diff options
| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:11:42 -0700 |
|---|---|---|
| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:11:42 -0700 |
| commit | 6651660be676b58c9cc1a570b8510805fd2eae34 (patch) | |
| tree | 71187a8ede3e12f6ab4a167929a7b0434af0b64b | |
| -rw-r--r-- | .gitattributes | 3 | ||||
| -rw-r--r-- | 39017-pdf.pdf | bin | 0 -> 1071568 bytes | |||
| -rw-r--r-- | 39017-pdf.zip | bin | 0 -> 1004612 bytes | |||
| -rw-r--r-- | 39017-t.zip | bin | 0 -> 523722 bytes | |||
| -rw-r--r-- | 39017-t/39017-t.tex | 6358 | ||||
| -rw-r--r-- | 39017-t/images/illo001.png | bin | 0 -> 124503 bytes | |||
| -rw-r--r-- | 39017-t/images/illo085.png | bin | 0 -> 113795 bytes | |||
| -rw-r--r-- | 39017-t/images/illo124.png | bin | 0 -> 57684 bytes | |||
| -rw-r--r-- | 39017-t/images/illo126.png | bin | 0 -> 58881 bytes | |||
| -rw-r--r-- | 39017-t/images/illo127.png | bin | 0 -> 24449 bytes | |||
| -rw-r--r-- | 39017-t/images/illo132.png | bin | 0 -> 66158 bytes | |||
| -rw-r--r-- | LICENSE.txt | 11 | ||||
| -rw-r--r-- | README.md | 2 |
13 files changed, 6374 insertions, 0 deletions
diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/39017-pdf.pdf b/39017-pdf.pdf Binary files differnew file mode 100644 index 0000000..a5f6c9c --- /dev/null +++ b/39017-pdf.pdf diff --git a/39017-pdf.zip b/39017-pdf.zip Binary files differnew file mode 100644 index 0000000..1ac2665 --- /dev/null +++ b/39017-pdf.zip diff --git a/39017-t.zip b/39017-t.zip Binary files differnew file mode 100644 index 0000000..51d3cc3 --- /dev/null +++ b/39017-t.zip diff --git a/39017-t/39017-t.tex b/39017-t/39017-t.tex new file mode 100644 index 0000000..53b3a3a --- /dev/null +++ b/39017-t/39017-t.tex @@ -0,0 +1,6358 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% Project Gutenberg's Eight Lectures on Theoretical Physics, by Max Planck% +% % +% 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: Eight Lectures on Theoretical Physics % +% Delivered at Columbia University in 1909 % +% % +% Author: Max Planck % +% % +% Translator: A. P. Wills % +% % +% Release Date: February 29, 2012 [EBook #39017] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK EIGHT LECTURES *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{39017} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Eight Lectures On Theoretical Physics %% +%% By Max Planck. %% +%% %% +%% Packages and substitutions: %% +%% %% +%% amsmath Basic AMS math package. %% +%% amssymb Basic AMS symbols %% +%% book Document class. %% +%% graphicx Basic graphics for images. %% +%% icomma Proper spacing around thousand-marker commas. %% +%% indentfirst indent first paragraph after section header %% +%% inputenc Encoding %% +%% setspace set line spacing %% +%% soul letterspacing/underlining %% +%% verbatim Preformated text %% +%% %% +%% PDF Pages: 99 %% +%% %% +%% 6 includegraphics calls included as illo???.png %% +%% %% +%% Compile sequence: %% +%% pdflatex x 2 %% +%% %% +%% Compile History: %% +%% %% +%% Feb 29: Laverock. Compiled with pdflatex: %% +%% [pdfeTeX, Version 3.1415926-1.40.10 (MiKTeX 2.8)] %% +%% %% +%% %% +%% February 2012: pglatex. %% +%% Compile this project with: %% +%% pdflatex 39017-t.tex ..... TWO times %% +%% %% +%% pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\documentclass[oneside,12pt]{book} +\usepackage[latin1]{inputenc}[2008/03/30] % encoding system +\usepackage{amsmath}[2000/07/18] % basic math support +\usepackage{amssymb}[2009/06/22] % math symbols +\usepackage{icomma}[2002/03/10] % allow comma separators for thousands +\usepackage{indentfirst}[1995/11/23] % indent first paragraph after section header +\usepackage{graphicx}[1999/02/16] % basic graphics for images. +\usepackage{setspace}[2011/12/19] % set line spacing +\usepackage{soul}[2003/11/17] % letterspacing/underlining +\usepackage{verbatim}[2003/08/22] % preformated text +\listfiles + +\addtolength{\oddsidemargin}{-.5in} +\addtolength{\evensidemargin}{-.25in} +\addtolength{\textwidth}{0.75in} +\addtolength{\topmargin}{-.875in} +\addtolength{\textheight}{1.75in} + +\newcount \Zw +% centred image - file name, width pixels +\newcommand\pngcent[2]{ +\Zw=#2 \divide \Zw by 4 +\begin{center} +\includegraphics*[width=\Zw pt]{images/#1} +\end{center}} + +\newcommand{\Chapter}[2]{% + \newpage\section*{\centering\normalfont\MakeUppercase{#1}} + \subsection*{\centering\normalfont\textsc{#2}} +} + +%temporary change of margins: from http://help-csli.stanford.edu/tex/latex-margins.shtml +\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]} +\let\endchangemargin=\endlist + +\newcommand{\advert}[2]{% +\begin{changemargin}{2em}{0in} +\begin{spacing}{0.9} +{\small\noindent\hspace{-2em} #1} +\end{spacing} +\begin{spacing}{0.75} +\textbf{\footnotesize #2} +\end{spacing} +\end{changemargin} +} + +\newcommand{\Section}[2]{\subsection*{\centering #1 \normalfont\textit{#2}}} + +\newcommand{\First}[1]{\textit{#1}} +\newcommand{\Eg}{E.\;g.} +\newcommand{\eg}{e.\;g.} +\newcommand{\ie}{i.\;e.} + +\newcommand{\Tag}[1]{\tag*{\ensuremath{#1}}} +\newcommand{\Eq}[1]{\ensuremath{#1}} + +\renewcommand{\cal}{\text{cal}} +\newcommand{\const}{\text{const}} +\renewcommand{\deg}{\text{deg}} +\newcommand{\erg}{\text{erg}} +\newcommand{\gr}{\text{gr}} +\newcommand{\cm}{\text{cm}} +\renewcommand{\sec}{\text{sec}} +\newcommand{\ts}[1]{\ensuremath{_{\textrm{#1}}}} + +\newcommand{\dd}{\partial} +\newcommand{\tsum}{{\textstyle\sum}} + +\renewcommand{\leq}{\leqq} +\renewcommand{\geq}{\geqq} +\renewcommand{\ln}{\stackrel{\text{e}}{\log}} +\newcommand{\logten}{\stackrel{\text{10}}{\log}} + +\renewcommand{\theta}{\vartheta} + +\DeclareMathOperator{\arctg}{arctg} +\DeclareMathOperator{\curl}{curl} + +\newcommand{\frakB}{\mathfrak{B}}% Magnetic induction +\newcommand{\frakD}{\mathfrak{D}}% Electric induction +\newcommand{\frakE}{\mathfrak{E}}% Electric field +\newcommand{\frakH}{\mathfrak{H}}% Magnetic field +\newcommand{\frakK}{\mathfrak{K}}% Intensity +\newcommand{\frakT}{\mathfrak{T}}% Large time + +\newcommand{\frakv}{\mathfrak{v}}% Velocity + +\newcommand{\DPnote}[1]{} +\newcommand{\DPtypo}[2]{#2} +\newcommand{\Hplus}{\overset{+}{H}} +\newcommand{\HOminus}{\overset{-}{HO}} +\DeclareInputText{176}{\ifmmode{{}^\circ}\else\textdegree\fi} +\DeclareInputText{183}{\ifmmode\cdot\else\textperiodcentered\fi} + +% For sensible insertion of boilerplate/licence +% overlong lines will wrap and be indented 0.25in +% and text is set in "small" size +\usepackage{verbatim} +\makeatletter +\def\@xobeysp{~\hfil\discretionary{}{\kern\z@}{}\hfilneg} +\renewcommand\verbatim@processline{\leavevmode + \null\kern-0.25in\the\verbatim@line\par} +\addto@hook\every@verbatim{\@totalleftmargin0.25in\small} +\makeatother + +\sodef{\so}{}{0.15em}{0.5em plus 0.5em}{0.5em plus 0.5em}% + +\begin{document} +\mainmatter +\pagestyle{empty} +\begin{verbatim} +Project Gutenberg's Eight Lectures on Theoretical Physics, by Max Planck + +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: Eight Lectures on Theoretical Physics + Delivered at Columbia University in 1909 + +Author: Max Planck + +Translator: A. P. Wills + +Release Date: February 29, 2012 [EBook #39017] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK EIGHT LECTURES *** + + +Produced by Brenda Lewis, Keith Edkins and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/Canadian Libraries) +\end{verbatim} +\pagestyle{empty} +\newpage + +%-----File: 001.png---\redacted\-------- +\begin{center} +{\small \so{COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK}} + +\vspace{0.5\baselineskip} + +{\footnotesize PUBLICATION NUMBER THREE\\ +OF THE ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH\\ +ESTABLISHED DECEMBER 17\textsc{th}, 1904} + +\rule{4in}{0.5pt} + +\vspace{-2.5ex} + +\rule{4in}{0.5pt} + +\vspace{\baselineskip} + +{\LARGE \textbf{EIGHT LECTURES}\\[1ex] +\textbf{ON THEORETICAL PHYSICS}} + +\vspace{\baselineskip} + +{\footnotesize DELIVERED AT COLUMBIA UNIVERSITY\\ +IN 1909} + +\vspace{\baselineskip} + +{\tiny BY} + +{\small MAX PLANCK} + +\vspace{0.5\baselineskip} + +{\tiny PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN\\[-1.5ex] +LECTURER IN MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1909} + +\vspace{\baselineskip} + +{\scriptsize TRANSLATED BY} + +{\footnotesize A. P. WILLS} + +{\tiny PROFESSOR OF MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY} + +\vspace{3\baselineskip} + +\pngcent{illo001.png}{534} + +\vspace{3\baselineskip} + +{\small NEW YORK\\ +COLUMBIA UNIVERSITY PRESS\\ +1915} +\end{center} + +{\scriptsize \noindent \textsc{Transcriber's Note:} \emph{A few typographical errors have been corrected - +these are noted at the end of the text.}} + +%-----File: 002.png---\redacted\-------- +\newpage +\begin{center} +\textsc{Translated and Published by Arrangement with\\ +S.~Hirzel, Leipzig, owner of the original copyright\\ +Copyright 1915 by Columbia University Press} + +\vspace{6in} + +\textsf{\footnotesize PRESS OF\\ +THE NEW ERA PRINTING COMPANY\\ +LANCASTER, PA.} + +{\small 1915} +\end{center} + +%-----File: 003.png---\redacted\-------- +\newpage +\begin{spacing}{0.9}{\small On the seventeenth day of December, nineteen hundred and four, Edward Dean +Adams, of New York, established in Columbia University ``The Ernest Kempton +Adams Fund for Physical Research'' as a memorial to his son, Ernest Kempton +Adams, who received the degrees of Electrical Engineering in~1897 and Master of +Arts in~1898, and who devoted his life to scientific research. The income of this +fund is, by the terms of the deed of gift, to be devoted to the maintenance of a +research fellowship and to the publication and distribution of the results of scientific +research on the part of the fellow. A generous interpretation of the terms of the +deed on the part of Mr.~Adams and of the Trustees of the University has made it +possible to issue these lectures as a publication of the Ernest Kempton Adams Fund.}\end{spacing} + +\begin{center} + +\vspace{-\baselineskip}\rule{4in}{0.5pt} + +\vspace{-2.5ex} + +\rule{4in}{0.5pt}\vspace{\baselineskip} + +\textbf{Publications of the\\ +Ernest Kempton Adams Fund for Physical Research} +\end{center} + +\advert{Number One. \textbf{Fields of Force.} By \textsc{Vilhelm Friman Koren Bjerknes}, Professor of Physics +in the University of Stockholm. A course of lectures delivered at Columbia University, +1905--6.} +{Hydrodynamic fields. Electromagnetic fields. Analogies between the two. Supplementary lecture on +application of hydrodynamics to meteorology. 160~pp.}\vspace{-1.5\baselineskip} + +\advert{Number Two. \textbf{The Theory of Electrons and its Application to the Phenomena of Light and +Radiant Heat.} By \textsc{H.~A. Lorentz}, Professor of Physics in the University of Leyden. +A course of lectures delivered at Columbia University, 1906--7. With added notes. +332~pp. Edition exhausted. Published in another edition by Teubner.}{}\vspace{-2.5\baselineskip} + +\advert{Number Three. \textbf{Eight Lectures on Theoretical Physics.} By \textsc{Max Planck}, Professor of +Theoretical Physics in the University of Berlin. A course of lectures delivered at +Columbia University in 1909, translated by \textsc{A.~P. Wills}, Professor of Mathematical +Physics in Columbia University.} +{Introduction: Reversibility and irreversibility. Thermodynamic equilibrium in dilute solutions. +Atomistic theory of matter. Equation of state of a monatomic gas. Radiation, electrodynamic theory. +Statistical theory. Principle of least work. Principle of relativity. 130~pp.}\vspace{-1.5\baselineskip} + +\advert{Number Four. \textbf{Graphical Methods.} By \textsc{C. Runge}, Professor of Applied Mathematics in the +University of Göttingen. A course of lectures delivered at Columbia University, +1909--10.} +{Graphical calculation. The graphical representation of functions of one or more independent variables. +The graphical methods of the differential and integral calculus. 148~pp.}\vspace{-1.5\baselineskip} + +\advert{Number Five. \textbf{Four Lectures on Mathematics.} By \textsc{J. Hadamard}, Member of the Institute, +Professor in the Collége de~France and in the École Polytechnique. A course of lectures +delivered at Columbia University in~1911.} +{Linear partial differential equations and boundary conditions. Contemporary researches in differential +and integral equations. Analysis situs. Elementary solutions of partial differential equations +and Green's functions. 53~pp.}\vspace{-1.5\baselineskip} + +\advert{Number Six. \textbf{Researches in Physical Optics, Part~I,} with especial reference to the radiation +of electrons. By \textsc{R.~W. Wood}, Adams Research Fellow, 1913, Professor of Experimental +Physics in the Johns Hopkins University. 134~pp. With 10~plates. Edition exhausted.}{}\vspace{-2.5\baselineskip} + +\advert{Number Seven. \textbf{Neuere Probleme der theoretischen Physik.} By \textsc{W.~Wien}, Professor of +Physics in the University of Würzburg. A course of six lectures delivered at Columbia +University in~1913.} +{Introduction: Derivation of the radiation equation. Specific heat theory of Debye. Newer radiation +theory of Planck. Theory of electric conduction in metals, electron theory for metals. The Einstein +fluctuations. Theory of Röntgen rays. Method of determining wave length. Photo-electric effect and +emission of light by canal ray particles. 76~pp.}\vspace{-1\baselineskip} + +\begin{spacing}{0.9}{\small These publications are distributed under the Adams Fund to many libraries +and to a limited number of individuals, but may also be bought at cost from the +Columbia University Press.}\end{spacing} + +%-----File: 004.png---\redacted\-------- +% [Blank Page] +%-----File: 005.png---\redacted\-------- + +\newpage +\Section{}{\emph{PREFACE TO ORIGINAL EDITION.}} + +The present book has for its object the presentation of the +lectures which I delivered as foreign lecturer at Columbia University +in the spring of the present year under the title: ``The +Present System of Theoretical Physics.'' The points of view +which influenced me in the selection and treatment of the +material are given at the beginning of the first lecture. Essentially, +they represent the extension of a theoretical physical +scheme, the fundamental elements of which I developed in an +address at Leyden entitled: ``The Unity of the Physical Concept +of the Universe.'' Therefore I regard it as advantageous to +consider again some of the topics of that lecture. The presentation +will not and can not, of course, claim to cover exhaustively +in all directions the principles of theoretical physics. + +\begin{flushright} + \textsc{The Author.\hspace*{1em}} +\end{flushright} + +\textsc{Berlin}, 1909 +%-----File: 006.png---\redacted\-------- +% [Blank Page] +%-----File: 007.png---\redacted\-------- + +\vspace{10ex} + +\Section{}{\emph{TRANSLATOR'S PREFACE.}} + +At the request of the Adams Fund Advisory Committee, and +with the consent of the author, the following translation of Professor +Planck's Columbia Lectures was undertaken. It is hoped +that the translation will be of service to many of those interested +in the development of theoretical physics who, in spite of +the inevitable loss, prefer a translated text in English to an +original text in German. Since the time of the publication of +the original text, some of the subjects treated, particularly that +of heat radiation, have received much attention, with the result +that some of the points of view taken at that time have undergone +considerable modifications. The author considers it desirable, +however, to have the translation conform to the original +text, since the nature and extent of these modifications can +best be appreciated by reference to the recent literature relating +to the matters in question. +\begin{flushright} + \textsc{A. P. Wills.}\hspace*{1em} +\end{flushright} +%-----File: 008.png---\redacted\-------- +% [Blank Page] +%-----File: 009.png---\redacted\-------- + +\newpage + +\begin{center}{\Large CONTENTS.}\end{center} + +\medskip\begin{center}\textsc{First Lecture.}\end{center} + +\vspace{-3ex}\hfill{\tiny PAGE} + +Introduction. Reversibility and Irreversibility \dotfill \pageref{Lect1} + +\medskip\begin{center}\textsc{Second Lecture.}\end{center} + +Thermodynamic States of Equilibrium in Dilute Solutions \dotfill \pageref{Lect2} + +\medskip\begin{center}\textsc{Third Lecture.}\end{center} + +Atomic Theory of Matter \dotfill \pageref{Lect3} + +\medskip\begin{center}\textsc{Fourth Lecture.}\end{center} + +Equation of State for a Monatomic Gas \dotfill \pageref{Lect4} + +\medskip\begin{center}\textsc{Fifth Lecture.}\end{center} + +Heat Radiation. Electrodynamic Theory \dotfill \pageref{Lect5} + +\medskip\begin{center}\textsc{Sixth Lecture.}\end{center} + +Heat Radiation. Statistical Theory \dotfill \pageref{Lect6} + +\medskip\begin{center}\textsc{Seventh Lecture.}\end{center} + +General Dynamics. Principle of Least Action \dotfill \pageref{Lect7} + +\medskip\begin{center}\textsc{Eighth Lecture.}\end{center} + +General Dynamics. Principle of Relativity \dotfill \pageref{Lect8} + +%-----File: 010.png---\redacted\-------- +% [Blank Page] +%-----File: 011.png---\redacted\-------- + +\Chapter{First Lecture.} +{Introduction: Reversibility and Irreversibility.}\label{Lect1} +\pagestyle{plain} + + +\First{Colleagues, ladies and gentlemen:} The cordial invitation, which +the President of Columbia University extended to me to +deliver at this prominent center of American science some +lectures in the domain of theoretical physics, has inspired in +me a sense of the high honor and distinction thus conferred +upon me and, in no less degree, a consciousness of the +special obligations which, through its acceptance, would be +imposed upon me. If I am to count upon meeting in some +measure your just expectations, I can succeed only through +directing your attention to the branches of my science with +which I myself have been specially and deeply concerned, thus +exposing myself to the danger that my report in certain respects +shall thereby have somewhat too subjective a coloring. + +From those points of view which appear to me the most +striking, it is my desire to depict for you in these lectures the +present status of the system of theoretical physics. I do not +say: the present status of theoretical physics; for to cover this +far broader subject, even approximately, the number of lecture +hours at my disposal would by no means suffice. Time limitations +forbid the extensive consideration of the details of this great +field of learning; but it will be quite possible to develop for you, in +bold outline, a representation of the system as a whole, that is, to +give a sketch of the fundamental laws which rule in the physics +of today, of the most important hypotheses employed, and of +the great ideas which have recently forced themselves into the +subject. I will often gladly endeavor to go into details, but not +in the sense of a thorough treatment of the subject, and only with +the object of making the general laws more clear, through appropriate +%-----File: 012.png---\redacted\-------- +specially chosen examples. I shall select these examples +from the most varied branches of physics. + +If we wish to obtain a correct understanding of the achievements +of theoretical physics, we must guard in equal measure +against the mistake of overestimating these achievements, and +on the other hand, against the corresponding mistake of underestimating +them. That the second mistake is actually often +made, is shown by the circumstance that quite recently voices +have been loudly raised maintaining the bankruptcy and, +débâcle of the whole of natural science. But I think such +assertions may easily be refuted by reference to the simple fact +that with each decade the number and the significance of the +means increase, whereby mankind learns directly through the +aid of theoretical physics to make nature useful for its own +purposes. The technology of today would be impossible without +the aid of theoretical physics. The development of the whole +of electro-technics from galvanoplasty to wireless telegraphy +is a striking proof of this, not to mention aerial navigation. On +the other hand, the mistake of overestimating the achievements +of theoretical physics appears to me to be much more dangerous, +and this danger is particularly threatened by those who have +penetrated comparatively little into the heart of the subject. +They maintain that some time, through a proper improvement +of our science, it will be possible, not only to represent completely +through physical formulae the inner constitution of the +atoms, but also the laws of mental life. I think that there is +nothing in the world entitling us to the one or the other of +these expectations. On the other hand, I believe that there is +much which directly opposes them. Let us endeavor then to +follow the middle course and not to deviate appreciably toward +the one side or the other. + +When we seek for a solid immovable foundation which is able +to carry the whole structure of theoretical physics, we meet +with the questions: What lies at the bottom of physics? What +is the material with which it operates? Fortunately, there is +%-----File: 013.png---\redacted\-------- +a complete answer to this question. The material with which +theoretical physics operates is measurements, and mathematics +is the chief tool with which this material is worked. All physical +ideas depend upon measurements, more or less exactly carried +out, and each physical definition, each physical law, possesses +a more definite significance the nearer it can be brought into +accord with the results of measurements. Now measurements +are made with the aid of the senses; before all with that of sight, +with hearing and with feeling. Thus far, one can say that the +origin and the foundation of all physical research are seated in +our sense perceptions. Through sense perceptions only do we +experience anything of nature; they are the highest court of +appeal in questions under dispute. This view is completely +confirmed by a glance at the historical development of physical +science. Physics grows upon the ground of sensations. The +first physical ideas derived were from the individual perceptions +of man, and, accordingly, physics was subdivided into: physics +of the eye (optics), physics of the ear (acoustics), and physics of +heat sensation (theory of heat). It may well be said that so +far as there was a domain of sense, so far extended originally +the domain of physics. Therefore it appears that in the beginning +the division of physics was based upon the peculiarities +of man. It possessed, in short, an anthropomorphic character. +This appears also, in that physical research, when not occupied +with special sense perceptions, is concerned with practical life, +and particularly with the practical needs of men. Thus, the +art of geodesy led to geometry, the study of machinery to mechanics, +and the conclusion lies near that physics in the last +analysis had only to do with the sense perceptions and needs +of mankind. + +In accordance with this view, the sense perceptions are the +essential elements of the world; to construct an object as opposed +to sense perceptions is more or less an arbitrary matter of will. +In fact, when I speak of a tree, I really mean only a complex of +sense perceptions: I can see it, I can hear the rustling of its +%-----File: 014.png---\redacted\-------- +branches, I can smell its fragrance, I experience pain if I knock +my head against it, but disregarding all of these sensations, +there remains nothing to be made the object of a measurement, +wherewith, therefore, natural science can occupy itself. This is +certainly true. In accordance with this view, the problem of +physics consists only in the relating of sense perceptions, in accordance +with experience, to fixed laws; or, as one may express +it, in the greatest possible economic accommodation of our ideas +to our sensations, an operation which we undertake solely +because it is of use to us in the general battle of existence. + +All this appears extraordinarily simple and clear and, in accordance +with it, the fact may readily be explained that +this positivist view is quite widely spread in scientific circles +today. It permits, so far as it is limited to the standpoint here +depicted (not always done consistently by the exponents of +positivism), no hypothesis, no metaphysics; all is clear and +plain. I will go still further; this conception never leads to an +actual contradiction. I may even say, it can lead to no contradiction. +But, ladies and gentlemen, this view has never contributed +to any advance in physics. If physics is to advance, in +a certain sense its problem must be stated in quite the inverse +way, on account of the fact that this conception is inadequate +and at bottom possesses only a formal meaning. + +The proof of the correctness of this assertion is to be found +directly from a consideration of the process of development +which theoretical physics has actually undergone, and which +one certainly cannot fail to designate as essential. Let us +compare the system of physics of today with the earlier and +more primitive system which I have depicted above. At the +first glance we encounter the most striking difference of all, that +in the present system, as well in the division of the various +physical domains as in all physical definitions, the historical +element plays a much smaller rôle than in the earlier system. +While originally, as I have shown above, the fundamental ideas +of physics were taken from the specific sense perceptions of man, +%-----File: 015.png---\redacted\-------- +the latter are today in large measure excluded from physical +acoustics, optics, and the theory of heat. The physical definitions +of tone, color, and of temperature are today in no +wise derived from perception through the corresponding senses; +but tone and color are defined through a vibration number or +wave length, and the temperature through the volume change +of a thermometric substance, or through a temperature scale +based on the second law of thermodynamics; but heat sensation +is in no wise mentioned in connection with the temperature. +With the idea of force it has not been otherwise. Without +doubt, the word force originally meant bodily force, corresponding +to the circumstance that the oldest tools, the ax, hammer, +and mallet, were swung by man's hands, and that the first +machines, the lever, roller, and screw, were operated by men +or animals. This shows that the idea of force was originally +derived from the sense of force, or muscular sense, and was, +therefore, a specific sense perception. Consequently, I regard +it today as quite essential in a lecture on mechanics to refer, at +any rate in the introduction, to the original meaning of the force +idea. But in the modern exact definition of force the specific +notion of sense perception is eliminated, as in the case of color +sense, and we may say, quite in general, that in modern theoretical +physics the specific sense perceptions play a much smaller rôle +in all physical definitions than formerly. In fact, the crowding +into the background of the specific sense elements goes so far +that the branches of physics which were originally completely +and uniquely characterized by an arrangement in accordance +with definite sense perceptions have fallen apart, in consequence +of the loosening of the bonds between different and widely +separated branches, on account of the general advance towards +simplification and coordination. The best example of this is +furnished by the theory of heat. Earlier, heat formed a separate +and unified domain of physics, characterized through the +perceptions of heat sensation. Today one finds in well nigh all +physics textbooks dealing with heat a whole domain, that of +%-----File: 016.png---\redacted\-------- +radiant heat, separated and treated under optics. The significance +of heat perception no longer suffices to bring together +the heterogeneous parts. + +In short, we may say that the characteristic feature of the entire +previous development of theoretical physics is a definite elimination +from all physical ideas of the anthropomorphic elements, particularly +those of specific sense perceptions. On the other hand, +as we have seen above, if one reflects that the perceptions form +the point of departure in all physical research, and that it is impossible +to contemplate their absolute exclusion, because we cannot +close the source of all our knowledge, then this conscious +departure from the original conceptions must always appear +astonishing or even paradoxical. There is scarcely a fact in the +history of physics which today stands out so clearly as this. +Now, what are the great advantages to be gained through such +a real obliteration of personality? What is the result for the +sake of whose achievement are sacrificed the directness and +succinctness such as only the special sense perceptions vouchsafe +to physical ideas? + +The result is nothing more than the attainment of unity +and compactness in our system of theoretical physics, and, in +fact, the unity of the system, not only in relation to all of its +details, but also in relation to physicists of all places, all times, +all peoples, all cultures. Certainly, the system of theoretical +physics should be adequate, not only for the inhabitants of this +earth, but also for the inhabitants of other heavenly bodies. +Whether the inhabitants of Mars, in case such actually exist, +have eyes and ears like our own, we do not know,---it is quite +improbable; but that they, in so far as they possess the necessary +intelligence, recognize the law of gravitation and the principle of +energy, most physicists would hold as self evident: and anyone +to whom this is not evident had better not appeal to the physicists, +for it will always remain for him an unsolvable riddle that the +same physics is made in the United States as in Germany. + +To sum up, we may say that the characteristic feature of the +%-----File: 017.png---\redacted\-------- +actual development of the system of theoretical physics is an +ever extending emancipation from the anthropomorphic elements, +which has for its object the most complete separation possible +of the system of physics and the individual personality of the +physicist. One may call this the objectiveness of the system +of physics. In order to exclude the possibility of any misunderstanding, +I wish to emphasize particularly that we have here +to do, not with an absolute separation of physics from the +physicist---for a physics without the physicist is unthinkable,---but +with the elimination of the individuality of the particular +physicist and therefore with the production of a common system +of physics for all physicists. + +\label{png17lab1}Now, how does this principle agree with the positivist conceptions +mentioned above? Separation of the system of physics +from the individual personality of the physicist? Opposed to +this principle, in accordance with those conceptions, each +particular physicist must have his special system of physics, in +case that complete elimination of all metaphysical elements is +effected; for physics occupies itself only with the facts discovered +through perceptions, and only the individual perceptions are +directly involved. That other living beings have sensations is, +strictly speaking, but a very probable, though arbitrary, conclusion +from analogy. The system of physics is therefore primarily an +individual matter and, if two physicists accept the same system, +it is a very happy circumstance in connection with their personal +relationship, but it is not essentially necessary. One can regard +this view-point however he will; in physics it is certainly quite +fruitless, and this is all that I care to maintain here. Certainly, +I might add, each great physical idea means a further advance +toward the emancipation from anthropomorphic ideas. This +was true in the passage from the Ptolemaic to the Copernican +cosmical system, just as it is true at the present time for the +apparently impending passage from the so-called classical mechanics +of mass points to the general dynamics originating in +the principle of relativity. In accordance with this, man and +%-----File: 018.png---\redacted\-------- +the earth upon which he dwells are removed from the centre +of the world. It may be predicted that in this century the +idea of time will be divested of the absolute character with +which men have been accustomed to endow it (cf.\ the final +lecture). Certainly, the sacrifices demanded by every such +revolution in the intuitive point of view are enormous; consequently, +the resistance against such a change is very great. But +the development of science is not to be permanently halted +thereby; on the contrary, its strongest impetus is experienced +through precisely those forces which attain success in the struggle +against the old points of view, and to this extent such a +struggle is constantly necessary and useful. + +Now, how far have we advanced today toward the unification +of our system of physics? The numerous independent domains +of the earlier physics now appear reduced to two; mechanics and +electrodynamics, or, as one may say: the physics of material +bodies and the physics of the ether. The former comprehends +acoustics, phenomena in material bodies, and chemical phenomena; +the latter, magnetism, optics, and radiant heat. But is +this division a fundamental one? Will it prove final? This +is a question of great consequence for the future development of +physics. For myself, I believe it must be answered in the +negative, and upon the following grounds: mechanics and electrodynamics +cannot be permanently sharply differentiated from +each other. Does the process of light emission, for example, +belong to mechanics or to electrodynamics? To which domain +shall be assigned the laws of motion of electrons? At first +glance, one may perhaps say: to electrodynamics, since with +the electrons ponderable matter does not play any rôle. But +let one direct his attention to the motion of free electrons in +metals. There he will find, in the study of the classical researches +of H.~A. Lorentz, for example, that the laws obeyed by +the electrons belong rather to the kinetic theory of gases than +to electrodynamics. In general, it appears to me that the +original differences between processes in the ether and processes +%-----File: 019.png---\redacted\-------- +in material bodies are to be considered as disappearing. Electrodynamics +and mechanics are not so remarkably far apart, as is +considered to be the case by many people, who already speak of a +conflict between the mechanical and the electrodynamic views +of the world. Mechanics requires for its foundation essentially +nothing more than the ideas of space, of time, and of that which +is moving, whether one considers this as a substance or a state. +The same ideas are also involved in electrodynamics. A sufficiently +generalized conception of mechanics can therefore also +well include electrodynamics, and, in fact, there are many indications +pointing toward the ultimate amalgamation of these two +subjects, the domains of which already overlap in some measure. + +If, therefore, the gulf between ether and matter be once bridged, +what is the point of view which in the last analysis will best +serve in the subdivision of the system of physics? The answer +to this question will characterize the whole nature of the further +development of our science. It is, therefore, the most important +among all those which I propose to treat today. But for the +purposes of a closer investigation it is necessary that we go somewhat +more deeply into the peculiarities of physical principles. + +We shall best begin at that point from which the first step was +made toward the actual realization of the unified system of +physics previously postulated by the philosophers only; at the +principle of conservation of energy. For the idea of energy is +the only one besides those of space and time which is common to +all the various domains of physics. In accordance with what I +have stated above, it will be apparent and quite self evident to +you that the principle of energy, before its general formularization +by Mayer, Joule, and Helmholz, also bore an anthropomorphic +character. The roots of this principle lay already in the recognition +of the fact that no one is able to obtain useful work from +nothing; and this recognition had originated essentially in the +experiences which were gathered in attempts at the solution of a +technical problem: the discovery of perpetual motion. To this +extent, perpetual motion has come to have for physics a far +%-----File: 020.png---\redacted\-------- +reaching significance, similar to that of alchemy for the chemist, +although it was not the positive, but rather the negative results +of these experiments, through which science was advanced. +Today we speak of the principle of energy quite without reference +to the technical viewpoint or to that of man. We say that the +total amount of energy of an isolated system of bodies is a +quantity whose amount can be neither increased nor diminished +through any kind of process within the system, and we no longer +consider the accuracy with which this law holds as dependent +upon the refinement of the methods, which we at present possess, +of testing experimentally the question of the realization of +perpetual motion. In this, strictly speaking, unprovable generalization, +impressed upon us with elemental force, lies the emancipation +from the anthropomorphic elements mentioned above. + +While the principle of energy stands before us as a complete +independent structure, freed from and independent of the accidents +appertaining to its historical development, this is by no +means true in equal measure in the case of that principle which +R.