&$ 1333 $&$1.329\bar{0} $&$ 1666 $ +\\ + .2 &$1.5123 $&$ 2335 $&$1.1518 $&$ 306\bar{6}$ + &$1.635\bar{3} $&$ 2654 $&$1.3090 $&$ 331\bar{5}$ +\\ + .3 &$1.474\bar{2} $&$ 347\bar{3}$&$1.1227 $&$ 4560 $ + &$1.594\bar{0} $&$ 3946 $&$1.276\bar{0} $&$ 493\bar{1}$ +\\ +&&&&&&&& \\ + .4 &$1.421\bar{3} $&$ 457\bar{6}$&$1.0824 $&$ .6009 $ + &$1.5368 $&$ 5201 $&$1.2302 $&$0.649\bar{8}$ +\\ + .5 &$1.354\bar{2} $&$ 5634 $&$1.031\bar{4} $&$ 739\bar{8}$ + &$1.464\bar{3} $&$ 6403 $&$1.1721 $&$0.7999 $ +\\ + .6 &$1.273\bar{6} $&$ 663\bar{6}$&$0.9699 $&$ 871\bar{8}$ + &$1.377\bar{1} $&$ 754\bar{2}$&$1.102\bar{4} $&$0.9421 $ +\\ + .7 &$1.1802 $&$ 757\bar{1}$&$0.8988 $&$ 994\bar{1}$ + &$1.276\bar{2} $&$ 8604 $&$1.021\bar{6} $&$1.074\bar{9}$ +\\ +&&&&&&&& \\ + .8 &$1.075\bar{1} $&$0.8430 $&$ .818\bar{8} $&$1.1069 $ + &$1.162\bar{5} $&$0.9581 $&$ .930\bar{6} $&$1.1969 $ +\\ + .9 &$0.9592 $&$0.920\bar{6}$&$ 7305 $&$1.2087 $ + &$1.037\bar{2} $&$1.0462 $&$ 8302 $&$1.3070 $ +\\ +1.0 &$0.8337 $&$0.9889 $&$ 635\bar{0} $&$1.298\bar{5}$ + &$0.9015 $&$1.1239 $&$ 721\bar{7} $&$1.4040 $ +\\ +1.1 &$0.6999 $&$1.0473 $&$ 533\bar{1} $&$1.375\bar{2}$ + &$0.7568 $&$1.1903 $&$ 6058 $&$1.487\bar{0}$ +\\ +&&&&&&&& \\ +1.2 &$ .559\bar{2} $&$1.0953 $&$ .4258 $&$1.4382 $ + &$ .6046 $&$1.244\bar{9} $&$ .484\bar{0} $&$1.5551 $ +\\ +1.3 &$ 5128 $&$1.132\bar{4} $&$ 314\bar{4} $&$1.486\bar{8}$ + &$ 4463 $&$1.287\bar{0} $&$ 357\bar{5} $&$1.6077 $ +\\ +1.4 &$ 262\bar{3} $&$1.158\bar{1} $&$ 199\bar{8} $&$1.5213 $ + &$ 2836 $&$1.3162 $&$ 2270 $&$1.6442 $ +\\ +1.5 &$ 109\bar{2} $&$1.172\bar{3} $&$ 0831 $&$1.5392 $ + &$ 1180 $&$1.332\bar{3} $&$ 094\bar{5} $&$1.6643 $ +\\ +&&&&&&&& \\ +$\tfrac{1}{2} \pi$ + &$ 0000 $&$1.1752 $&$ 0000 $&$1.543\bar{1}$ + &$ .0000 $&$1.3356 $&$ .0000 $&$1.6685 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} \normalsize + + +\newpage + +\subsubsection*{} %% RWD Nickalls : empty subsubsection{} command as a hook for the addcontentsline... +\textsc{Table II.---Values of $\cosh(x + iy)$ and $\sinh(x + iy)$.} +(\emph{continued}) \\ +\footnotesize +\bigskip +\begin{tabular}{r| rr| rr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = 1.2 $} & \multicolumn{4}{|c}{$ x = 1.3 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&& \\ + 0 &$1.810\bar{7} $&$ .0000 $&$1.509\bar{5}$&$ .0000 $ + &$1.9709 $&$ 0000 $&$1.698\bar{4}$&$ .0000 $ +\\ + .1 &$1.8016 $&$ 150\bar{7}$&$1.5019 $&$ 180\bar{8}$ + &$1.961\bar{1} $&$ 169\bar{6}$&$1.689\bar{9}$&$ 196\bar{8}$ +\\ + .2 &$1.774\bar{6} $&$ 299\bar{9}$&$1.479\bar{4}$&$ 