% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of The Theory of Heat Radiation, 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: The Theory of Heat Radiation                                     %
%                                                                         %
% Author: Max Planck                                                      %
%                                                                         %
% Translator: Morton Masius                                               %
%                                                                         %
% Release Date: June 18, 2012 [EBook #40030]                              %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF HEAT RADIATION ***
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The Project Gutenberg EBook of The Theory of Heat Radiation, 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: The Theory of Heat Radiation

Author: Max Planck

Translator: Morton Masius

Release Date: June 18, 2012 [EBook #40030]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF HEAT RADIATION ***
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Produced by Andrew D. Hwang, Brenda Lewis and the Online
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by The Internet Archive/American Libraries.)
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
%% -----File: 001.png---Folio xx-------
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% THE THEORY OF HEAT RADIATION
% PLANCK and MASIUS
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%% -----File: 002.png---Folio xx-------
% [Blank Page]
%% -----File: 003.png---Folio xx-------
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\Huge THE THEORY \\
\large OF \\
\Huge HEAT RADIATION
\vfill\vfill

\normalsize BY \\
\large DR. MAX PLANCK \\[4pt]
\scriptsize PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN
\vfill

\normalsize AUTHORISED TRANSLATION
\vfill

BY \\
\large MORTON MASIUS, M. A., Ph.\ D. (Leipzig) \\[4pt]
\scriptsize INSTRUCTOR IN PHYSICS IN THE WORCESTER POLYTECHNIC INSTITUTE
\vfill\vfill\vfill

\normalsize WITH 7 ILLUSTRATIONS
\vfill\vfill\vfill\vfill

PHILADELPHIA \\
\Large P. BLAKISTON'S SON \& CO. \\
\normalsize 1012 WALNUT STREET
\end{center}
%% -----File: 004.png---Folio xx-------
\newpage
\null\vfill
\begin{center}
\footnotesize\scshape Copyright, 1914, by P. Blakiston's Son \& Co.
\vfill\vfill

\scriptsize THE · MAPLE · PRESS · YORK · PA
\end{center}
%% -----File: 005.png---Folio xx-------


\Preface{Translator's Preface}

The present volume is a translation of the second edition of
Professor \Name{Planck's} \textsc{Waermestrahlung} (1913). The profoundly
original ideas introduced by \Name{Planck} in the endeavor to reconcile
the electromagnetic theory of radiation with experimental facts
have proven to be of the greatest importance in many parts of
physics. Probably no single book since the appearance of \Name{Clerk
Maxwell's} \textsc{Electricity and Magnetism} has had a deeper influence
on the development of physical theories. The great majority of
English-speaking physicists are, of course, able to read the work
in the language in which it was written, but I believe that many
will welcome the opportunity offered by a translation to study the
ideas set forth by \Name{Planck} without the difficulties that frequently
arise in attempting to follow a new and somewhat difficult line
of reasoning in a foreign language.

Recent developments of physical theories have placed the quantum
of action in the foreground of interest. Questions regarding
the bearing of the quantum theory on the law of equipartition of
energy, its application to the theory of specific heats and to
photoelectric effects, attempts to form some concrete idea of
the physical significance of the quantum, that is, to devise a
``model'' for it, have created within the last few years a large and
ever increasing literature. Professor \Name{Planck} has, however, in
this book confined himself exclusively to radiation phenomena
and it has seemed to me probable that a brief résumé of this
literature might prove useful to the reader who wishes to pursue
the subject further. I have, therefore, with Professor \Name{Planck's}
permission, given in an appendix a list of the most important
papers on the subjects treated of in this book and others closely
related to them. I have also added a short note on one or two
derivations of formulæ where the treatment in the book seemed
too brief or to present some difficulties.
%% -----File: 006.png---Folio xx-------

In preparing the translation I have been under obligation for
advice and helpful suggestions to several friends and colleagues
and especially to Professor A.~W.~Duff who has read the manuscript
and the galley proof.

\Signature{Morton Masius.}{Worcester, Mass.,}{February,}{1914.}
%% -----File: 007.png---Folio xx-------

\Preface{Preface to Second Edition}

Recent advances in physical research have, on the whole, been
favorable to the special theory outlined in this book, in particular
to the hypothesis of an elementary quantity of action. My radiation
formula especially has so far stood all tests satisfactorily,
including even the refined systematic measurements which have
been carried out in the Physikalisch-technische Reichsanstalt
at Charlottenburg during the last year. Probably the most
direct support for the fundamental idea of the hypothesis of
quanta is supplied by the values of the elementary quanta of
matter and electricity derived from it. When, twelve years ago,
I made my first calculation of the value of the elementary electric
charge and found it to be $4.69·10^{-10}$ electrostatic units, the value
of this quantity deduced by \Name{J.~J. Thomson} from his ingenious
experiments on the condensation of water vapor on gas ions,
namely $6.5·10^{-10}$ was quite generally regarded as the most
reliable value. This value exceeds the one given by me by $38$~per~cent.
Meanwhile the experimental methods, improved in
an admirable way by the labors of \Name{E.~Rutherford}, \Name{E.~Regener},
\Name{J.~Perrin}, \Name{R.~A. Millikan}, \Name{The Svedberg} and others, have without
exception decided in favor of the value deduced from the theory
of radiation which lies between the values of \Name{Perrin} and \Name{Millikan}.

To the two mutually independent confirmations mentioned,
there has been added, as a further strong support of the hypothesis
of quanta, the heat theorem which has been in the meantime
announced by \Name{W.~Nernst}, and which seems to point unmistakably
to the fact that, not only the processes of radiation, but also the
molecular processes take place in accordance with certain elementary
quanta of a definite finite magnitude. For the hypothesis
of quanta as well as the heat theorem of \Name{Nernst} may be reduced
to the simple proposition that the thermodynamic probability
(\Sec{120}) of a physical state is a definite integral number,
or, what amounts to the same thing, that the entropy of a state
has a quite definite, positive value, which, as a minimum, becomes
%% -----File: 008.png---Folio xx-------
zero, while in contrast therewith the entropy may, according to
the classical thermodynamics, decrease without limit to minus
infinity. For the present, I would consider this proposition as
the very quintessence of the hypothesis of quanta.

In spite of the satisfactory agreement of the results mentioned
with one another as well as with experiment, the ideas from which
they originated have met with wide interest but, so far as I am
able to judge, with little general acceptance, the reason probably
being that the hypothesis of quanta has not as yet been satisfactorily
completed. While many physicists, through conservatism,
reject the ideas developed by me, or, at any rate, maintain
an expectant attitude, a few authors have attacked them for the
opposite reason, namely, as being inadequate, and have felt compelled
to supplement them by assumptions of a still more radical
nature, for example, by the assumption that any radiant energy
whatever, even though it travel freely in a vacuum, consists of
indivisible quanta or cells. Since nothing probably is a greater
drawback to the successful development of a new hypothesis
than overstepping its boundaries, I have always stood for making
as close a connection between the hypothesis of quanta and the
classical dynamics as possible, and for not stepping outside of
the boundaries of the latter until the experimental facts leave no
other course open. I have attempted to keep to this standpoint
in the revision of this treatise necessary for a new edition.

The main fault of the original treatment was that it began with
the classical electrodynamical laws of emission and absorption,
whereas later on it became evident that, in order to meet the
demand of experimental measurements, the assumption of finite
energy elements must be introduced, an assumption which is in
direct contradiction to the fundamental ideas of classical electrodynamics.
It is true that this inconsistency is greatly reduced
by the fact that, in reality, only mean values of energy are taken
from classical electrodynamics, while, for the statistical calculation,
the real values are used; nevertheless the treatment must,
on the whole, have left the reader with the unsatisfactory feeling
that it was not clearly to be seen, which of the assumptions made
in the beginning could, and which could not, be finally retained.

In contrast thereto I have now attempted to treat the subject
from the very outset in such a way that none of the laws stated
%% -----File: 009.png---Folio xx-------
need, later on, be restricted or modified. This presents the
advantage that the theory, so far as it is treated here, shows no
contradiction in itself, though certainly I do not mean that it
does not seem to call for improvements in many respects, as
regards both its internal structure and its external form. To
treat of the numerous applications, many of them very important,
which the hypothesis of quanta has already found in other parts
of physics, I have not regarded as part of my task, still less to
discuss all differing opinions.

Thus, while the new edition of this book may not claim to
bring the theory of heat radiation to a conclusion that is satisfactory
in all respects, this deficiency will not be of decisive
importance in judging the theory. For any one who would make
his attitude concerning the hypothesis of quanta depend on
whether the significance of the quantum of action for the elementary
physical processes is made clear in every respect or may
be demonstrated by some simple dynamical model, misunderstands,
I believe, the character and the meaning of the hypothesis
of quanta. It is impossible to express a really new
principle in terms of a model following old laws. And, as regards
the final formulation of the hypothesis, we should not
forget that, from the classical point of view, the physics of
the atom really has always remained a very obscure, inaccessible
region, into which the introduction of the elementary
quantum of action promises to throw some light.

Hence it follows from the nature of the case that it will require
painstaking experimental and theoretical work for many years
to come to make gradual advances in the new field. Any one
who, at present, devotes his efforts to the hypothesis of quanta,
must, for the time being, be content with the knowledge that the
fruits of the labor spent will probably be gathered by a future
generation.

\Signature{The Author.}{Berlin,}{November,}{1912.}
%% -----File: 010.png---Folio xx-------
%[Blank Page]
%% -----File: 011.png---Folio xx-------

\Preface{Preface to First Edition}

\enlargethispage{\baselineskip}
In this book the main contents of the lectures which I gave at
the University of Berlin during the winter semester 1906--07 are
presented. My original intention was merely to put together
in a connected account the results of my own investigations,
begun ten years ago, on the theory of heat radiation; it soon became
evident, however, that it was desirable to include also the
foundation of this theory in the treatment, starting with Kirchhoff's
Law on emitting and absorbing power; and so I attempted
to write a treatise which should also be capable of serving as an
introduction to the study of the entire theory of radiant heat on
a consistent thermodynamic basis. Accordingly the treatment
starts from the simple known experimental laws of optics and
advances, by gradual extension and by the addition of the results
of electrodynamics and thermodynamics, to the problems of the
spectral distribution of energy and of irreversibility. In doing
this I have deviated frequently from the customary methods of
treatment, wherever the matter presented or considerations
regarding the form of presentation seemed to call for it, especially
in deriving Kirchhoff's laws, in calculating Maxwell's radiation
pressure, in deriving Wien's displacement law, and in generalizing
it for radiations of any spectral distribution of energy whatever.

I have at the proper place introduced the results of my own
investigations into the treatment. A list of these has been added
at the end of the book to facilitate comparison and examination
as regards special details.

I wish, however, to emphasize here what has been stated more
fully in the last paragraph of this book, namely, that the theory
thus developed does not by any means claim to be perfect or
complete, although I believe that it points out a possible way of
accounting for the processes of radiant energy from the same
point of view as for the processes of molecular motion.
%% -----File: 012.png---Folio xx-------
%[Blank Page]
%% -----File: 013.png---Folio xx-------
\TableOfContents
\iffalse
\Preface{TABLE OF CONTENTS}

\begin{tabular}{rlr}
\multicolumn{3}{c}{}\\
\multicolumn{3}{c}{\textbf{PART I}}\\
\multicolumn{3}{c}{\textbf{FUNDAMENTAL FACTS AND DEFINITIONS}}\\
\textsc{Chap} & & \textsc{Page}\\
I.& General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . &1\\
II.& Radiation at Thermodynamic Equilibrium. Kirchhoff's Law. & \\
   & Black Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . &22\\
\multicolumn{3}{c}{}\\
\multicolumn{3}{c}{\textbf{PART II}}\\
\multicolumn{3}{c}{\textbf{DEDUCTIONS FROM ELECTRODYNAMICS AND}}\\
\multicolumn{3}{c}{\textbf{THERMODYNAMICS}}\\
I. & Maxwell's Radiation Pressure . . . . . . . . . . . . . . . . . . . . . .  & 49\\
II. & Stefan-Boltzmann Law of Radiation . . . . . . . . . . . . . . . . . .  & 59\\
III. & Wien's Displacement Law . . . . . . . . . . . . . . . . . . . . . . . .   & 69\\
IV. & Radiation of any Arbitrary Spectral Distribution of Energy. . . . & \\
 & Entropy and Temperature of Monochromatic Radiation . . . . . .  & 87\\
V. & Electrodynamical Processes in a Stationary Field of Radiation  & 103\\
\multicolumn{3}{c}{}\\
\multicolumn{3}{c}{\textbf{PART III}}\\
\multicolumn{3}{c}{\textbf{ENTROPY AND PROBABILITY}}\\
I. & Fundamental Definitions and Laws. Hypothesis of Quanta  . . . .  &113\\
II. & Ideal Monatomic Gases . . . . . . . . . . . . . . . . . . . . . . . . .   & 127\\
III. & Ideal Linear Oscillators  . . . . . . . . . . . . . . . . . . . . . . . . . & 135\\
IV. & Direct Calculation of the Entropy in the Case of Thermodynamic& \\
 & Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  & 144\\
\multicolumn{3}{c}{}\\
\multicolumn{3}{c}{\textbf{PART IV}}\\
\multicolumn{3}{c}{\textbf{A SYSTEM OF OSCILLATORS IN A STATIONARY}}\\
\multicolumn{3}{c}{\textbf{FIELD OF RADIATION}}\\
I. & The Elementary Dynamical Law for the Vibrations of an Ideal& \\
 & Oscillator. Hypothesis of Emission of Quanta . . . . . . . . . . . . . & 151\\
II. & Absorbed Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  & 155\\
III. & Emitted Energy. Stationary State . . . . . . . . . . . . . . . . . . . . & 161\\
IV. & The Law of the Normal Distribution of Energy. Elementary& \\
 & Quanta of Matter and of Electricity . . . . . . . . . . . . . . . . . . .  & 167
\end{tabular}
\fi
%% -----File: 014.png---Folio xx-------
\iffalse
\begin{tabular}{rlr}
\multicolumn{3}{c}{\textbf{PART V}}\\
\multicolumn{3}{c}{}\\
\multicolumn{3}{c}{\textbf{IRREVERSIBLE RADIATION PROCESSES}}\\
\multicolumn{3}{c}{}\\
I. & Fields of Radiation in General  . . . . . . . . . . . . . . . . . . . . & 189\\
II. & One Oscillator in the Field of Radiation . . . . . . . . . . . . . . & 196\\
III. & A System of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . & 200\\
IV. &Conservation of Energy and Increase of Entropy. Conclusion.  & 205\\
 & List of Papers on Heat Radiation and the Hypothesis of Quanta  & \\
 & by the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & 216\\
 & Appendices  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &218\\
 & Errata  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &225
\end{tabular}
\fi
%% -----File: 015.png---Folio xx-------


\MainMatter
\Part{I}{Fundamental Facts and Definitions}
%% -----File: 016.png---Folio xx-------
%[Blank Page]
%% -----File: 017.png---Folio 1-------
%[** TN: Chapter macro prints text below]
% RADIATION OF HEAT

\Chapter{I}{General Introduction}

\Section{1.} Heat may be propagated in a stationary medium in two
entirely different ways, namely, by conduction and by radiation.
Conduction of heat depends on the temperature of the medium
in which it takes place, or more strictly speaking, on the non-uniform
distribution of the temperature in space, as measured by
the temperature gradient. In a region where the temperature
of the medium is the same at all points there is no trace of heat
conduction.

Radiation of heat, however, is in itself entirely independent of
the temperature of the medium through which it passes. It is
possible, for example, to concentrate the solar rays at a focus by
passing them through a converging lens of ice, the latter remaining
at a constant temperature of~$0°$, and so to ignite an inflammable
body. Generally speaking, radiation is a far more complicated
phenomenon than conduction of heat. The reason for this is
that the state of the radiation at a given instant and at a given
point of the medium cannot be represented, as can the flow of
heat by conduction, by a single vector (that is, a single directed
quantity). All heat rays which at a given instant pass through
the same point of the medium are perfectly independent of one
another, and in order to specify completely the state of the
radiation the intensity of radiation must be known in all the
directions, infinite in number, which pass through the point in
question; for this purpose two opposite directions must be
considered as distinct, because the radiation in one of them is
quite independent of the radiation in the other.
%% -----File: 018.png---Folio 2-------

\Section{2.} Putting aside for the present any special theory of heat
radiation, we shall state for our further use a law supported by a
large number of experimental facts. This law is that, so far as
their physical properties are concerned, heat rays are identical
with light rays of the same wave length. The term ``heat radiation,''
then, will be applied to all physical phenomena of the
same nature as light rays. Every light ray is simultaneously a
heat ray. We shall also, for the sake of brevity, occasionally
speak of the ``color'' of a heat ray in order to denote its wave
length or period. As a further consequence of this law we shall
apply to the radiation of heat all the well-known laws of experimental
optics, especially those of reflection and refraction, as
well as those relating to the propagation of light. Only the
phenomena of diffraction, so far at least as they take place in
space of considerable dimensions, we shall exclude on account of
their rather complicated nature. We are therefore obliged to
introduce right at the start a certain restriction with respect to
the size of the parts of space to be considered. Throughout the
following discussion it will be assumed that the linear dimensions
of all parts of space considered, as well as the radii of curvature
of all surfaces under consideration, are large compared with the
wave lengths of the rays considered. With this assumption we
may, without appreciable error, entirely neglect the influence of
diffraction caused by the bounding surfaces, and everywhere
apply the ordinary laws of reflection and refraction of light.
To sum up: We distinguish once for all between two kinds of
lengths of entirely different orders of magnitude---dimensions of
bodies and wave lengths. Moreover, even the differentials of the
former, \ie, elements of length, area and volume, will be regarded
as large compared with the corresponding powers of wave lengths.
The greater, therefore, the wave length of the rays we wish to
consider, the larger must be the parts of space considered. But,
inasmuch as there is no other restriction on our choice of size
of the parts of space to be considered, this assumption will not
give rise to any particular difficulty.

\Section{3.} Even more essential for the whole theory of heat radiation
than the distinction between large and small lengths, is the
distinction between long and short intervals of time. For the
definition of intensity of a heat ray, as being the energy transmitted
%% -----File: 019.png---Folio 3-------
by the ray per unit time, implies the assumption that the
unit of time chosen is large compared with the period of vibration
corresponding to the color of the ray. If this were not so, obviously
the value of the intensity of the radiation would, in general,
depend upon the particular phase of vibration at which the
measurement of the energy of the ray was begun, and the intensity
of a ray of constant period and amplitude would not be independent
of the initial phase, unless by chance the unit of time
were an integral multiple of the period. To avoid this difficulty,
we are obliged to postulate quite generally that the unit of time,
or rather that element of time used in defining the intensity, even
if it appear in the form of a differential, must be large compared
with the period of all colors contained in the ray in question.

The last statement leads to an important conclusion as to
radiation of variable intensity. If, using an acoustic analogy,
we speak of ``beats'' in the case of intensities undergoing periodic
changes, the ``unit'' of time required for a definition of
the instantaneous intensity of radiation must necessarily be small
compared with the period of the beats. Now, since from the
previous statement our unit must be large compared with a period
of vibration, it follows that the period of the beats must be large
compared with that of a vibration. Without this restriction it
would be impossible to distinguish properly between ``beats''
and simple ``vibrations.'' Similarly, in the general case of an
arbitrarily variable intensity of radiation, the vibrations must
take place very rapidly as compared with the relatively slower
changes in intensity. These statements imply, of course, a certain
far-reaching restriction as to the generality of the radiation
phenomena to be considered.

It might be added that a very similar and equally essential
restriction is made in the kinetic theory of gases by dividing the
motions of a chemically simple gas into two classes: visible,
coarse, or molar, and invisible, fine, or molecular. For, since the
velocity of a single molecule is a perfectly unambiguous quantity,
this distinction cannot be drawn unless the assumption be made
that the velocity-components of the molecules contained in sufficiently
small volumes have certain mean values, independent of
the size of the volumes. This in general need not by any means be
the case. If such a mean value, including the value zero, does not
%% -----File: 020.png---Folio 4-------
exist, the distinction between motion of the gas as a whole and
random undirected heat motion cannot be made.

Turning now to the investigation of the laws in accordance with
which the phenomena of radiation take place in a medium supposed
to be at rest, the problem may be approached in two ways:
We must either select a certain point in space and investigate the
different rays passing through this one point as time goes on, or
we must select one distinct ray and inquire into its history, that
is, into the way in which it was created, propagated, and finally
destroyed. For the following discussion, it will be advisable to
start with the second method of treatment and to consider
first the three processes just mentioned.

\Section[4.]{4. Emission.}---The creation of a heat ray is generally denoted
by the word emission. According to the principle of the conservation
of energy, emission always takes place at the expense of
other forms of energy (heat,\footnote
  {Here as in the following the German ``Körperwärme'' will be rendered simply as
  ``heat.''~(Tr.)}
chemical or electric energy,~etc.)\
and hence it follows that only material particles, not geometrical
volumes or surfaces, can emit heat rays. It is true that for the
sake of brevity we frequently speak of the surface of a body as
radiating heat to the surroundings, but this form of expression
does not imply that the surface actually emits heat rays. Strictly
speaking, the surface of a body never emits rays, but rather it
allows part of the rays coming from the interior to pass through.
The other part is reflected inward and according as the fraction
transmitted is larger or smaller the surface seems to emit more or
less intense radiations.

We shall now consider the interior of an emitting substance
assumed to be physically homogeneous, and in it we shall select
any volume-element~$d\tau$ of not too small size. Then the energy
which is emitted by radiation in unit time by all particles in this
volume-element will be proportional to~$d\tau$. Should we attempt
a closer analysis of the process of emission and resolve it into its
elements, we should undoubtedly meet very complicated conditions,
for then it would be necessary to consider elements of
space of such small size that it would no longer be admissible to
think of the substance as homogeneous, and we would have to
allow for the atomic constitution. Hence the finite quantity
%% -----File: 021.png---Folio 5-------
obtained by dividing the radiation emitted by a volume-element~$d\tau$
by this element~$d\tau$ is to be considered only as a certain mean
value. Nevertheless, we shall as a rule be able to treat the phenomenon
of emission as if all points of the volume-element~$d\tau$
took part in the emission in a uniform manner, thereby greatly
simplifying our calculation. Every point of~$d\tau$ will then be the
vertex of a pencil of rays diverging in all directions. Such a
pencil coming from one single point of course does not represent
a finite amount of energy, because a finite amount is emitted
only by a finite though possibly small volume, not by a single
point.

We shall next assume our substance to be isotropic. Hence
the radiation of the volume-element~$d\tau$ is emitted uniformly in
all directions of space. Draw a cone in an arbitrary direction,
having any point of the radiating element as vertex, and describe
around the vertex as center a sphere of unit radius. This sphere
intersects the cone in what is known as the solid angle of the cone,
and from the isotropy of the medium it follows that the radiation
in any such conical element will be proportional to its solid angle.
This holds for cones of any size. If we take the solid angle as infinitely
small and of size~$d\Omega$ we may speak of the radiation emitted
in a certain direction, but always in the sense that for the emission
of a finite amount of energy an infinite number of directions
are necessary and these form a finite solid angle.

\Section{5.} The distribution of energy in the radiation is in general
quite arbitrary; that is, the different colors of a certain radiation
may have quite different intensities. The color of a ray in experimental
physics is usually denoted by its wave length, because
this quantity is measured directly. For the theoretical treatment,
however, it is usually preferable to use the frequency~$\nu$ instead,
since the characteristic of color is not so much the wave length,
which changes from one medium to another, as the frequency,
which remains unchanged in a light or heat ray passing through
stationary media. We shall, therefore, hereafter denote a certain
color by the corresponding value of~$\nu$, and a certain interval
of color by the limits of the interval $\nu$~and~$\nu'$, where $\nu' > \nu$. The
radiation lying in a certain interval of color divided by the magnitude
$\nu' - \nu$ of the interval, we shall call the mean radiation in the
interval $\nu$~to~$\nu'$. We shall then assume that if, keeping $\nu$ constant,
%% -----File: 022.png---Folio 6-------
we take the interval $\nu' - \nu$ sufficiently small and denote it by~$d\nu$
the value of the mean radiation approaches a definite limiting
value, independent of the size of~$d\nu$, and this we shall briefly call
the ``radiation of frequency~$\nu$.'' To produce a finite intensity
of radiation, the frequency interval, though perhaps small, must
also be finite.

We have finally to allow for the polarization of the emitted
radiation. Since the medium was assumed to be isotropic the
emitted rays are unpolarized. Hence every ray has just twice
the intensity of one of its plane polarized components, which
could, \eg, be obtained by passing the ray through a \Name{Nicol's}
prism.

\Section{6.} Summing up everything said so far, we may equate the total
energy in a range of frequency from $\nu$ to~$\nu + d\nu$ emitted in the
time~$dt$ in the direction of the conical element~$d\Omega$ by a \DPchg{volume
element}{volume-element}~$d\tau$ to
\[
dt · d\tau · d\Omega · d\nu · 2\epsilon_{\nu}.
\Tag{(1)}
\]
The finite quantity~$\epsilon_{\nu}$ is called the coefficient of emission of the
medium for the frequency~$\nu$. It is a positive function of~$\nu$ and
refers to a plane polarized ray of definite color and direction. The
total emission of the volume-element~$d\tau$ may be obtained from
this by integrating over all directions and all frequencies. Since
$\epsilon_{\nu}$~is independent of the direction, and since the integral over all
conical elements~$d\Omega$ is~$4\pi$, we get:
\[
dt · d\tau · 8\pi \int_{0}^{\infty} \epsilon_{\nu}\, d\nu.
\Tag{(2)}
\]

\Section{7.} The coefficient of emission~$\epsilon$ depends, not only on the frequency~$\nu$,
but also on the condition of the emitting substance
contained in the volume-element~$d\tau$, and, generally speaking,
in a very complicated way, according to the physical and chemical
processes which take place in the elements of time and volume in
question. But the empirical law that the emission of any volume-element
depends entirely on what takes place inside of this element
holds true in all cases (\Name{Prevost's} principle). A body~$A$
at $100° \Celsius$ emits toward a body~$B$ at $0° \Celsius$ exactly the same
amount of radiation as toward an equally large and similarly
situated body~$B'$ at $1000°$~C\@. The fact that the body~$A$ is cooled
%% -----File: 023.png---Folio 7-------
by~$B$ and heated by~$B'$ is due entirely to the fact that $B$~is a
weaker, $B'$~a stronger emitter than~$A$.

We shall now introduce the further simplifying assumption
that the physical and chemical condition of the emitting substance
depends on but a single variable, namely, on its absolute
temperature~$T$. A necessary consequence of this is that the
coefficient of emission~$\epsilon$ depends, apart from the frequency~$\nu$
and the nature of the medium, only on the temperature~$T$.
The last statement excludes from our consideration a number
of radiation phenomena, such as fluorescence, phosphorescence,
electrical and chemical luminosity, to which \Name{E.~Wiedemann} has
given the common name ``phenomena of luminescence.'' We
shall deal with pure ``temperature radiation'' exclusively.

A special case of temperature radiation is the case of the
chemical nature of the emitting substance being invariable. In
this case the emission takes place entirely at the expense of the
heat of the body. Nevertheless, it is possible, according to what
has been said, to have temperature radiation while chemical
changes are taking place, provided the chemical condition is completely
determined by the temperature.

\Section[8.]{8. Propagation.}---The propagation of the radiation in a medium
assumed to be homogeneous, isotropic, and at rest takes place in
straight lines and with the same velocity in all directions, diffraction
phenomena being entirely excluded. Yet, in general, each
ray suffers during its propagation a certain weakening, because
a certain fraction of its energy is continuously deviated from its
original direction and scattered in all directions. This phenomenon
of ``scattering,'' which means neither a creation nor a
destruction of radiant energy but simply a change in distribution,
takes place, generally speaking, in all media differing from an
absolute vacuum, even in substances which are perfectly pure
chemically.\footnote
  {See, \eg, \Name{Lobry de~Bruyn} and \Name{L.~K. Wolff}, Rec.\ des Trav.\ Chim.\ des Pays-Bas~\textbf{23},
  p.~155, 1904.}
The cause of this is that no substance is homogeneous
in the absolute sense of the word. The smallest elements of
space always exhibit some discontinuities on account of their
atomic structure. Small impurities, as, for instance, particles of
dust, increase the influence of scattering without, however, appreciably
affecting its general character. Hence, so-called ``turbid''
%% -----File: 024.png---Folio 8-------
media, \ie, such as contain foreign particles, may be quite properly
regarded as optically homogeneous,\footnote
  {To restrict the word homogeneous to its absolute sense would mean that it could not be
  applied to any material substance.}
provided only that the
linear dimensions of the foreign particles as well as the distances
of neighboring particles are sufficiently small compared with the
wave lengths of the rays considered. As regards optical phenomena,
then, there is no fundamental distinction between chemically
pure substances and the turbid media just described. No space
is optically void in the absolute sense except a vacuum. Hence
a chemically pure substance may be spoken of as a vacuum made
turbid by the presence of molecules.

A typical example of scattering is offered by the behavior of
sunlight in the atmosphere. When, with a clear sky, the sun
stands in the zenith, only about two-thirds of the direct radiation
of the sun reaches the surface of the earth. The remainder is
intercepted by the atmosphere, being partly absorbed and
changed into heat of the air, partly, however, scattered and
changed into diffuse skylight. This phenomenon is produced
probably not so much by the particles suspended in the atmosphere
as by the air molecules themselves.

Whether the scattering depends on reflection, on diffraction, or
on a resonance effect on the molecules or particles is a point that
we may leave entirely aside. We only take account of the fact
that every ray on its path through any medium loses a certain
fraction of its intensity. For a very small distance,~$s$, this fraction
is proportional to~$s$, say
\[
\beta_{\nu} s
\Tag{(3)}
\]
where the positive quantity~$\beta_{\nu}$ is independent of the intensity of
radiation and is called the ``coefficient of scattering'' of the medium.
Inasmuch as the medium is assumed to be isotropic, $\beta_{\nu}$~is
also independent of the direction of propagation and polarization
of the ray. It depends, however, as indicated by the
subscript~$\nu$, not only on the physical and chemical constitution of
the body but also to a very marked degree on the frequency.
For certain values of~$\nu$, $\beta_{\nu}$~may be so large that the straight-line
propagation of the rays is virtually destroyed. For other values
of~$\nu$, however, $\beta_{\nu}$~may become so small that the scattering can
%% -----File: 025.png---Folio 9-------
be entirely neglected. For generality we shall assume a mean
value of~$\beta_{\nu}$. In the cases of most importance $\beta_{\nu}$~increases quite
appreciably as $\nu$~increases, \ie, the scattering is noticeably larger
for rays of shorter wave length;\footnote
  {\Name{Lord Rayleigh}, Phil.\ Mag., \textbf{47}, p.~379, 1899.}
hence the blue color of diffuse skylight.

The scattered radiation energy is propagated from the place
where the scattering occurs in a way similar to that in which the
emitted energy is propagated from the place of emission, since
it travels in all directions in space. It does not, however, have
the same intensity in all directions, and moreover is polarized
in some special directions, depending to a large extent on the
direction of the original ray. We need not, however, enter into
any further discussion of these questions.

\Section{9.} While the phenomenon of scattering means a continuous
modification in the interior of the medium, a discontinuous
change in both the direction and the intensity of a ray occurs
when it reaches the boundary of a medium and meets the surface
of a second medium. The latter, like the former, will be assumed
to be homogeneous and isotropic. In this case, the ray is in
general partly reflected and partly transmitted. The reflection
and refraction may be ``regular,'' there being a single reflected
ray according to the simple law of reflection and a single transmitted
ray, according to \Name{Snell's} law of refraction, or, they may be
``diffuse,'' which means that from the point of incidence on the
surface the radiation spreads out into the two media with intensities
that are different in different directions. We accordingly
describe the surface of the second medium as ``smooth'' or
``rough'' respectively. Diffuse reflection occurring at a rough
surface should be carefully distinguished from reflection at a
smooth surface of a turbid medium. In both cases part of the
incident ray goes back to the first medium as diffuse radiation.
But in the first case the scattering occurs on the surface, in the
second in more or less thick layers entirely inside of the second
medium.

\Section{10.} When a smooth surface completely reflects all incident
rays, as is approximately the case with many metallic surfaces,
it is termed ``reflecting.'' When a rough surface reflects all
incident rays completely and uniformly in all directions, it is
%% -----File: 026.png---Folio 10-------
called ``white.'' The other extreme, namely, complete transmission
of all incident rays through the surface never occurs with
smooth surfaces, at least if the two contiguous media are at all
optically different. A rough surface having the property of
completely transmitting the incident radiation is described as
``black.''

In addition to ``black surfaces'' the term ``black body'' is also
used. According to \Name{G.~Kirchhoff}\footnote
  {\Name{G.~Kirchhoff}, Pogg.\ Ann., \textbf{109}, p.~275, 1860. Gesammelte Abhandlungen, J.~A. Barth,
  Leipzig, 1882, p.~573. In defining a black body \Name{Kirchhoff} also assumes that the absorption
  of incident rays takes place in a layer ``infinitely thin.'' We do not include this in our
  definition.}
it denotes a body which has
the property of allowing all incident rays to enter without surface
reflection and not allowing them to leave again. Hence it is
seen that a black body must satisfy three independent conditions.
First, the body must have a black surface in order to allow the
incident rays to enter without reflection. Since, in general, the
properties of a surface depend on both of the bodies which are in
contact, this condition shows that the property of blackness as
applied to a body depends not only on the nature of the body
but also on that of the contiguous medium. A body which is
black relatively to air need not be so relatively to glass, and \textit{vice
versa}. Second, the black body must have a certain minimum
thickness depending on its absorbing power, in order to insure
that the rays after passing into the body shall not be able to
leave it again at a different point of the surface. The more absorbing
a body is, the smaller the value of this minimum thickness,
while in the case of bodies with vanishingly small absorbing
power only a layer of infinite thickness may be regarded as black.
Third, the black body must have a vanishingly small coefficient of
scattering (\Sec{8}). Otherwise the rays received by it would be
partly scattered in the interior and might leave again through
the surface.\footnote
  {For this point see especially \Name{A.~Schuster}, Astrophysical Journal, \textbf{21}, p.~1, 1905, who has
  pointed out that an infinite layer of gas with a black surface need by no means be a black
  body.}

\Section{11.} All the distinctions and definitions mentioned in the two
preceding paragraphs refer to rays of one definite color only.
It might very well happen that, \eg, a surface which is rough for a
certain kind of rays must be regarded as smooth for a different
kind of rays. It is readily seen that, in general, a surface shows
%% -----File: 027.png---Folio 11-------
decreasing degrees of roughness for increasing wave lengths\DPtypo{}{.}
Now, since smooth non-reflecting surfaces do not exist (\Sec{10}), it
follows that all approximately black surfaces which may be realized
in practice (lamp black, platinum black) show appreciable
reflection for rays of sufficiently long wave lengths.

\Section[12.]{12. Absorption.}---Heat rays are destroyed by ``absorption.''
According to the principle of the conservation of energy the
energy of heat radiation is thereby changed into other forms of
energy (heat, chemical energy). Thus only material particles
can absorb heat rays, not elements of surfaces, although sometimes
for the sake of brevity the expression absorbing surfaces
is used.

Whenever absorption takes place, the heat ray passing through
the medium under consideration is weakened by a certain fraction
of its intensity for every element of path traversed. For a
sufficiently small distance~$s$ this fraction is proportional to~$s$,
and may be written
\[
\alpha_{\nu}s\Add{.}
\Tag{(4)}
\]
Here $\alpha_{\nu}$~is known as the ``coefficient of absorption'' of the medium
for a ray of frequency~$\nu$. We assume this coefficient to be
independent of the intensity; it will, however, depend in general
in non-homogeneous and anisotropic media on the position of~$s$
and on the direction of propagation and polarization of the ray
(example: tourmaline). We shall, however, consider only homogeneous
isotropic substances, and shall therefore suppose that
$\alpha_{\nu}$~has the same value at all points and in all directions in the
medium, and depends on nothing but the frequency~$\nu$, the temperature~$T$,
and the nature of the medium.

Whenever $\alpha_{\nu}$~does not differ from zero except for a limited range
of the spectrum, the medium shows ``selective'' absorption. For
those colors for which $\alpha_{\nu} = 0$ and also the coefficient of scattering
$\beta_{\nu} = 0$ the medium is described as perfectly ``transparent'' or
``diathermanous.'' But the properties of selective absorption
and of diathermancy may for a given medium vary widely with
the temperature. In general we shall assume a mean value for~$\alpha_{\nu}$.
This implies that the absorption in a distance equal to a
single wave length is very small, because the distance~$s$, while
small, contains many wave lengths (\Sec{2}).
%% -----File: 028.png---Folio 12-------

\Section{13.}  The foregoing considerations regarding the emission, the
propagation, and the absorption of heat rays suffice for a mathematical
treatment of the radiation phenomena. The calculation
requires a knowledge of the value of the constants and the initial
and boundary conditions, and yields a full account of the changes
the radiation undergoes in a given time in one or more contiguous
media of the kind stated, including the  temperature changes
caused by it. The actual calculation is usually very complicated.
We shall, however, before entering upon the treatment of special
cases discuss the general radiation phenomena from a different
point of view, namely by fixing our attention not on a definite
ray, but on a definite position in space.

\Section{14.}  Let $d\sigma$~be an arbitrarily chosen, infinitely small element of
area in the interior of a medium through which radiation passes.
At a given instant rays are passing through this element in many
different directions. The energy radiated through it in an
element of time~$dt$ in a definite direction is proportional to the area~$d\sigma$,
the length of time~$dt$\Add{,} and to the cosine of the angle~$\theta$ made by
the normal of~$d\sigma$ with the direction of the radiation. If we make
$d\sigma$~sufficiently small, then, although this is only an approximation
to the actual state of affairs, we can think of all points in~$d\sigma$ as
being affected by the radiation in the same way. Then the
energy radiated through~$d\sigma$ in a definite direction must be proportional
to the solid angle in which $d\sigma$~intercepts that radiation
and this solid angle is measured by $d\sigma \cos\theta$. It is readily seen
that, when the direction of the element is varied relatively to the
direction of the radiation, the energy radiated through it vanishes
when
\[
\theta = \frac{\pi}{2}.
\]

Now in general a pencil of rays is propagated from every point
of the element~$d\sigma$ in all directions, but with different intensities
in different directions, and any two pencils emanating from two
points of the element are identical save for differences of higher
order. A single one of these pencils coming from a single point
does not represent a finite quantity of energy, because a finite
amount of energy is radiated only through a finite area. This
holds also for the passage of rays through a so-called focus. For
%% -----File: 029.png---Folio 13-------
example, when sunlight passes through a converging lens and is
concentrated in the focal plane of the lens, the solar rays do not
converge to a single point, but each pencil of parallel rays forms
a separate focus and all these foci together constitute a surface
representing a small but finite image of the sun. A finite amount
of energy does not pass through less than a finite portion of this
surface.

\Section{15.} Let us now consider quite generally the pencil, which is
propagated from a point of the element~$d\sigma$ as vertex in all directions
of space and on both sides of~$d\sigma$. A certain direction may
be specified by the angle~$\theta$ (between $0$ and~$\pi$), as already used,
and by an azimuth~$\phi$ (between $0$ and~$2\pi$). The intensity in this
direction is the energy propagated in an infinitely thin cone limited
by $\theta$ and $\theta + d\theta$ and $\phi$~and $\phi + d\phi$. The solid angle of this
cone is
\[
d\Omega = \sin\theta · d\theta · d\phi.
\Tag{(5)}
\]
Thus the energy radiated in time~$dt$ through the element of area~$d\sigma$
in the direction of the cone~$d\Omega$ is:
\[
dt\, d\sigma\, \cos\theta\, d\Omega\, K = K \sin\theta \cos\theta\, d\theta\, d\phi\, d\sigma\, dt.
\Tag{(6)}
\]

The finite quantity~$K$ we shall term the ``specific intensity''
or the ``brightness,'' $d\Omega$~the ``solid angle'' of the pencil emanating
from a point of the element~$d\sigma$ in the direction $(\theta, \phi)$. $K$~is a
positive function of position, time, and the angles $\theta$~and~$\phi$. In
general the specific intensities of radiation in different directions
are entirely independent of one another. For example, on substituting
$\pi - \theta$ for~$\theta$ and $\pi + \phi$ for~$\phi$ in the function~$K$, we obtain the
specific intensity of radiation in the diametrically opposite
direction, a quantity which in general is quite different from the
preceding one.

For the total radiation through the element of area~$d\sigma$ toward
one side, say the one on which $\theta$~is an acute angle, we get, by
integrating with respect to~$\phi$ from $0$ to~$2\pi$ and with respect to~$\theta$
from $0$ to~$\dfrac{\pi}{2}$
\[
\int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}} d\theta\, K\sin\theta \cos\theta\, d\sigma\, dt.
\]
%% -----File: 030.png---Folio 14-------
Should the radiation be uniform in all directions and hence $K$~be
a constant, the total radiation on one side will be
\[
\pi K\, d\sigma\, dt.
\Tag{(7)}
\]

\Section{16.} In speaking of the radiation in a definite direction
$(\theta, \phi)$ one should always keep in mind that the energy radiated in a
cone is not finite unless the angle of the cone is finite. No finite
radiation of light or heat takes place in one definite direction only,
or expressing it differently, in nature there is no such thing as
absolutely parallel light or an absolutely plane wave front.
From a pencil of rays called ``parallel'' a finite amount of energy of
radiation can only be obtained if the rays or wave normals of the
pencil diverge so as to form a finite though perhaps exceedingly
narrow cone.

\Section{17.} The specific intensity~$K$ of the whole energy radiated in
a certain direction may be further divided into the intensities of
the separate rays belonging to the different regions of the spectrum
which travel independently of one another. Hence we
consider the intensity of radiation within a certain range of frequencies,
say from $\nu$ to~$\nu'$. If the interval $\nu' - \nu$ be taken sufficiently
small and be denoted by~$d\nu$, the intensity of radiation
within the interval is proportional to~$d\nu$. Such radiation is called
homogeneous or monochromatic.

A last characteristic property of a ray of definite direction,
intensity, and color is its state of polarization. If we break up a
ray, which is in any state of polarization whatsoever and which
travels in a definite direction and has a definite frequency~$\nu$,
into two plane polarized components, the sum of the intensities
of the components will be just equal to the intensity of the ray
as a whole, independently of the direction of the two planes,
provided the two planes of polarization, which otherwise may be
taken at random, are at right angles to each other. If their position
be denoted by the azimuth~$\psi$ of one of the planes of vibration
(plane of the electric vector), then the two components of the
intensity may be written in the form
\[
\begin{alignedat}{2}
&\ssfK_{\nu} \cos^{2} \psi &&+ \ssfK_{\nu}' \sin^{2} \psi \\
\LeftText{and}
&\ssfK_{\nu} \sin^{2} \psi &&+ \ssfK_{\nu}' \cos^2 \psi\Add{.}
\end{alignedat}
\Tag{(8)}
\]
Herein $\ssfK$~is independent of~$\psi$. These expressions we shall call
%% -----File: 031.png---Folio 15-------
the ``components of the specific intensity of radiation of frequency~$\nu$.''
The sum is independent of~$\psi$ and is always equal to the
intensity of the whole ray $\ssfK_{\nu} + \ssfK_{\nu}'$. At the same time $\ssfK_{\nu}$ and
$\ssfK_{\nu}'$ represent respectively the largest and smallest values which
either of the components may have, namely, when $\psi = 0$ and $\psi = \dfrac{\pi}{2}$.
Hence we call these values the ``principal values of the intensities,''
or the ``principal intensities,'' and the corresponding planes
of vibration we call the ``principal planes of vibration'' of the
ray. Of course both, in general, vary with the time. Thus we
may write generally
\[
\ssfK = \int_{\DPtypo{c}{0}}^{\infty} d\nu\, (\ssfK_{\nu} + \ssfK_{\nu}')
\Tag{(9)}
\]
where the positive quantities $\ssfK_{\nu}$ and $\ssfK_{\nu}'$, the two principal values
of the specific intensity of the radiation (brightness) of frequency~$\nu$,
depend not only on~$\nu$ but also on their position, the time,
and on the angles $\theta$~and~$\phi$. By substitution in~\Eq{(6)} the energy
radiated in the time~$dt$ through the element of area~$d\sigma$ in the direction
of the conical element~$d\Omega$ assumes the value
\[
dt\, d\sigma \cos\theta\, d\Omega \int_{0}^{\infty} d\nu\, (\ssfK_{\nu} + \ssfK_{\nu}')
\Tag{(10)}
\]
and for monochromatic plane polarized radiation of brightness~$\ssfK_{\nu}$:
\[
dt\, d\sigma \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu
  = dt\, d\sigma \sin\theta \cos\theta\, d\theta\, d\phi\, \ssfK_{\nu}\, d\nu.
\Tag{(11)}
\]
For unpolarized rays $\ssfK_{\nu} = \ssfK_{\nu}'$, and hence
\[
K = 2 \int_{0}^{\infty} d\nu\, \ssfK_{\nu},
\Tag{(12)}
\]
and the energy of a monochromatic ray of frequency~$\nu$ will be:
\[
2\, dt\, d\sigma \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu
  = 2\, dt\, d\sigma \sin\theta \cos\theta\, d\theta\, d\phi\, \ssfK_{\nu}\, d\nu.
\Tag{(13)}
\]
When, moreover, the radiation is uniformly distributed in all
directions, the total radiation through~$d\sigma$ toward one side may be
found from \Eq{(7)}~and~\Eq{(12)}; it is
\[
2\pi\, d\sigma\, dt \int_{0}^{\infty} \ssfK_{\nu}\, d\nu.
\Tag{(14)}
\]
%% -----File: 032.png---Folio 16-------

\Section{18.} Since in nature $\ssfK_{\nu}$~can never be infinitely large, $K$~will not
have a finite value unless $\ssfK_{\nu}$~differs from zero over a finite range
of frequencies. Hence there exists in nature no absolutely
homogeneous or monochromatic radiation of light or heat. A
finite amount of radiation contains always a finite although possibly
very narrow range of the spectrum. This implies a fundamental
difference from the corresponding phenomena of acoustics,
where a finite intensity of sound may correspond to a single
definite frequency. This difference is, among other things, the
cause of the fact that the second law of thermodynamics has an
important bearing on light and heat rays, but not on sound waves.
This will be further discussed later on.

\Section{19.} From equation~\Eq{(9)} it is seen that the quantity~$\ssfK_{\nu}$, the
intensity of radiation of frequency~$\nu$, and the quantity~$K$, the
intensity of radiation of the whole spectrum, are of different
dimensions. Further it is to be noticed that, on subdividing
the spectrum according to wave lengths~$\lambda$, instead of frequencies~$\nu$,
the intensity of radiation~$E_{\lambda}$ of the wave lengths~$\lambda$ corresponding
to the frequency~$\nu$ is not obtained simply by replacing~$\nu$ in
the expression for~$\ssfK_{\nu}$ by the corresponding value of~$\lambda$ deduced
from
\[
\nu = \frac{q}{\lambda}
\Tag{(15)}
\]
where $q$~is the velocity of propagation. For if $d\lambda$~and~$d\nu$ refer to
the same interval of the spectrum, we have, not $E_{\lambda} = \ssfK_{\nu}$, but
$E_{\lambda}\, d\lambda = \ssfK_{\nu}\, d\nu$. By differentiating~\Eq{(15)} and paying attention
to the signs of corresponding values of $d\lambda$~and~$d\nu$ the equation
\[
d\nu = \frac{q\, d\lambda}{\lambda^{2}}
\]
is obtained. Hence we get by substitution:
\[
E_{\lambda} = \frac{q \ssfK_{\nu}}{\lambda^{2}}.
\Tag{(16)}
\]
This relation shows among other things that in a certain spectrum
the maxima of~$E_{\lambda}$ and~$\ssfK_{\nu}$ lie at different points of the spectrum.

\Section{20.} When the principal intensities $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ of all monochromatic
rays are given at all points of the medium and for all
directions, the state of radiation is known in all respects and all
%% -----File: 033.png---Folio 17-------
questions regarding it may be answered. We shall show this by
one or two applications to special cases. Let us first find the
amount of energy which is radiated through any element of area~$d\sigma$
toward any other element~$d\sigma'$. The distance~$r$ between the
two elements may be thought of as large compared with the
linear dimensions of the elements $d\sigma$ and~$d\sigma'$ but still so small
that no appreciable amount of radiation is absorbed or scattered
along it. This condition is, of course, superfluous for diathermanous
media.

From any definite point of~$d\sigma$ rays pass to all points of~$d\sigma'$.
These rays form a cone whose vertex lies in~$d\sigma$ and whose solid
angle is
\[
d\Omega = \frac{d\sigma' \cos (\DPchg{\nu'}{\Norm'}, r)}{r^{2}}
\]
%[** TN: \nu briefly denotes both a surface normal and frequency]
where $\DPchg{\nu'}{\Norm'}$~denotes the normal of~$d\sigma'$ and the angle $(\DPchg{\nu'}{\Norm'}, r)$ is to be
taken as an acute angle. This value of~$d\Omega$ is, neglecting small
quantities of higher order, independent of the particular position
of the vertex of the cone on~$d\sigma$.

If we further denote the normal to~$d\sigma$ by~$\DPchg{\nu}{\Norm}$ the angle~$\theta$ of~\Eq{(14)}
will be the angle $(\DPchg{\nu}{\Norm}, r)$ and hence from expression~\Eq{(6)} the energy of
radiation required is found to be:
\[
K · \frac{d\sigma\, d\sigma' \cos(\DPchg{\nu}{\Norm}, r) \DPchg{·}{} \cos(\DPchg{\nu'}{\Norm'}, r)}{r^{2}}\DPchg{\,}{·} dt.
\Tag{(17)}
\]
For monochromatic plane polarized radiation of frequency~$\nu$ the
energy will be, according to equation~\Eq{(11)},
\[
\ssfK_{\nu}\, d\nu · \frac{d\sigma\, d\sigma' \cos(\DPchg{\nu}{\Norm}, r) \cos(\DPchg{\nu'}{\Norm'}, r)}{r^{2}} · dt.
\Tag{(18)}
\]

The relative size of the two elements $d\sigma$~and~$d\sigma'$ may have any
value whatever. They may be assumed to be of the same or of a
different order of magnitude, provided the condition remains
satisfied that $r$~is large compared with the linear dimensions of
each of them. If we choose $d\sigma$~small compared with~$d\sigma'$, the rays
diverge from $d\sigma$ to~$d\sigma'$, whereas they converge from $d\sigma$ to~$d\sigma'$\DPtypo{,}{}
if we choose~$d\sigma$ large compared with~$d\sigma'$.