~Clausius introduced into physics; namely, the second law +of thermodynamics. This law plays a very peculiar rôle in the +development of physical science, to the extent that one is not +able to assert today that for it a generally recognized, and therefore +objective formularization, has been found. In our present +consideration it is therefore a matter of particular interest to +examine more closely its significance. + +In contrast to the first law of thermodynamics, or the energy +principle, the second law may be characterized as follows. While +the first law permits in all processes of nature neither the creation +nor destruction of energy, but permits of transformations only, +the second law goes still further into the limitation of the possible +processes of nature, in that it permits, not all kinds of transformations, +but only certain types, subject to certain conditions. +The second law occupies itself, therefore, with the +question of the kind and, in particular, with the direction of any +natural process. +%-----File: 021.png---\redacted\-------- + +At this point a mistake has frequently been made, which has +hindered in a very pronounced manner the advance of science +up to the present day. In the endeavor to give to the second +law of thermodynamics the most general character possible, it +has been proclaimed by followers of W.~Ostwald as the second +law of energetics, and the attempt made so to formulate it that +it shall determine quite generally the direction of every process +occurring in nature. Some weeks ago I read in a public academic +address of an esteemed colleague the statement that the import +of the second law consists in this, that a stone falls downwards, +that water flows not up hill, but down, that electricity flows from +a higher to a lower potential, and so on. This is a mistake which +at present is altogether too prevalent not to warrant mention +here. + +The truth is, these statements are false. A stone can just as +well rise in the air as fall downwards; water can likewise flow upwards, +as, for example, in a spring; electricity can flow very well +from a lower to a higher potential, as in the case of oscillating discharge +of a condenser. The statements are obviously quite correct, +if one applies them to a stone originally at rest, to water at +rest, to electricity at rest; but then they follow immediately from +the energy principle, and one does not need to add a special second +law. For, in accordance with the energy principle, the kinetic +energy of the stone or of the water can only originate at the +cost of gravitational energy, \ie, the center of mass must descend. +If, therefore, motion is to take place at all, it is necessary +that the gravitational energy shall decrease. That is, the +center of mass must descend. In like manner, an electric current +between two condenser plates can originate only at the +cost of electrical energy already present; the electricity must +therefore pass to a lower potential. If, however, motion and +current be already present, then one is not able to say, a priori, +anything in regard to the direction of the change; it can take +place just as well in one direction as the other. Therefore, there +is no new insight into nature to be obtained from this point of +view. +%-----File: 022.png---\redacted\-------- + +Upon an equally inadequate basis rests another conception of +the second law, which I shall now mention. In considering the circumstance +that mechanical work may very easily be transformed +into heat, as by friction, while on the other hand heat can only +with difficulty be transformed into work, the attempt has been +made so to characterize the second law, that in nature the transformation +of work into heat can take place completely, while +that of heat into work, on the other hand, only incompletely and +in such manner that every time a quantity of heat is transformed +into work another corresponding quantity of energy must necessarily +undergo at the same time a compensating transformation, +as, \eg, the passage of heat from a higher to a lower +temperature. This assertion is in certain special cases correct, +but does not strike in general at the true import of the matter, +as I shall show by a simple example. + +One of the most important laws of thermodynamics is, that +the total energy of an ideal gas depends only upon its temperature, +and not upon its volume. If an ideal gas be allowed to +expand while doing work, and if the cooling of the gas be prevented +through the simultaneous addition of heat from a heat reservoir +at higher temperature, the gas remains unchanged in temperature +and energy content, and one may say that the heat furnished +by the heat reservoir is completely transformed into work without +exchange of energy. Not the least objection can be urged +against this assertion. The law of incomplete transformation +of heat into work is retained only through the adoption of a +different point of view, but which has nothing to do with the +status of the physical facts and only modifies the way of looking +at the matter, and therefore can neither be supported nor contradicted +through facts; namely, through the introduction ad~hoc +of new particular kinds of energy, in that one divides the energy +of the gas into numerous parts which individually can depend +upon the volume. But it is a~priori evident that one can never +derive from so artificial a definition a new physical law, and it is +with such that we have to do when we pass from the first law, +the principle of conservation of energy, to the second law. +%-----File: 023.png---\redacted\-------- + +I desire now to introduce such a new physical law: ``It is not +possible to construct a periodically functioning motor which in +principle does not involve more than the raising of a load and the +cooling of a heat reservoir.'' It is to be understood, that in one +cycle of the motor quite arbitrary complicated processes may +take place, but that after the completion of one cycle there shall +remain no other changes in the surroundings than that the heat +reservoir is cooled and that the load is raised a corresponding +distance, which may be calculated from the first law. Such a +motor could of course be used at the same time as a refrigerating +machine also, without any further expenditure of energy and +materials. Such a motor would moreover be the most efficient +in the world, since it would involve no cost to run it; for the +earth, the atmosphere, or the ocean could be utilized as the heat +reservoir. We shall call this, in accordance with the proposal of +W.~Ostwald, perpetual motion of the second kind. Whether in +nature such a motion is actually possible cannot be inferred from +the energy principle, and may only be determined by special +experiments. + +Just as the impossibility of perpetual motion of the first kind +leads to the principle of the conservation of energy, the quite +independent principle of the impossibility of perpetual motion of +the second kind leads to the second law of thermodynamics, +and, if we assume this impossibility as proven experimentally, +the general law follows immediately: \emph{there are processes in +nature which in no possible way can be made completely reversible}. +For consider, \eg, a frictional process through which mechanical +work is transformed into heat with the aid of suitable +apparatus, if it were actually possible to make in some way such +complicated apparatus completely reversible, so that everywhere +in nature exactly the same conditions be reestablished as existed +at the beginning of the frictional process, then the apparatus +considered would be nothing more than the motor described +above, furnishing a perpetual motion of the second kind. This +appears evident immediately, if one clearly perceives what the +%-----File: 024.png---\redacted\-------- +apparatus would accomplish: transformation of heat into work +without any further outstanding change. + +We call such a process, which in no wise can be made completely +reversible, an irreversible process, and all other processes reversible +processes; and thus we strike the kernel of the second +law of thermodynamics when we say that irreversible processes +occur in nature. In accordance with this, the changes in nature +have a unidirectional tendency. With each irreversible process +the world takes a step forward, the traces of which under no +circumstances can be completely obliterated. Besides friction, +examples of irreversible processes are: heat conduction, diffusion, +conduction of electricity in conductors of finite resistance, +emission of light and heat radiation, disintegration of the atom +in radioactive substances, and so on. On the other hand, examples +of reversible processes are: motion of the planets, free +fall in empty space, the undamped motion of a pendulum, +the frictionless flow of liquids, the propagation of light and +sound waves without absorption and refraction, undamped +electrical vibrations, and so on. For all these processes are +already periodic or may be made completely reversible through +suitable contrivances, so that there remains no outstanding +change in nature; for example, the free fall of a body whereby +the acquired velocity is utilized to raise the body again to its +original height; a light or sound wave which is allowed in a suitable +manner to be totally reflected from a perfect mirror. + +What now are the general properties and criteria of irreversible +processes, and what is the general quantitative measure of +irreversibility? This question has been examined and answered +in the most widely different ways, and it is evident here again +how difficult it is to reach a correct formularization of a problem. +Just as originally we came upon the trail of the energy +principle through the technical problem of perpetual motion, so +again a technical problem, namely, that of the steam engine, +led to the differentiation between reversible and irreversible +processes. Long ago Sadi Carnot recognized, although he utilized +%-----File: 025.png---\redacted\-------- +an incorrect conception of the nature of heat, that irreversible +processes are less economical than reversible, or that in +an irreversible process a certain opportunity to derive mechanical +work from heat is lost. What then could have been +simpler than the thought of making, quite in general, the measure +of the irreversibility of a process the quantity of mechanical +work which is unavoidably lost in the process. For a reversible +process then, the unavoidably lost work is naturally to be set +equal to zero. This view, in accordance with which the import +of the second law consists in a dissipation of useful energy, has +in fact, in certain special cases, \eg, in isothermal processes, +proved itself useful. It has persisted, therefore, in certain of +its aspects up to the present day; but for the general case, however, +it has shown itself as fruitless and, in fact, misleading. The +reason for this lies in the fact that the question concerning the +lost work in a given irreversible process is by no means to be +answered in a determinate manner, so long as nothing further is +specified with regard to the source of energy from which the work +considered shall be obtained. + +An example will make this clear. Heat conduction is an +irreversible process, or as Clausius expresses it: Heat cannot +without compensation pass from a colder to a warmer body. +What now is the work which in accordance with definition is +lost when the quantity of heat~$Q$ passes through direct conduction +from a warmer body at the temperature~$T_{1}$ to a colder body, at +the temperature~$T_{2}$? In order to answer this question, we make +use of the heat transfer involved in carrying out a reversible +Carnot cyclical process between the two bodies employed as +heat reservoirs. \label{png25lab1}In this process a certain amount of work +would be obtained, and it is just the amount sought, since it is +that which would be lost in the direct passage by conduction; +but this has no definite value so long as we do not know whence +the work originates, whether, \eg, in the warmer body or in the +colder body, or from somewhere else. Let one reflect that the +heat given up by the warmer body in the reversible process is certainly +%-----File: 026.png---\redacted\-------- +not equal to the heat absorbed by the colder body, because +a certain amount of heat is transformed into work, and that we +can identify, with exactly the same right, the quantity of heat~$Q$ +transferred by the direct process of conduction with that which in +the cyclical process is given up by the warmer body, or with that +absorbed by the colder body. As one does the former or the latter, +he accordingly obtains for the quantity of lost work in the process +of conduction: +\[ +Q · \frac{T_{1} - T_{2}}{T_{1}} \quad \text{or} \quad +Q · \frac{T_{1} - T_{2}}{T_{2}}. +\] +We see, therefore, that the proposed method of expressing mathematically +the irreversibility of a process does not in general effect +its object, and at the same time we recognize the peculiar reason +which prevents its doing so. The statement of the question is +too anthropomorphic. It is primarily too much concerned with +the needs of mankind, in that it refers directly to the acquirement +of useful work. If one require from nature a determinate +answer, he must take a more general point of view, more disinterested, +less economic. We shall now seek to do this. + +Let us consider any typical process occurring in nature. This +will carry all bodies concerned in it from a determinate initial +state, which I designate as state~$A$, into a determinate final +state~$B$. The process is either reversible or irreversible. A +third possibility is excluded. But whether it is reversible or +irreversible depends solely upon the nature of the two states $A$ +and~$B$, and not at all upon the way in which the process has been +carried out; for we are only concerned with the answer to the +question as to whether or not, when the state~$B$ is once reached, a +complete return to~$A$ in any conceivable manner may be accomplished. +If now, the complete return from $B$ to~$A$ is not +possible, and the process therefore irreversible, it is obvious that +the state~$B$ may be distinguished in nature through a certain +property from state~$A$. Several years ago I ventured to express +this as follows: that nature possesses a greater ``preference'' for +state~$B$ than for state~$A$. In accordance with this mode of +%-----File: 027.png---\redacted\-------- +expression, all those processes of nature are impossible for whose +final state nature possesses a smaller preference than for the +original state. Reversible processes constitute a limiting case; +for such, nature possesses an equal preference for the initial and +for the final state, and the passage between them takes place as +well in one direction as the other. + +We have now to seek a physical quantity whose magnitude +shall serve as a general measure of the preference of nature for +a given state. This quantity must be one which is directly +determined by the state of the system considered, without +reference to the previous history of the system, as is the case with +the energy, with the volume, and with other properties of the +system. It should possess the peculiarity of increasing in all +irreversible processes and of remaining unchanged in all reversible +processes, and the amount of change which it experiences +in a process would furnish a general measure for the irreversibility +of the process. + +R.~Clausius actually found this quantity and called it +``entropy.'' Every system of bodies possesses in each of its +states a definite entropy, \label{png27lab1}and this entropy expresses the preference +of nature for the state in question. It can, in all the +processes which take place within the system, only increase and +never decrease. If it be desired to consider a process in which +external actions upon the system are present, it is necessary +to consider those bodies in which these actions originate as +constituting part of the system; then the law as stated in the +above form is valid. In accordance with it, the entropy of a +system of bodies is simply equal to the sum of the entropies of +the individual bodies, and the entropy of a single body is, in +accordance with Clausius, found by the aid of a certain reversible +process. Conduction of heat to a body increases its +entropy, and, in fact, by an amount equal to the ratio of the +quantity of heat given the body to its temperature. Simple +compression, on the other hand, does not change the entropy. + +Returning to the example mentioned above, in which the +%-----File: 028.png---\redacted\-------- +quantity of heat~$Q$ is conducted from a warmer body at the +temperature~$T_{1}$ to a colder body at the temperature~$T_{2}$, in +accordance with what precedes, the entropy of the warmer body +decreases in this process, while, on the other hand, that of the +colder increases, and the sum of both changes, that is, the change +of the total entropy of both bodies, is: +\[ +-\frac{Q}{T_{1}} + \frac{Q}{T_{2}} > 0. +\] + +This positive quantity furnishes, in a manner free from all +arbitrary assumptions, the measure of the irreversibility of the +process of heat conduction. Such examples may be cited +indefinitely. Every chemical process furnishes an increase of +entropy. + +We shall here consider only the most general case treated by +Clausius: an arbitrary reversible or irreversible cyclical process, +carried out with any physico-chemical arrangement, utilizing +an arbitrary number of heat reservoirs. Since the arrangement +at the conclusion of the cyclical process is the same as that at +the beginning, the final state of the process is to be distinguished +from the initial state solely through the different heat content +of the heat reservoirs, and in that a certain amount of mechanical +work has been furnished or consumed. Let $Q$~be the heat given +up in the course of the process by a heat reservoir at the temperature~$T$, +and let $A$~be the total work yielded (consisting, +\eg, in the raising of weights); then, in accordance with the first +law of thermodynamics: +\[ +\tsum Q = A. +\] +In accordance with the second law, the sum of the changes in +entropy of all the heat reservoirs is positive, or zero. It follows, +therefore, since the entropy of a reservoir is decreased by the +amount~$Q/T$ through the loss of heat~$Q$ that: +\[ +\tsum \frac{Q}{T} \leq 0. +\] +This is the well-known inequality of Clausius. +%-----File: 029.png---\redacted\-------- + +In an isothermal cyclical process, $T$~is the same for all reservoirs. +Therefore: +\[ +\tsum Q \leq 0, \quad \text{hence:}\quad A \leq 0. +\] +That is: in an isothermal cyclical process, heat is produced and +work is consumed. \label{png29lab1}In the limiting case, a reversible isothermal +cyclical process, the sign of equality holds, and therefore the +work consumed is zero, and also the heat produced. This law +plays a leading rôle in the application of thermodynamics to +physical chemistry. + +The second law of thermodynamics including all of its consequences +has thus led to the principle of increase of entropy. +You will now readily understand, having regard to the questions +mentioned above, why I express it as my opinion that in the +theoretical physics of the future the first and most important +differentiation of all physical processes will be into reversible +and irreversible processes. + +In fact, all reversible processes, whether they take place in +material bodies, in the ether, or in both together, show a much +greater similarity among themselves than to any irreversible +process. In the differential equations of reversible processes +the time differential enters only as an even power, corresponding +to the circumstance that the sign of time can be +reversed. This holds equally well for vibrations of the pendulum, +electrical vibrations, acoustic and optical waves, and +for motions of mass points or of electrons, if we only exclude +every kind of damping. But to such processes also +belong those infinitely slow processes of thermodynamics which +consist of states of equilibrium in which the time in general +plays no rôle, or, as one may also say, occurs with the zero power, +which is to be reckoned as an even power. As Helmholtz has +pointed out, all these reversible processes have the common +property that they may be completely represented by the principle +of least action, which gives a definite answer to all questions concerning +any such measurable process, and, to this extent, \label{png29lab2}the theory +of reversible processes may be regarded as completely established. +Reversible processes have, however, the disadvantage that +%-----File: 030.png---\redacted\-------- +singly and collectively they are only ideal: in actual nature there +is no such thing as a reversible process. Every natural process +involves in greater or less degree friction or conduction of heat. +But in the domain of irreversible processes the principle of least +action is no longer sufficient; for the principle of increase of +entropy brings into the system of physics a wholly new element, +foreign to the action principle, and which demands special +mathematical treatment. The unidirectional course of a process +in the attainment of a fixed final state is related to it. + +I hope the foregoing considerations have sufficed to make clear +to you that the distinction between reversible and irreversible +processes is much broader than that between mechanical and +electrical processes and that, therefore, this difference, with better +right than any other, may be taken advantage of in classifying +all physical processes, and that it may eventually play in the +theoretical physics of the future the principal rôle. + +However, the classification mentioned is in need of quite an +essential improvement, for it cannot be denied that in the form +set forth, the system of physics is still suffering from a strong +dose of anthropomorphism. In the definition of irreversibility, +as well as in that of entropy, reference is made to the possibility +of carrying out in nature certain changes, and this means, fundamentally, +nothing more than that the division of physical processes +is made dependent upon the manipulative skill of man in +the art of experimentation, which certainly does not always +remain at a fixed stage, but is continually being more and more +perfected. If, therefore, the distinction between reversible and +irreversible processes is actually to have a lasting significance +for all times, it must be essentially broadened and made independent +of any reference to the capacities of mankind. How this +may happen, I desire to state one week from tomorrow. The +lecture of tomorrow will be devoted to the problem of bringing +before you some of the most important of the great number of +practical consequences following from the entropy principle. +%-----File: 031.png---\redacted\-------- + + +\Chapter{SECOND LECTURE.}{% +Thermodynamic States of Equilibrium in Dilute +Solutions.}\label{Lect2} + +In the lecture of yesterday I sought to make clear the fact +that the essential, and therefore the final division of all processes +occurring in nature, is into reversible and irreversible processes, +and the characteristic difference between these two kinds of +processes, as I have further separated them, is that in irreversible +processes the entropy increases, while in all reversible processes +it remains constant. Today I am constrained to speak of some +of the consequences of this law which will illustrate its rich fruitfulness. +They have to do with the question of the laws of thermodynamic +equilibrium. Since in nature the entropy can only +increase, it follows that the state of a physical configuration +which is completely isolated, and in which the entropy of +the system possesses an absolute maximum, is necessarily a +state of stable equilibrium, since for it no further change is +possible. How deeply this law underlies all physical and chemical +relations has been shown by no one better and more completely +than by John Willard Gibbs, whose name, not only in +America, but in the whole world will be counted among those of +the most famous theoretical physicists of all times; to whom, to +my sorrow, it is no longer possible for me to tender personally +my respects. It would be gratuitous for me, here in the land +of his activity, to expatiate fully on the progress of his ideas, +but you will perhaps permit me to speak in the lecture of today +of some of the important applications in which thermodynamic +research, based on Gibbs works, can be advanced beyond +his results. + +These applications refer to the theory of dilute solutions, and +%-----File: 032.png---\redacted\-------- +we shall occupy ourselves today with these, while I show you +by a definite example what fruitfulness is inherent in thermodynamic +theory. I shall first characterize the problem quite +generally. It has to do with the state of equilibrium of a material +system of any number of arbitrary constituents in an arbitrary +number of phases, at a given temperature~$T$ and given +pressure~$p$. If the system is completely isolated, and therefore +guarded against all external thermal and mechanical +actions, then in any ensuing change the entropy of the system will +increase: +\[ +dS > 0. +\] +But if, as we assume, the system stands in such relation to +its surroundings that in any change which the system undergoes +the temperature~$T$ and the pressure~$p$ are maintained +constant, as, for instance, through its introduction into a calorimeter +of great heat capacity and through loading with a piston +of fixed weight, the inequality would suffer a change thereby. +We must then take account of the fact that the surrounding +bodies also, \eg, the calorimetric liquid, will be involved in the +change. If we denote the entropy of the surrounding bodies by~$S_{0}$, +then the following more general equation holds: +\[ +dS + dS_{0} > 0. +\] +In this equation +\[ +dS_{0} = -\frac{Q}{T}, +\] +if $Q$~denote the heat which is given up in the change by the +surroundings to the system. On the other hand, if $U$~denote +the energy, $V$~the volume of the system, then, in accordance +with the first law of thermodynamics, +\[ +Q = dU + p dV. +\] +Consequently, through substitution: +\[ +dS - \frac{dU + p dV}{T} > 0 +\] +%-----File: 033.png---\redacted\-------- +or, since $p$~and~$T$ are constant: +\[ +d \left(S - \frac{U + pV}{T} \right) > 0. +\] +If, therefore, we put: +\[ +S - \frac{U + pV}{T} = \Phi, +\Tag{(1)} +\] +then +\[ +d \Phi > 0, +\] +and we have the general law, that in every isothermal-isobaric +($T = \const.$, $p = \const.$) change of state of a physical system +the quantity~$\Phi$ increases. The absolutely stable state of +equilibrium of the system is therefore characterized through +the maximum of~$\Phi$: +\[ +\delta \Phi = 0. +\Tag{(2)} +\] +If the system consist of numerous phases, then, because $\Phi$, in +accordance with~\Eq{(1)}, is linear and homogeneous in $S$,~$U$ and~$V$, +the quantity~$\Phi$ referring to the whole system is the sum of the +quantities~$\Phi$ referring to the individual phases. If the expression +for~$\Phi$ is known as a function of the independent variables for +each phase of the system, then, from equation~\Eq{(2)}, all questions +concerning the conditions of stable equilibrium may be +answered. Now, within limits, this is the case for dilute solutions. +By ``solution'' in thermodynamics is meant each homogeneous +phase, in whatever state of aggregation, which is composed of a +series of different molecular complexes, each of which is represented +by a definite molecular number. If the molecular +number of a given complex is great with reference to all the +remaining complexes, then the solution is called dilute, and the +molecular complex in question is called the solvent; the remaining +complexes are called the dissolved substances. + +Let us now consider a dilute solution whose state is determined +by the pressure~$p$, the temperature~$T$, and the molecular numbers +$n_{0}$,~$n_{1}$, $n_{2}$, $n_{3}$,~$\cdots$, wherein the subscript zero refers to the solvent. +Then the numbers $n_{1}$,~$n_{2}$, $n_{3}$,~$\cdots$ are all small with respect to~$n_{0}$, +%-----File: 034.png---\redacted\-------- +and on this account the volume~$V$ and the energy~$U$ are linear +functions of the molecular numbers: +\begin{align*} + V &= n_{0}v_{0} + n_{1}v_{1} + n_{2}v_{2} + \cdots,\\ + U &= n_{0}u_{0} + n_{1}u_{1} + n_{2}u_{2} + \cdots, +\end{align*} +wherein the $v$'s and $u$'s depend upon $p$~and $T$ only. + +From the general equation of entropy: +\[ +dS = \frac{dU + p dV}{T}, +\] +in which the differentials depend only upon changes in $p$~and~$T$, +and not in the molecular numbers, there results therefore: +\[ +dS = n_{0} \frac{du_{0} + p dv_{0}}{T} + n_{1} \frac{du_{1} + p dv_{1}}{T} + \cdots, +\] +and from this it follows that the expressions multiplied by $n_{0}$,~$n_{1}$~$\cdots$, +dependent upon $p$~and $T$ only, are complete differentials. +We may therefore write: +\[ +\frac{du_{0} + p dv_{0}}{T} = ds_{0}, \quad +\frac{du_{1} + p dv_{1}}{T} = ds_{1},\ \cdots +\Tag{(3)} +\] +and by integration obtain: +\[ +S = n_{0}s_{0} + n_{1}s_{1} + n_{2}s_{2} + \cdots + C. +\] +The constant~$C$ of integration does not depend upon $p$~and~$T$, +but may depend upon the molecular numbers $n_{0}$,~$n_{1}$, $n_{2}$,~$\cdots$. +In order to express this dependence generally, it suffices to know +it for a special case, for fixed values of $p$~and~$T$. Now every +solution passes, through appropriate increase of temperature and +decrease of pressure, into the state of a mixture of ideal gases, +and for this case the entropy is fully known, the integration +constant being, in accordance with Gibbs: +\[ +C = - R (n_{0} \log c_{0} + n_{1} \log c_{1} + \cdots), +\] +wherein $R$~denotes the absolute gas constant and $c_{0}$,~$c_{1}$, $c_{2}$,~$\cdots$ +%-----File: 035.png---\redacted\-------- +denote the ``molecular concentrations'': +\[ +c_{0} = \frac{n_{0}}{n_{0} + n_{1} + n_{2} + \cdots}, \quad +c_{1} = \frac{n_{1}}{n_{0} + n_{1} + n_{2} + \cdots} ,\ \cdots. +\] +Consequently, quite in general, the entropy of a dilute solution is: +\[ +S = n_{0}(s_{0} - R \log c_{0}) + n_{1}(s_{1} - R \log c_{1}) + \cdots, +\] +and, finally, from this it follows by substitution in equation~\Eq{(1)} +that: +\[ +\Phi = n_{0}(\varphi_{0} - R \log c_{0}) + n_{1}(\varphi_{1} - R \log c_{1}) + \cdots, +\Tag{(4)} +\] +if we put for brevity: +\[ +\varphi_{0} = s_{0} - \frac{u_{0} + pv_{0}}{T}, \quad +\varphi_{1} = s_{1} - \frac{u_{1} + pv_{1}}{T},\ \cdots +\Tag{(5)} +\] +all of which quantities depend only upon $p$~and~$T$. + +With the aid of the expression obtained for~$\Phi$ we are enabled +through equation~\Eq{(2)} to answer the question with regard to +thermodynamic equilibrium. We shall first find the general +law of equilibrium and then apply it to a series of particularly +interesting special cases. + +Every material system consisting of an arbitrary number of +homogeneous phases may be represented symbolically in the +following way: +\[ +n_{0} m_{0},\ n_{1} m_{1},\ \cdots \mid +{n_{0}}' {m_{0}}',\ {n_{1}}' {m_{1}}',\ \cdots \mid +{n_{0}}''{m_{0}}'',\ {n_{1}}''{m_{1}}'',\ \cdots \mid \cdots. +\] +Here the molecular numbers are denoted by~$n$, the molecular +weights by~$m$, and the individual phases are separated from one +another by vertical lines. We shall now suppose that each +phase represents a dilute solution. This will be the case when +each phase contains only a single molecular complex and therefore +represents an absolutely pure substance; for then the concentrations +of all the dissolved substances will be zero. + +If now an isobaric-isothermal change in the system of such +kind is possible that the molecular numbers +\[ +n_{0},\ n_{1},\ n_{2},\ \cdots,\quad +{n_{0}}',\ {n_{1}}',\ {n_{2}}',\ \cdots,\quad +{n_{0}}'',\ {n_{1}}'',\ {n_{2}}'',\ \cdots +\] +%-----File: 036.png---\redacted\-------- +change simultaneously by the amounts +\[ +\delta n_{0},\ \delta n_{1},\ \delta n_{2}, \cdots,\quad +\delta {n_{0}}',\ \delta {n_{1}}',\ \delta {n_{2}}', \cdots,\quad +\delta {n_{0}}'',\ \delta {n_{1}}'',\ \delta {n_{2}}'', \cdots +\] +then, in accordance with equation~\Eq{(2)}, equilibrium obtains with +respect to the occurrence of this change if, when $T$~and~$p$ are held +constant, the function +\[ +\Phi + \Phi' + \Phi'' + \cdots +\] +is a maximum, or, in accordance with equation~\Eq{(4)}: +\[ +\tsum (\varphi_{0} - R \log c_{0})\delta n_{0} + + (\varphi_{1} - R \log c_{1})\delta n_{1} + \cdots = 0 +\] +(the summation~$\tsum$ being extended over all phases of the system). +Since we are only concerned in this equation with the ratios of +the~$\delta n$'s, we put +\begin{multline*} +\delta n_{0} : \delta n_{1} : \cdots : +\delta {n_{0}}' : \delta {n_{1}}' : \cdots : +\delta {n_{0}}'' : \delta {n_{1}}'' : \cdots \\ + = \nu_{0} : \nu_{1} : \cdots + : {\nu_{0}}' : {\nu_{1}}' : \cdots + : {\nu_{0}}'' : {\nu_{1}}'' : \cdots, +\end{multline*} +wherein we are to understand by the simultaneously changing~$\nu$'s, +in the variation considered, simple integer positive or negative +numbers, according as the molecular complex under consideration +is formed or disappears in the change. Then the condition +for equilibrium is: +\label{png36lab1} +\[ +\tsum \nu_{0} \log c_{0} + + \nu_{1} \log c_{1} + \cdots + = \frac{1}{R} \tsum \nu_{0} \varphi_{0} + \nu_{1} \varphi_{1} + \cdots + = \log K. +\Tag{(6)} +\] +$K$ and the quantities $\varphi_{0}$,~$\varphi_{1}$, $\varphi_{2}$,~$\cdots$\ depend only upon $p$~and~$T$, +and this dependence is to be found from the equations: +\begin{align*} +\frac{\dd \log K}{\dd p} &= \frac{1}{R} \tsum \nu_{0} \frac{\dd \varphi_{0}}{\dd p} + \nu_{1} \frac{\dd \varphi_{1}}{\dd p} + \cdots,\\ +\frac{\dd \log K}{\dd T} &= \frac{1}{R} \tsum \nu_{0} \frac{\dd \varphi_{0}}{\dd T} + \nu_{1} \frac{\dd \varphi_{1}}{\dd T} + \cdots. +\end{align*} +Now, in accordance with~\Eq{(5)}, for any infinitely small change of $p$~and~$T$: +\[ +d \varphi_{0} = ds_{0} - \frac{du_{0} + p dv_{0} + v_{0} dp}{T} + \frac{u_{0} + pv_{0}}{T^{2}} · dT, +\] +%-----File: 037.png---\redacted\-------- +and consequently, from~\Eq{(3)}: +\[ +d \varphi_{0} = \frac{u_{0} + pv_{0}}{T^{2}} dT - \frac{v_{0} dp}{T}, +\] +and hence: +\[ +\frac{\dd \varphi_{0}}{\dd p} = -\frac{v_{0}}{T},\quad +\frac{\dd \varphi_{0}}{\dd T} = \frac{u_{0} + pv_{0}}{T^{2}}. +\] +Similar equations hold for the other~$\varphi$'s, and therefore we get: +\begin{gather*} +\frac{\dd \log K}{\dd p} + = -\frac{1}{RT} \tsum \nu_{0}v_{0} + \nu_{1}v_{1} + \cdots, \\ +\frac{\dd \log K}{\dd T} + = -\frac{1}{RT^{2}} \tsum \nu_{0}u_{0} + \nu_{2}u_{2} + \cdots + p(\nu_{0}v_{0} + \nu_{1}v_{1} + \cdots) +\end{gather*} +or, more briefly: +\[ +\frac{\dd \log K}{\dd p} = -\frac{1}{RT} · \Delta V, \quad +\frac{\dd \log K}{\dd T} = \frac{\Delta Q}{RT^{2}}, +\Tag{(7)} +\] +if $\Delta V$~denote the change in the total volume of the system and +$\Delta Q$~the heat which is communicated to it from outside, during +the isobaric isothermal change considered. We shall now investigate +the import of these relations in a series of important +applications. + + +\Section{I.}{Electrolytic Dissociation of Water.} + +The system consists of a single phase: +\[ +n_{0}H_{2}O,\quad n_{1}\Hplus,\quad n_{2}\HOminus. +\] +The transformation under consideration +\[ +\nu_{0} : \nu_{1} : \nu_{2} = \delta n_{0} : \delta n_{1} : \delta n_{2} +\] +consists in the dissociation of a molecule~$H_{2}O$ into a molecule~$\Hplus$ +and a molecule~$\HOminus$, therefore: +\[ +\nu_{0} = -1,\quad \nu_{1} = 1,\quad \nu_{2} = 1. +\] +Hence, in accordance with~\Eq{(6)}, for equilibrium: +\[ +-\log c_{0} + \log c_{1} + \log c_{2} = \log K, +\] +%-----File: 038.png---\redacted\-------- +or, since $c_{1} = c_{2}$ and $c_{0} = 1$, approximately: +\[ +2 \log c_{1} = \log K. +\] +The dependence of the concentration~$c_{1}$ upon the temperature +now follows from~\Eq{(7)}: +\[ +2 \frac{\dd \log c_{1}}{\dd T} = \frac{\Delta Q}{R T^{2}} . +\] +$\Delta Q$,~the quantity of heat which it is necessary to supply for the +dissociation of a molecule of~$H_{2}O$ into the ions $\Hplus$~and~$\HOminus$, is, +in accordance with Arrhenius, equal to the heat of ionization in +the neutralization of a strong univalent base and acid in a +dilute aqueous solution, and, therefore, in accordance with the +recent measurements of Wörmann,\footnote + {Ad Heydweiller, Ann.\ d.~Phys.,~28, 506, 1909.} +\[ +\Delta Q = 27,857 - 48.5 T \ \gr.\ \cal. +\] +Using the number~$1.985$ for the ratio of the absolute gas constant~$R$ +to the mechanical equivalent of heat, it follows that: +\[ +\frac{\dd \log c_{1}}{\dd T} + = \frac{1}{2·1.985} \left(\frac{27,857}{T^{2}} - \frac{48.5}{T}\right), +\] +and by integration: +\[ +\logten c_{1} = - \frac{3047.3}{T} - 12.125 \logten T + \const. +\] +This dependence of the degree of dissociation upon the temperature +agrees very well with the measurements of the electric +conductivity of water at different temperatures by Kohlrausch +and Heydweiller, Noyes, and Lundén. + + +\Section{II.}{Dissociation of a Dissolved Electrolyte.