359\bar{8}$ + &$1.9316 $&$ 3374 $&$1.6645 $&$ 3916 $ +\\ + .3 &$1.729\bar{8} $&$ 446\bar{1}$&$1.4420 $&$ 535\bar{1}$ + &$1.882\bar{9} $&$ 5019 $&$1.6225 $&$ 5824 $ +\\ +&&&&&&& \\ + .4 &$1.6677 $&$ .5878 $&$1.3903 $&$0.7051 $ + &$1.8153 $&$ .661\bar{4}$&$1.5643 $&$0.7675 $ +\\ + .5 &$1.5890 $&$ 723\bar{7}$&$1.324\bar{7}$&$0.868\bar{1}$ + &$1.7296 $&$ 8142 $&$1.490\bar{5}$&$0.9449 $ +\\ + .6 &$1.4944 $&$ 8523 $&$1.2458 $&$1.022\bar{4}$ + &$1.626\bar{7} $&$ 959\bar{0}$&$1.4017 $&$1.1131 $ +\\ + .7 &$1.384\bar{9} $&$ 9724 $&$1.154\bar{5}$&$1.166\bar{5}$ + &$1.5074 $&$1.0941 $&$1.299\bar{0}$&$1.2697 $ +\\ +&&&&&&& \\ + .8 &$1.261\bar{5}$&$1.0828 $&$1.051\bar{7}$&$1.298\bar{9}$ + &$1.3731 $&$1.2183 $&$1.183\bar{3}$&$1.413\bar{9}$ +\\ + .9 &$1.1255 $&$1.182\bar{4} $&$0.938\bar{3}$&$1.4183 $ + &$1.2251 $&$1.330\bar{4} $&$1.0557 $&$1.543\bar{9}$ +\\ +1.0 &$0.9783 $&$1.270\bar{2} $&$0.815\bar{6}$&$1.5236 $ + &$1.064\bar{9}$&$1.4291 $&$0.9176 $&$1.658\bar{5}$ +\\ +1.1 &$0.8213 $&$1.3452 $&$0.684\bar{7}$&$1.613\bar{7}$ + &$0.8940 $&$1.5136 $&$0.770\bar{4}$&$1.756\bar{5}$ +\\ +&&&&&&& \\ +1.2 &$ .6561 $&$1.406\bar{9} $&$0.547\bar{0}$&$1.6876 $ + &$ .714\bar{2}$&$1.583\bar{0} $&$0.6154 $&$1.837\bar{0}$ +\\ +1.3 &$ 484\bar{4}$&$1.4544 $&$0.403\bar{8}$&$1.744\bar{7}$ + &$ 5272 $&$1.636\bar{5} $&$0.4543 $&$1.899\bar{1}$ +\\ +1.4 &$ 307\bar{8}$&$1.487\bar{5} $&$0.256\bar{6}$&$1.7843 $ + &$ 3350 $&$1.673\bar{7} $&$0.288\bar{7}$&$1.9422 $ +\\ +1.5 &$ 128\bar{1}$&$1.505\bar{7} $&$0.106\bar{8}$&$1.8061 $ + &$ 1394 $&$1.6941 $&$0.1201 $&$1.966\bar{0}$ +\\ +&&&&&&& \\ +$\frac{1}{2}\pi$ + &$ 0000 $&$1.509\bar{5} $&$ 0000 $&$1.810\bar{7}$ + &$ 0000 $&$1.698\bar{4} $&$ 0000 $&$1.9709 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} + +\bigskip +\begin{tabular}{r| rr| rr| rr| rr} +\hline + & \multicolumn{4}{|c|}{$ x = 1.4 $} & \multicolumn{4}{|c}{$ x = 1.5 $} +\\ +\cline{2-9} + $y$ & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c|}{$d$} + & \multicolumn{1}{c}{$a$}&\multicolumn{1}{c|}{$b$} + & \multicolumn{1}{c}{$c$}&\multicolumn{1}{c}{$d$} +\\ +\hline +&&&&&&& \\ + 0 &$2.150\bar{9}$&$ .0000 $&$1.9043 $&$ .0000 $ + &$2.3524$ &$ .0000 $&$2.129\bar{3}$&$ .0000 $ +\\ + .1 &$2.1401 $&$ 1901 $&$1.8948 $&$ 2147 $ + &$2.3413 $&$ 2126 $&$2.118\bar{7}$&$ 2348 $ +\\ + .2 &$2.1080 $&$ 3783 $&$1.8663 $&$ 4273 $ + &$2.3055 $&$ 4230 $&$2.0868 $&$ 4674 $ +\\ + .3 &$2.0548 $&$ 562\bar{8}$&$1.8192 $&$ 6356 $ + &$2.2473 $&$ 6292 $&$2.034\bar{2}$&$ 6951 $ +\\ +&&&&&&& \\ + .4 &$1.9811 $&$0.741\bar{6}$&$1.7540 $&$0.8376 $ + &$2.1667 $&$0.829\bar{2}$&$1.961\bar{2}$&$0.916\bar{1}$ +\\ + .5 &$1.887\bar{6}$&$0.913\bar{0}$&$1.671\bar{2}$&$1.031\bar{2}$ + &$2.0644 $&$1.0208 $&$1.8686 $&$1.1278 $ +\\ + .6 &$1.7752 $&$1.075\bar{3}$&$1.5713 $&$1.2145 $ + &$1.9415 $&$1.2023 $&$1.757\bar{4}$&$1.328\bar{3}$ +\\ + .7 &$1.6451 $&$1.228\bar{8}$&$1.4565 $&$1.3856 $ + &$1.7992 $&$1.3717 $&$1.628\bar{6}$&$1.515\bar{5}$ +\\ +&&&&&&& \\ + .8 &$1.4985 $&$1.3661 $&$1.326\bar{8}$&$1.543\bar{0}$ + &$1.6389 $&$1.527\bar{5}$&$1.483\bar{5}$&$1.6875 $ +\\ + .9 &$1.3370 $&$1.4917 $&$1.183\bar{8}$&$1.6849 $ + &$1.462\bar{3}$&$1.6679 $&$1.323\bar{6}$&$1.842\bar{7}$ +\\ +1.0 &$1.162\bar{2}$&$1.6024 $&$1.0289 $&$1.8099 $ + &$1.2710 $&$1.7917 $&$1.150\bar{5}$&$1.979\bar{5}$ +\\ +1.1 &$0.9756 $&$1.6971 $&$0.8638 $&$1.9168 $ + &$1.067\bar{1}$&$1.8976 $&$0.965\bar{9}$&$2.096\bar{5}$ +\\ +&&&&&&& \\ +1.2 &$ .7794 $&$1.774\bar{9}$&$ .6900 $&$2.0047 $ + &$ .8524 $&$1.984\bar{6}$&$ .771\bar{6}$&$2.1925 $ +\\ +1.3 &$ 5754 $&$1.8349 $&$ 5094 $&$2.0725 $ + &$ 629\bar{3}$&$2.051\bar{7}$&$ 569\bar{6}$&$2.266\bar{7}$ +\\ +1.4 &$ 365\bar{6}$&$1.876\bar{6}$&$ 323\bar{7}$&$2.1196 $ + &$ 3998 $&$2.0983 $&$ 3619 $&$2.318\bar{2}$ +\\ +1.5 &$ 152\bar{2}$&$1.8996 $&$ 1347 $&$2.1455 $ + &$ 1664 $&$2.1239 $&$ 1506 $&$2.3465 $ +\\ +&&&&&&& \\ +$\tfrac{1}{2}\pi$ + &$ .0000 $&$1.9043 $&$ 0000 $&$2.150\bar{9}$ + &$ .0000 $&$2.129\bar{3}$&$ .0000 $&$2.3524 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} \normalsize + + +\newpage + +\subsubsection*{} %% RWD Nickalls : empty subsubsection{} command as a hook for the addcontentsline... +\textsc{Table III.} \\ +\addcontentsline{lot}{table}{Table III.---Values of $\gd u$ and +$\theta^\circ$} +\scriptsize \medskip +\begin{tabular}{r|rr||r|rr||r|rr} +\hline +\multicolumn{1}{c|}{\rule[-5pt]{0pt}{12pt}$u$} + &\multicolumn{1}{c}{$\gd u$} + &\multicolumn{1}{c||}{$\theta^\circ$} + &\multicolumn{1}{c|}{$u$}&\multicolumn{1}{c}{$\gd u$} + &\multicolumn{1}{c||}{$\theta^\circ$} + &\multicolumn{1}{c|}{$u$}&\multicolumn{1}{c}{$\gd u$} + &\multicolumn{1}{c}{$\theta^\circ$} +\\ +\hline + & &\multicolumn{1}{c||}{$\circ$} +& & &\multicolumn{1}{c||}{$\circ$} +& & &\multicolumn{1}{c}{$\circ$} +\\ + 00 &$ .0000 $&$ 0.000 $ +& .60 &$ .5669 $&$ 32.483 $ +& 1.50 &$ 1.1317 $&$ 64.843 $ +\\ + .02 &$ 020\bar{0} $&$ 1.146 $ +& .62 &$ 583\bar{7} $&$ 33.444 $ +& 1.55 &$ 1.152\bar{5} $&$ 66.034 $ +\\ + .04 &$ 040\bar{0} $&$ 2.291 $ +& .64 &$ 600\bar{3} $&$ 34.395 $ +& 1.60 &$ 1.172\bar{4} $&$ 67.171 $ +\\ + .06 &$ 060\bar{0} $&$ 3.436 $ +& .66 &$ 6167 $&$ 35.336 $ +& 1.65 &$ 1.1913 $&$ 68.257 $ +\\ + .08 &$ 0799 $&$ 4.579 $ +& .68 &$ 6329 $&$ 36.265 $ +& 1.70 &$ 1.2094 $&$ 69.294 $ +\\ +&&&&&&&\\ + .10 &$ .0998 $&$ 5.720 $ +& .70 &$ .6489 $&$ 37.183 $ +& 1.75 &$ 1.226\bar{7} $&$ 70.284 $ +\\ + .12 &$ 1197 $&$ 6.859 $ +& .72 &$ 6648 $&$ 38.091 $ +& 1.80 &$ 1.243\bar{2} $&$ 71.228 $ +\\ + .14 &$ 1395 $&$ 7.995 $ +& .74 &$ 6804 $&$ 38.987 $ +& 1.85 &$ 1.258\bar{9} $&$ 72.128 $ +\\ + .16 &$ 1593 $&$ 9.128 $ +& .76 &$ 6958 $&$ 39.872 $ +& 1.90 &$ 1.273\bar{9} $&$ 72.987 $ +\\ + .18 &$ 1790 $&$ 10.258 $ +& .78 &$ 7111 $&$ 40.746 $ +& 1.95 &$ 1.2881 $&$ 73.805 $ +\\ + & & & & & & \hrulefill & & +\\ + .20 &$ .198\bar{7} $&$ 11.384 $ +& .80 &$ .7261 $&$ 41.608 $ +& 2.00 &$ 1.3017 $&$ 74.584 $ +\\ + .22 &$ 218\bar{3} $&$ 12.505 $ +& .82 &$ 7410 $&$ 42.460 $ +& 2.10 &$ 1.3271 $&$ 76.037 $ +\\ + .24 &$ 2377 $&$ 13.621 $ +& .84 &$ 755\bar{7} $&$ 43.299 $ +& 2.20 &$ 1.350\bar{1} $&$ 77.354 $ +\\ + .26 &$ 2571 $&$ 14.732 $ +& .86 &$ 770\bar{2} $&$ 44.128 $ +& 2.30 &$ 1.371\bar{0} $&$ 78.549 $ +\\ + .28 &$ 2764 $&$ 15.837 $ +& .88 &$ 7844 $&$ 44.944 $ +& 2.40 &$ 1.389\bar{9} $&$ 79.633 $ +\\ +&&&&&&&\\ + .30 &$ .2956 $&$ 16.937 $ +& .90 &$ .798\bar{5} $&$ 45.750 $ +& 2.50 &$ 1.407\bar{0} $&$ 80.615 $ +\\ + .32 &$ 314\bar{7} $&$ 18.030 $ +& .92 &$ 8123 $&$ 46.544 $ +& 2.60 &$ 1.422\bar{7} $&$ 81.513 $ +\\ + .34 &$ 3336 $&$ 19.116 $ +& .94 &$ 826\bar{0} $&$ 47.326 $ +& 2.70 &$ 1.436\bar{6} $&$ 82.310 $ +\\ + .36 &$ 352\bar{5} $&$ 20.195 $ +& .96 &$ 8394 $&$ 48.097 $ +& 2.80 &$ 1.4493 $&$ 83.040 $ +\\ + .38 &$ 371\bar{2} $&$ 21.267 $ +& .98 &$ 8528 $&$ 48.857 $ +& 2.90 &$ 1.460\bar{9} $&$ 83.707 $ +\\ + & & & \hrulefill & & & & +\\ + .40 &$ .3897 $&$ 22.331 $ +& 1.00 &$ .865\bar{8} $&$ 49.605 $ +& 3.00 &$ 1.4713 $&$ 84.301 $ +\\ + .42 &$ 408\bar{2} $&$ 23.386 $ +& 1.05 &$ 897\bar{6} $&$ 51.428 $ +& 3.10 &$ 1.4808 $&$ 84.841 $ +\\ + .44 &$ 4264 $&$ 24.434 $ +& 1.10 &$ 9281 $&$ 53.178 $ +& 3.20 &$ 1.4894 $&$ 85.336 $ +\\ + .46 &$ 444\bar{6} $&$ 25.473 $ +& 1.15 &$ 957\bar{5} $&$ 54.860 $ +& 3.30 &$ 1.497\bar{1} $&$ 80.715 $ +\\ + .48 &$ 462\bar{6} $&$ 26.503 $ +& 1.20 &$ 985\bar{7} $&$ 56.476 $ +& 3.40 &$ 1.504\bar{1} $&$ 86.177 $ +\\ +&&&&&&&\\ + .50 &$ .4804 $&$ 27.524 $ +& 1.25 &$ 1.0127 $&$ 58.026 $ +& 3.50 &$ 1.5104 $&$ 86.541 $ +\\ + .52 &$ 4980 $&$ 28.535 $ +& 1.30 &$ 1.038\bar{7} $&$ 59.511 $ +& 3.60 &$ 1.516\bar{2} $&$ 86.870 $ +\\ + .54 &$ 5155 $&$ 29.537 $ +& 1.35 &$ 1.063\bar{5} $&$ 60.933 $ +& 3.70 &$ 1.5214 $&$ 87.168 $ +\\ + .56 &$ 5328 $&$ 30.529 $ +& 1.40 &$ 1.087\bar{3} $&$ 62.295 $ +& 3.80 &$ 1.526\bar{1} $&$ 87.437 $ +\\ + .58 &$ 550\bar{0} $&$ 31.511 $ +& 1.45 &$ 1.110\bar{0} $&$ 63.598 $ +& 3.90 &$ 1.5303 $&$ 87.681 $ +\\ +&&&&&&&& \\ +\hline +\end{tabular} \\ \normalsize + +\bigskip +\textsc{Table IV.