\Section{21.} Since every point of~$d\sigma$ is the vertex of a cone spreading
out toward~$d\sigma'$, the whole pencil of rays here considered, which is
%% -----File: 034.png---Folio 18-------
defined by $d\sigma$~and~$d\sigma'$, consists of a double infinity of point pencils
or of a fourfold infinity of rays which must all be considered
equally for the energy radiation. Similarly the pencil of rays
may be thought of as consisting of the cones which, emanating
from all points of~$d\sigma$, converge in one point of~$d\sigma'$ respectively
as a vertex. If we now imagine the whole pencil of rays to be
cut by a plane at any arbitrary distance from the elements $d\sigma$
and~$d\sigma'$ and lying either between them or outside, then the
cross-sections of any two point pencils on this plane will not be
identical, not even approximately. In general they will partly
overlap and partly lie outside of each other, the amount of overlapping
being different for different intersecting planes. Hence
it follows that there is no definite cross-section of the pencil of
rays so far as the uniformity of radiation is concerned. If, however,
the intersecting plane coincides with either $d\sigma$ or~$d\sigma'$, then
the pencil has a definite cross-section. Thus these two planes
show an exceptional property. We shall call them the two
``focal planes'' of the pencil.

In the special case already mentioned above, namely, when one
of the two focal planes is infinitely small compared with the other,
the whole pencil of rays shows the character of a point pencil inasmuch
as its form is approximately that of a cone having its vertex
in that focal plane which is small compared with the other. In
that case the ``cross-section'' of the whole pencil at a definite
point has a definite meaning. Such a pencil of rays, which is
similar to a cone, we shall call an elementary pencil, and the
small focal plane we shall call the first focal plane of the elementary
pencil. The radiation may be either converging toward the
first focal plane or diverging from the first focal plane. All
the pencils of rays passing through a medium may be considered
as consisting of such elementary pencils, and hence we may base
our future considerations on elementary pencils only, which is a
great convenience, owing to their simple nature.

As quantities necessary to define an elementary pencil with a
given first focal plane~$d\sigma$, we may choose not the second focal
plane~$d\sigma'$ but the magnitude of that solid angle~$d\Omega$ under which
$d\sigma'$~is seen from~$d\sigma$. On the other hand, in the case of an arbitrary
pencil, that is, when the two focal planes are of the same
order of magnitude, the second focal plane in general cannot be
%% -----File: 035.png---Folio 19-------
replaced by the solid angle~$d\Omega$ without the pencil changing
markedly in character. For if, instead of $d\sigma'$~being given, the
magnitude and direction of~$d\Omega$, to be taken as constant for all
points of~$d\sigma$, is given, then the rays emanating from~$d\sigma$ do not
any longer form the original pencil, but rather an elementary
pencil whose first focal plane is~$d\sigma$ and whose second focal plane
lies at an infinite distance.

\Section{22.} Since the energy radiation is propagated in the medium
with a finite velocity~$q$, there must be in a finite space a finite
amount of energy. We shall therefore speak of the ``space density
of radiation,'' meaning thereby the ratio of the total quantity of
energy of radiation contained in a volume-element to the magnitude
of the latter. Let us now calculate the space density of
radiation~$u$ at any arbitrary point of the medium. When we
consider an infinitely small element of volume~$v$ at the point in
question, having any shape whatsoever, we must allow for all
rays passing through the volume-element~$v$. For this purpose
we shall construct about any point~$O$ of~$v$ as center a sphere
\Figure[2.5in]{1}
of radius~$r$, $r$~being large compared
with the linear dimensions of~$v$ but
still so small that no appreciable
absorption or scattering of the radiation
takes place in the distance~$r$
(\Fig{1}). Every ray which reaches~$v$
must then come from some point
on the surface of the sphere. If,
then, we at first consider only all the
rays that come from the points of an
infinitely small element of area~$d\sigma$
on the surface of the sphere, and
reach~$v$, and then sum up for all elements of the spherical surface,
we shall have accounted for all rays and not taken any one
more than once.

Let us then calculate first the amount of energy which is contributed
to the energy contained in~$v$ by the radiation sent from
such an element~$d\sigma$ to~$v$. We choose~$d\sigma$ so that its linear dimensions
are small compared with those of~$v$ and consider the cone of
rays which, starting at a point of~$d\sigma$, meets the volume~$v$. This
cone consists of an infinite number of conical elements with the
%% -----File: 036.png---Folio 20-------
common vertex at~$P$, a point of~$d\sigma$, each cutting out of the volume~$v$
a certain element of length, say~$s$. The solid angle of such a
conical element is $\dfrac{f}{r^{2}}$ where $f$~denotes the area of cross-section
normal to the axis of the cone at a distance~$r$ from the vertex.
The time required for the radiation to pass through the distance~$s$
is:
\[
\tau = \frac{s}{q}\Add{.}
\]
From expression~\Eq{(6)} we may find the energy radiated through a
certain element of area. In the present case $d\Omega = \dfrac{f}{r^{2}}$ and $\theta = 0$;
hence the energy is:
\[
\tau\, d\sigma\, \frac{f}{r^{2}} K = \frac{fs}{r^{2}q} · K\, d\sigma.
\Tag{(19)}
\]
This energy enters the conical element in~$v$ and spreads out into
the volume~$fs$. Summing up over all conical elements that start
from~$d\sigma$ and enter~$v$ we have
\[
\frac{K\, d\sigma}{r^{2}q} \sum fs = \frac{K\, d\sigma}{r^{2}q} v.
\]
This represents the entire energy of radiation contained in the
volume~$v$, so far as it is caused by radiation through the element~$d\sigma$.
In order to obtain the total energy of radiation contained
in~$v$ we must integrate over all elements~$d\sigma$ contained in the surface
of the sphere. Denoting by~$d\Omega$ the solid angle $\dfrac{d\sigma}{r^{2}}$ of a
cone which has its center in~$O$ and intersects in~$d\sigma$ the surface of
the sphere, we get for the whole energy:
\[
\frac{v}{q} \int K\, d\Omega.
\]
The volume density of radiation required is found from this by
dividing by~$v$. It is
\[
u = \frac{1}{q} \int K\, d\Omega.
\Tag{(20)}
\]
%% -----File: 037.png---Folio 21-------

Since in this expression $r$~has disappeared, we can think of~$K$
as the intensity of radiation at the point~$O$ itself. In integrating,
it is to be noted that $K$~in general depends on the direction $(\theta, \phi)$.
For radiation that is uniform in all directions $K$~is a constant and
on integration we get:
\[
u = \frac{4\pi K}{q}\Add{.}
\Tag{(21)}
\]

\Section{23.} A meaning similar to that of the volume density of the
total radiation~$u$ is attached to the volume density of radiation
of a definite frequency~$\ssfu_{\nu}$. Summing up for all parts of the spectrum
we get:
\[
u = \int_{0}^{\infty} \ssfu_{\nu}\, d\nu.
\Tag{(22)}
\]

Further by combining equations \Eq{(9)}~and~\Eq{(20)} we have:
\[
\ssfu_{\nu} = \frac{1}{q} \int (\ssfK_{\nu} + \ssfK_{\nu}')\, d\Omega,
\Tag{(23)}
\]
and finally for unpolarized radiation uniformly distributed in all
directions:
\[
\ssfu_{\nu} = \frac{8\pi \ssfK_{\nu}}{q}\Add{.}
\Tag{(24)}
\]
%% -----File: 038.png---Folio 22-------

\Chapter[Radiation at Thermodynamic Equilibrium]{II}{Radiation at Thermodynamic Equilibrium.
Kirchhoff's Law. Black Radiation}

\Section{24.} We shall now apply the laws enunciated in the last chapter
to the special case of thermodynamic equilibrium, and hence
we begin our consideration by stating a certain consequence of
the second principle of thermodynamics: A system of bodies of
arbitrary nature, shape, and position which is at rest and is surrounded
by a rigid cover impermeable to heat will, no matter
what its initial state may be, pass in the course of time into a
permanent state, in which the temperature of all bodies of the
system is the same. This is the state of thermodynamic equilibrium,
in which the entropy of the system has the maximum value
compatible with the total energy of the system as fixed by the
initial conditions. This state being reached, no further increase
in entropy is possible.

In certain cases it may happen that, under the given conditions,
the entropy can assume not only one but several maxima, of
which one is the absolute one, the others having only a relative
significance.\footnote
  {See, \eg, \Name{M.~Planck}, Vorlesungen über Thermodynamik, Leipzig, Veit and Comp., 1911
  (or English Translation, Longmans Green \&~Co.), Secs.\ 165 and~189, \textit{et~seq.}}
In these cases every state corresponding to a maximum
value of the entropy represents a state of thermodynamic
equilibrium of the system. But only one of them, the one corresponding
to the absolute maximum of entropy, represents the
absolutely stable equilibrium. All the others are in a certain
sense unstable, inasmuch as a suitable, however small, disturbance
may produce in the system a permanent change in the
equilibrium in the direction of the absolutely stable equilibrium.
An example of this is offered by supersaturated steam enclosed in
a rigid vessel or by any explosive substance. We shall also meet
such unstable equilibria in the case of radiation phenomena
(\Sec{52}).
%% -----File: 039.png---Folio 23-------

\Section{25.} We shall now, as in the previous chapter, assume that we
are dealing with homogeneous isotropic media whose condition
depends only on the temperature, and we shall inquire what laws
the radiation phenomena in them must obey in order to be consistent
with the deduction from the second principle mentioned
in the preceding section. The means of answering this inquiry
is supplied by the investigation of the state of thermodynamic
equilibrium of one or more of such media, this investigation to be
conducted by applying the conceptions and laws established in
the last chapter.

We shall begin with the simplest case, that of a single medium
extending very far in all directions of space, and, like all systems
we shall here consider, being surrounded by a rigid cover impermeable
to heat. For the present we shall assume that the
medium has finite coefficients of absorption, emission, and
scattering.

Let us consider, first, points of the medium that are far away
from the surface. At such points the influence of the surface is,
of course, vanishingly small and from the homogeneity and the
isotropy of the medium it will follow that in a state of thermodynamic
equilibrium the radiation of heat has everywhere and in all
directions the same properties. Then $\ssfK_{\nu}$, the specific intensity of
radiation of a plane polarized ray of frequency~$\nu$ (\Sec{17}), must be
independent of the azimuth of the plane of polarization as well as
of position and direction of the ray. Hence to each pencil of
rays starting at an element of area~$d\sigma$ and diverging within
a conical element~$d\Omega$ corresponds an exactly equal pencil of opposite
direction converging within the same conical element toward
the element of area.

Now the condition of thermodynamic equilibrium requires
that the temperature shall be everywhere the same and shall not
vary in time. Therefore in any given arbitrary time just as
much radiant heat must be absorbed as is emitted in each volume-element
of the medium. For the heat of the body depends
only on the heat radiation, since, on account of the uniformity in
temperature, no conduction of heat takes place. This condition
is not influenced by the phenomenon of scattering, because scattering
refers only to a change in direction of the energy radiated,
not to a creation or destruction of it. We shall, therefore, calculate
%% -----File: 040.png---Folio 24-------
the energy emitted and absorbed in the time~$dt$ by a
volume-element~$v$.

According to equation~\Eq{(2)} the energy emitted has the value
\[
dt\, v · 8\pi \int_{0}^{\infty} \epsilon_{\nu}\, d\nu
\]
where $\epsilon_{\nu}$, the coefficient of emission of the medium, depends only
on the frequency~$\nu$ and on the temperature in addition to the
chemical nature of the medium.

\Section{26.} For the calculation of the energy absorbed we shall employ
the same reasoning as was illustrated by \Fig{1} (\Sec{22}) and
shall retain the notation there used. The radiant energy
absorbed by the volume-element~$v$ in the time~$dt$ is found by considering
the intensities of all the rays passing through the element~$v$
and taking that fraction of each of these rays which is absorbed
in~$v$. Now, according to~\Eq{(19)}, the conical element that starts
from~$d\sigma$ and cuts out of the volume~$v$ a part equal to~$fs$ has the
intensity (energy radiated per unit time)
\[
d\sigma · \frac{f}{r^{2}} · K
\]
or, according to~\Eq{(12)}, by considering the different parts of the
spectrum separately:
\[
2\, d\sigma\, \frac{f}{r^{2}} \int_{0}^{\infty} \ssfK_{\nu}\, d\nu.
\]
Hence the intensity of a monochromatic ray is:
\[
2\, d\sigma\, \frac{f}{r^{2}} \ssfK_{\nu}\, d\nu.
\]
The amount of energy of this ray absorbed in the distance~$s$ in
the time~$dt$ is, according to~\Eq{(4)},
\[
dt\, \alpha_{\nu}s\, 2\, d\sigma\, \frac{f}{r^{2}} \ssfK_{\nu}\, d\nu.
\]
Hence the absorbed part of the energy of this small cone of rays,
as found by integrating over all frequencies, is:
\[
dt\, 2\, d\sigma\, \frac{fs}{r^{2}} \int_{0}^{\infty} \alpha_{\nu} \ssfK_{\nu}\, d\nu.
\]
%% -----File: 041.png---Folio 25-------
When this expression is summed up over all the different cross-sections~$f$
of the conical elements starting at~$d\sigma$ and passing
through~$v$, it is evident that $\sum fs = v$, and when we sum up over
all elements~$d\sigma$ of the spherical surface of radius~$r$ we have
\[
\int \frac{d\sigma}{r^{2}} = 4\pi.
\]
Thus for the total radiant energy absorbed in the time~$d$t by the
volume-element~$v$ the following expression is found:
\[
dt\, v\, 8\pi \int_{0}^{\infty} \alpha_{\nu} \ssfK_{\nu}\, d\nu.
\Tag{(25)}
\]
By equating the emitted and absorbed energy we obtain:
\[
\int_{0}^{\infty} \epsilon_{\nu}\, d\nu
  = \int_{0}^{\infty} \alpha_{\nu} \ssfK_{\nu}\, d\nu.
\]

A similar relation may be obtained for the separate parts of the
spectrum. For the energy emitted and the energy absorbed in the
state of thermodynamic equilibrium are equal, not only for the
entire radiation of the whole spectrum, but also for each monochromatic
radiation. This is readily seen from the following. The
magnitudes of $\epsilon_{\nu}$,~$\alpha_{\nu}$, and~$\ssfK_{\nu}$ are independent of position. Hence,
if for any single color the absorbed were not equal to the emitted
energy, there would be everywhere in the whole medium a continuous
increase or decrease of the energy radiation of that
particular color at the expense of the other colors. This would be
contradictory to the condition that $\ssfK_{\nu}$~for each separate frequency
does not change with the time. We have therefore for each
frequency the relation:
\begin{align*}
  \epsilon_{\nu} &= \alpha_{\nu} \ssfK_{\nu}, \textrm{ or}
\Tag{(26)}\\
  \ssfK_{\nu} &= \frac{\epsilon_{\nu}}{\alpha_{\nu}},
\Tag{(27)}
\end{align*}
%[** Semantic theorem]
\ie: \emph{in the interior of a medium in a state of thermodynamic equilibrium
the specific intensity of radiation of a certain frequency is
equal to the coefficient of emission divided by the coefficient of absorption
of the medium for this frequency}.
%% -----File: 042.png---Folio 26-------

\Section{27.} Since $\epsilon_{\nu}$~and~$\alpha_{\nu}$ depend only on the nature of the medium,
the temperature, and the frequency~$\nu$, the intensity of radiation of
a definite color in the state of thermodynamic equilibrium is
completely defined by the nature of the medium and the temperature.
An exceptional case  is when $\alpha_{\nu} = 0$, that is, when the
medium does not at all absorb the color in question. Since $\ssfK_{\nu}$~cannot
become infinitely large, a first consequence of this is that
in that case $\epsilon_{\nu} = 0$ also, that is, a medium does not emit any color
which it does not absorb. A second consequence is that if $\epsilon_{\nu}$~and~$\alpha_{\nu}$
both vanish, equation~\Eq{(26)} is satisfied by every value of~$\ssfK_{\nu}$.
%[** Theorem]
\emph{In a medium which is diathermanous for a certain color
thermodynamic equilibrium can exist for any intensity of radiation
whatever of that color.}

This supplies an immediate illustration of the cases spoken of
before (\Sec{24}), where, for a given value of the total energy of a
system enclosed by a rigid cover impermeable to heat, several
states of equilibrium can exist, corresponding to several relative
maxima of the entropy. That is to say, since the intensity of
radiation of the particular color in the state of thermodynamic
equilibrium is quite independent of the temperature of a medium
which is diathermanous for this color, the given total energy may
be arbitrarily distributed between radiation of that color and the
heat of the body, without making thermodynamic equilibrium
impossible. Among all these distributions there is one particular
one, corresponding to the absolute maximum of entropy, which
represents absolutely stable equilibrium. This one, unlike all the
others, which are in a certain sense unstable, has the property of
not being appreciably affected by a small disturbance. Indeed
we shall see later (\Sec{48}) that among the infinite number of
values, which the quotient $\dfrac{\epsilon_{\nu}}{\alpha_{\nu}}$ can have, if numerator and denominator
both vanish, there exists one particular one which depends
in a definite way on the nature of the medium, the frequency~$\nu$,
and the temperature. This distinct value of the fraction is
accordingly called the stable intensity of radiation~$\ssfK_{\nu}$, in the medium,
which at the temperature in question is diathermanous for
rays of the frequency~$\nu$.

Everything that has just been said of a medium which is diathermanous
for a certain kind of rays holds true for an absolute
%% -----File: 043.png---Folio 27-------
vacuum, which is a medium diathermanous for rays of all kinds,
the only difference being that one cannot speak of the heat and
the temperature of an absolute vacuum in any definite sense.

For the present we again shall put the special case of diathermancy
aside and assume that all the media considered have a
finite coefficient of absorption.

\Section{28.} Let us now consider briefly the phenomenon of scattering
at thermodynamic equilibrium. Every ray meeting the volume-element~$v$
suffers there, apart from absorption, a certain weakening
of its intensity because a certain fraction of its energy is
diverted in different directions. The value of the total energy
of scattered radiation received and diverted, in the time~$dt$ by
the volume-element~$v$ in all directions, may be calculated from
expression~\Eq{(3)} in exactly the same way as the value of the absorbed
energy was calculated in \Sec{26}. Hence we get an expression
similar to~\Eq{(25)}, namely,
\[
dt\, v\, 8\pi \int_{0}^{\infty} \beta_{\nu} \ssfK_{\nu}\, d\nu.
\Tag{(28)}
\]
The question as to what becomes of this energy is readily answered.
On account of the isotropy of the medium, the energy
scattered in~$v$ and given by~\Eq{(28)} is radiated uniformly in all directions
just as in the case of the energy entering~$v$. Hence that part
of the scattered energy received in~$v$ which is radiated out in a
cone of solid angle~$d\Omega$ is obtained by multiplying the last expression
by~$\dfrac{d\Omega}{4\pi}$. This gives
\[
2\, dt\, v\, d\Omega \int_{0}^{\infty} \beta_{\nu} \ssfK_{\nu}\, d\nu,
\]
and, for monochromatic plane polarized radiation,
\[
dt\, v\, d\Omega\, \beta_{\nu} \ssfK_{\nu}\, d\nu.
\Tag{(29)}
\]

Here it must be carefully kept in mind that this uniformity of
radiation in all directions holds only for all rays striking the element~$v$
taken together; a single ray, even in an isotropic medium,
is scattered in different directions with different intensities and
different directions of polarization. (See end of \Sec{8}.)
%% -----File: 044.png---Folio 28-------

It is thus found that, when thermodynamic equilibrium of radiation
exists inside of the medium, the process of scattering produces,
on the whole, no effect. The radiation falling on a volume-element
from all sides and scattered from it in all directions behaves
exactly as if it had passed directly through the volume-element
without the least modification. Every ray loses by
scattering just as much energy as it regains by the scattering of
other rays.

\WrapFigure[0.75in]{2}
\Section{29.} We shall now consider from a different point of view the
radiation phenomena in the interior of a very extended homogeneous
isotropic medium which is in thermodynamic
equilibrium. That is to say, we shall confine our
attention, not to a definite volume-element, but to a
definite pencil, and in fact to an elementary pencil
(\Sec{21}). Let this pencil be specified by the infinitely
small focal plane~$d\sigma$ at the point~$O$ (\Fig{2}), perpendicular
to the axis of the pencil, and by the solid
angle~$d\Omega$, and let the radiation take place toward the
focal plane in the direction of the arrow. We shall
consider exclusively rays which belong to this pencil.

The energy of monochromatic plane polarized radiation
of the pencil considered passing in unit time
through~$d\sigma$ is represented, according to~\Eq{(11)}, since in
this case $dt = 1$, $\theta = 0$, by
\[
d\sigma\, d\Omega\, \ssfK_{\nu}\, d\nu.
\Tag{(30)}
\]
The same value holds for any other cross-section of the pencil.
For first, $\ssfK_{\nu}\, d\nu$ has everywhere the same magnitude (\Sec{25}),
and second, the product of any right section of the pencil and
the solid angle at which the focal plane~$d\sigma$ is seen from this section
has the constant value~$d\sigma\, d\Omega$, since the magnitude of the
cross-section increases with the distance from the vertex~$O$ of
the pencil in the proportion in which the solid angle decreases.
Hence the radiation inside of the pencil takes place just as if the
medium were perfectly diathermanous.

On the other hand, the radiation is continuously modified along
its path by the effect of emission, absorption, and scattering. We
shall consider the magnitude of these effects separately.

\Section{30.} Let a certain volume-element of the pencil be bounded by
%% -----File: 045.png---Folio 29-------
two cross-sections at distances equal to~$r_{0}$ (of arbitrary length)
and $r_{0} + dr_{0}$ respectively from the vertex~$O$. The volume will be
represented by $dr_{0} · r_{0}^{2}\, d\Omega$. It emits in unit time toward the
focal plane~$d\sigma$ at~$O$ a certain quantity~$E$ of energy of monochromatic
plane polarized radiation. $E$~may be obtained from~\Eq{(1)}
by putting
\[
dt = 1,\quad
d\tau = dr_{0}\, r_{0}^{2}\, d\Omega,\quad
d\Omega = \frac{d\sigma}{r_{0}^{2}}
\]
and omitting the numerical factor~$2$. We thus get
\[
%[** Inconstsistent dot]
E = dr_{0} · d\Omega\, d\sigma\, \epsilon_{\nu}\, d\nu.
\Tag{(31)}
\]

Of the energy~$E$, however, only a fraction~$E_{0}$ reaches~$O$, since
in every infinitesimal element of distance~$s$ which it traverses
before reaching~$O$ the fraction $(\alpha_{\nu} + \beta_{\nu})s$ is lost by absorption and
scattering. Let $E_{r}$~represent that part of~$E$ which reaches a
cross-section at a distance~$r$ ($< r_{0}$) from~$O$. Then for a small
distance $s = dr$ we have
\[
E_{r + dr} - E_{r} = E_{r}(\alpha_{\nu} + \beta_{\nu})\, dr,
\]
or,
\[
\frac{dE_{r}}{dr} = E_{r}(\alpha_{\nu} + \beta_{\nu}),
\]
and, by integration,
\[
E_{r} = E e^{(\alpha_{\nu} + \beta_{\nu})(r-r_{0})}
\]
since, for $r = r_{0}$, $E_{r} = E$ is given by equation~\Eq{(31)}. From this, by
putting $r = 0$, the energy emitted by the volume-element at~$r_{0}$
which reaches~$O$ is found to be
\[
E_{0} = E e^{-(\alpha_{\nu} + \beta_{\nu})r_{0}}
  = dr_{0}\, d\Omega\, d\sigma\, \epsilon_{\nu} e^{-(\alpha_{\nu} + \beta_{\nu})r_{0}}\, d\nu.
\Tag{(32)}
\]
All volume-elements of the pencils combined produce by their
emission an amount of energy reaching~$d\sigma$ equal to
\[
d\Omega\, d\sigma\, d\nu\, \epsilon_{\nu} \int_{0}^{\infty} dr_{0}\, e^{-(\alpha_{\nu} + \beta_{\nu})r_{0}}
  = d\Omega\, d\sigma\, \frac{\epsilon_{\nu}}{\alpha_{\nu} + \beta_{\nu}}\, d\nu.
\Tag{(33)}
\]

\Section{31.} If the scattering did not affect the radiation, the total
energy reaching~$d\sigma$ would necessarily consist of the quantities of
energy emitted by the different volume-elements of the pencil,
allowance being made, however, for the losses due to absorption
%% -----File: 046.png---Folio 30-------
on the way. For $\beta_{\nu} = 0$ expressions \Eq{(33)} and \Eq{(30)} are identical,
as may be seen by comparison with~\Eq{(27)}. Generally, however,
\Eq{(30)}~is larger than~\Eq{(33)} because the energy reaching~$d\sigma$ contains
also some rays which were not at all emitted from elements inside
of the pencil, but somewhere else, and have entered later on by
scattering. In fact, the volume-elements of the pencil do not
merely scatter outward the radiation which is being transmitted
inside the pencil, but they also collect into the pencil rays coming
from without. The radiation~$E'$ thus collected by the volume-element
at~$r_{0}$ is found, by putting in~\Eq{(29)},
\[
dt = 1,\quad
\nu = dr_{0}\, d\Omega\, r_{0}^{2},\quad
d\Omega = \frac{d\sigma}{r_{0}^{2}},
\]
to be
\[
E' = dr_{0}\, d\Omega\, d\sigma\, \beta_{\nu} \ssfK_{\nu}\, d\nu.
\]

This energy is to be added to the energy~$E$ emitted by the volume-element,
which we have calculated in~\Eq{(31)}. Thus for the
total energy contributed to the pencil in the volume-element at~$r_{0}$
we find:
\[
E + E' = dr_{0}\, d\Omega\, d\sigma\, (\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu})\, d\nu.
\]
The part of this reaching~$O$ is, similar to~\Eq{(32)}:
\[
dr_{0}\, d\Omega\, d\sigma\, (\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu})\, d\nu\, e^{-r_{0} (\alpha_{\nu} + \beta_{\nu})}\Add{.}
\]
Making due allowance for emission and collection of scattered
rays entering on the way, as well as for losses by absorption and
scattering, all volume-elements of the pencil combined give for
the energy ultimately reaching~$d\sigma$
\[
d\Omega\, d\sigma\, (\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu})\, d\nu
  \int_{0}^{\infty} dr_{0}\, e^{-r_{0} (\alpha_{\nu} + \beta_{\nu})}
  = d\Omega\, d\sigma\, \frac{\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu}}{\alpha_{\nu} + \beta_{\nu}}\, d\nu,
\]
and this expression is really exactly equal to that given by~\Eq{(30)},
as may be seen by comparison with~\Eq{(26)}.

\Section{32.} The laws just derived for the state of radiation of a homogeneous
isotropic medium when it is in thermodynamic equilibrium
hold, so far as we have seen, only for parts of the medium
which lie very far away from the surface, because for such parts
only may the radiation be considered, by symmetry, as independent
of position and direction. A simple consideration, however,
%% -----File: 047.png---Folio 31-------
shows that the value of~$\ssfK_{\nu}$, which was already calculated and given
by~\Eq{(27)}, and which depends only on the temperature and the
nature of the medium, gives the correct value of the intensity of
radiation of the frequency considered for all directions up to
points directly below the surface of the medium. For in the state
of thermodynamic equilibrium every ray must have just the same
intensity as the one travelling in an exactly opposite direction,
since otherwise the radiation would cause a unidirectional transport
of energy. Consider then any ray coming from the surface
of the medium and directed inward; it must have the same
intensity as the opposite ray, coming from the interior. A
%[** Theorem]
further immediate consequence of this is \emph{that the total state of
radiation of the medium is the same on the surface as in the interior}.

\Section{33.} While the radiation that starts from a surface element and
is directed toward the interior of the medium is in every respect
equal to that emanating from an equally large parallel element of
area in the interior, it nevertheless has a different history. That
is to say, since the surface of the medium was assumed to be
impermeable to heat, it is produced only by reflection at the surface
of radiation coming from the interior. So far as special
details are concerned, this can happen in very different ways,
depending on whether the surface is assumed to be smooth, \ie,
in this case reflecting, or rough, \eg, white (\Sec{10}). In the first
case there corresponds to each pencil which strikes the surface
another perfectly definite pencil, symmetrically situated and
having the same intensity, while in the second case every incident
pencil is broken up into an infinite number of reflected pencils,
each having a different direction, intensity, and polarization.
While this is the case, nevertheless the rays that strike a surface-element
from all different directions with the same intensity~$\ssfK_{\nu}$
also produce, all taken together, a uniform radiation of the same
intensity~$\ssfK_{\nu}$, directed toward the interior of the medium.

\Section{34.} Hereafter there will not be the slightest difficulty in
dispensing with the assumption made in \Sec{25} that the medium
in question extends very far in all directions. For after thermodynamic
equilibrium has been everywhere established in our medium,
the equilibrium is, according to the results of the last
paragraph, in no way disturbed, if we assume any number of
rigid surfaces impermeable to heat and rough or smooth to be
%% -----File: 048.png---Folio 32-------
inserted in the medium. By means of these the whole system is
divided into an arbitrary number of perfectly closed separate
systems, each of which may be chosen as small as the general
restrictions stated in \Sec{2} permit. It follows from this that
the value of the specific intensity of radiation~$\ssfK_{\nu}$ given in~\Eq{(27)}
remains valid for the thermodynamic equilibrium of a substance
enclosed in a space as small as we please and of any shape whatever.

\Section{35.} From the consideration of a system consisting of a single
homogeneous isotropic substance we now pass on to the treatment
of a system consisting of two different homogeneous isotropic
substances contiguous to each other, the system being, as before,
enclosed by a rigid cover impermeable to heat. We consider the
state of radiation when thermodynamic equilibrium exists, at
first, as before, with the assumption that the media are of considerable
extent. Since the equilibrium is nowise disturbed, if we
think of the surface separating the two media as being replaced
for an instant by an area entirely impermeable to heat radiation,
the laws of the last paragraphs must hold for each of the two
substances separately. Let the specific intensity of radiation of
frequency~$\nu$ polarized in an arbitrary plane be $\ssfK_{\nu}$~in the first substance
(the upper one in \Fig{3}), and $\ssfK_{\nu}'$~in the second, and, in
general, let all quantities referring to the second substance be
indicated by the addition of an accent. Both of the quantities
$\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ depend, according to equation~\Eq{(27)}, only on the temperature,
the frequency~$\nu$, and the nature of the two substances,
and these values of the intensities of radiation hold up to very
small distances from the bounding surface of the substances, and
hence are entirely independent of the properties of this surface.

\Section{36.} We shall now suppose, to begin with, that the bounding
surface of the media is smooth (\Sec{9}). Then every ray coming
from the first medium and falling on the bounding surface is
divided into two rays, the reflected and the transmitted ray.
The directions of these two rays vary with the angle of incidence
and the color of the incident ray; the intensity also varies with
its polarization. Let us denote by~$\rho$ (coefficient of reflection) the
fraction of the energy reflected, then the fraction transmitted is~$(1 - \rho)$,
$\rho$~depending on the angle of incidence, the frequency, and
the polarization of the incident ray. Similar remarks apply to
%% -----File: 049.png---Folio 33-------
$\rho'$~the coefficient of reflection of a ray coming from the second
medium and falling on the bounding surface.

Now according to~\Eq{(11)} we have for the monochromatic plane
polarized radiation of frequency~$\nu$, emitted in time~$dt$ toward the
first medium (in the direction of the feathered arrow upper left
\Figure[4in]{3}
hand in \Fig{3}), from an element~$d\sigma$ of the bounding surface and
contained in the conical element~$d\Omega$,
\[
dt\, d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu,
\Tag{(34)}
\]
where
\[
d\Omega = \sin\theta\, d\theta\, d\phi.
\Tag{(35)}
\]
This energy is supplied by the two rays which come from the first
and the second medium and are respectively reflected from or
transmitted by the element~$d\sigma$ in the corresponding direction
(the unfeathered arrows). (Of the element~$d\sigma$ only the one point~$O$
is indicated.) The first ray, according to the law of reflection,
continues in the symmetrically situated conical element~$d\Omega$, the
second in the conical element
\[
d\Omega' = \sin\theta'\, d\theta'\, d\phi'
\Tag{(36)}
\]
where, according to the law of refraction,
\[
\phi' = \phi\quad \text{and}\quad
\frac{\sin\theta}{\sin\theta'} = \frac{q}{q'}\Add{.}
\Tag{(37)}
\]
%% -----File: 050.png---Folio 34-------

If we now assume the radiation~\Eq{(34)} to be polarized either in
the plane of incidence or at right angles thereto, the same will
be true for the two radiations of which it consists, and the
radiation coming from the first medium and reflected from~$d\sigma$
contributes the part
\[
\rho\, dt\, d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu
\Tag{(38)}
\]
while the radiation coming from the second medium and transmitted
through~$d\sigma$ contributes the part
\[
(1 - \rho')\, dt\, d\sigma\, \cos\theta'\, d\Omega'\, \ssfK_{\nu}'\, d\nu.
\Tag{(39)}
\]
The quantities $dt$,~$d\sigma$,~$\nu$ and~$d\nu$ are written without the accent,
because they have the same values in both media.

By adding \Eq{(38)}~and~\Eq{(39)} and equating their sum to the expression~\Eq{(34)}
we find
\[
\rho\, \cos\theta\, d\Omega\, \ssfK_{\nu} + (1 - \rho') \cos\theta'\, d\Omega'\, \ssfK_{\nu}'
  = \cos\theta\, d\Omega\, \ssfK_{\nu}.
\]

Now from~\Eq{(37)} we have
\[
\frac{\cos\theta\, d\theta}{q} = \frac{\cos\theta'\, d\theta'}{q'}
\]
and further by \Eq{(35)}~and~\Eq{(36)}
\[
d\Omega'\, \cos\theta' = \frac{d\Omega\, \cos\theta\, q'^{2}}{q^{2}}.
\]
Therefore we find
\[
\rho \ssfK_{\nu} + (1 - \rho') \frac{q'^{2}}{q^{2}} \ssfK_{\nu}' = \ssfK
\]
or
\[
\frac{\ssfK_{\nu}}{\ssfK_{\nu}'} · \frac{q^{2}}{q'^{2}} = \frac{1 - \rho'}{1 - \rho}.
\]

\Section{37.} In the last equation the quantity on the left side is independent
of the angle of incidence~$\theta$ and of the particular kind of
polarization; hence the same must be true for the right side.
Hence, whenever the value of this quantity is known for a single
angle of incidence and any definite kind of polarization, this
value will remain valid for all angles of incidence and all kinds
of polarization. Now in the special case when the rays are
polarized at right angles to the plane of incidence and strike the
%% -----File: 051.png---Folio 35-------
bounding surface at the angle of polarization, $\rho = 0$, and $\rho' = 0$.
The expression on the right side of the last equation then becomes~$1$;
hence it must always be~$1$ and we have the general relations:
\[
\rho = \rho'
\Tag{(40)}
\]
and
\[
q^{2} \ssfK_{\nu} = q'^{2} \ssfK_{\nu}'\Add{.}
\Tag{(41)}
\]

\Section{38.} The first of these two relations, which states that the
coefficient of reflection of the bounding surface is the same on
both sides, is a special case of a general law of reciprocity first
stated by \Name{Helmholtz}.\footnote
  {\Name{H.~v.\ Helmholtz}, Handbuch d.\ physiologischen Optik~1. Lieferung, Leipzig, Leop.\ Voss,
  1856, p.~169. See also \Name{Helmholtz}, Vorlesungen über die Theorie der Wärme herausgegeben
  von \Name{F.~Richarz}, Leipzig, J.~A. Barth, 1903, p.~161. The restrictions of the law of reciprocity
  made there do not bear on our problems, since we are concerned with temperature radiation
  only (\Sec{7}).}
According to this law the loss of intensity
which a ray of definite color and polarization suffers on its way
through any media by reflection, refraction, absorption, and
scattering is exactly equal to the loss suffered by a ray of the
same intensity, color, and polarization pursuing an exactly
opposite path. An immediate consequence of this law is that the
radiation striking the bounding surface of any two media is
always transmitted as well as reflected equally on both sides, for
every color, direction, and polarization.

\Section{39.} The second formula~\Eq{(41)} establishes a relation between the
intensities of radiation in the two media, for it states that, when
%[** Theorems]
thermodynamic equilibrium exists, \emph{the specific intensities of radiation
of a certain frequency in the two media are in the inverse
ratio of the squares of the velocities of propagation or in the direct
ratio of the squares of the indices of refraction}.\footnote
  {\Name{G.~Kirchhoff}, Gesammelte Abhandlungen, Leipzig, J.~A. Barth, 1882, p.~594.
  \Name{R.~Clausius}, Pogg.\ Ann.~\textbf{121}, p.~1, 1864.}

By substituting for~$\ssfK_{\nu}$ its value from~\Eq{(27)} we obtain the following
theorem: \emph{The quantity
\[
q^{2} \ssfK_{\nu} = q^{2} \frac{\epsilon_{\nu}}{\alpha_{\nu}}
\Tag{(42)}
\]
does not depend on the nature of the substance, and is, therefore,
a universal function of the temperature~$T$ and the frequency~$\nu$ alone.}

The great importance of this law lies evidently in the fact that
it states a property of radiation which is the same for all bodies
%% -----File: 052.png---Folio 36-------
in nature, and which need be known only for a single arbitrarily
chosen body, in order to be stated quite generally for all bodies.
We shall later on take advantage of the opportunity offered by
this statement in order actually to calculate this universal function
(\Sec{165}).

\Section{40.}  We now consider the other case, that in which the
bounding surface of the two media is rough. This case is much
more general than the one previously treated, inasmuch as the
energy of a pencil directed from an element of the bounding surface
into the first medium is no longer supplied by two definite
pencils, but by an arbitrary number, which come from both
media and strike the surface. Here the actual conditions may be
very complicated according to the peculiarities of the bounding
surface, which moreover may vary in any way from one element
to another. However, according to \Sec{35}, the values of the
specific intensities of radiation $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ remain always the
same in all directions in both media, just as in the case of a smooth
bounding surface. That this condition, necessary for thermodynamic
equilibrium, is satisfied is readily seen from \Name{Helmholtz's}
law of reciprocity, according to which, in the case of stationary
radiation, for each ray striking the bounding surface and
diffusely reflected from it on both sides, there is a corresponding
ray at the same point, of the same intensity and opposite direction,
produced by the inverse process at the same point on the
bounding surface, namely by the gathering of diffusely incident
rays into a definite direction, just as is the case in the interior of
each of the two media.

\Section{41.} We shall now further generalize the laws obtained.
First, just as in \Sec{34}, the assumption made above, namely,
that the two media extend to a great distance, may be abandoned
since we may introduce an arbitrary number of bounding surfaces
without disturbing the thermodynamic equilibrium. Thereby
we are placed in a position enabling us to pass at once to the case
of any number of substances of any size and shape. For when a
system consisting of an arbitrary number of contiguous substances
is in the state of thermodynamic equilibrium, the equilibrium is
in no way disturbed, if we assume one or more of the surfaces of
contact to be wholly or partly impermeable to heat. Thereby
we can always reduce the case of any number of substances to
%% -----File: 053.png---Folio 37-------
that of two substances in an enclosure impermeable to heat, and,
therefore, the law may be stated quite generally, that, when any
arbitrary system is in the state of thermodynamic equilibrium,
the specific intensity of radiation~$\ssfK_{\nu}$ is determined in each
separate substance by the universal function~\Eq{(42)}.

\Section{42.} We shall now consider a system in a state of thermodynamic
equilibrium, contained within an enclosure impermeable
to heat and consisting of $n$~emitting and absorbing adjacent bodies
of any size and shape whatever. As in \Sec{36}, we again confine
our attention to a monochromatic plane polarized pencil
which proceeds from an element~$d\sigma$ of the bounding surface of the
two media in the direction toward the first medium (\Fig{3},
feathered arrow) within the conical element~$d\Omega$. Then, as in~\Eq{(34)},
the energy supplied by the pencil in unit time is
\[
d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu = I.
\Tag{(43)}
\]
This energy of radiation~$I$ consists of a part coming from the first
medium by regular or diffuse reflection at the bounding surface
and of a second part coming through the bounding surface from
the second medium. We shall, however, not stop at this mode of
division, but shall further subdivide~$I$ according to that one of
the $n$~media from which the separate parts of the radiation~$I$
have been emitted. This point of view is distinctly different
from the preceding, since, \eg, the rays transmitted from the
second medium through the bounding surface into the pencil
considered have not necessarily been emitted in the second
medium, but may, according to circumstances, have traversed a
long and very complicated path through different media and may
have undergone therein the effect of refraction, reflection, scattering,
and partial absorption any number of times. Similarly
the rays of the pencil, which coming from the first medium are
reflected at~$d\sigma$, were not necessarily all emitted in the first
medium. It may even happen that a ray emitted from a certain
medium, after passing on its way through other media, returns to
the original one and is there either absorbed or emerges from this
medium a second time.

We shall now, considering all these possibilities, denote that
part of~$I$ which has been emitted by volume-elements of the first
medium by~$I_{1}$ no matter what paths the different constituents
%% -----File: 054.png---Folio 38-------
have pursued, that which has been emitted by volume-elements
of the second medium by~$I_{2}$, etc. Then since every part of~$I$
must have been emitted by an element of some body, the following
equation must hold,
\[
I = I_{1} + I_{2} + I_{3} + \dots I_{n}.
\Tag{(44)}
\]

\Section{43.} The most adequate method of acquiring more detailed
information as to the origin and the paths of the different rays
of which the radiations $I_{1}$,~$I_{2}$, $I_{3}$,~$\dots I_{n}$ consist, is to
pursue the opposite course and to inquire into the future fate of
that pencil, which travels exactly in the opposite direction to
the pencil~$I$ and which therefore comes from the first medium in
the cone~$d\Omega$ and falls on the surface element~$d\sigma$ of the second medium.
For since every optical path may also be traversed in the
opposite direction, we may obtain by this consideration all paths
along which rays can pass into the pencil~$I$, however complicated
they may otherwise be. Let $J$ represent the intensity of this
inverse pencil, which is directed toward the bounding surface
and is in the same state of polarization. Then, according to
\Sec{40},
\[
J = I.
\Tag{(45)}
\]

At the bounding surface~$d\sigma$ the rays of the pencil~$J$ are partly
reflected and partly transmitted regularly or diffusely, and
thereafter, travelling in both media, are partly absorbed, partly
scattered, partly again reflected or transmitted to different
media,~etc., according to the configuration of the system. But
finally the whole pencil~$J$ after splitting into many separate rays
will be completely absorbed in the $n$~media. Let us denote that
part of~$J$ which is finally absorbed in the first medium by~$J_{1}$, that
which is finally absorbed in the second medium by~$J_{2}$, etc., then
we shall have
\[
J = J_{1} + J_{2} + J_{3} + \dots + J_{n}.
\]

Now the volume-elements of the $n$~media, in which the absorption
of the rays of the pencil~$J$ takes place, are precisely the same
as those in which takes place the emission of the rays constituting
the pencil~$I$, the first one considered above. For, according to
\Name{Helmholtz's} law of reciprocity, no appreciable radiation of the pencil~$J$
can enter a volume-element which contributes no appreciable
radiation to the pencil~$I$ and \textit{vice versa}.
%% -----File: 055.png---Folio 39-------

Let us further keep in mind that the absorption of each volume-element
is, according to~\Eq{(42)}, proportional to its emission and that,
according to \Name{Helmholtz's} law of reciprocity, the decrease which
the energy of a ray suffers on any path is always equal to the decrease
suffered by the energy of a ray pursuing the opposite path.
It will then be clear that the volume-elements considered absorb
the rays of the pencil~$J$ in just the same ratio as they contribute
by their emission to the energy of the opposite pencil~$I$. Since,
moreover, the sum~$I$ of the energies given off by emission by all
volume-elements is equal to the sum~$J$ of the energies absorbed
by all elements, the quantity of energy absorbed by each separate
volume-element from the pencil~$J$ must be equal to the quantity
of energy emitted by the same element into the pencil~$I$. In
%[** Theorem]
other words: \emph{the part of a pencil~$I$ which has been emitted from a
certain volume of any medium is equal to the part of the pencil~$J$
\($= I$\) oppositely directed, which is absorbed in the same volume}.

Hence not only are the sums $I$~and~$J$ equal, but their constituents
are also separately equal or
\[
J_{1} = I_{1},\quad
J_{2} = I_{2},\ \dots\quad
J_{n} = I_{n}.
\Tag{(46)}
\]

\Section{44.} Following \Name{G.~Kirchhoff}\footnote
  {\Name{G.~Kirchhoff}, Gesammelte Abhandlungen, 1882, p.~574.}
we call the quantity~$I_{2}$, \ie,~the
intensity of the pencil emitted from the second medium into the
first, the \emph{emissive power~$E$} of the second medium, while we call
the ratio of $J_{2}$ to~$J$, \ie,~that fraction of a pencil incident on the
second medium which is absorbed in this medium, the \emph{absorbing
power~$A$} of the second medium. Therefore
\[
E = I_{2}\ (\leq I),\quad
A = \frac{J_{2}}{J}\ (\leq 1).
\Tag{(47)}
\]

The quantities $E$~and~$A$ depend (a)~on the nature of the two
media, (b)~on the temperature, the frequency~$\nu$, and the direction
and the polarization of the radiation considered, (c)~on the nature
of the bounding surface and on the magnitude of the surface
element~$d\sigma$ and that of the solid angle~$d\Omega$, (d)~on the geometrical
extent and the shape of the total surface of the two media, (e)~on
the nature and form of all other bodies of the system. For a ray
may pass from the first into the second medium, be partly transmitted
by the latter, and then, after reflection somewhere else,
%% -----File: 056.png---Folio 40-------
may return to the second medium and may be there entirely
absorbed.

With these assumptions, according to equations \Eq{(46)},~\Eq{(45)},
and~\Eq{(43)}, \Name{Kirchhoff's} law holds,
\[
\frac{E}{A} = I = d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu,
\Tag{(48)}
\]
%[** Theorem]
\ie, \emph{the ratio of the emissive power to the absorbing power of any body
is independent of the nature of the body}. For this ratio is equal to
the intensity of the pencil passing through the \emph{first} medium,
which, according to equation~\Eq{(27)}, does not depend on the second
medium at all. The value of this ratio does, however, depend on
the nature of the first medium, inasmuch as, according to~\Eq{(42)},
it is not the quantity~$\ssfK_{\nu}$ but the quantity~$q^{2}\ssfK_{\nu}$, which is a universal
function of the temperature and frequency. The proof of this
%[** TN: "l. c." italicized in the original]
law given by \Name{G.~Kirchhoff} \lc\ was later greatly simplified by
\Name{E.~Pringsheim}.\footnote
  {\Name{E.~Pringsheim}, Verhandlungen der Deutschen Physikalischen Gesellschaft, \textbf{3}, p.~81, 1901.}

\Section{45.} When in particular the second medium is a black body
(\Sec{10}) it absorbs all the incident radiation. Hence in that case
%[** Theorems]
$J_{2} = J$, $A = 1$, and $E = \DPtypo{A}{I}$, \ie, \emph{the emissive power of a black body is
independent of its nature. Its emissive power is larger than that
of any other body at the same temperature and, in fact, is just equal to
the intensity of radiation in the contiguous medium}.

\Section{46.} We shall now add, without further proof, another general
law of reciprocity, which is closely connected with that stated at
the end of \Sec{43} and which may be stated thus: \emph{When any
emitting and absorbing bodies are in the state of thermodynamic
equilibrium, the part of the energy of definite color emitted by a body~$A$,
which is absorbed by another body~$B$, is equal to the part of the
energy of the same color emitted by~$B$ which is absorbed by~$A$.} Since
a quantity of energy emitted causes a decrease of the heat of the
body, and a quantity of energy absorbed an increase of the heat of
the body, it is evident that, when thermodynamic equilibrium
exists, any two bodies or elements of bodies selected at random
exchange by radiation equal amounts of heat with each other.
Here, of course, care must be taken to distinguish between the
radiation emitted and the total radiation which reaches one body
from the other.
%% -----File: 057.png---Folio 41-------

\Section{47.} The law holding for the quantity~\Eq{(42)} can be expressed in a
different form, by introducing, by means of~\Eq{(24)}, the volume
density~$\ssfu_{\nu}$ of monochromatic radiation instead of the intensity
of radiation~$\ssfK_{\nu}$. We then obtain the law that, for radiation in
a state of thermodynamic equilibrium, the quantity
\[
\ssfu_{\nu} q^{3}
\Tag{(49)}
\]
is a function of the temperature~$T$ and the frequency~$\nu$, and is
the same for all substances.\footnote
  {In this law it is assumed that the quantity~$q$ in~\Eq{(24)} is the same as in~\Eq{(37)}. This does
  not hold for strongly dispersing or absorbing substances. For the generalization applying
  to such cases see \Name{M.~Laue}, Annalen d.\ Physik\DPchg{,}{} \textbf{32}, p.~1085, 1910.}
This law becomes clearer if we
consider that the quantity
\[
\ssfu_{\nu}\, d\nu\, \frac{q^{3}}{\nu^{3}}
\Tag{(50)}
\]
also is a universal function of $T$,~$\nu$, and~$\nu + d\nu$, and that the
product $\ssfu_{\nu}\, d\nu$ is, according to~\Eq{(22)}, the volume density of the
radiation whose frequency lies between $\nu$ and $\nu + d\nu$, while the
quotient~$\dfrac{q}{\nu}$ represents the wave length of a ray of frequency~$\nu$ in
the medium in question. The law then takes the following simple
%[** Theorem]
form: \emph{When any bodies whatever are in thermodynamic equilibrium,
the energy of monochromatic radiation of a definite frequency,
contained in a cubical element of side equal to the wave length, is
the same for all bodies.}

\Section{48.} We shall finally take up the case of diathermanous (\Sec{12})
media, which has so far not been considered. In \Sec{27} we
saw that, in a medium which is diathermanous for a given color
and is surrounded by an enclosure impermeable to heat, there can
be thermodynamic equilibrium for any intensity of radiation
of this color. There must, however, among all possible intensities
of radiation be a definite one, corresponding to the absolute
maximum of the total entropy of the system, which designates
the absolutely stable equilibrium of radiation. In fact, in equation~\Eq{(27)}
the intensity of radiation~$\ssfK_{\nu}$ for $\alpha_{\nu} = 0$ and $\epsilon_{\nu} = 0$
assumes the value~$\dfrac{0}{0}$, and hence cannot be calculated from this
equation. But we see also that this indeterminateness is removed
by equation~\Eq{(41)}, which states that in the case of thermodynamic
%% -----File: 058.png---Folio 42-------
equilibrium the product~$q^{2}\ssfK_{\nu}$ has the same value for all substances.
From this we find immediately a definite value of~$\ssfK_{\nu}$
which is thereby distinguished from all other values. Furthermore
the physical significance of this value is immediately seen
by considering the way in which that equation was obtained.
It is that intensity of radiation which exists in a diathermanous
medium, if it is in thermodynamic equilibrium when in contact
with an arbitrary absorbing and emitting medium. The volume
and the form of the second medium do not matter in the least,
in particular the volume may be taken as small as we please.
%[** Thm]
Hence we can formulate the following law: \emph{Although generally
speaking thermodynamic equilibrium can exist in a diathermanous
medium for any intensity of radiation whatever, nevertheless there
exists in every diathermanous medium for a definite frequency at a
definite temperature an intensity of radiation defined by the universal
function~\Eq{(42)}. This may be called the stable intensity, inasmuch
as it will always be established, when the medium is exchanging
stationary radiation with an arbitrary emitting and absorbing
substance.}

\Section{49.} According to the law stated in \Sec{45}, the intensity of a
pencil, when a state of stable heat radiation exists in a diathermanous
medium, is equal to the emissive power~$E$ of a black
body in contact with the medium. On this fact is based the
possibility of measuring the emissive power of a black body,
although absolutely black bodies do not exist in nature.\footnote
  {\Name{W.~Wien} and \Name{O.~Lummer}, Wied.\ Annalen, \textbf{56}, p.~451, 1895.}
A
diathermanous cavity is enclosed by strongly emitting walls\footnote
  {The strength of the emission influences only the time required to establish stationary
  radiation, but not its character. It is essential, however, that the walls transmit no radiation
  to the exterior.}
and the walls kept at a certain constant temperature~$T$. Then
the radiation in the cavity, when thermodynamic equilibrium is
established for every frequency~$\nu$, assumes the intensity corresponding
to the velocity of propagation~$q$ in the diathermanous
medium, according to the universal function~\Eq{(42)}. Then any
element of area of the walls radiates into the cavity just as if the
wall were a black body of temperature~$T$. The amount lacking
in the intensity of the rays actually emitted by the walls as
compared with the emission of a black body is supplied by rays
%% -----File: 059.png---Folio 43-------
which fall on the wall and are reflected there. Similarly every
element of area of a wall receives the same radiation.