} + +\label{png38lab1}Let the system consists of an aqueous solution of acetic acid: +\[ +n_{0}H_{2}O,\quad n_{1}H_{4}C_{2}O_{2},\quad n_{2}\Hplus,\quad n_{3}\overset{-}{H_{3}C_{2}O_{2}}. +\] +The change under consideration consists in the dissociation of a +%-----File: 039.png---\redacted\-------- +molecule $H_{4}C_{2}O_{2}$ into its two ions, therefore +\[ +\nu_{0} = 0, \quad \nu_{1} = -1, \quad \nu_{2} = 1, \quad \nu_{3} = 1. +\] +Hence, for the state of equilibrium, in accordance with~\Eq{(6)}: +\[ +-\log c_{1} + \log c_{2} + \log c_{3} = \log K, +\] +or, since $c_{2} = c_{3}$: +\[ +\frac{{c_{2}}^{2}}{c_{1}} = K. +\] +Now the sum $c_{1} + c_{2} = c$ is to be regarded as known, since the +total number of the undissociated and dissociated acid molecules +is independent of the degree of dissociation. Therefore $c_{1}$~and~$c_{2}$ +may be calculated from $K$~and~$c$. An experimental test of the +equation of equilibrium is possible on account of the connection +between the degree of dissociation and electrical conductivity of +the solution. In accordance with the electrolytic dissociation +theory of Arrhenius, the ratio of the molecular conductivity~$\lambda$ of +the solution in any dilution to the molecular conductivity~$\lambda_{\infty}$ +of the solution in infinite dilution is: +\[ +\frac{\lambda}{\lambda_{\infty}} = \frac{c_{2}}{c_{1} + c_{2}} = \frac{c_{2}}{c}, +\] +since electric conduction is accounted for by the dissociated molecules +only. It follows then, with the aid of the last equation, that: +\[ +\frac{\lambda^{2} c}{\lambda_{\infty} - \lambda} = K · \lambda_{\infty} = \const. +\] +With unlimited decreasing~$c$, $\lambda$~increases to~$\lambda_{\infty}$. This ``law of +dilution'' for binary electrolytes, first enunciated by Ostwald, has +been confirmed in numerous cases by experiment, as in the case +of acetic acid. + +Also, the dependence of the degree of dissociation upon the +temperature is indicated here in quite an analogous manner to +that in the example considered above, of the dissociation of water. +%-----File: 040.png---\redacted\-------- + + +\Section{III.}{Vaporization or Solidification of a Pure Liquid.} + +In equilibrium the system consists of two phases, one liquid, +and one gaseous or solid: +\[ +n_{0}m_{0} \mid {n_{0}}'{m_{0}}'. +\] + +Each phase contains only a single molecular complex (the +solvent), but the molecules in both phases do not need to be the +same. Now, if a liquid molecule evaporates or solidifies, then +in our notation +\[ +\nu_{0} = - 1,\quad {\nu_{0}}' = \frac{m_{0}}{{m_{0}}'},\quad c_{0} = 1,\quad {c_{0}}' = 1, +\] +and consequently the condition for equilibrium, in accordance +with~\Eq{(6)}, is: +\[ +0 = \log K. +\Tag{(8)} +\] +Since $K$ depends only upon $p$~and~$T$, this equation therefore +expresses a definite relation between $p$~and~$T$: the law of dependence +of the pressure of vaporization (or melting pressure) +upon the temperature, or vice versa. The import of this law is +obtained through the consideration of the dependence of the +quantity~$K$ upon $p$~and~$T$. If we form the complete differential +of the last equation, there results: +\[ +0 = \frac{\dd \log K}{\dd p} dp + \frac{\dd \log K}{\dd T} dT, +\] +or, in accordance with~\Eq{(7)}: +\[ +0 = -\frac{\Delta V}{T} dp + \frac{\Delta Q}{T^2} dT. +\] +If $v_{0}$~and~${v_{0}}'$ denote the molecular volumes of the two phases, then: +\[ +\Delta V = \frac{m_{0}{v_{0}}'}{{m_{0}}'} - v_{0}, +\] +consequently: +\[ +\Delta Q = T\left(\frac{m_{0}{v_{0}}'}{{m_{0}}'} - v_{0}\right) \frac{dp}{dT}, +\] +%-----File: 041.png---\redacted\-------- +or, referred to unit mass: +\[ +\frac{\Delta Q}{m_{0}} + = T \left(\frac{{v_{0}}'}{{m_{0}}'} - \frac{v_{0}}{m_{0}}\right) · \frac{dp}{dT}, +\] +the well-known formula of Carnot and Clapeyron. + + +\Section{IV.}{The Vaporization or Solidification of a Solution of Non-Volatile +Substances.} + +Most aqueous salt solutions afford examples. The symbol of +the system in this case is, since the second phase (gaseous or solid) +contains only a single molecular complex: +\[ +n_{0}m_{0},\ n_{1}m_{1},\ n_{2}m_{2},\ \cdots \mid {n_{0}}'{m_{0}}'. +\] +The change is represented by: +\[ +\nu_{0} = -1,\quad +\nu_{1} = 0,\quad +\nu_{2} = 0,\quad \cdots\quad +{\nu_{0}}' = \frac{m_{0}}{{m_{0}}'}, +\] +and hence the condition of equilibrium, in accordance with~\Eq{(6)}, is: +\[ +-\log c_{0} = \log K, +\] +or, since to small quantities of higher order: +\begin{align*} +c_{0} = \frac{n_{0}}{n_{0} + n_{1} + n_{2} + \cdots} + &= 1 - \frac{n_{1} + n_{2} + \cdots}{n_{0}},\\[1ex] +\frac{n_{1} + n_{2} + \cdots}{n_{0}} &= \log K. +\Tag{(9)} +\end{align*} + +A comparison with formula~\Eq{(8)}, found in example~III, shows +that through the solution of a foreign substance there is involved +in the total concentration a small proportionate departure from +the law of vaporization or solidification which holds for the pure +solvent. One can express this, either by saying: at a fixed pressure~$p$, +the boiling point or the freezing point~$T$ of the solution +is different than that~($T_{0}$) for the pure solvent, or: \label{png41lab1}at a fixed +temperature~$T$ the vapor pressure or solidification pressure~$p$ of the +solution is different from that~($p_{0}$) of the pure solvent. Let us +calculate the departure in both cases. +%-----File: 042.png---\redacted\-------- + +1. If $T_{0}$~be the boiling (or freezing temperature) of the pure +solvent at the pressure~$p$, then, in accordance with~\Eq{(8)}: +\[ +(\log K)_{T = T_{0}} = 0, +\] +and by subtraction of~\Eq{(9)} there results: +\[ +\log K - (\log K)_{T = T_{0}} = \frac{n_{1} + n_{2} + \cdots}{n_{0}}. +\] +Now, since $T$~is little different from~$T_{0}$, we may write in place of +this equation, with the aid of~\Eq{(7)}: +\[ +\frac{\dd \log K}{\dd T} (T - T_{0}) + = \frac{\Delta Q}{RT_{0}^{2}} (T - T_{0}) + = \frac{n_{1} + n_{2} + \cdots}{n_{0}}, +\] +and from this it follows that: +\[ +T - T_{0} = \frac{n_{1} + n_{2} + \cdots}{n_{0}} · \frac{RT_{0}^{2}}{\Delta Q}. +\Tag{(10)} +\] + +This is the law for the raising of the boiling point or for the +lowering of the freezing point, first derived by van't~Hoff: in the +case of freezing $\Delta Q$~(the heat taken from the surroundings during +the freezing of a liquid molecule) is negative. Since $n_{0}$~and~$\Delta Q$ +occur only as a product, it is not possible to infer anything from +this formula with regard to the molecular number of the liquid +solvent. + +2. If $p_{0}$~be the vapor pressure of the pure solvent at the +temperature~$T$, then, in accordance with~\Eq{(8)}: +\[ +(\log K)_{p = p_{0}} = 0, +\] +and by subtraction of~\Eq{(9)} there results: +\[ +\log K - (\log K)_{p = p_{0}} = \frac{n_{1} + n_{2} + \cdots}{n_{0}}. +\] +Now, since $p$~and~$p_{0}$ are nearly equal, with the aid of~\Eq{(7)} we may +write: +\[ +\frac{\dd \log K}{\dd p} (p - p_{0}) + = - \frac{\Delta V}{RT} (p - p _{0}) + = \frac{n_{1} + n_{2} + \cdots}{n_{0}}, +\] +%-----File: 043.png---\redacted\-------- +and from this it follows, if $\Delta V$~be placed equal to the volume of +the gaseous molecule produced in the vaporization of a liquid +molecule: +\begin{gather*} +\Delta V = \frac{m_{0}}{{m_{0}}'} \frac{RT}{p}, \\ +\frac{p_{0} - p}{p} = \frac{{m_{0}}'}{m_{0}} · \frac{n_{1} + n_{2} + \cdots}{n_{0}}. +\end{gather*} +This is the law of relative depression of the vapor pressure, +first derived by van't~Hoff. Since $n_{0}$~and~$m_{0}$ occur only as a +product, it is not possible to infer from this formula anything +with regard to the molecular weight of the liquid solvent. Frequently +the factor~${m_{0}}'/m_{0}$ is left out in this formula; but this is +not allowable when $m_{0}$~and~${m_{0}}'$ are unequal (as, \eg, in the +case of water). + + +\Section{V.}{Vaporization of a Solution of Volatile Substances.} + +\begin{center}(\textit{\Eg., a Sufficiently Dilute Solution of Propyl Alcohol in Water.})\end{center} + +The system, consisting of two phases, is represented by the +following symbol: +\[ +n_{0} m_{0},\ n_{1} m_{1},\ n_{2} m_{2},\ \cdots \mid +{n_{0}}'{m_{0}}',\ {n_{1}}'{m_{1}}',\ {n_{2}}'{m_{2}}',\ \cdots, +\] +wherein, as above, the figure~$0$ refers to the solvent and the +figures $1$,~$2$, $3$~$\cdots$ refer to the various molecular complexes of +the dissolved substances. By the addition of primes in the case +of the molecular weights (${m_{0}}'$,~${m_{1}}'$, ${m_{2}}'$~$\cdots$) the possibility is +left open that the various molecular complexes in the vapor +may possess a different molecular weight than in the liquid. + +Since the system here considered may experience various sorts +of changes, there are also various conditions of equilibrium to +fulfill, each of which relates to a definite sort of transformation. +Let us consider first that change which consists in the vaporization +of the solvent. In accordance with our scheme of notation, +the following conditions hold:\label{png43lab1} +\[ +\nu_{0} = - 1,\ \nu_{1} = 0,\ \nu_{2} = 0,\ \cdots\ +\nu_{0}' = \frac{m_{0} }{ {m_{0}}'},\ {\nu_{1}}' = 0,\ {\nu_{2}}' = 0,\ \cdots, +\] +%-----File: 044.png---\redacted\-------- +and, therefore, the condition of equilibrium~\Eq{(6)} becomes: +\[ +-\log c_{0} + \frac{m_{0}}{{m_{0}}'} \log {c_{0}}' = \log K, +\] +or, if one substitutes: +\begin{gather*} +c_{0} = 1 - \frac{n_{1} + n_{2} + \cdots}{n_{0}} \quad \text{and} \quad +{c_{0}}' = 1 - \frac{{n_{1}}' + {n_{2}}' + \cdots}{{n_{0}}'},\\ +\frac{n_{1} + n_{2} + \cdots}{n_{0}} - \frac{m_{0}}{{m_{0}}'} · \frac{{n_{1}}' + {n_{2}}' + \cdots}{{n_{0}}'} = \log K. +\end{gather*} +If we treat this equation upon equation~\Eq{(9)} as a model, there +results an equation similar to~\Eq{(10)}: +\[ +T - T_{0} + = \left(\frac{n_{1} + n_{2} + \cdots}{n_{0}m_{0}} + - \frac{{n_{1}}' + {n_{2}}' + \cdots}{{n_{0}}'{m_{0}}'}\right) \frac{RT_{0}^{2}m_{0}}{\Delta Q}. +\] + +Here $\Delta Q$~is the heat effect in the vaporization of one molecule +of the solvent and, therefore, $\Delta Q/m_{0}$~is the heat effect in the +vaporization of a unit mass of the solvent. + +We remark, once more, that the solvent always occurs in the +formula through the mass only, and not through the molecular +number or the molecular weight, while, on the other hand, in the +case of the dissolved substances, the molecular state is characteristic +on account of their influence upon vaporization. Finally, the +formula contains a generalization of the law of van't~Hoff, stated +above, for the raising of the boiling point, in that here in place +of the number of dissolved molecules in the liquid, the difference +between the number of dissolved molecules in unit mass of the +liquid and in unit mass of the vapor appears. According as the +unit mass of liquid or the unit mass of vapor contains more +dissolved molecules, there results for the solution a raising or +lowering of the boiling point; in the limiting case, when both +quantities are equal, and the mixture therefore boils without +changing, the change in boiling point becomes equal to zero. +Of course, there are corresponding laws holding for the change +in the vapor pressure. +%-----File: 045.png---\redacted\-------- + +Let us consider now a change which consists in the vaporization +of a dissolved molecule. For this case we have in our notation +\[ +\nu_{0} = 0,\ \nu_{1} = -1,\ \nu_{2} = 0\ \cdots, \ +{\nu_{0}}' = 0,\ {\nu_{1}}' = \frac{m_{1}}{{m_{1}}'},\ {\nu_{2}}' = 0,\ \cdots +\] +and, in accordance with~\Eq{(6)}, for the condition of equilibrium: +\[ +-\log c_{1} + \frac{m_{1}}{{m_{1}}'} \log {c_{1}}' = \log K +\] +or: +\[ +\frac{{{c_{1}}'}^{\frac{m_{1}}{{m_{1}}'}}}{c_{1}} = K. +\] +This equation expresses the Nernst law of distribution. If +the dissolved substance possesses in both phases the same +molecular weight ($m_{1} = {m_{1}}'$), then, in a state of equilibrium a +fixed ratio of the concentrations $c_{1}$~and~${c_{1}}'$ in the liquid and in the +vapor exists, which depends only upon the pressure and temperature. +But, if the dissolved substance polymerises somewhat in +the liquid, then the relation demanded in the last equation appears +in place of the simple ratio. + + +\Section{VI.}{The Dissolved Substance only Passes over into the Second +Phase.} + +This case is in a certain sense a special case of the one preceding. +To it belongs that of the solubility of a slightly soluble salt, +first investigated by van't~Hoff, \eg, succinic acid in water. The +symbol of this system is: +\[ +n_{0}H_{2}O,\ n_{1}H_{6}C_{4}O_{4} \mid {n_{0}}'H_{6}C_{4}O_{4}, +\] +in which we disregard the small dissociation of the acid solution. +The concentrations of the individual molecular complexes are: +\[ +c_{0} = \frac{n_{0}}{n_{0} + n_{1}}, \quad +c_{1} = \frac{n_{1}}{n_{0} + n_{1}}, \quad +{c_{0}}' = \frac{{n_{0}}'}{{n_{0}}'} = 1. +\] +For the precipitation of solid succinic acid we have: +\[ +\nu_{0} = 0, \quad \nu_{1} = -1, \quad {\nu_{0}}' = 1, +\] +%-----File: 046.png---\redacted\-------- +and, therefore, from the condition of equilibrium~\Eq{(6)}: +\[ +-\log c_{1} = \log K, +\] +hence, from~\Eq{(7)}: +\[ +\Delta Q = - RT^{2} \frac{\dd \log c_{1}}{\dd T}. +\] +By means of this equation van't~Hoff calculated the heat of +solution~$\Delta Q$ from the solubility of succinic acid at~$0°$ and at $8.5°$~C. +The corresponding numbers were $2.88$ and $4.22$ in an arbitrary +unit. Approximately, then: +\[ +\frac{\dd \log c_{1}}{\dd T} = \frac{\ln 4.22 - \ln 2.88}{8.5} = 0.04494, +\] +from which for $T = 273$: +\[ +\Delta Q = -1.98 · 273^{2} · 0.04494 = -6,600\ \cal., +\] +that is, in the precipitation of a molecule of succinic acid, $6,600~\cal.$ +are given out to the surroundings. Berthelot found, however, +through direct measurement, $6,700$~calories for the heat +of solution. + +The absorption of a gas also comes under this head, \eg\ +carbonic acid, in a liquid of relatively unnoticeable smaller +vapor pressure, \eg, water at not too high a temperature. The +symbol of the system is then +\[ +n_{0}H_{2}O,\ n_{1}CO_{2} \mid {n_{0}}'CO_{2}. +\] +The vaporization of a molecule~$CO_{2}$ corresponds to the values +\[ +\nu_{0} = 0,\quad \nu_{1} = -1,\quad {\nu_{0}}' = 1. +\] +The condition of equilibrium is therefore again: +\[ +-\log c_{1} = \log K, +\] +\ie, at a fixed temperature and a fixed pressure the concentration~$c_{1}$ +of the gas in the solution is constant. The change of the concentration +%-----File: 047.png---\redacted\-------- +with $p$~and~$T$ is obtained through substitution in equation~\Eq{(7)}. +It follows from this that: +\[ +\frac{\dd \log c_{1}}{\dd p} = \frac{\Delta V}{RT} ,\quad +\frac{\dd \log c_{1}}{\dd T} = -\frac{\Delta Q}{RT^{2}}. +\] + +$\Delta V$~is the change in volume of the system which occurs in the +isobaric-isothermal vaporization of a molecule of~$CO_{2}$, $\Delta Q$~the +quantity of heat absorbed in the process from outside. Now, +since $\Delta V$~represents approximately the volume of a molecule of +gaseous carbonic acid, we may put approximately: +\[ +\Delta V = \frac{RT}{p}, +\] +and the equation gives: +\[ +\frac{\dd \log c_{1}}{\dd p} = \frac{1}{p}, +\] +which integrated, gives: +\[ +\log c_{1} = \log p + \const., \quad c_{1} = C · p, +\] +\ie, the concentration of the dissolved gas is proportional to the +pressure of the free gas above the solution (law of Henry and +Bunsen). The factor of proportionality~$C$, which furnishes a measure +of the solubility of the gas, depends upon the heat effect in +quite the same manner as in the example previously considered. + +A number of no less important relations are easily derived as +by-products of those found above, \eg, the Nernst laws concerning +the influence of solubility, the Arrhenius theory of isohydric +solutions,~etc. All such may be obtained through the +application of the general condition of equilibrium~\Eq{(6)}. In +conclusion, there is one other case that I desire to treat here. +In the historical development of the theory this has played a +particularly important rôle. + + +\Section{VII.}{Osmotic Pressure.} + +We consider now a dilute solution separated by a membrane +(permeable with regard to the solvent but impermeable as +regards the dissolved substance) from the pure solvent (in the +%-----File: 048.png---\redacted\-------- +same state of aggregation), and inquire as to the condition of +equilibrium. The symbol of the system considered we may again +take as +\[ +n_{0}m_{0},\ n_{1}m_{1},\ n_{2}m_{2},\ \cdots \mid {n_{0}}'m_{0}. +\] + +The condition of equilibrium is also here again expressed by +equation~\Eq{(6)}, valid for a change of state in which the temperature +and the pressure in each phase is maintained constant. The +only difference with respect to the cases treated earlier is this, +that here, in the presence of a separating membrane between +two phases, the pressure~$p$ in the first phase may be different from +the pressure~$p'$ in the second phase, whereby by ``pressure,'' as +always, is to be understood the ordinary hydrostatic or manometric +pressure. + +The proof of the applicability of equation~\Eq{(6)} is found in the +same way as this equation was derived above, proceeding from the +principle of increase of entropy. One has but to remember that, +in the somewhat more general case here considered, the external +work in a given change is represented by the sum~$p dV + p' dV'$, +where $V$~and~$V'$ denote the volumes of the two individual phases, +while before $V$~denoted the total volume of all phases. Accordingly, +we use, instead of~\Eq{(7)}, to express the dependence of the +constant~$K$ in~\Eq{(6)} upon the pressure: +\[ +\frac{\dd \log K}{\dd p} = -\frac{\Delta V}{RT}, \quad +\frac{\dd \log K}{\dd p'} = -\frac{\Delta V'}{RT}. +\Tag{(11)} +\] +We have here to do with the following change: +\[ +\nu_{0} = -1,\quad \nu_{1} = 0,\quad \nu_{2} = 0,\quad \cdots,\quad {\nu_{0}}' = 1, +\] +whereby is expressed, that a molecule of the solvent passes out +of the solution through the membrane into the pure solvent. +Hence, in accordance with~\Eq{(6)}: +\[ +-\log c_{0} = \log K, +\] +or, since +\[ +c_{0} = 1 - \frac{n_{1} + n_{2} + \cdots}{n_{0}}, \quad +\frac{n_{1} + n_{2} + \cdots}{n_{0}} = \log K. +\] +%-----File: 049.png---\redacted\-------- +Here $K$~depends only upon $T$,~$p$ and~$p'$. If a pure solvent were +present upon both sides of the membrane, we should have +$c_{0} = 1$, and $p = p'$; consequently: +\[ +(\log K)_{p = p'} = 0, +\] +and by subtraction of the last two equations: +\[ +\frac{n_{1} + n_{2} + \cdots}{n_{0}} + = \log K - (\log K)_{p = p'} + = \frac{\dd \log K}{\dd p} (p - p') +\] +and in accordance with~\Eq{(11)}: +\[ +\frac{n_{1} + n_{2} + \cdots}{n_{0}} = -(p - p') · \frac{\Delta V}{RT}. +\] +Here $\Delta V$~denotes the change in volume of the solution due to the +loss of a molecule of the solvent ($\nu_{0} = -1$). Approximately +then: +\[ +-\Delta V · n_{0} = V, +\] +the volume of the whole solution, and +\[ +\frac{n_{1} + n_{2} + \cdots}{n_{0}} = (p - p') · \frac{V}{RT}. +\] +If we call the difference $p - p'$, the osmotic pressure of the +solution, this equation contains the well known law of osmotic +pressure, due to van't~Hoff. + +The equations here derived, which easily permit of multiplication +and generalization, have, of course, for the most part not been +derived in the ways described above, but have been derived, +either directly from experiment, or theoretically from the consideration +of special reversible isothermal cycles to which the +thermodynamic law was applied, that in such a cyclic process +not only the algebraic sum of the work produced and the heat +produced, but that also each of these two quantities separately, is +equal to zero (first lecture, p.~\pageref{png29lab1}). The employment of a cyclic +process has the advantage over the procedure here proposed, +%-----File: 050.png---\redacted\-------- +that in it the connection between the directly measurable quantities +and the requirements of the laws of thermodynamics +succinctly appears in each case; but for each individual case a +satisfactory cyclic process must be imagined, and one has not +always the certain assurance that the thermodynamic realization +of the cyclic process also actually supplies all the conditions +of equilibrium. Furthermore, in the process of calculation +certain terms of considerable weight frequently appear as +empty ballast, since they disappear at the end in the summation +over the individual phases of the process. + +On the other hand, the significance of the process here employed +consists therein, that the necessary and sufficient conditions +of equilibrium for each individually considered case appear +collectively in the single equation~\Eq{(6)}, and that they are derived +collectively from it in a direct manner through an unambiguous +procedure. The more complicated the systems considered are, +the more apparent becomes the advantage of this method, and +there is no doubt in my mind that in chemical circles it will be +more and more employed, especially, since in general it is now +the custom to deal directly with the energies, and not with cyclic +processes, in the calculation of heat effects in chemical changes. +%-----File: 051.png---\redacted\-------- + +\Chapter{THIRD LECTURE.}{The Atomic Theory of Matter.}\label{Lect3} + +The problem with which we shall be occupied in the present +lecture is that of a closer investigation of the atomic theory of +matter. It is, however, not my intention to introduce this +theory with nothing further, and to set it up as something apart +and disconnected with other physical theories, but I intend above +all to bring out the peculiar significance of the atomic theory as +related to the present general system of theoretical physics; for +in this way only will it be possible to regard the whole system +as one containing within itself the essential compact unity, and +thereby to realize the principal object of these lectures. + +Consequently it is self evident that we must rely on that sort +of treatment which we have recognized in last week's lecture as +fundamental. That is, the division of all physical processes into +reversible and irreversible processes. Furthermore, we shall be +convinced that the accomplishment of this division is only possible +through the atomic theory of matter, or, in other words, +that irreversibility leads of necessity to atomistics. + +I have already referred at the close of the first lecture to the +fact that in pure thermodynamics, which knows nothing of an +atomic structure and which regards all substances as absolutely +continuous, the difference between reversible and irreversible +processes can only be defined in one way, which a priori carries +a provisional character and does not withstand penetrating analysis. +This appears immediately evident when one reflects that +the purely thermodynamic definition of irreversibility which +proceeds from the impossibility of the realization of certain +changes in nature, as, \eg, the transformation of heat into +work without compensation, has at the outset assumed a definite +limit to man's mental capacity, while, however, such a +%-----File: 052.png---\redacted\-------- +limit is not indicated in reality. On the contrary: mankind is +making every endeavor to press beyond the present boundaries +of its capacity, and we hope that later on many things will be +attained which, perhaps, many regard at present as impossible +of accomplishment. Can it not happen then that a process, +which up to the present has been regarded as irreversible, may +be proved, through a new discovery or invention, to be reversible? +In this case the whole structure of the second law would undeniably +collapse, for the irreversibility of a single process conditions +that of all the others. + +It is evident then that the only means to assure to the second +law real meaning consists in this, that the idea of irreversibility +be made independent of any relationship to man and especially of +all technical relations. + +Now the idea of irreversibility harks back to the idea of entropy; +for a process is irreversible when it is connected with an increase +of entropy. The problem is hereby referred back to a proper +improvement of the definition of entropy. In accordance with +the original definition of Clausius, the entropy is measured by +means of a certain reversible process, and the weakness of this +definition rests upon the fact that many such reversible processes, +strictly speaking all, are not capable of being carried out in +practice. With some reason it may be objected that we have +here to do, not with an actual process and an actual physicist, +but only with ideal processes, so-called thought experiments, and +with an ideal physicist who operates with all the experimental +methods with absolute accuracy. But at this point the difficulty +is encountered: How far do the physicist's ideal measurements +of this sort suffice? It may be understood, by passing to the +limit, that a gas is compressed by a pressure which is equal to +the pressure of the gas, and is heated by a heat reservoir which +possesses the same temperature as the gas, but, for example, +that a saturated vapor shall be transformed through isothermal +compression in a reversible manner to a liquid without at any +time a part of the vapor being condensed, as in certain thermodynamic +%-----File: 053.png---\redacted\-------- +considerations is supposed, must certainly appear +doubtful. Still more striking, however, is the liberty as regards +thought experiments, which in physical chemistry is granted the +theorist. With his semi-permeable membranes, which in reality +are only realizable under certain special conditions and then +only with a certain approximation, he separates in a reversible +manner, not only all possible varieties of molecules, whether or +not they are in stable or unstable conditions, but he also separates +the oppositely charged ions from one another and from the +undissociated molecules, and he is disturbed, neither by the +enormous electrostatic forces which resist such a separation, nor +by the circumstance that in reality, from the beginning of the +separation, the molecules become in part dissociated while the +ions in part again combine. But such ideal processes are necessary +throughout in order to make possible the comparison of +the entropy of the undissociated molecules with the entropy of +the dissociated molecules; for the law of thermodynamic equilibrium +does not permit in general of derivation in any other way, +in case one wishes to retain pure thermodynamics as a basis. It +must be considered remarkable that all these ingenious thought +processes have so well found confirmation of their results in +experience, as is shown by the examples considered by us in the +last lecture. + +If now, on the other hand, one reflects that in all these results +every reference to the possibility of actually carrying out each +ideal process has disappeared---there are certainly left relations +between directly measurable quantities only, such as temperature, +heat effect, concentration,~etc.---the presumption forces +itself upon one that perhaps the introduction as above of such +ideal processes is at bottom a round-about method, and that +the peculiar import of the principle of increase of entropy with +all its consequences can be evolved from the original idea of +irreversibility or, just as well, from the impossibility of perpetual +motion of the second kind, just as the principle of conservation +of energy has been evolved from the law of impossibility of +perpetual motion of the first kind. +%-----File: 054.png---\redacted\-------- + +This step: to have completed the emancipation of the entropy +idea from the experimental art of man and the elevation of the +second law thereby to a real principle, was the scientific life's +work of Ludwig Boltzmann. Briefly stated, it consisted in +general of referring back the idea of entropy to the idea of +probability. Thereby is also explained, at the same time, the +significance of the above (p.~\pageref{png27lab1}) auxiliary term used by me; +``preference'' of nature for a definite state. Nature prefers the +more probable states to the less probable, because in nature +processes take place in the direction of greater probability. Heat +goes from a body at higher temperature to a body at lower +temperature because the state of equal temperature distribution +is more probable than a state of unequal temperature distribution. + +Through this conception the second law of thermodynamics +is removed at one stroke from its isolated position, the mystery +concerning the preference of nature vanishes, and the entropy +principle reduces to a well understood law of the calculus of +probability. + +The enormous fruitfulness of so ``objective'' a definition of +entropy for all domains of physics I shall seek to demonstrate +in the following lectures. But today we have principally to do +with the proof of its admissibility; for on closer consideration we +shall immediately perceive that the new conception of entropy +at once introduces a great number of questions, new requirements +and difficult problems. The first requirement is the introduction +of the atomic hypothesis into the system of physics. For, if one +wishes to speak of the probability of a physical state, \ie, if he +wishes to introduce the probability for a given state as a definite +quantity into the calculation, this can only be brought about, as +in cases of all probability calculations, by referring the state back +to a variety of possibilities; \ie,~by considering a finite number +of a~priori equally likely configurations (complexions) through +each of which the state considered may be realized. The greater +the number of complexions, the greater is the probability of the +state. Thus, \eg, the probability of throwing a total of four +%-----File: 055.png---\redacted\-------- +with two ordinary six-sided dice is found through counting the +complexions by which the throw with a total of four may be +realized. Of these there are three complexions: +\begin{center} +with the first die, $1$, with the second die, $3$,\\ +with the first die, $2$, with the second die, $2$,\\ +with the first die, $3$, with the second die, $1$. +\end{center} +On the other hand, the throw of two is only realized through +a single complexion. Therefore, the probability of throwing a +total of four is three times as great as the probability of throwing +a total of two. + +Now, in connection with the physical state under consideration, +in order to be able to differentiate completely from one another +the complexions realizing it, and to associate it with a definite +reckonable number, there is obviously no other means than to +regard it as made up of numerous discrete homogeneous elements---for +in perfectly continuous systems there exist no reckonable +elements---and hereby the atomistic view is made a fundamental +requirement. We have, therefore, to regard all bodies in nature, +in so far as they possess an entropy, as constituted of atoms, and +we therefore arrive in physics at the same conception of matter as +that which obtained in chemistry for so long previously. + +But we can immediately go a step further yet. The conclusions +reached hold, not only for thermodynamics of material +bodies, but also possess complete validity for the processes of +heat radiation, which are thus referred back to the second law +of thermodynamics. That radiant heat also possesses an entropy +follows from the fact that a body which emits radiation into a surrounding +diathermanous medium experiences a loss of heat and, +therefore, a decrease of entropy. Since the total entropy of +a physical system can only increase, it follows that one part +of the entropy of the whole system, consisting of the body and the +diathermanous medium, must be contained in the radiated heat. +If the entropy of the radiant heat is to be referred back to the +notion of probability, we are forced, in a similar way as above, to +%-----File: 056.png---\redacted\-------- +the conclusion that for radiant heat the atomic conception +possesses a definite meaning. But, since radiant heat is not +directly connected with matter, it follows that this atomistic conception +relates, not to matter, but only to energy, and hence, +that in heat radiation certain energy elements play an essential +rôle. Even though this conclusion appears so singular and even +though in many circles today vigorous objection is strongly urged +against it, in the long run physical research will not be able +to withhold its sanction from it, and the less, since it is confirmed +by experience in quite a satisfactory manner. We shall return +to this point in the lectures on heat radiation. I desire here +only to mention that the novelty involved by the introduction +of atomistic conceptions into the theory of heat radiation is by no +means so revolutionary as, perhaps, might appear at the first +glance. For there is, in my opinion at least, nothing which makes +necessary the consideration of the heat processes in a complete +vacuum as atomic, and it suffices to seek the atomistic features at +the source of radiation, \ie, in those processes which have +their play in the centres of emission and absorption of radiation. +Then the Maxwellian electrodynamic differential equations can +retain completely their validity for the vacuum, and, besides, +the discrete elements of heat radiation are relegated exclusively +to a domain which is still very mysterious and where there is +still present plenty of room for all sorts of hypotheses. + +Returning to more general considerations, the most important +question comes up as to whether, with the introduction of atomistic +conceptions and with the reference of entropy to probability, +the content of the principle of increase of entropy is exhaustively +comprehended, or whether still further physical hypotheses are required +in order to secure the full import of that principle. If this +important question had been settled at the time of the introduction +of the atomic theory into thermodynamics, then the +atomistic views would surely have been spared a large number of +conceivable misunderstandings and justifiable attacks. For it +turns out, in fact---and our further considerations will confirm +%-----File: 057.png---\redacted\-------- +this conclusion---that there has as yet nothing been done with +atomistics which in itself requires much more than an essential +generalization, in order to guarantee the validity of the +second law. + +We must first reflect that, in accordance with the central +idea laid down in the first lecture (p.