} \\ +\addcontentsline{lot}{table}{Table IV.---Values of $\gd u, \log\sinh +u, \log\cosh u$} +\medskip \scriptsize +\begin{tabular}{r|r|r|r||r|r|r|r} +\hline \multicolumn{1}{c|}{\rule[-5pt]{0pt}{12pt}$u$} + &\multicolumn{1}{c|}{$\gd u$} + &\multicolumn{1}{c|}{$\log\sinh u$}&\multicolumn{1}{c||}{$\log\cosh u$} + &\multicolumn{1}{c|}{$u$}&\multicolumn{1}{c|}{$\gd u$} + &\multicolumn{1}{c|}{$\log\sinh u$}&\multicolumn{1}{c}{$\log\cosh u$} +\\ +\hline +&&&&&&&\\ + 4.0 &$ 1.534\bar{2} $&$ 1.4360 $&$ 1.4363 $ +& 5.5 &$ 1.5626 $&$ 2.08758 $&$ 2.0876\bar{0} $ +\\ + 4.1 &$ 1.537\bar{7} $&$ 1.4795 $&$ 1.4797 $ +& 5.6 &$ 1.5634 $&$ 2.13101 $&$ 2.1310\bar{3} $ +\\ + 4.2 &$ 1.5408 $&$ 1.5229 $&$ 1.5231 $ +& 5.7 &$ 1.5641 $&$ 2.17444 $&$ 2.17445 $ +\\ + 4.3 &$ 1.543\bar{7} $&$ 1.5664 $&$ 1.5665 $ +& 5.8 &$ 1.5648 $&$ 2.21787 $&$ 2.21788 $ +\\ + 4.4 &$ 1.5462 $&$ 1.6098 $&$ 1.6099 $ +& 5.9 &$ 1.5653 $&$ 2.36130 $&$ 2.26131 $ +\\ + & & & & \hrulefill & & & +\\ + 4.5 &$ 1.548\bar{6} $&$ 1.6532 $&$ 1.6533 $ +& 6.0 &$ 1.5658 $&$ 2.30473 $&$ 2.3047\bar{4} $ +\\ + 4.6 &$ 1.550\bar{7} $&$ 1.6967 $&$ 1.6968 $ +& 6.2 &$ 1.5667 $&$ 2.39159 $&$ 2.3916\bar{0} $ +\\ + 4.7 &$ 1.5526 $&$ 1.7401 $&$ 1.7402 $ +& 6.4 &$ 1.567\bar{5} $&$ 2.47845 $&$ 2.47846 $ +\\ + 4.8 &$ 1.5543 $&$ 1.7836 $&$ 1.7836 $ +& 6.6 &$ 1.568\bar{1} $&$ 2.56531 $&$ 2.56531 $ +\\ + 4.9 &$ 1.5559 $&$ 1.8270 $&$ 1.8270 $ +& 6.8 &$ 1.568\bar{6} $&$ 2.65217 $&$ 2.65217 $ +\\ + & & & & \hrulefill & & & +\\ + 5.0 &$ 1.5573 $&$ 1.8704 $&$ 1.870\bar{5} $ +& 7.0 &$ 1.569\bar{0} $&$ 2.73903 $&$ 2.73903 $ +\\ + 5.1 &$ 1.5586 $&$ 1.913\bar{9} $&$ 1.913\bar{9} $ +& 7.5 &$ 1.569\bar{7} $&$ 2.9561\bar{8} $&$ 3.9561\bar{8} $ +\\ + 5.2 &$ 1.559\bar{8} $&$ 1.957\bar{3} $&$ 1.9573 $ +& 8.0 &$ 1.570\bar{1} $&$ 3.1733\bar{3} $&$ 3.1733\bar{3} $ +\\ + 5.3 &$ 1.5608 $&$ 2.0007 $&$ 2.0007 $ +& 8.5 &$ 1.570\bar{4} $&$ 3.39047 $&$ 3.39047 $ +\\ + 5.4 &$ 1.561\bar{8} $&$ 2.044\bar{2} $&$ 2.044\bar{2} $ +& 9.0 &$ 1.5705 $&$ 3.60762 $&$ 3.60762 $ +\\ + & & & & +\multicolumn{1}{c|}{$\infty $} + &$ 1.570\bar{8} $&\multicolumn{1}{c|}{$\infty $} + &\multicolumn{1}{c}{$\infty $} +\\ +&&&&&&& \\ +\hline +\end{tabular} \index{Hyperbolic functions!tables of|)} \normalsize +\end{center} + +\chapter{Appendix.