In fact, the radiation~$I$ starting from an element of area of a
wall consists of the radiation~$E$ emitted by the element of area and
of the radiation reflected from the element of area from the
incident radiation~$I$, \ie,~the radiation which is not absorbed
$(1 - A)I$. We have, therefore, in agreement with \Name{Kirchhoff's}
law~\Eq{(48)},
\[
I = E + (1 - A)I.
\]

If we now make a hole in one of the walls of a size~$d\sigma$, so small
that the intensity of the radiation directed toward the hole is
not changed thereby, then radiation passes through the hole to
the exterior where we shall suppose there is the same diathermanous
medium as within. This radiation has exactly the same
properties as if $d\sigma$~were the surface of a black body, and this
radiation may be measured for every color together with the
temperature~$T$.

\Section{50.} Thus far all the laws derived in the preceding sections for
diathermanous media hold for a definite frequency, and it is to
be kept in mind that a substance may be diathermanous for one
color and adiathermanous for another. Hence the radiation of a
medium completely enclosed by absolutely reflecting walls is,
when thermodynamic equilibrium has been established for all
colors for which the medium has a finite coefficient of absorption,
always the stable radiation corresponding to the temperature
of the medium such as is represented by the emission of a black
body. Hence this is briefly called ``black'' radiation.\footnote
  {\Name{M.~Thiesen}, Verhandlungen d.\ Deutschen Physikal.\ Gesellschaft, \textbf{2}, p.~65, 1900.}
On the
other hand, the intensity of colors for which the medium is diathermanous
is not necessarily the stable black radiation, unless
the medium is in a state of stationary exchange of radiation with
an absorbing substance.

There is but one medium that is diathermanous for all kinds of
rays, namely, the absolute vacuum, which to be sure cannot be
produced in nature except approximately. However, most gases,
\eg, the air of the atmosphere, have, at least if they are not too
dense, to a sufficient approximation the optical properties of a
vacuum with respect to waves of not too short length. So far as
%% -----File: 060.png---Folio 44-------
this is the case the velocity of propagation~$q$ may be taken as the
same for all frequencies, namely,
\[
c = 3 × 10^{10}\, \frac{\cm}{\sec}\Add{.}
\Tag{(51)}
\]

\Section{51.} Hence in a vacuum bounded by totally reflecting walls any
state of radiation may persist. But as soon as an arbitrarily
small quantity of matter is introduced into the vacuum, a stationary
state of radiation is gradually established. In this the
radiation of every color which is appreciably absorbed by the
substance has the intensity~$\ssfK_{\nu}$ corresponding to the temperature
of the substance and determined by the universal function~\Eq{(42)}
for $q = c$, the intensity of radiation of the other colors remaining
indeterminate. If the substance introduced is not diathermanous
for any color, \eg, a piece of carbon however small, there
exists at the stationary state of radiation in the whole vacuum for
all colors the intensity~$\ssfK_{\nu}$ of black radiation corresponding to the
temperature of the substance. The magnitude of~$\ssfK_{\nu}$ regarded as
a function of~$\nu$ gives the spectral distribution of black radiation in
a vacuum, or the so-called \emph{normal energy spectrum}, which depends
on nothing but the temperature. In the normal spectrum,
since it is the spectrum of emission of a black body, the intensity
of radiation of every color is the largest which a body can emit at
that temperature at all.

\Section{52.} It is therefore possible to change a perfectly arbitrary
radiation, which exists at the start in the evacuated cavity with
perfectly reflecting walls under consideration, into black radiation
by the introduction of a minute particle of carbon. The characteristic
feature of this process is that the heat of the carbon particle
may be just as small as we please, compared with the energy
of radiation contained in the cavity of arbitrary magnitude.
Hence, according to the principle of the conservation of energy,
the total energy of radiation remains essentially constant during
the change that takes place, because the changes in the heat of the
carbon particle may be entirely neglected, even if its changes in
temperature should be finite. Herein the carbon particle exerts
only a releasing (auslösend) action. Thereafter the intensities
of the pencils of different frequencies originally present and having
different frequencies, directions, and different states of polarization
%% -----File: 061.png---Folio 45-------
change at the expense of one another, corresponding to
the passage of the system from a less to a more stable state of
radiation or from a state of smaller to a state of larger entropy.
From a thermodynamic point of view this process is perfectly
analogous, since the time necessary for the process is not essential,
to the change produced by a minute spark in a quantity of oxy-hydrogen
gas or by a small drop of liquid in a quantity of supersaturated
vapor. In all these cases the magnitude of the disturbance
is exceedingly small and cannot be compared with the
magnitude of the energies undergoing the resultant changes, so
that in applying the two principles of thermodynamics the cause
of the disturbance of equilibrium, \viz, the carbon particle, the
spark, or the drop, need not be considered. It is always a case of
a system passing from a more or less unstable into a more stable
state, wherein, according to the first principle of thermodynamics,
the energy of the system remains constant, and, according to the
second principle, the entropy of the system increases.
%% -----File: 062.png---Folio 46-------
% [Blank Page]
%% -----File: 063.png---Folio 47-------

\Part[Deductions From Electrodynamics]{II}{Deductions From Electrodynamics
And Thermodynamics}
%% -----File: 064.png---Folio 48-------
%[Blank Page]
%% -----File: 065.png---Folio 49-------

\Chapter{I}{Maxwell's Radiation Pressure}

\Section{53.} While in the preceding part the phenomena of radiation
have been presented with the assumption of only well known
elementary laws of optics summarized in \Sec{2}, which are common
to all optical theories, we shall hereafter make use of the
electromagnetic theory of light and shall begin by deducing a
consequence characteristic of that theory. We shall, namely,
calculate the magnitude of the mechanical force, which is exerted
by a light or heat ray passing through a vacuum on striking a
reflecting (\Sec{10}) surface assumed to be at rest.

For this purpose we begin by stating Maxwell's general equations
for an electromagnetic process in a vacuum. Let the vector~$\ssfE$
denote the electric field-strength (intensity of the electric field)
in electric units and the vector~$\ssfH$ the magnetic field-strength in
magnetic units. Then the equations are, in the abbreviated
notation of the vector calculus,
\[
\begin{aligned}
  \dot{\ssfE} &= c \curl\ssfH & \dot{\ssfH} &= -c \curl\ssfE\\
  \div \ssfE &= 0             & \div \ssfH &= 0\Add{.}
\end{aligned}
\Tag{(52)}
\]
Should the reader be unfamiliar with the symbols of this notation,
he may readily deduce their meaning by working backward from
the subsequent equations~\Eq{(53)}.

\Section{54.} In order to pass to the case of a plane wave in any direction
we assume that all the quantities that fix the state depend only
on the time~$t$ and on one of the coordinates $x'$,~$y'$,~$z'$, of an orthogonal
right-handed system of coordinates, say on~$x'$. Then the
equations~\Eq{(52)} reduce to
\[
\begin{aligned}
\frac{\dd\ssfE_{x'}}{\dd t} &= 0 &
\frac{\dd\ssfH_{x'}}{\dd t} &= 0 \\[4pt]
%
\frac{\dd\ssfE_{y'}}{\dd t} &=     -c \frac{\dd \ssfH_{z'}}{\dd x'} \qquad &
\frac{\dd\ssfH_{y'}}{\dd t} &= \Neg c \frac{\dd \ssfE_{z'}}{\dd x'} \\[4pt]
%% -----File: 066.png---Folio 50-------
\frac{\dd\ssfE_{x'}}{\dd t}  &= \Neg c\frac{\dd \ssfH_{y'}}{\dd x'} &
\frac{\dd\ssfH_{z'}}{\dd t}  &=     -c\frac{\dd \ssfE_{y'}}{\dd x'} \\[4pt]
\frac{\dd\ssfE_{x'}}{\dd x'} &= 0 &
\frac{\dd\ssfH_{x'}}{\dd x'} &= 0\Add{.}
\end{aligned}
\Tag{(53)}
\]
Hence the most general expression for a plane wave passing
through a vacuum in the direction of the positive $x'$-axis is
\[
\begin{aligned}
\ssfE_{x'} &= 0 &
\ssfH_{x'} &= 0 \\
\ssfE_{y'} &=  f\left(t - \frac{x'}{c}\right) &
\ssfH_{y'} &= -g\left(t - \frac{x'}{c}\right) \\
\ssfE_{z'} &= g\left(t - \frac{x'}{c}\right) &
\ssfH_{z'} &= \Neg f\left(t - \frac{x'}{c}\right)
\end{aligned}
\Tag{(54)}
\]
where $f$~and~$g$ represent two arbitrary functions of the same
argument.

\Section{55.} Suppose now that this wave strikes a reflecting surface,
\eg, the surface of an absolute conductor (metal) of infinitely
\Figure[2.25in]{4}
large conductivity. In such a
conductor even an infinitely
small electric field-strength produces
a finite conduction current;
hence the electric field-strength~$\ssfE$
in it must be always
and everywhere infinitely small.
For simplicity we also suppose
the conductor to be non-magnetizable,
\ie, we assume the
magnetic induction~$\ssfB$ in it to be
equal to the magnetic field-strength~$\ssfH$,
just as is the case
in a vacuum.

%[** F2: Comma-less coordinate system notation starts here]
If we place the $x$-axis of a
right-handed coordinate system
$(xyz)$ along the normal of the surface
directed toward the interior
of the conductor, the $x$-axis is the normal of incidence. We
place the $(x'y')$~plane in the plane of incidence and take this as
the plane of the figure (\Fig{4}). Moreover, we can also, without
%% -----File: 067.png---Folio 51-------
any restriction of generality, place the $y$-axis in the plane of the
figure, so that the $z$-axis coincides with the $z'$-axis (directed from
the figure toward the observer). Let the common origin~$O$ of
the two coordinate systems lie in the surface. If finally $\theta$~represents
the angle of incidence, the coordinates with and without
accent are related to each other by the following equations:
\begin{align*}
 x &= x' \cos\theta - y' \sin\theta & x' &= \Neg x \cos\theta + y \sin\theta\\
 y &= x' \sin\theta + y' \cos\theta & y' &= -x \sin\theta + y \cos\theta\\
 z &= z'                            & z' &=  z\Add{.}
\end{align*}

By the same transformation we may pass from the components
of the electric or magnetic field-strength in the first coordinate
system to their components in the second system. Performing
this transformation the following values are obtained from~\Eq{(54)}
for the components of the electric and magnetic field-strengths
of the incident wave in the coordinate system without accent,
\[
\begin{aligned}
  \ssfE_{x} &=     -\sin\theta · f \qquad & \ssfH_{x} &= \Neg\sin\theta · g \\
  \ssfE_{y} &=  \Neg\cos\theta · f        & \ssfH_{y} &=    -\cos\theta · g \\
  \ssfE_{z} &= g                          & \ssfH_{z} &= f\Add{.}
\end{aligned}
\Tag{(55)}
\]
Herein the argument of the functions $f$~and~$g$ is
\[
t - \frac{x'}{c} = t - \frac{x \cos\theta + y \sin\theta}{c}\Add{.}
\Tag{(56)}
\]

\Section{56.} In the surface of separation of the two media $x = 0$. According
to the general electromagnetic boundary conditions the
components of the field-strengths in the surface of separation,
\ie, the four quantities $\ssfE_{y}$,~$\ssfE_{z}$, $\ssfH_{y}$,~$\ssfH_{z}$ must be equal to each
other on the two sides of the surface of separation for this value
of~$x$. In the conductor the electric field-strength~$\ssfE$ is infinitely
small in accordance with the assumption made above. Hence
$\ssfE_{y}$~and~$\ssfE_{z}$ must vanish also in the vacuum for $x = 0$. This condition
cannot be satisfied unless we assume in the vacuum,
besides the incident, also a reflected wave superposed on the former
in such a way that the components of the electric field of the
two waves in the $y$~and~$z$ direction just cancel at every instant
and at every point in the surface of separation. By this assumption
and the condition that the reflected wave is a plane wave
returning into the interior of the vacuum, the other four components
%% -----File: 068.png---Folio 52-------
of the reflected wave are also completely determined. They
are all functions of the single argument
\[
t - \frac{-x \cos\theta + y \sin\theta}{c}.
\Tag{(57)}
\]
The actual calculation yields as components of the total electromagnetic
field produced in the vacuum by the superposition of
the two waves, the following expressions valid for points of the
surface of separation $x = 0$,
\[
\begin{aligned}
  \ssfE_{x} &=    -\sin\theta · f - \sin\theta · f = -2\sin\theta · f\\
  \ssfE_{y} &= \Neg\cos\theta · f - \cos\theta · f = 0\\
  \ssfE_{z} &=  g - g = 0 \\
  \ssfH_{x} &= \Neg\sin\theta · g - \sin\theta · g = 0\\
  \ssfH_{y} &=    -\cos\theta · g - \cos\theta · g = -2\cos\theta · g\\
  \ssfH_{z} &= f + f = 2f.
\end{aligned}
\Tag{(58)}
\]
In these equations the argument of the functions $f$~and~$g$ is, according
to \Eq{(56)}~and~\Eq{(57)},
\[
t - \frac{y \sin\theta}{c}\Add{.}
\]
From these values the electric and magnetic field-strength within
the conductor in the immediate neighborhood of the separating
surface $x = 0$ is obtained:
\[
\begin{aligned}
  \ssfE_{x} &=0 \qquad & \ssfH_{x} &=  0\\
  \ssfE_{y} &=0        & \ssfH_{y} &= -2\cos\theta · g\\
  \ssfE_{z} &=0        & \ssfH_{z} &=  2f
\end{aligned}
\Tag{(59)}
\]
where again the argument $t - \dfrac{y \sin\theta}{c}$ is to be substituted in the
functions $f$~and~$g$. For the components of~$\ssfE$ all vanish in an absolute
conductor and the components $\ssfH_{x}$,~$\ssfH_{y}$,~$\ssfH_{z}$ are all continuous
at the separating surface, the two latter since they are tangential
components of the field-strength, the former since it is the normal
component of the magnetic induction~$\ssfB$ (\Sec{55}), which likewise
remains continuous on passing through any surface of separation.

On the other hand, the normal component of the electric field-strength~$\ssfE_{x}$
is seen to be discontinuous; the discontinuity shows
%% -----File: 069.png---Folio 53-------
the existence of an electric charge on the surface, the surface
density of which is given in magnitude and sign as follows:
\[
\frac{1}{4\pi} 2\sin\theta · f = \frac{1}{2\pi} \sin\theta · f.
\Tag{(60)}
\]
In the interior of the conductor at a finite distance from the
bounding surface, \ie, for $x > 0$, all six field components are infinitely
small. Hence, on increasing~$x$, the values of $\ssfH_{y}$~and~$\ssfH_{z}$,
which are finite for $x = 0$, approach the value~$0$ at an infinitely
rapid rate.

\Section{57.} A certain mechanical force is exerted on the substance of
the conductor by the electromagnetic field considered. We shall
calculate the component of this force normal to the surface. It
is partly of electric, partly of magnetic, origin. Let us first consider
the former,~$\ssfF_{e}$. Since the electric charge existing on the
surface of the conductor is in an electric field, a mechanical force
equal to the product of the charge and the field-strength is exerted
on it. Since, however, the field-strength is discontinuous, having
the value $-2\sin\theta f$ on the side of the vacuum and $0$~on the side
of the conductor, from a well-known law of electrostatics the magnitude
of the mechanical force~$\ssfF_{e}$ acting on an element of surface~$d\sigma$
of the conductor is obtained by multiplying the electric charge
of the element of area calculated in~\Eq{(60)} by the arithmetic mean
of the electric field-strength on the two sides. Hence
\[
\ssfF_{e} = \frac{\sin\theta}{2\pi} f\, d\sigma\, (-\sin\theta f)
  = -\frac{\sin^{2}\theta}{2\pi} f^{2}\, d\sigma\Add{.}
\]
This force acts in the direction toward the vacuum and therefore
exerts a tension.

\Section{58.} We shall now calculate the mechanical force of magnetic
origin~$\ssfF_{m}$. In the interior of the conducting substance there are
certain conduction currents, whose intensity and direction are
determined by the vector~$\ssfI$ of the current density
\[
\ssfI = \frac{c}{4\pi} \curl \ssfH.
\Tag{(61)}
\]
A mechanical force acts on every element of space~$d\tau$ of the conductor
through which a conduction current flows, and is given by
the vector product
\[
\frac{d\tau}{c} \DPchg{[\ssfI \ssfH]}{[\ssfI × \ssfH]}\Add{.}
\Tag{(62)}
\]
%% -----File: 070.png---Folio 54-------
Hence the component of this force normal to the surface of the
conductor $x = 0$ is equal to
\[
\frac{d\tau}{c} (\ssfI_{y} \ssfH_{z} - \ssfI_{z} \ssfH_{y}).
\]
On substituting the values of $\ssfI_{y}$~and~$\ssfI_{z}$ from~\Eq{(61)} we obtain
\[
\frac{d\tau}{4\pi}
  \left[\ssfH_{z} \left(\frac{\dd\ssfH_{x}}{\dd z} - \frac{\dd\ssfH_{z}}{\dd x}\right)
      - \ssfH_{y} \left(\frac{\dd\ssfH_{y}}{\dd x} - \frac{\dd\ssfH_{x}}{\dd y}\right)\right].
\]
In this expression the differential coefficients with respect to
$y$~and~$z$ are negligibly small in comparison to those with respect to~$x$,
according to the remark at the end of \Sec{56}; hence the expression
reduces to
\[
-\frac{d\tau}{4\pi}
  \left(\ssfH_{y} \frac{\dd\ssfH_{y}}{\dd x}
      + \ssfH_{z} \frac{\dd\ssfH_{z}}{\dd x}\right).
\]
Let us now consider a cylinder cut out of the conductor perpendicular
to the surface with the cross-section~$d\sigma$, and extending
from $x = 0$ to $x = \infty$. The entire mechanical force of magnetic
origin acting on this cylinder in the direction of the $x$-axis, since
$d\tau = d\sigma\, x$, is given by
\[
\ssfF_{m} = -\frac{d\sigma}{4\pi} \int_{0}^{\infty} dx
  \left(\ssfH_{y} \frac{\dd\ssfH_{y}}{\dd x}
      + \ssfH_{z} \frac{\dd\ssfH_{z}}{\dd x}\right).
\]
On integration, since $\ssfH$~vanishes for $x = \infty$, we obtain
\[
\ssfF_{m} = \frac{d\sigma}{8\pi} \left(\ssfH_{y}^{2} + \ssfH_{z}^{2}\right)_{x = 0}
\]
or by equation~\Eq{(59)}
\[
\ssfF_{m} = \frac{d\sigma}{2\pi} (\cos^{2} \theta · g^{2} + f^{2}).
\]

By adding $\ssfF_{e}$~and~$\ssfF_{m}$ the total mechanical force acting on the
cylinder in question in the direction of the $x$-axis is found to be
\[
\ssfF = \frac{d\sigma}{2\pi} \cos^{2} \theta (f^{2} + g^{2}).
\Tag{(63)}
\]
This force exerts on the surface of the conductor a pressure, which
acts in a direction normal to the surface toward the interior and is
%% -----File: 071.png---Folio 55-------
called ``\Name{Maxwell's} radiation pressure.'' The existence and the
magnitude of the radiation pressure as predicted by the theory
was first found by delicate measurements with the radiometer by
\Name{P.~Lebedew.}\footnote
  {\Name{P. Lebedew}, Annalen d.\ Phys.\DPchg{,}{}\ \textbf{6}, p.~433, 1901. See also \Name{E.~F. Nichols} and \Name{G.~F. Hull},
  Annalen d.\ Phys.\DPchg{,}{}\ \textbf{12}, p.~225, 1903.}

\Section{59.} We shall now establish a relation between the radiation
pressure and the energy of radiation~$I\, dt$ falling on the surface
element~$d\sigma$ of the conductor in a time element~$dt$. The latter
from \Name{Poynting's} law of energy flow is
\[
I\, dt = \frac{c}{4\pi} (\ssfE_{y} \ssfH_{z} - \ssfE_{z} \ssfH_{y})\, d\sigma\, dt,
\]
hence from~\Eq{(55)}
\[
I\, dt = \frac{c}{4\pi} \cos\theta (f^{2} + g^{2})\, d\sigma\, dt.
\]
By comparison with~\Eq{(63)} we obtain
\[
\ssfF = \frac{2\cos\theta}{c} I.
\Tag{(64)}
\]

From this we finally calculate the total pressure~$p$, \ie, that
mechanical force, which an arbitrary radiation proceeding from
the vacuum and totally reflected upon incidence on the conductor
exerts in a normal direction on a unit surface of the conductor.
The energy radiated in the conical element
\[
d\Omega = \sin\theta\, d\theta\, d\phi
\]
in the time~$dt$ on the element of area~$d\sigma$ is, according to~\Eq{(6)},
\[
I\, dt = K \cos\theta\, d\Omega\, d\sigma\, dt,
\]
where $K$~represents the specific intensity of the radiation in the
direction~$d\Omega$ toward the reflector. On substituting this in~\Eq{(64)} and
integrating over~$d\Omega$ we obtain for the total pressure of all pencils
which fall on the surface and are reflected by it
\[
p = \frac{2}{c} \int K \cos^{2} \theta\, d\Omega,
\Tag{(65)}
\]
the integration with respect to~$\phi$ extending from $0$ to~$2\pi$ and with
respect to~$\theta$ from $0$ to~$\dfrac{\pi}{2}$.
%% -----File: 072.png---Folio 56-------

In case $K$~is independent of direction as in the case of black
radiation, we obtain for the radiation pressure
\[
p = \frac{2K}{c} \int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}} d\theta \cos^{2} \theta \sin\theta
  = \frac{4\pi K}{3c}
\]
or, if we introduce instead of~$K$ the volume density of radiation~$u$
from~\Eq{(21)}
\[
p = \frac{u}{3}.
\Tag{(66)}
\]

This value of the radiation pressure holds only when the reflection
of the radiation occurs at the surface of an absolute non-magnetizable
conductor. Therefore we shall in the thermodynamic
deductions of the next chapter make use of it only in such
cases. Nevertheless it will be shown later on (\Sec{66}) that
equation~\Eq{(66)} gives the pressure of uniform radiation against any
totally reflecting surface, no matter whether it reflects uniformly
or diffusely.

\Section{60.} In view of the extraordinarily simple and close relation
between the radiation pressure and the energy of radiation, the
question might be raised whether this relation is really a special
consequence of the electromagnetic theory, or whether it might
not, perhaps, be founded on more general energetic or thermodynamic
considerations. To decide this question we shall calculate
the radiation pressure that would follow by Newtonian
mechanics from \Name{Newton's} (emission) theory of light, a theory
which, in itself, is quite consistent with the energy principle.
According to it the energy radiated onto a surface by a light ray
passing through a vacuum is equal to the kinetic energy of the
light particles striking the surface, all moving with the constant
velocity~$c$. The decrease in intensity of the energy radiation
with the distance is then explained simply by the decrease of the
volume density of the light particles.

Let us denote by $n$ the number of the light particles contained
in a unit volume and by $m$ the mass of a particle. Then for a
beam of parallel light the number of particles impinging in unit
time on the element~$d\sigma$ of a reflecting surface at the angle of
incidence~$\theta$ is
\[
nc \cos\theta\, d\sigma.
\Tag{(67)}
\]
%% -----File: 073.png---Folio 57-------

Their kinetic energy is given according to Newtonian mechanics
by
\[
I = nc \cos\theta\, d\sigma\, \frac{mc^{2}}{2} = nm \cos\theta \frac{c^{3}}{2}\, d\sigma.
\Tag{(68)}
\]
Now, in order to determine the normal pressure of these particles
on the surface, we may note that the normal component of the
velocity $c \cos\theta$ of every particle is changed on reflection into a
component of opposite direction. Hence the normal component
of the momentum of every particle (impulse-coordinate) is
changed through reflection by $-2mc \cos\theta$. Then the change
in momentum for all particles considered will be, according to~\Eq{(67)},
\[
-2nm \cos^{2} \theta\, c^{2}\, d\sigma.
\Tag{(69)}
\]

Should the reflecting body be free to move in the direction of
the normal of the reflecting surface and should there be no force
acting on it except the impact of the light particles, it would be
set into motion by the impacts. According to the law of action
and reaction the ensuing motion would be such that the momentum
acquired in a certain interval of time would be equal and
opposite to the change in momentum of all the light particles
reflected from it in the same time interval. But if we allow a
separate constant force to act from outside on the reflector, there
is to be added to the change in momenta of the light particles
the impulse of the external force, \ie, the product of the force
and the time interval in question.

Therefore the reflector will remain continuously at rest, whenever
the constant external force exerted on it is so chosen that its
impulse for any time is just equal to the change in momentum
of all the particles reflected from the reflector in the same time.
Thus it follows that the force~$\ssfF$ itself which the particles exert
by their impact on the surface element~$d\sigma$ is equal and opposite
to the change of their momentum in unit time as expressed in~\Eq{(69)}
\[
\ssfF = 2nm \cos^{2} \theta\, c^{2}\, d\sigma
\]
and by making use of~\Eq{(68)},
\[
\ssfF = \frac{4\cos\theta}{c} I.
\]

On comparing this relation with equation~\Eq{(64)} in which all
symbols have the same physical significance, it is seen that
%% -----File: 074.png---Folio 58-------
\Name{Newton's} radiation pressure is twice as large as \Name{Maxwell's} for the
same energy radiation. A necessary consequence of this is that
the magnitude of \Name{Maxwell's} radiation pressure cannot be deduced
from general energetic considerations, but is a special feature of
the electromagnetic theory and hence all deductions from \Name{Maxwell's}
radiation pressure are to be regarded as consequences of the
electromagnetic theory of light and all confirmations of them
are confirmations of this special theory.
%% -----File: 075.png---Folio 59-------

\Chapter{II}{Stefan-Boltzmann Law of Radiation}

\Section{61.} For the following we imagine a perfectly evacuated hollow
cylinder with an absolutely tight-fitting piston free to move in a
vertical direction with no friction. A part of the walls of the
cylinder, say the rigid bottom, should consist of a black body,
whose temperature~$T$ may be regulated arbitrarily from the outside.
The rest of the walls including the inner surface of the piston
may be assumed as totally reflecting. Then, if the piston
remains stationary and the temperature,~$T$, constant, the radiation
in the vacuum will, after a certain time, assume the character
of black radiation (\Sec{50}) uniform in all directions. The
specific intensity,~$K$, and the volume density,~$u$, depend only on
the temperature,~$T$, and are independent of the volume,~$V$, of
the vacuum and hence of the position of the piston.

If now the piston is moved downward, the radiation is compressed
into a smaller space; if it is moved upward the radiation
expands into a larger space. At the same time the temperature
of the black body forming the bottom may be arbitrarily changed
by adding or removing heat from the outside. This always
causes certain disturbances of the stationary state. If, however,
the arbitrary changes in $V$~and~$T$ are made sufficiently slowly, the
departure from the conditions of a stationary state may always be
kept just as small as we please. Hence the state of radiation in
the vacuum may, without appreciable error, be regarded as a
state of thermodynamic equilibrium, just as is done in the thermodynamics
of ordinary matter in the case of so-called infinitely
slow processes, where, at any instant, the divergence from the
state of equilibrium may be neglected, compared with the changes
which the total system considered undergoes as a result of the
entire process.

If, \eg, we keep the temperature~$T$ of the black body forming
the bottom constant, as can be done by a suitable connection
%% -----File: 076.png---Folio 60-------
between it and a heat reservoir of large capacity, then, on raising
the piston, the black body will emit more than it absorbs, until
the newly made space is filled with the same density of radiation
as was the original one. \textit{Vice versa}, on lowering the piston the
black body will absorb the superfluous radiation until the original
radiation corresponding to the temperature~$T$ is again established.
Similarly, on raising the temperature~$T$ of the black body, as
can be done by heat conduction from a heat reservoir which is
slightly warmer, the density of radiation in the vacuum will be
correspondingly increased by a larger emission,~etc. To accelerate
the establishment of radiation equilibrium the reflecting
mantle of the hollow cylinder may be assumed white (\Sec{10}),
since by diffuse reflection the predominant directions of radiation
that may, perhaps, be produced by the direction of the motion
of the piston, are more quickly neutralized. The reflecting
surface of the piston, however, should be chosen for the present as
a perfect metallic reflector, to make sure that the radiation pressure~\Eq{(66)}
on the piston is \Name{Maxwell's}. Then, in order to produce
mechanical equilibrium, the piston must be loaded by a weight
equal to the product of the radiation pressure~$p$ and the cross-section
of the piston. An exceedingly small difference of the
loading weight will then produce a correspondingly slow motion
of the piston in one or the other direction.

Since the effects produced from the outside on the system in
question, the cavity through which the radiation travels, during
the processes we are considering, are partly of a mechanical
nature (displacement of the loaded piston), partly of a thermal
nature (heat conduction away from and toward the reservoir),
they show a certain similarity to the processes usually considered
in thermodynamics, with the difference that the system here
considered is not a material system, \eg,~a gas, but a purely energetic
one. If, however, the principles of thermodynamics hold
quite generally in nature, as indeed we shall assume, then they
must also hold for the system under consideration. That is to
say, in the case of any change occurring in nature the energy of
all systems taking part in the change must remain constant
(first principle), and, moreover, the entropy of all systems taking
part in the change must increase, or in the limiting case of reversible
processes must remain constant (second principle).
%% -----File: 077.png---Folio 61-------

\Section{62.} Let us first establish the equation of the first principle for
an infinitesimal change of the system in question. That the
cavity enclosing the radiation has a certain energy we have
already (\Sec{22}) deduced from the fact that the energy radiation
is propagated with a finite velocity. We shall denote the energy
by~$U$. Then we have
\[
U = Vu,
\Tag{(70)}
\]
where $u$~the volume density of radiation depends only on the
temperature \DPtypo{of~$T$}{$T$~of} the black body at the bottom.

The work done by the system, when the volume~$V$ of the cavity
increases by~$dV$ against the external forces of pressure (weight of
the loaded piston), is~$p\, dV$, where $p$~represents \Name{Maxwell's} radiation
pressure~\Eq{(66)}. This amount of mechanical energy is therefore
gained by the surroundings of the system, since the weight is
raised. The error made by using the radiation pressure on a
stationary surface, whereas the reflecting surface moves during
the volume change, is evidently negligible, since the motion may
be thought of as taking place with an arbitrarily small velocity.

If, moreover, $Q$~denotes the infinitesimal quantity of heat in
mechanical units, which, owing to increased emission, passes
from the black body at the bottom to the cavity containing the
radiation, the bottom or the heat reservoir connected to it loses
this heat~$Q$, and its internal energy is decreased by that amount.
Hence, according to the first principle of thermodynamics, since
the sum of the energy of radiation and the energy of the material
bodies remains constant, we have
\[
dU + p\, dV - Q = 0.
\Tag{(71)}
\]

According to the second principle of thermodynamics the cavity
containing the radiation also has a definite entropy. For
when the heat~$Q$ passes from the heat reservoir into the cavity,
the entropy of the reservoir decreases, the change being
\[
-\frac{Q}{T}\Add{.}
\]

Therefore, since no changes occur in the other bodies---inasmuch
as the rigid absolutely reflecting piston with the weight on
it does not change its internal condition with the motion---there
%% -----File: 078.png---Folio 62-------
must somewhere in nature occur a compensation of entropy having
at least the value~$\dfrac{Q}{T}$, by which the above diminution is compensated,
and this can be nowhere except in the entropy of the
cavity containing the radiation. Let the entropy of the latter be
denoted by~$S$.

Now, since the processes described consist entirely of states
of equilibrium, they are perfectly reversible and hence there is no
increase in entropy. Then we have
\[
dS - \frac{Q}{T} = 0,
\Tag{(72)}
\]
or from~\Eq{(71)}
\[
dS = \frac{dU + p\, dV}{T}\Add{.}
\Tag{(73)}
\]

In this equation the quantities $U$,~$p$, $V$,~$S$ represent certain
properties of the heat radiation, which are completely defined by
the instantaneous state of the radiation. Therefore the quantity~$T$
is also a certain property of the state of the radiation, \ie,~the
black radiation in the cavity has a certain temperature~$T$ and
this temperature is that of a body which is in heat equilibrium
with the radiation.

\Section{63.} We shall now deduce from the last equation a consequence
which is based on the fact that the state of the system considered,
and therefore also its entropy, is determined by the values of two
independent variables. As the first variable we shall take~$V$, as
the second either $T$,~$u$, or~$p$ may be chosen. Of these three quantities
any two are determined by the third together with~$V$.
We shall take the volume~$V$ and the temperature~$T$ as independent
variables. Then by substituting from \Eq{(66)}~and~\Eq{(70)} in~\Eq{(73)}
we have
\[
dS = \frac{V}{T}\, \frac{du}{dT}\, dT + \frac{4u}{3T}\, dV.
\Tag{(74)}
\]
From this we obtain
\[
\left(\frac{\dd S}{\dd T}\right)_{V} = \frac{V}{T}\, \frac{du}{dT}\qquad
\left(\frac{\dd S}{\dd V}\right)_{T} = \frac{4u}{3T}.
\]
%% -----File: 079.png---Folio 63-------
On partial differentiation of these equations, the first with respect
to~$V$, the second with respect to~$T$, we find
\[
\frac{\dd^{2} S}{\dd T\, \dd V}
  = \frac{1}{T}\, \frac{du}{dT}
  = \frac{4}{3T}\, \frac{du}{dT} - \frac{4u}{3T^{2}}
\]
or
\[
\frac{du}{dT} = \frac{4u}{T}
\]
and on integration
\[
u = aT^{4}
\Tag{(75)}
\]
and from~\Eq{(21)} for the specific intensity of black radiation
\[
K = \frac{c}{4\pi} · u = \frac{ac}{4\pi} T^{4}.
\Tag{(76)}
\]
Moreover for the pressure of black radiation
\[
p = \frac{a}{3} T^{4},
\Tag{(77)}
\]
and for the total radiant energy
\[
U = aT^{4} · V.
\Tag{(78)}
\]
This law, which states that the volume density and the specific
intensity of black radiation are proportional to the fourth power
of the absolute temperature, was first established by \Name{J.~Stefan}\footnote
  {\Name{J.~Stefan}, Wien.\ Berichte, \textbf{79}, p.~391, 1879.}
on
a basis of rather rough measurements. It was later deduced
by \Name{L.~Boltzmann}\footnote
  {\Name{L.~Boltzmann}, Wied.\ Annalen, \textbf{22}, p.~291, 1884.}
on a thermodynamic basis from \Name{Maxwell's}
radiation pressure and has been more recently confirmed by
\Name{O.~Lummer} and \Name{E.~Pringsheim}\footnote
  {\Name{O.~Lummer} und \Name{E.~Pringsheim}, Wied.\ Annalen, \textbf{63}, p.~395, 1897.  Annalen d.\ Physik\DPchg{,}{} \textbf{3},
  p.~159, 1900.}
by exact measurements between
$100°$ and $1300° \Celsius$, the temperature being defined by the gas
thermometer. In ranges of temperature and for requirements
of precision for which the readings of the different gas thermometers
no longer agree sufficiently or cannot be obtained at all, the
\Name{Stefan-Boltzmann} law of radiation can be used for an absolute
definition of temperature independent of all substances.

\Section{64.} The numerical value of the constant~$a$ is obtained from
measurements made by \Name{F.~Kurlbaum}.\footnote
  {\Name{F.~Kurlbaum}, Wied.\ Annalen, \textbf{65}, p.~759, 1898.}
According to them, if
%% -----File: 080.png---Folio 64-------
we denote by~$S_{t}$ the total energy radiated in one second into air
by a square centimeter of a black body at a temperature of $t° \Celsius$,
the following equation holds
\[
S_{100} - S_{0}
  = 0.0731\, \frac{\watt}{\cm^{2}}
  = 7.31 × 10^{5}\, \frac{\erg}{\cm^{2}\, \sec}\Add{.}
%[** TN: No equation number in the original]
\Tag{(79)}
\]
Now, since the radiation in air is approximately identical with
the radiation into a vacuum, we may according to \Eq{(7)}~and~\Eq{(76)}
put
\[
S_{t} = \pi K = \frac{ac}{4} (273 + t)^{4}
\]
and from this
\[
S_{100} - S_{0} = \frac{ac}{4} (373^{4} - 273^{4}),
\]
therefore
\[
a = \frac{4 × 7.31 × 10^{5}}{3 × 10^{10} × (373^{4} - 273^{4})}
  = 7.061 × 10^{-15}\, \frac{\erg}{\cm^{3}\, \degree^{4}}\Add{.}
\]

Recently \Name{Kurlbaum} has increased the value measured by him
by $2.5$~per cent.,\footnote
  {\Name{F. Kurlbaum}, Verhandlungen d.\ Deutsch.\ physikal.\ Gesellschaft, \textbf{14}, p.~580, 1912.}
on account of the bolometer used being not
perfectly black, whence it follows that $a = 7.24 · 10^{-15}$.

Meanwhile the radiation constant has been made the object
of as accurate measurements as possible in various places. Thus
it was measured by \Name{Féry}, \Name{Bauer} and \Name{Moulin}, \Name{Valentiner}, \Name{Féry} and
\Name{Drecq}, \Name{Shakespear}, \Name{Gerlach}, with in some cases very divergent
results, so that a mean value may hardly be formed.

For later computations we shall use the most recent determination
made in the physical laboratory of the University of Berlin\footnote
  {According to private information kindly furnished by my colleague \Name{H.~Rubens} (July,
  \Label{64}%[** TN: Page label]
  1912). (These results have since been published. See \Name{W.~H. Westphal}, Verhandlungen d.\
  Deutsch.\ physikal.\ Gesellschaft, \textbf{14}, p.~987, 1912,~Tr.)}
\[
\frac{ac}{4}
  = \sigma
  = 5.46 · 10^{-12}\, \frac{\watt}{\cm^{2}\, \degree^{4}}\Add{.}
\]
From this $a$~is found to be
\[
a = \frac{4 · 5.46 · 10^{-12} · 10^{7}}{3 · 10^{10}}
  = 7.28 · 10^{-15}\, \frac{\erg}{\cm^{3}\, \degree^{4}}
\]
which agrees rather closely with \Name{Kurlbaum's} corrected value.
%% -----File: 081.png---Folio 65-------

\Section{65.} The magnitude of the entropy~$S$ of black radiation found
by integration of the differential equation~\Eq{(73)} is
\[
S = \frac{4}{3} aT^{3}V.
\Tag{(80)}
\]

In this equation the additive constant is determined by a choice
that readily suggests itself, so that at the zero of the absolute
scale of temperature, that is to say, when $u$~vanishes, $S$~shall
become zero. From this the entropy of unit volume or the
volume density of the entropy of black radiation is obtained,
\[
\frac{S}{V} = s = \frac{4}{3} aT^{3}.
\Tag{(81)}
\]

\Section{66.} We shall now remove a restricting assumption made in
order to enable us to apply the value of \Name{Maxwell's} radiation
pressure, calculated in the preceding chapter. Up to now we
have assumed the cylinder to be fixed and only the piston to be
free to move. We shall now think of the whole of the vessel,
consisting of the cylinder, the black bottom, and the piston, the
latter attached to the walls in a definite height above the bottom,
as being free to move in space. Then, according to the principle
of action and reaction, the vessel as a whole must remain constantly
at rest, since no external force acts on it. This is the
conclusion to which we must necessarily come, even without,
in this case, admitting \textit{a~priori} the validity of the principle of
action and reaction. For if the vessel should begin to move,
the kinetic energy of this motion could originate only at the expense
of the heat of the body forming the bottom or the energy of
radiation, as there exists in the system enclosed in a rigid cover
no other available energy; and together with the decrease of
energy the entropy of the body or the radiation would also decrease,
an event which would contradict the second principle,
since no other changes of entropy occur in nature. Hence the
vessel as a whole is in a state of mechanical equilibrium. An
immediate consequence of this is that the pressure of the radiation
on the black bottom is just as large as the oppositely directed
pressure of the radiation on the reflecting piston. Hence the
pressure of black radiation is the same on a black as on a reflecting
body of the same temperature and the same may be readily proven
%% -----File: 082.png---Folio 66-------
for any completely reflecting surface whatsoever, which we may
assume to be at the bottom of the cylinder without in the least
disturbing the stationary state of radiation. Hence we may also
in all the foregoing considerations replace the reflecting metal
by any completely reflecting or black body whatsoever, at the
same temperature as the body forming the bottom, and it may
be stated as a quite general law that the radiation pressure
depends only on the properties of the radiation passing to and
fro, not on the properties of the enclosing substance.

\Section{67.} If, on raising the piston, the temperature of the black body
forming the bottom is kept constant by a corresponding addition
of heat from the heat reservoir, the process takes place isothermally.
Then, along with the temperature~$T$ of the black body,
the energy density~$u$, the radiation pressure~$p$, and the density of
the entropy~$s$ also remain constant; hence the total energy of
radiation increases from $U = uV$ to $U' = uV'$, the entropy from
$S = sV$ to $S' = sV'$ and the heat supplied from the heat reservoir
is obtained by integrating~\Eq{(72)} at constant~$T$,
\[
Q = T(S'- S) = Ts(V' - V)
\]
or, according to \Eq{(81)}~and~\Eq{(75)},
\[
Q = \frac{4}{3} aT^{4}(V' - V) = \frac{4}{3} (U' - U).
\]

Thus it is seen that the heat furnished from the outside exceeds
the increase in energy of radiation $(U' - U)$ by $\frac{1}{3}(U' - U)$.
This excess in the added heat is necessary to do the external work
accompanying the increase in the volume of radiation.

\Section{68.} Let us also consider a reversible adiabatic process. For
this it is necessary not merely that the piston and the mantle but
also that the bottom of the cylinder be assumed as completely
reflecting, \eg,~as white. Then the heat furnished on compression
or expansion of the volume of radiation is $Q = 0$ and the energy
of radiation changes only by the value~$p\, dV$ of the external work.
To insure, however, that in a finite adiabatic process the radiation
shall be perfectly stable at every instant, \ie, shall have the character
of black radiation, we may assume that inside the evacuated
cavity there is a carbon particle of minute size. This particle,
which may be assumed to possess an absorbing power differing
%% -----File: 083.png---Folio 67-------
from zero for all kinds of rays, serves merely to produce stable
equilibrium of the radiation in the cavity (\Sec{51} \textit{et~seq.})\ and
thereby to insure the reversibility of the process, while its heat
contents may be taken as so small compared with the energy of
radiation,~$U$, that the addition of heat required for an appreciable
temperature change of the particle is perfectly negligible. Then
at every instant in the process there exists absolutely stable
equilibrium of radiation and the radiation has the temperature of
the particle in the cavity. The volume, energy, and entropy of
the particle may be entirely neglected.

On a reversible adiabatic change, according to~\Eq{(72)}, the entropy~$S$
of the system remains constant. Hence from~\Eq{(80)} we have as
a condition for such a process
\[
T^{3}V = \const.,
\]
or, according to~\Eq{(77)},
\[
pV^{\efrac{4}{3}} = \const.,
\]
\ie, on an adiabatic compression the temperature and the pressure
of the radiation increase in a manner that may be definitely
stated. The energy of the radiation,~$U$, in such a case varies
according to the law
\[
\frac{U}{T} = \frac{3}{4} S = \const.,
\]
\ie, it increases in proportion to the absolute temperature, although
the volume becomes smaller.

\Section{69.} Let us finally, as a further example, consider a simple case
of an irreversible process. Let the cavity of volume~$V$, which is
everywhere enclosed by absolutely reflecting walls, be uniformly
filled with black radiation. Now let us make a small hole
through any part of the walls, \eg,~by opening a stopcock, so
that the radiation may escape into another completely evacuated
space, which may also be surrounded by rigid, absolutely reflecting
walls. The radiation will at first be of a very irregular character;
after some time, however, it will assume a stationary condition
and will fill both communicating spaces uniformly, its total
volume being, say,~$V'$. The presence of a carbon particle will
cause all conditions of black radiation to be satisfied in the new
%% -----File: 084.png---Folio 68-------
state. Then, since there is neither external work nor addition of
heat from the outside, the energy of the new state is, according
to the first principle, equal to that of the original one, or $U' = U$
and hence from~\Eq{(78)}
\begin{gather*}
T'^{4} V' = T^{4} V \\
\frac{T'}{T} = \sqrt[4]{\frac{V}{V'}}
\end{gather*}
which defines completely the new state of equilibrium. Since
$V' > V$ the temperature of the radiation has been lowered by the
process.

According to the second principle of thermodynamics the
entropy of the system must have increased, since no external
changes have occurred; in fact we have from~\Eq{(80)}
\[
\frac{S'}{S} = \frac{T'^{3}V'}{T^{3}V} = \sqrt[4]{\frac{V'}{V}} > 1.
\Tag{(82)}
\]

\Section{70.} If the process of irreversible adiabatic expansion of the
radiation from the volume~$V$ to the volume~$V'$ takes place as
just described with the single difference that there is no carbon
particle present in the vacuum, after the stationary state of radiation
is established, as will be the case after a certain time on
account of the diffuse reflection from the walls of the cavity, the
radiation in the new volume~$V'$ will not any longer have the
character of black radiation, and hence no definite temperature.
Nevertheless the radiation, like every system in a definite physical
state, has a definite entropy, which, according to the second principle,
is larger than the original~$S$, but not as large as the $S'$ given
in~\Eq{(82)}. The calculation cannot be performed without the use
of laws to be taken up later (see \Sec{103}). If a carbon particle
is afterward introduced into the vacuum, absolutely stable
equilibrium is established by a second irreversible process, and,
the total energy as well as the total volume remaining constant,
the radiation assumes the normal energy distribution of black
radiation and the entropy increases to the maximum value~$S'$
given by~\Eq{(82)}.
%% -----File: 085.png---Folio 69-------

\Chapter{III}{Wien's Displacement Law}

\Section{71.} Though the manner in which the volume density~$u$ and the
specific intensity~$K$ of black radiation depend on the temperature
is determined by the \Name{Stefan-Boltzmann} law, this law is of comparatively
little use in finding the volume density~$\ssfu_{\nu}$ corresponding
to a definite frequency~$\nu$, and the specific intensity of radiation~$\ssfK_{\nu}$
of monochromatic radiation, which are related to each other
by equation~\Eq{(24)} and to $u$~and~$K$ by equations \Eq{(22)}~and~\Eq{(12)}.
There remains as one of the principal problems of the theory of
heat radiation the problem of determining the quantities $\ssfu_{\nu}$~and~$\ssfK_{\nu}$
for black radiation in a vacuum and hence, according to~\Eq{(42)},
in any medium whatever, as functions of $\nu$~and~$T$, or, in other
words, to find the distribution of energy in the normal spectrum
for any arbitrary temperature. An essential step in the solution
of this problem is contained in the so-called ``displacement
law'' stated by \Name{W.~Wien},\footnote
  {\Name{W.~Wien}, Sitzungsberichte d.\ Akad.\ d.\ Wissensch.\ Berlin, Febr.~9, 1893, p.~55. Wiedemann's
  Annal., \textbf{52}, p.~132, 1894. See also among others \Name{M.~Thiesen}, Verhandl.\ d.\ Deutsch.\
  phys.\ Gesellsch\DPtypo{}{.}, \textbf{2}, p.~65, 1900. \Name{H.~A. Lorentz}, Akad.\ d.\ Wissensch.\ Amsterdam, May~18,
  1901, p.~607. \Name{M.~Abraham}, Annal.\ d.\ Physik\DPtypo{.}{}\ \textbf{14}, p.~236, 1904.}
the importance of which lies in the
fact that it reduces the functions $\ssfu_{\nu}$~and~$\ssfK_{\nu}$ of the two arguments
$\nu$~and~$T$ to a function of a single argument.

The starting point of \Name{Wien's} displacement law is the following
theorem. If the black radiation contained in a perfectly evacuated
cavity with absolutely reflecting walls is compressed or
expanded adiabatically and infinitely slowly, as described above
%[** Theorem]
in \Sec{68}, \emph{the radiation always retains the character of black radiation,
even without the presence of a carbon particle}. Hence the
process takes place in an absolute vacuum just as was calculated
in \Sec{68} and the introduction, as a precaution, of a carbon
particle is shown to be superfluous. But this is true only in this
special case, not at all in the case described in \Sec{70}.

The truth of the proposition stated may be shown as follows:
%% -----File: 086.png---Folio 70-------
Let the completely evacuated hollow cylinder, which is at the
start filled with black radiation, be compressed adiabatically
and infinitely slowly to a finite fraction of the original volume.
If, now, the compression being completed, the radiation were no
longer black, there would be no stable thermodynamic equilibrium
of the radiation (\Sec{51}). It would then be possible to
produce a finite change at constant volume and constant total
energy of radiation, namely, the change to the absolutely stable
state of radiation, which would cause a finite increase of entropy.
This change could be brought about by the introduction of a
carbon particle, containing a negligible amount of heat as compared
with the energy of radiation. This change, of course,
refers only to the spectral density of radiation~$\ssfu_{\nu}$, whereas the
total density of energy~$u$ remains constant. After this has been
accomplished, we could, leaving the carbon particle in the space,
allow the hollow cylinder to return adiabatically and infinitely
slowly to its original volume and then remove the carbon particle.
The system will then have passed through a cycle without any
external changes remaining. For heat has been neither added
nor removed, and the mechanical work done on compression has
been regained on expansion, because the latter, like the radiation
pressure, depends only on the total density~$u$ of the energy of radiation,
not on its spectral distribution. Therefore, according to
the first principle of thermodynamics, the total energy of radiation
is at the end just the same as at the beginning, and hence
also the temperature of the black radiation is again the same.
The carbon particle and its changes do not enter into the calculation,
for its energy and entropy are vanishingly small compared
with the corresponding quantities of the system. The
process has therefore been reversed in all details; it may be
repeated any number of times without any permanent change
occurring in nature. This contradicts the assumption, made
above, that a finite increase in entropy occurs; for such a finite
increase, once having taken place, cannot in any way be completely
reversed. Therefore no finite increase in entropy can have
been produced by the introduction of the carbon particle in the
space of radiation, but the radiation was, before the introduction
and always, in the state of stable equilibrium.