~\pageref{png17lab1}), the second law must +possess validity as an objective physical law, independently of +the individuality of the physicist. There is nothing to hinder +us from imagining a physicist---we shall designate him a ``microscopic'' +observer---whose senses are so sharpened that he +is able to recognize each individual atom and to follow it in +its motion. For this observer each atom moves exactly in +accordance with the elementary laws which general dynamics +lays down for it, and these laws allow, so far as we know, of an +inverse performance of every process. Accordingly, here again +the question is neither one of probability nor of entropy and its +increase. Let us imagine, on the other hand, another observer, +designated a ``macroscopic'' observer, who regards an +ensemble of atoms as a homogeneous gas, say, and consequently +applies the laws of thermodynamics to the mechanical and thermal +processes within it. Then, for such an observer, in accordance +with the second law, the process in general is an irreversible +process. Would not now the first observer be justified in saying: +``The reference of the entropy to probability has its origin in +the fact that irreversible processes ought to be explained through +reversible processes. At any rate, this procedure appears to me +in the highest degree dubious. In any case, I declare each change +of state which takes place in the ensemble of atoms designated +a gas, as reversible, in opposition to the macroscopic observer.'' +There is not the slightest thing, so far as I know, that one can +urge against the validity of these statements. But do we not +thereby place ourselves in the painful position of the judge who +declared in a trial the correctness of the position of each separately +of two contending parties and then, when a third contends that +only one of the parties could emerge from the process victorious, +%-----File: 058.png---\redacted\-------- +was obliged to declare him also correct? Fortunately we find ourselves +in a more favorable position. We can certainly mediate +between the two parties without its being necessary for one or +the other to give up his principal point of view. For closer +consideration shows that the whole controversy rests upon a misunderstanding---a +new proof of how necessary it is before one +begins a controversy to come to an understanding with his +opponent concerning the subject of the quarrel. Certainly, a +given change of state cannot be both reversible and irreversible. +But the one observer connects a wholly different idea with the +phrase ``change of state'' than the other. What is then, in +general, a change of state? The state of a physical system cannot +well be otherwise defined than as the aggregate of all those physical +quantities, through whose instantaneous values the time +changes of the quantities, with given boundary conditions, are +uniquely determined. If we inquire now, in accordance with +the import of this definition, of the two observers as to what +they understand by the state of the collection of atoms or the +gas considered, they will give quite different answers. The +microscopic observer will mention those quantities which determine +the position and the velocities of all the individual atoms. +There are present in the simplest case, namely, that in which +the atoms may be considered as material points, six times as many +quantities as atoms, namely, for each atom the three coordinates +and the three velocity components, and in the case of combined +molecules, still more quantities. For him the state and the +progress of a process is then first determined when all these +various quantities are individually given. We shall designate +the state defined in this way the ``micro-state.'' The macroscopic +observer, on the other hand, requires fewer data. He will +say that the state of the homogeneous gas considered by him is +determined by the density, the visible velocity and the temperature +at each point of the gas, and he will expect that, when these +quantities are given, their time variations and, therefore, the progress +of the process, to be completely determined in accordance +%-----File: 059.png---\redacted\-------- +with the two laws of thermo-dynamics, and therefore accompanied +by an increase in entropy. In this connection he can call upon +all the experience at his disposal, which will fully confirm his expectation. +If we call this state the ``macro-state,'' it is clear that +the two laws: ``the micro-changes of state are reversible'' and +``the macro-changes of state are irreversible,'' lie in wholly +different domains and, at any rate, are not contradictory. + +But now how can we succeed in bringing the two observers to +an understanding? This is a question whose answer is obviously +of fundamental significance for the atomic theory. First of all, +it is easy to see that the macro-observer reckons only with mean +values; for what he calls density, visible velocity and temperature +of the gas are, for the micro-observer, certain mean values, statistical +data, which are derived from the space distribution and from +the velocities of the atoms in an appropriate manner. But the +micro-observer cannot operate with these mean values alone, for, +if these are given at one instant of time, the progress of the process +is not determined throughout; on the contrary: he can easily +find with given mean values an enormously large number of +individual values for the positions and the velocities of the atoms, +all of which correspond with the same mean values and which, in +spite of this, lead to quite different processes with regard to the +mean values. It follows from this of necessity that the micro-observer +must either \label{png59lab1}give up the attempt to understand the unique +progress, in accordance with experience, of the macroscopic +changes of state---and this would be the end of the atomic theory---or +that he, through the introduction of a special physical +hypothesis, restrict in a suitable manner the manifold of micro-states +considered by him. There is certainly nothing to prevent +him from assuming that not all conceivable micro-states are +realizable in nature, and that certain of them are in fact thinkable, +but never actually realized. In the formularization of such a +hypothesis, there is of course no point of departure to be found +from the principles of dynamics alone; for pure dynamics leaves +this case undetermined. But on just this account any dynamical +%-----File: 060.png---\redacted\-------- +hypothesis, which involves nothing further than a closer specification +of the micro-states realized in nature, is certainly permissible. +Which hypothesis is to be given the preference can only +be decided through comparison of the results to which the +different possible hypotheses lead in the course of experience. + +In order to limit the investigation in this way, we must obviously +fix our attention only upon all imaginable configurations and +velocities of the individual atoms which are compatible with +determinate values of the density, the velocity and the temperature +of the gas, or in other words: we must consider all the +micro-states which belong to a determinate macro-state, and +must investigate the various kinds of processes which follow in +accordance with the fixed laws of dynamics from the different +micro-states. Now, precise calculation has in every case always +led to the important result that an enormously large number of +these different micro-processes relate to one and the same macro-process, +and that only proportionately few of the same, which are +distinguished by quite special exceptional conditions concerning +the positions and the velocities of neighboring atoms, furnish +exceptions. Furthermore, it has also shown that one of the +resulting macro-processes is that which the macroscopic observer +recognizes, so that it is compatible with the second law +of thermodynamics. + +Here, manifestly, the bridge of understanding is supplied. The +micro-observer needs only to assimilate in his theory the physical +hypothesis that all those special cases in which special exceptional +conditions exist among the neighboring configurations of interacting +atoms do not occur in nature, or, in other words, that the +micro-states are in elementary disorder. Then the uniqueness +of the macroscopic process is assured and with it, also, the fulfillment +of the principle of increase of entropy in all directions. + +Therefore, it is not the atomic distribution, but rather the +hypothesis of elementary disorder, which forms the real kernel of +the principle of increase of entropy and, therefore, the preliminary +condition for the existence of entropy. Without elementary +%-----File: 061.png---\redacted\-------- +disorder there is neither entropy nor irreversible process.\footnote + {To those physicists who, in spite of all this, regard the hypothesis of + elementary disorder as gratuitous or as incorrect, I wish to refer the simple + fact that in every calculation of a coefficient of friction, of diffusion, or of heat + conduction, from molecular considerations, the notion of elementary disorder + is employed, whether tacitly or otherwise, and that it is therefore essentially + more correct to stipulate this condition instead of ignoring or concealing it. But + he who regards the hypothesis of elementary disorder as self-evident, should + be reminded that, in accordance with a law of H.~Poincaré, the precise investigation + concerning the foundation of which would here lead us too far, + the assumption of this hypothesis for all times is unwarranted for a closed + space with absolutely smooth walls,---an important conclusion, against which + can only be urged the fact that absolutely smooth walls do not exist in nature.} +Therefore, a single atom can never possess an entropy; for we +cannot speak of disorder in connection with it. But with a +fairly large number of atoms, say $100$ or~$1,000$, the matter is +quite different. Here, one can certainly speak of a disorder, in +case that the values of the coordinates and the velocity components +are distributed among the atoms in accordance with the +laws of accident. Then it is possible to calculate the probability +for a given state. But how is it with regard to the increase of +entropy? May we assert that the motion of $100$~atoms is irreversible? +Certainly not; but this is only because the state of +$100$~atoms cannot be defined in a thermodynamic sense, since the +process does not proceed in a unique manner from the standpoint +of a macro-observer, and this requirement forms, as we have seen +above, the foundation and preliminary condition for the definition +of a thermodynamic state. + +If one therefore asks: How many atoms are at least necessary +in order that a process may be considered irreversible?, the answer +is: so many atoms that one may form from them definite mean +values which define the state in a macroscopic sense. One must +reflect that to secure the validity of the principle of increase of +entropy there must be added to the condition of elementary disorder +still another, namely, that the number of the elements +under consideration be sufficiently large to render possible the +formation of definite mean values. The second law has a +meaning for these mean values only; but for them, it is quite +%-----File: 062.png---\redacted\-------- +exact, just as exact as the law of the calculus of probability, that +the mean value, so far as it may be defined, of a sufficiently large +number of throws with a six-sided die, is~$3\frac{1}{2}$. + +These considerations are, at the same time, capable of throwing +light upon questions such as the following: Does the principle of +increase of entropy possess a meaning for the so-called Brownian +molecular movement of a suspended particle? Does the kinetic +energy of this motion represent useful work or not? The entropy +principle is just as little valid for a single suspended particle as +for an atom, and therefore is not valid for a few of them, but +only when there is so large a number that definite mean values +can be formed. That one is able to see the particles and not +the atoms makes no material difference; because the progress of a +process does not depend upon the power of an observing instrument. +The question with regard to useful work plays no rôle +in this connection; strictly speaking, this possesses, in general, no +objective physical meaning. For it does not admit of an answer +without reference to the scheme of the physicist or technician +who proposes to make use of the work in question. The second +law, therefore, has fundamentally nothing to do with the idea of +useful work (cf.\ first lecture, p.~\pageref{png25lab1}). + +But, if the entropy principle is to hold, a further assumption is +necessary, concerning the various disordered elements,---an +assumption which tacitly is commonly made and which we +have not previously definitely expressed. It is, however, not +less important than those referred to above. The elements must +actually be of the same kind, or they must at least form a number +of groups of like kind, \eg, constitute a mixture in which each +kind of element occurs in large numbers. For only through the +similarity of the elements does it come about that order and law +can result in the larger from the smaller. If the molecules of a +gas be all different from one another, the properties of a gas can +never show so simple a law-abiding behavior as that which is +indicated by thermodynamics. In fact, the calculation of the +probability of a state presupposes that all complexions which +%-----File: 063.png---\redacted\-------- +correspond to the state are a priori equally likely. Without +this condition one is just as little able to calculate the probability +of a given state as, for instance, the probability of a given throw +with dice whose sides are unequal in size. In summing up we +may therefore say: the second law of thermodynamics in its +objective physical conception, freed from anthropomorphism, +relates to certain mean values which are formed from a large +number of disordered elements of the same kind. + +The validity of the principle of increase of entropy and of the +irreversible progress of thermodynamic processes in nature is +completely assured in this formularization. After the introduction +of the hypothesis of elementary disorder, the microscopic +observer can no longer confidently assert that each process considered +by him in a collection of atoms is reversible; for the +motion occurring in the reverse order will not always obey the +requirements of that hypothesis. In fact, the motions of single +atoms are always reversible, and thus far one may say, as before, +that the irreversible processes appear reduced to a reversible +process, but the phenomenon as a whole is nevertheless irreversible, +because upon reversal the disorder of the numerous +individual elementary processes would be eliminated. Irreversibility +is inherent, not in the individual elementary processes +themselves, but solely in their irregular constitution. It is +this only which guarantees the unique change of the macroscopic +mean values. + +Thus, for example, the reverse progress of a frictional process +is impossible, in that it would presuppose elementary arrangement +of interacting neighboring molecules. For the collisions between +any two molecules must thereby possess a certain distinguishing +character, in that the velocities of two colliding molecules +depend in a definite way upon the place at which they meet. +In this way only can it happen that in collisions like directed +velocities ensue and, therefore, visible motion. + +Previously we have only referred to the principle of elementary +disorder in its application to the atomic theory of matter. But +%-----File: 064.png---\redacted\-------- +it may also be assumed as valid, as I wish to indicate at this +point, on quite the same grounds as those holding in the case of +matter, for the theory of radiant heat. Let us consider, \eg, +two bodies at different temperatures between which exchange of +heat occurs through radiation. We can in this case also imagine +a microscopic observer, as opposed to the ordinary macroscopic +observer, who possesses insight into all the particulars +of electromagnetic processes which are connected with emission +and absorption, and the propagation of heat rays. The microscopic +observer would declare the whole process reversible +because all electrodynamic processes can also take place in the +reverse direction, and the contradiction may here be referred +back to a difference in definition of the state of a heat ray. Thus, +while the macroscopic observer completely defines a monochromatic +ray through direction, state of polarization, color, and +intensity, the microscopic observer, in order to possess a complete +knowledge of an electromagnetic state, necessarily requires the +specification of all the numerous irregular variations of amplitude +and phase to which the most homogeneous heat ray is actually +subject. That such irregular variations actually exist follows +immediately from the well known fact that two rays of the same +color never interfere, except when they originate in the same source +of light. But until these fluctuations are given in all particulars, +the micro-observer can say nothing with regard to the progress +of the process. He is also unable to specify whether the exchange +of heat radiation between the two bodies leads to a decrease or +to an increase of their difference in temperature. The principle +of elementary disorder first furnishes the adequate criterion of +the tendency of the radiation process, \ie, the warming of the +colder body at the expense of the warmer, just as the same principle +conditions the irreversibility of exchange of heat through conduction. +However, in the two cases compared, there is indicated +an essential difference in the kind of the disorder. While in +heat conduction the disordered elements may be represented +as associated with the various molecules, in heat radiation there +%-----File: 065.png---\redacted\-------- +are the numerous vibration periods, connected with a heat ray, +among which the energy of radiation is irregularly distributed. +In other words: the disorder among the molecules is a material +one, while in heat radiation it is one of energy distribution. This +is the most important difference between the two kinds of disorder; +a common feature exists as regards the great number of +uncoordinated elements required. Just as the entropy of a body +is defined as a function of the macroscopic state, only when the +body contains so many atoms that from them definite mean +values may be formed, so the entropy principle only possesses +a meaning with regard to a heat ray when the ray comprehends +so many periodic vibrations, \ie, persists for so long a time, that +a definite mean value for the intensity of the ray may be obtained +from the successive irregular fluctuating amplitudes. + +Now, after the principle of elementary disorder has been +introduced and accepted by us as valid throughout nature, the +fundamental question arises as to the calculation of the probability +of a given state, and the actual derivation of the entropy +therefrom. From the entropy all the laws of thermodynamic +states of equilibrium, for material substances, and also for +energy radiation, may be uniquely derived. With regard to +the connection between entropy and probability, this is inferred +very simply from the law that the probability of two independent +configurations is represented by the product of the individual +probabilities: +\[ +W = W_{1} · W_{2}, +\] +while the entropy~$S$ is represented by the sum of the individual +entropies: +\[ +S = S_{1} + S_{2}. +\] +Accordingly, the entropy is proportional to the logarithm of the +probability: +\[ +S = k \log W. +\Tag{(12)} +\] +$k$~is a universal constant. In particular, it is the same for atomic +as for radiation configurations, for there is nothing to prevent +%-----File: 066.png---\redacted\-------- +us assuming that the configuration designated by~$1$ is atomic, +while that designated by~$2$ is a radiation configuration. If $k$~has +been calculated, say with the aid of radiation measurements, +then $k$~must have the same value for atomic processes. Later +we shall follow this procedure, in order to utilize the laws of heat +radiation in the kinetic theory of gases. Now, there remains, as +the last and most difficult part of the problem, the calculation of +the probability~$W$ of a given physical configuration in a given +macroscopic state. We shall treat today, by way of preparation +for the quite general problem to follow, the simple problem: to +specify the probability of a given state for a single moving +material point, subject to given conservative forces. Since the +state depends upon $6$~variables: the $3$~generalized coordinates +$\varphi_{1}$,~$\varphi_{2}$,~$\varphi_{3}$, and the three corresponding velocity components +$\dot{\varphi}_{1}$,~$\dot{\varphi}_{2}$,~$\dot{\varphi}_{3}$, and since all possible values of these $6$~variables constitute +a continuous manifold, the probability sought is, that +these $6$~quantities shall lie respectively within certain infinitely +small intervals, or, if one thinks of these $6$~quantities as the +rectilinear orthogonal coordinates of a point in an ideal six-dimensional +space, that this ideal ``state point'' shall fall within +a given, infinitely small ``state domain.'' Since the domain is +infinitely small, the probability will be proportional to the magnitude +of the domain and therefore proportional to +\[ +\int d\varphi_{1} · d\varphi_{2} · d\varphi_{3} · d\dot{\varphi}_{1} · d\dot{\varphi}_{2} · d\dot{\varphi}_{3}. +\] + +But this expression cannot serve as an absolute measure of +the probability, because in general it changes in magnitude with +the time, if each state point moves in accordance with the laws +of motion of material points, while the probability of a state +which follows of necessity from another must be the same for +the one as the other. Now, as is well known, another integral +quite similarly formed, may be specified in place of the one +above, which possesses the special property of not changing in +value with the time. It is only necessary to employ, in addition +to the general coordinates $\varphi_{1}$,~$\varphi_{2}$,~$\varphi_{3}$, the three so-called momenta +%-----File: 067.png---\redacted\-------- +$\psi_{1}$,~$\psi_{2}$,~$\psi_{3}$, in place of the three velocities $\dot{\varphi}_{1}$,~$\dot{\varphi}_{2}$,~$\dot{\varphi}_{3}$ as the determining +coordinates of the state. These are defined in the +following way: +\label{png67lab1} +\[ +\psi_{1} = \left(\frac{\dd H}{\dd \dot{\varphi}_{1}}\right)_{\varphi},\quad +\psi_{2} = \left(\frac{\dd H}{\dd \dot{\varphi}_{2}}\right)_{\varphi},\quad +\psi_{3} = \left(\frac{\dd H}{\dd \dot{\varphi}_{3}}\right)_{\varphi}, +\] +wherein $H$~denotes the kinetic potential (Helmholz). Then, in +Hamiltonian form, the equations of motion are: +\label{png67lab2} +\[ +\dot{\psi}_{1} = \frac{d\psi_{1}}{dt} = -\left(\frac{\dd E}{\dd \varphi_{1}}\right)_{\psi},\ \cdots,\quad +\dot{\varphi}_{1} = \frac{d\varphi_{1}}{dt} = \left(\frac{\dd E}{\dd \psi_{1}}\right)_{\varphi},\ \cdots, +\] +($E$~is the energy), and from these equations follows the ``condition +of incompressibility'': +\[ +\frac{\dd \dot{\varphi}_{1}}{\dd \varphi_{1}} + \frac{\dd \dot{\psi}_{1}}{\dd \psi_{1}} + \cdots = 0. +\] +Referring to the six-dimensional space represented by the coordinates +$\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$, $\psi_{1}$, $\psi_{2}$, $\psi_{3}$, this equation states that the magnitude +of an arbitrarily chosen state domain,~viz.: +\label{png67lab3} +\[ +\int d\varphi_{1} · d\varphi_{2} · d\varphi_{3} · d\psi_{1} · d\psi_{2} · d\psi_{3} +\] +does not change with the time, when each point of the domain +changes its position in accordance with the laws of motion of +material points. Accordingly, it is made possible to take the +magnitude of this domain as a direct measure for the probability +that the state point falls within the domain. + +From the last expression, which can be easily generalized for +the case of an arbitrary number of variables, we shall \label{png67lab4}calculate +later the probability of a thermodynamic state, for the +case of radiant energy as well as that for material substances. +%-----File: 068.png---\redacted\-------- + + +\Chapter{Fourth Lecture.}{The Equation of State for a Monatomic Gas.}\label{Lect4} + +My problem today is to utilize the general fundamental laws +concerning the concept of irreversibility, which we established +in the lecture of yesterday, in the solution of a definite problem: +the calculation of the entropy of an ideal monatomic gas in a +given state, and the derivation of all its thermodynamic properties. +The way in which we have to proceed is prescribed for us +by the general definition of entropy: +\[ +S = k \log W. +\Tag{(13)} +\] +The chief part of our problem is the calculation of~$W$ for a given +state of the gas, and in this connection there is first required a +more precise investigation of that which is to be understood as +the state of the gas. Obviously, the state is to be taken here +solely in the sense of the conception which we have called macroscopic +in the last lecture. Otherwise, a state would possess +neither probability nor entropy. Furthermore, we are not +allowed to assume a condition of equilibrium for the gas. For +this is characterized through the further special condition +that the entropy for it is a maximum. Thus, an unequal distribution +of density may exist in the gas; also, there may be +present an arbitrary number of different currents, and in general +no kind of equality between the various velocities of the molecules +is to be assumed. The velocities, as the coordinates of the +molecules, are rather to be taken a~priori as quite arbitrarily +given, but in order that the state, considered in a macroscopic +sense, may be assumed as known, certain mean values of the +densities and the velocities must exist. Through these mean +%-----File: 069.png---\redacted\-------- +values the state from a macroscopic standpoint is completely +characterized. + +The conditions mentioned will all be fulfilled if we consider +the state as given in such manner that the number of molecules +in a sufficiently small macroscopic space, but which, however, +contains a very large number of molecules, is given, and furthermore, +that the (likewise great) number of these molecules is +given, which are found in a certain macroscopically small velocity +domain, \ie, whose velocities lie within certain small intervals. +If we call the coordinates $x$,~$y$,~$z$, and the velocity components +$\dot{x}$,~$\dot{y}$,~$\dot{z}$, then this number will be proportional to\footnote + {We can call $\sigma$ a ``macro-differential'' in contradistinction to the micro-differentials + which are infinitely small with reference to the dimensions of a + molecule. I prefer this terminology for the discrimination between ``physical'' + and ``mathematical'' differentials in spite of the inelegance of phrasing, because + the macro-differential is also just as much mathematical as physical and the + micro-differential just as much physical as mathematical.} +\[ +dx · dy · dz · d\dot{x} · d\dot{y} · d\dot{z} = \sigma. +\] +It will depend, besides, upon a finite factor of proportionality +which may be an arbitrarily given function $f(x, y, z, \dot{x}, \dot{y}, \dot{z})$ of +the coordinates and the velocities, and which has only the one +condition to fulfill that +\[ +\tsum f · \sigma = N, +\Tag{(14)} +\] +where $N$~denotes the total number of molecules in the gas. +We are now concerned with the calculation of the probability~$W$ +of that state of the gas which corresponds to the arbitrarily +given distribution function~$f$. + +The probability that a given molecule possesses such coordinates +and such velocities that it lies within the domain~$\sigma$ is +expressed, in accordance with the final result of the previous lecture, +by the magnitude of the corresponding elementary domain: +\[ +d\varphi_{1} · d\varphi_{2} · d\varphi_{3} · d\psi_{1} · d\psi_{2} · d\psi_{3}, +\] +therefore, since here +\[ +\varphi_{1} = x,\quad \varphi_{2} = y,\quad \varphi_{3} = z,\quad +\psi_{1} = m\dot{x},\quad \psi_{2} = m\dot{y},\quad \psi_{3} = m\dot{z}, +\] +%-----File: 070.png---\redacted\-------- +($m$~the mass of a molecule) by +\[ +m^{3} \sigma. +\] +Now we divide the whole of the six dimensional ``state domain'' +containing all the molecules into suitable equal elementary +domains of the magnitude~$m^{3} \sigma$. Then the probability that a +given molecule fall in a given elementary domain is equally +great for all such domains. Let $P$~denote the number of these +equal elementary domains. Next, let us imagine as many dice +as there are molecules present, \ie,~$N$, and each die to be +provided with $P$~equal sides. Upon these $P$~sides we imagine +numbers $1$,~$2$, $3$,~$\cdots$ to~$P$, so that each of the $P$~sides indicates +a given elementary domain. Then each throw with the $N$~dice +corresponds to a given state of the gas, while the number of +dice which show a given number corresponds to the molecules +which lie in the elementary domain considered. In accordance +with this, each single die can indicate with the same probability +each of the numbers from $1$ to~$P$, corresponding to the circumstance +that each molecule may fall with equal probability in any +one of the $P$~elementary domains. The probability~$W$ sought, +of the given state of the molecules, corresponds, therefore, to +the number of different kinds of throws (complexions) through +which is realized the given distribution~$f$. Let us take, \eg, +$N$~equal to $10$~molecules (dice) and $P = 6$ elementary domains +(sides) and let us imagine the state so given that there are +\begin{center} +\begin{tabular}{l@{\ }l@{\ }l} +3 molecules & in 1st & elementary domain \\ +4 molecules & in 2d & elementary domain \\ +0 molecules & in 3d & elementary domain \\ +1 molecule & in 4th & elementary domain \\ +0 molecules & in 5th & elementary domain \\ +2 molecules & in 6th & elementary domain, +\end{tabular} +\end{center} +then this state, \eg, may be realized through a throw for which +the 10 dice indicate the following numbers: +\[ +\begin{array}[b]{@{\qquad}*{10}{c}} +\rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & +\rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} \\[-2ex] +1\text{st} & 2\text{d} & 3\text{d} & 4\text{th} & 5\text{th} & 6\text{th} & 7\text{th} & 8\text{th} & 9\text{th} & 10\text{th} \\ +2 & 6 & 2 & 1 & 1 & 2 & 6 & 2 & 1 & 4. +\end{array} \Tag{(15)} +\] +%-----File: 071.png---\redacted\-------- +Under each of the characters representing the ten dice stands +the number which the die indicates in the throw. In fact, +\begin{center} +3 dice show the figure 1\phantom{.} \\ +4 dice show the figure 2\phantom{.} \\ +0 dice show the figure 3\phantom{.} \\ +1 die shows the figure 4\phantom{.} \\ +0 dice show the figure 5\phantom{.} \\ +2 dice show the figure 6. +\end{center} +The state in question may likewise be realized through many other +complexions of this kind. The number sought of all possible +complexions is now found through consideration of the number +series indicated in~\Eq{(15)}. For, since the number of molecules +(dice) is given, the number series contains a fixed number of +elements ($10 = N$). Furthermore, since the number of molecules +falling in an elementary domain is given, each number, in all +permissible complexions, appears equally often in the series. +Finally, each change of the number configuration conditions a +new complexion. The number of possible complexions or the +probability~$W$ of the given state is therefore equal to the number +of possible permutations with repetition under the conditions +mentioned. In the simple example chosen, in accordance with +a well known formula, the probability is +\[ +\frac{10!}{3!\; 4!\; 0!\; 1!\; 0!\; 2!\;} = 12,600. +\] +Therefore, in the general case: +\[ +W = \frac{N!}{\prod(f · \sigma)!}. +\] +The sign~$\prod$ denotes the product extended over all of the $P$~elementary +domains. + +From this there results, in accordance with equation~\Eq{(13)}, for +the entropy of the gas in the given state: +\[ +S = k \log N! - k \tsum \log (f · \sigma)!. +\] +%-----File: 072.png---\redacted\-------- +The summation is to be extended over all domains~$\sigma$. Since +$f · \sigma$ is a large quantity, Stirling's formula may be employed for +its factorial, which for a large number~$n$ is expressed by: +\[ +n! = \left(\frac{n}{e}\right)^{n} \sqrt{2 \pi n}, +\Tag{(16)} +\] +therefore, neglecting unimportant terms: +\[ +\log n! = n (\log n - 1); +\] +and hence: +\[ +S = k \log N! - k \tsum f \sigma (\log [f · \sigma] - 1), +\] +or, if we note that $\sigma$~and $N = \tsum f \sigma$ remain constant in all changes +of state: +\[ +S = \const - k \tsum f · \log f · \sigma. +\Tag{(17)} +\] +This quantity is, to the universal factor~$(-k)$, the same as that +which L.~Boltzmann denoted by~$H$, and which he showed to +vary in one direction only for all changes of state. + +In particular, we will now determine the entropy of a gas in a +state of equilibrium, and inquire first as to that form of the law of +distribution which corresponds to thermodynamic equilibrium. +In accordance with the second law of thermodynamics, a state +of equilibrium is characterized by the condition that with given +values of the total volume~$V$ and the total energy~$E$, the entropy~$S$ +assumes its maximum value. If we assume the total volume +of the gas +\[ +V = \int dx · dy · dz, +\] +and the total energy +\[ +E = \frac{m}{2} \tsum (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2} )f \sigma +\Tag{(18)} +\] +as given, then the condition: +\[ +\delta S = 0 +\] +must hold for the state of equilibrium, or, in accordance with~\Eq{(17)}: +\[ +\tsum (\log f + 1) · \delta f · \sigma = 0, +\Tag{(19)} +\] +%-----File: 073.png---\redacted\-------- +wherein the variation~$\delta f$ refers to an arbitrary change in the +law of distribution, compatible with the given values of $N$,~$V$ +and~$E$. + +Now we have, on account of the constancy of the total number +of molecules $N$, in accordance with~\Eq{(14)}: +\[ +\tsum \delta f · \sigma = 0 +\] +and, on account of the constancy of the total energy, in accordance +with~\Eq{(18)}: +\[ +\tsum (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2}) · \delta f · \sigma = 0. +\] +Consequently, for the fulfillment of condition~\Eq{(19)} for all permissible +values of~$\delta f$, it is sufficient and necessary that +\[ +\log f + \beta (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2}) = \const, +\] +or: +\[ +f = \alpha e^{-\beta (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2})}, +\] +wherein $\alpha$~and~$\beta$ are constants. In the state of equilibrium, +therefore, the space distribution of molecules is uniform, \ie, +independent of $x$,~$y$,~$z$, and the distribution of velocities is the +well known Maxwellian distribution. + +The values of the constants $\alpha$~and~$\beta$ are to be found from those +of $N$,~$V$ and~$E$. For the substitution of the value found for~$f$ +in~\Eq{(14)} leads to: +\[ +N = V \alpha \left(\frac{\pi}{\beta}\right)^{\tfrac{3}{2}}, +\] +and the substitution of~$f$ in~\Eq{(18)} leads to: +\[ +E = \tfrac{3}{4} Vm \frac{\alpha}{\beta}\left(\frac{\pi}{\beta}\right)^{\tfrac{3}{2}}. +\] +From these equations it follows that: +\[ +\alpha = \frac{N}{V} · \left(\frac{3mN}{4\pi E}\right)^{\tfrac{3}{2}},\quad +\beta = \frac{3mN}{4E}, +\] +and hence finally, in accordance with~\Eq{(17)}, the expression for the +%-----File: 074.png---\redacted\-------- +entropy~$S$ of the gas in a state of equilibrium with given values +for $N$,~$V$ and~$E$ is: +\[ +S = \const + kN (\tfrac{3}{2} \log E + \log V). +\Tag{(20)} +\] +The additive constant contains terms in $N$~and~$m$, but not in +$E$~and~$V$. + +The determination of the entropy here carried out permits +now the specification directly of the complete thermodynamic +behavior of the gas, viz., of the equation of state, and of the +values of the specific heats. From the general thermodynamic +definition of entropy: +\[ +dS = \frac{dE + p dV}{T} +\] +are obtained the partial differential quotients of~$S$ with regard +to $E$~and~$V$ respectively: +\label{png74lab1} +\[ +\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{1}{T},\quad +\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{p}{T}. +\] +Consequently, with the aid of~\Eq{(20)}: +\label{png74lab2} +\[ +\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{3}{2} \frac{kN}{E} = \frac{1}{T}, +\Tag{(21)} +\] +and +\[ +\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{kN}{V} = \frac{p}{T}. +\Tag{(22)} +\] +The second of these equations: +\[ +p = \frac{kNT}{V} +\] +contains the laws of Boyle, Gay~Lussac and Avogadro, the latter +because the pressure depends only upon the number~$N$, and not +upon the constitution of the molecules. Writing it in the +ordinary form: +\[ +p = \frac{RnT}{V}, +\] +%-----File: 075.png---\redacted\-------- +where $n$~denotes the number of gram molecules or mols of the +gas, referred to $O_{2} = 32g$, and $R$~the absolute gas constant: +\[ +R = 8.315 · 10^{7} \frac{\erg}{\deg}, +\] +we obtain by comparison: +\[ +k = \frac{Rn}{N}. +\Tag{(23)} +\] +If we denote the ratio of the mol number to the molecular +number by~$\omega$, or, what is the same thing, the ratio of the +molecular mass to the mol mass: +\[ +\omega = \frac{n}{N}, +\] +and hence: +\[ +k = \omega R. +\Tag{(24)} +\] +From this, if $\omega$~is given, we can calculate the universal constant~$k$, +and conversely. + +The equation~\Eq{(21)} gives: +\[ +E = \tfrac{3}{2} kNT. +\Tag{(25)} +\] +Now since the energy of an ideal gas is given by: +\[ +E = Anc_{v} T, +\] +wherein $c_{v}$~denotes in calories the heat capacity at constant +volume of a mol, $A$~the mechanical equivalent of heat: +\[ +A = 4.19 · 10^{7} \frac{\erg}{\cal}, +\] +it follows that: +\[ +c_{v} = \frac{3kN}{2An}, +\] +and, having regard to~\Eq{(23)}, we obtain: +\[ +c_{v} = \frac{3}{2} \frac{R}{A} = 3.0, +\Tag{(26)} +\] +%-----File: 076.png---\redacted\-------- +the mol heat in calories of any monatomic gas at constant volume. + +For the mol heat~$c_{p}$ at constant pressure we have from the +first law of thermodynamics +\[ +c_{p} - c_{v} = \frac{R}{A}, +\] +and, therefore, having regard to~\Eq{(26)}: +\[ +c_{p} = 5,\quad \frac{c_{p}}{c_{v}} = \tfrac{5}{3}, +\] +a known result for monatomic gases. + +The mean kinetic energy~$L$ of a molecule is obtained from~\Eq{(25)}: +\[ +L = \frac{E}{N} = \tfrac{3}{2} kT. +\Tag{(27)} +\] +You notice that we have derived all these relations through the +identification of the mechanical with the thermodynamic expression +for the entropy, and from this you recognize the fruitfulness +of the method here proposed. + +But a method can first demonstrate fully its usefulness when +we utilize it, not only to derive laws which are already known, +but when we apply it in domains for whose investigation there +at present exist no other methods. In this connection its +application affords various possibilities. Take the case of a +monatomic gas which is not sufficiently attenuated to have the +properties of the ideal state; there are here, as pointed out by +J.