} + +\section{Historical and Bibliographical.} + +What is probably the earliest suggestion of the analogy between the +sector of the circle and that of the hyperbola is found in Newton's +Principia (Bk.~2, prop.~8 et seq.) in connection with the solution +of a dynamical problem.\index{Newton, reference to} On the +analytical side, the first hint of the modified sine and cosine is +seen in Roger Cotes' Harmonica Mensurarum (1722), where he suggests +the possibility of modifying the expression for the area of the +prolate spheroid so as to give that of the oblate one, by a certain +use of the operator $\sqrt{-1}$.\index{Cotes, reference to} The +actual inventor of the hyperbolic trigonometry was Vincenzo Riccati, +S.J.\ (Opuscula ad res Phys.\ et Math.\ pertinens, Bononi\ae{}, +1757).\index{Riccati's place in the history} He adopted the notation +$\mathrm{Sh.}\phi$, $\mathrm{Ch.}\phi$ for the hyperbolic functions, +and $\mathrm{Sc.}\phi$, $\mathrm{Cc.}\phi$ for the circular ones. He +proved the addition theorem geometrically and derived a construction +for the solution of a cubic equation. Soon after, Daviet de Foncenex +showed how to interchange circular and hyperbolic functions by the +use of $\sqrt{-1}$, and gave the analogue of De Moivre's theorem, +the work resting more on analogy, however, than on clear definition +(Reflex.\ sur les quant.\ imag., Miscel.\ Turin Soc., +Tom.~1).\index{Foncenex, reference to} Johann Heinrich Lambert +systematized the subject, and gave the serial developments and the +exponential expressions. He adopted the notation $\sinh u$, etc., +and introduced the transcendent angle, now called the gudermanian, +using it in computation and in the construction of tables (l.~c.\ +page 30).\index{Lambert's!place in the history} The important place +occupied by Gudermann in the history of the subject is indicated on +page~\pageref{gudermanian}.\index{Gudermanian!function} + +The analogy of the circular and hyperbolic trigonometry naturally +played a considerable part in the controversy regarding the doctrine +of imaginaries, which occupied so much attention in the eighteenth +century, and which gave birth to the modern theory of functions of +the complex variable. In the growth of the general complex theory, +the importance of the ``singly periodic functions'' became still +clearer, and was gradually developed by such writers as Ferroni +(Magnit. expon.\ log.\ et trig., Florence, 1782)% +\index{Ferroni, reference to}; Dirksen (Organon der tran.