\Section{72.} In order to bring out more clearly the essential part of
%% -----File: 087.png---Folio 71-------
this important proof, let us point out an analogous and more or
less obvious consideration. Let a cavity containing originally
a vapor in a state of saturation be compressed adiabatically and
infinitely slowly.

%[** Quoted; semantics?]
``Then on an arbitrary adiabatic compression the vapor remains
always just in the state of saturation. For let us suppose that it
becomes supersaturated on compression. After the compression
to an appreciable fraction of the original volume has taken place,
condensation of a finite amount of vapor and thereby a change
into a more stable state, and hence a finite increase of entropy of
the system, would be produced at constant volume and constant
total energy by the introduction of a minute drop of liquid, which
has no appreciable mass or heat capacity. After this has been
done, the volume could again be increased adiabatically and
infinitely slowly until again all liquid is evaporated and thereby
the process completely reversed, which contradicts the assumed
increase of entropy.''

Such a method of proof would be erroneous, because, by the
process described, the change that originally took place is not
at all completely reversed. For since the mechanical work
expended on the compression of the supersaturated steam is not
equal to the amount gained on expanding the saturated steam,
there corresponds to a definite volume of the system when it is
being compressed an amount of energy different from the one
during expansion and therefore the volume at which all liquid is
just vaporized cannot be equal to the original volume. The
supposed analogy therefore breaks down and the statement made
above in quotation marks is incorrect.

\Section{73.} We shall now again suppose the reversible adiabatic process
described in \Sec{68} to be carried out with the black radiation
contained in the evacuated cavity with white walls and white
bottom, by allowing the piston, which consists of absolutely
reflecting metal, to move downward infinitely slowly, with the
single difference that now there shall be no carbon particle in the
cylinder. The process will, as we now know, take place exactly
as there described, and, since no absorption or emission of radiation
takes place, we can now give an account of the changes of
color and intensity which the separate pencils of the system
undergo. Such changes will of course occur only on reflection
%% -----File: 088.png---Folio 72-------
from the moving metallic reflector, not on reflection from the
stationary walls and the stationary bottom of the cylinder.

If the reflecting piston moves down with the constant, infinitely
small, velocity~$v$, the monochromatic pencils striking it during
the motion will suffer on reflection a change of color, intensity,
and direction. Let us consider these different influences in order.\footnote
  {The complete solution of the problem of reflection of a pencil from a moving absolutely
  reflecting surface including the case of an arbitrarily large velocity of the surface may be
  found in the paper by \Name{M.~Abraham} quoted in \Sec{71}. See also the text-book by the same
  author. Electromagnetische Theorie der Strahlung, 1908 (Leipzig, B.~G. Teubner).}

\Section{74.} To begin with, we consider the \emph{change of color} which a monochromatic
ray suffers by reflection from the reflector, which is
\Figure[2in]{5}
moving with an infinitely small velocity.
For this purpose we consider
first the case of a ray which falls
normally from below on the reflector
and hence is reflected normally downward.
Let the plane~$A$ (\Fig{5}) represent
the position of the reflector at the
time~$t$, the plane~$A'$ the position at
the time $t + \delta t$, where the distance~$AA'$
equals $v\, \delta t$, $v$~denoting the velocity
of the reflector. Let us now suppose a stationary plane~$B$ to be
placed parallel to~$A$ at a suitable distance and let us denote by~$\lambda$
the wave length of the ray incident on the reflector and by~$\lambda'$
the wave length of the ray reflected from it. Then at a time~$t$
there are in the interval~$AB$ in the vacuum containing the radiation
$\dfrac{AB}{\lambda}$ waves of the incident and $\dfrac{AB}{\lambda'}$ waves of the reflected ray,
as can be seen, \eg, by thinking of the electric field-strength as
being drawn at the different points of each of the two rays at
the time~$t$ in the form of a sine curve. Reckoning both incident
and reflected ray there are at the time~$t$
\[
AB\left(\frac{1}{\lambda} + \frac{1}{\lambda'}\right)
\]
waves in the interval between $A$~and~$B$. Since this is a large number,
it is immaterial whether the number is an integer or not.
%% -----File: 089.png---Folio 73-------

Similarly at the time $t + \delta t$, when the reflector is at~$A'$, there are
\[
A'B\left(\frac{1}{\lambda} + \frac{1}{\lambda'}\right)
\]
waves in the interval between $A'$~and~$B$ all told.

The latter number will be smaller than the former, since in the
shorter distance~$A'B$ there is room for fewer waves of both kinds
than in the longer distance~$AB$. The remaining waves must have
been expelled in the time~$\delta t$ from the space between the stationary
plane~$B$ and the moving reflector, and this must have taken place
through the plane~$B$ downward; for in no other way could a
wave disappear from the space considered.

Now $\nu\, \delta t$~waves pass in the time~$\delta t$ through the stationary
plane~$B$ in an upward direction and $\nu'\, \delta t$~waves in a downward
direction; hence we have for the difference
\[
(\nu' - \nu)\, \delta t = (AB - A'B) \left(\frac{1}{\lambda} + \frac{1}{\lambda'}\right)
\]
or, since
\[
AB - A'B = v\, \delta t,
\]
and
\begin{gather*}
\lambda  = \frac{c}{\nu}\quad
\lambda' = \frac{c}{\nu'} \\
\nu' = \frac{c + v}{c - v}  \nu
\end{gather*}
or, since $v$~is infinitely small compared with~$c$,
\[
\nu' = \nu \left(1 + \frac{2v}{c}\right)\Add{.}
\]

\Section{75.} When the radiation does not fall on the reflector normally
but at an acute angle of incidence~$\theta$, it is possible to pursue a very
similar line of reasoning, with the difference that then~$A$, the
point of intersection of a definite ray~$BA$ with the reflector at
the time~$t$, has not the same position on the reflector as the point
of intersection,~$A'$, of the same ray with the reflector at the time
$t + \delta t$ (\Fig{6}). The number of waves which lie in the interval~$BA$
at the time~$t$ is~$\dfrac{BA}{\lambda}$. Similarly, at the time~$t$ the number of
waves in the interval~$AC$ representing the distance of the point~$A$
%% -----File: 090.png---Folio 74-------
from a wave plane~$CC'$, belonging to the reflected ray and
stationary in the vacuum, is~$\dfrac{AC}{\lambda'}$.

Hence there are, all told, at the time~$t$ in the interval~$BAC$
\[
\frac{BA}{\lambda} + \frac{AC}{\lambda'}
\]
waves of the ray under consideration. We may further note
that the angle of reflection~$\theta'$ is not exactly equal to the angle
\Figure[4in]{6}
of incidence, but is a little smaller as can be shown by a simple
geometric consideration based on \Name{Huyghens'} principle. The
difference of $\theta$ and~$\theta'$, however, will be shown to be non-essential
for our calculation. Moreover there are at the time $t + \delta t$, when
the reflector passes through~$A'$,
\[
\frac{BA'}{\lambda} + \frac{A'C'}{\lambda'}
\]
waves in the distance~$BA'C'$. The latter number is smaller than
the former and the difference must equal the total number of
waves which are expelled in the time~$\delta t$ from the space which is
bounded by the stationary planes $BB'$~and~$CC'$.

Now $\nu\, \delta t$~waves enter into the space through the plane~$BB'$ in
the time~$\delta t$ and $\nu'\, \delta t$~waves leave the space through the plane~$CC'$\Add{.}
Hence we have
\[
(\nu' - \nu)\, \delta t
  = \left(\frac{BA}{\lambda} + \frac{AC}{\lambda'}\right)
  - \left(\frac{BA'}{\lambda} + \frac{A'C'}{\lambda'}\right)
\]
%% -----File: 091.png---Folio 75-------
but
\begin{gather*}
BA - BA' = AA' = \frac{v\, \delta t}{\cos\theta} \\
AC - A'C' = AA' \cos(\theta + \theta') \\
\lambda  = \frac{c}{\nu},\quad
\lambda' = \frac{c}{\nu'}.
\end{gather*}
Hence
\[
\nu' = \frac{c \cos\theta + v}{c \cos\theta - v \cos(\theta + \theta')} \nu.
\]

This relation holds for any velocity~$v$ of the moving reflector.
Now, since in our case $v$~is infinitely small compared with~$c$, we
have the simpler expression
\[%[** Size of parentheses]
\nu' = \nu (1 + \frac{v}{c \cos\theta} [1 + \cos(\theta + \theta')])\Add{.}
\]
The difference between the two angles $\theta$~and~$\theta'$ is in any case of
the order of magnitude~$\dfrac{v}{c}$; hence we may without appreciable
error replace $\theta'$ by~$\theta$, thereby obtaining the following expression
for the frequency of the reflected ray for oblique incidence
\[
\nu' = \nu \left(1 + \frac{2v \cos\theta}{c}\right)\Add{.}
\Tag{(83)}
\]

\Section{76.} From the foregoing it is seen that the frequency of all rays
which strike the moving reflector are increased on reflection, when
the reflector moves toward the radiation, and decreased, when the
reflector moves in the direction of the incident rays ($v < 0$).
However, the total radiation of a definite frequency~$\nu$ striking the
moving reflector is by no means reflected as monochromatic radiation
but the change in color on reflection depends also essentially
on the angle of incidence~$\theta$. Hence we may not speak of a certain
spectral ``displacement'' of color except in the case of a single
pencil of rays of definite direction, whereas in the case of the
entire monochromatic radiation we must refer to a spectral
``dispersion.'' The change in color is the largest for normal incidence
and vanishes entirely for grazing incidence.

\Section{77.} Secondly, let us calculate the \emph{change in energy}, which the
%% -----File: 092.png---Folio 76-------
moving reflector produces in the incident radiation, and let us
consider from the outset the general case of oblique incidence.
Let a monochromatic, infinitely thin, unpolarized pencil of rays\DPtypo{.}{,}
which falls on a surface element of the reflector at the angle of
incidence~$\theta$, transmit the energy~$I\, \delta t$ to the reflector in the time~$\delta t$.
Then, ignoring vanishingly small quantities, the mechanical
pressure of the pencil of rays normally to the reflector is, according
to equation~\Eq{(64)},
\[
\ssfF = \frac{2 \cos\theta}{c} I,
\]
and to the same degree of approximation the work done from the
outside on the incident radiation in the time~$\delta t$ is
\[
\ssfF v\, \delta t = \frac{2v \cos\theta}{c} I\, \delta t.
\Tag{(84)}
\]
According to the principle of the conservation of energy this
amount of work must reappear in the energy of the reflected radiation.
Hence the reflected pencil has a larger intensity than the
incident one. It produces, namely, in the time~$\delta t$ the energy\footnote
  {It is clear that the change in intensity of the reflected radiation caused by the motion of
  the reflector can also be derived from purely electrodynamical considerations, since electrodynamics
  are consistent with the energy principle. This method is somewhat lengthy,
  but it affords a deeper insight into the details of the phenomenon of reflection.}
\[
I\, \delta t + \ssfF v\, \delta t
  = I\left(1 + \frac{2v \cos\theta}{c}\right) \delta t
  = I'\, \delta t.
\Tag{(85)}
\]
Hence we may summarize as follows: By the reflection of a
monochromatic unpolarized pencil, incident at an angle~$\theta$ on a
reflector moving toward the radiation with the infinitely small
velocity~$v$, the radiant energy~$I\, \delta t$, whose frequencies extend from
$\nu$ to $\nu + d\nu$, is in the time~$\delta t$ changed into the radiant energy
$I'\, \delta t$ with the interval of frequency $(\nu', \nu' + d\nu')$, where $I'$~is given
by~\Eq{(85)}, $\nu'$~by~\Eq{(83)}, and accordingly $d\nu'$, the spectral breadth of
the reflected pencil, by
\[
d\nu' = d\nu \left(1 + \frac{2v \cos\theta}{c}\right)\Add{.}
\Tag{(86)}
\]
A comparison of these values shows that
\[
\frac{I'}{I} = \frac{\nu'}{\nu} = \frac{d\nu'}{d\nu}\Add{.}
\Tag{(87)}
\]
%% -----File: 093.png---Folio 77-------
The absolute value of the radiant energy which has disappeared
in this change is, from equation~\Eq{(13)},
\[
I\, \delta t = 2\ssfK_{\nu}\, d\sigma \cos\theta\, d\Omega\, d\nu\, \delta t,
\Tag{(88)}
\]
and hence the absolute value of the radiant energy which has
been formed is, according to~\Eq{(85)},
\[
I'\, \delta t = 2\ssfK_{\nu}\, d\sigma \cos\theta\, d\Omega\, d\nu \left(1 + \frac{2v \cos\theta}{c}\right) \delta t.
\Tag{(89)}
\]

Strictly speaking these last two expressions would require an
infinitely small correction, since the quantity~$I$ from equation~\Eq{(88)}
represents the energy radiation on a stationary element of area~$d\sigma$,
while, in reality, the incident radiation is slightly increased
by the motion of~$d\sigma$ toward the incident pencil. \Erratum{The additional
terms resulting therefrom may, however, be omitted here without
appreciable error.}{The corresponding additional terms may, however, be omitted
here without appreciable error, since the correction caused by
them would consist merely of the addition to the energy change
here calculated of a comparatively infinitesimal energy change of
the same kind with an external work that is infinitesimal of the
second order.}

\Section{78.} As regards finally the \emph{changes in direction}, which are imparted
to the incident ray by reflection from the moving reflector,
we need not calculate them at all at this stage. For if the motion
of the reflector takes place sufficiently slowly, all irregularities
in the direction of the radiation are at once equalized by further
reflection from the walls of the vessel. We may, indeed, think of
the whole process as being accomplished in a very large number of
short intervals, in such a way that the piston, after it has moved
a very small distance with very small velocity, is kept at rest for
a while, namely, until all irregularities produced in the directions
of the radiation have disappeared as the result of the reflection
from the white walls of the hollow cylinder. If this procedure
be carried on sufficiently long, the compression of the radiation
may be continued to an arbitrarily small fraction of the original
volume, and while this is being done, the radiation may be always
regarded as uniform in all directions. This continuous process
of equalization refers, of course, only to difference in the direction
of the radiation; for changes in the color or intensity of the
radiation of however small size, having once occurred, can
evidently never be equalized by reflection from totally reflecting
stationary walls but continue to exist forever.

\Section{79.} With the aid of the theorems established we are now in a
position to calculate the change of the density of radiation for
%% -----File: 094.png---Folio 78-------
every frequency for the case of infinitely slow adiabatic compression
of the perfectly evacuated hollow cylinder, which is filled
with uniform radiation. For this purpose we consider the radiation
at the time~$t$ in a definite infinitely small interval of frequencies,
from $\nu$ to $\nu + d\nu$, and inquire into the change which
the total energy of radiation contained in this definite constant
interval suffers in the time~$\delta t$.

At the time~$t$ this radiant energy is, according to \Sec{23}, $V\ssfu\, d\nu$;
at the time $t + \delta t$ it is $\bigl(V\ssfu + \delta (V\ssfu)\bigr)\, d\nu$, hence the change to be
calculated is
\[
\delta (V\ssfu)\, d\nu.
\Tag{(90)}
\]
In this the density of monochromatic radiation~$\ssfu$ is to be regarded
as a function of the mutually independent variables $\nu$~and~$t$, the
differentials of which are distinguished by the symbols $d$~and~$\delta$.

The change of the energy of monochromatic radiation is produced
only by the reflection from the moving reflector, that is
to say, firstly by certain rays, which at the time~$t$ belong to the
interval $(\nu, d\nu)$, leaving this interval on account of the change in
color suffered by reflection, and secondly by certain rays, which at
the time~$t$ do not belong to the interval $(\nu, d\nu)$, coming into this
interval on account of the change in color suffered on reflection.
Let us calculate these influences in order. The calculation is
greatly simplified by taking the width of this interval~$d\nu$ so small
that
\[
d\nu \text{ is small compared with } \frac{v}{c} \nu,
\Tag{(91)}
\]
a condition which can always be satisfied, since $d\nu$~and~$v$ are
mutually independent.

\Section{80.} The rays which at the time~$t$ belong to the interval $(\nu, d\nu)$
and leave this interval in the time~$\delta t$ on account of reflection from
the moving reflector, are simply those rays which strike the
moving reflector in the time~$\delta t$. For the change in color which
such a ray undergoes is, from \Eq{(83)}~and~\Eq{(91)}, large compared with~$d\nu$,
the width of the whole interval. Hence we need only calculate
the energy, which in the time~$\delta t$ is transmitted to the reflector
by the rays in the interval $(\nu, d\nu)$.

For an elementary pencil, which falls on the element~$d\sigma$ of the
%% -----File: 095.png---Folio 79-------
reflecting surface at the angle of incidence~$\theta$, this energy is,
according to \Eq{(88)}~and~\Eq{(5)},
\[
I\, \delta t = 2\ssfK_{\nu}\, d\sigma \cos\theta\, d\Omega\, d\nu\, \delta t
  = 2\ssfK_{\nu}\, d\sigma \sin\theta \cos\theta\, d\theta\, d\phi\, d\nu\, \delta t.
\]
Hence we obtain for the total monochromatic radiation, which
falls on the whole surface~$F$ of the reflector, by integration with
respect to~$\phi$ from $0$ to~$2\pi$, with respect to~$\theta$ from $0$ to~$\dfrac{\pi}{2}$, and with
respect to~$d\sigma$ from $0$ to~$F$,
\[
2\pi F \ssfK_{\nu}\, d\nu\, \delta t.
\Tag{(92)}
\]
Thus this radiant energy leaves, in the time~$\delta t$, the interval of
frequencies $(\nu, d\nu)$ considered.

\Section{81.} In calculating the radiant energy which enters the interval
$(\nu, d\nu)$ in the time~$\delta t$ on account of reflection from the moving
reflector, the rays falling on the reflector at different angles of
incidence must be considered separately. Since in the case of a
positive~$v$, the frequency is increased by the reflection, the rays
which must be considered have, at the time~$t$, the frequency
$\nu_{1} < \nu$. If we now consider at the time~$t$ a monochromatic pencil
of frequency $(\nu_{1}, d\nu_{1})$, falling on the reflector at an angle of incidence~$\theta$,
a necessary and sufficient condition for its entrance, by
reflection, into the interval $(\nu, d\nu)$ is
\[
\nu = \nu_{1} \left(1 + \frac{2v \cos\theta}{c}\right) \quad\text{and}\quad
d\nu = d\nu_{1} \left(1 + \frac{2v \cos\theta}{c}\right)\Add{.}
\]
These relations are obtained by substituting $\nu_{1}$~and $\nu$ respectively
in the equations \Eq{(83)}~and~\Eq{(86)} in place of the frequencies before
and after reflection $\nu$~and~$\nu'$.

The energy which this pencil carries into the interval $(\nu_{1}, d\nu)$
in the time~$\delta t$ is obtained from~\Eq{(89)}, likewise by substituting $\nu_{1}$
for~$\nu$. It is
\[
2\ssfK_{\nu_{1}}\, d\sigma \cos\theta\, d\Omega\, d\nu_{1} \left(1 + \frac{2v \cos\theta}{c}\right) \delta t
  = 2\ssfK_{\nu_{1}}\, d\sigma \cos\theta\, d\Omega\, d\nu\, \delta t.
\]
Now we have
\[
\ssfK_{\nu_{1}} = \ssfK_{\nu} + (\nu_{1} - \nu) \frac{\dd \ssfK}{\dd \nu} + \dots
\]
where we shall assume $\dfrac{\dd \ssfK}{\dd \nu}$ to be finite.
%% -----File: 096.png---Folio 80-------

Hence, neglecting small quantities of higher order,
\[
\ssfK_{\nu_{1}} = \ssfK_{\nu} - \frac{2\nu v \cos\theta}{c}\, \frac{\dd \ssfK}{\dd \nu}\Add{.}
\]
Thus the energy required becomes
\[
2\, d\sigma \left(\ssfK_{\nu} - \frac{2\nu v \cos\theta}{c}\, \frac{\dd \ssfK}{\dd \nu}\right) \sin\theta \cos\theta\, d\theta\, d\phi\, d\nu\, \delta t,
\]
and, integrating this expression as above, with respect to $d\sigma$,~$\phi$,
and~$\theta$, the total radiant energy which enters into the interval
$(\nu, d\nu)$ in the time~$\delta t$ becomes
\[
2\pi F \left(\ssfK_{\nu} - \frac{4}{3}\, \frac{\nu v}{c}\, \frac{\dd \ssfK}{\dd \nu}\right) d\nu\, \delta t.
\Tag{(93)}
\]

\Section{82.} The difference of the two expressions \Eq{(93)}~and~\Eq{(92)} is equal
to the whole change~\Eq{(90)}, hence
\[
-\frac{8\pi}{3} F \frac{\nu v}{c}\, \frac{\dd \ssfK}{\dd \nu}\, \delta t = \delta (V\ssfu),
\]
or, according to~\Eq{(24)},
\[
-\frac{1}{3} F\nu v \frac{\dd \ssfu}{\dd \nu}\, \delta t = \delta (V\ssfu),
\]
or, finally, since $Fv\, \delta t$ is equal to the decrease of the volume~$V$,
\[
\frac{1}{3} \nu \frac{\dd \ssfu}{\dd \nu}\, \delta V
  = \delta (V\ssfu)
  = \ssfu\, \delta V + V\, \delta \ssfu,
\Tag{(94)}
\]
whence it follows that
\[
\delta \ssfu = \left(\frac{\nu}{3}\, \frac{\dd \ssfu}{\dd \nu} - \ssfu\right) \frac{\delta V}{V}.
\Tag{(95)}
\]
This equation gives the change of the energy density of any
definite frequency~$\nu$, which occurs on an infinitely slow adiabatic
compression of the radiation. It holds, moreover, not only for
black radiation, but also for \emph{radiation originally of a perfectly
arbitrary distribution of energy}, as is shown by the method of
derivation.

Since the changes taking place in the state of the radiation in
the time~$\delta t$ are proportional to the infinitely small velocity~$v$ and
are reversed on changing the sign of the latter, this equation
holds for any sign of~$\delta V$; \emph{hence the process is reversible}.
%% -----File: 097.png---Folio 81-------

\Section{83.} Before passing on to the general integration of equation~\Eq{(95)}
let us examine it in the manner which most easily suggests
itself. According to the energy principle, the change in the
radiant energy
\[
U = Vu = V \int_{0}^{\infty} \ssfu\, d\nu,
\]
occurring on adiabatic compression, must be equal to the external
work done against the radiation pressure
\[
-p\, \delta V = -\frac{u}{3}\, \delta V
  = -\frac{\delta V}{3} \int_{0}^{\infty} \ssfu\, d\nu.
\Tag{(96)}
\]
Now from~\Eq{(94)} the change in the total energy is found to be
\[
\delta U = \int_{0}^{\infty} d\nu\, \delta(V\ssfu)
  = \frac{\delta V}{3} \int_{0}^{\infty} \nu \frac{\dd \ssfu}{\dd \nu}\, d\nu,
\]
or, by partial integration,
\[
\delta U = \frac{\delta V}{3} \bigl(
  [\nu \ssfu]\Strut_{0}^{\infty} - \int_{0}^{\infty} \ssfu\, d\nu
\bigr),
\]
and this expression is, in fact, identical with~\Eq{(96)}, since the product~$\nu\ssfu$
vanishes for $\nu = 0$ as well as for $\nu = \infty$. The latter might
at first seem doubtful; but it is easily seen that, if $\nu\ssfu$ for $\nu = \infty$
had a value different from zero, the integral of~$\ssfu$ with respect to~$\nu$
taken from $0$ to~$\infty$ could not have a finite value, which, however,
certainly is the case.

\Section{84.} We have already emphasized (\Sec{79}) that $\ssfu$~must be
regarded as a function of two independent variables, of which we
have taken as the first the frequency~$\nu$ and as the second the time~$t$.
Since, now, in equation~\Eq{(95)} the time~$t$ does not explicitly
appear, it is more appropriate to introduce the volume~$V$, which
depends only on~$t$, as the second variable instead of $t$~itself. Then
equation~\Eq{(95)} may be written as a partial differential equation as
follows:
\[
V \frac{\dd \ssfu}{\dd V} = \frac{\nu}{3}\, \frac{\dd \ssfu}{\dd \nu} - \ssfu.
\Tag{(97)}
\]
From this equation, if, for a definite value of~$V$, $\ssfu$~is known as a
function of~$\nu$, it may be calculated for all other values of~$V$ as a
%% -----File: 098.png---Folio 82-------
function of~$\nu$. The general integral of this differential equation,
as may be readily seen by substitution, is
\[
\ssfu = \frac{1}{V} \phi(\nu^{3} V),
\Tag{(98)}
\]
where $\phi$~denotes an arbitrary function of the single argument
$\nu^{3} V$. Instead of this we may, on substituting $\nu^{3} V \phi(\nu^{3} V)$ for
$\phi(\nu^{3} V)$, write
\[
\ssfu = \nu^{3} \phi(\nu^{3} V).
\Tag{(99)}
\]
Either of the last two equations is the general expression of
\Name{Wien's} displacement law.

If for a definitely given volume~$V$ the spectral distribution of
energy is known (\ie, $\ssfu$~as a function of~$\nu$), it is possible to deduce
therefrom the dependence of the function~$\phi$ on its argument, and
thence the distribution of energy for any other volume~$V'$, into
which the radiation filling the hollow cylinder may be brought by
a reversible adiabatic process.

\Section{84a.} The characteristic feature of this new distribution of
energy may be stated as follows: If we denote all quantities
referring to the new state by the addition of an accent, we have
the following equation in addition to~\Eq{(99)}
\[
\ssfu' = \nu'^{3} \phi(\nu'^{3} V').
\]
Therefore, if we put
\[
\nu'^{3} V' = \nu^{3} V,
\Tag{(99a)}
\]
we shall also have
\[
\frac{\ssfu'}{\nu'^{3}} = \frac{\ssfu}{\nu^{3}} \quad\text{and}\quad
\ssfu' V' = \ssfu V,
\Tag{(99b)}
\]
\ie, if we coordinate with every frequency~$\nu$ in the original state
that frequency~$\nu'$ which is to~$\nu$ in the inverse ratio of the cube
roots of the respective volumes, the corresponding energy
densities $\ssfu'$~and~$\ssfu$ will be in the inverse ratio of the volumes.

The meaning of these relations will be more clearly seen, if we
write
\[
\frac{V'}{\lambda'^{3}} = \frac{V}{\lambda^{3}}\Add{.}
\]
This is the number of the cubes of the wave lengths, which
correspond to the frequency~$\nu$ and are contained in the volume
%% -----File: 099.png---Folio 83-------
of the radiation. Moreover $\ssfu\, d\nu\, V = \ssfU\, d\nu$ denotes the radiant
energy lying between the frequencies $\nu$ and $\nu + d\nu$, which is contained
in the volume~$V$. Now since, according to~\Eq{(99a)},
\[
\sqrt[3]{V'}\, d\nu' = \sqrt[3]{V}\, d\nu \quad\text{or}\quad
\frac{d\nu'}{\nu'} = \frac{d\nu}{\nu}
\Tag{(99c)}
\]
we have, taking account of~\Eq{(99b)},
\[
\ssfU' \frac{d\nu'}{\nu'} = \ssfU \frac{d\nu}{\nu}\Add{.}
\]
These results may be summarized thus: On an infinitely slow
reversible adiabatic change in volume of radiation contained in
a cavity and uniform in all directions, the frequencies change in
such a way that the number of cubes of wave lengths of every
frequency contained in the total volume remains unchanged, and
the radiant energy of every infinitely small spectral interval
changes in proportion to the frequency.
\Erratum{}{These laws hold for any original distribution of energy whatever;
hence, \eg, an originally monochromatic radiation remains
monochromatic during the process described, its color changing
in the way stated.}

\Section{85.} Returning now to the discussion of \Sec{73} we introduce
the assumption that at first the spectral distribution of energy is
the normal one, corresponding to black radiation. Then, according
to the law there proven, the radiation retains this property
without change during a reversible adiabatic change of volume
and the laws derived in \Sec{68} hold for the process. The radiation
then possesses in every state a definite temperature~$T$, which
depends on the volume~$V$ according to the equation derived in
that paragraph,
\[
T^{3}V = \const. = T'^{3}V'.
\Tag{(100)}
\]
Hence we may now write equation~\Eq{(99)} as follows:
\[
\ssfu = \nu^{3} \phi\left(\frac{\nu^{3}}{T^{3}}\right)
\]
or
\[
\ssfu = \nu^{3} \phi \left(\frac{T}{\nu}\right)\Add{.}
\]
Therefore, if for a single temperature the spectral distribution
of black radiation, \ie, $\ssfu$~as a function of~$\nu$, is known, the dependence
of the function~$\phi$ on its argument, and hence the spectral
distribution for any other temperature, may be deduced
therefrom.
%% -----File: 100.png---Folio 84-------

If we also take into account the law proved in \Sec{47}, that,
for the black radiation of a definite temperature, the product~$\ssfu q^{3}$
has for all media the same value, we may also write
\[
\ssfu = \frac{\nu^{3}}{c^{3}} F\left(\frac{T}{\nu}\right)
\Tag{(101)}
\]
where now the function~$F$ no longer contains the velocity of
propagation.

\Section{86.} For the total radiation density in space of the black radiation
in the vacuum we find
\[
u = \int_{0}^{\infty} \ssfu\, d\nu
  = \frac{1}{c^{3}} \int_{0}^{\infty} \nu^{3} F\left(\frac{T}{\nu}\right) d\nu,
\Tag{(102)}
\]
or, on introducing $\dfrac{T}{\nu} = x$ as the variable of integration instead
of~$\nu$,
\[
u = \frac{T^{4}}{c^{3}} \int_{0}^{\infty} \frac{F(x)}{x^{5}}\, dx.
\Tag{(103)}
\]
If we let the absolute constant
\[
\frac{1}{c^{3}} \int_{0}^{\infty} \frac{F(x)}{x^{5}}\, dx = a
\Tag{(104)}
\]
the equation reduces to the form of the \Name{Stefan-Boltzmann} law of
radiation expressed in equation~\Eq{(75)}.

\Section{87.} If we combine equation~\Eq{(100)} with equation~\Eq{(99a)} we
obtain
\[
\frac{\nu'}{T'} = \frac{\nu}{T}\Add{.}
\Tag{(105)}
\]

Hence the laws derived at the end of \Sec{84a} assume the following
form: On infinitely slow reversible adiabatic change in
volume of black radiation contained in a cavity, the temperature~$T$
varies in the inverse ratio of the cube root of the volume~$V$,
the frequencies~$\nu$ vary in proportion to the temperature, and
the radiant energy~$\ssfU\, d\nu$ of an infinitely small spectral interval
varies in the same ratio. Hence the total radiant energy~$U$ as
the sum of the energies of all spectral intervals varies also in
proportion to the temperature, a statement which agrees with the
%% -----File: 101.png---Folio 85-------
conclusion arrived at already at the end of \Sec{68}, while the
space density of radiation, $u = \dfrac{U}{V}$, varies in proportion to the
fourth power of the temperature, in agreement with the \Name{Stefan-Boltzmann}
law.

\Section{88.} \Name{Wien's} displacement law may also in the case of black
radiation be stated for the specific intensity of radiation~$\ssfK_{\nu}$ of
a plane polarized monochromatic ray. In this form it reads
according to~\Eq{(24)}
\[
\ssfK_{\nu} = \frac{\nu^{3}}{\DPtypo{c}{c^{2}}} F\left(\frac{T}{\nu}\right)\Add{.}
\Tag{(106)}
\]
If, as is usually done in experimental physics, the radiation intensity
is referred to wave lengths~$\lambda$ instead of frequencies~$\nu$, according
to~\Eq{(16)}, namely
\[
\ssfE_{\lambda} = \frac{c \ssfK_{\nu}}{\lambda^{2}}\Add{,}
\]
equation~\Eq{(106)} takes the following form:
\[
E_{\lambda} = \frac{c^{2}}{\lambda^{5}} F\left(\frac{\lambda T}{c}\right).
\Tag{(107)}
\]
This form of \Name{Wien's} displacement law has usually been the starting-point
for an experimental test, the result of which has in all
cases been a fairly accurate verification of the law.\footnote
  {\Eg, \Name{F.~Paschen}, Sitzungsber.\ d.\ Akad.\ d.\ Wissensch.\ Berlin, pp.\ 405 and 959, 1899.
  \Name{O.~Lummer} und \Name{E.~Pringsheim}, Verhandlungen d.\ Deutschen physikalischen Gesellschaft~\textbf{1},
  pp.\ 23 and 215, 1899. Annal.\ d.\ Physik~\textbf{6}, p.~192, 1901.}

\Section{89.} Since $E_{\lambda}$~vanishes for $\lambda = 0$ as well as for $\lambda = \infty$, $E_{\lambda}$~must
have a maximum with respect to~$\lambda$, which is found from the
equation
\[
\frac{dE_{\lambda}}{d\lambda} = 0
  = -\frac{5}{\lambda^{6}} F\left(\frac{\lambda T}{c}\right)
   + \frac{1}{\lambda^{5}}\, \frac{T}{c} \dot{F}\left(\frac{\lambda T}{c}\right)
\]
where $\dot{F}$~denotes the differential coefficient of~$F$ with respect to
its argument. Or
\[
\frac{\lambda T}{c} \dot{F}\left(\frac{\lambda T}{c}\right)
  - 5F\left(\frac{\lambda T}{c}\right) = 0.
\Tag{(108)}
\]
This equation furnishes a definite value for the argument $\dfrac{\lambda T}{c}$, so
%% -----File: 102.png---Folio 86-------
that for the wave length~$\lambda_{m}$ corresponding to the maximum of the
radiation intensity~$E_{\lambda}$ the relation holds
\[
\lambda_{m}T = b.
\Tag{(109)}
\]
With increasing temperature the maximum of radiation is
therefore displaced in the direction of the shorter wave lengths.

The numerical value of the constant~$b$ as determined by
\Name{Lummer} and \Name{Pringsheim}\footnote
  {\Name{O.~Lummer} und \Name{E.~Pringsheim}, \lc}
is
\[
b = 0.294\, \cm\DPchg{.}{}\, \degree.
\Tag{(110)}
\]

\Name{Paschen}\footnote
  {\Name{F.~Paschen}, Annal.\ d.\ Physik\DPchg{,}{} \textbf{6}, p.~657, 1901.}
has found a slightly smaller value, about $0.292$.

We may emphasize again at this point that, according to
\Sec{19}, the maximum of~$E_{\lambda}$ does not by any means occur at the
same point in the spectrum as the maximum of~$\ssfK_{\nu}$ and that hence
the significance of the constant~$b$ is essentially dependent on the
fact that the intensity of monochromatic radiation is referred to
wave lengths, not to frequencies.

\Section{90.} The value also of the maximum of~$E_{\lambda}$ is found from~\Eq{(107)}
by putting $\lambda = \lambda_{m}$. Allowing for~\Eq{(109)} we obtain
\[
E_{\max} = \const.\; T^{5},
\Tag{(111)}
\]
\ie, the value of the maximum of radiation in the spectrum of the
black radiation is proportional to the fifth power of the absolute
temperature.

Should we measure the intensity of monochromatic radiation
not by~$E_{\lambda}$ but by~$\ssfK_{\nu}$, we would obtain for the value of the radiation
maximum a quite different law, namely,
\[
\ssfK_{max} = \const.\;T^{3}.
\Tag{(112)}
\]
%% -----File: 103.png---Folio 87-------

\Chapter[Spectral Distribution of Energy]
{IV}{Radiation of Any Arbitrary Spectral Distribution
of Energy. Entropy and Temperature
of Monochromatic Radiation}

\Section{91.} We have so far applied \Name{Wien's} displacement law only to
the case of black radiation; it has, however, a much more general
importance. For equation~\Eq{(95)}, as has already been stated, gives,
for any original spectral distribution of the energy radiation contained
in the evacuated cavity and radiated uniformly in all directions,
the change of this energy distribution accompanying a
reversible adiabatic change of the total volume. Every state of
radiation brought about by such a process is perfectly stationary
and can continue infinitely long, subject, however, to the condition
that no trace of an emitting or absorbing substance exists
in the radiation space. For otherwise, according to \Sec{51}, the
distribution of energy would, in the course of time, change
through the releasing action of the substance irreversibly, \ie,
with an increase of the total entropy, into the stable distribution
\DPtypo{correponding}{corresponding} to black radiation.

The difference of this general case from the special one dealt
with in the preceding chapter is that we can no longer, as in the
case of black radiation, speak of a definite temperature of the
radiation. Nevertheless, since the second principle of thermodynamics
is supposed to hold quite generally, the radiation, like
every physical system which is in a definite state, has a definite
entropy, $S = Vs$. This entropy consists of the entropies of the
monochromatic radiations, and, since the separate kinds of rays
are independent of one another, may be obtained by addition.
Hence
\[
s = \int_{0}^{\infty} \ssfs\, d\nu,\qquad
S = V \int_{0}^{\infty} \ssfs\, d\nu,
\Tag{(113)}
\]
where $\ssfs\, d\nu$ denotes the entropy of the radiation of frequencies
between $\nu$ and $\nu + d\nu$ contained in unit volume. $\ssfs$~is a definite
%% -----File: 104.png---Folio 88-------
function of the two independent variables $\nu$~and~$\ssfu$ and in the
following will always be treated as such.

\Section{92.} If the analytical expression of the function~$\ssfs$ were known,
the law of energy distribution in the normal spectrum could
immediately be deduced from it; for the normal spectral distribution
of energy or that of black radiation is distinguished from
all others by the fact that it has the maximum of the entropy of
radiation~$S$.

Suppose then we take $\ssfs$ to be a known function of $\nu$~and~$\ssfu$.
Then as a condition for black radiation we have
\[
\delta S = 0,
\Tag{(114)}
\]
for any variations of energy distribution, which are possible
with a constant total volume~$V$ and constant total energy of
radiation~$U$. Let the variation of energy distribution be characterized
by making an infinitely small change~$\delta \ssfu$ in the energy~$\ssfu$
of every separate definite frequency~$\nu$. Then we have as fixed
conditions
\[
\delta V = 0 \quad\text{and}\quad
\int_{0}^{\infty} \delta \ssfu\, d\nu = 0.
\Tag{(115)}
\]
The changes $d$~and~$\delta$ are of course quite independent of each
other.

Now since $\delta V = 0$, we have from \Eq{(114)}~and~\Eq{(113)}
\[
\int_{0}^{\infty} \delta \ssfs\, d\nu = 0,
\]
or, since $\nu$~remains unvaried
\[
\int_{0}^{\infty} \frac{\dd \ssfs}{\dd \ssfu}\, \delta \ssfu\, d\nu = 0,
\]
and, by allowing for~\Eq{(115)}, the validity of this equation for all
values of $\delta \ssfu$ whatever requires that
\[
\frac{\dd \ssfs}{\dd \ssfu} = \const.
\Tag{(116)}
\]
for all different frequencies. This equation states the law of
energy distribution in the case of black radiation.

\Section{93.} The constant of equation~\Eq{(116)} bears a simple relation to
the temperature of black radiation. For if the black radiation,
%% -----File: 105.png---Folio 89-------
by conduction into it of a certain amount of heat at constant volume~$V$,
undergoes an infinitely small change in energy~$\delta U$, then,
according to~\Eq{(73)}, its change in entropy is
\[
\delta S = \frac{\delta U}{T}.
\]
However, from \Eq{(113)}~and~\Eq{(116)},
\[
\delta S = V \int_{0}^{\infty} \frac{\dd \ssfs}{\dd \ssfu}\, \delta \ssfu\, d\nu
  = \frac{\dd \ssfs}{\dd \ssfu} V \int_{0}^{\infty} \delta\ssfu\, d\nu
  = \frac{\dd\ssfs}{\dd\ssfu}\, \delta U
\]
hence
\[
\frac{\dd\ssfs}{\dd\ssfu} = \frac{1}{T}
\Tag{(117)}
\]
and the above quantity, which was found to be the same for all
frequencies in the case of black radiation, is shown to be the reciprocal
of the temperature of black radiation.

Through this law the concept of temperature gains significance
also for radiation of a quite arbitrary distribution of
%[** Thm]
energy. For since $\ssfs$~depends only on $\ssfu$~and~$\nu$, \emph{monochromatic
radiation, which is uniform in all directions and has a definite
energy density~$\ssfu$, has also a definite temperature given by~\Eq{(117)},
and, among all conceivable distributions of energy, the normal one
is characterized by the fact that the radiations of all frequencies
have the same temperature}.

Any change in the energy distribution consists of a passage of
energy from one monochromatic radiation into another, and, if
the temperature of the first radiation is higher, the energy
transformation causes an increase of the total entropy and is
hence possible in nature without compensation; on the other hand,
if the temperature of the second radiation is higher, the total
entropy decreases and therefore the change is impossible in nature,
unless compensation occurs simultaneously, just as is the case
with the transfer of heat between two bodies of different temperatures.

\Section{94.} Let us now investigate \Name{Wien's} displacement law with regard
to the dependence of the quantity~$\ssfs$ on the variables $\ssfu$~and~$\nu$.
%% -----File: 106.png---Folio 90-------
From equation~\Eq{(101)} it follows, on solving for~$T$ and substituting
the value given in~\Eq{(117)}, that
\[
\frac{1}{T}
  = \frac{1}{\nu} F\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)
  = \frac{\dd \ssfs}{\dd \ssfu}
\Tag{(118)}
\]
where again $F$~represents a function of a single argument and the
constants do not contain the velocity of propagation~$c$. On
integration with respect to the argument we obtain
\[
\ssfs = \frac{\nu^{2}}{c^{3}} F_{1}\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)
\Tag{(119)}
\]
the notation remaining the same. In this form \Name{Wien's} displacement
law has a significance for every separate monochromatic
radiation and hence also for radiations of any arbitrary energy
distribution.

\Section{95.} According to the second principle of thermodynamics, the
total entropy of radiation of quite arbitrary distribution of
energy must remain constant on adiabatic reversible compression.
We are now able to give a direct proof of this proposition on the
basis of equation~\Eq{(119)}. For such a process, according to
equation~\Eq{(113)}, the relation holds:
\begin{align*}
\delta S &= \int_{0}^{\infty} d\nu\, (V\, \delta \ssfs + \ssfs\, \delta V) \\
  &= \int_{0}^{\infty} d\nu\, (V \frac{\dd \ssfs}{\dd \ssfu}\, \delta \ssfu + \ssfs\, \delta V).
\Tag{(120)}
\end{align*}
Here, as everywhere, $\ssfs$~should be regarded as a function of $\ssfu$~and~$\nu$,
and $\delta \nu = 0$.

Now for a reversible adiabatic change of state the relation~\Eq{(95)}
holds. Let us take from the latter the value of~$\delta \ssfu$ and substitute.
Then we have
\[
\delta S = \delta V \int_{0}^{\infty} d\nu \left\{\frac{\dd \ssfs}{\dd \ssfu} \left(\frac{\nu\, d\ssfu}{3\, d\nu} - \ssfu\right) + \ssfs\right\}.
\]
In this equation the differential coefficient of~$\ssfu$ with respect to~$\nu$
refers to the spectral distribution of energy originally assigned
arbitrarily and is therefore, in contrast to the partial differential
coefficients, denoted by the letter~$d$.
%% -----File: 107.png---Folio 91-------

Now the complete differential is:
\[
\frac{d\ssfs}{d\nu} = \frac{\dd \ssfs}{\dd \ssfu}\, \frac{d\ssfu}{d\nu} + \frac{\dd \ssfs}{\dd \nu}\Add{.}
\]
Hence by substitution:
\[
\delta S = \delta V \int_{0}^{\infty} d\nu \left\{\frac{\nu}{3} \left(\frac{d\ssfs}{d\nu} -\frac{\dd \ssfs}{\dd \nu}\right) - \ssfu \frac{\dd \ssfs}{\dd \ssfu} + \ssfs\right\}.
\Tag{(121)}
\]
But from equation~\Eq{(119)} we obtain by differentiation
\[
\frac{\dd \ssfs}{\dd \ssfu} = \frac{1}{\nu} \dot{F}\left(\frac{c^3\ssfu}{\nu^3}\right)
\quad\text{and}\quad
\frac{\dd \ssfs}{\dd \nu}
  = \frac{2\nu}{c^{3}} F\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)
  - \frac{3\ssfu}{\nu^{2}} \dot{F}\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)\Add{.}
\Tag{(122)}
\]
Hence
\[
\DPchg{\frac{\nu\, \dd \ssfs}{\dd \nu}}{\nu \frac{\dd \ssfs}{\dd \nu}}
   = 2\ssfs - 3\ssfu \frac{\dd \ssfs}{\dd \ssfu}\Add{.}
\Tag{(123)}
\]
On substituting this in~\Eq{(121)}, we obtain
\[
\delta S = \delta V \int_{0}^{\infty} d\nu \left(\frac{\nu}{3}\, \frac{d\ssfs}{d\nu} + \frac{1}{3} \ssfs\right)
\Tag{(124)}
\]
or,
\[
\delta S = \frac{\delta V}{3} [\nu\ssfs]\Strut_{0}^{\infty} = 0,
\]
as it should be. That the product~$\nu\ssfs$ vanishes also for $\nu = \infty$
may be shown just as was done in \Sec{83} for the product~$\nu\ssfu$.

\Section{96.} By means of equations \Eq{(118)}~and~\Eq{(119)} it is possible to give
to the laws of reversible adiabatic compression a form in which
their meaning is more clearly seen and which is the generalization
of the laws stated in \Sec{87} for black radiation and a supplement
to them. It is, namely, possible to derive~\Eq{(105)} again from \Eq{(118)}~and~\Eq{(99b)}.
Hence the laws deduced in \Sec{87} for the change of
frequency and temperature of the monochromatic radiation
energy remain valid for a radiation of an originally quite arbitrary
distribution of energy. The only difference as compared with
the black radiation consists in the fact that now every frequency
has its own distinct temperature.

Moreover it follows from \Eq{(119)}~and~\Eq{(99b)} that
\[
\frac{\ssfs'}{\nu'^{2}} = \frac{\ssfs}{\nu^{2}}\Add{.}
\Tag{(125)}
\]
%% -----File: 108.png---Folio 92-------
Now $\ssfs\, d\nu\, V = \ssfS\, d\nu$ denotes the radiation entropy between the
frequencies $\nu$ and $\nu + d\nu$ contained in the volume~$V$. Hence on
account of \Eq{(125)},~\Eq{(99a)}, and~\Eq{(99c)}
\[
\ssfS'\, d\nu' = \ssfS\, d\nu,
\Tag{(126)}
\]
\ie, the radiation entropy of an infinitely small spectral interval
remains constant. This is another statement of the fact that the
total entropy of radiation, taken as the sum of the entropies of all
monochromatic radiations contained therein, remains constant.

\Section{97.} We may go one step further, and, from the entropy~$\ssfs$
and the temperature~$T$ of an unpolarized monochromatic radiation
which is uniform in all directions, draw a certain conclusion
regarding the entropy and temperature of a single, plane polarized,
monochromatic pencil. That every separate pencil also has
a certain entropy follows by the second principle of thermodynamics
from the phenomenon of emission. For since, by the
act of emission, heat is changed into radiant heat, the entropy
of the emitting body decreases during emission, and, along with
this decrease, there must be, according to the principle of increase
of the total entropy, an increase in a different form of entropy as
a compensation. This can only be due to the energy of the
emitted radiation. Hence every separate, plane polarized, monochromatic
pencil has its definite entropy, which can depend only
on its energy and frequency and which is propagated and
spreads into space with it. We thus gain the idea of entropy
radiation, which is measured, as in the analogous case of energy
radiation, by the amount of entropy which passes in unit time
through unit area in a definite direction. Hence statements,
exactly similar to those made in \Sec{14} regarding energy radiation,
will hold for the radiation of entropy, inasmuch as every
pencil possesses and conveys, not only its energy, but also its
entropy. Referring the reader to the discussions of \Sec{14},
we shall, for the present, merely enumerate the most important
laws for future use.

\Section{98.} In a space filled with any radiation whatever the entropy
radiated in the time~$dt$ through an element of area~$d\sigma$ in the
direction of the conical element~$d\Omega$ is given by an expression of
the form
\[
dt\, d\sigma \cos\theta\, d\Omega\, L = L \sin\theta \cos\theta\, d\theta\, d\phi\, d\sigma\, dt.
\Tag{(127)}
\]
%% -----File: 109.png---Folio 93-------
The positive quantity~$L$ we shall call the ``specific intensity of
entropy radiation'' at the position of the element of area~$d\sigma$
in the direction of the solid angle~$d\Omega$. $L$~is, in general, a function
of position, time, and direction.