~D. van~der Waals, two things to consider: (1)~the finite size of +the atoms, (2)~the forces which act among the atoms. Taking +account of these involves a change in the value of the probability +and in the energy of the gas as well, and, so far as can now be +shown, the corresponding change in the conditions for thermodynamic +equilibrium leads to an equation of state which agrees +with that of van~der Waals. Certainly there is here a rich field +for further investigations, of greater promise when experimental +tests of the equation of state exist in larger number. +%-----File: 077.png---\redacted\-------- + +Another important application of the theory has to do with +heat radiation, with which we shall be occupied the coming +week. We shall proceed then in a similar way as here, and shall +be able from the expression for the entropy of radiation to derive +the thermodynamic properties of radiant heat. + +Today we will refer briefly to the treatment of polyatomic +gases. I have previously, upon good grounds, limited the treatment +to monatomic molecules; for up to the present real difficulties +appear to stand in the way of a generalization, from +the principles employed by us, to include polyatomic molecules; in +fact, if we wish to be quite frank, we must say that a satisfactory +mechanical theory of polyatomic gases has not yet been found. +Consequently, at present we do not know to what place in the +system of theoretical physics to assign the processes within a +molecule---the intra-molecular processes. We are obviously confronted +by puzzling problems. A noteworthy and much discussed +beginning was, it is true, made by Boltzmann, who introduced +the most plausible assumption that for intra-molecular +processes simple laws of the same kind hold as for the motion of +the molecules themselves, \textit{\ie}, the general equations of dynamics. +It is easy then, in fact, to proceed to the proof that for a monatomic +gas the molecular heat~$c_{v}$ must be greater than~$3$ and that +consequently, since the difference $c_{p} - c_{v}$ is always equal to~$2$, +the ratio is +\[ +\frac{c_{p}}{c_{v}} = \frac{c_{v} + 2}{c_{v}} < \tfrac{5}{3} . +\] +This conclusion is completely confirmed by experience. But this +in itself does not confirm the assumption of Boltzmann; for, +indeed, the same conclusion is reached very simply from the +assumption that there exists intra-molecular energy which +increases with the temperature. For then the molecular heat +of a polyatomic gas must be greater by a corresponding amount +than that of a monatomic gas. + +Nevertheless, up to this point the Boltzmann theory never leads +%-----File: 078.png---\redacted\-------- +to contradiction with experience. But so soon as one seeks to +draw special conclusions concerning the magnitude of the specific +heats hazardous difficulties arise; I will refer to only one of them. +If one assumes the Hamiltonian equations of mechanics as +applicable to intra-molecular motions, he arrives of necessity at +\label{png78lab1}the law of ``uniform distribution of energy,'' which asserts that +under certain conditions, not essential to consider here, in a +thermodynamic state of equilibrium the total energy of the gas +is distributed uniformly among all the individual energy phases +corresponding to the independent variables of state, or, as +one may briefly say; the same amount of energy is associated +with every independent variable of state. Accordingly, the +mean energy of motion of the molecules~$\frac{1}{2} kT$, corresponding to a +given direction in space, is the same as for any other direction, +and, moreover, the same for all the different kinds of molecules, +and ions; also for all suspended particles (dust) in the gas, of +whatever size, and, furthermore, the same for all kinds of motions +of the constituents of a molecule relative to its centroid. If +one now reflects that a molecule commonly contains, so far as +we know, quite a large number of different freely moving +constituents, certainly, that a normal molecule of a monatomic +gas, \eg, mercury, possesses numerous freely moving +electrons, then, in accordance with the law of uniform energy +distribution, the intra-molecular energy must constitute a much +larger fraction of the whole specific heat of the gas, and therefore +$c_{p}/c_{v}$~must turn out much smaller, than is consistent with the +measured values. Thus, \eg, for an atom of mercury, in +accordance with the measured value of $c_{p}/c_{v} = 5/3$, no part +whatever of the heat added may be assigned to the intra-molecular +energy. Boltzmann and others, in order to eliminate this contradiction, +have fixed upon the possibility that, within the time +of observation of the specific heats, the vibrations of the constituents +(of a molecule) do not change appreciably with respect +to one another, and come later with their progressive motion so +slowly into heat equilibrium that this process is no longer capable +%-----File: 079.png---\redacted\-------- +of detection through observation. Up to now no such delay in +the establishment of a state of equilibrium has been observed. +Perhaps it would be productive of results if in delicate measurements +special attention were paid the question as to whether +observations which take a longer time lead to a greater value of +the mol-heat, or, what comes to the same thing, a smaller value +of~$c_{p}/c_{v}$, than observations lasting a shorter time. + +If one has been made mistrustful through these considerations +concerning the applicability of the law of uniform energy distribution +to intra-molecular processes, the mistrust is accentuated +upon the inclusion of the laws of heat radiation. I shall make +mention of this in a later lecture. + +When we pass from stable atoms to the unstable atoms of +radioactive substances, the principles following from the kinetic +gas theory lose their validity completely. For the striking +failure of all attempts to find any influence of temperature +upon radioactive phenomena shows us that an application here of +the law of uniform energy distribution is certainly not warranted. +It will, therefore, be safest meanwhile to offer no definite conjectures +with regard to the nature and the laws of these noteworthy +phenomena, and to leave this field for further development +to experimental research alone, which, I may say, with every +day throws new light upon the subject. +%-----File: 080.png---\redacted\-------- + + +\Chapter{Fifth Lecture.}{Heat Radiation. Electrodynamic Theory.}\label{Lect5} + +Last week I endeavored to point out that we find in the +atomic theory a complete explanation for the whole content of +the two laws of thermodynamics, if we, with Boltzmann, define +the entropy by the probability, and I have further shown, in the +example of an ideal monatomic gas, how the calculation of the +probability, without any additional special hypothesis, enables +us not only to find the properties of gases known from thermodynamics, +but also to reach conclusions which lie essentially +beyond those of pure thermodynamics. Thus, \eg, +the law of Avogadro in pure thermodynamics is only a definition, +while in the kinetic theory it is a necessary consequence; +furthermore, the value of~$c_{v}$, the mol-heat of a gas, is +completely undetermined by pure thermodynamics, but from the +kinetic theory it is of equal magnitude for all monatomic gases +and, in fact, equal to~$3$, corresponding to our experimental +knowledge. Today and tomorrow we shall be occupied with +the application of the theory to radiant heat, and it will appear +that we reach in this apparently quite isolated domain conclusions +which a thorough test shows are compatible with experience. +Naturally, we take as a basis the electro-magnetic +theory of heat radiation, which regards the rays as electro-magnetic +waves of the same kind as light rays. + +We shall utilize the time today in developing in bold outline +the important consequences which follow from the electro-magnetic +theory for the characteristic quantities of heat radiation, +and tomorrow seek to answer, through the calculation of the +entropy, the question concerning the dependence of these quantities +%-----File: 081.png---\redacted\-------- +upon the temperature, as was done last week for ideal +gases. Above all, we are concerned here with the determination +of those quantities which at any place in a medium traversed +by heat rays determine the state of the radiant heat. The state +of radiation at a given place will not be represented by a vector +which is determined by three components; for the energy flowing +in a given direction is quite independent of that flowing in any +other direction. In order to know the state of radiation, we +must be able to specify, moreover, the energy which in the time~$dt$ +flows through a surface element~$d\sigma$ for every direction in +space. This will be proportional to the magnitude of~$d\sigma$, to +the time~$dt$, and to the cosine of the angle~$\theta$ which the direction +considered makes with the normal to~$d\sigma$. But the quantity to +be multiplied by $d\sigma · dt · \cos \theta$ will not be a finite quantity; +for since the radiation through any point of~$d\sigma$ passes in all directions, +therefore the quantity will also depend upon the magnitude +of the solid angle~$d\Omega$, which we shall assume as the same for all +points of~$d\sigma$. In this manner we obtain for the energy which in +the time~$dt$ flows through the surface element~$d\sigma$ in the direction +of the elementary cone~$d\Omega$, the expression: +\[ +K d\sigma dt · \cos \theta · d\Omega. +\Tag{(28)} +\] +$K$~is a positive function of place, of time and of direction, and is +for unpolarized light of the following form: +\[ +K = 2 \int_{0}^{\infty} \frakK_{\nu} d\nu +\Tag{(29)} +\] +where $\nu$~denotes the frequency of a color of wave length~$\lambda$ and +whose velocity of propagation is~$q$: +\[ +\nu = \frac{q}{\lambda}, +\] +and $\frakK_{\nu}$~denotes the corresponding intensity of spectral radiation +of the plane polarized light. +%-----File: 082.png---\redacted\-------- + +From the value of~$K$ is to be found the space density of radiation~$\epsilon$, +\ie, the energy of radiation contained in unit volume. The +point~$0$ in question forms the centre of a sphere whose radius~$r$ +we take so small that in the distance~$r$ no appreciable absorption +of radiation takes place. Then each element~$d\sigma$ of the surface +of the sphere furnishes, by virtue of the radiation traversing the +same, the following contribution to the radiation density at~$0$: +\[ +\frac{d\sigma · dt · K · d\Omega}{r^{2} d\Omega · q dt} = \frac{d\sigma · K}{r^{2} q}. +\] +For the radiation cone of solid angle~$d\Omega$ proceeding from a point +of~$d\sigma$ in the direction toward~$0$ has at the distance~$r$ from~$d\sigma$ the +cross-section~$r^{2} d\Omega$ and the energy passing in the time~$dt$ through +this cross-section distributes itself along the distance~$q dt$. By +integration over all of the surface elements~$d\sigma$ we obtain the +total space density of radiation at~$0$: +\[ +\epsilon = \int \frac{d\sigma K}{r^{2} q} = \frac{1}{q} \int K d\Omega, +\] +wherein $d\Omega$~denotes the solid angle of an elementary cone whose +vertex is~$0$. For uniform radiation we obtain: +\[ +\epsilon = \frac{4\pi K}{q} = \frac{8\pi}{q} · \int_{0}^{\infty} \frakK_{\nu} d\nu. +\Tag{(30)} +\] + +The production of radiant heat is a consequence of the act of +emission, and its destruction is the result of absorption. Both +processes, emission and absorption, have their origin only in +material particles, atoms or electrons, not at the geometrical +bounding surface; although one frequently says, for the sake of +brevity, that a surface element emits or absorbs. In reality a +surface element of a body is a place of entrance for the radiation +falling upon the body from without and which is to be +absorbed; or a place of exit for the radiation emitted from +within the body and passing through the surface in the outward +%-----File: 083.png---\redacted\-------- +direction. The capacity for emission and the capacity for +absorption of an element of a body depend only upon its own +condition and not upon that of the surrounding elements. If, +therefore, as we shall assume in what follows, the state of the +body varies only with the temperature, then the capacity for +emission and the capacity for absorption of the body will also +vary only with the temperature. The dependence upon the +temperature can of course be different for each wave length. + +We shall now introduce that result following from the second +law of thermodynamics which will serve us as a basis +in all subsequent considerations: ``a system of bodies at rest +of arbitrary nature, form and position, which is surrounded by a +fixed shell impervious to heat, passes in the course of time from +an arbitrarily chosen initial state to a permanent state in which +the temperature of all bodies of the system is the same.'' +This is the thermodynamic state of equilibrium in which the +entropy of the system, among all those values which it may assume +compatible with the total energy specified by the initial conditions, +has a maximum value. Let us now apply this law to a +single homogeneous isotropic medium which is of great extent +in all directions of space and which, as in all cases subsequently +considered, is surrounded by a fixed shell, perfectly reflecting as +regards heat rays. The medium possesses for each frequency~$\nu$ +of the heat rays a finite capacity for emission and a finite capacity +for absorption. Let us consider, now, such regions of the medium +as are very far removed from the surface. Here the influence +of the surface will be in any case vanishingly small, because no +rays from the surface reach these regions, and on account of the +homogeneity and isotropy of the medium we must conclude that +the heat radiation is in thermodynamic equilibrium everywhere +and has the same properties in all directions, so that $\frakK_{\nu}$,~the +specific intensity of radiation of a plane polarized ray, is independent +of the frequency~$\nu$, of the azimuth of polarization, of the +direction of the ray, and of location. Thus, there will correspond +to each diverging bundle of rays in an elementary cone~$d\Omega$, +%-----File: 084.png---\redacted\-------- +proceeding from a surface element~$d\sigma$, an exactly equal bundle +oppositely directed, within the same elemental cone converging +toward the surface element. This law retains its validity, as a +simple consideration shows, right up to the surface of the medium. +For in thermodynamic equilibrium each ray must possess +exactly the same intensity as that of the directly opposite ray, +otherwise, more energy would flow in one direction than in +the opposite direction. Let us fix our attention upon a ray +proceeding inwards from the surface, this must have the +same intensity as that of the directly opposite ray coming +from within, and from this it follows immediately that the +state of radiation of the medium at all points on the surface is +the same as that within. The nature of the bounding surface +and the spacial extent of the medium are immaterial, and in a +stationary state of radiation~$\frakK_{\nu}$ is completely determined by the +nature of the medium for each temperature. + +This law suffers a modification, however, in the special case +that the medium is absolutely diathermanous for a definite +frequency~$\nu$. It is then clear that the capacity for absorption +and also that for emission must be zero, because otherwise no +stationary state of radiation could exist, \ie, a medium emits +no color which it does not absorb. But equilibrium can then obviously +exist for every intensity of radiation of the frequency considered, +\ie, $\frakK_{\nu}$~is now undetermined and cannot be found without +knowledge of the initial conditions. An important example of +this is furnished by an absolute vacuum, which is diathermanous +for all frequencies. In a complete vacuum thermodynamic +equilibrium can therefore exist for each arbitrary intensity of +radiation and for each frequency, \ie, for each arbitrary distribution +of the spectral energy. From a general thermodynamic +point of view this indeterminateness of the properties of thermodynamic +states of equilibrium is explained through the presence +of numerous different relative maxima of the entropy, as in the +case of a vapor which is in a state of supersaturation. But +among all the different maxima there is a special maximum, the +%-----File: 085.png---\redacted\-------- +absolute, which indicates stable equilibrium. In fact, we shall +see that in a diathermanous medium for each temperature there +exists a quite definite intensity of radiation, which is designated +as the stable intensity of radiation of the frequency~$\nu$ considered. +But for the present we shall assume for all frequencies +a finite capacity for absorption and for emission. + +We consider now two homogeneous isotropic media in thermodynamic +equilibrium separated from each other by a plane +surface. Since the equilibrium will not be disturbed if one +imagines for the moment the surface of separation between the +two substances to be replaced by a surface quite non-transparent +to heat radiation, all of the foregoing laws hold for each of the +% [Illustration: Fig. 1.] +two substances individually. Let the specific intensity of radiation +of frequency~$\nu$, polarized in any arbitrary plane within the +first substance (the upper in Fig.~1)\footnote + {From my lectures upon the theory of heat radiation (Leipzig, J.~A. Barth), + wherein are to be found the details of the above somewhat abbreviated + calculations.}, +be~$\frakK_{\nu}$ and that within the +second substance~${\frakK_{\nu}}'$ (we shall in general designate with a dash +%-----File: 086.png---\redacted\-------- +those quantities which refer to the second substance). Both +quantities $\frakK_{\nu}$~and~${\frakK_{\nu}}'$, besides depending upon the temperature +and the frequency, depend only upon the nature of the two substances, +and, in fact, these values of the intensity of radiation +hold quite up to the boundary surface between the substances, +and are therefore independent of the properties of this surface. + +\vspace{2\baselineskip} + +\pngcent{illo085.png}{1263} + +\vspace{2\baselineskip} + +Each ray from the first medium is split into two rays at the +boundary surface: the reflected and the transmitted. The directions +of these two rays vary according to the angle of incidence +and the color of the incident ray, and, in addition, the +intensity varies according to its polarization. If we denote +by~$\rho$ (the reflection coefficient) the amount of the reflected +energy of radiation and consequently by~$1 - \rho$ the amount of +transmitted energy with respect to the incident energy, then $\rho$~depends +upon the angle of incidence, upon the frequency and +upon the polarization of the incident ray. Similar remarks hold +for~$\rho'$, the reflection coefficient for a ray from the second +medium, upon meeting the boundary surface. + +Now the energy of a monochromatic plane polarized ray of +frequency~$\nu$ proceeding from an element~$d\sigma$ of the boundary +surface within the elementary cone~$d\Omega$ in a direction toward the +first medium (see the feathered arrow at the left in Fig.~1) is +for the time~$dt$, in accordance with \Eq{(28)}~and~\Eq{(29)}: +\[ +dt · d\sigma · \cos \theta · d\Omega · \frakK_{\nu} d\nu, +\Tag{(31)} +\] +where +\[ +d\Omega = \sin \theta d\theta d\varphi. +\Tag{(32)} +\] +This energy is furnished by the two rays which, approaching the +surface from the first and the second medium respectively, are +reflected and transmitted respectively at the surface element~$d\sigma$ +in the same direction. (See the unfeathered arrows. The surface +element~$d\sigma$ is indicated only by the point~$0$.) The first ray proceeds +in accordance with the law of reflection within the symmetrically +drawn elementary cone~$d\Omega$: the second approaches +the surface within the elementary cone +%-----File: 087.png---\redacted\-------- +\[ +d\Omega' = \sin \theta' d\theta' d\varphi', +\Tag{(33)} +\] +where, in accordance with the law of refraction, +\[ +\varphi' = \varphi\quad \text{and}\quad +\frac{\sin \theta}{\sin \theta'} = \frac{q}{q'}. +\Tag{(34)} +\] +We now assume that the ray is either polarized in the plane of +incidence or perpendicular to this plane, and likewise for the +two radiations out of whose energies it is composed. The radiation +coming from the first medium and reflected from~$d\sigma$ contributes +the energy: +\[ +\rho · dt · d\sigma \cos \theta · d\Omega · \frakK_{\nu} d\nu, +\Tag{(35)} +\] +and the radiation coming from the second medium and transmitted +through $d\sigma$ contributes the energy: +\[ +(1 - \rho') · dt · d\sigma \cos \theta' · d\Omega' · {\frakK_{\nu}}' d\nu. +\Tag{(36)} +\] +The quantities $dt$,~$d\sigma$,~$\nu$, and~$d\nu$ are here written without the +accent, since they have the same values in both media. + +Adding the expressions \Eq{(35)}~and~\Eq{(36)} and placing the sum +equal to the expression~\Eq{(31)}, we obtain: +\[ +\rho \cos \theta d\Omega \frakK_{\nu} + + (1 - \rho') \cos \theta' d\Omega' {\frakK_{\nu}}' + = \cos \theta d\Omega \frakK_{\nu}. +\] +Now, in accordance with~\Eq{(34)}: +\[ +\frac{\cos \theta d\theta}{q} = \frac{\cos \theta' d\theta'}{q'}, +\] +and further, taking note of \Eq{(32)}~and~\Eq{(33)}: +\[ +d\Omega' \cos \theta' = d\Omega \cos \theta · \frac{q'^{2}}{q^{2}}, +\] +and it follows that: +\[ +\rho \frakK_{\nu} + (1 - \rho') \frac{q'^{2}}{q^{2}} {\frakK_{\nu}}' = \frakK_{\nu} +\] +or: +\[ +\frac{\frakK_{\nu}}{{\frakK_{\nu}}'} · \frac{q^{2}}{q'^{2}} = \frac{1 - \rho'}{1 - \rho}. +\] +%-----File: 088.png---\redacted\-------- + +In the last equation the quantity on the left is independent +of the angle of incidence~$\theta$ and of the kind of polarization, consequently +the quantity upon the right side must also be independent +of these quantities. If one knows the value of these +quantities for a single angle of incidence and for a given kind of +polarization, then this value is valid for all angles of incidence +and for all polarizations. Now, in the particular case that the +rays are polarized at right angles to the plane of incidence and +meet the bounding surface at the angle of polarization, +\[ +\rho = 0\quad \text{and}\quad \rho' = 0. +\] +Then the expression on the right will be equal to~$1$, and therefore +it is in general equal to~$1$, and we have always: +\[ +\rho = \rho',\quad q^{2} \frakK_{\nu} = q'^{2} {\frakK_{\nu}}'. +\Tag{(37)} +\] +The first of these two relations, which asserts that the coefficient +of reflection is the same for both sides of the boundary surface, +constitutes the special expression of a general reciprocal law, +first announced by Helmholz, whereby the loss of intensity which +a ray of given color and polarization suffers on its path through +any medium in consequence of reflection, refraction, absorption, +and dispersion is exactly equal to the loss of intensity which a ray +of corresponding intensity, color and polarization suffers in +passing over the directly opposite path. It follows immediately +from this that the radiation meeting a boundary surface between +two media is transmitted or reflected equally well from both +sides, for every color, direction and polarization. + +The second relation,~\Eq{(37)}, brings into connection the radiation +intensities originating in both substances. It asserts that in +thermodynamic equilibrium the specific intensities of radiation +of a definite frequency in both media vary inversely as the square +of the velocities of propagation, or directly as the squares of the +refractive indices. We may therefore write +\[ +q^{2} \frakK_{\nu} = F(\nu, T), +\] +%-----File: 089.png---\redacted\-------- +wherein $F$~denotes a universal function depending only upon $\nu$~and~$T$, +the discovery of which is one of the chief problems of the +theory. + +Let us fix our attention again on the case of a diathermanous +medium. We saw above that in a medium surrounded by a +non-transparent shell which for a given color is diathermanous +equilibrium can exist for any given intensity of radiation of this +color. But it follows from the second law that, among all the +intensities of radiation, a definite one, namely, that corresponding +to the absolute maximum of the total entropy of the system, +must exist, which characterizes the absolutely stable equilibrium +of radiation. We now see that this indeterminateness is eliminated +by the last equation, which asserts that in thermodynamic +equilibrium the product~$q^{2}\frakK_{\nu}$ is a universal function. For it +results immediately therefrom that there is a definite value of~$\frakK_{\nu}$ +for every diathermanous medium which is thus differentiated +from all other values. The physical meaning of this value is +derived directly from a consideration of the way in which this +equation was derived: it is that intensity of radiation which +exists in the diathermanous medium when it is in thermodynamic +equilibrium while in contact with a given absorbing and emitting +medium. The volume and the form of the second medium is +immaterial; in particular, the volume may be taken arbitrarily +small. + +For a vacuum, the most diathermanous of all media, in which +the velocity of propagation $q = c$ is the same for all rays, we can +therefore express the following law: The quantity +\[ +\frakK_{\nu} = \frac{1}{c^{2}} F(\nu, T) +\Tag{(38)} +\] +denotes that intensity of radiation which exists in any complete +vacuum when it is in a stationary state as regards exchange of +radiation with any absorbing and emitting substance, whose +amount may be arbitrarily small. This quantity~$\frakK_{\nu}$ regarded +as a function of~$\nu$ gives the so-called normal energy spectrum. +%-----File: 090.png---\redacted\-------- + +Let us consider, therefore, a vacuum surrounded by given +emitting and absorbing bodies of uniform temperature. Then, +in the course of time, there is established therein a normal energy +radiation~$\frakK_{\nu}$ corresponding to this temperature. If now $\rho_{\nu}$~be +the reflection coefficient of a wall for the frequency~$\nu$, then of +the radiation~$\frakK_{\nu}$ falling upon the wall, the part~$\rho_{\nu} \frakK_{\nu}$ will be reflected. +On the other hand, if we designate by~$E_{\nu}$ the emission +coefficient of the wall for the same frequency~$\nu$, the total radiation +proceeding from the wall will be: +\[ +\rho_{\nu} \frakK_{\nu} + E_{\nu} = \frakK_{\nu}, +\] +since each bundle of rays possesses in a stationary state the intensity~$\frakK_{\nu}$. +From this it follows that: +\[ +\frakK_{\nu} = \frac{E_{\nu}}{1 - \rho_{\nu}}, +\Tag{(39)} +\] +\ie, the ratio of the emission coefficient~$E_{\nu}$ to the capacity for +absorption $(1-\rho_{\nu})$ of a given substance is the same for all +substances and equal to the normal intensity of radiation for +each frequency (Kirchoff). For the special case that $\rho_{\nu}$~is equal +to~$0$, \ie, that the wall shall be perfectly black, we have: +\[ +\frakK_{\nu} = E_{\nu}, +\] +that is, the normal intensity of radiation is exactly equal to the +emission coefficient of a black body. Therefore the normal +radiation is also called ``black radiation.'' Again, for any given +body, in accordance with~\Eq{(39)}, we have: +\[ +E_{\nu} < \frakK_{\nu}, +\] +\ie, the emission coefficient of a body in general is smaller than +that of a black body. Black radiation, thanks to W.~Wien and +O.~Lummer, has been made possible of measurement, through +a small hole bored in the wall bounding the space considered. + +We proceed now to the treatment of the problem of determining +the specific intensity~$\frakK_{\nu}$ of black radiation in a vacuum, +%-----File: 091.png---\redacted\-------- +as regards its dependence upon the frequency~$\nu$ and the temperature~$T$. +In the treatment of this problem it will be necessary +to go further than we have previously done into those processes +which condition the production and destruction of heat rays; +that is, into the question regarding the act of emission and that +of absorption. On account of the complicated nature of these +processes and the difficulty of bringing some of the details into +connection with experience, it is certainly quite out of the question +to obtain in this manner any reliable results if the following +law cannot be utilized as a dependable guide in this domain: a +vacuum surrounded by reflecting walls in which arbitrary +emitting and absorbing bodies are distributed in any given +arrangement assumes in the course of time the stationary state +of black radiation, which is completely determined by a single +parameter, the temperature, and which, in particular, does not +depend upon the number, the properties and the arrangement of +the bodies. In the investigation of the properties of the state +of black radiation the nature of the bodies which are supposed +to be in the vacuum is therefore quite immaterial, and it is certainly +immaterial whether such bodies actually exist anywhere +in nature, so long as their existence and their properties are +compatible throughout with the laws of electrodynamics and of +thermodynamics. As soon as it is possible to associate with +any given special kind and arrangement of emitting and absorbing +bodies a state of radiation in the surrounding vacuum which +is characterized by absolute stability, then this state can be no +other than that of black radiation. Making use of the freedom +furnished by this law, we choose among all the emitting and +absorbing systems conceivable, the most simple, namely, a single +oscillator at rest, consisting of two poles charged with equal +quantities of electricity of opposite sign which are movable +relative to each other in a fixed straight line, the axis of the +oscillator. The state of the oscillator is completely determined +by its moment,~$f(t)$; \ie,~by the product of the electric charge of +the pole on the positive side of the axis into the distance between +%-----File: 092.png---\redacted\-------- +the poles, and by its differential quotient with regard to the time: +\[ +\frac{df(t)}{dt} = \dot{f}(t). +\] +The energy of the oscillator is of the following simple form: +\[ +U = \tfrac{1}{2} Kf^{2} + \tfrac{1}{2} L \dot{f}^{2}, +\Tag{(40)} +\] +wherein $K$~and~$L$ denote positive constants which depend upon +the nature of the oscillator in some manner into which we need +not go further at this time. + +If, in the vibrations of the oscillator, the energy~$U$ remain absolutely +constant, we should have: $dU = 0$ or: +\[ +K f(t) + L \ddot{f}(t) = 0, +\] +and from this there results, as a general solution of the differential +equation, a pure periodic vibration: +\[ +f = C \cos (2\pi \nu_{0} t - \theta), +\] +wherein $C$~and~$\theta$ denote the integration constants and $\nu_{0}$~the +number of vibrations per unit of time: +\[ +\nu_{0} = \frac{1}{2\pi} \sqrt{\frac{K}{L}}. +\Tag{(41)} +\] +Such an oscillator vibrating periodically with constant energy +would neither be influenced by the electromagnetic field surrounding +it, nor would it exert any external actions due to radiation. +It could therefore have no sort of influence on the heat +radiation in the surrounding vacuum. + +In accordance with the theory of Maxwell, the energy of +vibration~$U$ of the oscillator by no means remains constant in +general, but an oscillator by virtue of its vibrations sends out +spherical waves in all directions into the surrounding field and, +in accordance with the principle of conservation of energy, if no +actions from without are exerted upon the oscillator, there must +%-----File: 093.png---\redacted\-------- +necessarily be a loss in the energy of vibration and, therefore, a +damping of the amplitude of vibration is involved. In order to +find the amount of this damping we calculate the quantity of +energy which flows out through a spherical surface with the +oscillator at the center, in accordance with the law of Poynting. +However, we may not place the energy flowing outwards in +accordance with this law through the spherical surface in an +infinitely small interval of time~$dt$ equal to the energy radiated +in the same time interval from the oscillator. For, in general, +the electromagnetic energy does not always flow in the outward +direction, but flows alternately outwards and inwards, and +we should obtain in this manner for the quantity of the radiation +outwards, values which are alternately positive and negative, +and which also depend essentially upon the radius of the +supposed sphere in such manner that they increase toward +infinity with decreasing radius---which is opposed to the fundamental +conception of radiated energy. This energy will, moreover, +be only found independent of the radius of the sphere +when we calculate the total amount of energy flowing outwards +through the surface of the sphere, not for the time element~$dt$, +but for a sufficiently large time. If the vibrations are purely +periodic, we may choose for the time a period; if this is not +the case, which for the sake of generality we must here assume, +it is not possible to specify a~priori any more general criterion +for the least possible necessary magnitude of the time than that +which makes the energy radiated essentially independent of the +radius of the supposed sphere. + +In this way we succeed in finding for the energy emitted from +the oscillator in the time from $t$ to $t + \frakT$ the following expression: +\[ +\frac{2}{3c^{3}} \int_{t}^{t + \frakT} \ddot{f}^{2}(t) dt. +\] +If now, the oscillator be in an electromagnetic field which has the +electric component~$\frakE_{z}$ at the oscillator in the direction of its axis, +%-----File: 094.png---\redacted\-------- +then the energy absorbed by the oscillator in the same time is: +\[ +\int_{t}^{t + \frakT} \frakE_{z} \dot{f} · dt. +\] +Hence, the principle of conservation of energy is expressed in +the following form: +\[ +\int_{t}^{t + \frakT} \left(\frac{dU}{dt} + \frac{2}{3c^{3}} \ddot{f}^{2} - \frakE_{z} \dot{f}\right) dt = 0. +\] +This equation, together with the assumption that the constant +\[ +\frac{4\pi^{2} \nu_{0}}{3c^{3} L} = \sigma +\Tag{(42)} +\] +is a small number, leads to the following linear differential equation +for the vibrations of the oscillator: +\[ +Kf + L\ddot{f} - \frac{2}{3c^{3}} \dddot{f} = \frakE_{z}. +\Tag{(43)} +\] +In accordance with what precedes, in so far as the oscillator is +excited into vibrations by an external field~$\frakE_{z}$, one may designate +it as a resonator which possesses the natural period~$\nu_{0}$ and the +small logarithmic decrement~$\sigma$. The same equation may be +obtained from the electron theory, but I have considered it an +advantage to derive it in a manner independent of any hypothesis +concerning the nature of the resonator. + +Now, let the resonator be in a vacuum filled with stationary +black radiation of specific intensity~$\frakK_{\nu}$. How, then, does the +mean energy~$U$ of the resonator in a state of stationary vibration +depend upon the specific intensity of radiation~$\frakK_{\nu_{0}}$ with the natural +period~$\nu_{0}$ of the corresponding color? It is this question which +we have still to consider today. Its answer will be found by expressing +on the one hand the energy of the resonator~$U$ and on +the other hand the intensity of radiation~$\frakK_{\nu_{0}}$ by means of the +component~$\frakE_{z}$ of the electric field exciting the resonator. Now +however complicated this quantity may be, it is capable of +%-----File: 095.png---\redacted\-------- +development in any case for a very large time interval, from +$t = 0$ to $t = \frakT$, in the Fourier's series: +\[ +\frakE_{z} = \sum\limits_{n = 1}^{n = \infty} C_{n} \cos \left(\frac{2\pi n t}{\frakT} - \theta_{n}\right), +\Tag{(44)} +\] +and for this same time interval~$\frakT$ the moment of the resonator +in the form of a Fourier's series may be calculated as a function +of~$t$ from the linear differential equation~\Eq{(43)}. The initial +condition of the resonator may be neglected if we only consider +such times~$t$ as are sufficiently far removed from the origin of +time $t = 0$. + +If it be now recalled that in a stationary state of vibration +the mean energy~$U$ of the resonator is given, in accordance with +\Eq{(40)},~\Eq{(41)} and~\Eq{(42)}, by: +\[ +U = K \bar{f}^{2} = \frac{16\pi^{4} \nu_{0}{}^{3}}{3 \sigma c^{3}} \bar{f}^{2}, +\] +it appears after substitution of the value of~$f$ obtained from the +differential equation~\Eq{(43)} that: +\[ +U = \frac{3 c^{3}}{64\pi^{2} \nu_{0}{}^{2}} \frakT \bar{C}_{n0}{}^{2}, +\Tag{(45)} +\] +wherein $\bar{C}_{n0}{}^{2}$~denotes the mean value of~$C_{n}$ for all the series of +numbers~$n$ which lie in the neighborhood of the value~$\nu_{0} \frakT$, \ie, +for which $\nu_{0} \frakT$~is approximately~$= 1$. + +Now let us consider on the other hand the intensity of black +radiation, and for this purpose proceed from the space density +of the total radiation. In accordance with~\Eq{(30)}, this is: +\[ +\epsilon = \frac{8\pi}{c} \int_{0}^{\infty} \frakK_{\nu} d\nu + = \frac{1}{8\pi} (\bar{\frakE}_{x}{}^{2} + \bar{\frakE}_{y}{}^{2} + \bar{\frakE}_{z}{}^{2} + + \bar{\frakH}_{x}{}^{2} + \bar{\frakH}_{y}{}^{2} + \bar{\frakH}_{z}{}^{2}), +\Tag{(46)} +\] +and therefore, since the radiation is isotropic, in accordance with~\Eq{(44)}: +\[ +\frac{8\pi}{c} \int_{0}^{\infty} \frakK_{\nu} d\nu + = \frac{3}{4\pi} \bar{\frakE}_{z}{}^{2} + = \frac{3}{8\pi} \sum\limits_{n = 1}^{n = \infty} C_{n}{}^{2}. +\] +%-----File: 096.png---\redacted\-------- +If we write $\Delta n/\frakT$ on the left instead of~$d\nu$, where $\Delta n$~is a large +number, we get: +\[ +\frac{8\pi}{c} \sum\limits_{n = 1}^{n = \infty} \frakK_{v} \frac{\Delta n}{\frakT} + = \frac{3}{8\pi} \sum\limits_{n = 1}^{n = \infty} C_{n}{}^{2}, +\] +and obtain then by ``spectral'' division of this equation: +\[ +\frac{8\pi}{c} \frakK_{\nu_{0}} \frac{\Delta n}{\frakT} + = \frac{3}{8\pi} \sum\limits_{n_{0} - (\Delta n/2)}^{n_{0} + (\Delta n/2)} C_{n}{}^{2}, +\] +and, if we introduce again the mean value +\[ +\frac{1}{\Delta n} · \sum\limits_{n_{0} - (\Delta n/2)}^{n_{0} + (\Delta n/2)} C_{n}{}^{2} = \bar{C}_{n0}{}^{2}, +\] +we then get: +\[ +\frakK_{\nu_{0}} = \frac{3 c \frakT}{64\pi^{2}} · \bar{C}_{n 0}. +\] +By comparison with~\Eq{(45)} the relation sought is now found: +\[ +\frakK_{\nu_{0}} = \frac{\nu_{0}{}^{2}}{c^{2}} U, +\Tag{(47)} +\] +which is striking on account of its simplicity and, in particular, +because it is quite independent of the damping constant~$\sigma$ of the +resonator. + +This relation, found in a purely electrodynamic manner, +between the spectral intensity of black radiation and the energy +of a vibrating resonator will furnish us in the next lecture, with +the aid of thermodynamic considerations, the necessary means of +attack in deriving the temperature of black radiation together +with the distribution of energy in the normal spectrum. +%-----File: 097.png---\redacted\-------- + + +\Chapter{Sixth Lecture.}{Heat Radiation. Statistical Theory.}\label{Lect6} + +Following the preparatory considerations of the last lecture +we shall treat today the problem which we have come to recognize +as one of the most important in the theory of heat radiation: +the establishment of that universal function which governs the +energy distribution in the normal spectrum. The means for the +solution of this problem will be furnished us through the calculation +of the entropy~$S$ of a resonator placed in a vacuum filled +with black radiation and thereby excited into stationary vibrations. +Its energy~$U$ is then connected with the corresponding +specific intensity~$\frakK_{\nu}$ and its natural frequency~$\nu$ in the radiation +of the surrounding field through equation~\Eq{(47)}: +\[ +\frakK_{\nu} = \frac{\nu^{2}}{c^{2}} U. +\Tag{(48)} +\] +When $S$~is found as a function of~$U$, the temperature~$T$ of the +resonator and that of the surrounding radiation will be given by: +\[ +\frac{dS}{dU} = \frac{1}{T}, +\Tag{(49)} +\] +and by elimination of~$U$ from the last two equations, we then +find the relationship among $\frakK_{\nu}$,~$T$ and~$\nu$. + +In order to find the entropy~$S$ of the resonator we will utilize +the general connection between entropy and probability, which +we have extensively discussed in the previous lectures, and inquire +then as to the existing probability that the vibrating resonator +possesses the energy~$U$. In accordance with what we have seen +in connection with the elucidation of the second law through +%-----File: 098.png---\redacted\-------- +atomistic ideas, the second law is only applicable to a physical +system when we consider the quantities which determine the +state of the system as mean values of numerous disordered +individual values, and the probability of a state is then equal +to the number of the numerous, a~priori equally probable, complexions +which make possible the realization of the state. Accordingly, +we have to consider the energy~$U$ of a resonator +placed in a stationary field of black radiation as a constant mean +value of many disordered independent individual values, and +this procedure agrees with the fact that every measurement of +the intensity of heat radiation is extended over an enormous +number of vibration periods. The entropy of a resonator is +then to be calculated from the existing probability that the energy +of the radiator possesses a definite mean value~$U$ within a certain +time interval. + +In order to find this probability, we inquire next as to the +existing probability that the resonator at any fixed time possesses +a given energy, or in other words, that that point (the +state point) which through its coordinates indicates the state of +the resonator falls in a given ``state domain.'' At the conclusion +of the third lecture (p.