\ Anal., +Berlin, 1845)\index{Dirksen's Organon}; Schellbach (Die einfach.\ +period.\ funkt., Crelle, 1854)\index{Schellback, reference to}; Ohm +(Versuch eines volk.\ conseq.\ Syst.\ der Math., Nürnberg, +1855)\index{Ohm, reference to}; Hoüel (Theor.\ des quant.\ complex, +Paris, 1870).\index{Hoüel's notation, etc.} Many other writers have +helped in systematizing and tabulating these functions, and in +adapting them to a variety of applications. The following works may +be especially mentioned: Gronau (Tafeln, 1862, Theor.\ und Anwend., +1865)\index{Gronau's!Tafeln}\index{Gronau's!Theor.\ und Anwend.}; +Forti (Tavoli e teoria, 1870)\index{Forti's Tavoli e teoria}; +Laisant (Essai, 1874)\index{Laisant's Essai, etc.}; Gunther (Die +Lehre ..., 1881)\index{Gunther's Die Lehre, etc.}. The last-named +work contains a very full history and bibliography with numerous +applications. Professor A.~G.\ Greenhill, in various places in his +writings, has shown the importance of both the direct and inverse +hyperbolic functions, and has done much to popularize their use (see +Diff.\ and Int.\ Calc., 1891).\index{Greenhill's!Calculus} The +following articles on fundamental conceptions should be noticed: +Macfarlane, On the definitions of the trigonometric functions +(Papers on Space Analysis, N.~Y., 1894)\index{Macfarlane on +definitions}; Haskell, On the introduction of the notion of +hyperbolic functions (Bull.\ N.~Y.\ M.\ Soc., 1895).\index{Haskell +on fundamental notions} Attention has been called in Arts.\ 30 and +37 to the work of Arthur E.\ Kennelly in applying the hyperbolic +complex theory to the plane vectors which present themselves in the +theory of alternating currents; and his chart has been described on +page~\pageref{period-hyp-funct} as a useful +substitute for a numerical complex table (Proc. A.~I.~E.~E., 1895). +It may be worth mentioning in this connection that the present +writer's complex table in Art.\ 39 is believed to be the only one of +its kind for any function of the general argument $x+iy$. + +\medskip +\section{Exponential Expressions as Definitions.}% +\index{Exponential expressions} + +For those who wish to start with the exponential expressions as the +definitions of $\sinh u$ and $\cosh u$, as indicated on +page~\pageref{def hyper as exp}, it is here proposed to show how +these definitions can be easily brought into direct geometrical +relation with the hyperbolic sector in the form $\frac{x}{a}=\cosh +\frac{S}{K}$, $\frac{y}{b} = \sinh \frac{S}{K}$, by making use of +the identity $\cosh^2 u - \sinh^2 u = 1$, and the differential +relations $d \cosh u = \sinh u\, du$, $d \sinh u = \cosh u\, du$, +which are themselves immediate consequences of those exponential +definitions. Let $OA$, the initial radius of the hyperbolic sector, +be taken as axis of $x$, and its conjugate radius $OB$ as axis of +$y$; let $OA = a$, $OB = b$, angle $AOB = \omega$, and area of +triangle $AOB = K$, then $K = \frac{1}{2}ab \sin \omega$. Let the +coordinates of a point $P$ on the hyperbola be $x$ and $y$, then +$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Comparison of this equation +with the identity $\cosh^2 u - \sinh^2 u = 1$ permits the two +assumptions $\frac{x}{a} = \cosh u$ and $\frac{y}{b} = \sinh u$, +wherein $u$ is a single auxiliary variable; and it now remains to +give a geometrical interpretation to $u$, and to prove that $u = +\frac{S}{K}$, wherein $S$ is the area of the sector $OAP$. Let the +coordinates of a second point $Q$ be $x + \Delta x$ and $y + \Delta +y$, then the area of the triangle $POQ$ is, by analytic geometry, +$\frac{1}{2}(x \Delta y - y \Delta x) \sin \omega$. Now the sector +$POQ$ bears to the triangle $POQ$ a ratio whose limit is unity, +hence the differential of the sector $S$ may be written $dS = +\frac{1}{2}(x dy - y dx) \sin \omega = \frac{1}{2} ab \sin \omega +(\cosh^2 u-\sinh^2 u) du = K du$. By integration $S = Ku$, hence $u= +\frac{S}{K}$, the sectorial measure (p.~\pageref{sectoral-measures}); +this establishes the fundamental geometrical relations +$\frac{x}{a}=\cosh \frac{S}{K}, \frac{y}{b} = \sinh \frac{S}{K}$. + +%%=================================================================== +%% RWD Nickalls(dick@nickalls.org)(June2018) +%% use \backmatter command to disable chapter numbering +\backmatter + +\printindex %% just inputs the refashioned .ind file +%%================================================================== + +\newpage +\chapter{PROJECT GUTENBERG "SMALL PRINT"} +\small +\pagenumbering{gobble} +\begin{verbatim} + + + + + +End of the Project Gutenberg EBook Hyperbolic Functions, by James McMahon + +*** END OF THIS PROJECT GUTENBERG EBOOK HYPERBOLIC FUNCTIONS *** + +***** This file should be named 13692-t.tex or 13692-t.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/1/3/6/9/13692/ + +Produced by David Starner, Joshua Hutchinson, John Hagerson, +and the Project Gutenberg On-line Distributed Proofreading Team. + +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. 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