The total radiation of entropy through the element of area~$d\sigma$
toward one side, say the one where $\theta$~is an acute angle, is obtained
by integration with respect to~$\phi$ from $0$ to~$2\pi$ and with
respect to~$\theta$ from $0$ to~$\dfrac{\pi}{2}$. It is
\[
d\sigma\, dt \int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}} d\theta\, L \sin\theta \cos\theta.
\]
When the radiation is uniform in all directions, and hence $L$~constant,
the entropy radiation through~$d\sigma$ toward one side is
\[
\pi L\, d\sigma\, dt.
\Tag{(128)}
\]

The specific intensity~$L$ of the entropy radiation in every direction
consists further of the intensities of the separate rays belonging
to the different regions of the spectrum, which are propagated
independently of one another. Finally for a ray of definite color
and intensity the nature of its polarization is characteristic.
When a monochromatic ray of frequency~$\nu$ consists of two
mutually independent\footnote
  {``Independent'' in the sense of ``\DPchg{noncoherent}{non-coherent}.'' If, \eg, a ray with the principal intensities
  $\ssfK$~and~$\ssfK'$ is elliptically polarized, its entropy is not equal to~$\ssfL + \ssfL'$, but equal to the
  entropy of a plane polarized ray of intensity $\ssfK + \ssfK'$. For an elliptically polarized ray may
  be transformed at once into a plane polarized one, \eg, by total reflection. For the entropy
  of a ray with coherent components see below \Sec{104}, \textit{et~seq.}}
components, polarized at right angles to
each other, with the principal intensities of energy radiation
(\Sec{17}) $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$, the specific intensity of entropy radiation
is of the form
\[
L = \int_{0}^{\infty} d\nu\, (\ssfL_{\nu} + \ssfL_{\nu}').
\Tag{(129)}
\]

The positive quantities $L_{\nu}$~and~$L_{\nu}'$ in this expression, the
principal intensities of entropy radiation of frequency~$\nu$, are
determined by the values of $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$. By substitution in~\Eq{(127)},
this gives for the entropy which is radiated in the time
%% -----File: 110.png---Folio 94-------
%[** tie]
$dt$ through the element of area~$d\sigma$ in the direction of the conical
element~$d\Omega$ the expression
\[
dt\, d\sigma \cos\theta\, d\Omega \int_{0}^{\infty} d\nu\, (\ssfL_{\nu} + \ssfL_{\nu}'),
\]
and, for monochromatic plane polarized radiation,
\[
dt\, d\sigma \cos\theta\, d\Omega\, \ssfL_{\nu}\, d\nu
  = \ssfL_{\nu}\, d\nu\, \sin\theta \cos\theta\, d\theta\, d\phi\, d\sigma\, dt.
\Tag{(130)}
\]
For unpolarized rays $\ssfL_{\nu} = \ssfL_{\nu}'$ and \Eq{(129)}~becomes
\[
L = 2 \int_{0}^{\infty} \ssfL_{\nu}\, d\nu.
\]
For radiation which is uniform in all directions the total entropy
radiation toward one side is, according to~\Eq{(128)},
\[
2\pi\, d\sigma\, dt \int_{0}^{\infty} \ssfL_{\nu}\, d\nu.
\]

\Section{99.} From the intensity of the propagated entropy radiation
the expression for the \emph{space density} of the radiant entropy may also
be obtained, just as the space density of the radiant energy
follows from the intensity of the propagated radiant energy.
(Compare \Sec{22}.) In fact, in analogy with equation~\Eq{(20)}, the
space density,~$s$, of the entropy of radiation at any point in a
vacuum is
\[
s = \frac{1}{c} \int L\, d\Omega,
\Tag{(131)}
\]
where the integration is to be extended over the conical elements
which spread out from the point in question in all directions.
$L$~is constant for uniform radiation and we obtain
\[
s = \frac{4\pi L}{c}\Add{.}
\Tag{(132)}
\]
By spectral resolution of the quantity~$L$, according to equation~\Eq{(129)},
we obtain from~\Eq{(131)} also the space density of the monochromatic
radiation entropy:
\[
\ssfs = \frac{1}{c} \int (\ssfL + \ssfL')\, d\Omega,
\]
and for unpolarized radiation, which is uniform in all directions
\[
\ssfs = \frac{8\pi \ssfL}{c}\Add{.}
\Tag{(133)}
\]
%% -----File: 111.png---Folio 95-------

\Section{100.} As to how the entropy radiation~$\ssfL$ depends on the energy
radiation~$\ssfK$ \Name{Wien's} displacement law in the form of~\Eq{(119)} affords
immediate information. It follows, namely, from it, considering
\Eq{(133)}~and~\Eq{(24)}, that
\[
\ssfL = \frac{\nu^{2}}{c^{2}} F\left(\frac{c^{2}\ssfK}{\nu^{3}}\right)
\Tag{(134)}
\]
and, moreover, on taking into account~\Eq{(118)},
\[
\frac{\dd \ssfL}{\dd \ssfK} = \frac{\dd \ssfs}{\dd \ssfu} = \frac{1}{T}\Add{.}
\Tag{(135)}
\]
Hence also
\[
T = \nu F_{1}\left(\frac{c^{2} \ssfK}{\nu^{3}}\right)
\Tag{(136)}
\]
or
\[
\ssfK = \frac{\nu^{3}}{c^{2}} F_{2}\left(\frac{T}{\nu}\right)\Add{.}
\Tag{(137)}
\]

It is true that these relations, like the equations \Eq{(118)} and~\Eq{(119)},
were originally derived for radiation which is unpolarized
and uniform in all directions. They hold, however, generally in
the case of any radiation whatever for each separate monochromatic
plane polarized ray. For, since the separate rays behave
and are propagated quite independently of one another, the intensity,~$\ssfL$,
of the entropy radiation of a ray can depend only on the
intensity of the energy radiation,~$\ssfK$, of the same ray. Hence
every separate monochromatic ray has not only its energy but
also its entropy defined by~\Eq{(134)} and its temperature defined by~\Eq{(136)}.

\Section{101.} The extension of the conception of temperature to a
single monochromatic ray, just discussed, implies that at the
same point in a medium, through which any rays whatever pass,
there exist in general an infinite number of temperatures, since
every ray passing through the point has its separate temperature,
and, moreover, even the rays of different color traveling in the
same direction show temperatures that differ according to the
spectral distribution of energy. In addition to all these temperatures
there is finally the temperature of the medium itself, which
at the outset is entirely independent of the temperature of the
radiation. This complicated method of consideration lies in the
%% -----File: 112.png---Folio 96-------
nature of the case and corresponds to the complexity of the
physical processes in a medium through which radiation travels
in such a way. It is only in the case of stable thermodynamic
equilibrium that there is but one temperature, which then is
common to the medium itself and to all rays of whatever color
crossing it in different directions.

In practical physics also the necessity of separating the conception
of radiation temperature from that of body temperature
has made itself felt to a continually increasing degree. Thus it
has for some time past been found advantageous to speak, not
only of the real temperature of the sun, but also of an ``apparent''
or ``effective'' temperature of the sun, \ie, that temperature
which the sun would need to have in order to send to the earth
the heat radiation actually observed, if it radiated like a black
body. Now the apparent temperature of the sun is obviously
nothing but the actual temperature of the solar rays,\footnote
  {On the average, since the solar rays of different color do not have exactly the same
  temperature.}
depending
entirely on the nature of the rays, and hence a property of the
rays and not a property of the sun itself. Therefore it would be,
not only more convenient, but also more correct, to apply this
notation directly, instead of speaking of a fictitious temperature
of the sun, which can be made to have a meaning only by the
introduction of an assumption that does not hold in reality.

Measurements of the brightness of monochromatic light have
recently led \Name{L.~Holborn} and \Name{F.~Kurlbaum}\footnote
  {\Name{L.~Holborn} und \Name{F.~Kurlbaum}, Annal.\ d.\ Physik\DPtypo{.}{}\DPchg{,}{} \textbf{10}, p.~229, 1903.}
to the introduction of
the concept of ``black'' temperature of a radiating surface. The
black temperature of a radiating surface is measured by the
brightness of the rays which it emits. It is in general a separate
one for each ray of definite color, direction, and polarization,
which the surface emits, and, in fact, merely represents the
temperature of such a ray. It is, according to equation~\Eq{(136)},
determined by its brightness (specific intensity),~$\ssfK$, and its
frequency,~$\nu$, without any reference to its origin and previous
states. The definite numerical form of this equation will be
given below in \Sec{166}. Since a black body has the maximum
emissive power, the temperature of an emitted ray can never be
higher than that of the emitting body.
%% -----File: 113.png---Folio 97-------

\Section{102.} Let us make one more simple application of the laws just
found to the special case of black radiation. For this, according
to~\Eq{(81)}, the total space density of entropy is
\[
s = \frac{4}{3} a^{3} T.
\Tag{(138)}
\]
Hence, according to~\Eq{(132)}, the specific intensity of  the total
entropy radiation in any direction is
\[
L = \frac{c}{3\pi} a T^{3},
\Tag{(139)}
\]
and the total entropy radiation through an element of area~$d\sigma$
toward one side is, according to~\Eq{(128)},
\[
\frac{c}{3} aT^{3}\, d\sigma\, dt.
\Tag{(140)}
\]
As a special example we shall now apply the two principles of
thermodynamics to the case in which the surface of a black body
of temperature~$T$ and of infinitely large heat capacity is struck
by black radiation of temperature~$T'$ coming from all directions.
Then, according to \Eq{(7)}~and~\Eq{(76)}, the black body emits per unit
area and unit time the energy
\[
\pi K = \frac{ac}{4} T^{4},
\]
and, according to~\Eq{(140)}, the entropy
\[
\frac{ac}{3} T^{3}.
\]
On the other hand, it absorbs the energy
\[
\frac{ac}{4} T'^{4}
\]
and the entropy
\[
\frac{ac}{3} T'^{3}.
\]
Hence, according to the first principle, the total heat added to the
body, positive or negative according as $T'$~is larger or smaller
than~$T$, is
\[
Q = \frac{ac}{4} T'^{4} - \frac{ac}{4} T^{4} = \frac{ac}{4} (T'^{4} - T^{4}),
\]
%% -----File: 114.png---Folio 98-------
and, according to the second principle, the change of the entire
entropy is positive or zero. Now the entropy of the body changes
by~$\dfrac{Q}{T}$, the entropy of the radiation in the vacuum by
\[
\frac{ac}{3} (T^{3} - T'^{3}).
\]
Hence the change per unit time and unit area of the entire entropy
of the system considered is
\[
\frac{ac}{4}\, \frac{T'^{4} - T^{4}}{T} + \frac{ac}{3} (T^{3} - T'^{3}) \geq 0.
\]
In fact this relation is satisfied for all values of $T$~and~$T'$. The
minimum value of the expression on the left side is zero; this value
is reached when $T = T'$. In that case the process is reversible.
If, however, $T$~differs from~$T'$, we have an appreciable increase
of entropy; hence the process is irreversible. In particular we
find that if $T = 0$ the increase in entropy is~$\infty$, \ie,~the absorption
of heat radiation by a black body of vanishingly small temperature
is accompanied by an infinite increase in entropy and
cannot therefore be reversed by any finite compensation. On the
other hand for $T' = 0$, the increase in entropy is only equal to~$\dfrac{ac}{12} T^{3}$,
\ie,~the emission of a black body of temperature~$T$ without
simultaneous absorption of heat radiation is irreversible without
compensation, but can be reversed by a compensation of at least
the stated finite amount. For example, if we let the rays emitted
by the body fall back on it, say by suitable reflection, the body,
while again absorbing these rays, will necessarily be at the same
time emitting new rays, and this is the compensation required by
the second principle.

Generally we may say: Emission without simultaneous absorption
is irreversible, while the opposite process, absorption without
emission, is impossible in nature.

\Section{103.} A further example of the application of the two principles
of thermodynamics is afforded by the irreversible expansion of
originally black radiation of volume~$V$ and temperature~$T$ to
the larger volume~$V'$ as considered above in \Sec{70}, but in the
absence of any absorbing or emitting substance whatever. Then
%% -----File: 115.png---Folio 99-------
not only the total energy but also the energy of every separate
frequency~$\nu$ remains constant; hence, when on account of diffuse
reflection from the walls the radiation has again become uniform
in all directions, $\ssfu_{\nu}V = \ssfu_{\nu}'V'$; moreover by this relation, according
to~\Eq{(118)}, the temperature~$T_{\nu}$' of the monochromatic radiation of
frequency~$\nu$ in the final state is determined. The actual calculation,
however, can be performed only with the help of equation~\Eq{(275)}
(see below). The total entropy of radiation, \ie,~the sum
of the entropies of the radiations of all frequencies,
\[
V'\int_{0}^{\infty} \ssfs_{\nu}'\, d\nu,
\]
must, according to the second principle, be larger in the final state
than in the original state. Since $T_{\nu}'$~has different values for the
different frequencies~$\nu$, the final radiation is no longer black.
Hence, on subsequent introduction of a carbon particle into the
cavity, a finite change of the distribution of energy is obtained,
and simultaneously the entropy increases further to the value~$S'$
calculated in~\Eq{(82)}.

\Section{104.} In \Sec{98} we have found the intensity of entropy radiation
of a definite frequency in a definite direction by adding the
entropy radiations of the two independent components $\ssfK$~and~$\ssfK'$,
polarized at right angles to each other, or
\[
\ssfL(\ssfK) + \ssfL(\ssfK'),
\Tag{(141)}
\]
where $\ssfL$~denotes the function of~$\ssfK$ given in equation~\Eq{(134)}.
This method of procedure is based on the general law that the
entropy of two mutually independent physical systems is equal
to the sum of the entropies of the separate systems.

If, however, the two components of a ray, polarized at right
angles to each other, are not independent of each other, this
method of procedure no longer remains correct. This may be
seen, \eg, on resolving the radiation intensity, not with reference
to the two principal planes of polarization with the principal
intensities $\ssfK$~and~$\ssfK'$, but with reference to any other two planes
at right angles to each other, where, according to equation~\Eq{(8)},
the intensities of the two components assume the following
values
\[
\begin{alignedat}{3}
&\ssfK \cos^{2} \psi &&+ \ssfK' \sin^{2} \psi &&= \ssfK'' \\
&\ssfK \sin^{2} \psi &&+ \ssfK' \cos^{2} \psi &&= \ssfK'''.
\end{alignedat}
\Tag{(142)}
\]
%% -----File: 116.png---Folio 100-------
In that case, of course, the entropy radiation is not equal to
$\ssfL(\ssfK'') + \ssfL(\ssfK''')$.

Thus, while the energy radiation is always obtained by the
summation of any two components which are polarized at right
angles to each other, no matter according to which azimuth the
resolution is performed, since always
\[
\ssfK'' + \ssfK''' = \ssfK + \ssfK',
\Tag{(143)}
\]
a corresponding equation does not hold in general for the entropy
radiation. The cause of this is that the two components, the
intensities of which we have denoted by $\ssfK''$~and~$\ssfK'''$, are, unlike
$\ssfK$~and~$\ssfK'$, not independent or \DPchg{noncoherent}{non-coherent} in the optic sense.
In such a case
\[
\ssfL(\ssfK'') + \ssfL(\ssfK''') > \ssfL(\ssfK) + \ssfL(\ssfK'),
\Tag{(144)}
\]
as is shown by the following consideration.

Since in the state of thermodynamic equilibrium all rays of
the same frequency have the same intensity of radiation, the
intensities of radiation of any two plane polarized rays will tend
to become equal, \ie, the passage of energy between them will
be accompanied by an increase of entropy, when it takes place
in the direction from the ray of greater intensity toward that of
smaller intensity. Now the left side of the inequality~\Eq{(144)}
represents the entropy radiation of two \DPchg{noncoherent}{non-coherent} plane polarized
rays with the intensities $\ssfK''$ and $\ssfK'''$, and the right side the
entropy radiation of two \DPchg{noncoherent}{non-coherent} plane polarized rays with the
intensities $\ssfK$ and~$\ssfK'$. But, according to~\Eq{(142)}, the values of $\ssfK''$
and $\ssfK'''$ lie between $\ssfK$ and~$\ssfK'$; therefore the inequality~\Eq{(144)}
holds.

At the same time it is apparent that the error committed, when
the entropy of two coherent rays is calculated as if they were
\DPchg{noncoherent}{non-coherent}, is always in such a sense that the entropy found is
too large. The radiations $\ssfK''$ and $\ssfK'''$ are called ``partially
coherent,'' since they have some terms in common. In the
special case when one of the two principal intensities $\ssfK$ and $\ssfK'$
vanishes entirely, the radiations $\ssfK''$ and $\ssfK'''$ are said to be
``completely coherent,'' since in that case the expression for one
radiation may be completely reduced to that for the other. The
entropy of two completely coherent plane polarized rays is equal
%% -----File: 117.png---Folio 101-------
to the entropy of a single plane polarized ray, the energy of which
is equal to the sum of the two separate energies.

\Section{105.} Let us for future use solve also the more general problem
of calculating the entropy radiation of a ray consisting of an
arbitrary number of plane polarized \DPchg{noncoherent}{non-coherent} components
$\ssfK_{1}$,~$\ssfK_{2}$, $\ssfK_{3},~\dots$, the planes of vibration (planes of
the electric vector) of which are given by the azimuths $\psi_{1}$,~$\psi_{2}$,
$\psi_{3},~\dots$. This problem amounts to finding the principal
intensities $\ssfK_{0}$ and $\ssfK_{0}'$ of the whole ray; for the ray behaves in
every physical respect as if it consisted of the \DPchg{noncoherent}{non-coherent} components
$\ssfK_{0}$ and~$\ssfK_{0}'$. For this purpose we begin by establishing
the value~$\ssfK_{\psi}$ of the component of the ray for an azimuth~$\psi$
taken arbitrarily. Denoting by~$f$ the electric vector of the ray
in the direction~$\psi$, we obtain this value~$\ssfK_{\psi}$ from the equation
\[
f = f_{1} \cos(\psi_{1} - \psi)
  + f_{2} \cos(\psi_{2} - \psi)
  + f_{3} \cos(\psi_{3} - \psi) + \dots,
\]
where the terms on the right side denote the projections of the
vectors of the separate components in the direction~$\psi$, by squaring
and averaging and taking into account the fact that $f_{1}$,~$f_{2}$, $f_{3},~\dots$
are \DPchg{noncoherent}{non-coherent}
\[
\begin{aligned}
\ssfK_{\psi}
  &= \ssfK_{1} \cos^{2}(\psi_{1} - \psi) + \ssfK_{2} \cos^{2}(\psi_{2} - \psi) + \dots \\
\LeftText[\qquad]{or}
\ssfK_{\psi} &= A\cos^{2}\psi + B\sin^{2}\psi + C\sin\psi \cos\psi \\
\LeftText[\qquad]{where}
A &= \ssfK_{1} \cos^{2}\psi_{1} + \ssfK_{2} \cos^{2}\psi_{2} + \dots \\
B &= \ssfK_{1} \sin^{2}\psi_{1} + \ssfK_{2} \sin^{2}\psi_{2} + \dots \\
C &= 2(\ssfK_{1} \sin\psi_{1} \cos\psi_{1} + \ssfK_{2} \sin\psi_{2} \cos\psi_{2} + \dots).
\end{aligned}
\Tag{(145)}
\]

The principal intensities $\ssfK_{0}$ and $\ssfK_{0}'$ of the ray follow from this
expression as the maximum and the minimum value of~$\ssfK_{\psi}$
according to the equation
\[
\frac{d\ssfK_{\psi}}{d\psi} = 0 \quad\text{or,}\quad
\tan 2\psi = \frac{C}{A - B}\Add{.}
\]
Hence it follows that the principal intensities are
\[
\left.
\begin{aligned}
&\ssfK_{0}  \\
&\ssfK_{0}'
\end{aligned}
\right\} = \tfrac{1}{2}(A + B ± \sqrt{(A - B)^{2} + C^{2}}),
\Tag{(146)}
\]
or, by taking~\Eq{(145)} into account,
\begin{multline*}
\left.
\begin{aligned}
&\ssfK_{0} \\
&\ssfK_{0}'
\end{aligned}
\right\} = \frac{1}{2}\Biggl(\ssfK_{1} + \ssfK_{2} + \dots \\
  ± \sqrt{
    \begin{aligned}
      (\ssfK_{1} \cos 2\psi_{1} &+ \ssfK_{2} \cos 2\psi_{2} + \dots)^{2} \\
      &\quad + (\ssfK_{1} \sin 2\psi_{1} + \ssfK_{2} \sin 2\psi_{2} + \dots)^{2}
    \end{aligned}
  }\DPtypo{.}{}\,\Biggr)\DPtypo{}{.}
\Tag{(147)}
\end{multline*}
%% -----File: 118.png---Folio 102-------
Then the entropy radiation required becomes:
\[
\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}').
\Tag{(148)}
\]

\Section{106.} When two ray components $\ssfK$ and~$\ssfK'$, polarized at right
angles to each other, are \DPchg{noncoherent}{non-coherent}, $\ssfK$~and $\ssfK'$ are also the principal
intensities, and the entropy radiation is given by~\Eq{(141)}.
The converse proposition, however, does not hold in general, that
is to say, the two components of a ray polarized at right angles to
each other, which correspond to the principal intensities $\ssfK$ and~$\ssfK'$,
are not necessarily \DPchg{noncoherent}{non-coherent}, and hence the entropy radiation
is not always given by~\Eq{(141)}.

This is true, \eg, in the case of elliptically polarized light.
There the radiations $\ssfK$ and $\ssfK'$ are completely coherent and their
entropy is equal to~$\ssfL(\ssfK + \ssfK')$. This is caused by the fact that
it is possible to give the two ray components an arbitrary displacement
of phase in a reversible manner, say by total reflection.
Thereby it is possible to change elliptically polarized light to
plane polarized light and \textit{vice versa}.

The entropy of completely or partially coherent rays has been
investigated most thoroughly by \Name{M.~Laue}.\footnote
  {\Name{M.~Laue}, Annalen d.\ Phys.\DPchg{,}{}\ \textbf{23}, p.~1, 1907.}
For the significance
of optical coherence for thermodynamic probability see the next
part, \Sec{119}.
%% -----File: 119.png---Folio 103-------

\Chapter[Stationary Field of Radiation]
{V}{Electrodynamical Processes in a Stationary
Field of Radiation}

\Section{107.} We shall now consider from the standpoint of pure electrodynamics
the processes that take place in a vacuum, which
is bounded on all sides by reflecting walls and through which
heat radiation passes uniformly in all directions, and shall then
inquire into the relations between the electrodynamical and the
thermodynamic quantities.

The electrodynamical state of the field of radiation is determined
at every instant by the values of the electric field-strength~$\ssfE$
and the magnetic field-strength~$\ssfH$ at every point in the field,
and the changes in time of these two vectors are completely
determined by \Name{Maxwell's} field equations~\Eq{(52)}, which we have
already used in \Sec{53}, together with the boundary conditions,
which hold at the reflecting walls. In the present case, however,
we have to deal with a solution of these equations of much greater
complexity than that expressed by~\Eq{(54)}, which corresponds to a
plane wave. For a plane wave, even though it be periodic with
a wave length lying within the optical or thermal spectrum, can
never be interpreted as heat radiation. For, according to \Sec{16},
a finite intensity~$K$ of heat radiation requires a finite solid angle
of the rays and, according to \Sec{18}, a spectral interval of finite
width. But an absolutely plane, absolutely periodic wave has a
zero solid angle and a zero spectral width. Hence in the case of
a plane periodic wave there can be no question of either entropy
or temperature of the radiation.

\Section{108.} Let us proceed in a perfectly general way to consider the
components of the field-strengths $\ssfE$~and~$\ssfH$ as functions of the
time at a definite point, which we may think of as the origin of
the coordinate system. Of these components, which are produced
by all rays passing through the origin, there are six; we
select one of them, say~$\ssfE_{z}$, for closer consideration. However
%% -----File: 120.png---Folio 104-------
complicated it may be, it may under all circumstances be written
as a \Name{Fourier's} series for a limited time interval, say from $t = 0$
to $t = \ssfT$; thus
\[
\ssfE_{z} = \sum_{n=1}^{n=\infty} C_{n} \cos\left(\frac{2\pi nt}{\ssfT} - \theta_{n}\right)
\Tag{(149)}
\]
where the summation is to extend over all positive integers~$n$,
while the constants $C_{n}$ (positive) and~$\theta_{n}$ may vary arbitrarily
from term to term. The time interval~$\ssfT$, the fundamental
period of the \Name{Fourier's} series, we shall choose so large that all
times~$t$ which we shall consider hereafter are included in this
time interval, so that $0 < t < \ssfT$. Then we may regard $\ssfE_{z}$ as
identical in all respects with the \Name{Fourier's} series, \ie,~we may
regard $\ssfE_{z}$ as consisting of ``partial vibrations,'' which are strictly
periodic and of frequencies given by
\[
\nu = \frac{n}{\ssfT}\Add{.}
\]

Since, according to \Sec{3}, the time differential~$dt$ required for
the definition of the intensity of a heat ray is necessarily large
compared with the periods of vibration of all colors contained
in the ray, a single time differential~$dt$ contains a large number of
vibrations, \ie, the product~$\nu\, dt$ is a large number. Then it
follows \textit{a~fortiori} that $\nu t$ and, still more,
\[%[** Attn]
\nu \ssfT = n \text{ is enormously large}
\Tag{(150)}
\]
for all values of~$\nu$ entering into consideration. From this we
must conclude that all amplitudes~$C_{n}$ with a moderately large
value for the ordinal number~$n$ do not appear at all in the
\Name{Fourier's} series, that is to say, they are negligibly small.

\Section{109.} Though we have no detailed special information about
the function~$\ssfE_{z}$, nevertheless its relation to the radiation of heat
affords some important information as to a few of its general
properties. Firstly, for the space density of radiation in a vacuum
we have, according to Maxwell's theory,
\[
u = \frac{1}{8\pi} (\bar{\ssfE_{x}^{2}} + \bar{\ssfE_{y}^{2}} + \bar{\ssfE_{z}^{2}}
                  + \bar{\ssfH_{x}^{2}} + \bar{\ssfH_{y}^{2}} + \bar{\ssfH_{z}^{2}}).
\]
Now the radiation is uniform in all directions and in the stationary
%% -----File: 121.png---Folio 105-------
state, hence the six mean values named are all equal to one
another, and it follows that
\[
u = \frac{3}{4\pi} \bar{\ssfE_{z}^{2}}.
\Tag{(151)}
\]
Let us substitute in this equation the value of~$\ssfE_{z}$ as given by~\Eq{(149)}.
Squaring the latter and integrating term by term through a
time interval, from $0$ to~$t$, assumed large in comparison with all
periods of vibration~$\dfrac{1}{\nu}$ but otherwise arbitrary, and then dividing
by~$t$, we obtain, since the radiation is perfectly stationary,
\[
u = \frac{3}{8\pi} \sum C_{n}^{2}.
\Tag{(152)}
\]

From this relation we may at once draw an important conclusion
as to the nature of~$\ssfE_{z}$ as a function of time. Namely,
since the \Name{Fourier's} series~\Eq{(149)} consists, as we have seen, of a
great many terms, the squares,~$C_{n}^{2}$, of the separate amplitudes
of vibration the sum of which gives the space density of radiation,
must have exceedingly small values. Moreover in the integral of
the square of the \Name{Fourier's} series the terms which depend on the
time~$t$ and contain the products of any two different amplitudes
all cancel; hence the amplitudes~$C_{n}$ and the phase-constants~$\theta_{n}$
must vary from one ordinal number to another in a quite irregular
manner. We may express this fact by saying that the separate
partial vibrations of the series are very small and in a ``chaotic''\footnote
  {Compare footnote to \PageRef{page}{116} (Tr.).}
state.

For the specific intensity of the radiation travelling in any
direction whatever we obtain from~\Eq{(21)}
\[
K = \frac{cu}{4\pi} = \frac{3c}{32\pi^{2}} \sum C_{n}^{2}.
\Tag{(153)}
\]

\Section{110.} Let us now perform the spectral resolution of the last two
equations. To begin with we have from~\Eq{(22)}:
\[
u = \int_{0}^{\infty} \ssfu_{\nu}\, d\nu = \frac{3}{8\pi} \sum_{1}^{\infty} C_{n}^{2}.
\Tag{(154)}
\]
On the right side of the equation the sum~$\sum$ consists of separate
%% -----File: 122.png---Folio 106-------
terms, every one of which corresponds to a separate ordinal
number $n$ and to a simple periodic partial vibration. Strictly
speaking this sum does not represent a continuous sequence of
frequencies $\nu$, since $n$~is an integral number. But $n$~is, according
to~\Eq{(150)}, so enormously large for all frequencies which need be
considered that the frequencies~$\nu$ corresponding to the successive
values of~$n$ lie very close together. Hence the interval~$d\nu$,
though infinitesimal compared with~$\nu$, still contains a large
number of partial vibrations, say~$n'$, where
\[
d\nu = \frac{n'}{\ssfT}\Add{.}
\Tag{(155)}
\]
If now in~\Eq{(154)} we equate, instead of the total energy densities,
the energy densities corresponding to the interval $d\nu$ only,
which are independent of those of the other spectral regions, we
obtain
\[
\ssfu_{\nu}\, d\nu = \frac{3}{8\pi} \sum_{n}^{n + n'} C_{n}^{2},
\]
or, according to~\Eq{(155)},
\[
\ssfu_{\nu}
  = \frac{3\ssfT}{8\pi} · \frac{1}{n'} \sum_{n}^{n + n'} C_{n}^{2}
  = \frac{3\ssfT}{8\pi} · \bar{C_{n}^{2}},
\Tag{(156)}
\]
where we denote by~$\bar{C_{n}^{2}}$ the average value of~$C_{n}^{2}$ in the interval
from $n$ to~$n + n'$. The existence of such an average value, the
magnitude of which is independent of~$n$, provided $n'$~be taken
small compared with~$n$, is, of course, not self-evident at the
outset, but is due to a special property of the function~$\ssfE_{z}$ which is
peculiar to stationary heat radiation. On the other hand, since
many terms contribute to the mean value, nothing can be said
either about the magnitude of a separate term~$C_{n}^{2}$, or about the
connection of two consecutive terms, but they are to be regarded
as perfectly independent of each other.

In a very similar manner, by making use of~\Eq{(24)}, we find for
the specific intensity of a monochromatic plane polarized ray,
travelling in any direction whatever,
\[
\ssfK_{\nu} = \frac{3c\ssfT}{64\pi^{2}} \bar{C_{n}^{2}}.
\Tag{(157)}
\]
%% -----File: 123.png---Folio 107-------

From this it is apparent, among other things, that, according
to the electromagnetic theory of radiation, a monochromatic
light or heat ray is represented, not by a simple periodic wave, but
by a superposition of a large number of simple periodic waves,
the mean value of which constitutes the intensity of the ray. In
accord with this is the fact, known from optics, that two rays of
the same color and intensity but of different origin never interfere
with each other, as they would, of necessity, if every ray were a
simple periodic one.

Finally we shall also perform the spectral resolution of the mean
value of~$\ssfE_{z}^{2}$, by writing
\[
\ssfE_{z}^{2} = J = \int_{0}^{\infty} \ssfJ_{\nu}\, d\nu\Add{.}
\Tag{(158)}
\]

Then by comparison with \Eq{(151)},~\Eq{(154)}, and~\Eq{(156)} we find
\[
\ssfJ_{\nu} = \frac{4\pi}{3} \ssfu_{\nu} = \frac{\ssfT}{2} \bar{C_{n}^{2}}\Add{.}
\Tag{(159)}
\]
According to~\Eq{(157)}, $\ssfJ_{\nu}$~is related to~$\ssfK_{\nu}$, the specific intensity of
radiation of a plane polarized ray, as follows:
\[
\ssfK_{\nu} = \frac{3c}{32\pi^{2}} \ssfJ_{\nu}.
\Tag{(160)}
\]

\Section{111.} Black radiation is frequently said to consist of a large
number of regular periodic vibrations. This method of expression
is perfectly justified, inasmuch as it refers to the resolution
of the total vibration in a \Name{Fourier's} series, according to equation~\Eq{(149)},
and often is exceedingly well adapted for convenience and
clearness of discussion. It should, however, not mislead us into
believing that such a ``regularity'' is caused by a special physical
property of the elementary processes of vibration. For the
resolvability into a Fourier's series is mathematically self-evident
and hence, in a physical sense, tells us nothing new. In fact, it
is even always possible to regard a vibration which is damped
to an arbitrary extent as consisting of a sum of regular periodic
partial vibrations with constant amplitudes and constant phases.
On the contrary, it may just as correctly be said that in all nature
there is no process more complicated than the vibrations of black
%% -----File: 124.png---Folio 108-------
radiation. In particular, these vibrations do not depend in any
characteristic manner on the special processes that take place
in the centers of emission of the rays, say on the period or the
damping of the emitting particles; for the normal spectrum is
distinguished from all other spectra by the very fact that all
individual differences caused by the special nature of the emitting
substances are perfectly equalized and effaced. Therefore to
attempt to draw conclusions concerning the special properties
of the particles emitting the rays from the elementary vibrations
in the rays of the normal spectrum would be a hopeless
undertaking.

In fact, black radiation may just as well be regarded as consisting,
not of regular periodic vibrations, but of absolutely
irregular separate impulses. The special regularities, which we
observe in monochromatic light resolved spectrally, are caused
merely by the special properties of the spectral apparatus used,
\eg, the dispersing prism (natural periods of the molecules), or
the diffraction grating (width of the slits). Hence it is also incorrect
to find a characteristic difference between light rays and
Roentgen\DPnote{**F2: No italics; check other instances?} rays (the latter assumed as an electromagnetic process
in a vacuum) in the circumstance that in the former the vibrations
take place with greater regularity. Roentgen rays may,
under certain conditions, possess more selective properties than
light rays. The resolvability into a \Name{Fourier's} series of partial
vibrations with constant amplitudes and constant phases exists
for both kinds of rays in precisely the same manner. What
especially distinguishes light vibrations from Roentgen vibrations
is the much smaller frequency of the partial vibrations of the
former. To this is due the possibility of their spectral resolution,
and probably also the far greater regularity of the changes of the
radiation intensity in every region of the spectrum in the course of
time, which, however, is not caused by a special property of the
elementary processes of vibration, but merely by the constancy
of the mean values.

\Section{112.} The elementary processes of radiation exhibit regularities
only when the vibrations are restricted to a narrow spectral region,
that is to say in the case of spectroscopically resolved light, and
especially in the case of the natural spectral lines. If, \eg, the
amplitudes~$C_{n}$ of the \Name{Fourier's} series~\Eq{(149)} differ from zero only
%% -----File: 125.png---Folio 109-------
between the ordinal numbers $n = n_{0}$ and $n = n_{1}$, where $\dfrac{n_{1} - n_{0}}{n_{0}}$
is small, we may write
\[
\ssfE_{z} = C_{0} \cos\left(\frac{2\pi n_{0}t}{\ssfT} - \theta_{0}\right),
\Tag{(161)}
\]
where
\begin{align*}
C_{0} \cos\theta_{0} &= \sum_{n_{0}}^{n_{1}} C_{n} \cos\left(\frac{2\pi(n - n_{0})t}{\ssfT}-\theta_{n}\right)\\
C_{0} \sin\theta_{0} &=-\sum_{n_{0}}^{n_{1}} C_{n} \sin\left(\frac{2\pi(n - n_{0})t}{\ssfT}-\theta_{n}\right)
\end{align*}
and $\ssfE_{z}$~may be regarded as a single approximately periodic vibration
of frequency $\nu_{0} = \dfrac{n_{0}}{\ssfT}$ with an amplitude~$C_{0}$ and a phase-constant~$\theta_{0}$
which vary slowly and irregularly.

The smaller the spectral region, and accordingly the smaller
$\dfrac{n_{1} - n_{0}}{n_{0}}$, the slower are the fluctuations (``Schwankungen'') of
$C_{0}$~and~$\theta_{0}$, and the more regular is the resulting vibration and also
the larger is the difference of path for which radiation can interfere
with itself. If a spectral line were absolutely sharp, the
radiation would have the property of being capable of interfering
with itself for differences of path of any size whatever. This
case, however, according to \Sec{18}, is an ideal abstraction, never
occurring in reality.
%% -----File: 126.png---Folio 110-------
% [Blank Page]
%% -----File: 127.png---Folio 111-------

\Part{III}{Entropy and Probability}
%% -----File: 128.png---Folio 112-------
% [Blank Page]
%% -----File: 129.png---Folio 113-------

\Chapter[Fundamental Definitions and Laws]
{I}{Fundamental Definitions and Laws.
Hypothesis of Quanta}

\Section{113.} Since a wholly new element, entirely unrelated to the
fundamental principles of electrodynamics, enters into the range
of investigation with the introduction of probability considerations
into the electrodynamic theory of heat radiation, the question
arises at the outset, whether such considerations are justifiable
and necessary. At first sight we might, in fact, be inclined
to think that in a purely electrodynamical theory there would be
no room at all for probability calculations. For since, as is well
known, the electrodynamic equations of the field together with
the initial and boundary conditions determine uniquely the way
in which an electrodynamical process takes place, in the course
of time, considerations which lie outside of the equations of the
field would seem, theoretically speaking, to be uncalled for and in
any case dispensable. For either they lead to the same results
as the fundamental equations of electrodynamics and then they
are superfluous, or they lead to different results and in this case
they are wrong.

In spite of this apparently unavoidable dilemma, there is a
flaw in the reasoning. For on closer consideration it is seen
that what is understood in electrodynamics by ``initial and
boundary'' conditions, as well as by the ``way in which a process
takes place in the course of time,'' is entirely different from what
is denoted by the same words in thermodynamics. In order to
make this evident, let us consider the case of radiation \textit{in~vacuo},
uniform in all directions, which was treated in the last chapter.

From the standpoint of thermodynamics the state of radiation
is completely determined, when the intensity of monochromatic
radiation~$\ssfK_{\nu}$ is given for all frequencies~$\nu$. The electrodynamical
observer, however, has gained very little by this single statement;
because for him a knowledge of the state requires that every one
%% -----File: 130.png---Folio 114-------
of the six components of the electric and magnetic field-strength
be given at all points of the space; and, while from the thermodynamic
point of view the question as to the way in which the
process takes place in time is settled by the constancy of the
intensity of radiation~$\ssfK_{\nu}$, from the electrodynamical point of
view it would be necessary to know the six components of the
field at every point as functions of the time, and hence the amplitudes~$C_{n}$
and the phase-constants~$\theta_{n}$ of all the several partial
vibrations contained in the radiation would have to be calculated.
This, however, is a problem whose solution is quite impossible,
for the data obtainable from the measurements are by no
means sufficient. The thermodynamically measurable quantities,
looked at from the electrodynamical standpoint, represent
only certain mean values, as we saw in the special case of
stationary radiation in the last chapter.

We might now think that, since in thermodynamic measurements
we are always concerned with mean values only, we need
consider nothing beyond these mean values, and, therefore, need
not take any account of the particular values at all. This method
is, however, impracticable, because frequently and that too just
in the most important cases, namely, in the cases of the processes
of emission and absorption, we have to deal with mean values
which cannot be calculated unambiguously by electrodynamical
methods from the measured mean values. For example, the
mean value of~$C_{n}$ cannot be calculated from the mean value of~$C_{n}^{2}$,
if no special information as to the particular values of~$C_{n}$ is
available.

Thus we see that the electrodynamical state is not by any
means determined by the thermodynamic data and that in cases
where, according to the laws of thermodynamics and according
to all experience, an unambiguous result is to be expected, a purely
electrodynamical theory fails entirely, since it admits not one
definite result, but an infinite number of different results.

\Section{114.} Before entering on a further discussion of this fact and
of the difficulty to which it leads in the electrodynamical theory
of heat radiation, it may be pointed out that exactly the same case
and the same difficulty are met with in the mechanical theory of
heat, especially in the kinetic theory of gases. For when, for
example, in the case of a gas flowing out of an opening at the time
%% -----File: 131.png---Folio 115-------
$t = 0$, the velocity, the density, and the temperature are given
at every point, and the boundary conditions are completely
known, we should expect, according to all experience, that these
data would suffice for a unique determination of the way in which
the process takes place in time. This, however, from a purely
mechanical point of view is not the case at all; for the positions
and velocities of all the separate molecules are not at all given
by the visible velocity, density, and temperature of the gas, and
they would have to be known exactly, if the way in which the
process takes place in time had to be completely calculated from
the equations of motion. In fact, it is easy to show that, with
given initial values of the visible velocity, density, and temperature,
an infinite number of entirely different processes is mechanically
possible, some of which are in direct contradiction to the
principles of thermodynamics, especially the second principle.

\Section{115.} From these considerations we see that, if we wish to calculate
the way in which a thermodynamic process takes place
in time, such a formulation of initial and boundary conditions
as is perfectly sufficient for a unique determination of the process
in thermodynamics, does not suffice for the mechanical theory of
heat or for the electrodynamical theory of heat radiation. On
the contrary, from the standpoint of pure mechanics or electrodynamics
the solutions of the problem are infinite in number.
Hence, unless we wish to renounce entirely the possibility of
representing the thermodynamic processes mechanically or electrodynamically,
there remains only one way out of the difficulty,
namely, to supplement the initial and boundary conditions by
special hypotheses of such a nature that the mechanical or
electrodynamical equations will lead to an unambiguous result
in agreement with experience. As to how such an hypothesis
is to be formulated, no hint can naturally be obtained from the
principles of mechanics or electrodynamics, for they leave the
question entirely open. Just on that account any mechanical or
electrodynamical hypothesis containing some further specialization
of the given initial and boundary conditions, which cannot
be tested by direct measurement, is admissible \textit{a~priori}. What
hypothesis is to be preferred can be decided only by testing the
results to which it leads in the light of the thermodynamic principles
based on experience.
%% -----File: 132.png---Folio 116-------

\Section{116.}  Although, according to the statement just made, a decisive
test of the different admissible hypotheses can be made only
\textit{a~posteriori}, it is nevertheless worth while noticing that it is possible
to obtain \textit{a~priori}, without relying in any way on thermodynamics,
a definite hint as to the nature of an admissible hypothesis.
Let us again consider a flowing gas as an illustration (\Sec{114}).
The mechanical state of all the separate gas molecules is not at
all completely defined by the thermodynamic state of the gas,
as has previously been pointed out. If, however, we consider all
conceivable positions and velocities of the separate gas molecules,
consistent with the given values of the visible velocity, density,
and temperature, and calculate for every combination of them the
mechanical process, assuming some simple law for the impact
of two molecules, we shall arrive at processes, the vast majority
of which agree completely in the mean values, though perhaps
not in all details. Those cases, on the other hand, which show
appreciable  deviations, are vanishingly few, and only occur
when certain very special and far-reaching conditions between the
coordinates and velocity-components of the molecules are
satisfied. Hence, if the assumption be made that such special
conditions do not exist, however different the mechanical details
may be in other respects, a form of flow of gas will be found,
which may be called quite definite with respect to all measurable
mean values---and they are the only ones which can be tested
experimentally---although it will not, of course, be quite definite
in all details. And the remarkable feature of this is that it is
just the motion obtained in this manner that satisfies the postulates
of the second principle of thermodynamics.

\Section{117.}  From these considerations it is evident that the hypotheses
whose introduction was proven above to be necessary completely
answer their purpose, if they state nothing more than that
exceptional cases, corresponding to special conditions which exist
between the separate quantities determining the state and which
cannot be tested directly, do not occur in nature. In mechanics
this is done by the hypothesis\footnote
  {\Name{L.~Boltzmann}, Vorlesungen über Gastheorie \textbf{1}, p.~21, 1896. Wiener Sitzungsberichte
  \textbf{78}, Juni, 1878, at the end. Compare also \Name{S.~H. Burbury}, Nature, \textbf{51}, p.~78, 1894.}
that the heat motion is a ``molecular
chaos'';\footnote
  {Hereafter \Name{Boltzmann's} ``Unordnung'' will be rendered by chaos, ``ungeordnet'' by
  \Label{116}% [** TN: Page label]
  chaotic (Tr.).}
in electrodynamics the same thing is accomplished
%% -----File: 133.png---Folio 117-------
by the hypothesis of ``natural radiation,'' which states that
there exist between the numerous different partial vibrations~\Eq{(149)}
of a ray no other relations than those caused by the measurable
mean values (compare below, \Sec{148}). If, for brevity, we
denote any condition or process for which such an hypothesis
%[** Thm]
holds as an ``elemental chaos,'' the principle, \emph{that in nature any
state or any process containing numerous elements not in themselves
measurable is an elemental chaos}, furnishes the necessary condition
for a unique determination of the measurable processes in mechanics
as well as in electrodynamics and also for the validity of the
second principle of thermodynamics. This must also serve as a
mechanical or electrodynamical explanation of the conception of
entropy, which is characteristic of the second law and of the
closely allied concept of temperature.\footnote
  {To avoid misunderstanding I must emphasize that the question, whether the hypothesis
  of elemental chaos is really everywhere satisfied in nature, is not touched upon by the preceding
  considerations. I intended only to show at this point that, wherever this hypothesis
  does not hold, the natural processes, if viewed from the thermodynamic (macroscopic) point
  of view, do not take place unambiguously.}
It also follows from this
that the significance of entropy and temperature is, according to
their nature, connected with the condition of an elemental
chaos. The terms entropy and temperature do not apply to a
purely periodic, perfectly plane wave, since all the quantities in
such a wave are in themselves measurable, and hence cannot be
an elemental chaos any more than a single rigid atom in motion
can. The necessary condition for the hypothesis of an elemental
chaos and with it for the existence of entropy and temperature
can consist only in the irregular simultaneous effect of
very many partial vibrations of different periods, which are
propagated in the different directions in space independent
of one another, or in the irregular flight of a multitude of
atoms.

\Section{118.} But what mechanical or electrodynamical quantity
represents the entropy of a state? It is evident that this quantity
depends in some way on the ``probability'' of the state.
For since an elemental chaos and the absence of a record of any
individual element forms an essential feature of entropy, the
tendency to neutralize any existing temperature differences,
which is connected with an increase of entropy, can mean nothing
for the mechanical or electrodynamical observer but that uniform
%% -----File: 134.png---Folio 118-------
distribution of elements in a chaotic state is more probable than
any other distribution.

Now since the concept of entropy as well as the second principle
of thermodynamics are of universal application, and since
on the other hand the laws of probability have no less universal
validity, it is to be expected that the connection between entropy
and probability should be very close. Hence we make the
following proposition the foundation of our further discussion:
%[** Thm]
\emph{The entropy of a physical system in a definite state depends solely
on the probability of this state.} The fertility of this law will be
seen later in several cases. We shall not, however, attempt to
give a strict general proof of it at this point. In fact, such an
attempt evidently would have no meaning at this point. For,
so long as the ``probability'' of a state is not numerically defined,
the correctness of the proposition cannot be quantitatively
tested. One might, in fact, suspect at first sight that on this
account the proposition has no definite physical meaning. It
may, however, be shown by a simple deduction that it is possible
by means of this fundamental proposition to determine quite
generally the way in which entropy depends on probability,
without any further discussion of the probability of a state.

\Section{119.} For let $S$ be the entropy, $W$~the probability of a physical
system in a definite state; then the \DPtypo{propositon}{proposition}
states that
\[
S = f(W)
\Tag{(162)}
\]
where $f(W)$~represents a universal function of the argument~$W$.
In whatever way $W$~may be defined, it can be safely inferred from
the mathematical concept of probability that the probability of
a system which consists of two entirely independent\footnote
  {It is well known that the condition that the two systems be independent of each other is
  essential for the validity of the expression~\Eq{(163)}. That it is also a necessary condition for the
  additive combination of the entropy was proven first by \Name{M.~Laue} in the case of optically
  coherent rays. Annalen d.\ Physik\DPchg{,}{} \textbf{20}, p.~365, 1906.}
systems
is equal to the product of the probabilities of these two systems
separately. If we think, \eg, of the first system as any body
whatever on the earth and of the second system as a cavity containing
radiation on Sirius, then the probability that the terrestrial
body be in a certain state~$1$ and that simultaneously the
radiation in the cavity in a definite state~$2$ is
\[
W = W_{1}W_{2},
\Tag{(163)}
\]
%% -----File: 135.png---Folio 119-------
where $W_{1}$~and~$W_{2}$ are the probabilities that the systems involved
are in the states in question.

If now $S_{1}$~and~$S_{2}$ are the entropies of the separate systems in
the two states, then, according to~\Eq{(162)}, we have
\[
S_{1} = f(W_{1})\qquad
S_{2} = f(W_{2}).
\]
But, according to the second principle of thermodynamics, the
total entropy of the two systems, which are independent (see
\DPchg{footnote to preceding page}{preceding footnote}) of each other, is $S = S_{1} + S_{2}$ and hence
from \Eq{(162)}~and~\Eq{(163)}
\[
f(W_{1}W_{2}) = f(W_{1}) + f(W_{2}).
\]

From this functional equation $f$~can be determined. For on
differentiating both sides with respect to~$W_{1}$, $W_{2}$~remaining constant,
we obtain
\[
W_{2} \dot{f}(W_{1}W_{2}) = \dot{f}(W_{1}).
\]
On further differentiating with respect to~$W_{2}$, $W_{1}$~now remaining
constant, we get
\[
\dot{f}(W_{1}W_{2}) + W_{1}W_{2} \ddot{f}(W_{1}W_{2}) = 0
\]
or
\[
\dot{f}(W) + W \ddot{f}(W) = 0.
\]
The general integral of this differential equation of the second
order is
\[
f(W) = k \log W + \const.
\]
Hence from~\Eq{(162)} we get
\[
S = k \log W + \const.,
%[** TN: No equation number in the original]
\Tag{(164)}
\]
an equation which determines the general way in which the entropy
depends on the probability. The universal constant of
integration~$k$ is the same for a terrestrial as for a cosmic system,
and its value, having been determined for the former, will remain
valid for the latter. The second additive constant of integration
may, without any restriction as regards generality, be included
as a constant multiplier in the quantity~$W$, which here has not yet
been completely defined, so that the equation reduces to
\[
S = k \log W.
\]

\Section{120.} The logarithmic connection between entropy and probability
was first stated by \Name{L.~Boltzmann}\footnote
  {\Name{L.~Boltzmann}, Vorlesungen über Gastheorie, \textbf{1}, \Sec{6}.}
in his kinetic theory of
%% -----File: 136.png---Folio 120-------
gases. Nevertheless our equation~\Eq{(164)} differs in its meaning
from the corresponding one of Boltzmann in two essential points.

Firstly, \Name{Boltzmann's} equation lacks the factor~$k$, which is due
to the fact that \Name{Boltzmann} always used gram-molecules, not the
molecules themselves, in his calculations. Secondly, and this is
of greater consequence, \Name{Boltzmann} leaves an additive constant
undetermined in the entropy~$S$ as is done in the whole of classical
thermodynamics, and accordingly there is a constant factor of
proportionality, which remains undetermined in the value of the
probability~$W$.

In contrast with this we assign a definite absolute value to the
entropy~$S$. This is a step of fundamental importance, which
can be justified only by its consequences. As we shall see later,
this step leads necessarily to the ``hypothesis of quanta'' and
moreover it also leads, as regards radiant heat, to a definite law
of distribution of energy of black radiation, and, as regards heat
energy of bodies, to \Name{Nernst's} heat theorem.

From \Eq{(164)} it follows that with the entropy~$S$ the probability~$W$
is, of course, also determined in the absolute sense. We shall
designate the quantity~$W$ thus defined as the ``thermodynamic
probability,'' in contrast to the ``mathematical probability,'' to
which it is proportional but not equal. For, while the mathematical
probability is a proper fraction, the thermodynamic
probability is, as we shall see, always an integer.

\Section{121.} The relation~\Eq{(164)} contains a general method for calculating
the entropy~$S$ by probability considerations. This,
however, is of no practical value, unless the thermodynamic
probability~$W$ of a system in a given state can be expressed
numerically. The problem of finding the most general and most
precise definition of this quantity is among the most important
problems in the mechanical or electrodynamical theory of heat.
It makes it necessary to discuss more fully what we mean by the
``state'' of a physical system.

By the state of a physical system at a certain time we mean the
aggregate of all those mutually independent quantities, which
determine uniquely the way in which the processes in the system
take place in the course of time for given boundary conditions.
Hence a knowledge of the state is precisely equivalent to a knowledge
of the ``initial conditions.'' If we now take into account
%% -----File: 137.png---Folio 121-------
the considerations stated above in \Sec{113}, it is evident that we
must distinguish in the theoretical treatment two entirely different
kinds of states, which we may denote as ``microscopic'' and
``macroscopic'' states. The microscopic state is the state as
described by a mechanical or electrodynamical observer; it contains
the separate values of all coordinates, velocities, and field-strengths.
The microscopic processes, according to the laws of
mechanics and electrodynamics, take place in a perfectly unambiguous
way; for them entropy and the second principle of thermodynamics
have no significance. The macroscopic state,
however, is the state as observed by a thermodynamic observer;
any macroscopic state contains a large number of microscopic
ones, which it unites in a mean value. Macroscopic processes
take place in an unambiguous way in the sense of the second
principle, when, and only when, the hypothesis of the elemental
chaos (\Sec{117}) is satisfied.

\Section{122.} If now the calculation of the probability~$W$ of a state is
in question, it is evident that the state is to be thought of in the
macroscopic sense. The first and most important question is
now: How is a macroscopic state defined? An answer to it will
dispose of the main features of the whole problem.