~\pageref{png67lab3}) we saw in general that this probability +is simply measured through the magnitude of the corresponding +state domain: +\[ +\int d\varphi · d\psi, +\] +in case one employs as coordinates of state the general coordinate~$\varphi$ +and the corresponding momentum~$\psi$. Now in general, the +energy of the resonator, in accordance with~\Eq{(40)}, is: +\[ +U = \tfrac{1}{2} Kf^{2} + \tfrac{1}{2} L \dot{f}^{2}. +\] +If we choose $f$ as the general coordinate~$\varphi$ and put, therefore, +$\varphi = f$, then the corresponding impulse~$\psi$ is equal +\[ +\frac{\dd U}{\dd \dot{f}} = L \dot{f}, +\] +%-----File: 099.png---\redacted\-------- +and the energy~$U$ expressed as a function of $\varphi$~and~$\psi$ is: +\[ +U = \tfrac{1}{2} K\varphi^{2} + \frac{1}{2} \frac{\psi^{2}}{L}. +\] +If now we desire to find the existing probability that the energy +of a resonator shall lie between $U$ and $U + \Delta U$, we have to +calculate the magnitude of that state domain in the $(\varphi, \psi)$-plane +which is bounded by the curves $U = \const.$\ and $U + \Delta U = \const.$ +These two curves are similar and similarly placed ellipses and +the portion of surface bounded by them is equal to the difference +of the areas of the two ellipses. The areas are respectively $U/\nu$ +and $(U + \Delta U)/\nu$; consequently, the magnitude sought for the +state domain is:~$\Delta U/\nu.$ Let us now consider the whole state +plane so divided into elementary portions by a large number of +ellipses, such that the annular areas between consecutive ellipses +are equal to each other; \ie, so that: +\[ +\frac{\Delta U}{\nu} = \const = h. +\] +We thus obtain those portions~$\Delta U$ of the energy which correspond +to equal probabilities and which are therefore to be designated +as the energy elements: +\[ +\epsilon = \Delta U = h \nu. +\Tag{(50)} +\] + +If the determination of the elementary domains is effected in +a manner quite similar to that employed in the kinetic gas theory, +there exist, with respect to the relationships there found, very +notable differences. In the first place, the state of the physical +system considered here, the resonator, does not depend as there +upon the coordinates and the velocities, but upon the energy +only, and this circumstance necessitates that the entropy of a +state depend, not upon the distribution of the state quantities +$\varphi$~and~$\psi$, but only upon the energy~$U$. A further difference +consists in this, that we have to do in the case of molecules with +spacial mean values, but in the case of radiation with mean values +%-----File: 100.png---\redacted\-------- +as regards time. But this distinction may be disregarded when +we reflect that the mean time value of the energy~$U$ of a given +resonator is obviously identical with the mean space value at a +given instant of time of a great number~$N$ of similar resonators +distributed in the same stationary field of radiation. Of course +these resonators must be placed sufficiently far apart in order +not directly to influence one another. Then the total energy of +all the resonators: +\[ +U_{N} = NU +\Tag{(51)} +\] +is quite irregularly distributed among all the individual resonators, +and we have referred back the disorder as regards time to a +disorder as regards space. + +We are now concerned with the probability~$W$ of the state +determined by the energy~$U_{N}$ of the $N$~resonators placed in the +same stationary field of radiation; \ie,~with the number of +individual arrangements or complexions which correspond to the +distribution of energy~$U_{N}$ among the $N$~resonators. With this +in view, we subdivide the given total energy~$U_{N}$ into its elements~$\epsilon$ +so that: +\[ +U_{N} = P \epsilon. +\Tag{(52)} +\] +These $P$~energy elements are to be distributed in every possible +manner among the $N$~resonators. Let us consider, then, the +$N$~resonators to be numbered and the figures written beside +one another in a series, and in such manner that the number +of times each figure appears is equal to the number of energy +elements which fall upon the corresponding resonator. Then +we obtain through such a number series a representation of a +fixed complexion, in which with each individual resonator there +is associated a definite energy. For example, if there are $N = 4$ +resonators and $P = 6$ energy elements present, then one of +the possible complexions is represented by the number series +\[ +1\quad 1\quad 3\quad 3\quad 3\quad 4 +\] +which asserts that the first resonator contains two, the second~$0$, +%-----File: 101.png---\redacted\-------- +the third~$3$, and the fourth $1$~energy element. The totality of +numbers in the series is~$6$, equal to the number of the energy +elements present. The arrangement of figures in the series is +immaterial for any complexion, since the mere interchange of +figures does not change the energy of a given resonator. The +number of all the possible different complexions is therefore +equal to the number of possible ``combinations with repetition'' +of $4$~elements with $6$~classes: +\[ +W = \frac{(4 + 6 - 1)!}{(4 - 1)!\;6!} = \frac{9!}{3!\;6!} = 84, +\] +or, in our general case the probability sought is: +\[ +W = \frac{(N + P - 1)!}{(N - 1)!\;P!}. +\] +We obtain, therefore, for the entropy~$S_{N}$ of the resonator system, +in accordance with equation~\Eq{(12)}, since $N$~and~$P$ are large +numbers, +\[ +S_{N} = k \log \frac{(N + P)!}{N!\;P!} +\] +and with the aid of Sterling's formula~\Eq{(16)}: +\[ +S_{N} = k \{(N + P) \log (N + P) - N \log N - P \log P\}. +\] +If, in accordance with~\Eq{(52)}, we now write $U_{N}/\epsilon$ for~$P$, $NU$~for $U_{N}$ +in accordance with~\Eq{(51)}, and $h\nu$~for~$\epsilon$, in accordance with~\Eq{(50)}, +we obtain, after an easy transformation, for the mean entropy +of a single resonator: +\[ +\frac{S_{N}}{N} = S + = k \left\{\left(1 + \frac{U}{h\nu}\right) \log \left(1 + \frac{U}{h\nu}\right) + - \frac{U}{h\nu} \log \frac{U}{h\nu}\right\} +\] +as the solution of the problem in hand. + +We will now introduce the temperature~$T$ of the resonator, +and will express through $T$ the energy~$U$ of the resonator and +also the intensity~$\frakK_{\nu}$ of the heat radiation related to it through a +%-----File: 102.png---\redacted\-------- +stationary state of energy exchange. For this purpose we utilize +equation~\Eq{(49)} and obtain then for the energy of the resonator: +\[ +U = \frac{h\nu}{e^{h\nu/kT} - 1}. +\] +It is to be observed that we have not here to do with a uniform +distribution of energy (cf.\ p.~\pageref{png78lab1}) among the various resonators. + +For the specific intensity of the monochromatic plane polarized +ray of frequency~$\nu$, we have, in accordance with~\Eq{(48)}: +\[ +\frakK_{\nu} = \frac{h\nu^{3}}{c^{2}} · \frac{1}{e^{h\nu/kT} - 1}. +\Tag{(53)} +\] +This expression furnishes for each temperature~$T$ the energy +distribution in the normal spectrum of a black body. A comparison +with equation~\Eq{(38)} of the last lecture furnishes us then +with the universal function: +\[ +F(\nu, T) = \frac{h\nu^{3}}{e^{h\nu/kT} - 1}. +\] + +If we refer the specific intensity of a monochromatic ray, not to +the frequency~$\nu$, but, as is commonly done in experimental physics, +to the wave length~$\lambda$, then, since between the absolute values of +$d\nu$~and~$d\lambda$ the relation exists: +\[ +|d\nu| = \frac{c · |d\lambda|}{\lambda^{2}}, +\] +we obtain from +\[ +E_{\lambda} |d\lambda| = \frakK_{\nu} |d\nu|, +\] +the relation: +\[ +E_{\lambda} = \frac{c^{2}h}{\lambda^{5}} · \frac{1}{e^{ch/k \lambda T} - 1} +\Tag{(54)} +\] +as the intensity of a monochromatic plane polarized ray of wave +length~$\lambda$ is emitted normally to the surface of a black +body in a vacuum at temperature~$T$. For small values of~$\lambda T$ +%-----File: 103.png---\redacted\-------- +\Eq{(54)}~reduces to: +\[ +E_{\lambda} = \frac{c^{2} h}{\lambda^{5}} · e^{-(ch/k\lambda T)}, +\Tag{(55)} +\] +which expresses Wien's Displacement Law. For large values of~$\lambda T$ +on the other hand, there results from~\Eq{(54)}: +\[ +E_{\lambda} = \frac{ckT}{\lambda^{4}}, +\Tag{(56)} +\] +a relation first established by Lord Rayleigh and which we may +here designate as the Rayleigh Law of Radiation. + +From equation~\Eq{(30)}, taking account of~\Eq{(53)}, we obtain for the +space density of black radiation in a \label{png103lab1}vacuum: +\[ +\epsilon = \frac{48\pi h}{c^{3}} \left(\frac{kT}{h}\right)^{4} · \alpha = aT^{4}, +\] +wherein +\[ +\alpha = 1 + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \frac{1}{4^{4}} + \cdots = 1.0823. +\] +The Stefan-Boltzmann law is hereby expressed. In accordance +with the measurements of Kurlbaum, we have the constant +\[ +a = \frac{48\pi k^{4}}{c^{3} h^{3}} · \alpha = 7.061 · 10^{-15} \frac{\erg}{\cm^{3} \deg^{4}}. +\] + +For that wave length~$\lambda_{m}$ which corresponds in the spectrum +of black radiation to the maximum intensity of radiation~$E_{\lambda}$ +we have from equation~\Eq{(54)}: +\[ +\left(\frac{dE_{\lambda}}{d\lambda}\right)_{\lambda = \lambda_{m}} = 0. +\] +Carrying out the differentiation, we get, after putting for brevity: +\[ +\frac{ch}{k\lambda_{m} T} = \beta,\quad +e^{-\beta} + \frac{\beta}{5} - 1 = 0. +\] +The root of this transcendental equation is: +\[ +\beta = 4.9651; +\] +%-----File: 104.png---\redacted\-------- +and $\lambda_{m} T = ch/k\beta = b$ is a constant (Wien's Displacement Law). +In accordance with the measurements of O.~Lummer and E.~Pringsheim, +\[ +b = 0.294\ \cm · \deg. +\] +From this there follow the numerical values +\[ +k = 1.346 · 10^{-16} \frac{\erg}{\deg},\quad \text{and}\quad +h = 6.548 · 10^{-27} \erg · \sec. +\] +The value found for~$k$ easily permits of the specification numerically, +in the C.G.S. system, of the general connection between +entropy and probability, as expressed through the universal +equation~\Eq{(12)}. Thus, quite in general, the entropy of a physical +system is: +\[ +S = 1.346 · 10^{-16} \ln W. +\] + +In the application to the kinetic gas theory we obtain from +equation~\Eq{(24)} for the ratio of the molecular mass to the mol mass: +\label{png104lab1} +\[ +\omega = \frac{k}{R} = 1.62 · 10^{-24}, +\] +\ie, to one mol there corresponds $1/\omega = 6.175 · 10^{23}$ molecules, +where it is supposed that the mol of oxygen +\[ +O_{2} = 32\text{g}. +\] +Accordingly, the number of molecules contained in $1$~cu.~cm.\ of +an ideal gas at $0°$~Cels.\ and at atmospheric pressure is: +\[ +N = 2.76 · 10^{19}. +\] +The mean kinetic energy of the progressive motion of a molecule +at the absolute temperature $T = 1$ in the absolute C.G.S. system, +in accordance with~\Eq{(27)}, is: +\[ +L = \tfrac{3}{2} k = 2.02 · 10^{-16}. +\] +In general, the mean kinetic energy of progressive motion of a +%-----File: 105.png---\redacted\-------- +molecule is expressed by the product of this number and the +absolute temperature~$T$. + +The elementary quantum of electricity, or the free electric +charge of a monovalent ion or electron, in electrostatic measure is: +\[ +e = \omega · 9658 · 3 · 10^{10} = 4.69 · 10^{-10}. +\] +This result stands in noteworthy agreement with the results of +the latest direct measurements of the electric elementary quantum +made by E.~Rutherford and H.~Geiger, and E.~Regener. + +Even if the radiation formula~\Eq{(54)} here derived had shown itself +as valid with respect to all previous tests, the theory would still +require an extension as regards a certain point; for in it the +physical meaning of the universal constant~$h$ remains quite +unexplained. All previous attempts to derive a radiation formula +upon the basis of the known laws of electron theory, among which +the theory of J.~H. Jeans is to be considered as the most general +and exact, have led to the conclusion that $h$~is infinitely small, +so that, therefore, the radiation formula of Rayleigh possesses +general validity, but, in my opinion, there can be no doubt that +this formula loses its validity for short waves, and that the pains +which Jeans has taken to place\footnote + {In that the walls used in the measurements of hollow space radiations + must be diathermanous for the shortest waves.} +the blame for the contradiction +between theory and experiment upon the latter are unwarranted. + +Consequently, there remains only the one conclusion, that +previous electron theories suffer from an essential incompleteness +which demands a modification, but how deeply this modification +should go into the structure of the theory is a question upon +which views are still widely divergent. J.~J. Thompson inclines +to the most radical view, as do J.~Larmor, A.~Einstein, and +with him I.~Stark, who even believe that the propagation of +electromagnetic waves in a pure vacuum does not occur precisely +in accordance with the Maxwellian field equations, but in +definite energy quanta~$h\nu$. I am of the opinion, on the other +hand, that at present it is not necessary to proceed in so revolutionary +%-----File: 106.png---\redacted\-------- +a manner, and that one may come successfully through by +seeking the significance of the energy quantum~$h\nu$ solely in +the mutual actions with which the resonators influence one +another.\footnote + {It is my intention to give a complete presentation of these relations in + Volume~31 of the Annalen der~Physik.} +A definite decision with regard to these important +questions can only be brought about as a result of further +experience. +%-----File: 107.png---\redacted\-------- + + +\Chapter{Seventh Lecture.}{General Dynamics. Principle of Least Action.}\label{Lect7} + +Since I began three weeks ago today to depict for you the +present status of the system of theoretical physics and its +probable future development, I have continually sought to +bring out that in the theoretical physics of the future the most +important and the final division of all physical processes would +likely be into reversible and irreversible processes. In succeeding +lectures, with the aid of the calculus of probability and with the +introduction of the hypothesis of elementary disorder, we have +seen that all irreversible processes may be considered as reversible +elementary processes: in other words, that irreversibility does +not depend upon an elementary property of a physical process, +but rather depends upon the ensemble of numerous disordered +elementary processes of the same kind, each one of which individually +is completely reversible, and upon the introduction +of the macroscopic method of treatment. From this standpoint +one can say quite correctly that in the final analysis all processes +in nature are reversible. That there is herein contained no contradiction +to the principle regarding the irreversibility of processes +expressed in terms of the mean values of elementary processes +of macroscopic changes of state, I have demonstrated fully in +the third lecture. Perhaps it will be appropriate at this place +to interject a more general statement. We are accustomed in +physics to seek the explanation of a natural process by the method +of division of the process into elements. We regard each complicated +process as composed of simple elementary processes, +and seek to analyse it through thinking of the whole as the sum +of the parts. This method, however, presupposes that through +%-----File: 108.png---\redacted\-------- +this division the character of the whole is not changed; in somewhat +similar manner each measurement of a physical process +presupposes that the progress of the phenomena is not influenced +by the introduction of the measuring instrument. We have +here a case in which that supposition is not warranted, and where +a direct conclusion with regard to the parts applied to the whole +leads to quite false results. If we divide an irreversible process +into its elementary constituents, the disorder and along with it +the irreversibility vanishes; an irreversible process must remain +beyond the understanding of anyone who relies upon the fundamental +law: that all properties of the whole must also be recognizable +in the parts. It appears to me as though a similar difficulty +presents itself in most of the problems of intellectual life. + +Now after all the irreversibility in nature thus appears in a +certain sense eliminated, it is an illuminating fact that general +elementary dynamics has only to do with reversible processes. +Therefore we shall occupy ourselves in what follows with reversible +processes exclusively. That which makes this procedure +so valuable for the theory is the circumstance that all known +reversible processes, be they mechanical, electrodynamical or +thermal, may be brought together under a single principle which +answers unambiguously all questions regarding their behavior. +This principle is not that of conservation of energy; this holds, it +is true, for all these processes, but does not determine unambiguously +their behavior; it is the more comprehensive principle +of least action. + +The principle of least action has grown upon the ground of +mechanics where it enjoys equal rank and regard with numerous +other principles; the principle of d'Alembert, the principle of +virtual displacement, Gauss's principle of least constraint, the +Lagrangian Equations of the first and second kind. All these +principles are equivalent to one another and therefore at bottom +are only different formularizations of the same laws; sometimes +one and sometimes another is the most convenient to use. But +the principle of least action has the decided advantage over all +%-----File: 109.png---\redacted\-------- +the other principles mentioned in that it connects together in a +single equation the relations between quantities which possess, +not only for mechanics, but also for electrodynamics and for +thermodynamics, direct significance, namely, the quantities: +space, time and potential. This is the reason why one may +directly apply the principle of least action to processes other +than mechanical, and the result has shown that such applications, +as well in electrodynamics as in thermodynamics, lead to +the appropriate laws holding in these subjects. Since a representation +of a unified system of theoretical physics such as we +have here in mind must lay the chief emphasis upon as general +an interpretation as possible of physical laws, it is self evident +that in our treatment the principle of least action will be called +upon to play the principal rôle. I desire now to show how it is +applied in simple individual cases. + +The general formularization of the principle of least action in +the interpretation given to it by Helmholz is as follows: among +all processes which may carry a certain arbitrarily given physical +system subject to given external actions from a given initial +position into a given final position in a given time, the process +which actually takes place in nature is that which is distinguished +by the condition that the integral +\[ +\int_{t_{0}}^{t_{1}} (\delta H + A) dt = 0, +\Tag{(57)} +\] +wherein an arbitrary displacement of the independent coordinates +(and velocities) is denoted by the sign~$\delta$, and $A$~denotes the +infinitely small increase in energy (external work) which the +system experiences in the displacement~$\delta$. The function~$H$ +is the kinetic potential. When we speak here of the positions, +the coordinates, and the velocities of the configuration, we understand +thereby, not only those special ones corresponding to mechanical +ideas, but also all the so-called generalized coordinates +with the quantities derived therefrom; and these may represent +equally well quantities of electricity, volumes, and the like. +%-----File: 110.png---\redacted\-------- + +In the applications which we shall now make of the principle +of least action, we must first decide as to whether the generalized +coordinates which determine the state of the system considered +are present in finite number or form a continuous infinite +manifold. We shall distinguish the examples here considered +in accordance with this viewpoint. + + +\Section{1.}{The Position (Configuration) is Determined by a Finite Number +of Coordinates.} + +In ordinary mechanics this is actually the case in every system +of a finite number of material points or rigid bodies among whose +coordinates there exist arbitrary fixed equations of condition. +If we call the independent coordinates $\varphi_{1}$,~$\varphi_{2}$,~$\cdots$, then the +external work is: +\[ +A = \Phi_{1} \delta \varphi_{1} + \Phi_{2} \delta \varphi_{2} + \cdots = \delta E, +\Tag{(58)} +\] +wherein $\Phi_{1}$,~$\Phi_{2}$,~$\cdots$ are the ``external force components'' which +correspond to the individual coordinates, and $E$~denotes the +energy of the system. Then the principle of least action is +expressed by: +\label{png110lab1} +\[ +\int_{t_{0}}^{t_{1}} dt · \sum\limits_{1, 2, \cdots} \left( + \frac{\dd H}{\dd \varphi_{1}} \delta \varphi_{1} + + \frac{\dd H}{\dd \dot{\varphi}_{1}} \delta \dot{\varphi}_{1} + + \Phi_{1} \delta \varphi_{1}\right) = 0. +\] +From this follow the equations of motion: +\[ +\Phi_{1} - \frac{d}{dt} \left(\frac{\dd H}{\dd \dot{\varphi}_{1}}\right) + + \frac{\dd H}{\dd \varphi_{1}} = 0, +\Tag{(59)} +\] +and so on for all the indices, $1$,~$2$,~$\cdots$. Through multiplication +of the individual equations by $\dot{\varphi}_{1}$,~$\dot{\varphi}_{2}$,~$\cdots$ addition and integration +with respect to time, there results the equation of conservation +of energy, whereby the energy~$E$ is given by the expression: +\[ +E = \sum\limits_{1, 2, \cdots} \dot{\varphi}_{1} \frac{\dd H}{\dd \dot{\varphi}_{1}} - H. +\Tag{(60)} +\] +In ordinary mechanics $H = L - U$, if $L$~denote the kinetic and +%-----File: 111.png---\redacted\-------- +$U$~the potential energy. Since $L$~is a homogeneous function of +the second degree with respect to the~$\dot{\varphi}$'s, it follows from~\Eq{(60)} +that: +\[ +E = 2L - H = L + U. +\] +But this expression holds by no means in general. + +We pass now to the consideration of the quasi-stationary +motion of a system of linear conductors carrying simple closed +galvanic currents. The state of the system is given by the +position and the velocities of the conductors and by the current +densities in each of the same. The coordinates referring +to the position of the first conductor may be represented by +$\varphi_{1}$,~${\varphi_{1}}'$, ${\varphi_{1}}''$,~$\cdots$, corresponding designations holding for the +remaining conductors. We inquire now as to the increase of +energy or the external work,~$A$, which corresponds to a virtual +displacement of all coordinates. Energy may be conveyed to +the system through mechanical actions and through electromagnetic +induction as well. The former corresponds to mechanical +work, the latter to electromotive work. The former will +be of the familiar form: +\label{png111lab1} +\[ +\Phi_{1} \delta\varphi_{1} + + {\Phi_{1}}' \delta\varphi_{1} + \cdots + + \Phi_{2} \delta\varphi_{2} + \cdots. +\] +If we denote by $E_{1}$,~$E_{2}$,~$\cdots$ the electromotive forces which +are induced in the individual conductors through external +agencies (\eg,~moving magnets which do not belong to the +system), then the electromotive work done from outside upon +the currents in the conductors of the system is: +\[ +E_{1} \delta\epsilon_{1} + E_{2} \delta\epsilon_{2} + \cdots, +\] +if $\delta\epsilon_{1}$,~$\delta\epsilon_{2}$,~$\cdots$ denote the quantities of electricity which pass +through cross sections of the conductors due to infinitely small +virtual currents. The finite current densities will then be denoted +by $\dot{\epsilon}_{1}$,~$\dot{\epsilon}_{2}$,~$\cdots$. The electrical state of the first conductor is +thus determined in general by the current density~$\dot{\epsilon}_{1}$, the +mechanical state (position and velocity) by the coordinates +%-----File: 112.png---\redacted\-------- +$\varphi_{1}$,~${\varphi_{1}}'$, ${\varphi_{1}}''$,~$\cdots$ and the corresponding velocities $\dot{\varphi}_{1}$,~$\dot{\varphi}_{1}'$, $\dot{\varphi}_{1}''$,~$\cdots$. +The coordinates $\epsilon_{1}$,~$\epsilon_{2}$,~$\cdots$ are so-called ``cyclical'' coordinates, +since the state does not depend upon their momentary values, +but only upon their differential quotients with respect to time, +just as, for example, the state of a body rotatable about an axis +of symmetry depends only upon the angular velocity, and not +upon the angle of rotation. The scheme of notation adopted +permits of the direct application of the above formularization +of the principle of least action to the case here considered. +Thus $H = H_{\phi} + H_{\epsilon}$, where $H_{\phi}$, the mechanical potential, depends +only upon the $\varphi$'s~and~$\dot{\varphi}$'s, while the electrokinetic potential~$H_{\epsilon}$ +takes the following form: +\[ +H_{\epsilon} + = \tfrac{1}{2} L_{11} \dot{\epsilon}_{1}{}^{2} + + L_{12} \dot{\epsilon}_{1} \dot{\epsilon}_{2} + + L_{13} \dot{\epsilon}_{1} \dot{\epsilon}_{3} + \cdots + + \tfrac{1}{2} L_{22} \dot{\epsilon}_{2}{}^{2} + \cdots. +\] +The quantities $L_{11}$,~$L_{12}$,~$L_{13}$~$\cdots$ $L_{22}$,~$\cdots$ the coefficients of self +induction and mutual induction depend, however, in a definite +manner upon the coordinates of position $\varphi_{1}$,~${\varphi_{1}}'$, ${\varphi_{1}}''$,~$\cdots$, $\varphi_{2}$,~${\varphi_{2}}'$, +${\varphi_{2}}''$,~$\cdots$. + +In accordance with~\Eq{(59)}, we have for the motion of the first +conductor:\label{png112lab1} +\[ +\Phi_{1} - \frac{d}{dt} \left(\frac{\dd H_{\phi}}{\dd \dot{\varphi}_{1}}\right) + + \frac{\dd H_{\phi}}{\dd \varphi_{1}} + + \frac{\dd H_{\epsilon}}{\dd \varphi_{1}} = 0, +\] +with corresponding equations for ${\varphi_{1}}'$,~${\varphi_{1}}''$,~$\cdots$, and for the electric +current in it: +\[ +E_{1} - \frac{d}{dt} \left(\frac{\dd H_{\epsilon}}{\dd \dot{\epsilon}_{1}}\right) = 0. +\] + +The laws for the mechanical (ponderomotive) actions may be +condensed into the statement that, in addition to the ordinary +force upon the first conductor expressed by~$\Phi_{1}$, there is a mechanical +force +\[ +\frac{\dd H_{\epsilon}}{\dd \varphi_{1}} + = \frac{1}{2} \frac{\dd L_{11}}{\dd \varphi_{1}} \dot{\epsilon}_{1}{}^{2} + + \frac{\dd L_{12}}{\dd \varphi_{1}} \dot{\epsilon}_{1} \dot{\epsilon}_{2} + + \frac{\dd L_{13}}{\dd \varphi_{1}} \dot{\epsilon}_{1} \dot{\epsilon}_{3} + \cdots, +\] +which is composed of an action of the current upon itself (first +term) and of the actions of the remaining currents upon it +(following terms). +%-----File: 113.png---\redacted\-------- + +The laws of electrical action, on the other hand, are expressed +by the statement, that to the external electromotive force~$E_{1}$ +in the first conductor there is added the electromotive force +\label{png113lab1} +\[ +-\frac{d}{dt} \left(\frac{\dd H_{\epsilon}}{\dd \dot{\epsilon}_{1}}\right) + = -\frac{d}{dt} (L_{11} \dot{\epsilon}_{1} + L_{12} \dot{\epsilon}_{2} + L_{13} \dot{\epsilon}_{3} + \cdots) +\] +which likewise is composed of an action of the current upon itself +(self induction) and of the inducing actions of the remaining +currents, and that these two forces compensate each other. + +The galvanic conductance or the galvanic resistance is not +contained in these equations because the corresponding energy, +Joule heat, is produced in an irreversible manner, and irreversible +processes are not represented by the principle of least action. +One can formally include this action, likewise any other irreversible +action, in accordance with the procedure of Helmholz, +by introducing it as an external force, in the present case as +the electromotive force due to the resistance~$w$, which operates +to cause a diminution in the energy of the system. For an +infinitely small element of time, the amount of this energy change +is: +\[ + -(w_{1} \dot{\epsilon}_{1}{}^{2} + + w_{2} \dot{\epsilon}_{2}{}^{2} + + w_{3} \dot{\epsilon}_{3}{}^{2} + \cdots) · dt + = -(w_{1} \dot{\epsilon}_{1} d\epsilon_{1} + + w_{2} \dot{\epsilon}_{2} d\epsilon_{2} + \cdots). +\] +Consequently, since the external work $E_{1} d\epsilon_{1} + E_{2} d\epsilon_{2} + \cdots$ now +includes the Joule heat, the external force components $E_{1}$,~$E_{2}$,~$\cdots$ +in the electromotive equations must be increased by the additional +terms $-w_{1} \dot{\epsilon}_{1}$,~$-w_{2} \dot{\epsilon}_{2}$,~$\cdots$. + +The application of the principle of least action to thermodynamic +processes is of special interest, because the importance +of the question relating to the fixing of the generalized coordinates, +which determine the state of the system, here becomes +prominent. From the standpoint of pure thermodynamics, +the variables which determine the state of a body can certainly +be quite arbitrarily chosen, \eg, in the case of a gas of invariable +constitution any two of the following quantities may be chosen +%-----File: 114.png---\redacted\-------- +as independent variables and all others expressed through them: +volume~$V$, temperature~$T$, pressure~$P$, energy~$E$, entropy~$S$. In +the present case, the matter is quite different. If we inquire, in +order to apply the principle of least action, with regard to the +energy change or the total work~$A$ which will be done upon the +gas from without in an infinitely small virtual displacement, it +may be written in the form: +\[ +A = -p · \delta V + T · \delta S. +\] +$T \delta S$ is the heat added from without, $-p \delta V$~the mechanical work +furnished from without. In order to bring this into agreement +with the general formula for external work~\Eq{(58)}: +\[ +A = \Phi_{1} \delta \varphi_{1} + \Phi_{2} \delta \varphi_{2} +\] +it becomes necessary now to choose $V$~and~$S$ as the generalized +coordinates of state and, therefore, to identify with them the +previously employed quantities $\varphi_{1}$~and~$\varphi_{2}$. Then $-p$~and~$T$ +are the generalized force components $\Phi_{1}$~and~$\Phi_{2}$. Now, since in +thermodynamics every reversible change of state proceeds with +infinite slowness, the velocity components $\dot{V}$~and~$\dot{S}$, and in general +all differential coefficients with respect to time, are to be placed +equal to zero, and the principle of least action~\Eq{(59)} reduces to: +\[ +\Phi + \frac{\dd H}{\dd \varphi} = 0, +\] +and, therefore, in our case: +\[ +-p + \left(\frac{\dd H}{\dd V}\right)_{S} = 0\quad \text{and}\quad + T + \left(\frac{\dd H}{\dd S}\right)_{V} = 0. +\] +Further, in accordance with~\Eq{(60)}: +\[ +E = -H. +\] +Now these equations are actually valid, since they only present +other forms of the relation +\[ +dS = \frac{dE + p dV}{T}. +\] +%-----File: 115.png---\redacted\-------- + +The view here presented is fundamentally that which is given +in the energetics of Mach, Ostwald, Helm, and Wiedeburg. The +generalized coordinates $V$~and~$S$ are in this theory the ``capacity +factors,'' $-p$~and~$T$ the ``intensity factors.''\footnote + {The breaking up of the energy differentials into two factors by the exponents + of energetics is by no means associated with a special property of + energy, but is simply an expression for the elementary law that the differential + of a function~$F(x)$ is equal to the product of the differential~$dx$ by the derivative~$\dot{F}(x)$.} +So long as +one limits himself to an irreversible process, nothing stands in +the way of carrying out this method completely, nor of a generalization +to include chemical processes. + +In opposition to it there is an essentially different method of regarding +thermodynamic processes, which in its complete generality +was first introduced into physics by Helmholtz. In accordance +with this method, one generalized coordinate is~$V$, and the other +is not~$S$, but a certain cyclical coordinate---we shall denote it, +as in the previous example, by~$\epsilon$---which does not appear itself +in the expression for the kinetic potential~$H$ and only appears +through its differential coefficient,~$\dot{\epsilon}$; and this differential coefficient +is the temperature~$T$. Accordingly, $H$~is dependent only +upon $V$~and~$T$. The equation for the total external work, in +accordance with~\Eq{(58)}, is: +\[ +A = -p \delta V + E \delta\epsilon, +\] +and agreement with thermodynamics is obviously found if we +set: +\[ +E \delta\epsilon = T \delta S,\quad \text{and also:}\quad +E d\epsilon = T dS,\quad +E dt = dS. +\] +The equations~\Eq{(59)} for the principle of least action become:\label{png115lab1} +\[ +-p + \left(\frac{\dd H}{\dd V}\right)_{T} = 0\quad \text{and}\quad + E - \frac{d}{dt} \left(\frac{\dd H}{\dd T}\right)_{V} = 0, +\] +or +\[ +d\left(\frac{\dd H}{\dd T}\right)_{V} = E dt = dS, +\] +%-----File: 116.png---\redacted\-------- +or by integration: +\[ +\left(\frac{\dd H}{\dd T}\right)_{V} = S, +\] +to an additive constant, which we may set equal to~$0$. For the +energy there results, in accordance with~\Eq{(60)}: +\[ +E = \dot{\epsilon} \frac{\dd H}{\dd \dot{\epsilon}} - H + = T \left(\frac{\dd H}{\dd T}\right)_{V} - H, +\] +and consequently: +\[ +H = -(E - TS). +\] +$H$~is therefore equal to the negative of the function which +Helmholz has called the ``free energy'' of the system, and the +above equations are known from thermodynamics. + +Furthermore, the method of Helmholz permits of being carried +through consistently, and so long as one limits himself to the +consideration of reversible processes, it is in general quite impossible +to decide in favor of the one method or the other. However, +the method of Helmholz possesses a distinct advantage +over the other which I desire to emphasize here. It lends itself +better to the furtherance of our endeavor toward the unification +of the system of physics. In accordance with the purely energetic +method, the independent variables $V$~and~$S$ have absolutely +nothing to do with each other; heat is a form of energy which is +distinguished in nature from mechanical energy and which in +no way can be referred back to it. In accordance with Helmholz, +heat energy is reduced to motion, and this certainly indicates an +advance which is to be placed, perhaps, upon exactly the same +footing as the advance which is involved in the consideration of +light waves as electromagnetic waves. + +To be sure, the view of Helmholz is not broad enough to include +irreversible processes; with regard to this, as we have earlier +stated in detail, the introduction of the calculus of probability +is necessary in order to throw light on the question. At the +same time, this is also the real reason that the exponents of +%-----File: 117.png---\redacted\-------- +energetics will have nothing to do with the strict observance +of irreversible processes, and they either declare them as doubtful +or ignore them completely. In reality, the facts of the case are +quite the reverse; irreversible processes are the only processes +occurring in nature. Reversible processes form only an ideal +abstraction, which is very valuable for the theory, but which is +never completely realized in nature. + + +\Section{II.}{The Generalized Coordinates of State Form a Continuous +Manifold.