For the sake of simplicity, let us first consider a special case,
that of a very large number,~$N$, of simple similar molecules. Let
the problem be solely the distribution of these molecules in space
within a given volume,~$V$, irrespective of their velocities, and further
the definition of a certain macroscopic distribution in space.
The latter cannot consist of a statement of the coordinates of all
the separate molecules, for that would be a definite microscopic
distribution. We must, on the contrary, leave the positions of
the molecules undetermined to a certain extent, and that can be
done only by thinking of the whole volume~$V$ as being divided
into a number of small but finite \emph{space elements},~$G$, each containing
a specified number of molecules. By any such statement a
definite macroscopic distribution in space is defined. The manner
in which the molecules are distributed within every separate
space element is immaterial, for here the hypothesis of elemental
chaos (\Sec{117}) provides a supplement, which insures the unambiguity
of the macroscopic state, in spite of the microscopic
indefiniteness. If we distinguish the space elements in order by
%% -----File: 138.png---Folio 122-------
the numbers $1$,~$2$, $3,~\dots$ and, for any particular macroscopic
distribution in space, denote the number of the molecules
lying in the separate space elements by $N_{1}$,~$N_{2}$, $N_{3}~\dots$,
then to every definite system of values $N_{1}$,~$N_{2}$, $N_{3}~\dots$,
there corresponds a definite macroscopic distribution in space.
We have of course always:
\[
N_{1} + N_{2} + N_{3} + \dots = N
\Tag{(165)}
\]
or if
\begin{gather*}
\frac{N_{1}}{N} = w_{1}\Add{,}\quad
\frac{N_{2}}{N} = w_{2},\ \dots
\Tag{(166)} \\
w_{1} + w_{2} + w_{3} + \dots = 1.
\Tag{(167)}
\end{gather*}
The quantity~$w_{i}$ may be called the density of distribution of the
molecules, or the mathematical probability that any molecule
selected at random lies in the $i$th~space element.

If we now had, \eg, only $10$~molecules and $7$~space elements, a
definite space distribution would be represented by the values:
\[
N_{1} = 1,\ N_{2} = 2,\ N_{3} = 0,\ N_{4} = 0,\ N_{5} = 1,\ N_{6} = 4,\ N_{7} = 2,
\Tag{(168)}
\]
which state that in the seven space elements there lie respectively
$1$,~$2$, $0$, $0$, $1$, $4$, $2$~molecules.

\Section{123.} The definition of a macroscopic distribution in space may
now be followed immediately by that of its thermodynamic
probability~$W$. The latter is founded on the consideration that
a certain distribution in space may be realized in many different
ways, namely, by many different individual coordinations or
``complexions,'' according as a certain molecule considered will
happen to lie in one or the other space element. For, with a
given distribution of space, it is of consequence only how many, not
which, molecules lie in every space element.

The number of all complexions which are possible with a given
distribution in space we equate to the thermodynamic probability~$W$
of the space distribution.

In order to form a definite conception of a certain complexion,
we can give the molecules numbers, write these numbers in
order from $1$ to~$N$, and place below the number of every molecule
the number of that space element to which the molecule in question
belongs in that particular complexion. Thus the following
%% -----File: 139.png---Folio 123-------
table represents one particular complexion, selected at random,
for the distribution in the preceding illustration
\[
\begin{array}{r*{9}{>{\quad}r}}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\
6 & 1 & 7 & 5 & 6 & 2 & 2 & 6 & 6 &  7
\end{array}
\Tag{(169)}
\]
By this the fact is exhibited that the

%[** Attn]
Molecule~$2$ lies in space element~$1$.

Molecules $6$~and~$7$ lie in space element~$2$.

Molecule~$4$ lies in space element~$5$.

Molecules $1$,~$5$,~$8$, and~$9$ lie in space element~$6$.

Molecules $3$~and~$10$ lie in space element~$7$.

As becomes evident on comparison with~\Eq{(168)}, this complexion
does, in fact, correspond in every respect to the space
distribution given above, and in a similar manner it is easy to
exhibit many other complexions, which also belong to the same
space distribution. The number of all possible complexions
required is now easily found by inspecting the lower of the two
lines of figures in~\Eq{(169)}. For, since the number of the molecules
is given, this line of figures contains a definite number of places.
Since, moreover, the distribution in space is also given, the number
of times that every figure (\ie,~every space element) appears
in the line is equal to the number of molecules which lie in that
particular space element. But every change in the table gives
a new particular coordination between molecules and space
elements and hence a new complexion. Hence the number of
the possible complexions, or the thermodynamic probability,~$W$,
of the given space distribution, is equal to the number of ``permutations
with repetition'' possible under the given conditions.
In the simple numerical example chosen, we get for~$W$, according
to a well-known formula, the expression
\[
\frac{10!}{1!\, 2!\, 0!\, 0!\, 1!\, 4!\, 2!} = 37,800.
\]

The form of this expression is so chosen that it may be applied
easily to the general case. The numerator is equal to factorial~$N$,
$N$~being the total number of molecules considered, and the
denominator is equal to the product of the factorials of the numbers,
$N_{1}$,~$N_{2}$, $N_{3},~\dots$ of the molecules, which lie in every
separate space element and which, in the general case, must be
%% -----File: 140.png---Folio 124-------
thought of as large numbers. Hence we obtain for the required
probability of the given space distribution
\[
W = \frac{N!}{N_{1}!\, N_{2}!\, N_{3}!\, \dots}\Add{.}
\Tag{(170)}
\]

Since all the $N$'s are large numbers, we may apply to their
factorials Stirling's formula, which for a large number may be
abridged\footnote
  {Abridged in the sense that factors which in the logarithmic expression~\Eq{(173)} would give
  \Label{124}% [** TN: Page label]
  rise to small additive terms have been omitted at the outset. A brief derivation of equation~\Eq{(173)}
  may be found on \PageRef{p.}{218} (Tr.).}
to\footnote
  {See for example \Name{E.~Czuber}, Wahrscheinlichkeitsrechnung (Leipzig, B.~G. Teubner)
  p.~22, 1903; \Name{H.~Poincaré}, Calcul des Probabilités (Paris, Gauthier-Villars), p.~85, 1912.}
\[
n! = \left(\frac{n}{e}\right)^{n}\Add{.}
\Tag{(171)}
\]
Hence, by taking account of~\Eq{(165)}, we obtain
\[
W = \left(\frac{N}{N_{1}}\right)^{N_{1}}
    \left(\frac{N}{N_{2}}\right)^{N_{2}}
    \left(\frac{N}{N_{3}}\right)^{N_{3}} \dots\Add{.}
\Tag{(172)}
\]

\Section{124.} Exactly the same method as in the case of the space distribution
just considered may be used for the definition of a
macroscopic state and of the thermodynamic probability in the
general case, where not only the coordinates but also the velocities,
the electric moments,~etc., of the molecules are to be dealt
with. Every thermodynamic state of a system of $N$~molecules
is, in the macroscopic sense, defined by the statement of the
number of molecules, $N_{1}$,~$N_{2}$, $N_{3},~\dots$, which are contained
in the region elements $1$,~$2$, $3,~\dots$ of the ``state
space.'' This state space, however, is not the ordinary three-dimensional
space, but an ideal space of as many dimensions as
there are variables for every molecule. In other respects the
definition and the calculation of the thermodynamic probability~$W$
are exactly the same as above and the entropy of the state is
accordingly found from~\Eq{(164)}, taking \Eq{(166)} also into account, to
be
\[
S = -kN \sum w_{1} \log w_{1},%[** F2: Probably w_{i}]
\Tag{(173)}
\]
where the sum $\sum$ is to be taken over all region elements. It is
obvious from this expression that the entropy is in every case a
\emph{positive quantity}.

\Section{125.} By the preceding developments the calculation of the
%% -----File: 141.png---Folio 125-------
entropy of a system of $N$~molecules in a given thermodynamic
state is, in general, reduced to the single problem of finding the
magnitude~$G$ of the region elements in the state space. That
such a definite finite quantity really exists is a characteristic
feature of the theory we are developing, as contrasted with that
due to \Name{Boltzmann}, and forms the content of the so-called \emph{hypothesis
of quanta}. As is readily seen, this is an immediate consequence
of the proposition of \Sec{120} that the entropy~$S$ has an
absolute, not merely a relative, value; for this, according to~\Eq{(164)},
necessitates also an absolute value for the magnitude of the thermodynamic
probability~$W$, which, in turn, according to \Sec{123},
is dependent on the number of complexions, and hence also
on the number and size of the region elements which are used.
Since all different complexions contribute uniformly to the value
of the probability~$W$, the region elements of the state space
represent also \emph{regions of equal probability}. If this were not so,
the complexions would not be all equally probable.

However, not only the magnitude, but also the shape and position
of the region elements must be perfectly definite. For since,
in general, the distribution density~$w$ is apt to vary appreciably
from one region element to another, a change in the shape of a
region element, the magnitude remaining unchanged, would, in
general, lead to a change in the value of~$w$ and hence to a change
in~$S$. We shall see that only in special cases, namely, when the
distribution densities~$w$ are very small, may the absolute magnitude
of the region elements become physically unimportant, inasmuch
as it enters into the entropy only through an additive constant.
This happens, \eg, at high temperatures, large volumes,
slow vibrations (state of an ideal gas, \Sec{132}, \Name{Rayleigh's} radiation
law, \DPtypo{Sec.~195}{\Sec{159}}). Hence it is permissible for such limiting
cases to assume, without appreciable error, that $G$~is infinitely
small in the macroscopic sense, as has hitherto been the practice
in statistical mechanics. As soon, however, as the distribution
densities~$w$ assume appreciable values, the classical statistical
mechanics fail.

\Section{126.} If now the problem be to determine the magnitude~$G$
of the region elements of equal probability, the laws of the classical
statistical mechanics afford a certain hint, since in certain
limiting cases they lead to correct results.
%% -----File: 142.png---Folio 126-------

Let $\phi_{1}$,~$\phi_{2}$, $\phi_{3},~\dots$ be the ``generalized coordinates,''
$\psi_{1}$,~$\psi_{2}$, $\psi_{3},~\dots$ the corresponding ``impulse coordinates''
or ``moments,'' which determine the microscopic state of a certain
molecule; then the state space contains as many dimensions
as there are coordinates~$\phi$ and moments~$\psi$ for every molecule.
Now the region element of probability, according to classical
statistical mechanics, is identical with the infinitely small element
of the state space (in the macroscopic sense)\footnote
  {Compare, for example, \Name{L.~Boltzmann}, Gastheorie, \textbf{2}, p.~62 \textit{et~seq.}, 1898, or \Name{J.~W.~Gibbs},
  Elementary principles in statistical mechanics, Chapter~I, 1902.}
\[
d\phi_{1}\, d\phi_{2}\, d\phi_{3} \dots d\psi_{1}\, d\psi_{2}\, d\psi_{3} \dots\Add{.}
\Tag{(174)}
\]

According to the hypothesis of quanta, on the other hand,
every region element of probability has a definite finite magnitude
\[
G = \int d\phi_{1}\, d\phi_{2}\, d\phi_{3} \dots d\psi_{1}\, d\psi_{2}\, d\psi_{3} \dots
\Tag{(175)}
\]
whose value is the same for all different region elements and, moreover,
depends on the nature of the system of molecules considered.
The shape and position of the separate region elements are determined
by the limits of the integral and must be determined anew
in every separate case.
%% -----File: 143.png---Folio 127-------

\Chapter{II}{Ideal Monatomic Gases}

\Section{127.} In the preceding chapter it was proven that the introduction
of probability considerations into the mechanical and
electrodynamical theory of heat is justifiable and necessary, and
from the general connection between entropy~$S$ and probability~$W$,
as expressed in equation~\Eq{(164)}, a method was derived for calculating
the entropy of a physical system in a given state. Before
we apply this method to the determination of the entropy of
radiant heat we shall in this chapter make use of it for calculating
the entropy of an ideal monatomic gas in an arbitrarily given
state. The essential parts of this calculation are already contained
in the investigations of \Name{L.~Boltzmann}\footnote
  {\Name{L.~Boltzmann}, Sitzungsber.\ d.\ Akad.\ d.\ Wissensch.\ zu Wien~(II) \textbf{76}, p.~373, 1877. Compare
  also Gastheorie, \textbf{1}, p.~38, 1896.}
on the mechanical
theory of heat; it will, however, be advisable to discuss this
simple case in full, firstly to enable us to compare more readily
the method of calculation and physical significance of mechanical
entropy with that of radiation entropy, and secondly, what is
more important, to set forth clearly the differences as compared
with \Name{Boltzmann's} treatment, that is, to discuss the meaning of
the universal constant~$k$ and of the finite region elements~$G$. For
this purpose the treatment of a special case is sufficient.

\Section{128.} Let us then take $N$~similar monatomic gas molecules in
an arbitrarily given thermodynamic state and try to find the
corresponding entropy. The state space is six-dimensional,
with the three coordinates $x$,~$y$,~$z$, and the three corresponding
moments $m\xi$,~$m\eta$,~$m\zeta$, of a molecule, where we denote the mass
by~$m$ and velocity components by $\xi$,~$\eta$,~$\zeta$. Hence these quantities
are to be substituted for the $\phi$~and~$\psi$ in \Sec{126}. We thus obtain
for the size of a region element~$G$ the sextuple integral
\[
G = m^{3} \int d\sigma,
\Tag{(176)}
\]
where, for brevity
\[
dx\, dy\, dz\, d\xi\, d\eta\, d\zeta = d\sigma\Add{.}
\Tag{(177)}
\]
%% -----File: 144.png---Folio 128-------

If the region elements are known, then, since the macroscopic
state of the system of molecules was assumed as known, the
numbers $N_{1}$,~$N_{2}$, $N_{3},~\dots$ of the molecules which lie in
the separate region elements are also known, and hence the distribution
densities $w_{1}$,~$w_{2}$, $w_{3},~\dots$ \Eq{(166)} are given and the
entropy of the state follows at once from~\Eq{(173)}.

\Section{129.} The theoretical determination of~$G$ is a problem as difficult
as it is important. Hence we shall at this point restrict ourselves
from the very outset to the special case in which the distribution
density varies but slightly from one region element to the next---the
characteristic feature of the state of an ideal gas. Then the
summation over all region elements may be replaced by the integral
over the whole state space. Thus we have from \Eq{(176)}~and~\Eq{(167)}
\[
%[** Again, w_{i}?]
\sum w_{1}
  = \sum w_{1} \frac{m^{3}}{G} \int d\sigma
  = \frac{m^{3}}{G} \int w\, d\sigma = 1,
\Tag{(178)}
\]
in which $w$~is no longer thought of as a discontinuous function
of the ordinal number,~$i$, of the region element, where $i = 1$,
$2$, $3,~\dots~n$, but as a continuous function of the variables,
$x$,~$y$,~$z$, $\xi$,~$\eta$,~$\zeta$, of the state space. Since the whole state region
contains very many region elements, it follows, according to~\Eq{(167)}
and from the fact that the distribution density~$w$ changes
slowly, that $w$ has everywhere a small value.

Similarly we find for the entropy of the gas from~\Eq{(173)}:
\[
S = -kN \sum w_{1} \log w_{1} = -kN \frac{m^{3}}{G} \int w \log w\, d\sigma.
\Tag{(179)}
\]
Of course the whole energy~$E$ of the gas is also determined by the
distribution densities~$w$. If $w$~is sufficiently small in every
region element, the molecules contained in any one region
element are, on the average, so far apart that their energy depends
only on the velocities. Hence:
\begin{align*}
E = &\sum N_{1}\, \frac{1}{2} m(\xi_{1}^{2} + \eta_{1}^{2} + \zeta_{1}^{2}) + E_{0}\\
  = N&\sum w_{1}\, \frac{1}{2} m(\xi_{1}^{2} + \eta_{1}^{2} + \zeta_{1}^{2}) + E_{0},
\Tag{(180)}
\end{align*}
where $\xi_{1}\, \eta_{1}\, \zeta_{1}$ denotes any velocity lying within the region element~$1$
and $E_{0}$~denotes the internal energy of the stationary molecules,
%% -----File: 145.png---Folio 129-------
which is assumed constant. In place of the latter expression we
may write, again according to~\Eq{(176)},
\[
E = \frac{m^{4}N}{2G} \int (\xi^{2} + \eta^{2} + \zeta^{2}) w\, d\sigma + E_{0}.
\Tag{(181)}
\]

\Section{130.} Let us consider the state of thermodynamic equilibrium.
According to the second principle of thermodynamics this state
is distinguished from all others by the fact that, for a given volume~$V$
and a given energy~$E$ of the gas, the entropy~$S$ is a maximum.
Let us then regard the volume
\[
V = \iiint dx\, dy\, dz
\Tag{(182)}
\]
and the energy~$E$ of the gas as given. The condition for equilibrium
is $\delta S = 0$, or, according to~\Eq{(179)},
\[
\sum (\log w_{1} + 1)\, \delta w_{1} = 0,
\]
and this holds for any variations of the distribution densities
whatever, provided that, according to \Eq{(167)}~and~\Eq{(180)}, they
satisfy the conditions
\[
\sum \delta w_{1} = 0 \quad\text{and}\quad
\sum (\xi_{1}^{2} + \eta_{1}^{2} + \zeta_{1}^{2})\, \delta w_{1} = 0.
\]
This gives us as the necessary and sufficient condition for thermodynamic
equilibrium for every separate distribution density~$w$:
\[
\log w + \beta(\xi^{2} + \eta^{2} + \zeta^{2}) + \const. = 0
\]
or
\[
w = \alpha e^{-\beta(\xi^{2} + \eta^{2} + \zeta^{2})},
\Tag{(183)}
\]
where $\alpha$~and~$\beta$ are constants. Hence in the state of equilibrium
the distribution of the molecules in space is independent of
$x$,~$y$,~$z$, that is, macroscopically uniform, and the distribution of
velocities is the well-known one of \Name{Maxwell}.

\Section{131.} The values of the constants $\alpha$~and~$\beta$ may be found from
those of $V$~and~$E$. For, on substituting the value of~$w$ just
found in~\Eq{(178)} and taking account of \Eq{(177)}~and~\Eq{(182)}, we get
\[%[** Attn: notation for integrating over space]
\frac{G}{m^{3}}
  = \alpha V \iiint_{-\infty}^{\infty} e^{-\beta(\xi^{2} + \eta^{2} + \zeta^{2})}\, d\xi\, d\eta\, d\zeta
  = \alpha V \left(\frac{\pi}{\beta}\right)^{\efrac{3}{2}},
\]
%% -----File: 146.png---Folio 130-------
and on substituting $w$ in~\Eq{(181)} we get
\[
E = E_{0} + \frac{\alpha m^{4} NV}{2G} \iiint_{-\infty}^{\infty} (\xi^{2} + \eta^{2} + \zeta^{2}) e^{-\beta(\xi^{2} + \eta^{2} + \zeta^{2})}\, d\xi\, d\eta\, d\zeta,
\]
or
\[
E = E_{0} + \frac{3\alpha m^{4} NV}{4G}\, \frac{1}{\beta} \left(\frac{\pi}{\beta}\right)^{\efrac{3}{2}}.
\]
Solving for $\alpha$~and~$\beta$ we have
\begin{align*}
\alpha &= \frac{G}{V} \left(\frac{3N}{4\pi m(E - E_{0})}\right)^{\efrac{3}{2}}
\Tag{(184)} \\
\beta &= \frac{3}{4}\, \frac{Nm}{E - E_{0}}.
\Tag{(185)}
\end{align*}
From this finally we find, as an expression for the entropy~$S$ of
the gas in the state of equilibrium with given values of $N$,~$V$,
and~$E$,
\[
S = kN \log \left\{\frac{V}{G} \left(\frac{4\pi em(E - E_{0})}{3N}\right)^{\efrac{3}{2}}\right\}.
\Tag{(186)}
\]

\Section{132.} This determination of the entropy of an ideal monatomic
gas is based solely on the general connection between entropy and
probability as expressed in equation~\Eq{(164)}; in particular, we have
at no stage of our calculation made use of any special law of the
theory of gases. It is, therefore, of importance to see how the
entire thermodynamic behavior of a monatomic gas, especially
the equation of state and the values of the specific heats, may be
deduced from the expression found for the entropy directly by
means of the principles of thermodynamics. From the general
thermodynamic equation defining the entropy, namely,
\[
dS = \frac{dE + p\, dV}{T},
\Tag{(187)}
\]
the partial differential coefficients of~$S$ with respect to $E$~and~$V$
are found to be
\[
\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{1}{T}, \quad
\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{p}{T}\Add{.}
\]
%% -----File: 147.png---Folio 131-------
Hence, by using~\Eq{(186)}, we get for our gas
\[
\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{3}{2}\, \frac{kN}{E - E_{0}} = \frac{1}{T}
\Tag{(188)}
\]
and
\[
\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{kN}{V} = \frac{p}{T}.
\Tag{(189)}
\]
The second of these equations
\[
p = \frac{kNT}{V}
\Tag{(190)}
\]
contains the laws of \Name{Boyle}, \Name{Gay Lussac}, and \Name{Avogadro}, the last
named because the pressure depends only on the number~$N$, not
on the nature of the molecules. If we write it in the customary
form:
\[
p = \frac{RnT}{V},
\Tag{(191)}
\]
where $n$~denotes the number of gram molecules or mols of the gas,
referred to $O_{2} = 32\, \DPchg{g}{\gr.}$, and $R$~represents the absolute gas constant
\[
R = 831 × 10^{5}\, \frac{\erg}{\degree},
\Tag{(192)}
\]
we obtain by comparison
\[
k = \frac{Rn}{N}.
\Tag{(193)}
\]
If we now call the ratio of the number of mols to the number of
molecules~$\omega$, or, what is the same thing, the ratio of the mass of a
molecule to that of a mol, $\omega = \dfrac{n}{N}$, we shall have
\[
k = \omega R.
\Tag{(194)}
\]
From this the universal constant~$k$ may be calculated, when $\omega$~is
given, and \textit{vice versa}. According to~\Eq{(190)} this constant~$k$ is
nothing but the absolute gas constant, if it is referred to molecules
instead of mols.

From equation~\Eq{(188)}
\[
E - E_{0} = \tfrac{3}{2} kNT.
\Tag{(195)}
\]
%% -----File: 148.png---Folio 132-------
Now, since the energy of an ideal gas is also given by
\[
E = Anc_{v}T + E_{0}
\Tag{(196)}
\]
where $c_{v}$~is the heat capacity of a mol at constant volume in
calories and $A$~is the mechanical equivalent of heat:
\[
A = 419 × 10^{5}\, \frac{\erg}{\cal}
\Tag{(197)}
\]
it follows that
\[
c_{v} = \frac{3}{2}\, \frac{kN}{An}
\]
and further, by taking account of~\Eq{(193)}
\[
c_{v} = \frac{3}{2}\, \frac{R}{A}
  = \frac{3}{2}\DPchg{\,}{·} \frac{831 × 10^{5}}{419 × 10^{5}} = 3.0
\Tag{(198)}
\]
as an expression for the heat capacity per mol of any monatomic
gas at constant volume in calories.\footnote
  {Compare \Name{F.~Richarz}, Wiedemann's Annal., \textbf{67}, p.~705, 1899.}

For the heat capacity per mol at constant pressure,~$c_{p}$, we
have as a consequence of the first principle of thermodynamics:
\[
c_{p} - c_{v} = \frac{R}{A}
\]
and hence by~\Eq{(198)}
\[
c_{p} = \frac{5}{2}\, \frac{R}{A},\qquad
\frac{c_{p}}{c_{v}} = \frac{5}{3},
\Tag{(199)}
\]
as is known to be the case for monatomic gases. It follows from~\Eq{(195)}
that the kinetic energy~$L$ of the gas molecules is equal to
\[
L = E - E_{0} = \DPchg{\frac{3}{2}}{\tfrac{3}{2}} NkT\Add{.}
\Tag{(200)}
\]

\Section{133.} The preceding relations, obtained simply by identifying
the mechanical expression of the entropy~\Eq{(186)} with its thermodynamic
expression~\Eq{(187)}, show the usefulness of the theory
developed. In them an additive constant in the expression for
the entropy is immaterial and hence the size~$G$ of the region element
of probability does not matter. The hypothesis of quanta,
however, goes further, since it fixes the absolute value of the
entropy and thus leads to the same conclusion as the heat theorem
%% -----File: 149.png---Folio 133-------
of \Name{Nernst}. According to this theorem the ``characteristic function''
of an ideal gas\footnote
  {\Eg, \Name{M.~Planck}, Vorlesungen über Thermodynamik, Leipzig, Veit und Comp., 1911,
  Sec.~287, equation~267.}
is in our notation
\[
\Phi = S - \frac{E + pV}{T} = n\left(Ac_{p} \log T - R \log p + a - \frac{b}{T}\right),
\]
where $a$~denotes \Name{Nernst's} chemical constant, and $b$~the energy
constant.

On the other hand, the preceding formulæ \Eq{(186)},~\Eq{(188)}, and~\Eq{(189)}
give for the same function~$\Phi$ the following expression:
\[
\Phi = N\left(\frac{5}{2} k \log T - k \log p + a'\right) - \frac{E_{0}}{T}
\]
where for brevity $a'$~is put for:
\[
a' = k \log \left\{\frac{kN}{eG} (2\pi mk)^{\efrac{3}{2}}\right\}\Add{.}
\]

From a comparison of the two expressions for~$\Phi$ it is seen, by
taking account of \Eq{(199)}~and~\Eq{(193)}, that they agree completely,
provided
\[
% [** TN: Set on two lines in the original]
a = \frac{N}{n} a'
  = R \log \left\{\frac{Nk^{\efrac{5}{2}}}{eG} (2\pi m)^{\efrac{3}{2}}\right\},
\qquad
b = \frac{E_{0}}{n}\Add{.}
\Tag{(201)}
\]
This expresses the relation between the chemical constant~$a$ of
the gas and the region element~$G$ of the probability.\footnote
  {Compare also \Name{O.~Sackur}, Annal.\ d.\ Physik\DPchg{,}{} \textbf{36}, p.~958, 1911, Nernst-Festschrift, p.~405,
  1912, and \Name{H.~Tetrode}, Annal.\ d.\ Physik\DPchg{,}{} \textbf{38}, p.~434, 1912.}

It is seen that $G$~is proportional to the total number,~$N$, of the
molecules. Hence, if we put $G = Ng$, we see that $g$, the molecular
region element, depends only on the chemical nature of the gas.

Obviously the quantity~$g$ must be closely connected with the
law, so far unknown, according to which the molecules act microscopically
on one another. Whether the value of~$g$ varies with
the nature of the molecules or whether it is the same for all
kinds of molecules, may be left undecided for the present.
%% -----File: 150.png---Folio 134-------

If $g$~were known, \Name{Nernst's} chemical constant,~$a$, of the gas
could be calculated from~\Eq{(201)} and the theory could thus be
tested. For the present the reverse only is feasible, namely, to
calculate $g$ from~$a$. For it is known that $a$~may be measured
directly by the tension of the saturated vapor, which at sufficiently
low temperatures satisfies the simple equation\footnote
  {\Name{M.~Planck}, \lc, Sec.~288, equation~271.}
\[
\log p = \frac{5}{2} \log T - \frac{Ar_{0}}{RT} + \frac{a}{R}
\Tag{(202)}
\]
(where $r_{0}$ is the heat of vaporization of a mol at~$0°$ in calories).
When $a$~has been found by measurement, the size~$g$ of the molecular
region element is found from~\Eq{(201)} to be
\[
g = (2\pi m)^{\efrac{3}{2}} k^{\efrac{5}{2}} e^{-\efrac{a}{R} - 1}\Add{.}
\Tag{(203)}
\]
Let us consider the dimensions of~$g$.

According to~\Eq{(176)} $g$~is of the dimensions $[\erg^{3}\, \sec^{3}]$. The
same follows from the present equation, when we consider that the
dimension of the chemical constant~$a$ is not, as might at first be
thought, that of~$R$, but, according to~\Eq{(202)}, that of $R \log \dfrac{p}{T^{\efrac{5}{2}}}$\Add{.}

\Section{134.} To this we may at once add another quantitative relation.
All the preceding calculations rest on the assumption that
the distribution density~$w$ and hence also the constant~$\alpha$ in~\Eq{(183)}
are small (\Sec{129}). Hence, if we take the value of~$\alpha$
from~\Eq{(184)} and take account of \Eq{(188)},~\Eq{(189)} and~\Eq{(201)}, it follows
that
\[
\frac{p}{T^{\efrac{5}{2}}} e^{-\efrac{a}{R} - 1} \quad\text{must be small.}
\]
When this relation is not satisfied, the gas cannot be in the ideal
state. For the saturated vapor it follows then from~\Eq{(202)} that
$e^{-\efrac{Ar_{0}}{RT}}$ is small. In order, then, that a saturated vapor may be
assumed to be in the state of an ideal gas, the temperature~$T$
must certainly be less than $\dfrac{A}{R}r_{0}$ or~$\dfrac{r_{0}}{2}$. Such a restriction is unknown
to the classical thermodynamics.
%% -----File: 151.png---Folio 135-------

\Chapter{III}{Ideal Linear Oscillators}

\Section{135.} The main problem of the theory of heat radiation is to
determine the energy distribution in the normal spectrum of
black radiation, or, what amounts to the same thing, to find the
function which has been left undetermined in the general expression
of \Name{Wien's} displacement law~\Eq{(119)}, the function which connects
the entropy of a certain radiation with its energy. The
purpose of this chapter is to develop some preliminary theorems
leading to this solution. Now since, as we have seen in \Sec{48},
the normal energy distribution in a diathermanous medium cannot
be established unless the medium exchanges radiation with
an emitting and absorbing substance, it will be necessary for the
treatment of this problem to consider more closely the processes
which cause the creation and the destruction of heat rays, that is,
the processes of emission and absorption. In view of the complexity
of these processes and the difficulty of acquiring knowledge of
any definite details regarding them, it would indeed be quite
hopeless to expect to gain any certain results in this way, if it
were not possible to use as a reliable guide in this obscure region
the law of \Name{Kirchhoff} derived in \Sec{51}. This law states that a
vacuum completely enclosed by reflecting walls, in which any
emitting and absorbing bodies are scattered in any arrangement
whatever, assumes in the course of time the stationary state of
black radiation, which is completely determined by one parameter
only, namely, the temperature, and in particular does not
depend on the number, the nature, and the arrangement of the
material bodies present. Hence, for the investigation of the
properties of the state of black radiation the nature of the bodies
which are assumed to be in the vacuum is perfectly immaterial.
In fact, it does not even matter whether such bodies really exist
somewhere in nature, provided their existence and their properties
are consistent with the laws of thermodynamics and electrodynamics.
%% -----File: 152.png---Folio 136-------
If, for any special arbitrary assumption regarding the
nature and arrangement of emitting and absorbing systems, we
can find a state of radiation in the surrounding vacuum which is
distinguished by absolute stability, this state can be no other
than that of black radiation.

Since, according to this law, we are free to choose any system
whatever, we now select from all possible emitting and absorbing
systems the simplest conceivable one, namely, one consisting
of a large number~$N$ of similar stationary oscillators, each consisting
of two poles, charged with equal quantities of electricity of
opposite sign, which may move relatively to each other on a fixed
straight line, the axis of the oscillator.

It is true that it would be more general and in closer accord with
the conditions in nature to assume the vibrations to be those of an
oscillator consisting of two poles, each of which has three degrees
of freedom of motion instead of one, \ie, to assume the vibrations
as taking place in space instead of in a straight line only. Nevertheless
we may, according to the fundamental principle stated
above, restrict ourselves from the beginning to the treatment of
one single component, without fear of any essential loss of
generality of the conclusions we have in view.

It might, however, be questioned as a matter of principle,
whether it is really permissible to think of the centers of mass
of the oscillators as stationary, since, according to the kinetic
theory of gases, all material particles which are contained in
substances of finite temperature and free to move possess a certain
finite mean kinetic energy of translatory motion. This
objection, however, may also be removed by the consideration
that the velocity is not fixed by the kinetic energy alone. We
need only think of an oscillator as being loaded, say at its positive
pole, with a comparatively large inert mass, which is perfectly
neutral electrodynamically, in order to decrease its velocity for a
given kinetic energy below any preassigned value whatever. Of
course this consideration remains valid also, if, as is now frequently
done, all inertia is reduced to electrodynamic action. For this
action is at any rate of a kind quite different from the one to be
considered in the following, and hence cannot influence it.

Let the state of such an oscillator be completely determined
by its moment~$f(t)$, that is, by the product of the electric charge
%% -----File: 153.png---Folio 137-------
of the pole situated on the positive side of the axis and the pole
distance, and by the derivative of~$f$ with respect to the time or
\[
\frac{df(t)}{dt} = \dot{f}(t).
\Tag{(204)}
\]
Let the energy of the oscillator be of the following simple form:
\[
U = \tfrac{1}{2} Kf^{2} + \tfrac{1}{2} L \dot{f}^{2},
\Tag{(205)}
\]
where $K$~and~$L$ denote positive constants, which depend on the
nature of the oscillator in some way that need not be discussed
at this point.

If during its vibration an oscillator neither absorbed nor
emitted any energy, its energy of vibration,~$U$, would remain
constant, and we would have:
\[
dU = Kf\, df + L \dot{f}\, d\dot{f} = 0,
\Tag{(205a)}
\]
or, on account of~\Eq{(204)},
\[
Kf(t) + L \ddot{f}(t) = 0.
\Tag{(206)}
\]
The general solution of this differential equation is found to be a
purely periodical vibration:
\[
f = C \cos(2\pi \nu t - \theta)
\Tag{(207)}
\]
where $C$~and~$\theta$ denote the integration constants and $\nu$~the number
of vibrations per unit time:
\[
\nu = \frac{1}{2\pi} \sqrt{\frac{K}{L}}\Add{.}
\Tag{(208)}
\]

\Section{136.} If now the assumed system of oscillators is in a space
traversed by heat rays, the energy of vibration,~$U$, of an oscillator
will not in general remain constant, but will be always changing
by absorption and emission of energy. Without, for the present,
considering in detail the laws to which these processes are subject,
let us consider any one arbitrarily given thermodynamic state
of the oscillators and calculate its entropy, irrespective of the
surrounding field of radiation. In doing this we proceed entirely
according to the principle advanced in the two preceding chapters,
allowing, however, at every stage for the conditions caused by
the peculiarities of the case in question.

The first question is: What determines the thermodynamic
state of the system considered? For this purpose, according to
%% -----File: 154.png---Folio 138-------
\Sec{124}, the numbers $N_{1}$,~$N_{2}$, $N_{3},~\dots$ of the oscillators,
which lie in the region elements $1$,~$2$, $3,~\dots$ of the ``state
space'' must be given. The state space of an oscillator contains
those coordinates which determine the microscopic state of an
oscillator. In the case in question these are only two in number,
namely, the moment~$f$ and the rate at which it varies,~$\dot{f}$, or instead
of the latter the quantity
\[
\psi = L \dot{f},
\Tag{(209)}
\]
which is of the dimensions of an impulse. The region element
of the state plane is, according to the hypothesis of quanta
(\Sec{126}), the double integral
\[
\iint df\, d\psi = h.
\Tag{(210)}
\]
The quantity~$h$ is the same for all region elements. \textit{A~priori},
it might, however, depend also on the nature of the system considered,
for example, on the frequency of the oscillators. The
following simple consideration, however, leads to the assumption
that $h$~is a universal constant. We know from the generalized
displacement law of \Name{Wien} (equation~\DPtypo{119}{\Eq{(119)}}) that in the universal
function, which gives the entropy radiation as dependent on the
energy radiation, there must appear a universal constant of the
dimension $\dfrac{c^{3} \ssfu}{\nu^{3}}$ and this is of the dimension of a quantity of action\footnote
  {The quantity from which the principle of \emph{least action} takes its name. (Tr.)}
($\erg\, \sec.$). Now, according to~\Eq{(210)}, the quantity~$h$ has precisely
this dimension, on which account we may denote it as ``element
of action'' or ``quantity element of action.'' Hence, unless a
second constant also enters, $h$~cannot depend on any other physical
quantities.

\Section{137.} The principal difference, compared with the calculations
for an ideal gas in the preceding chapter, lies in the fact that we
do not now assume the distribution densities $w_{1}$,~$w_{2}$, $w_{3}~\dots$
of the oscillators among the separate region elements to vary but
little from region to region as was assumed in \Sec{129}. Accordingly
the $w$'s are not small, but finite proper fractions, and the
summation over the region elements cannot be written as an
integration.
%% -----File: 155.png---Folio 139-------

In the first place, as regards the shape of the region elements,
the fact that in the case of undisturbed vibrations of an oscillator
the phase is always changing, whereas the amplitude remains
constant, leads to the conclusion that, for the macroscopic state
of the oscillators, the amplitudes only, not the phases, must be
considered, or in other words the region elements in the $f\psi$~plane
are bounded by the curves $C = \const.$, that is, by ellipses, since
from \Eq{(207)}~and~\Eq{(209)}
\[
\left(\frac{f}{C}\right)^{2} + \left(\frac{\psi}{2\pi\nu LC}\right)^{2} = 1.
\Tag{(211)}
\]

The semi-axes of such an ellipse are:
\[
a = C \quad\text{and}\quad
b = 2\pi\nu LC.
\Tag{(212)}
\]
Accordingly the region elements $1$,~$2$, $3,~\dots$ $n~\dots$\DPnote{** regularize comas?}
are the concentric, similar, and similarly situated elliptic rings,
which are determined by the increasing values of~$C$:
\[
0,\ C_{1},\ C_{2},\ C_{3},\ \dots\ C_{n-1},\ C_{n}\ \dots.
\Tag{(213)}
\]
{\Loosen The $n$th~region element is that which is bounded by the ellipses
$C = C_{n-1}$ and $C = C_{n}$. The first region element is the full
ellipse~$C_{1}$. All these rings have the same area~$h$, which is found
by subtracting the area of the full ellipse~$C_{n-1}$ from that of the
full ellipse~$C_{n}$; hence}
\[
h = (a_{n}b_{n} - a_{n-1}b_{n-1})\pi
\]
or, according to~\Eq{(212)},
\[
h = (C_{n}^{2} - C_{n-1}^{2}) 2\pi^{2}\nu L,
\]
where $n = 1$, $2$, $3,~\dots$.

From the additional fact that $C_{0} = 0$, it follows that:
\[
C_{n}^{2} = \frac{nh}{2\pi^{2}\nu L}.
\Tag{(214)}
\]
Thus the semi-axes of the bounding ellipses are in the ratio of
the square roots of the integral numbers.

\Section{138.} The thermodynamic state of the system of oscillators
is fixed by the fact that the values of the distribution densities
$w_{1}$,~$w_{2}$, $w_{3},~\dots$ of the oscillators among the separate
region elements are given. \emph{Within} a region element the distribution
of the oscillators is according to the law of elemental
chaos (\Sec{122}), \ie,~it is approximately \emph{uniform}.
%% -----File: 156.png---Folio 140-------

These data suffice for calculating the entropy~$S$ as well as the
energy~$E$ of the system in the given state, the former quantity
directly from~\Eq{(173)}, the latter by the aid of~\Eq{(205)}. It must be
kept in mind in the calculation that, since the energy varies
appreciably within a region element, the energy~$E_{n}$ of all those
oscillators which lie in the $n$th~region element is to be found by an
integration. Then the whole energy~$E$ of the system is:
\[
E = E_{1} + E_{2} + \dots E_{n} + \dots.
\Tag{(215)}
\]
$E_{n}$~may be calculated with the help of the law that within every
region element the oscillators are uniformly distributed. If the
$n$th~region element contains, all told, $N_{n}$~oscillators, there are per
unit area $\dfrac{N_{n}}{h}$~oscillators and hence $\dfrac{N_{n}}{h}\, df · d\psi$ per element of area.
Hence we have:
\[
E_{n} = \frac{N_{n}}{h} \iint U\, df\, d\psi.
\]
In performing the integration, instead of $f$~and~$\psi$ we take $C$~and~$\phi$,
as new variables, and since according to~\Eq{(211)},
\[
f = C \cos\phi\qquad \psi = 2\pi\nu LC \sin\phi
\Tag{(216)}
\]
we get:
\[
E_{n} = 2\pi \nu L \frac{N_{n}}{h} \iint U C\, dC\, d\phi
\]
to be integrated with respect to~$\phi$ from $0$ to~$2\pi$ and with respect
to~$C$ from $C_{n-1}$ to~$C_{n}$. If we substitute from \Eq{(205)},~\Eq{(209)}
and~\Eq{(216)}
\[
U = \tfrac{1}{2} KC^{2},
\Tag{(217)}
\]
we obtain by integration
\[
E_{n} = \frac{\pi^{2}}{2} \nu LK \frac{N_{n}}{h} (C_{n}^{4} - C_{n-1}^{4})
\]
and from \Eq{(214)}~and~\Eq{(208)}:
\[
E_{n} = N_{n}(n - \tfrac{1}{2})h\nu = Nw_{n}(n - \tfrac{1}{2})h\nu,
%[** TN: No equation number in the original]
\Tag{(218)}
\]
{\Loosen that is, the mean energy of an oscillator in the $n$th~region element
is $(n - \frac{1}{2})h\nu$. This is exactly the arithmetic mean of the energies
$(n - 1)h\nu$ and~$nh\nu$ which correspond to the two ellipses $C = C_{n-1}$
and $C = C_{n}$ bounding the region, as may be seen from~\Eq{(217)}, if
the values of $C_{n-1}$~and~$C_{n}$ are therein substituted from~\Eq{(214)}.}
%% -----File: 157.png---Folio 141-------

The total energy~$E$ is, according to~\Eq{(215)},
\[
E = Nh\nu \sum_{n=1}^{n=\infty} (n - \tfrac{1}{2}) w_{n}.
\Tag{(219)}
\]

\Section{139.} Let us now consider the state of thermodynamic equilibrium
of the oscillators. According to the second principle of
thermodynamics, the entropy~$S$ is in that case a maximum for a
given energy~$E$. Hence we assume $E$ in~\Eq{(219)} as given. Then
from~\Eq{(179)} we have for the state of equilibrium:
\[
\delta S = 0 = \sum_{1}^{\infty} (\log w_{n} + 1)\, \delta w_{n},
\]
where according to \Eq{(167)}~and~\Eq{(219)}
\[
\sum_{1}^{\infty} \delta w_{n} = 0 \quad\text{and}\quad
\sum_{1}^{\infty} (n - \tfrac{1}{2})\, \delta w_{n} = 0\Add{.}
\]
From these relations we find:
\[
\log w_{n} + \beta n + \const. = 0
\]
or
\[
w_{n} = \alpha \gamma^{n}.
\Tag{(220)}
\]
The values of the constants $\alpha$~and~$\gamma$ follow from equations \Eq{(167)}
and~\Eq{(219)}:
\[
\alpha = \frac{2Nh\nu}{2E - Nh\nu}\qquad
\gamma = \frac{2E - Nh\nu}{2E + Nh\nu}.
\Tag{(221)}
\]
Since $w_{n}$~is essentially positive it follows that equilibrium is not
possible in the system of oscillators considered unless the total
energy~$E$ has a greater value than~$\dfrac{Nh\nu}{2}$, that is unless the mean
energy of the oscillators is at least~$\dfrac{h\nu}{2}$. This, according to~\Eq{(218)},
is the mean energy of the oscillators lying in the first
region element. In fact, in this extreme case all $N$ oscillators
lie in the first region element, the region of smallest energy;
within this element they are arranged uniformly.

The entropy~$S$ of the system, which is in thermodynamic
equilibrium, is found by combining \Eq{(173)} with \Eq{(220)}~and~\Eq{(221)}
\begin{multline*}
S = kN\biggl\{\left(\frac{E}{Nh\nu} + \frac{1}{2}\right) \log\left(\frac{E}{Nh\nu} + \frac{1}{2}\right) \\
           - \left(\frac{E}{Nh\nu} - \frac{1}{2}\right) \log\left(\frac{E}{Nh\nu} - \frac{1}{2}\right)\biggr\}\Add{.}
\Tag{(222)}
\end{multline*}
%% -----File: 158.png---Folio 142-------

\Section{140.} The connection between energy and entropy just obtained
allows furthermore a certain conclusion as regards the temperature.
For from the equation of the second principle of thermodynamics,
$dS = \dfrac{dE}{T}$ and from differentiation of~\Eq{(222)} with respect
to~$E$ it follows that
\[
E = N \frac{h\nu}{2}\, \frac{1 + e^{-\efrac{h\nu}{kT}}}{1 - e^{-\efrac{h\nu}{kT}}}
  = Nh\nu \left(\frac{1}{2} + \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\right)\Add{.}
%[** TN: No equation number in the original]
\Tag{(223)}
\]
Hence, for the zero point of the absolute temperature~$E$ becomes,
not~$0$, but~$N\dfrac{h\nu}{2}$. This is the extreme case discussed in the preceding
paragraph, which just allows thermodynamic equilibrium
to exist. That the oscillators are said to perform vibrations even
at the temperature zero, the mean energy of which is as large as~$\dfrac{h\nu}{2}$
and hence may become quite large for rapid vibrations, may
at first sight seem strange. It seems to me, however, that certain
facts point to the existence, inside the atoms, of vibrations
independent of the temperature and supplied with appreciable
energy, which need only a small suitable excitation to become
evident externally. For example, the velocity, sometimes very
large, of secondary cathode rays produced by Roentgen rays,
and that of electrons liberated by photoelectric effect are independent
of the temperature of the metal and of the intensity of
the exciting radiation. Moreover the radioactive energies are
also independent of the temperature. It is also well known that
the close connection between the inertia of matter and its energy
as postulated by the relativity principle leads to the assumption
of very appreciable quantities of intra-atomic energy even at the
zero of absolute temperature.

For the extreme case, $T = \infty$, we find from~\Eq{(223)} that
\[
E = NkT,
\Tag{(224)}
\]
\ie, the energy is proportional to the temperature and independent
of the size of the quantum of action,~$h$, and of the nature of
the oscillators. It is of interest to compare this value of the
energy of vibration~$E$ of the system of oscillators, which holds at
high temperatures, with the kinetic energy~$L$ of the molecular
%% -----File: 159.png---Folio 143-------
motion of an ideal monatomic gas at the same temperature as
calculated in~\Eq{(200)}. From the comparison it follows that
\[
E = \tfrac{2}{3} L\Add{.}
\Tag{(225)}
\]
This simple relation is caused by the fact that for high temperatures
the contents of the hypothesis of quanta coincide with
those of the classical statistical mechanics. Then the absolute
magnitude of the region element, $G$~or $h$ respectively, becomes
physically unimportant (compare \Sec{125}) and we have the
simple law of equipartition of the energy among all variables in
question (see below \Sec{169}). The factor~$\frac{2}{3}$ in equation~\Eq{(225)}
is due to the fact that the kinetic energy of a moving molecule
depends on three variables ($\xi$,~$\eta$,~$\zeta$,) and the energy of a vibrating
oscillator on only two ($f$,~$\psi$).

The heat capacity of the system of oscillators in question is,
from~\Eq{(223)},
\[
\frac{dE}{dT} = Nk\left(\frac{h\nu}{kT}\right)^{2} \frac{e^{\efrac{h\nu}{kT}}}{(e^{\efrac{h\nu}{kT}} - 1)^{2}}\Add{.}
\Tag{(226)}
\]
It vanishes for $T = 0$ and becomes equal to~$Nk$ for $T = \infty$.
\Name{A.~Einstein}\footnote
  {\Name{A.~Einstein}, Ann.\ d.\ Phys.\ \textbf{22}, p.~180, 1907. Compare also \Name{M.~Born} und \Name{Th.\ von~Kárman},
  Phys.\ Zeitschr.\ \textbf{13}, p.~297, 1912.}
has made an important application of this equation
to the heat capacity of solid bodies, but a closer discussion of
this would be beyond the scope of the investigations to be made
in this book.

For the constants $\alpha$~and~$\gamma$ in the expression~\Eq{(220)} for the distribution
density~$w$ we find from~\Eq{(221)}:
\[
\alpha = e^{\efrac{h\nu}{kT}} - 1\qquad
\gamma = e^{-\efrac{h\nu}{kT}}
\Tag{(227)}
\]
and finally for the entropy~$S$ of our system as a function of temperature:
\[
S = kN \left\{\frac{\dfrac{h\nu}{kT}}{e^{\efrac{h\nu}{kT}-1}} - \log\left(1 - e^{-\efrac{h\nu}{kT}}\right)\right\}\Add{.}
\Tag{(228)}
\]
%% -----File: 160.png---Folio 144-------

\Chapter[Direct Calculation of the Entropy]
{IV}{Direct Calculation of the Entropy in The
Case of Thermodynamic Equilibrium}

\Section{141.} In the calculation of the entropy of an ideal gas and of a
system of resonators, as carried out in the preceding chapters, we
proceeded in both cases, by first determining the entropy for an
arbitrarily given state, then introducing the special condition of
thermodynamic equilibrium, \ie, of the maximum of entropy,
and then deducing for this special case an expression for the
entropy.

If the problem is only the determination of the entropy in the
case of thermodynamic equilibrium, this method is a roundabout
one, inasmuch as it requires a number of calculations, namely,
the determination of the separate distribution densities $w_{1}$,~$w_{2}$,
$w_{3},~\dots$ which do not enter separately into the final
result. It is therefore useful to have a method which leads
directly to the expression for the \emph{entropy} of a system in the state
of thermodynamic equilibrium, without requiring any consideration
of the \emph{state} of thermodynamic equilibrium. This method
is based on an important general property of the thermodynamic
probability of a state of equilibrium.

We know that there exists between the entropy~$S$ and the thermodynamic
probability~$W$ in any state whatever the general
relation~\Eq{(164)}. In the state of thermodynamic equilibrium both
quantities have maximum values; hence, if we denote the maximum
values by a suitable index:
\[
S_{m} = k \log W_{m}.
\Tag{(229)}
\]
It follows from the two equations that:
\[
\frac{W_{m}}{W} = e^{\efrac{S_{m} - S}{k}}\Add{.}
\]
Now, when the deviation from thermodynamic equilibrium is at
all appreciable, $\dfrac{S_{m} - S}{k}$ is certainly a very large number. Accordingly
%% -----File: 161.png---Folio 145-------
$W_{m}$~is not only large but of a very high order large, compared
with~$W$, that is to say: The thermodynamic probability
of the state of equilibrium is enormously large compared with the
thermodynamic probability of all states which, in the course of
time, change into the state of equilibrium.

This proposition leads to the possibility of calculating~$W_{m}$
with an accuracy quite sufficient for the determination of~$S_{m}$,
without the necessity of introducing the special condition of
equilibrium. According to \Sec{123}, \textit{et~seq.}, $W_{m}$~is equal to the
number of all different complexions possible in the state of thermodynamic
equilibrium. This number is so enormously large compared
with the number of complexions of all states deviating from
equilibrium that we commit no appreciable error if we think of
the number of complexions of all states, which as time goes on
change into the state of equilibrium, \ie,~all states which are at
all possible under the given external conditions, as being included
in this number. The total number of all possible complexions
may be calculated much more readily and directly than the
number of complexions referring to the state of equilibrium only.

\Section{142.} We shall now use the method just formulated to calculate
the entropy, in the state of equilibrium, of the system of ideal
linear oscillators considered in the last chapter, when the total
energy~$E$ is given. The notation remains the same as above.

We put then $W_{m}$ equal to the number of complexions of all
stages which are at all possible with the given energy~$E$ of the
system. Then according to~\Eq{(219)} we have the condition:
\[
E = h\nu \sum_{n=1}^{\infty} (n - \tfrac{1}{2}) N_{n}.
\Tag{(230)}
\]
Whereas we have so far been dealing with the number of complexions
with given~$N_{n}$, now the~$N_{n}$ are also to be varied in all ways
consistent with the condition~\Eq{(230)}.

The total number of all complexions is obtained in a simple
way by the following consideration. We write, according to~\Eq{(165)},
the condition~\Eq{(230)} in the following form:
\[
\frac{E}{h\nu} - \frac{N}{2} = \sum_{n=1}^{\infty} (n - 1) N_{n}
\]
%% -----File: 162.png---Folio 146-------
or
\[
%[** TN: Set on two lines in the original]
0 · N_{1} + 1 · N_{2} + 2 · N_{3} + \dots + (n - 1)N_{n} + \dots
  = \frac{E}{h\nu} - \frac{N}{2} = P.
\Tag{(231)}
\]
$P$~is a given large positive number, which may, without
restricting the generality, be taken as an integer.

According to \Sec{123} a complexion is a definite assignment of
every individual oscillator to a definite region element $1$,~$2$,
$3,~\dots$ of the state plane $(f, \psi)$. Hence we may characterize
a certain complexion by thinking of the $N$~oscillators as
being numbered from $1$ to~$N$ and, when an oscillator is assigned
to the $n$th~region element, writing down the number of the
oscillator $(n - 1)$~times. If in any complexion an oscillator is
assigned to the first region element its number is not put down at
all. Thus every complexion gives a certain row of figures, and
\textit{vice versa} to every row of figures there corresponds a certain complexion.
The position of the figures in the row is immaterial.

What makes this form of representation useful is the fact that
according to~\Eq{(231)} the number of figures in such a row is always
equal to~$P$. Hence we have ``combinations with repetitions of
$N$~elements taken $P$~at a time,'' whose total number is
\[%[** Attn alignment of factors]
\frac{N(N + 1)(N + 2) \dots (N + P - 1)}
  {\PadTo{N}{1} \PadTo{(N+1)}{2} \PadTo{(N+2)}{3} \dots \PadTo{(N + P - 1)}{P}}
  = \frac{(N + P - 1)!}{(N - 1)!\, P!}\Add{.}
\Tag{(232)}
\]
If for example we had $N = 3$ and $P = 4$ all possible complexions
would be represented by the rows of figures:
\[
\begin{array}{c*{2}{>{\qquad}c}}
1111 & 1133  &  2222 \\
1112 & 1222  &  2223 \\
1113 & 1223  &  2233 \\
1122 & 1233  &  2333 \\
1123 & 1333  &  3333
\end{array}
\]

The first row denotes that complexion in which the first oscillator
lies in the $5$th~region element and the two others in the first.
The number of complexions in this case is~$15$, in agreement with
the formula.

\Section{143.} For the entropy~$S$ of the system of oscillators which is
%% -----File: 163.png---Folio 147-------
in the state of thermodynamic equilibrium we thus obtain from
equation~\Eq{(229)} since $N$~and~$P$ are large numbers:
\[
S = k \log\frac{(N + P)!}{N!\, P!}
\]
and by making use of Stirling's formula~\Eq{(171)}\footnote
  {Compare footnote to \PageRef{page}{124}. See also \PageRef{page}{218}.}
\[
S = kN \left\{\left(\frac{P}{N} + 1\right) \log\left(\frac{P}{N} + 1\right) - \frac{P}{N} \log \frac{P}{N}\right\}.
\]
If we now replace $P$ by~$E$ from~\Eq{(231)} we find for the entropy
exactly the same value as given by~\Eq{(222)} and thus we have
demonstrated in a special case both the admissibility and the
practical usefulness of the method employed.\footnote
  {A complete mathematical discussion of the subject of this chapter has been given by
  \Name{H.~A. Lorentz}. Compare, \eg, Nature, \textbf{92}, p.~305, Nov.~6, 1913. (Tr.)}
%% -----File: 164.png---Folio 148-------
%[Blank Page]
%% -----File: 165.png---Folio 149-------

\Part[A System of Oscillators]{IV}{A System of Oscillators in a Stationary
Field of Radiation}
%% -----File: 166.png---Folio 150-------
%[Blank Page]
%% -----File: 167.png---Folio 151-------

\Chapter[The Elementary Dynamical Law]
{I}{The Elementary Dynamical Law for The
Vibrations of an Ideal Oscillator.
Hypothesis of Emission of Quanta}

\Section{144.} All that precedes has been by way of preparation. Before
taking the final step, which will lead to the law of distribution of
energy in the spectrum of black radiation, let us briefly put
together the essentials of the problem still to be solved. As we
have already seen in \Sec{93}, the whole problem amounts to the
determination of the temperature corresponding to a monochromatic
radiation of given intensity. For among all conceivable
distributions of energy the normal one, that is, the one
peculiar to black radiation, is characterized by the fact that in it
the rays of all frequencies have the same temperature. But the
temperature of a radiation cannot be determined unless it be
brought into thermodynamic equilibrium with a system of molecules
or oscillators, the temperature of which is known from other
sources. For if we did not consider any emitting and absorbing
matter there would be no possibility of defining the entropy and
temperature of the radiation, and the simple propagation of free
radiation would be a reversible process, in which the entropy and
temperature of the separate pencils would not undergo any
change. (Compare below \Sec{166}.)

Now we have deduced in the preceding section all the characteristic
properties of the thermodynamic equilibrium of a system
of ideal oscillators. Hence, if we succeed in indicating a state of
radiation which is in thermodynamic equilibrium with the system
of oscillators, the temperature of the radiation can be no other
than that of the oscillators, and therewith the problem is solved.

\Section{145.} Accordingly we now return to the considerations of \Sec{135}
and assume a system of ideal linear oscillators in a stationary
field of radiation. In order to make progress along the line
proposed, it is necessary to know the elementary dynamical law,
%% -----File: 168.png---Folio 152-------
according to which the mutual action between an oscillator and the
incident radiation takes place, and it is moreover easy to see that
this law cannot be the same as the one which the classical electrodynamical
theory postulates for the vibrations of a linear Hertzian
oscillator. For, according to this law, all the oscillators, when
placed in a stationary field of radiation, would, since their
properties are exactly similar, assume the same energy of vibration,
if we disregard certain irregular variations, which, however,
will be smaller, the smaller we assume the damping constant of
the oscillators, that is, the more pronounced their natural vibration
is. This, however, is in direct contradiction to the
definite discrete values of the distribution densities $w_{1}$,~$w_{2}$,
$w_{3},~\dots$ which we have found in \Sec{139} for the stationary
state of the system of oscillators. The latter allows us to conclude
with certainty that in the dynamical law to be established the
quantity element of action~$h$ must play a characteristic part.
Of what nature this will be cannot be predicted \textit{a~priori}; this much,
however, is certain, that the only type of dynamical law admissible
is one that will give for the stationary state of the oscillators
exactly the distribution densities~$w$ calculated previously. It is in
this problem that the question of the dynamical significance of the
quantum of action~$h$ stands for the first time in the foreground,
a question the answer to which was unnecessary for the calculations
of the preceding sections, and this is the principal reason
why in our treatment the preceding section was taken up first.

\Section{146.} In establishing the dynamical law, it will be rational to
proceed in such a way as to make the deviation from the laws of
classical electrodynamics, which was recognized as necessary, as
slight as possible. Hence, as regards the influence of the field of
radiation on an oscillator, we follow that theory closely. If the
oscillator vibrates under the influence of any external electromagnetic
field whatever, its energy~$U$ will not in general remain
constant, but the energy equation~\Eq{(205a)} must be extended to
include the work which the external electromagnetic field does on
the oscillator, and, if the axis of the electric doublet coincides with
the $z$-axis, this work is expressed by the term $\ssfE_{z}\, df = \ssfE_{z} \dot{f}\, dt$.
Here $\ssfE_{z}$~denotes the $z$~component of the external electric field-strength
at the position of the oscillator, that is, that electric
field-strength which would exist at the position of the oscillator,
%% -----File: 169.png---Folio 153-------
if the latter were not there at all. The other components of
the external field have no influence on the vibrations of the
oscillator.

Hence the complete energy equation reads:
\[
Kf\, df + L \dot{f}\, d\dot{f} = \ssfE_{z}\, df
\]
or:
\[
Kf + L \ddot{f} = \ssfE_{z},
\Tag{(233)}
\]
and the energy absorbed by the oscillator during the time element~$dt$
is:
\[
\ssfE_{z} \dot{f}\, dt\Add{.}
\Tag{(234)}
\]

\Section{147.} While the oscillator is absorbing it must also be emitting,
for otherwise a stationary state would be impossible. Now, since
in the law of absorption just assumed the hypothesis of quanta
has as yet found no room, it follows that it must come into play
in some way or other in the emission of the oscillator, and this is
provided for by the introduction of the hypothesis of emission of
quanta. That is to say, we shall assume that the emission does
not take place continuously, as does the absorption, but that it
occurs only at certain definite times, suddenly, in pulses, and in
particular we assume that an oscillator can emit energy only at
the moment when its energy of vibration,~$U$, is an integral multiple~$n$
of the quantum of energy, $\epsilon = h\nu$. Whether it then really
emits or whether its energy of vibration increases further by
absorption will be regarded as a matter of chance. This will not
be regarded as implying that there is no causality for emission;
but the processes which cause the emission will be assumed to be
of such a concealed nature that for the present their laws cannot
be obtained by any but statistical methods. Such an assumption
is not at all foreign to physics; it is, \eg, made in the atomistic
theory of chemical reactions and the disintegration theory of
radioactive substances.

It will be assumed, however, that if emission does take place,
the entire energy of vibration,~$U$, is emitted, so that the vibration
of the oscillator decreases to zero and then increases again by
further absorption of radiant energy.

It now remains to fix the law which gives the probability that
an oscillator will or will not emit at an instant when its energy has
reached an integral multiple of~$\epsilon$. For it is evident that the statistical
state of equilibrium, established in the system of oscillators
%% -----File: 170.png---Folio 154-------
by the assumed alternations of absorption and emission
will depend on this law; and evidently the mean energy~$U$ of the
oscillators will be larger, the larger the probability that in such a
critical state no emission takes place. On the other hand, since
the mean energy~$U$ will be larger, the larger the intensity of the
field of radiation surrounding the oscillators, we shall state the
%[** Thm]
law of emission as follows: \emph{The ratio of the probability that no
emission takes place to the probability that emission does take place
is proportional to the intensity~$\ssfI$} of the vibration which excites the
oscillator and which was defined in equation~\Eq{(158)}. The value
of the constant of proportionality we shall determine later on by
the application of the theory to the special case in which the
energy of vibration is very large. For in this case, as we know,
the familiar formulæ of the classical dynamics hold for any period
of the oscillator whatever, since the quantity element of action~$h$
may then, without any appreciable error, be regarded as infinitely
small.

These statements define completely the way in which the
radiation processes considered take place, as time goes on, and
the properties of the stationary state. We shall now, in the
first place, consider in the second chapter the absorption, and,
then, in the third chapter the emission and the stationary distribution
of energy, and, lastly, in the fourth chapter we shall
compare the stationary state of the system of oscillators thus
found with the thermodynamic state of equilibrium which was
derived directly from the hypothesis of quanta in the preceding
part. If we find them to agree, the hypothesis of emission of
quanta may be regarded as admissible.

It is true that we shall not thereby prove that this hypothesis
represents the only possible or even the most adequate expression
of the elementary dynamical law of the vibrations of the oscillators.
On the contrary I think it very probable that it may be
greatly improved as regards form and contents. There is, however,
no method of testing its admissibility except by the investigation
of its consequences, and as long as no contradiction in
itself or with experiment is discovered in it, and as long as no
more adequate hypothesis can be advanced to replace it, it may
justly claim a certain importance.
%% -----File: 171.png---Folio 155-------

\Chapter{II}{Absorbed Energy}

\Section{148.} Let us consider an oscillator which has just completed an
emission and which has, accordingly, lost all its energy of vibration.
If we reckon the time~$t$ from this instant then for $t = 0$ we
have $f = 0$ and $\DPchg{df/dt}{\dfrac{df}{dt}} = 0$, and the vibration takes place according
to equation~\Eq{(233)}. Let us write~$\ssfE_{z}$ as in~\Eq{(149)} in the form of a
Fourier's series:
\[
\ssfE_{z} = \sum_{n=1}^{n=\infty}
  \left[A_{n} \cos\frac{2\pi nt}{\ssfT}
      + B_{n} \sin\frac{2\pi nt}{\ssfT}\right],
\Tag{(235)}
\]
where $\ssfT$~may be chosen very large, so that for all times~$t$ considered
$t < \ssfT$. Since we assume the radiation to be stationary,
the constant coefficients $A_{n}$~and~$B_{n}$ depend on the ordinal numbers~$n$
in a wholly irregular way, according to the hypothesis of
natural radiation (\Sec{117}). The partial vibration with the
ordinal number~$n$ has the frequency~$\nu$, where
\[
\omega = 2\pi \nu = \frac{2\pi n}{\ssfT},
\Tag{(236)}
\]
while for the frequency~$\nu_{0}$ of the natural period of the oscillator
\[
\omega_{0} = 2\pi \nu_{0} = \sqrt{\frac{K}{L}}.
\]

Taking the initial condition into account, we now obtain as
the solution of the differential equation~\Eq{(233)} the expression
\[
f = \sum_{1}^{\infty}
   [a_{n} (\cos \omega t - \cos \omega_{0} t)
  + b_{n} (\sin \omega t - \frac{\omega}{\omega_{0}} \sin \omega_{0}t)],
\Tag{(237)}
\]
where
\[
a_{n} = \frac{A_{n}}{L(\omega_{0}^{2} - \omega^{2})},\qquad
b_{n} = \frac{B_{n}}{L(\omega_{0}^{2} - \omega^{2})}.
\Tag{(238)}
\]
%% -----File: 172.png---Folio 156-------
This represents the vibration of the oscillator up to the instant
when the next emission occurs.

The coefficients $a_{n}$~and~$b_{n}$ attain their largest values when $\omega$~is
nearly equal to~$\omega_{0}$. (The case $\omega = \omega_{0}$ may be excluded by
assuming at the outset that $\nu_{0} \ssfT$~is not an integer.)

\Section{149.} Let us now calculate the total energy which is absorbed
by the oscillator in the time from $t = 0$ to $t = \tau$, where
\[
\omega_{0} \tau \text{ is large}.
\Tag{(239)}
\]
According to equation~\Eq{(234)}, it is given by the integral
\[
\int_{0}^{\tau} \ssfE_{z} \frac{df}{dt}\, dt,
\Tag{(240)}
\]
the value of which may be obtained from the known expression
for~$\ssfE_{z}$ \Eq{(235)} and from
\[
\frac{df}{dt} = \sum_{1}^{\infty}
    [a_{n}(-\omega \sin \omega t + \omega_{0} \sin \omega_{0} t)
   + b_{n}( \omega \cos \omega t - \omega     \cos \omega_{0} t)].
\Tag{(241)}
\]
By multiplying out, substituting for $a_{n}$~and~$b_{n}$ their values from~\Eq{(238)},
and leaving off all terms resulting from the multiplication
of two constants $A_{n}$ and~$B_{n}$, this gives for the absorbed energy
the following value:
\begin{multline*}
%[** TN: Moved + from end of first line to beginning of second line]
\frac{1}{L} \int_{0}^{\tau} dt \sum_{1}^{\infty}\biggl[
  \frac{A_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \cos \omega t (-\omega \sin \omega t + \omega_{0} \sin \omega_{0} t) \\
  + \frac{B_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \sin \omega t ( \omega \cos \omega t - \omega     \cos \omega_{0} t)\biggr].
\Tag{(241a)}
\end{multline*}
In this expression the integration with respect to~$t$ may be performed
term by term. Substituting the limits $\tau$ and~$0$ it gives
\begin{align*}
  &\frac{1}{L} \sum_{1}^{\infty} \frac{A_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \left[-\frac{\sin^{2} \omega \tau}{2} + \omega_{0} \left(\frac{\sin^{2}\dfrac{\omega_{0} + \omega}{2} \tau}{\omega_{0} + \omega} + \frac{\sin^{2}\dfrac{\omega_{0} - \omega}{2} \tau}{\omega_{0} - \omega}\right)\right]\\
%
+ &\frac{1}{L} \sum_{1}^{\infty} \frac{B_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \left[\Neg \frac{\sin^{2} \omega \tau}{2} - \omega     \left(\frac{\sin^{2}\dfrac{\omega_{0} + \omega}{2} \tau}{\omega_{0} + \omega} - \frac{\sin^{2}\dfrac{\omega_{0} - \omega}{2} \tau}{\omega_{0} - \omega}\right)\right].
\end{align*}
%% -----File: 173.png---Folio 157-------
In order to separate the terms of different order of magnitude, this
expression is to be transformed in such a way that the difference
$\omega_{0} - \omega$ will appear in all terms of the sum. This gives
\begin{multline*}
% [** TN: Reformatted slightly from the original]
\frac{1}{L} \sum_{1}^{\infty} \frac{A_{n}^{2}}{\omega_{0}^{2} - \omega^{2}}
\biggl[\frac{\omega_{0} - \omega}{2(\omega_{0} + \omega)} \sin^{2} \omega \tau
     + \frac{\omega_{0}}{\omega_{0} + \omega} \sin \frac{\omega_{0} - \omega}{2} \tau · \sin \frac{\omega_{0} + 3\omega}{2} \tau \\
     \shoveright{+ \frac{\omega_{0}}{\omega_{0} - \omega} \sin^{2} \frac{\omega_{0} - \omega}{2} \tau \biggr]\phantom{.}} \\
%
+ \frac{1}{L} \sum_{1}^{\infty} \frac{B_{n}^{2}}{\omega_{0}^{2} - \omega^{2}}
\biggl[\frac{\omega_{0} - \omega}{2(\omega_{0} + \omega)} \sin^{2} \omega \tau
     - \frac{\omega}{\omega_{0} + \omega} \sin \frac{\omega_{0} - \omega}{2} \tau · \sin \frac{\omega_{0} + 3\omega}{2} \tau \\
     + \frac{\omega}{\omega_{0} - \omega} \sin^{2} \frac{\omega_{0} - \omega}{2} \tau \biggr].
\end{multline*}
The summation with respect to the ordinal numbers~$n$ of the
Fourier's series may now be performed. Since the fundamental
period~$\ssfT$ of the series is extremely large, there corresponds to
the difference of two consecutive ordinal numbers, $\Delta n = 1$ only
a very small difference of the corresponding values of $\omega$,~$d\omega$,
namely, according to~\Eq{(236)},
\[
\Delta n = 1 = \ssfT\, d\nu = \frac{\ssfT\, d\omega}{2\pi},
\Tag{(242)}
\]
and the summation with respect to~$n$ becomes an integration with
respect to~$\omega$.

The last summation with respect to~$A_{n}$ may be rearranged as
the sum of three series, whose orders of magnitude we shall first
compare. So long as only the order is under discussion we may
disregard the variability of the~$A_{n}^{2}$ and need only compare the
three integrals
\begin{gather*}
\int_{0}^{\infty} d\omega\, \frac{\sin^{2} \omega\tau}{2(\omega_{0} + \omega)^{2}} = J_{1}, \\
\int_{0}^{\infty} d\omega\, \frac{\omega_{0}}{(\omega_{0} + \omega)^{2} (\omega_{0} - \omega)}
  \sin\frac{\omega_{0} - \omega}{2} \tau · \sin\frac{\omega_{0} + 3\omega}{2} \tau = J_{2}, \\
\intertext{and}
\int_{0}^{\infty} d\omega\, \frac{\omega_{0}}{(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}}
  \sin^{2} \frac{\omega_{0} - \omega}{2} \tau = J_{3}.
\end{gather*}
%% -----File: 174.png---Folio 158-------
The evaluation of these integrals is greatly simplified by the fact
that, according to~\Eq{(239)}, $\omega_{0}\tau$~and therefore also~$\omega\tau$ are large numbers,
at least for all values of~$\omega$ which have to be considered.
Hence it is possible to replace the expression $\sin^{2} \omega \tau$ in the integral~$J_{1}$
by its mean value~$\frac{1}{2}$ and thus we obtain:
\[
J_{1} = \frac{1}{4 \omega_{0}}\Add{.}
\]
It is readily seen that, on account of the last factor, we obtain
\[
J_{2} = 0
\]
for the second integral.

In order finally to calculate the third integral~$J_{3}$ we shall lay
off in the series of values of~$\omega$ on both sides of~$\omega_{0}$ an interval
extending from $\omega_{1}$ ($< \omega_{0}$) to $\omega_{2}$ ($> \omega_{0}$) such that
\[
\frac{\omega_{0} - \omega_{1}}{\omega_{0}} \quad\text{and}\quad
\frac{\omega_{2} - \omega_{0}}{\omega_{0}} \quad\text{are small,}
\Tag{(243)}
\]
and simultaneously
\[
(\omega_{0} - \omega_{1}) \tau \quad\text{and}\quad
(\omega_{2} - \omega_{0}) \tau \quad\text{are large.}
\Tag{(244)}
\]
This can always be done, since $\omega_{0}\tau$~is large. If we now break up
the integral~$J_{3}$ into three parts, as follows:
\[
J_{3} = \int_{0}^{\infty}
  = \int_{0}^{\omega_{1}}
  + \int_{\omega_{1}}^{\omega_{2}}
  + \int_{\omega_{2}}^{\infty},
\]
it is seen that in the first and third partial integral the expression
$\sin^{2} \dfrac{\omega_{0} - \omega}{2} \tau$ may, because of the condition~\Eq{(244)}, be replaced by its
mean value~$\frac{1}{2}$. Then the two partial integrals become:
\[
\int_{0}^{\omega_{1}} \frac{\omega_{0}\, d\omega}{2(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}}
\quad\text{and}\quad
\int_{\omega_{2}}^{\infty} \frac{\omega_{0}\, d\omega}{2(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}}.
\Tag{(245)}
\]
These are certainly smaller than the integrals:
\[
\int_{0}^{\omega_{1}} \frac{d\omega}{2(\omega_{0} - \omega)^{2}}
\quad\text{and}\quad
\int_{\omega_{2}}^{\infty} \frac{d\omega}{2(\omega_{0} - \omega)^{2}}
\]
%% -----File: 175.png---Folio 159-------
which have the values
\[
\frac{1}{2}\, \frac{\omega_{1}}{\omega_{0} (\omega_{0} - \omega_{1})}
\quad\text{and}\quad
\frac{1}{2(\omega_{2} - \omega_{0})}
\Tag{(246)}
\]
respectively. We must now consider the middle one of the three
partial integrals:
\[
\int_{\omega_{1}}^{\omega_{2}} d\omega\, \frac{\omega_{0}}{(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}} · \sin^{2} \frac{\omega_{0} - \omega}{2} \tau.
\]
Because of condition~\Eq{(243)} we may write instead of this:
\[
\int_{\omega_{1}}^{\omega_{2}} d\omega · \frac{\sin^{2} \dfrac{\omega_{0} - \omega}{2} \tau}{2(\omega_{0} - \omega)^{2}}
\]
and by introducing the variable of integration~$x$, where
\[
x = \frac{\omega - \omega_{0}}{2} \tau
\]
and taking account of condition~\Eq{(244)} for the limits of the integral,
we get:
\[
\frac{\tau}{4} \int_{-\infty}^{+\infty} \frac{\sin^{2} x\, dx}{x^{2}}
  = \frac{\tau}{4}\, \pi.
\]
This expression is of a higher order of magnitude than the expressions~\Eq{(246)}
and hence of still higher order than the partial integrals~\Eq{(245)}
and the integrals $J_{1}$~and~$J_{2}$ given above. Thus for
our calculation only those values of~$\omega$ will contribute an appreciable
part which lie in the interval between $\omega_{1}$~and~$\omega_{2}$, and hence
we may, because of~\Eq{(243)}, replace the separate coefficients $A_{n}^{2}$~and~$B_{n}^{2}$
in the expression for the total absorbed energy by their mean
values $A_{0}^{2}$~and~$B_{0}^{2}$ in the neighborhood of~$\omega_{0}$ and thus, by taking
account of~\Eq{(242)}, we shall finally obtain for the total value of the
energy absorbed by the oscillator in the time~$\tau$:
\[
\frac{1}{L}\, \frac{\tau}{8} (A_{0}^{2} + B_{0}^{2}) \ssfT\Add{.}
\Tag{(247)}
\]
If we now, as in \Eq{(158)}, define $\ssfI$, the ``intensity of the vibration
%% -----File: 176.png---Folio 160-------
exciting the oscillator,'' by spectral resolution of the mean value
of the square of the exciting field-strength~$\ssfE_{z}$:
\[
\bar{\ssfE_{z}^{2}} = \int_{0}^{\infty} \ssfI_{\nu}\, d\nu
\Tag{(248)}
\]
we obtain from \Eq{(235)}~and~\Eq{(242)}:
\[
\bar{\ssfE_{z}^{2}}
  = \tfrac{1}{2} \sum_{1}^{\infty} (A_{n}^{2} + B_{n}^{2})
  = \tfrac{1}{2} \int_{0}^{\infty} (A_{n}^{2} + B_{n}^{2}) \ssfT\, d\nu,
\]
and by comparison with~\Eq{(248)}:
\[
\ssfI = \tfrac{1}{2} (A_{0}^{2} + B_{0}^{2}) \ssfT.
\]
Accordingly from~\Eq{(247)} the energy absorbed in the time~$\tau$ becomes:
\[
\frac{\ssfI}{4L} \tau,
\]
%[** Thm]
that is, \emph{in the time between two successive emissions, the energy~$U$
of the oscillator increases uniformly with the time}, according to the
law
\[
\frac{dU}{dt} = \frac{\ssfI}{4L} = a.
\Tag{(249)}
\]
Hence the energy absorbed by all $N$ oscillators in the time~$dt$ is:
\[
\frac{N \ssfI}{4L}\, dt = Na\, dt.
\Tag{(250)}
\]
%% -----File: 177.png---Folio 161-------

\Chapter{III}{Emitted Energy. Stationary State}

\Section{150.} Whereas the absorption of radiation by an oscillator
takes place in a perfectly continuous way, so that the energy of
the oscillator increases continuously and at a constant rate, for
its emission we have, in accordance with \Sec{147}, the following
law: The oscillator emits in irregular intervals, subject to the
laws of chance; it emits, however, only at a moment when its
energy of vibration is just equal to an integral multiple~$n$ of the
elementary quantum $\epsilon = h\nu$, and then it always emits its whole
energy of vibration~$n\epsilon$.

We may represent the whole process by the following figure in
which the abscissæ represent the time~$t$ and the ordinates the
energy
\[
U = n\epsilon + \rho,\quad (\rho < \epsilon)
\Tag{(251)}
\]
\Figure[3.5in]{7}
of a definite oscillator under consideration. The oblique parallel
lines indicate the continuous increase of energy at a constant
rate\DPtypo{.}{}
\[
\frac{dU}{dt} = \frac{d\rho}{dt} = a,
\Tag{(252)}
\]
%% -----File: 178.png---Folio 162-------
which is, according to~\Eq{(249)}, caused by absorption at a constant
rate. Whenever this straight line intersects one of the parallels
to the axis of abscissæ  $U = \epsilon$, $U = 2\epsilon,~\dots$ emission may
possibly take place, in which case the curve drops down to zero
at that point and immediately begins to rise again.

\Section{151.} Let us now calculate the most important properties of
the state of statistical equilibrium thus produced. Of the $N$~oscillators
situated in the field of radiation the number of those
whose energy at the time~$t$ lies in the interval between $U = n\epsilon + \rho$
and $U + dU = n\epsilon + \rho + d\rho$ may be represented by
\[
NR_{n, \rho}\, d\rho,
\Tag{(253)}
\]
where $R$~depends in a definite way on the integer~$n$ and the quantity~$\rho$
which varies continuously between $0$ and~$\epsilon$.

After a time $dt = \dfrac{d\rho}{a}$ all the oscillators will have their energy increased
by~$d\rho$ and hence they will all now lie outside of the energy
interval considered. On the other hand, during the same time~$dt$,
all oscillators whose energy at the time~$t$ was between $n\epsilon + \rho - d\rho$
and $n\epsilon + \rho$ will have entered that interval. The number of all
these oscillators is, according to the notation used above,
\[
NR_{n, \rho - d\rho}\, d\rho.
\Tag{(254)}
\]
Hence this expression gives the number of oscillators which are
at the time $t + dt$ in the interval considered.

Now, since we assume our system to be in a state of statistical
equilibrium, the distribution of energy is independent of the time
and hence the expressions \Eq{(253)}~and~\Eq{(254)} are equal,~\ie,
\[
R_{n, \rho - d\rho} = R_{n, \rho} = R_{n}.
\Tag{(255)}
\]
Thus $R_{n}$ does not depend on $\rho$.

This consideration must, however, be modified for the special
case in which $\rho = 0$. For, in that case, of the oscillators,
$N = R_{n-1}\, d\rho$ in number, whose energy at the time~$t$ was between
$n\epsilon$ and $n\epsilon - d\rho$, during the time $dt = \dfrac{d\rho}{a}$ some enter into the energy
interval (from $U = n\epsilon$ to $U + dU = n\epsilon + d\rho$) considered; but all of
them do not necessarily enter, for an oscillator may possibly emit
all its energy on passing through the value $U = n\epsilon$. If the probability
%% -----File: 179.png---Folio 163-------
that emission takes place be denoted by~$\eta$ ($<1$) the number
of oscillators which pass through the critical value without
emitting will be
\[
NR_{n-1}(1 - \eta)\, d\rho,
\Tag{(256)}
\]
and by equating \Eq{(256)}~and~\Eq{(253)} it follows that
\[
R_{n} = R_{n-1}(1 - \eta),
\]
and hence, by successive reduction,
\[
R_{n} = R_{0}(1 - \eta)^{n}.
\Tag{(257)}
\]
To calculate~$R_{0}$ we repeat the above process for the special case
when $n = 0$ and $\rho = 0$. In this case the energy interval in question
extends from $U = 0$ to $dU = d\rho$. Into this interval enter in the
time $dt = \dfrac{d\rho}{a}$ all the oscillators which perform an emission during
this time, namely, those whose energy at the time~$t$ was between
$\epsilon - d\rho$ and~$\epsilon$, $2\epsilon - d\rho$ and~$2\epsilon$, $3\epsilon - d\rho$ and~$3\epsilon~\dots$.
The numbers of these oscillators are respectively
\[
NR_{0}\, d\rho,\quad
NR_{1}\, d\rho,\quad
NR_{2}\, d\rho,
\]
hence their sum multiplied by~$\eta$ gives the desired number of
emitting oscillators, namely,
\[
N\eta(R_{0} + R_{1} + R_{2} + \dots)\, d\rho,
\Tag{(258)}
\]
and this number is equal to that of the oscillators in the energy
interval between $0$ and~$d\rho$ at the time~$t + dt$, which is~$NR_{0}\, d\rho$.
Hence it follows that
\[
R_{0} = \eta (R_{0} + R_{1} + R_{2} + \dots).
\Tag{(259)}
\]

Now, according to~\Eq{(253)}, the whole number of all the oscillators
is obtained by integrating with respect to~$\rho$ from $0$ to~$\epsilon$, and
summing up with respect to~$n$ from $0$ to~$\infty$. Thus
\[
N = N \sum_{n=0}^{n=\infty} \int_{0}^{\epsilon} R_{n, \rho}\, d\rho
  = N \sum R_{n}\epsilon
\Tag{(260)}
\]
and
\[
\sum R_{n} = \frac{1}{\epsilon}.
\Tag{(261)}
\]
Hence we get from \Eq{(257)}~and~\Eq{(259)}
\[
R_{0} = \frac{\eta}{\epsilon},\quad
R_{n} = \frac{\eta}{\epsilon} (1 - \eta)^{n}.
\Tag{(262)}
\]
%% -----File: 180.png---Folio 164-------

\Section{152.} The total energy emitted in the time element $dt = \dfrac{d\rho}{a}$
is found from~\Eq{(258)} by considering that every emitting oscillator
expends all its energy of vibration and is
\begin{align*}
  &N \eta\, d\rho\, (R_{0} + 2R_{1} + 3R_{2} + \dots) \epsilon\\
= &N \eta\, d\rho\, \eta(1 + 2(1 - \eta) + 3(1 - \eta)^{2} + \dots)\\
= &N\, d\rho  = Na\, dt.
\end{align*}
It is therefore equal to the energy absorbed in the same time by
all oscillators~\Eq{(250)}, as is necessary, since the state is one of
statistical equilibrium.

Let us now consider the \emph{mean energy}~$\bar{U}$ of an oscillator. It is
evidently given by the following relation, which is derived in the
same way as~\Eq{(260)}:
\[
N \bar{U} = N \sum_{0}^{\infty} \int_{0}^{\epsilon} (n\epsilon + \rho) R_{n}\, d\rho.
\Tag{(263)}
\]
From this it follows by means of~\Eq{(262)}, that
\[
\bar{U}
  = \left(\frac{1}{\eta} - \frac{1}{2}\right) \epsilon
  = \left(\frac{1}{\eta} - \frac{1}{2}\right) h\nu\Add{.}
\Tag{(264)}
\]
Since $\eta < 1$, $\bar{U}$~lies between $\dfrac{h\nu}{2}$ and~$\infty$. Indeed, it is immediately
evident that $\bar{U}$~can never become less than $\dfrac{h\nu}{2}$ since the energy
of \emph{every} oscillator, however small it may be, will assume the value
$\epsilon = h\nu$ within a time limit, which can be definitely stated.

\Section{153.} The probability constant~$\eta$ contained in the formulæ for
the stationary state is determined by the law of emission enunciated
in \Sec{147}. According to this, the ratio of the probability
that no emission takes place to the probability that emission does
take place is proportional to the intensity~$\ssfI$ of the vibration
exciting the oscillator, and hence
\[
\frac{1 - \eta}{\eta} = p \ssfI
\Tag{(265)}
\]
where the constant of proportionality is to be determined in
%% -----File: 181.png---Folio 165-------
such a way that for very large energies of vibration the familiar
formulæ of classical dynamics shall hold.

Now, according to~\Eq{(264)}, $\eta$~becomes small for large values of~$\bar{U}$
and for this special case the equations \Eq{(264)} and~\Eq{(265)} give
\[
\bar{U} = ph\nu \ssfI,
\]
and the energy emitted or absorbed respectively in the time~$dt$ by
all $N$ oscillators becomes, according to~\Eq{(250)},
\[
\frac{N \ssfI}{4L}\, dt = \frac{N \bar{U}}{4Lph\nu}\, dt.
\Tag{(266)}
\]

On the other hand, \Name{H.~Hertz} has already calculated from
\Name{Maxwell's} theory the energy emitted by a linear oscillator
vibrating periodically. For the energy emitted in the time of
one-half of one vibration he gives the expression\footnote
  {\Name{H.~Hertz}, Wied.\ Ann.\ \textbf{36}, p.~12, 1889.}
\[
\frac{\pi^{4} E^{2} l^{2}}{3\lambda^{3}}
\]
where $\lambda$~denotes half the wave length, and the product~$El$ (the $C$
of our notation) denotes the amplitude of the moment~$f$
(\Sec{135}) of the vibrations. This gives for the energy emitted
in the time of a whole vibration
\[
\frac{16\pi^{4} C^{2}}{3\lambda^{3}}
\]
where $\lambda$~denotes the whole wave length, and for the energy
emitted by $N$~similar oscillators in the time~$dt$
\[
N \frac{16\pi^{4} C^{2} \nu^{4}}{3c^{3}}\, dt
\]
since $\lambda = \dfrac{c}{\nu}$. On introducing into this expression the energy~$U$ of
an oscillator from \Eq{(205)},~\Eq{(207)}, and~\Eq{(208)}, namely
\[
U = 2\pi^{2} \nu^{2} LC^{2},
\]
we have for the energy emitted by the system of oscillators
\[
N \frac{8\pi^{2} \nu^{2} U}{3c^{3}L}\, dt
\Tag{(267)}
\]
%% -----File: 182.png---Folio 166-------
and by equating the expressions \Eq{(266)}~and~\Eq{(267)} we find for the
factor of proportionality~$p$
\[
p = \frac{3c^{3}}{32\pi^{2} h\nu^{3}}.
\Tag{(268)}
\]

\Section{154.} By the determination of~$p$ the question regarding the
properties of the state of statistical equilibrium between the
system of the oscillators and the vibration exciting them receives
a general answer. For from~\Eq{(265)} we get
\[
\eta = \frac{1}{1 + p \ssfI}
\]
and further from~\Eq{(262)}
\[
R_{n} = \frac{1}{\epsilon}\, \frac{(p \ssfI)^{n}}{(1 + p \ssfI)^{n+1}}.
\Tag{(269)}
\]
Hence in the state of stationary equilibrium the number of
oscillators whose energy lies between $nh\nu$ and $(n + 1)h\nu$ is, from
equation~\Eq{(253)},
\[
N \int_{0}^{\epsilon} R_{n}\, d\rho
  = NR_{n}\epsilon
  = N \frac{(p \ssfI)^{n}}{(1 + p \ssfI)^{n+1}}
\Tag{(270)}
\]
where $n = 0$, $1$, $2$, $3,~\dots$.
%% -----File: 183.png---Folio 167-------

\Chapter[Law of the Normal Distribution Of Energy]
{IV}{The Law of the Normal Distribution Of
Energy. Elementary Quanta Of
Matter and Electricity}

\Section{155.} In the preceding chapter we have made ourselves familiar
with all the details of a system of oscillators exposed to uniform
radiation. We may now develop the idea put forth at the end of
\Sec{144}. That is to say, we may identify the stationary state
of the oscillators just found with the state of maximum entropy
of the system of oscillators which was derived directly from the
hypothesis of quanta in the preceding part, and we may then
equate the temperature of the radiation to the temperature of
the oscillators. It is, in fact, possible to obtain perfect agreement
of the two states by a suitable coordination of their corresponding
quantities.

According to \Sec{139}, the ``distribution density''~$w$ of the
oscillators in the state of statistical equilibrium changes abruptly
from one region element to another, while, according to \Sec{138},
the distribution within a single region element is uniform. The
region elements of the state plane~$(f\psi)$ are bounded by concentric
similar and similarly situated ellipses which correspond to those
values of the energy~$U$ of an oscillator which are integral multiples
of~$h\nu$. We have found exactly the same thing for the stationary
state of the oscillators when they are exposed to uniform radiation,
and the distribution density~$w_{n}$ in the $n$th~region element
may be found from~\Eq{(270)}, if we remember that the $n$th~region
element contains the energies between $(n - 1)h \nu$ and~$nh \nu$. Hence:
\[
w_{n} = \frac{(p \ssfI)^{n-1}}{(1 + p \ssfI)^{n}}
  = \frac{1}{p \ssfI} \left(\frac{p \ssfI}{1 + p \ssfI}\right)^{n}.
\Tag{(271)}
\]
This is in perfect agreement with the previous value~\Eq{(220)} of~$w_{n}$
if we put
\[
\alpha = \frac{1}{p \ssfI} \quad\text{and}\quad
\gamma = \frac{p \ssfI}{1 + p \ssfI},
\]
%% -----File: 184.png---Folio 168-------
and each of these two equations leads, according to~\Eq{(221)}, to the
following relation between the intensity of the exciting vibration~$\ssfI$
and the total energy~$E$ of the $N$~oscillators:
\[
p \ssfI = \frac{E}{Nh \nu} - \frac{1}{2}.
\Tag{(272)}
\]

\Section{156.} If we finally introduce the temperature~$T$ from~\Eq{(223)}, we
get from the last equation, by taking account of the value~\Eq{(268)} of
the factor of proportionality~$p$,
\[
\ssfI = \frac{32\pi^{2} h\nu^{3}}{3c^{3}}\,
  \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
\Tag{(273)}
\]
Moreover the specific intensity~$\ssfK$ of a monochromatic plane
polarized ray of frequency~$\nu$ is, according to equation~\Eq{(160)},
\[
\ssfK = \frac{h\nu^{3}}{c^{2}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}
\Tag{(274)}
\]
and the space density of energy of uniform monochromatic unpolarized
radiation of frequency~$\nu$ is, from~\Eq{(159)},
\[
\ssfu = \frac{8\pi h\nu^{3}}{c^{3}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
\Tag{(275)}
\]
Since, among all the forms of radiation of differing constitutions,
black radiation is distinguished by the fact that all monochromatic
rays contained in it have the same temperature (\Sec{93})
 these equations also give the law of distribution of energy in
the normal spectrum, \ie, in the emission spectrum of a body
which is black with respect to the vacuum.

If we refer the specific intensity of a monochromatic ray not to
the frequency~$\nu$ but, as is usually done in experimental physics,
to the wave length~$\lambda$, by making use of \Eq{(15)}~and~\Eq{(16)} we obtain the
expression
\[%[** F2: Check meaning of subscripted "c"s]
E_{\lambda}
  = \frac{c^{2} h}{\lambda^{5}}\, \frac{1}{e^{\efrac{ch}{k\lambda T}} - 1}
  = \frac{c_{1}}{\lambda^{5}}\,
      \frac{1}{e^{\efrac{c_{2}}{\lambda T}} - 1}\Add{.}
\Tag{(276)}
\]
This is the specific intensity of a monochromatic plane polarized
ray of the wave length~$\lambda$ which is emitted from a black body at the
temperature~$T$ into a vacuum in a direction perpendicular to the
%% -----File: 185.png---Folio 169-------
surface. The corresponding space density of unpolarized radiation
is obtained by multiplying~$E_{\lambda}$ by~$\dfrac{8\pi}{c}$.

Experimental tests have so far confirmed equation~\Eq{(276)}.\footnote
  {See among others \Name{H.~Rubens} und \Name{F.~Kurlbaum}, Sitz.\ Ber.\ d.\ Akad.\ d.\ Wiss.\ zu Berlin
  vom 25.~Okt., 1900, p.~929. Ann.\ d.\ Phys.\ \textbf{4}, p.~649, 1901. \Name{F.~Paschen}, Ann.\ d.\ Phys.\ \textbf{4},
  p.~277, 1901. \Name{O.~Lummer} und \Name{E.~Pringsheim}, Ann.\ d.\ Phys.\ \textbf{6}, p.~210, 1901. Tätigkeitsbericht
  der Phys.-Techn.\ Reichsanstalt vom J.~1911, Zeitschr.\ f.\ Instrumentenkunde, 1912,
  April, p.~134~ff.}
According to the most recent measurements made in the Physikalisch-technische
Reichsanstalt\footnote
  {According to private information kindly furnished by the president, \Name{Mr.~Warburg}.}
the value of the second radiation
constant~$c_{2}$ is approximately
\[
c_{2} = \frac{ch}{k} = 1.436\, \cm\, \degree.
%[** TN: No equation number in the original]
\Tag{(277)}
\]

More detailed information regarding the history of the equation
of radiation is to be found in the original papers and in the
first edition of this book. At this point it may merely be added
that equation~\Eq{(276)} was not simply extrapolated from radiation
measurements, but was originally found in a search after a
connection between the entropy and the energy of an oscillator
vibrating in a field, a connection which would be as simple as
possible and consistent with known measurements.

\Section{157.} The entropy of a ray is, of course, also determined
by its temperature. In fact, by combining equations \Eq{(138)}
and~\Eq{(274)} we readily obtain as an expression for the entropy
radiation~$\ssfL$ of a monochromatic plane polarized ray of the
specific intensity of radiation~$\ssfK$ and the frequency~$\nu$,
\[
\ssfL = \frac{k\nu^{2}}{c^{2}} \left\{
  \left(1 + \frac{c^{2}\ssfK}{h\nu^{3}}\right) \log\left(1 + \frac{c^{2}\ssfK}{h\nu^{3}}\right)
  - \frac{c^{2}\ssfK}{h\nu^{3}} \log\frac{c^{2}\ssfK}{h\nu^{3}}
\right\}
\Tag{(278)}
\]
which is a more definite statement of equation~\Eq{(134)} for \Name{Wien's}
displacement law.

Moreover it follows from~\Eq{(135)}, by taking account of~\Eq{(273)},
that the space density of the entropy~$\ssfs$ of uniform monochromatic
unpolarized radiation as a function of the space density of
energy~$\ssfu$ is
\[
\Squeeze[0.95]{$\displaystyle
\ssfs = \frac{8\pi k\nu^{2}}{c^{3}} \biggl\{
  \biggl(1 + \frac{c^{3}\ssfu}{8\pi h\nu^{3}}\biggr) \log\biggl(1 + \frac{c^{3}\ssfu}{8\pi h\nu^{3}}\biggr)
  - \frac{c^{3}\ssfu}{8\pi h\nu^{3}} \log\frac{c^{3}\ssfu}{8\pi h\nu^{3}}\biggr\}\Add{.}$}
\Tag{(279)}
\]
This is a more definite statement of equation~\Eq{(119)}.
%% -----File: 186.png---Folio 170-------

%[** Size of ()]
\Section{158.} For \emph{small} values of~$\lambda T$ (\ie,~small compared with the
constant $\dfrac{ch}{k}$) equation~\Eq{(276)} becomes
\[
E_{\lambda} = \frac{c^{2}h}{\lambda^{5}} e^{-\efrac{ch}{k\lambda T}}
\Tag{(280)}
\]
an equation which expresses \Name{Wien's}\footnote
  {\Name{W.~Wien}, Wied.\ Ann.\ \textbf{58}, p.~662, 1896.}
law of energy distribution.

The specific intensity of radiation~$\ssfK$ then becomes, according to~\Eq{(274)},
\[
\ssfK = \frac{h\nu^{3}}{c^{2}} e^{-\efrac{h\nu}{kT}}
\Tag{(281)}
\]
and the space density of energy~$\ssfu$ is, from~\Eq{(275)},
\[
\ssfu = \frac{8\pi h\nu^{3}}{c^{3}} e^{-\efrac{h\nu}{kT}}\Add{.}
\Tag{(282)}
\]

\Section{159.} On the other hand, for \emph{large} values of~$\lambda T$ \Eq{(276)}~becomes
\[
E_{\lambda} = \frac{ckT}{\lambda^{4}}\Add{,}
\Tag{(283)}
\]
a relation which was established first by \Name{Lord Rayleigh}\footnote
  {\Name{Lord Rayleigh}, Phil.\ Mag.\ \textbf{49}, p.~539, 1900.}
and which
we may, therefore, call ``\Name{Rayleigh's} law of radiation.''

We then find for the specific intensity of radiation~$\ssfK$ from~\Eq{(274)}
\[
\ssfK = \frac{k\nu^{2}T}{c^{2}}
\Tag{(284)}
\]
and from~\Eq{(275)} for the space density of monochromatic radiation
we get
\[
\ssfu = \frac{8\pi k\nu^{2}T}{c^{3}}\Add{.}
\Tag{(285)}
\]

\Name{Rayleigh's} law of radiation is of very great theoretical interest,
since it represents that distribution of energy which is obtained
for radiation in statistical equilibrium with material molecules
by means of the classical dynamics, and without introducing the
hypothesis of quanta.\footnote
  {\Name{J.~H. Jeans}, Phil.\ Mag.\ Febr.,~1909, p.~229, \Name{H.~A. Lorentz}, Nuovo Cimento~V, vol.~\textbf{16},
  1908.}
This may also be seen from the fact
that for a vanishingly small value of the quantity element of
action,~$h$, the general formula~\Eq{(276)} degenerates into \Name{Rayleigh's}
formula~\Eq{(283)}. See also below, \Sec{168} \textit{et~seq.}
%% -----File: 187.png---Folio 171-------

\Section{160.} For the total space density,~$u$, of black radiation at any
temperature~$T$ we obtain, from~\Eq{(275)},
\[
u = \int_{0}^{\infty} \ssfu\, d\nu
  = \frac{8\pi h}{c^{3}} \int_{0}^{\infty} \frac{\nu^{3}\, d\nu}{e^{\efrac{h\nu}{kT}} - 1}
\]
or
\[
u = \frac{8\pi h}{c^{3}} \int_{0}^{\infty} \left(
    e^{-\efrac{h\nu}{kT}}
  + e^{-\efrac{2h\nu}{kT}}
  + e^{-\efrac{3h\nu}{kT}} + \dots\right) \nu^{3}\, d\nu
\]
and, integrating term by term,
\[
u = \frac{48\pi h}{c^{3}}\left(\frac{kT}{h}\right)^{4}\alpha
\Tag{(286)}
\]
where $\alpha$~is an abbreviation for
\[
\alpha = 1 + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \frac{1}{4^{4}} + \dots = 1.0823.
\Tag{(287)}
\]

This relation expresses the \Name{Stefan-Boltzmann} law~\Eq{(75)} and it
also tells us that the constant of this law is given by
\[
a = \frac{48\pi\alpha k^{4}}{c^{3}h^{3}}.
\Tag{(288)}
\]

\Section{161.} For that wave length~$\lambda_{m}$ the maximum of the
intensity of radiation corresponds in the spectrum of black radiation,
we find from~\Eq{(276)}
\[
\left(\frac{dE_{\lambda}}{d\lambda}\right)_{\lambda = \lambda_{m}} = 0.
\]
On performing the differentiation and putting as an abbreviation
\[
\frac{ch}{k\lambda_{m}T} = \beta,
\]
we get
\[
e^{-\beta} + \frac{\beta}{5} - 1 = 0.
\]
The root of this transcendental equation is
\[
\beta = 4.9651,
\Tag{(289)}
\]
and accordingly $\lambda_{m}T = \dfrac{ch}{\beta k}$, and this is a constant, as demanded
%% -----File: 188.png---Folio 172-------
by \Name{Wien's} displacement law. By comparison with~\Eq{(109)} we
find the meaning of the constant~$b$, namely,
\[
b = \frac{ch}{\beta k},
\Tag{(290)}
\]
and, from~\Eq{(277)},
\[
b = \frac{c_{2}}{\beta}
  = \frac{1.436}{4.9651}
  = 0.289\, \cm \DPchg{·}{}\, \degree,
\Tag{(291)}
\]
while \Name{Lummer} and \Name{Pringsheim} found by measurements $0.294$ and
\Name{Paschen} $0.292.$

\Section{162.} By means of the measured values\footnote
  {Here as well as later on the value given above~\Eq{(79)} has been replaced by $a =
  7.39 · 10^{-15}$, obtained from $\sigma = \DPchg{ac/4}{\dfrac{ac}{4}} = 5.54 · 10^{-5}$. This is the final result of the newest measurements
  made by \Name{W.~Westphal}, according to information kindly furnished by him and
  \Name{Mr.~H. Rubens}.\DPnote{** "Mr." not ital in orig} (Nov.,~1912). [Compare \PageRef{p.}{64}, footnote. Tr.]}
of $a$~and~$c_{2}$ the universal
constants $h$~and~$k$ may be readily calculated. For it follows from
equations \Eq{(277)}~and~\Eq{(288)} that
\[
h = \frac{ac_{2}^{4}}{48\pi \alpha c}\qquad
k = \frac{ac_{2}^{3}}{48\pi \alpha}\Add{.}
\Tag{(292)}
\]
Substituting the values of the constants $a$,~$c_{2}$, $\alpha$,~$c$, we get
\[
h = 6.415 · 10^{-27}\, \erg\, \sec.,\qquad
k = 1.34 · 10^{-16}\, \frac{\erg}{\degree}\Add{.}
\Tag{(293)}
\]

\Section{163.} To ascertain the full physical significance of the quantity
element of action,~$h$, much further research work will be required.
On the other hand, the value obtained for~$k$ enables us readily
to state numerically in the \CGS\ system the general connection
between the entropy~$S$ and the thermodynamic probability~$W$
as expressed by the universal equation~\Eq{(164)}. The general
expression for the entropy of a physical system is
\[
S = 1.34 · 10^{-16} \log W\, \frac{\erg}{\degree}\Add{.}
\Tag{(294)}
\]

\emph{This equation may be regarded as the most general definition of
entropy.} Herein the thermodynamic probability~$W$ is an integral
number, which is completely defined by the macroscopic state of
the system. Applying the result expressed in~\Eq{(293)} to the kinetic
%% -----File: 189.png---Folio 173-------
theory of gases, we obtain from equation~\Eq{(194)} for the ratio of the
mass of a molecule to that of a mol,
\[
\omega = \frac{k}{R} = \frac{1.34 × 10^{-16}}{831 × 10^{5}} = 1.61 × 10^{-24},
\Tag{(295)}
\]
that is to say, there are in one mol
\[
\frac{1}{\omega} = 6.20 × 10^{23}
\]
molecules, where the mol of oxygen, $O_{2}$, is always assumed as
$32\, \gr$. Hence, for example, the absolute mass of a hydrogen
atom ($\frac{1}{2}H_{2} = 1.008$) equals $1.62 × 10^{-24}\, \gr$. With these numerical
values the number of molecules contained in $1\, \cm\DPchg{.}{}^{3}$ of an
ideal gas at $0° \Celsius$ and $1$~atmosphere pressure becomes
\[
N = \frac{76 · 13.6 · 981}{831 · 10^{5} · 273 \omega} = 2.77 · 10^{19}.
\Tag{(296)}
\]
The mean kinetic energy of translatory motion of a molecule
at the absolute temperature $T = 1$ is, in the absolute \CGS\
system, according to~\Eq{(200)},
\[
\DPchg{\frac{3}{2}}{\tfrac{3}{2}} k = 2.01 · 10^{-16}\Add{.}
\Tag{(297)}
\]
In general the mean kinetic energy of translatory motion of a
molecule is expressed by the product of this number and the
absolute temperature~$T$.