} + +The laws of infinitely small motions of perfectly elastic bodies +furnish us with the simplest example. The coordinates of state +are then the displacement components, $\frakv_{x}$,~$\frakv_{y}$,~$\frakv_{z}$, of a material +point from its position of equilibrium $(x, y, z)$, considered as a +function of the coordinates $x$,~$y$,~$z$. The external work is given +by a surface integral: +\[ +A = \int d\sigma (X_{\nu} \delta \frakv_{x} + Y_{\nu} \delta \frakv_{y} + Z_{\nu} \delta \frakv_{z}) +\] +($d\sigma$,~surface element; $\nu$,~inner normal). The kinetic potential +is again given by the difference of the kinetic energy~$L$ and the +potential energy~$U$: +\[ +H = L - U. +\] +The kinetic energy is: +\[ +L = \int \frac{d\tau k}{2} (\dot{\frakv}_{x}^{2} + \dot{\frakv}_{y}^{2} + \dot{\frakv}_{z}^{2}), +\] +wherein $d\tau$~denotes a volume element, $k$~the volume density. +The potential energy~$U$ is likewise a space integral of a homogeneous +quadratic function~$f$ which specifies the potential energy +of a volume element. This depends, as is seen from purely +geometrical considerations, only upon the $6$ ``strain coefficients:'' +\begin{gather*} +\frac{\dd \frakv_{x}}{\dd x} = x_{x},\quad +\frac{\dd \frakv_{y}}{\dd y} = y_{y},\quad +\frac{\dd \frakv_{z}}{\dd z} = z_{z}, \\ +\frac{\dd \frakv_{y}}{\dd z} + \frac{\dd \frakv_{z}}{\dd y} = y_{z} = z_{y},\quad +\frac{\dd \frakv_{z}}{\dd x} + \frac{\dd \frakv_{x}}{\dd z} = z_{x} = x_{z},\quad +\frac{\dd \frakv_{x}}{\dd y} + \frac{\dd \frakv_{y}}{\dd x} = x_{y} = y_{x}. +\end{gather*} +%-----File: 118.png---\redacted\-------- +In general, therefore, the function~$f$ contains $21$~independent +constants, which characterize the whole elastic behavior of the +substance. For isotropic substances these reduce on grounds +of symmetry to~$2$. Substituting these values in the expression +for the principle of least action~\Eq{(57)} we obtain: +\begin{multline*} +\int dt \biggl\{ \int d\tau k (\dot{\frakv}_{x} \delta\dot{\frakv}_{x} + \cdots) + - \int d\tau \left(\frac{\dd f}{\dd x_{x}} \delta x_{x} + + \frac{\dd f}{\dd x_{y}} \delta x_{y} + \cdots\right)\\ + + \int d\sigma (X_{\nu} \delta\frakv_{x} + \cdots) \biggr\} = 0. +\end{multline*} +If we put for brevity: +\begin{align*} +-\frac{\dd f}{\dd x_{x}} &= X_{x}, &-\frac{\dd f}{\dd y_{y}} &= Y_{y}, &-\frac{\dd f}{\dd z_{z}} &= Z_{z},\\ +-\frac{\dd f}{\dd y_{z}} &= Y_{z} = Z_{y}, &-\frac{\dd f}{\dd z_{x}} &= Z_{x} = X_{z}, &-\frac{\dd f}{\dd x_{y}} &= X_{y} = Y_{x}, +\end{align*} +it turns out, as the result of purely mathematical operations in +which the variations $\delta\dot{\frakv}_{x}$,~$\delta\dot{\frakv}_{y}$,~$\cdots$ and likewise the variations +$\delta x_{x}$,~$\delta x_{y}$,~$\cdots$ are reduced through suitable partial integration with +respect to the variations $\delta\frakv_{x}$,~$\delta\frakv_{y}$,~$\cdots$, that the conditions within +the body are expressed by: +\[ +k \ddot{\frakv}_{x} + + \frac{\dd X_{x}}{\dd x} + + \frac{\dd X_{y}}{\dd y} + + \frac{\dd X_{z}}{\dd z} = 0,\ \cdots +\] +and at the surface, by: +\[ +X_{\nu} = X_{x} \cos \nu x + X_{y} \cos \nu y + X_{z} \cos \nu z,\ \cdots +\] +as is known from the theory of elasticity. The mechanical significance +of the quantities $X_{x}$,~$Y_{y}$,~$\cdots$ as surface forces follows +from the surface conditions. + +For the last application of the principle of least action we will +take a special case of electrodynamics, namely, electrodynamic +processes in a homogeneous isotropic non-conductor at rest, \eg, +a vacuum. The treatment is analogous to that carried out in the +foregoing example. The only difference lies in the fact that in +%-----File: 119.png---\redacted\-------- +electrodynamics the dependence of the potential energy~$U$ upon +the generalized coordinate~$\frakv$ is somewhat different than in elastic +phenomena. + +We therefore again put for the external work: +\[ +A = \int d\sigma (X_{\nu} \delta\frakv_{x} + Y_{\nu} \delta\frakv_{y} + Z_{\nu} \delta\frakv_{z}), +\Tag{(61)} +\] +and for the kinetic potential: +\[ +H = L - U, +\] +wherein again: +\[ +L = \int d\tau \frac{k}{2} (\dot{\frakv}_{x}{}^{2} + \dot{\frakv}_{y}{}^{2} + \dot{\frakv}_{z}{}^{2}) + = \int d\tau \frac{k}{2} (\dot{\frakv})^{2}. +\] +On the other hand, we write here: +\[ +U = \int d\tau \frac{h}{2} (\curl \frakv)^{2}. +\] +Through these assumptions the dynamical equations including +the boundary conditions are now completely determined. The +principle of least action~\Eq{(57)} furnishes: +\begin{multline*}\textstyle +\int dt \{ \int d\tau k (\dot{\frakv}_{x} \delta\dot{\frakv}_{x} + \cdots) + - \int d\tau h (\curl_{x} \frakv \delta\curl_{x} \frakv + \cdots)\\ +\textstyle + \int d\sigma (X_{\nu} \delta\frakv_{x} + \cdots) \} = 0. +\end{multline*} +From this follow, in quite an analogous way to that employed +above in the theory of elasticity, first, for the interior of the +non-conductor: +\[ +k \ddot{\frakv}_{x} + = h\left(\frac{\dd \curl_{y} \frakv}{\dd z} + - \frac{\dd \curl_{z} \frakv}{\dd y}\right),\ \cdots +\] +or more briefly +\[ +k \ddot{\frakv} = -h \curl \curl \frakv, +\Tag{(62)} +\] +and secondly, for the surface: +\[ +X_{\nu} = h(\curl_{z} \frakv · \cos \nu y - \curl_{y} \frakv · \cos \nu z),\ \cdots +\Tag{(63)} +\] +These equations are identical with the known electrodynamical +equations, if we identify $L$~with the electric, and $U$~with the +%-----File: 120.png---\redacted\-------- +magnetic energy (or conversely). If we put +\[ +L = \frac{1}{8\pi} \int d\tau · \epsilon \frakE^{2} \quad\text{and}\quad +U = \frac{1}{8\pi} \int d\tau · \mu \frakH^{2}, +\] +($\frakE$~and~$\frakH$, the field strengths, $\epsilon$,~the dielectric constant, $\mu$,~the +permeability) and compare these values with the above expressions +for $L$~and~$U$ we may write: +\[ +\dot{\frakv} = -\frakE · \sqrt{\frac{\epsilon}{4\pi k}},\quad +\curl \frakv = \frakH \sqrt{\frac{\mu}{4\pi h}}. +\Tag{(64)} +\] +It follows then, by elimination of~$\frakv$, that: +\[ +\dot{\frakH} = -\sqrt{\frac{\epsilon h}{\mu k}} · \curl \frakE, +\] +and further, by substitution of $\dot{\frakv}$~and~$\curl \frakv$ in equation~\Eq{(62)} found +above for the interior of the non-conductor, that: +\[ +\dot{\frakE} = \sqrt{\frac{\mu h}{\epsilon k}} \curl \frakH. +\] +Comparison with the known electrodynamical equations expressed +in Gaussian units: +\[ +\mu \dot{\frakH} = -c \curl \frakE,\quad +\epsilon \dot{\frakE} = c \curl \frakH +\] +($c$,~velocity of light in vacuum) results in a complete agreement, +if we put: +\[ +\frac{c}{\mu} = \sqrt{\frac{\epsilon h}{\mu k}} \quad\text{and}\quad +\frac{c}{\epsilon} = \sqrt{\frac{\mu h}{\epsilon k}}. +\] +From either of these two equations it follows that: +\[ +\frac{h}{k} = \frac{c^{2}}{\epsilon \mu}, +\] +the square of the velocity of propagation. + +We obtain from~\Eq{(61)} for the energy entering the system from +without: +\[ +\textstyle dt · \int d\sigma (X_{\nu} \dot{\frakv}_{x} + Y_{\nu} \dot{\frakv}_{y} + Z_{\nu} \dot{\frakv}_{z}), +\] +%-----File: 121.png---\redacted\-------- +or, taking account of the surface equation~\Eq{(63)}: +\[ +\textstyle dt · \int d\sigma h \{(\curl_{z} \frakv \cos \nu y - \curl_{y} \frakv \cos \nu z) \dot{\frakv}_{x} + \cdots\}, +\] +an expression which, upon substitution of the values of $\dot{\frakv}$ and~$\curl \frakv$ +from~\Eq{(64)}, turns out to be identical with the Poynting energy +current. + +We have thus by an application of the principle of least action +with a suitably chosen expression for the kinetic potential~$H$ +arrived at the known Maxwellian field equations. + +Are, then, the electromagnetic processes thus referred back to +mechanical processes? By no means; for the vector~$\frakv$ employed +here is certainly not a mechanical quantity. It is moreover not +possible in general to interpret~$\frakv$ as a mechanical quantity, for +instance, $\frakv$~as a displacement, $\dot{\frakv}$~as a velocity, $\curl \frakv$~as a rotation. +Thus, \eg, in an electrostatic field $\dot{\frakv}$~is constant. Therefore, +$\frakv$~increases with the time beyond all limits, and $\curl \frakv$~can +no longer signify a rotation.\label{png121lab1}\footnote + {With regard to the impossibility of interpreting electrodynamic processes + in terms of the motions of a continuous medium, cf.\ particularly, H.~Witte: + ``Über den gegenwärtigen Stand der Frage nach einer mechanischen Erklärung + der elektrischen Erscheinungen'' Berlin, 1906 (E.~Ebering).} +While from these considerations +the possibility of a mechanical explanation of electrical phenomena +is not proven, it does appear, on the other hand, to be undoubtedly +true that the significance of the principle of least +action may be essentially extended beyond ordinary mechanics +and that this principle can therefore also be utilized as the +foundation for general dynamics, since it governs all known reversible +processes. +%-----File: 122.png---\redacted\-------- + + +\Chapter{Eighth Lecture.}{General Dynamics. Principle of Relativity.}\label{Lect8} + +In the lecture of yesterday we saw, by means of examples, +that all continuous reversible processes of nature may be represented +as consequences of the principle of least action, and +that the whole course of such a process is uniquely determined +as soon as we know, besides the actions which are exerted upon +the system from without, the kinetic potential~$H$ as a function +of the generalized coordinates and their differential coefficients +with respect to time. The determination of this function +remains then as a special problem, and we recognize here a +rich field for further theories and hypotheses. It is my purpose +to discuss with you today an hypothesis which represents a magnificent +attempt to establish quite generally the dependency of +the kinetic potential~$H$ upon the velocities, and which is commonly +designated as the principle of relativity. The gist of this principle +is: it is in no wise possible to detect the motion of a +body relative to empty space; in fact, there is absolutely +no physical sense in speaking of such a motion. If, therefore, +two observers move with uniform but different velocities, then +each of the two with exactly the same right may assert that with +respect to empty space he is at rest, and there are no physical +methods of measurement enabling us to decide in favor of the one +or the other. The principle of relativity in its generalized form +is a very recent development. The preparatory steps were taken +by H.~A. Lorentz, it was first generally formulated by A.~Einstein, +and was developed into a finished mathematical system by +H.~Minkowski. However, traces of it extend quite far back +into the past, and therefore it seems desirable first to say something +concerning the history of its development. +%-----File: 123.png---\redacted\-------- + +The principle of relativity has been recognized in mechanics +since the time of Galilee and Newton. It is contained in the +form of the simple equations of motion of a material point, since +these contain only the acceleration and not the velocity of +the point. If, therefore, we refer the motion of the point, +first to the coordinates $x$,~$y$,~$z$, and again to the coordinates +$x'$,~$y'$,~$z'$ of a second system, whose axes are directed parallel +to the first and which moves with the velocity~$\nu$ in the direction +of the positive $x$-axis: +\[ +x' = x - \nu t,\quad y' = y,\quad z' = z, +\Tag{(65)} +\] +and the form of the equations of motion is not changed in the +slightest. Nothing short of the assumption of the general validity +of the relativity principle in mechanics can justify the inclusion +by physics of the Copernican cosmical system, since through +it the independence of all processes upon the earth of the progressive +motion of the earth is secured. If one were obliged to take +account of this motion, I should have, \eg, to admit that the piece +of chalk in my hand possesses an enormous kinetic energy, corresponding +to a velocity of something like $30$~kilometers per~second. + +It was without doubt his conviction of the absolute validity +of the principle of relativity which guided Heinrich Hertz +in the establishment of his fundamental equations for the electrodynamics +of moving bodies. The electrodynamics of Hertz +is, in fact, wholly built upon the principle of relativity. It recognizes +no absolute motion with regard to empty space. It speaks +only of motions of material bodies relative to one another. In +accordance with the theory of Hertz, all electrodynamic processes +occur in material bodies; if these move, then the electrodynamic +processes occurring therein move with them. To speak +of an independent state of motion of a medium outside of material +bodies, such as the ether, has just as little sense in the theory of +Hertz as in the modern theory of relativity. + +\label{png123lab1}\pngcent{illo124.png}{1350} + +But the theory of Hertz has led to various contradictions with +experience. I will refer here to the most important of these. +%-----File: 124.png---\redacted\-------- +Fizeau brought (1851) into parallelism a bundle of rays originating +in a light source~$L$ by means of a lens and then brought it +to a focus by means of a second lens upon a screen~$S$ (Fig.~2). +In the path of the parallel light rays between the two lenses he +placed a tube system of such sort that a transparent liquid could +be passed through it, and in such manner that in one half (the +upper) the light rays would pass in the direction of flow of the +liquid while in the other half (the lower), the rays would pass in +the opposite direction. + +If now a liquid or a gas flow through the tube system with the +velocity~$\nu$, then, in accordance with the theory of Hertz, since +light must be a process in the substance, the light waves must +be transported with the velocity of the liquid. The velocity +of light relative to $L$ and $S$ is, therefore, in the upper part +$q_{0} + \nu$, and the lower part $q_{0} - \nu$, if $q_{0}$~denote the velocity +of light relative to the liquid. The difference of these two +velocities,~$2\nu$, should be observable at~$S$ through corresponding +interference of the lower and the upper light rays, and quite independently +of the nature of the flowing substance. Experiment +did not confirm this conclusion. Moreover, it showed in gases +generally no trace of the expected action; \ie,~light is propagated +in a flowing gas in the same manner as in a gas at rest. On the +other hand, in the case of liquids an effect was certainly indicated, +%-----File: 125.png---\redacted\-------- +but notably smaller in amount than that demanded by the theory +of Hertz. Instead of the expected velocity difference~$2\nu$, the +difference $2\nu(1 - 1/n^{2})$ only was observed, where $n$~is the refractive +index of the liquid. The factor $(1 - 1/n^{2})$ is called +the Fresnel coefficient. There is contained (for $n = 1$) in this +expression the result obtained in the case of gases. + +It follows from the experiment of Fizeau that, as regards +electrodynamic processes in a gas, the motion of the gas is +practically immaterial. If, therefore, one holds that electrodynamic +processes require for their propagation a substantial +carrier, a special medium, then it must be concluded that this +medium, the ether, remains at rest when the gas moves in an arbitrary +manner. This interpretation forms the basis of the electrodynamics +of Lorentz, involving an absolutely quiescent ether. +In accordance with this theory, electrodynamic phenomena have +only indirectly to do with the motion of matter. Primarily all +electrodynamical actions are propagated in ether at rest. Matter +influences the propagation only in a secondary way, so far as it +is the cause of exciting in greater or less degree resonant vibrations +in its smallest parts by means of the electrodynamic waves +passing through it. Now, since the refractive properties of substances +are also influenced through the resonant vibrations of its +smallest particles, there results from this theory a definite connection +between the refractive index and the coefficient of Fresnel, +and this connection is, as calculation shows, exactly that demanded +by measurements. So far, therefore, the theory of +Lorentz is confirmed through experience, and the principle of +relativity is divested of its general significance. + +The principle of relativity was immediately confronted by +a new difficulty. The theory of a quiescent ether admits the +idea of an absolute velocity of a body, namely the velocity +relative to the ether. Therefore, in accordance with this theory, +of two observers $A$~and~$B$ who are in empty space and who +move relatively to each other with the uniform velocity~$\nu$, it would +be at best possible for only one rightly to assert that he is at +%-----File: 126.png---\redacted\-------- +rest relative to the ether. If we assume, \eg, that at the moment +at which the two observers meet an instantaneous optical signal, +a flash, is made by each, then an infinitely thin spherical wave +spreads out from the place of its origin in all directions through +empty space. If, therefore, the observer~$A$ remain at the center +of the sphere, the observer~$B$ will not remain at the center and, +as judged by the observer~$B$, the light in his own direction of +motion must travel (with the velocity $c - \nu$) more slowly than +in the opposite direction (with the velocity $c + \nu$), or than in a +perpendicular direction (with the velocity $\sqrt{c^{2} - \nu^{2}}$) (cf.\ Fig.~3). +Under suitable conditions the observer~$B$ should be able to +detect and measure this sort of effect. + +\label{png126lab1}\pngcent{illo126.png}{914} + +This elementary consideration led to the celebrated attempt +of Michelson to measure the motion of the earth relative to the +ether. A parallel beam of rays proceeding from~$L$ (Fig.~4) +falls upon a transparent plane parallel plate~$P$ inclined at~$45°$, +by which it is in part transmitted and in part reflected. The +transmitted and reflected beams are brought into interference +by reflection from suitable metallic mirrors $S_{1}$~and~$S_{2}$, which are +removed by the same distance~$l$ from~$P$. If, now, the earth with +the whole apparatus moves in the direction~$PS_{1}$ with the velocity~$\nu$, +then the time which the light needs in order to go from $P$ to +$S_{1}$ and back is: +%-----File: 127.png---\redacted\-------- +\[ +\frac{l}{c - \nu} + \frac{l}{c + \nu} + = \frac{2l}{c} \left(1 + \frac{\nu^{2}}{c^{2}} + \cdots\right). +\] +On the other hand, the time which the light needs in order to pass +from $P$ to $S_{2}$ and back to~$P$ is: +\[ +\frac{l}{\sqrt{c^{2} - \nu^{2}}} + \frac{l}{\sqrt{c^{2} - \nu^{2}}} + = \frac{2l}{c} \left(1 + \frac{1}{2} \frac{\nu^{2}}{c^{2}} + \cdots\right). +\] +If, now, the whole apparatus be turned through a right angle, a +noticeable displacement of the interference bands should result, +since the time for the passage over the path~$PS_{2}$ is now longer. +No trace was observed of the marked effect to be expected. + +\pngcent{illo127.png}{1002} + +Now, how will it be possible to bring into line this result, +established by repeated tests with all the facilities of modern +experimental art? E.~Cohn has attempted to find the necessary +compensation in a certain influence of the air in which +the rays are propagated. But for anyone who bears in mind the +great results of the atomic theory of dispersion and who does +not renounce the simple explanation which this theory gives for +the dependence of the refractive index upon the color, without +introducing something else in its place, the idea that a moving +%-----File: 128.png---\redacted\-------- +absolutely transparent medium, whose refractive index is absolutely~$= 1$, +shall yet have a notable influence upon the velocity +of propagation of light, as the theory of Cohn demands, is not +possible of assumption. For this theory distinguishes essentially +a transparent medium, whose refractive index is~$= 1$, from a +perfect vacuum. For the former the velocity of propagation of +light in the direction of the velocity~$\nu$ of the medium with relation +to an observer at rest is +\[ +q = c + \frac{\nu^{2}}{c}, +\] +for a vacuum, on the other hand, $q = c$. In the former medium, +Cohn's theory of the Michelson experiment predicts no effect, +but, on the other hand, the Michelson experiment should give +a positive effect in a vacuum. + +In opposition to E.~Cohn, H.~A. Lorentz and FitzGerald +ascribe the necessary compensation to a contraction of the whole +optical apparatus in the direction of the earth's motion of the +order of magnitude~$\nu^{2}/c^{2}$. This assumption allows better of the +introduction again of the principle of relativity, but it can first +completely satisfy this principle when it appears, not as a necessary +hypothesis made to fit the present special case, but as a +consequence of a much more general postulate. We have to +thank A.~Einstein for the framing of this postulate and H.~Minkowski +for its further mathematical development. + +Above all, the general principle of relativity demands the +renunciation of the assumption which led H.~A. Lorentz to the +framing of his theory of a quiescent ether; the assumption +of a substantial carrier of electromagnetic waves. For, when +such a carrier is present, one must assume a definite velocity of a +ponderable body as definable with respect to it, and this is exactly +that which is excluded by the relativity principle. Thus the +ether drops out of the theory and with it the possibility of +mechanical explanation of electrodynamic processes, \ie, of referring +them to motions. The latter difficulty, however, does +%-----File: 129.png---\redacted\-------- +not signify here so much, since it was already known before, +that no mechanical theory founded upon the continuous motions +of the ether permits of being completely carried through (cf.\ p.~\pageref{png121lab1}). +In place of the so-called free ether there is now substituted +the absolute vacuum, in which electromagnetic energy is independently +propagated, like ponderable atoms. I believe it follows +as a consequence that no physical properties can be consistently +ascribed to the absolute vacuum. The dielectric constant and the +magnetic permeability of a vacuum have no absolute meaning, +only relative. If an electrodynamic process were to occur in a +ponderable medium as in a vacuum, then it would have absolutely +no sense to distinguish between field strength and induction. +In fact, one can ascribe to the vacuum any arbitrary value of the +dielectric constant, as is indicated by the various systems of +units. But how is it now with regard to the velocity of propagation +of light? This also is not to be regarded as a property of +the vacuum, but as a property of electromagnetic energy which +is present in the vacuum. Where there is no energy there can +exist no velocity of propagation. + +With the complete elimination of the ether, the opportunity is +now pre\-sent for the framing of the principle of relativity. Obviously, +we must, as a simple consideration shows, introduce +something radically new. In order that the moving observer~$B$ +mentioned above (Fig.~3, p.~\pageref{png126lab1}) shall not see the light +signal given by him travelling more slowly in his own direction +of motion (with the velocity $c - \nu$) than in the opposite direction +(with the velocity $c + \nu$), it is necessary that he shall not identify +the instant of time at which the light has covered the distance +$c - \nu$ in the direction of his own motion with the instant of time at +which the light has covered the distance $c + \nu$ in the opposite +direction, but that he regard the latter instant of time as later. +In other words: the observer~$B$ measures time differently from +the observer~$A$. This is a~priori quite permissible; for the +relativity principle only demands that neither of the two observers +shall come into contradiction with himself. However, the +%-----File: 130.png---\redacted\-------- +possibility is left open that the specifications of time of both +observers may be mutually contradictory. + +It need scarcely be emphasized that this new conception of the +idea of time makes the most serious demands upon the capacity +of abstraction and the projective power of the physicist. It +surpasses in boldness everything previously suggested in speculative +natural phenomena and even in the philosophical theories +of knowledge: non-euclidean geometry is child's play in comparison. +And, moreover, the principle of relativity, unlike non-euclidean +geometry, which only comes seriously into consideration +in pure mathematics, undoubtedly possesses a real physical +significance. The revolution introduced by this principle into +the physical conceptions of the world is only to be compared in +extent and depth with that brought about by the introduction +of the Copernican system of the universe. + +Since it is difficult, on account of our habitual notions concerning +the idea of absolute time, to protect ourselves, without +special carefully considered rules, against logical mistakes in the +necessary processes of thought, we shall adopt the mathematical +method of treatment. Let us consider then an electrodynamic +process in a pure vacuum; first, from the standpoint of an observer~$A$; +secondly, from the standpoint of an observer~$B$, who +moves relatively to observer~$A$ with a velocity~$\nu$ in the direction +of the $x$-axis. Then, if $A$~employ the system of reference $x$,~$y$,~$z$,~$t$, +and $B$~the system of reference $x'$,~$y'$,~$z'$,~$t'$, our first problem is to +find the relations among the primed and the unprimed quantities. +Above all, it is to be noticed that since both systems of reference, +the primed and the unprimed, are to be like directed, the equations +of transformation between corresponding quantities in the +two systems must be so established that it is possible through +a transformation of exactly the same kind to pass from the first +system to the second, and conversely, from the second back to +the first system. It follows immediately from this that the velocity +of light~$c'$ in a vacuum for the observer~$B$ is exactly the same +as for the observer~$A$. Thus, if $c'$~and~$c$ are different, $c' > c$, +%-----File: 131.png---\redacted\-------- +say, it would follow that: if one passes from one observer~$A$ to +another observer~$B$ who moves with respect to~$A$ with uniform +velocity, then he would find the velocity of propagation of light +for~$B$ greater than for~$A$. This conclusion must likewise hold +quite in general independently of the direction in which $B$ moves +with respect to~$A$, because all directions in space are equivalent +for the observer~$A$. On the same grounds, in passing from~$B$ to~$A$, +$c$~must be greater than~$c'$, for all directions in space for the +observer~$B$ are now equivalent. Since the two inequalities contradict, +therefore $c'$~must be equal to~$c$. Of course this important +result may be generalized immediately, so that the totality +of the quantities independent of the motion, such as the +velocity of light in a vacuum, the constant of gravitation +between two bodies at rest, every isolated electric charge, \label{png131lab1}and +the entropy of any physical system possess the same values for +both observers. On the other hand, this law does not hold for +quantities such as energy, volume, temperature,~etc. For these +quantities depend also upon the velocity, and a body which is +at rest for~$A$ is for~$B$ a moving body. + +We inquire now with regard to the form of the equations +of transformation between the unprimed and the primed coordinates. +For this purpose let us consider, returning to the +previous example, the propagation, as it appears to the two +observers $A$~and~$B$, of an instantaneous signal creating an infinitely +thin light wave which, at the instant at which the observers +meet, begins to spread out from the common origin of +coordinates. For the observer~$A$ the wave travels out as a +spherical wave: +\[ +x^{2} + y^{2} + z^{2} - c^{2}t^{2} = 0. +\Tag{(66)} +\] +For the second observer~$B$ the same wave also travels as a +spherical wave with the same velocity: +\[ +x'^{2} + y'^{2} + z'^{2} - c^{2}t'^{2} = 0; +\Tag{(67)} +\] +for the first observer has no advantage over the second observer. +%-----File: 132.png---\redacted\-------- +$B$~can exactly with the same right as~$A$ assert that he is at rest +at the center of the spherical wave, and for~$B$, after unit time, the +wave appears as in Fig.~5, while its appearance for the observer~$A$ +after unit time, is represented by Fig.~3 (p.~\pageref{png126lab1}).\footnote + {The circumstance that the signal is a finite one, however small the time + may be, has significance only as regards the thickness of the spherical layer + and not for the conclusions here under consideration.} + +\pngcent{illo132.png}{944} + +The equations of transformation must therefore fulfill the +condition that the two last equations, which represent the same +physical process, are compatible with each other; and furthermore: +the passage from the unprimed to the primed quantities +must in no wise be distinguished from the reverse passage from +the primed to the unprimed quantities. In order to satisfy +these conditions, we generalize the equations of transformation~\Eq{(65)}, +set up at the beginning of this lecture for the old mechanical +principle of relativity, in the following manner: +\[ +x' = \kappa(x - \nu t),\quad y' = \lambda y,\quad z' = \mu z,\quad t' = \nu t + \rho x. +\] +Here $\nu$~denotes, as formerly, the velocity of the observer~$B$ relative +to~$A$ and the constants $\kappa$,~$\lambda$, $\mu$, $\nu$,~$\rho$ are yet to be determined. We +must have: +\[ +x = \kappa' (x' - \nu' t'),\quad y = \lambda' y',\quad z = \mu' z',\quad t = \nu' t' + \rho' x'. +\] +It is now easy to see that $\lambda$~and~$\lambda'$ must both~$= 1$. For, if, \eg, +%-----File: 133.png---\redacted\-------- +$\lambda$~be greater than~$1$, then $\lambda'$~must also be greater than~$1$; for the +two transformations are equivalent with regard to the $y$~axis. +In particular, it is impossible that $\lambda$~and~$\lambda'$ depend upon the +direction of motion of the other observer. But now, since, in +accordance with what precedes, $\lambda = 1/\lambda'$, each of the two +inequalities contradict and therefore $\lambda = \lambda' = 1$; likewise, +$\mu = \mu' = 1$. The condition for identity of the two spherical +waves then demands that the expression~\Eq{(66)}: +\[ +x^{2} + y^{2} + z^{2} - c^{2}t^{2} +\] +become, through the transformation of coordinates, identical with +the expression~\Eq{(67)}: +\[ +x'^{2} + y'^{2} + z'^{2} - c^{2}t'^{2}, +\] +and from this the equations of transformation follow without +ambiguity: +\[ +x' = \kappa (x - \nu t),\quad y' = y,\quad z' = z,\quad t' = \kappa \left(t - \frac{\nu}{c^{2}} x\right), +\Tag{(68)} +\] +wherein +\[ +\kappa = \frac{c}{\sqrt{c^{2} - \nu^{2}}}. +\] +Conversely:\label{png133lab1} +\[ +x = \kappa (x' + \nu t'),\quad y = y',\quad z = z',\quad t = \kappa \left(t' + \frac{\nu}{c^{2}} x'\right). +\Tag{(69)} +\] + +These equations permit quite in general of the passage from the +system of reference of one observer to that of the other (H.~A. +Lorentz), and the principle of relativity asserts that all processes +in nature occur in accordance with the same laws and with the +same constants for both observers (A.~Einstein). Mathematically +considered, the equations of transformation correspond to +a rotation in the four dimensional system of reference $(x, y, z, ict)$ +through the imaginary angle $\arctg (i(\nu/c))$ (H.~Minkowski). +Accordingly, the principle of relativity simply teaches that there +is in the four dimensional system of space and time no special +characteristic direction, and any doubts concerning the general +%-----File: 134.png---\redacted\-------- +validity of the principle are of exactly the same kind as those +concerning the existence of the antipodians upon the other side +of the earth. + +We will first make some applications of the principle of +relativity to processes which we have already treated above. +That the result of the Michelson experiment is in agreement +with the principle of relativity, is immediately evident; for, in +accordance with the relativity principle, the influence of a +uniform motion of the earth upon processes on the earth can +under no conditions be detected. + +We consider now the Fizeau experiment with the flowing +liquid (see p.~\pageref{png123lab1}). If the velocity of propagation of light in +the liquid at rest be again~$q_{0}$, then, in accordance with the +relativity principle, $q_{0}$~is also the velocity of the propagation +of light in the flowing liquid for an observer who moves with +the liquid, in case we disregard the dispersion of the liquid; +for the color of the light is different for the moving observer. If +we call this observer~$B$ and the velocity of the liquid as above,~$\nu$, +we may employ immediately the above formulae in the calculation +of the velocity of propagation of light in the flowing +liquid, judged by an observer~$A$ at the screen~$S$. We have only +to put +\[ +\frac{dx'}{dt'} = x' = q_{0}, +\] +to seek the corresponding value of +\[ +\frac{dx}{dt} = \dot{x}. +\] +For this obviously gives the velocity sought. + +Now it follows directly from the equations of transformation~\Eq{(69)} +that: +\[ +\frac{dx}{dt} = \dot{x} = \frac{\dot{x}' + \nu}{1 + \dfrac{\nu \dot{x}'}{c^{2}}}, +\] +%-----File: 135.png---\redacted\-------- +and, therefore, through appropriate substitution, the velocity +sought in the upper tube, after neglecting higher powers in $\nu/c$ +and~$\nu/q_{0}$, is: +\[ +\dot{x} = \frac{q_{0} + \nu}{1 + \dfrac{\nu q_{0}}{c^{2}}} + = q_{0} + \nu \left(1 - \frac{q_{0}^2}{c^{2}}\right), +\] +and the corresponding velocity in the lower tube is: +\[ +q_{0} - \nu \left(1 - \frac{q_{0}^{2}}{c^{2}}\right). +\] +The difference of the two velocities is +\[ +2\nu \left(1 - \frac{q_{0}^{2}}{c^{2}}\right) = 2\nu \left(1 - \frac{1}{n^{2}}\right), +\] +which is the Fresnel coefficient, in agreement with the measurements +of Fizeau. + +The significance of the principle of relativity extends, not only +to optical and other electrodynamic phenomena, but also to +all processes of ordinary mechanics; but the familiar expression~($\frac{1}{2} mq^{2}$) +for the kinetic energy of a mass point moving with +the velocity~$q$ is incompatible with this principle. + +But, on the other hand, since all mechanics as well as the +rest of physics is governed by the principle of least action, the +significance of the relativity principle extends at bottom only to +the particular form which it prescribes for the kinetic potential~$H$, +and this form, though I will not stop to prove it, is characterized +by the simple law that the expression +\[ +H · dt +\] +for every space element of a physical system is an invariant +\[ += H' · dt' +\] +with respect to the passage from one observer~$A$ to the other +%-----File: 136.png---\redacted\-------- +observer~$B$ or, what is the same thing, the expression $H/\sqrt{c^{2} - q^{2}}$ +is in this passage an invariant $= H'/\sqrt{c^{2} - q'^{2}}$. + +Let us now make some applications of this very general law, +first to the dynamics of a single mass point in a vacuum, whose +state is determined by its velocity~$q$. Let us call the kinetic +potential of the mass point for $q = 0$, $H_{0}$, and consider now the +point at an instant when its velocity is~$q$. For an observer~$B$ +who moves with the velocity~$q$ with respect to the observer~$A$, +$q' = 0$ at this instant, and therefore $H' = H_{0}$. But now +since in general: +\[ +\frac{H}{\sqrt{c^{2} - q^{2}}} = \frac{H'}{\sqrt{c^{2} - q'^{2}}}, +\] +we have after substitution: +\[ +H = \sqrt{1 - \frac{q^{2}}{c^{2}}} · H_{0} + = \sqrt{1 - \frac{\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2}}{c^{2}}} · H_{0}. +\] +With this value of~$H$, the Lagrangian equations of motion~\Eq{(59)} +of the previous lecture are applicable. + +In accordance with~\Eq{(60)}, the kinetic energy of the mass point +amounts to: +\[ +E = \dot{x} \frac{\dd H}{\dd \dot{x}} + + \dot{y} \frac{\dd H}{\dd \dot{y}} + + \dot{z} \frac{\dd H}{\dd \dot{z}} - H + = q \frac{\dd H}{\dd q} - H + = - \frac{H_{0}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}}, +\] +and the momentum to: +\[ +G = \frac{\dd H}{\dd q} = -\frac{q H_{0}}{c \sqrt{c^{2} - q^{2}}}. +\] +$G/q$~is called the transverse mass~$m_{t}$, and $dG/dq$~the longitudinal +mass~$m_{l}$ of the point; accordingly: +\[ +m_{t} = -\frac{H_{0}}{c \sqrt{c^{2} - q^{2}}}, \quad +m_{l} = -\frac{c H_{0}}{(c^{2} - q^{2})^{3/2}}. +\] +%-----File: 137.png---\redacted\-------- +For $q = 0$, we have +\[ +m_{t} = m_{l} = m_{0} = -\frac{H_{0}}{c^{2}}. +\] +It is apparent, if one replaces in the above expressions the constant~$H_{0}$ +by the constant~$m_{0}$, that the momentum is: +\[ +G = \frac{m_{0}q}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}} +\] +and the transverse mass: +\[ +m_{t} = \frac{m_{0}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}}, +\] +and the longitudinal mass: +\[ +m_{l} = \frac{m_{0}}{\left(1 - \dfrac{q^{2}}{c^{2}}\right)^{3/2}}, +\] +and, finally, that the kinetic energy is: +\[ +E = \frac{m_{0} c^{2}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}} + = m_{0}c^{2} + \tfrac{1}{2} m_{0}q^{2} + \cdots. +\] +The familiar value of ordinary mechanics~$\frac{1}{2} m_{0}q^{2}$ appears here +therefore only as an approximate value. These equations have +been experimentally tested and confirmed through the measurements +of A.~H.~Bucherer and of E.~Hupka upon the magnetic +deflection of electrons. + +A further example of the invariance of $H · dt$ will be taken from +electrodynamics. Let us consider in any given medium any +electromagnetic field. For any volume element~$V$ of the medium, +the law holds that $V · dt$~is invariant in the passage from the one +to the other observer. It follows from this that $H/V$~is invariant; +%-----File: 138.png---\redacted\-------- +\ie, the kinetic potential of a unit volume or the ``\textit{space density +of kinetic potential}'' is invariant. + +Hence the following relation exists; +\[ +\frakE \frakD - \frakH \frakB = \frakE' \frakD' - \frakH' \frakB', +\] +wherein $\frakE$~and~$\frakH$ denote the field strengths and $\frakD$~and~$\frakB$ the +corresponding inductions. Obviously a corresponding law for +the space energy density $\frakE \frakD + \frakH \frakB$ will not hold. + +A third example is selected from thermodynamics. If we +take the velocity~$q$ of a moving body, the volume~$V$ and the +temperature~$T$ as independent variables, then, as I have shown +in the previous lecture (p.~\pageref{png115lab1}), we shall have for the pressure~$p$ +and the entropy~$S$ the following relations: +\[ +\frac{\dd H}{\dd V} = p \quad\text{and}\quad \frac{\dd H}{\dd T} = S. +\] +Now since $V/\sqrt{c^{2} - q^{2}}$ is invariant, and $S$~likewise invariant +(see p.~\pageref{png131lab1}), it follows from the invariance of $H/\sqrt{c^{2} - q^{2}}$ +that $p$~is invariant and also that $T/\sqrt{c^{2} - q^{2}}$ is invariant, and +hence that: +\[ +p = p' \quad\text{and}\quad +\frac{T}{\sqrt{c^{2} - q^{2}}} = \frac{T'}{\sqrt{c^{2} - q'^{2}}}. +\] + +The two observers $A$~and~$B$ would estimate the pressure of a +body as the same, but the temperature of the body as different. + +A special case of this example is supplied when the body +considered furnishes a black body radiation. The black body +radiation is the only physical system whose dynamics (for quasi-stationary +processes) is known with absolute accuracy. That the +black body radiation possesses inertia was first pointed out by +F.~Hasenöhrl. For black body radiation at rest the energy +$E_{0} = a T^{4}V$ is given by the Stefan-Boltzmann law, and the entropy +$S_{0} = \int (dE_{0}/T) = \frac{4}{3} aT^{3}V$, and the pressure $p_{0} = (a/3)T^{4}$, and, +therefore, in accordance with the above relations, the kinetic +%-----File: 139.png---\redacted\-------- +potential is: +\[ +H_{0} = \frac{a}{3} T^{4} V. +\] +Let us imagine now a black body radiation moving with the +velocity~$q$ with respect to the observer~$A$ and introduce an +observer~$B$ who is at rest ($q = 0$) with reference to the black body +radiation; then: +\[ +\frac{H}{\sqrt{c^{2} - q^{2}}} = \frac{H'}{\sqrt{c^{2} - q'^{2}}} = \frac{H'_{0}}{c}, +\] +wherein +\[ +H'_{0} = \frac{a}{3} T'^{4} V'. +\] +Taking account of the above general relations between $T'$~and~$T$, +$V'$~and~$V$, this gives for the moving black body radiation the +kinetic potential: +\[ +H = \frac{a}{3} \frac{T^{4} V}{\left(1 - \dfrac{q^{2}}{c^{2}}\right)^{2}}, +\] +from which all the remaining thermodynamic quantities: the +pressure~$p$, the energy~$E$, the momentum~$G$, the longitudinal and +transverse masses $m_{l}$~and~$m_{t}$ of the moving black body radiation +are uniquely determined. + +Colleagues, ladies and gentlemen, I have arrived at the conclusion +of my lectures. I have endeavored to bring before +you in bold outline those characteristic advances in the present +system of physics which in my opinion are the most important. +Another in my place would perhaps have made another and better +choice and, at another time, it is quite likely that I myself +should have done so. The principle of relativity holds, not only +for processes in physics, but also for the physicist himself, in +that a fixed system of physics exists in reality only for a given +physicist and for a given time. But, as in the theory of relativity, +there exist invariants in the system of physics: ideas and +%-----File: 140.png---\redacted\-------- +laws which retain their meaning for all investigators and for +all times, and to discover these invariants is always the real +endeavor of physical research. We shall work further in this +direction in order to leave behind for our successors where possible---lasting +results. For if, while engaged in body and mind +in patient and often modest individual endeavor, one thought +strengthens and supports us, it is this, that we in physics work, +not for the day only and for immediate results, but, so to speak, +for eternity. + +I thank you heartily for the encouragement which you have +given me. I thank you no less for the patience with which you +have followed my lectures to the end, and I trust that it may be +possible for many among you to furnish in the direction indicated +much valuable service to our beloved science. + +\newpage +\pagestyle{empty} +\begin{center}\Large % make the heading a bit more noticeable +\textsc{Typographical Errors corrected\\in Project Gutenberg +edition}\end{center} + +\noindent p.~\pageref{png29lab2}.~In `the theory of reversible processes', `the' omitted (before line-break `the-ory'). + +\vspace{\baselineskip} + +\noindent p.~\pageref{png36lab1}.~Eqn.~(6), first term $\tsum \nu_{0} \log c_{0}$ +was printed as $\tsum \nu_{0} \log c_{1}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png38lab1}.~`Let the system consist'--`consists' in text. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png41lab1}.~`at a fixed temperature~$T$' was printed `at a fixed pressure~$T$'. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png43lab1}.~$\displaystyle\nu_{0}' = \frac{m_{0} }{ {m_{0}}'}$ was printed $\displaystyle v_{0}' = \frac{m_{0} }{ {m_{0}}'}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png59lab1}.~In `give up the attempt to understand'--`undertand' in text. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png67lab1}.~Definitions of $\psi$, third expression $\displaystyle\psi_{3} = \cdots$ +printed as $\displaystyle\dot{\varphi}_{3} = \cdots$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png67lab2}.~Equation beginning $\displaystyle\dot{\psi}_{1} = \frac{d\psi_{1}}{dt}$, this term +was printed as $\displaystyle\psi_{1} = \frac{d\psi_{1}}{dt}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png67lab4}.~In `we shall calculate later'--`calulate' in text. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png74lab1}.~Eqn. before (21), the term $\displaystyle\left(\frac{\dd S}{\dd E}\right)_{V}$ +was printed as $\displaystyle\left(\frac{dS}{\dd E}\right)_{V}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png74lab2}.~Eqn. (21), the term $\displaystyle\left(\frac{\dd S}{\dd E}\right)_{V}$ +was printed as $\displaystyle\left(\frac{\dd S}{dE}\right)_{V}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png103lab1}.~`black radiation in a vacuum'--`vaccuum' in text. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png104lab1}.~The mass ratio symbol $\omega$ was consistently printed as $\infty$ on this page. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png110lab1}.~Eqn. after (58), the term $\displaystyle\frac{\dd H}{\dd \varphi_{1}}$ +was printed as $\displaystyle\frac{\dd H}{\delta \varphi_{1}}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png111lab1}.~Term ${\Phi_{1}}' \delta{\varphi_{1}}'$ was printed as ${\Phi_{1}}' \delta\varphi_{1}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png112lab1}.~Term $\displaystyle\left(\frac{\dd H_{\phi}}{\dd \dot{\varphi}_{1}}\right)$ +was printed as $\displaystyle\left(\frac{dH_{\phi}}{\dd \dot{\varphi}_{1}}\right)$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png113lab1}.~Eqn. after `The laws of electrical action', +the term $\displaystyle\frac{\dd H_{\epsilon}}{\dd \dot{\epsilon}_{1}}$ +was printed as $\displaystyle\frac{dH_{\epsilon}}{\dd \dot{\epsilon}_{1}}$. + +\vspace{\baselineskip} + +\noindent p.~\pageref{png133lab1}.~Eqn. (69), the term $\kappa (x' + \nu t')$ +was printed as $\kappa (x' + vt')$. + +\vspace{\baselineskip} + +\newpage + +\begin{verbatim} +End of the Project Gutenberg EBook of Eight Lectures on Theoretical Physics, by +Max Planck + +*** END OF THIS PROJECT GUTENBERG EBOOK EIGHT LECTURES *** + +***** This file should be named 39017-pdf.pdf or 39017-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/9/0/1/39017/ + +Produced by Brenda Lewis, Keith Edkins and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/Canadian Libraries) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. Special rules, +set forth in the General Terms of Use part of this license, apply to +copying and distributing Project Gutenberg-tm electronic works to +protect the PROJECT GUTENBERG-tm concept and trademark. Project +Gutenberg is a registered trademark, and may not be used if you +charge for the eBooks, unless you receive specific permission. If you +do not charge anything for copies of this eBook, complying with the +rules is very easy. You may use this eBook for nearly any purpose +such as creation of derivative works, reports, performances and +research. They may be modified and printed and given away--you may do +practically ANYTHING with public domain eBooks. Redistribution is +subject to the trademark license, especially commercial +redistribution. + + + +*** START: FULL LICENSE *** + +THE FULL PROJECT GUTENBERG LICENSE +PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK + +To protect the Project Gutenberg-tm mission of promoting the free +distribution of electronic works, by using or distributing this work +(or any other work associated in any way with the phrase "Project +Gutenberg"), you agree to comply with all the terms of the Full Project +Gutenberg-tm License (available with this file or online at +http://gutenberg.org/license). + + +Section 1. General Terms of Use and Redistributing Project Gutenberg-tm +electronic works + +1.A. By reading or using any part of this Project Gutenberg-tm +electronic work, you indicate that you have read, understand, agree to +and accept all the terms of this license and intellectual property +(trademark/copyright) agreement. If you do not agree to abide by all +the terms of this agreement, you must cease using and return or destroy +all copies of Project Gutenberg-tm electronic works in your possession. +If you paid a fee for obtaining a copy of or access to a Project +Gutenberg-tm electronic work and you do not agree to be bound by the +terms of this agreement, you may obtain a refund from the person or +entity to whom you paid the fee as set forth in paragraph 1.E.8. + +1.B. "Project Gutenberg" is a registered trademark. It may only be +used on or associated in any way with an electronic work by people who +agree to be bound by the terms of this agreement. There are a few +things that you can do with most Project Gutenberg-tm electronic works +even without complying with the full terms of this agreement. See +paragraph 1.C below. There are a lot of things you can do with Project +Gutenberg-tm electronic works if you follow the terms of this agreement +and help preserve free future access to Project Gutenberg-tm electronic +works. See paragraph 1.E below. + +1.C. The Project Gutenberg Literary Archive Foundation ("the Foundation" +or PGLAF), owns a compilation copyright in the collection of Project +Gutenberg-tm electronic works. Nearly all the individual works in the +collection are in the public domain in the United States. If an +individual work is in the public domain in the United States and you are +located in the United States, we do not claim a right to prevent you from +copying, distributing, performing, displaying or creating derivative +works based on the work as long as all references to Project Gutenberg +are removed. Of course, we hope that you will support the Project +Gutenberg-tm mission of promoting free access to electronic works by +freely sharing Project Gutenberg-tm works in compliance with the terms of +this agreement for keeping the Project Gutenberg-tm name associated with +the work. You can easily comply with the terms of this agreement by +keeping this work in the same format with its attached full Project +Gutenberg-tm License when you share it without charge with others. + +1.D. The copyright laws of the place where you are located also govern +what you can do with this work. Copyright laws in most countries are in +a constant state of change. If you are outside the United States, check +the laws of your country in addition to the terms of this agreement +before downloading, copying, displaying, performing, distributing or +creating derivative works based on this work or any other Project +Gutenberg-tm work. The Foundation makes no representations concerning +the copyright status of any work in any country outside the United +States. + +1.E. Unless you have removed all references to Project Gutenberg: + +1.E.1. The following sentence, with active links to, or other immediate +access to, the full Project Gutenberg-tm License must appear prominently +whenever any copy of a Project Gutenberg-tm work (any work on which the +phrase "Project Gutenberg" appears, or with which the phrase "Project +Gutenberg" is associated) is accessed, displayed, performed, viewed, +copied or distributed: + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + +1.E.2. If an individual Project Gutenberg-tm electronic work is derived +from the public domain (does not contain a notice indicating that it is +posted with permission of the copyright holder), the work can be copied +and distributed to anyone in the United States without paying any fees +or charges. If you are redistributing or providing access to a work +with the phrase "Project Gutenberg" associated with or appearing on the +work, you must comply either with the requirements of paragraphs 1.E.1 +through 1.E.7 or obtain permission for the use of the work and the +Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or +1.E.9. + +1.E.3. If an individual Project Gutenberg-tm electronic work is posted +with the permission of the copyright holder, your use and distribution +must comply with both paragraphs 1.E.1 through 1.E.7 and any additional +terms imposed by the copyright holder. Additional terms will be linked +to the Project Gutenberg-tm License for all works posted with the +permission of the copyright holder found at the beginning of this work. + +1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm +License terms from this work, or any files containing a part of this +work or any other work associated with Project Gutenberg-tm. + +1.E.5. Do not copy, display, perform, distribute or redistribute this +electronic work, or any part of this electronic work, without +prominently displaying the sentence set forth in paragraph 1.E.1 with +active links or immediate access to the full terms of the Project +Gutenberg-tm License. + +1.E.6. You may convert to and distribute this work in any binary, +compressed, marked up, nonproprietary or proprietary form, including any +word processing or hypertext form. However, if you provide access to or +distribute copies of a Project Gutenberg-tm work in a format other than +"Plain Vanilla ASCII" or other format used in the official version +posted on the official Project Gutenberg-tm web site (www.gutenberg.org), +you must, at no additional cost, fee or expense to the user, provide a +copy, a means of exporting a copy, or a means of obtaining a copy upon +request, of the work in its original "Plain Vanilla ASCII" or other +form. Any alternate format must include the full Project Gutenberg-tm +License as specified in paragraph 1.E.1. + +1.E.7. Do not charge a fee for access to, viewing, displaying, +performing, copying or distributing any Project Gutenberg-tm works +unless you comply with paragraph 1.E.8 or 1.E.9. + +1.E.8. You may charge a reasonable fee for copies of or providing +access to or distributing Project Gutenberg-tm electronic works provided +that + +- You pay a royalty fee of 20% of the gross profits you derive from + the use of Project Gutenberg-tm works calculated using the method + you already use to calculate your applicable taxes. The fee is + owed to the owner of the Project Gutenberg-tm trademark, but he + has agreed to donate royalties under this paragraph to the + Project Gutenberg Literary Archive Foundation. Royalty payments + must be paid within 60 days following each date on which you + prepare (or are legally required to prepare) your periodic tax + returns. Royalty payments should be clearly marked as such and + sent to the Project Gutenberg Literary Archive Foundation at the + address specified in Section 4, "Information about donations to + the Project Gutenberg Literary Archive Foundation." + +- You provide a full refund of any money paid by a user who notifies + you in writing (or by e-mail) within 30 days of receipt that s/he + does not agree to the terms of the full Project Gutenberg-tm + License. You must require such a user to return or + destroy all copies of the works possessed in a physical medium + and discontinue all use of and all access to other copies of + Project Gutenberg-tm works. + +- You provide, in accordance with paragraph 1.F.3, a full refund of any + money paid for a work or a replacement copy, if a defect in the + electronic work is discovered and reported to you within 90 days + of receipt of the work. + +- You comply with all other terms of this agreement for free + distribution of Project Gutenberg-tm works. + +1.E.9. If you wish to charge a fee or distribute a Project Gutenberg-tm +electronic work or group of works on different terms than are set +forth in this agreement, you must obtain permission in writing from +both the Project Gutenberg Literary Archive Foundation and Michael +Hart, the owner of the Project Gutenberg-tm trademark. Contact the +Foundation as set forth in Section 3 below. + +1.F. + +1.F.1. Project Gutenberg volunteers and employees expend considerable +effort to identify, do copyright research on, transcribe and proofread +public domain works in creating the Project Gutenberg-tm +collection. Despite these efforts, Project Gutenberg-tm electronic +works, and the medium on which they may be stored, may contain +"Defects," such as, but not limited to, incomplete, inaccurate or +corrupt data, transcription errors, a copyright or other intellectual +property infringement, a defective or damaged disk or other medium, a +computer virus, or computer codes that damage or cannot be read by +your equipment. + +1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right +of Replacement or Refund" described in paragraph 1.F.3, the Project +Gutenberg Literary Archive Foundation, the owner of the Project +Gutenberg-tm trademark, and any other party distributing a Project +Gutenberg-tm electronic work under this agreement, disclaim all +liability to you for damages, costs and expenses, including legal +fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT +LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE +PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE +TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE +LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR +INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH +DAMAGE. + +1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a +defect in this electronic work within 90 days of receiving it, you can +receive a refund of the money (if any) you paid for it by sending a +written explanation to the person you received the work from. If you +received the work on a physical medium, you must return the medium with +your written explanation. The person or entity that provided you with +the defective work may elect to provide a replacement copy in lieu of a +refund. If you received the work electronically, the person or entity +providing it to you may choose to give you a second opportunity to +receive the work electronically in lieu of a refund. If the second copy +is also defective, you may demand a refund in writing without further +opportunities to fix the problem. + +1.F.4. Except for the limited right of replacement or refund set forth +in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER +WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO +WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE. + +1.F.5. Some states do not allow disclaimers of certain implied +warranties or the exclusion or limitation of certain types of damages. +If any disclaimer or limitation set forth in this agreement violates the +law of the state applicable to this agreement, the agreement shall be +interpreted to make the maximum disclaimer or limitation permitted by +the applicable state law. The invalidity or unenforceability of any +provision of this agreement shall not void the remaining provisions. + +1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the +trademark owner, any agent or employee of the Foundation, anyone +providing copies of Project Gutenberg-tm electronic works in accordance +with this agreement, and any volunteers associated with the production, +promotion and distribution of Project Gutenberg-tm electronic works, +harmless from all liability, costs and expenses, including legal fees, +that arise directly or indirectly from any of the following which you do +or cause to occur: (a) distribution of this or any Project Gutenberg-tm +work, (b) alteration, modification, or additions or deletions to any +Project Gutenberg-tm work, and (c) any Defect you cause. + + +Section 2. Information about the Mission of Project Gutenberg-tm + +Project Gutenberg-tm is synonymous with the free distribution of +electronic works in formats readable by the widest variety of computers +including obsolete, old, middle-aged and new computers. It exists +because of the efforts of hundreds of volunteers and donations from +people in all walks of life. + +Volunteers and financial support to provide volunteers with the +assistance they need, are critical to reaching Project Gutenberg-tm's +goals and ensuring that the Project Gutenberg-tm collection will +remain freely available for generations to come. In 2001, the Project +Gutenberg Literary Archive Foundation was created to provide a secure +and permanent future for Project Gutenberg-tm and future generations. +To learn more about the Project Gutenberg Literary Archive Foundation +and how your efforts and donations can help, see Sections 3 and 4 +and the Foundation web page at http://www.pglaf.org. + + +Section 3. Information about the Project Gutenberg Literary Archive +Foundation + +The Project Gutenberg Literary Archive Foundation is a non profit +501(c)(3) educational corporation organized under the laws of the +state of Mississippi and granted tax exempt status by the Internal +Revenue Service. The Foundation's EIN or federal tax identification +number is 64-6221541. Its 501(c)(3) letter is posted at +http://pglaf.org/fundraising. Contributions to the Project Gutenberg +Literary Archive Foundation are tax deductible to the full extent +permitted by U.S. federal laws and your state's laws. + +The Foundation's principal office is located at 4557 Melan Dr. S. +Fairbanks, AK, 99712., but its volunteers and employees are scattered +throughout numerous locations. Its business office is located at +809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email +business@pglaf.org. Email contact links and up to date contact +information can be found at the Foundation's web site and official +page at http://pglaf.org + +For additional contact information: + Dr. Gregory B. Newby + Chief Executive and Director + gbnewby@pglaf.org + + +Section 4. Information about Donations to the Project Gutenberg +Literary Archive Foundation + +Project Gutenberg-tm depends upon and cannot survive without wide +spread public support and donations to carry out its mission of +increasing the number of public domain and licensed works that can be +freely distributed in machine readable form accessible by the widest +array of equipment including outdated equipment. Many small donations +($1 to $5,000) are particularly important to maintaining tax exempt +status with the IRS. + +The Foundation is committed to complying with the laws regulating +charities and charitable donations in all 50 states of the United +States. Compliance requirements are not uniform and it takes a +considerable effort, much paperwork and many fees to meet and keep up +with these requirements. We do not solicit donations in locations +where we have not received written confirmation of compliance. To +SEND DONATIONS or determine the status of compliance for any +particular state visit http://pglaf.org + +While we cannot and do not solicit contributions from states where we +have not met the solicitation requirements, we know of no prohibition +against accepting unsolicited donations from donors in such states who +approach us with offers to donate. + +International donations are gratefully accepted, but we cannot make +any statements concerning tax treatment of donations received from +outside the United States. U.S. laws alone swamp our small staff. + +Please check the Project Gutenberg Web pages for current donation +methods and addresses. Donations are accepted in a number of other +ways including checks, online payments and credit card donations. +To donate, please visit: http://pglaf.org/donate + + +Section 5. General Information About Project Gutenberg-tm electronic +works. + +Professor Michael S. Hart is the originator of the Project Gutenberg-tm +concept of a library of electronic works that could be freely shared +with anyone. For thirty years, he produced and distributed Project +Gutenberg-tm eBooks with only a loose network of volunteer support. + + +Project Gutenberg-tm eBooks are often created from several printed +editions, all of which are confirmed as Public Domain in the U.S. +unless a copyright notice is included. Thus, we do not necessarily +keep eBooks in compliance with any particular paper edition. + + +Most people start at our Web site which has the main PG search facility: + + http://www.gutenberg.org + +This Web site includes information about Project Gutenberg-tm, +including how to make donations to the Project Gutenberg Literary +Archive Foundation, how to help produce our new eBooks, and how to +subscribe to our email newsletter to hear about new eBooks. + +\end{verbatim} +% Syntax is control word, for each argument: 1=Mandatory/0=Optional, 1=Keep/0=Delete, opening replacement, closing replacement, ... +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% End of the Project Gutenberg EBook of Eight Lectures on Theoretical Physics, by +% Max Planck % +% % +% *** END OF THIS PROJECT GUTENBERG EBOOK EIGHT LECTURES *** % +% % +% ***** This file should be named 39017-t.tex or 39017-t.zip ***** % +% This and all associated files of various formats will be found in: % +% http://www.gutenberg.org/3/9/0/1/39017/ % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\end{document} +### lprep configuration +@ControlwordReplace = ( + ['\eg',"e. g."], + ['\ie',"i. e."], + ['\Eg',"E. g."], + ); +@ControlwordArguments = ( + ['\\pngcent',1,0,"\n[Illustration","]\n",1,0,'',''], + ['\\Chapter',1,1,''," ",1,1,"","\n\n"], + ['\\Section',1,1,''," ",1,1,"","\n\n"], + ['\\begin{spacing}',1,0,'',"",1,1,"",""], + ); + +### +This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) (format=pdflatex 2011.9.6) 29 FEB 2012 08:18 +entering extended mode + %&-line parsing enabled. +**39017-t.tex +(./39017-t.tex +LaTeX2e <2009/09/24> +Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh +yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov +ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 +9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san +skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn +orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian +, estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po +rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. +(/usr/share/texmf-texlive/tex/latex/base/book.cls +Document Class: book 2007/10/19 v1.4h Standard LaTeX document class +(/usr/share/texmf-texlive/tex/latex/base/bk12.clo +File: bk12.clo 2007/10/19 v1.4h Standard LaTeX file (size option) +) +\c@part=\count79 +\c@chapter=\count80 +\c@section=\count81 +\c@subsection=\count82 +\c@subsubsection=\count83 +\c@paragraph=\count84 +\c@subparagraph=\count85 +\c@figure=\count86 +\c@table=\count87 +\abovecaptionskip=\skip41 +\belowcaptionskip=\skip42 +\bibindent=\dimen102 +) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty +Package: inputenc 2008/03/30 v1.1d Input encoding file +\inpenc@prehook=\toks14 +\inpenc@posthook=\toks15 +(/usr/share/texmf-texlive/tex/latex/base/latin1.def +File: latin1.def 2008/03/30 v1.1d Input encoding file +)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty +Package: amsmath 2000/07/18 v2.13 AMS math features +\@mathmargin=\skip43 +For additional information on amsmath, use the `?' option. +(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 +(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 +\@emptytoks=\toks16 +\ex@=\dimen103 +)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d +\pmbraise@=\dimen104 +) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty +Package: amsopn 1999/12/14 v2.01 operator names +) +\inf@bad=\count88 +LaTeX Info: Redefining \frac on input line 211. +\uproot@=\count89 +\leftroot@=\count90 +LaTeX Info: Redefining \overline on input line 307. +\classnum@=\count91 +\DOTSCASE@=\count92 +LaTeX Info: Redefining \ldots on input line 379. +LaTeX Info: Redefining \dots on input line 382. +LaTeX Info: Redefining \cdots on input line 467. +\Mathstrutbox@=\box26 +\strutbox@=\box27 +\big@size=\dimen105 +LaTeX Font Info: Redeclaring font encoding OML on input line 567. +LaTeX Font Info: Redeclaring font encoding OMS on input line 568. +\macc@depth=\count93 +\c@MaxMatrixCols=\count94 +\dotsspace@=\muskip10 +\c@parentequation=\count95 +\dspbrk@lvl=\count96 +\tag@help=\toks17 +\row@=\count97 +\column@=\count98 +\maxfields@=\count99 +\andhelp@=\toks18 +\eqnshift@=\dimen106 +\alignsep@=\dimen107 +\tagshift@=\dimen108 +\tagwidth@=\dimen109 +\totwidth@=\dimen110 +\lineht@=\dimen111 +\@envbody=\toks19 +\multlinegap=\skip44 +\multlinetaggap=\skip45 +\mathdisplay@stack=\toks20 +LaTeX Info: Redefining \[ on input line 2666. +LaTeX Info: Redefining \] on input line 2667. +) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2009/06/22 v3.00 +(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2009/06/22 v3.00 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 96. +)) (/usr/share/texmf-texlive/tex/latex/was/icomma.sty +Package: icomma 2002/03/10 v2.0 (WaS) +) (/usr/share/texmf-texlive/tex/latex/tools/indentfirst.sty +Package: indentfirst 1995/11/23 v1.03 Indent first paragraph (DPC) +) (/usr/share/texmf-texlive/tex/latex/graphics/graphicx.sty +Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) +(/usr/share/texmf-texlive/tex/latex/graphics/keyval.sty +Package: keyval 1999/03/16 v1.13 key=value parser (DPC) +\KV@toks@=\toks21 +) (/usr/share/texmf-texlive/tex/latex/graphics/graphics.sty +Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) +(/usr/share/texmf-texlive/tex/latex/graphics/trig.sty +Package: trig 1999/03/16 v1.09 sin cos tan (DPC) +) (/etc/texmf/tex/latex/config/graphics.cfg +File: graphics.cfg 2009/08/28 v1.8 graphics configuration of TeX Live +) +Package graphics Info: Driver file: pdftex.def on input line 91. +(/usr/share/texmf-texlive/tex/latex/pdftex-def/pdftex.def +File: pdftex.def 2009/08/25 v0.04m Graphics/color for pdfTeX +\Gread@gobject=\count100 +)) +\Gin@req@height=\dimen112 +\Gin@req@width=\dimen113 +) (/usr/share/texmf-texlive/tex/latex/setspace/setspace.sty +Package: setspace 2000/12/01 6.7 Contributed and Supported LaTeX2e package +Package: `setspace' 6.7 <2000/12/01> +) + +LaTeX Warning: You have requested, on input line 74, version + `2011/12/19' of package setspace, + but only version + `2000/12/01 6.7 Contributed and Supported LaTeX2e package' + is available. + +(/usr/share/texmf-texlive/tex/latex/soul/soul.sty +Package: soul 2003/11/17 v2.4 letterspacing/underlining (mf) +\SOUL@word=\toks22 +\SOUL@lasttoken=\toks23 +\SOUL@cmds=\toks24 +\SOUL@buffer=\toks25 +\SOUL@token=\toks26 +\SOUL@spaceskip=\skip46 +\SOUL@ttwidth=\dimen114 +\SOUL@uldp=\dimen115 +\SOUL@ulht=\dimen116 +) (/usr/share/texmf-texlive/tex/latex/tools/verbatim.sty +Package: verbatim 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements +\every@verbatim=\toks27 +\verbatim@line=\toks28 +\verbatim@in@stream=\read1 +) +\Zw=\count101 +LaTeX Info: Redefining \so on input line 172. +(./39017-t.aux) +\openout1 = `39017-t.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 174. +LaTeX Font Info: ... okay on input line 174. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 174. +LaTeX Font Info: ... okay on input line 174. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 174. +LaTeX Font Info: ... okay on input line 174. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 174. +LaTeX Font Info: ... okay on input line 174. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 174. +LaTeX Font Info: ... okay on input line 174. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 174. +LaTeX Font Info: ... okay on input line 174. +(/usr/share/texmf/tex/context/base/supp-pdf.mkii +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count102 +\scratchdimen=\dimen117 +\scratchbox=\box28 +\nofMPsegments=\count103 +\nofMParguments=\count104 +\everyMPshowfont=\toks29 +\MPscratchCnt=\count105 +\MPscratchDim=\dimen118 +\MPnumerator=\count106 +\everyMPtoPDFconversion=\toks30 +) [1 + + +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] <images/illo001.png, id=6, + 192.9609pt x 185.7339pt> +File: images/illo001.png Graphic file (type png) +<use images/illo001.png> [2 <./images/illo001.png (PNG copy)>] [3] [4] [5] [6] +[7] [8] [9] [10] [11] [12] [13] [14] [15] +LaTeX Font Info: Try loading font information for U+msa on input line 1007. +(/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2009/06/22 v3.00 AMS symbols A +) +LaTeX Font Info: Try loading font information for U+msb on input line 1007. +(/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2009/06/22 v3.00 AMS symbols B +) [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [3 +1] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [ +47] [48] [49] [50] [51] +LaTeX Font Info: Try loading font information for U+euf on input line 3295. +(/usr/share/texmf-texlive/tex/latex/amsfonts/ueuf.fd +File: ueuf.fd 2009/06/22 v3.00 Euler Fraktur +) [52] [53] [54] <images/illo085.png, id=198, 456.38506pt x 430.36784pt> +File: images/illo085.png Graphic file (type png) +<use images/illo085.png> [55 <./images/illo085.png (PNG copy)>] [56] [57] [58] +[59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] +[75] [76] [77] [78] [79] <images/illo124.png, id=281, 487.8225pt x 280.4076pt> +File: images/illo124.png Graphic file (type png) +<use images/illo124.png> [80 <./images/illo124.png (PNG copy)>] <images/illo126 +.png, id=287, 330.2739pt x 299.1978pt> +File: images/illo126.png Graphic file (type png) +<use images/illo126.png> [81 <./images/illo126.png (PNG copy)>] <images/illo127 +.png, id=293, 362.0727pt x 341.1144pt> +File: images/illo127.png Graphic file (type png) +<use images/illo127.png> [82 <./images/illo127.png (PNG copy)>] [83] [84] <imag +es/illo132.png, id=305, 341.1144pt x 291.9708pt> +File: images/illo132.png Graphic file (type png) +<use images/illo132.png> [85 <./images/illo132.png (PNG copy)>] [86] [87] [88] +[89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] (./39017-t.aux) + + *File List* + book.cls 2007/10/19 v1.4h Standard LaTeX document class + bk12.clo 2007/10/19 v1.4h Standard LaTeX file (size option) +inputenc.sty 2008/03/30 v1.1d Input encoding file + latin1.def 2008/03/30 v1.1d Input encoding file + amsmath.sty 2000/07/18 v2.13 AMS math features + amstext.sty 2000/06/29 v2.01 + amsgen.sty 1999/11/30 v2.0 + amsbsy.sty 1999/11/29 v1.2d + amsopn.sty 1999/12/14 v2.01 operator names + amssymb.sty 2009/06/22 v3.00 +amsfonts.sty 2009/06/22 v3.00 Basic AMSFonts support + icomma.sty 2002/03/10 v2.0 (WaS) +indentfirst.sty 1995/11/23 v1.03 Indent first paragraph (DPC) +graphicx.sty 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) + keyval.sty 1999/03/16 v1.13 key=value parser (DPC) +graphics.sty 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) + trig.sty 1999/03/16 v1.09 sin cos tan (DPC) +graphics.cfg 2009/08/28 v1.8 graphics configuration of TeX Live + pdftex.def 2009/08/25 v0.04m Graphics/color for pdfTeX +setspace.sty 2000/12/01 6.7 Contributed and Supported LaTeX2e package + soul.sty 2003/11/17 v2.4 letterspacing/underlining (mf) +verbatim.sty 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements +supp-pdf.mkii +images/illo001.png + umsa.fd 2009/06/22 v3.00 AMS symbols A + umsb.fd 2009/06/22 v3.00 AMS symbols B + ueuf.fd 2009/06/22 v3.00 Euler Fraktur +images/illo085.png +images/illo124.png +images/illo126.png +images/illo127.png +images/illo132.png + *********** + + ) +Here is how much of TeX's memory you used: + 2265 strings out of 493848 + 27433 string characters out of 1152824 + 91911 words of memory out of 3000000 + 5508 multiletter control sequences out of 15000+50000 + 18385 words of font info for 69 fonts, out of 3000000 for 9000 + 714 hyphenation exceptions out of 8191 + 26i,11n,24p,262b,291s stack positions out of 5000i,500n,10000p,200000b,50000s +</usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/sha +re/texmf-texlive/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texmf-te +xlive/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/share/texmf-texlive/font +s/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texmf-texlive/fonts/type1/pub +lic/amsfonts/cm/cmmi10.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfont +s/cm/cmmi12.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmmi6. +pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/sh +are/texmf-texlive/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texmf-te +xlive/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texmf-texlive/fonts/ +type1/public/amsfonts/cm/cmr17.pfb></usr/share/texmf-texlive/fonts/type1/public +/amsfonts/cm/cmr6.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/ +cmr7.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmr8.pfb></us +r/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmss10.pfb></usr/share/tex +mf-texlive/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texmf-texlive/ +fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/share/texmf-texlive/fonts/type1/ +public/amsfonts/cm/cmsy8.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfo +nts/cm/cmti12.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmti +8.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr +/share/texmf-texlive/fonts/type1/public/amsfonts/euler/eufm10.pfb></usr/share/t +exmf-texlive/fonts/type1/public/amsfonts/symbols/msam10.pfb> +Output written on 39017-t.pdf (99 pages, 1071568 bytes). +PDF statistics: + 430 PDF objects out of 1000 (max. 8388607) + 0 named destinations out of 1000 (max. 500000) + 67 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/39017-t/images/illo001.png b/39017-t/images/illo001.png Binary files differnew file mode 100644 index 0000000..2851b13 --- /dev/null +++ b/39017-t/images/illo001.png diff --git a/39017-t/images/illo085.png b/39017-t/images/illo085.png Binary files differnew file mode 100644 index 0000000..bfd1dd1 --- /dev/null +++ b/39017-t/images/illo085.png diff --git a/39017-t/images/illo124.png b/39017-t/images/illo124.png Binary files differnew file mode 100644 index 0000000..e76e755 --- /dev/null +++ b/39017-t/images/illo124.png diff --git a/39017-t/images/illo126.png b/39017-t/images/illo126.png Binary files differnew file mode 100644 index 0000000..2ef7e3f --- /dev/null +++ b/39017-t/images/illo126.png diff --git a/39017-t/images/illo127.png b/39017-t/images/illo127.png Binary files differnew file mode 100644 index 0000000..97150fb --- /dev/null +++ b/39017-t/images/illo127.png diff --git a/39017-t/images/illo132.png b/39017-t/images/illo132.png Binary files differnew file mode 100644 index 0000000..2c491db --- /dev/null +++ b/39017-t/images/illo132.png diff --git a/LICENSE.txt b/LICENSE.txt new file mode 100644 index 0000000..6312041 --- /dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..4787c17 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #39017 (https://www.gutenberg.org/ebooks/39017) |