The elementary quantity of electricity or the free charge of a
monovalent ion or electron is, in electrostatic units,
\[
e = \omega · 9654 · 3 · 10^{10} = 4.67 · 10^{-10}.
\Tag{(298)}
\]
Since absolute accuracy is claimed for the formulæ here employed,
the degree of approximation to which these numbers
represent the corresponding physical constants depends only on
the accuracy of the measurements of the two radiation constants
$a$~and~$c_{2}$.

\Section[164.]{164. Natural Units.}---All the systems of units which have
hitherto been employed, including the so-called absolute \CGS\
system, owe their origin to the coincidence of accidental circumstances,
%% -----File: 190.png---Folio 174-------
inasmuch as the choice of the units lying at the base of
every system has been made, not according to general points of
view which would necessarily retain their importance for all
places and all times, but essentially with reference to the special
needs of our terrestrial civilization.

Thus the units of length and time were derived from the present
dimensions and motion of our planet, and the units of mass
and temperature from the density and the most important
temperature points of water, as being the liquid which plays the
most important part on the surface of the earth, under a pressure
which corresponds to the mean properties of the atmosphere
surrounding us. It would be no less arbitrary if, let us say, the
invariable wave length of Na-light were taken as unit of length.
For, again, the particular choice of Na from among the many
chemical elements could be justified only, perhaps, by its common
occurrence on the earth, or by its double line, which is in
the range of our vision, but is by no means the only one of its
kind. Hence it is quite conceivable that at some other time,
under changed external conditions, every one of the systems of
units which have so far been adopted for use might lose, in part
or wholly, its original natural significance.

In contrast with this it might be of interest to note that, with
the aid of the two constants $h$~and~$k$ which appear in the universal
law of radiation, we have the means of establishing units of length,
mass, time, and temperature, which are independent of special
bodies or substances, which necessarily retain their significance
for all times and for all environments, terrestrial and human or
otherwise, and which may, therefore, be described as ``natural
units.''

The means of determining the four units of length, mass, time,
and temperature, are given by the two constants $h$~and~$k$ mentioned,
together with the magnitude of the velocity of propagation
of light in a vacuum,~$c$, and that of the constant of gravitation,~$f$.
Referred to centimeter, gram, second, and degrees
Centigrade, the numerical values of these four constants are as
follows:
\begin{align*}
h &= 6.415 · 10^{-27}\, \frac{\DPchg{g}{\gr}\, \cm^{2}}{\sec} \displaybreak[0] \\
%% -----File: 191.png---Folio 175-------
k &= 1.34 · 10^{-16}\, \frac{\DPchg{g}{\gr}\, \cm^{2}}{\sec^{2}\, \degree} \displaybreak[0] \\
c &= 3 · 10^{10}\, \frac{\cm}{\sec} \displaybreak[0] \\
f &= 6.685 · 10^{-8}\, \frac{\cm^{3}}{\DPchg{g}{\gr}\, \sec^{2}}\Add{\;.}\footnotemark
\end{align*}
\footnotetext{\Name{F.~Richarz} and \Name{O.~Krigar-Menzel},  Wied.\ Ann.\ \textbf{66}, p.~190, 1898.}%
If we now choose the natural units so that in the new system of
measurement each of the four preceding constants assumes the
value~$1$, we obtain, as unit of length, the quantity
\[
\sqrt\frac{fh}{c^{3}} = 3.99 · 10^{-33}\, \cm,
\]
as unit of mass
\[
\sqrt\frac{ch}{f} = 5.37 · 10^{-5}\, \DPchg{g}{\gr},
\]
as unit of time
\[
\sqrt\frac{fh}{c^{5}} = 1.33 · 10^{-43}\, \sec,
\]
as unit of temperature
\[
\frac{1}{k} \sqrt\frac{c^{5} h}{f} = 3.60 · 10^{32}\, \degree.
\]
These quantities retain their natural significance as long as
the law of gravitation and that of the propagation of light in
a vacuum and the two principles of thermodynamics remain
valid; they therefore must be found always the same, when
measured by the most widely differing intelligences according to
the most widely differing methods.

\Section{165.} The relations between the intensity of radiation and the
temperature expressed in \Sec{156} hold for radiation in a pure
vacuum. If the radiation is in a medium of refractive index~$n$,
the way in which the intensity of radiation depends on the
frequency and the temperature is given by the proposition of
\Sec{39}, namely, the product of the specific intensity of radiation~$\ssfK_{\nu}$
and the square of the velocity of propagation of the radiation
%% -----File: 192.png---Folio 176-------
has the same value for all substances. The form of this universal
function~\Eq{(42)} follows directly from~\Eq{(274)}
\[
\ssfK q^{2}
  = \frac{\epsilon_{\nu}}{\alpha_{\nu}} q^{2}
  = \frac{h\nu^{3}}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
\Tag{(299)}
\]
Now, since the refractive index~$n$ is inversely proportional to the
velocity of propagation, equation~\Eq{(274)} is, in the case of a medium
with the index of refraction~$n$, replaced by the more general relation
\[
\ssfK_{\nu} = \frac{h\nu^{3} n^{2}}{c^{2}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}
\Tag{(300)}
\]
and, similarly, in place of~\Eq{(275)} we have the more general relation
\[
\ssfu = \frac{8\pi h\nu^{3} n^{3}}{c_{3}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
\Tag{(301)}
\]
These expressions hold, of course, also for the emission of a body
which is black with respect to a medium with an index of refraction~$n$.

\Section{166.} We shall now use the laws of radiation we have obtained
to calculate the temperature of a monochromatic unpolarized
radiation of given intensity in the following case. Let the light
pass normally through a small area (slit) and let it fall on an
arbitrary system of diathermanous media separated by spherical
surfaces, the centers of which lie on the same line, the axis of
the system. Such radiation consists of homocentric pencils and
hence forms behind every refracting surface a real or virtual
image of the emitting surface, the image being likewise normal
to the axis. To begin with, we assume the last as well as the first
medium to be a pure vacuum. Then, for the determination of
the temperature of the radiation according to equation~\Eq{(274)},
we need calculate only the specific intensity of radiation~$\ssfK_{\nu}$ in
the last medium, and this is given by the total intensity of the
monochromatic radiation~$I_{\nu}$, the size of the area of the image~$F$,
and the solid angle~$\Omega$ of the cone of rays passing through a point
of the image. For the specific intensity of radiation~$\ssfK_{\nu}$ is,
according to~\Eq{(13)}, determined by the fact that an amount
\[
2\ssfK_{\nu}\, d\sigma\, d\Omega\, d\nu\, dt
\]
%% -----File: 193.png---Folio 177-------
of energy of unpolarized light corresponding to the interval of
frequencies from $\nu$ to $\nu + d\nu$ is, in the time~$dt$, radiated in a normal
direction through an element of area~$d\sigma$ within the conical element~$d\Omega$.
If now $d\sigma$~denotes an element of the area of the surface
image in the last medium, then the total monochromatic radiation
falling on the image has the intensity
\[
I_{\nu} = 2\ssfK_{\nu} \int d\sigma \int d\Omega.
\]
$I_{\nu}$~is of the dimensions of energy, since the product $d\nu\, dt$ is a mere
number. The first integral is the whole area,~$F$, of the image,
the second is the solid angle,~$\Omega$, of the cone of rays passing
through a point of the surface of the image. Hence we get
\[
I_{\nu} = 2\ssfK_{\nu} F\Omega,
\Tag{(302)}
\]
and, by making use of~\Eq{(274)}, for the temperature of the radiation
\[
T = \frac{h\nu}{k}
  · \frac{1}{\log\left(\dfrac{2h\nu^{3} F\Omega}{c^{2}I_{\nu}} + 1 \right)}\Add{.}
\Tag{(303)}
\]
If the diathermanous medium considered is not a vacuum but
has an index of refraction~$n$, \Eq{(274)}~is replaced by the more general
relation~\Eq{(300)}, and, instead of the last equation, we obtain
\[
T = \frac{h\nu}{k}\,
    \frac{1}{\log\left(\dfrac{2h\nu^{3} F\Omega n^{2}}{c^{2}I_{\nu}} + 1 \right)}
\Tag{(304)}
\]
or, on substituting the numerical values of $c$,~$h$, and~$k$,
\[
T = \frac{0.479 · 10^{-10}\nu}{\log\left(\dfrac{1.43 · 10^{-47}\nu^{3} F\Omega n^{2}}{I_{\nu}} + 1 \right)} \text{ degree Centigrade}.
\]
In this formula the natural logarithm is to be taken, and $I_{\nu}$~is
to be expressed in ergs, $\nu$~in ``reciprocal seconds,'' \ie, $(\text{seconds})^{-1}$,
$F$~in square centimeters. In the case of visible rays the second
term,~$1$, in the denominator may usually be omitted.

The temperature thus calculated is retained by the radiation
considered, so long as it is propagated without any disturbing
%% -----File: 194.png---Folio 178-------
influence in the diathermanous medium, however great the distance
to which it is propagated or the space in which it spreads.
For, while at larger distances an ever decreasing amount of energy
is radiated through an element of area of given size, this is contained
in a cone of rays starting from the element, the angle of
the cone continually decreasing in such a way that the value of~$\ssfK$
remains entirely unchanged. Hence the free expansion of radiation
is a perfectly reversible process. (Compare above, \Sec{144}.)
It may actually be reversed by the aid of a suitable concave mirror
or a converging lens.

Let us next consider the temperature of the radiation in the
other media, which lie between the separate refracting or reflecting
spherical surfaces. In every one of these media the radiation
has a definite temperature, which is given by the last formula
when referred to the real or virtual image formed by the radiation
in that medium.

The frequency~$\nu$ of the monochromatic radiation is, of course,
the same in all media; moreover, according to the laws of geometrical
optics, the product $n^{2}F\Omega$ is the same for all media. Hence,
if, in addition, the total intensity of radiation~$I_{\nu}$ remains constant
on refraction (or reflection), $T$~also remains constant, or in other
words: The temperature of a homocentric pencil is not changed
by regular refraction or reflection, unless a loss in energy of
radiation occurs. Any weakening, however, of the total intensity~$I_{\nu}$
by a subdivision of the radiation, whether into two or
into many different directions, as in the case of diffuse reflection,
leads to a lowering of the temperature of the pencil. In fact, a
certain loss of energy by refraction or reflection does occur, in
general, on a refraction or reflection, and hence also a lowering of
the temperature takes place. In these cases a fundamental
difference appears, depending on whether the radiation is weakened
merely by free expansion or by subdivision or absorption.
In the first case the temperature remains constant, in the second
it decreases.\footnote
  {Nevertheless regular refraction and reflection are not irreversible processes; for the
  refracted and the reflected rays are coherent and the entropy of two coherent rays is not
  equal to the sum of the entropies of the separate rays. (Compare above, \Sec{104}.) On
  the other hand, diffraction is an irreversible process. \Name{M.~Laue}, Ann.\ d.\ Phys.\ \textbf{31}, p.~547,
  1910.}

\Section{167.} The laws of emission of a black body having been determined,
%% -----File: 195.png---Folio 179-------
it is possible to calculate, with the aid of \Name{Kirchhoff's} law~\Eq{(48)},
the emissive power~$E$ of any body whatever, when its
absorbing power~$A$ or its reflecting power $1 - A$ is known. In the
case of metals this calculation becomes especially simple for long
waves, since \Name{E.~Hagen} and \Name{H.~Rubens}\footnote
  {\Name{E.~Hagen} und \Name{H.~Rubens}, Ann.\ d.\ \DPtypo{Phs.y}{Phys.}\ \textbf{11}, p.~873, 1903.}
have shown experimentally
that the reflecting power and, in fact, the entire optical behavior
of the metals in the spectral region mentioned is represented by
the simple equations of \Name{Maxwell} for an electromagnetic field with
homogeneous conductors and hence depends only on the specific
conductivity for steady electric currents. Accordingly, it is
possible to express completely the emissive power of a metal for
long waves by its electric conductivity combined with the formul{\ae}
for black radiation.\footnote
  {\Name{E.~Aschkinass}, Ann.\ d.\ Phys.\ \textbf{17}, p.~960, 1905.}

\Section{168.} There is, however, also a method, applicable to the case
of long waves, for the direct theoretical determination of the electric
conductivity and, with it, of the absorbing power,~$A$, as well
as the emissive power,~$E$, of metals. This is based on the ideas
of the electron theory, as they have been developed for the thermal
and electrical processes in metals by \Name{E.~Riecke}\footnote
  {\Name{E.~Riecke}, Wied.\ Ann.\ \textbf{66}, p.~353, 1898.}
and especially
by \Name{P.~Drude}.\footnote
  {\Name{P.~Drude}, Ann.\ d.\ Phys.\ \textbf{1}, p.~566, 1900.}
According to these, all such processes are based on
the rapid irregular motions of the negative electrons, which fly
back and forth between the positively charged molecules of matter
(here of the metal) and rebound on impact with them as well
as with one another, like gas molecules when they strike a rigid
obstacle or one another. The velocity of the heat motions of the
material molecules may be neglected compared with that of the
electrons, since in the stationary state the mean kinetic energy of
motion of a material molecule is equal to that of an electron, and
since the mass of a material molecule is more than a thousand
times as large as that of an electron. Now, if there is an electric
field in the interior of the metal, the oppositely charged particles
are driven in opposite directions with average velocities depending
on the mean free path, among other factors, and this explains
the conductivity of the metal for the electric current. On the
other hand, the emissive power of the metal for the radiant heat
follows from the calculation of the impacts of the electrons. For,
%% -----File: 196.png---Folio 180-------
so long as an electron flies with constant speed in a constant
direction, its kinetic energy remains constant and there is no
radiation of energy; but, whenever it suffers by impact a change
of its velocity components, a certain amount of energy, which
may be calculated from electrodynamics and which may always
be represented in the form of a \Name{Fourier's} series, is radiated into the
surrounding space, just as we think of \emph{Roentgen rays}\DPnote{cf. Roentgen non-italic elsewhere} as being
caused by the impact on the anticathode of the electrons ejected
from the cathode. From the standpoint of the hypothesis of
quanta this calculation cannot, for the present, be carried out
without ambiguity except under the assumption that, during the
time of a partial vibration of the \Name{Fourier} series, a large number of
impacts of electrons occurs, \ie, for comparatively long waves,
for then the fundamental law of impact does not essentially
matter.

Now this method may evidently be used to derive the laws of
black radiation in a new way, entirely independent of that previously
employed. For if the emissive power,~$E$, of the metal,
thus calculated, is divided by the absorbing power,~$A$, of the same
metal, determined by means of its electric conductivity, then,
according to \Name{Kirchhoff's} law~\Eq{(48)}, the result must be the emissive
power of a black body, irrespective of the special substance used
in the determination. In this manner \Name{H.~A. Lorentz}\footnote
  {\Name{H.~A. Lorentz}, Proc.\ Kon.\ Akad.\ v.\ Wet.\ Amsterdam, 1903, p.~666.}
has, in a
profound investigation, derived the law of radiation of a black
body and has obtained a result the contents of which agree exactly
with equation~\Eq{(283)}, and where also the constant~$k$ is related to
the gas constant~$R$ by equation~\Eq{(193)}. It is true that this method
of establishing the laws of radiation is, as already said, restricted
to the range of long waves, but it affords a deeper and very important
insight into the mechanism of the motions of the electrons
and the radiation phenomena in metals caused by them. At the
same time the point of view described above in \Sec{111}, according
to which the normal spectrum may be regarded as consisting of a
large number of quite irregular processes as elements, is expressly
confirmed.

\Section{169.} A further interesting confirmation of the law of radiation
of black bodies for long waves and of the connection of the
radiation constant~$k$ with the absolute mass of the material
%% -----File: 197.png---Folio 181-------
molecules was found by \Name{J.~H. Jeans}\footnote
  {\Name{J.~H. Jeans}, Phil.\ Mag.\ \textbf{10}, p.~91, 1905.}
by a method previously
used by \Name{Lord Rayleigh},\footnote
  {\Name{Lord Rayleigh}, Nature \textbf{72}, p.~54 and p.~243, 1905.}
which differs essentially from the
one pursued here, in the fact that it entirely avoids making
use of any special mutual action between matter (molecules,
oscillators) and the ether and considers essentially only the
processes in the vacuum through which the radiation passes.
The starting point for this method of treatment is given by the
following proposition of statistical mechanics. (Compare above,
\Sec{140}.) When irreversible processes take place in a system,
which satisfies \Name{Hamilton's} equations of motion, and whose state
is determined by a large number of independent variables and
whose total energy is found by addition of different parts depending
on the squares of the variables of state, they do so, on the
average, in such a sense that the partial energies corresponding
to the separate independent variables of state tend to equality,
so that finally, on reaching statistical equilibrium, their mean
values have become equal. From this proposition the stationary
distribution of energy in such a system may be found, when the
independent variables which determine the state are known.

Let us now imagine a perfect vacuum, cubical in form, of
edge~$l$, and with metallically reflecting sides. If we take the
origin of coordinates at one corner of the cube and let the axes of
coordinates coincide with the adjoining edges, an electromagnetic
process which may occur in this cavity is represented by the
following system of equations:
\[
\begin{aligned}
\ssfE_{x} &= \cos\frac{\ssfa \pi x}{l} \sin\frac{\ssfb \pi y}{l} \sin\frac{\ssfc \pi z}{l} (e_{1} \cos 2\pi \nu t + e_{1}' \sin 2\pi \nu t), \\
\ssfE_{y} &= \sin\frac{\ssfa \pi x}{l} \cos\frac{\ssfb \pi y}{l} \sin\frac{\ssfc \pi z}{l} (e_{2} \cos 2\pi \nu t + e_{2}' \sin 2\pi \nu t), \\
\ssfE_{z} &= \sin\frac{\ssfa \pi x}{l} \sin\frac{\ssfb \pi y}{l} \cos\frac{\ssfc \pi z}{l} (e_{3} \cos 2\pi \nu t + e_{3}' \sin 2\pi \nu t), \\
\ssfH_{x} &= \sin\frac{\ssfa \pi x}{l} \cos\frac{\ssfb \pi y}{l} \cos\frac{\ssfc \pi z}{l} (h_{1} \sin 2\pi \nu t - h_{1}' \cos 2\pi \nu t), \\
%% -----File: 198.png---Folio 182-------
\ssfH_{y} &= \cos\frac{\ssfa \pi x}{l} \sin\frac{\ssfb \pi y}{l} \cos\frac{\ssfc \pi z}{l} (h_{2} \sin 2\pi \nu t - h_{2}' \cos 2\pi \nu t), \\
\ssfH_{z} &= \cos\frac{\ssfa \pi x}{l} \cos\frac{\ssfb \pi y}{l} \sin\frac{\ssfc \pi z}{l} (h_{3} \sin 2\pi \nu t - h_{3}' \cos 2\pi \nu t),
\end{aligned}
%[** TN: No equation number in the original]
\Tag{(305)}
\]
where $\ssfa$,~$\ssfb$,~$\ssfc$ represent any three positive integral numbers.
The boundary conditions in these expressions are satisfied by the
fact that for the six bounding surfaces $x = 0$, $x = l$, $y = 0$, $y = l$,
$z = 0$, $z = l$ the tangential components of the electric field-strength~$\ssfE$
vanish. Maxwell's equations of the field~\Eq{(52)} are also satisfied,
as may be seen on substitution, provided there exist certain conditions
between the constants which may be stated in a single
proposition as follows: Let $a$~be a certain positive constant, then
there exist between the nine quantities written in the following
square:
\[
\begin{array}{c*{2}{>{\qquad}c}}
\dfrac{\ssfa c}{2l \nu} & \dfrac{\ssfb c}{2l \nu} & \dfrac{\ssfc c}{2l \nu} \\
\Strut[28pt]\dfrac{h_{1}}{a} & \dfrac{h_{2}}{a} & \dfrac{h_{3}}{a} \\
\Strut[28pt]\dfrac{e_{1}}{a} & \dfrac{e_{2}}{a} & \dfrac{e_{3}}{a}
\end{array}
\]
all the relations which are satisfied by the nine so-called ``direction
cosines'' of two orthogonal right-handed coordinate systems,
\ie,~the cosines of the angles of any two axes of the systems.

Hence the sum of the squares of the terms of any horizontal
or vertical row equals~$1$, for example,
\[
\begin{gathered}
\frac{c^{2}}{4l^{2}} \nu^{2} (\ssfa^{2} + \ssfb^{2} + \ssfc^{2}) = 1 \\[4pt]
h_{1}^{2} + h_{2}^{2} + h_{3}^{2} = a^{2} = e_{1}^{2} + e_{2}^{2} + e_{3}^{2}.
\end{gathered}
\Tag{(306)}
\]
Moreover the sum of the products of corresponding terms in any
two parallel rows is equal to zero, for example,
\[
\begin{alignedat}{3}
\ssfa e_{1} &+ \ssfb e_{2} &&+ \ssfc e_{3} &&= 0 \\
\ssfa h_{1} &+ \ssfb h_{2} &&+ \ssfc h_{3} &&= 0.
\end{alignedat}
\Tag{(307)}
\]
%% -----File: 199.png---Folio 183-------
Moreover there are relations of the following form:
\[
\frac{h_{1}}{a}
  = \frac{e_{2}}{a} · \frac{\ssfc c}{2l \nu}
  - \frac{e_{3}}{a}\, \frac{\ssfb c}{2l \nu}, %[** Inconsistent dots]
\]
and hence
\[
h_{1} = \frac{c}{2l \nu} (\ssfc e_{2} - \ssfb e_{3}), \text{ etc.}
\Tag{(308)}
\]
If the integral numbers $\ssfa$,~$\ssfb$,~$\ssfc$ are given, then the frequency~$\nu$ is
immediately determined by means of~\Eq{(306)}. Then among the
six quantities $e_{1}$,~$e_{2}$,~$e_{3}$, $h_{1}$,~$h_{2}$,~$h_{3}$, only two may be chosen arbitrarily,
the others then being uniquely determined by them by
linear homogeneous relations. If, for example, we assume $e_{1}$
and $e_{2}$ arbitrarily, $e_{3}$~follows from~\Eq{(307)} and the values of $h_{1}$,~$h_{2}$,~$h_{3}$
are then found by relations of the form~\Eq{(308)}. Between the
quantities with accent $e_{1}'$,~$e_{2}'$,~$e_{3}'$, $h_{1}'$,~$h_{2}'$,~$h_{3}'$ there exist exactly
the same relations as between those without accent, of which
they are entirely independent. Hence two also of them, say
$h_{1}'$ and~$h_{2}'$, may be chosen arbitrarily so that in the equations
given above for given values of $\ssfa$,~$\ssfb$,~$\ssfc$ four constants remain
undetermined. If we now form, for all values of $\ssfa$~$\ssfb$~$\ssfc$\DPnote{** commas?} whatever,
expressions of the type~\Eq{(305)} and add the corresponding field
components, we again obtain a solution for \Name{Maxwell's} equations
of the field and the boundary conditions, which, however, is now
so general that it is capable of representing any electromagnetic
process possible in the hollow cube considered. For it is always
possible to dispose of the constants $e_{1}$,~$e_{2}$, $h_{1}'$,~$h_{2}'$ which have
remained undetermined in the separate particular solutions in
such a way that the process may be adapted to any initial state
($t = 0$) whatever.

If now, as we have assumed so far, the cavity is entirely void
of matter, the process of radiation with a given initial state is
uniquely determined in all its details. It consists of a set of
stationary vibrations, every one of which is represented by one
of the particular solutions considered, and which take place
entirely independent of one another. Hence in this case there
can be no question of irreversibility and hence also none of any
tendency to equality of the partial energies corresponding to the
separate partial vibrations. As soon, however, as we assume the
%% -----File: 200.png---Folio 184-------
presence in the cavity of only the slightest trace of matter which
can influence the electrodynamic vibrations, \eg,~a few gas
molecules, which emit or absorb radiation, the process becomes
chaotic and a passage from less to more probable states will take
place, though perhaps slowly. Without considering any further
details of the electromagnetic constitution of the molecules, we
may from the law of statistical mechanics quoted above draw
the conclusion that, among all possible processes, that one in
which the energy is distributed uniformly among all the independent
variables of the state has the stationary character.

From this let us determine these independent variables. In
the first place there are the velocity components of the gas molecules.
In the stationary state to every one of the three mutually
independent velocity components of a molecule there corresponds
on the average the energy $\frac{1}{3}\bar{L}$ where $\bar{L}$~represents the mean energy
of a molecule and is given by~\Eq{(200)}. Hence the partial energy,
which on the average corresponds to any one of the independent
variables of the electromagnetic system, is just as large.

Now, according to the above discussion, the \DPchg{electro-magnetic}{electromagnetic}
state of the whole cavity for every stationary vibration corresponding
to any one system of values of the numbers $\ssfa$~$\ssfb$~$\ssfc$\DPnote{** commas?} is
determined, at any instant, by four mutually independent quantities.
Hence for the radiation processes the number of independent
variables of state is four times as large as the number
of the possible systems of values of the positive integers $\ssfa$, $\ssfb$,~$\ssfc$.

We shall now calculate the number of the possible systems of
values $\ssfa$, $\ssfb$,~$\ssfc$, which correspond to the vibrations within a certain
small range of the spectrum, say between the frequencies $\nu$ and
$\nu + d \nu$. According to~\Eq{(306)}, these systems of values satisfy the
inequalities
\[
\left(\frac{2l \nu}{c}\right)^{2}
  < \ssfa^{2} + \ssfb^{2} + \ssfc^{2}
  < \left(\frac{2l (\nu + d \nu)}{c}\right)^{2},
\Tag{(309)}
\]
where not only $\dfrac{2l \nu}{c}$ but also $\dfrac{2l\, d\nu}{c}$ is to be thought of as a large
number. If we now represent every system of values of $\ssfa$, $\ssfb$,~$\ssfc$
graphically by a point, taking $\ssfa$,~$\ssfb$,~$\ssfc$ as coordinates in an orthogonal
coordinate system, the points thus obtained occupy one
octant of the space of infinite extent, and condition~\Eq{(309)} is
%% -----File: 201.png---Folio 185-------
equivalent to requiring that the distance of any one of these
points from the origin of the coordinates shall lie between $\dfrac{2l\nu}{c}$
and $\dfrac{2l (\nu + d\nu)}{c}$. Hence the required number is equal to the
number of points which lie between the two spherical surface-octants
corresponding to the radii $\dfrac{2l \nu}{c}$ and $\dfrac{2l (\nu + d\nu)}{c}$. Now since
to every point there corresponds a cube of volume~$1$ and \textit{vice
versa}, that number is simply equal to the space between the two
spheres mentioned, and hence equal to
\[
\frac{1}{8}\, 4\pi \left(\frac{2l \nu}{c}\right)^{2} \frac{2l\, d\nu}{c},
\]
and the number of the independent variables of state is four times
as large or
\[
\frac{16\pi l^{3} \nu^{2}\, d\nu}{c^{3}}\Add{.}
\]

Since, moreover, the partial energy~$\dfrac{\bar{L}}{3}$ corresponds on the average
to every independent variable of state in the state of equilibrium,
the total energy falling in the interval from $\nu$ to $\nu + d\nu$
becomes
\[
\frac{16\pi l^{3} \nu^{2}\, d\nu}{3c^{3}} \bar{L}.
\]
Since the volume of the cavity is~$l^{3}$, this gives for the space
density of the energy of frequency~$\nu$
\[
\ssfu\, d\nu = \frac{16\pi \nu^{2}\, d\nu}{3c^{3}} \bar{L},
\]
and, by substitution of the value of $\bar{L} = \dfrac{L}{N}$ from~\Eq{(200)},
\[
\ssfu = \frac{8\pi \nu^{2} kT}{c^{3}},
\Tag{(310)}
\]
which is in perfect agreement with \Name{Rayleigh's} formula~\Eq{(285)}.

If the law of the equipartition of energy held true in all
%% -----File: 202.png---Folio 186-------
cases, \Name{Rayleigh's} law of radiation would, in consequence, hold for
all wave lengths and temperatures. But since this possibility
is excluded by the measurements at hand, the only possible
conclusion is that the law of the equipartition of energy and,
with it, the system of \Name{Hamilton's} equations of motion does not
possess the general importance attributed to it in classical dynamics.
Therein lies the strongest proof of the necessity of a fundamental
modification of the latter.
%% -----File: 203.png---Folio 187-------

\Part{V}{Irreversible Radiation Processes}
%% -----File: 204.png---Folio 188-------
%[Blank Page]
%% -----File: 205.png---Folio 189-------

\Chapter{I}{Fields of Radiation in General}

\Section{170.} According to the theory developed in the preceding section,
the nature of heat radiation within an isotropic medium,
when the state is one of stable thermodynamic equilibrium, may
be regarded as known in every respect. The intensity of the
radiation, uniform in all directions, depends for all wave lengths
only on the temperature and the velocity of propagation, according
to equation~\Eq{(300)}, which applies to black radiation in any
medium whatever. But there remains another problem to be
solved by the theory. It is still necessary to explain how and by
what processes the radiation which is originally present in the
medium and which may be assigned in any way whatever,
passes gradually, when the medium is bounded by walls impermeable
to heat, into the stable state of black radiation, corresponding
to the maximum of entropy, just as a gas which is
enclosed in a rigid vessel and in which there are originally currents
and temperature differences assigned in any way whatever
gradually passes into the state of rest and of uniform distribution
of temperature.

To this much more difficult question only a partial answer can,
at present, be given. In the first place, it is evident from the
extensive discussion in the first chapter of the third part that,
since irreversible processes are to be dealt with, the principles of
pure electrodynamics alone will not suffice. For the second principle
of thermodynamics or the principle of increase of entropy is
foreign to the contents of pure electrodynamics as well as of pure
mechanics. This is most immediately shown by the fact that the
fundamental equations of mechanics as well as those of electrodynamics
allow the direct reversal of every process as regards
time, which contradicts the principle of increase of entropy.
Of course all kinds of friction and of electric conduction of currents
%% -----File: 206.png---Folio 190-------
must be assumed to be excluded; for these processes, since
they are always connected with the production of heat, do not
belong to mechanics or electrodynamics proper.

This assumption being made, the time~$t$ occurs in the fundamental
equations of mechanics only in the components of
acceleration; that is, in the form of the square of its differential.
Hence, if instead of~$t$ the quantity~$-t$ is introduced as time variable
in the equations of motion, they retain their form without change,
and hence it follows that if in any motion of a system of material
points whatever the velocity components of all points are suddenly
reversed at any instant, the process must take place
in the reverse direction. For the electrodynamic processes in
a homogeneous non-conducting medium a similar statement
holds. If in \Name{Maxwell's} equations of the electrodynamic field
$-t$~is written everywhere instead of~$t$, and if, moreover, the sign of
the magnetic field-strength~$\ssfH$ is reversed, the equations remain
unchanged, as can be readily seen, and hence it follows that if in
any electrodynamic process whatever the magnetic field-strength
is everywhere suddenly reversed at a certain instant, while the
electric field-strength keeps its value, the whole process must take
place in the opposite sense.

If we now consider any radiation processes whatever, taking
place in a perfect vacuum enclosed by reflecting walls, it is found
that, since they are completely determined by the principles of
classical electrodynamics, there can be in their case no question of
irreversibility of any kind. This is seen most clearly by considering
the perfectly general formulæ~\Eq{(305)}, which hold for a
cubical cavity and which evidently have a periodic, \ie, reversible
character. Accordingly we have frequently (Sec.\ \SecNo{144}~and~\SecNo{166})
pointed out that the simple propagation of free radiation
represents a reversible process. An irreversible element is
introduced by the addition of emitting and absorbing substance.

\Section{171.} Let us now try to define for the general case the state of
radiation in the thermodynamic-macroscopic sense as we did
above in \Sec{107}, \textit{et~seq.}, for a stationary radiation. Every one
of the three components of the electric field-strength, \eg,~$\ssfE_{z}$ may,
for the long time interval from $t = 0$ to $t = \ssfT$, be represented at
every point, \eg, at the origin of coordinates, by a \Name{Fourier's}
%% -----File: 207.png---Folio 191-------
integral, which in the present case is somewhat more convenient
than the \Name{Fourier's} series~\Eq{(149)}:
\[
\ssfE_{z} = \int_{0}^{\infty} d\nu\, C_{\nu} \cos (2\pi \nu t - \theta_{\nu}),
\Tag{(311)}
\]
where $C_{\nu}$ (positive) and $\theta_{\nu}$ denote certain functions of the positive
variable of integration~$\nu$. The values of these functions are
not wholly determined by the behavior of~$\ssfE_{z}$ in the time interval
mentioned, but depend also on the manner in which $\ssfE_{z}$~varies
as a function of the time beyond both ends of that interval.
Hence the quantities $C_{\nu}$~and~$\theta_{\nu}$ possess separately no definite
physical significance, and it would be quite incorrect to think
of the vibration~$\ssfE_{z}$ as, say, a continuous spectrum of periodic
vibrations with the constant amplitudes~$C_{\nu}$. This may, by the
way, be seen at once from the fact that the character of the vibration~$\ssfE_{z}$
may vary with the time in any way whatever. How the
spectral resolution of the vibration~$\ssfE_{z}$ is to be performed and to
what results it leads will be shown below (\Sec{174}).

\Section{172.} We shall, as heretofore~\Eq{(158)}, define~$J$, the ``intensity of
the exciting vibration,''\footnote
  {Not to be confused with the ``field intensity'' (field-strength)~$\ssfE_{z}$ of the exciting vibration.}
as a function of the time to be the mean
value of~$\ssfE_{z}^{2}$ in the time interval from $t$ to $t + \tau$, where $\tau$~is taken
as large compared with the time~$\DPchg{1/\nu}{\dfrac{1}{\nu}}$, which is the duration of one
of the periodic partial vibrations contained in the radiation, but
as small as possible compared with the time~$\ssfT$. In this statement
there is a certain indefiniteness, from which results the fact that
$J$~will, in general, depend not only on~$t$ but also on~$\tau$. If this is
the case one cannot speak of the intensity of the exciting vibration
at all. For it is an essential feature of the conception of the
intensity of a vibration that its value should change but unappreciably
within the time required for a single vibration. (Compare
above, \Sec{3}.) Hence we shall consider in future only those
processes for which, under the conditions mentioned, there exists
a mean value of~$\ssfE_{z}^{2}$ depending only on~$t$. We are then obliged
to assume that the quantities~$C_{\nu}$ in~\Eq{(311)} are negligible for all
values of~$\nu$ which are of the same order of magnitude as~$\dfrac{1}{\tau}$ or
smaller,~\ie,
\[
\nu\tau \text{ is large.}
\Tag{(312)}
\]
%% -----File: 208.png---Folio 192-------

In order to calculate~$J$ we now form from~\Eq{(311)} the value of~$\ssfE_{z}^{2}$
and determine the mean value~$\bar{\ssfE_{z}^{2}}$ of this quantity by integrating
with respect to~$t$ from $t$ to $t + \tau$, then dividing by~$\tau$ and
passing to the limit by decreasing $\tau$ sufficiently. Thus we get
\[
\ssfE_{z}^{2} = \int_{0}^{\infty} \int_{0}^{\infty} d\nu'\, d\nu\, C_{\nu'} C_{\nu} \cos(2\pi \nu' t - \theta_{\nu'}) \cos(2\pi \nu t - \theta_{\nu}).
\]
If we now exchange the values of $\nu$~and~$\nu'$, the function under
the sign of integration does not change; hence we assume
\[
\nu' > \nu
\]
and write:
\[
\ssfE_{z}^{2} = 2 \iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \cos(2\pi \nu' t - \theta_{\nu'}) \cos(2\pi \nu t - \theta_{\nu}),
\]
or
\begin{multline*}
\ssfE_{z}^{2} = \iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \{\cos[2\pi(\nu' - \nu)t - \theta_{\nu'} + \theta_{\nu}] \\
  + \cos[2\pi(\nu' + \nu)t - \theta_{\DPtypo{v'}{\nu'}} - \theta_{\nu}]\}.
\end{multline*}
And hence
\begin{multline*}
J = \bar{\ssfE_{z}^{2}}
  = \frac{1}{\tau} \int_{t}^{t+\tau} \ssfE_{z}^{2}\, dt \\
  = \iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \biggl\{
    \frac{\sin \pi(\nu' - \nu)\tau · \cos[\pi(\nu' - \nu)(2t + \tau) - \theta_{\nu'} + \theta_{\nu}]}{\pi(\nu' - \nu) \tau} \\
  + \frac{\sin \pi(\nu' + \nu)\tau · \cos[\pi(\nu' + \nu)(2t + \tau) - \theta_{\nu'} - \theta_{\nu}]}{\pi(\nu' + \nu) \tau}\biggr\}.
\end{multline*}

If we now let $\tau$~become smaller and smaller, since $\nu\tau$~remains
large, the denominator $(\nu' + \nu)\tau$ of the second fraction remains
large under all circumstances, while that of the first fraction
$(\nu' - \nu)\tau$ may decrease with decreasing value of~$\tau$ to less than any
finite value. Hence for sufficiently small values of $(\nu' - \nu)$ the integral
reduces to
\[
\iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \cos[2\pi(\nu' - \nu)t - \theta_{\nu'} + \theta_{\nu}]
\]
which is in fact independent of~$\tau$. The remaining terms of the
double integral, which correspond to larger values of $\nu' - \nu$, \ie,
to more rapid changes with the time, depend in general on~$\tau$ and
%% -----File: 209.png---Folio 193-------
therefore must vanish, if the intensity~$J$ is not to depend on~$\tau$.
Hence in our case on introducing as a second variable of integration
instead of~$\nu$
\[
\mu = \nu' - \nu\ (> 0)
\]
we have
\[
J = \iint d\mu\, d\nu\, C_{\nu+\mu} C_{\nu} \cos(2\pi \mu t - \theta_{\nu+\mu} + \theta_{\nu})
\Tag{(313)}
\]
or
\begin{align*}
J &= \int d\mu\, (A_{\mu} \cos 2\pi \mu t + B_{\mu} \sin 2\pi \mu t) \\
\LeftText{where}
A_{\mu} &= \int d\nu\, C_{\nu+\mu} C_{\nu} \cos(\theta_{\nu+\mu} - \theta_{\nu})
\Tag{(314)}\\
B_{\mu} &= \int d\nu\, C_{\nu+\mu} C_{\nu} \sin(\theta_{\nu+\mu} - \theta_{\nu})\Add{.}
\end{align*}

By this expression the intensity~$J$ of the exciting vibration,
if it exists at all, is expressed by a function of the time in the form
of a \Name{Fourier's} integral.

\Section{173.} The conception of the intensity of vibration~$J$ necessarily
contains the assumption that this quantity varies much more
slowly with the time~$t$ than the vibration $\ssfE_{z}$ itself. The same
follows from the calculation of~$J$ in the preceding paragraph.
For there, according to~\Eq{(312)}, $\nu\tau$~and $\nu'\tau$ are large, but $(\nu' - \nu)\tau$
is small for all pairs of values $C_{\nu}$~and~$C_{\nu'}$ that come into consideration;
hence, \textit{a~fortiori},
\[
\frac{\nu' - \nu}{\nu} = \frac{\mu}{\nu} \quad\text{is small},
\Tag{(315)}
\]
and accordingly the \Name{Fourier's} integrals~$\ssfE_{z}$ in~\Eq{(311)} and $J$ in~\Eq{(314)}
vary with the time in entirely different ways. Hence in the
following we shall have to distinguish, as regards dependence on
time, two kinds of quantities, which vary in different ways:
Rapidly varying quantities, as~$\ssfE_{z}$, and slowly varying quantities
as $J$~and~$\ssfI$ the spectral intensity of the exciting vibration, whose
value we shall calculate in the next paragraph. Nevertheless
this difference in the variability with respect to time of the quantities
%% -----File: 210.png---Folio 194-------
named is only relative, since the absolute value of the differential
coefficient of~$J$ with respect to time depends on the value of
the unit of time and may, by a suitable choice of this unit, be
made as large as we please. It is, therefore, not proper to speak
of $J(t)$ simply as a slowly varying function of~$t$. If, in the
following, we nevertheless employ this mode of expression for
the sake of brevity, it will always be in the relative sense, namely,
with respect to the different behavior of the function~$\ssfE_{z}(t)$.

On the other hand, as regards the dependence of the phase
constant~$\theta_{\nu}$ on its index~$\nu$ it necessarily possesses the property
of rapid variability in the \emph{absolute} sense. For, although $\mu$~is
small compared with~$\nu$, nevertheless the difference $\theta_{\nu+\mu} - \theta_{\nu}$
is in general not small, for if it were, the quantities $A_{\mu}$~and~$B_{\mu}$
in~\Eq{(314)} would have too special values and hence it follows that
$\DPchg{(\dd \theta_{\nu}/\dd \nu)}{\dfrac{\dd \theta_{\nu}}{\dd \nu}} · \nu$ must be large. This is not essentially modified by
changing the unit of time or by shifting the origin of time.

Hence the rapid variability of the quantities~$\theta_{\nu}$ and also $C_{\nu}$
with~$\nu$ is, in the absolute sense, a necessary condition for the
existence of a definite intensity of vibration~$J$, or, in other words,
for the possibility of dividing the quantities depending on the
time into those which vary rapidly and those which vary slowly---a
distinction which is also made in other physical theories and
upon which all the following investigations are based.

\Section{174.} The distinction between rapidly variable and slowly
variable quantities introduced in the preceding section has,
at the present stage, an important physical aspect, because in
the following we shall assume that only slow variability with
time is capable of direct measurement. On this assumption we
approach conditions as they actually exist in optics and heat
radiation. Our problem will then be to establish relations between
slowly variable quantities exclusively; for these only can
be compared with the results of experience. Hence we shall now
determine the most important one of the slowly variable quantities
to be considered here, namely, the ``spectral intensity''~$\ssfI$ of
the exciting vibration. This is effected as in~\Eq{(158)} by means of
the equation
\[
J = \int_{0}^{\infty} \ssfI\, d\nu.
\]
%% -----File: 211.png---Folio 195-------
By comparison with~\DPtypo{313}{\Eq{(313)}} we obtain:
\[%[** Attn]
\left\{
\begin{gathered}
\ssfI = \int d\mu\, (\ssfA_{\mu} \cos 2 \pi \mu t + \ssfB_{\mu} \sin 2 \pi \mu t) \\
\text{where\qquad}
\begin{aligned}[t]
\ssfA_{\mu} &= \bar{C_{\nu + \mu} C_{\nu} \cos (\theta_{\nu + \mu} - \theta_{\nu})} \\
\ssfB_{\mu} &= \bar{C_{\nu + \mu} C_{\nu} \sin (\theta_{\nu + \mu} - \theta_{\nu})}.
\end{aligned}
\end{gathered}\right.
\Tag{(316)}
\]

By this expression the spectral intensity,~$\ssfI$, of the exciting vibration
at a point in the spectrum is expressed as a slowly variable
function of the time~$t$ in the form of a \Name{Fourier's} integral. The
dashes over the expressions on the right side denote the mean
values extended over a narrow spectral range for a given value
of~$\mu$. If such mean values do not exist, there is no definite spectral
intensity.
%% -----File: 212.png---Folio 196-------


\Chapter{II}{One Oscillator in the Field of Radiation}

\Section{175.} If in any field of radiation whatever we have an ideal
oscillator of the kind assumed above (\Sec{135}), there will take
place between it and the radiation falling on it certain mutual
actions, for which we shall again assume the validity of the
elementary dynamical law introduced in the preceding section.
The question is then, how the processes of emission and absorption
will take place in the case now under consideration.

In the first place, as regards the emission of radiant energy by
the oscillator, this takes place, as before, according to the hypothesis
of emission of quanta (\Sec{147}), where the probability
quantity~$\eta$ again depends on the corresponding spectral intensity~$\ssfI$
through the relation~\Eq{(265)}.

On the other hand, the absorption is calculated, exactly as
above, from~\Eq{(234)}, where the vibrations of the oscillator also
take place according to the equation~\Eq{(233)}. In this way, by
calculations analogous to those performed in the second chapter
of the preceding part, with the difference only that instead
of the Fourier's series~\Eq{(235)} the Fourier's integral~\Eq{(311)} is used,
we obtain for the energy absorbed by the oscillator in the time~$\tau$
the expression
\[
\frac{\tau}{4L} \int d\mu\, (\ssfA_{\mu} \cos 2\pi \mu t + \ssfB_{\mu} \sin 2\pi \mu t),
\]
where the constants $\ssfA_{\mu}$~and~$\ssfB_{\mu}$ denote the mean values expressed
in~\Eq{(316)}, taken for the spectral region in the neighborhood of
the natural frequency~$\nu_{0}$ of the oscillator. Hence the law of
absorption will again be given by equation~\Eq{(249)}, which now
holds also for an intensity of vibration~$\ssfI$ varying with the time.

\Section{176.} There now remains the problem of deriving the expression
for~$\ssfI$, the spectral intensity of the vibration exciting the oscillator,
when the thermodynamic state of the field of radiation at
%% -----File: 213.png---Folio 197-------
the oscillator is given in accordance with the statements made in
\Sec{17}.

Let us first calculate the total intensity $J = \bar{{\ssfE_{z}^{2}}}$ of the vibration
exciting an oscillator, from the intensities of the heat rays striking
the oscillator from all directions.

For this purpose we must also allow for the polarization of the
monochromatic rays which strike the oscillator. Let us begin
by considering a pencil which strikes the oscillator within a conical
element whose vertex lies in the oscillator and whose solid
angle,~$d\Omega$, is given by~\Eq{(5)}, where the angles $\theta$~and~$\phi$, polar coordinates,
designate the direction of the propagation of the rays.
The whole pencil consists of a set of monochromatic pencils,
one of which may have the principal values of intensity $\ssfK$~and~$\ssfK'$
(\Sec{17}). If we now denote the angle which the plane of
vibration belonging to the principal intensity~$\ssfK$ makes with the
plane through the direction of the ray and the $z$-axis (the axis of
the oscillator) by~$\psi$, no matter in which quadrant it lies, then,
according to~\Eq{(8)}, the specific intensity of the monochromatic
pencil may be resolved into the two plane polarized components
at right angles with each other,
\begin{align*}
  \ssfK\cos^{2}\psi &+\ssfK'\sin^{2} \psi\\
  \ssfK\sin^{2}\psi &+\ssfK'\cos^{2} \psi,
\end{align*}
the first of which vibrates in a plane passing through the $z$-axis
and the second in a plane perpendicular thereto.

The latter component does not contribute anything to the
value of~$\ssfE_{z}^{2}$, since its electric field-strength is perpendicular to
the axis of the oscillator. Hence there remains only the first
component whose electric field-strength makes the angle $\dfrac{\pi}{2} - \theta$
with the $z$-axis. Now according to \Name{Poynting's} law the intensity of
a plane polarized ray in a vacuum is equal to the product of $\dfrac{c}{4\pi}$
and the mean square of the electric field-strength. Hence the
mean square of the electric field-strength of the pencil here
considered is
\[
\frac{4\pi}{c} (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\, d\nu\, d\Omega,
\]
%% -----File: 214.png---Folio 198-------
and the mean square of its component in the direction of the
$z$-axis is
\[
\frac{4\pi}{c} (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi) \sin^{2} \theta\, d\nu\, d\Omega.
\Tag{(317)}
\]
By integration over all frequencies and all solid angles we then
obtain the value required
\[
\bar{\ssfE_{z}^{2}}
  = \frac{4\pi}{c} \int \sin^{2} \theta\, d\Omega
      \int d\nu\, (\ssfK_{\nu} \cos^{2} \psi + \ssfK_{\nu}' \sin^{2} \psi) = J.
\Tag{(318)}
\]

The space density~$u$ of the electromagnetic energy at a point
of the field is
\[
u = \frac{1}{8\pi}
   (\bar{\ssfE_{x}^{2}} + \bar{\ssfE_{y}^{2}} + \bar{\ssfE_{z}^{2}}
  + \bar{\ssfH_{x}^{2}} + \bar{\ssfH_{y}^{2}} + \bar{\ssfH_{z}^{2}}),
\]
where $\ssfE_{x}^{2}$,~$\ssfE_{y}^{2}$,~$\ssfE_{z}^{2}$, $\ssfH_{x}^{2}$,~$\ssfH_{y}^{2}$,~$\ssfH_{z}^{2}$ denote the squares of the
field-strengths, regarded as ``slowly variable'' quantities, and are
hence supplied with the dash to denote their mean value. Since
for every separate ray the mean electric and magnetic energies
are equal, we may always write
\[
u = \frac{1}{4\pi} = (\bar{\ssfE_{x}^{2}} + \bar{\ssfE_{y}^{2}} + \bar{\ssfE_{z}^{2}}).
\Tag{(319)}
\]
If, in particular, all rays are unpolarized and if the intensity of
radiation is constant in all directions, $\ssfK_{\nu} = \ssfK_{\nu}'$ and, since
\[
\begin{gathered}
\int \sin^{2} \theta\, d\Omega = \iint \sin^{3} \theta\, d\theta\, d\phi = \frac{8\pi}{3} \\
\bar{\ssfE_{z}^{2}} = \frac{32\pi^{2}}{3c} \int \ssfK_{\nu}\, d\nu = \bar{\ssfE_{x}^{2}} = \bar{\ssfE_{y}^{2}}
\end{gathered}
\Tag{(319a)}
\]
and, by substitution in~\Eq{(319)},
\[
u = \frac{8\pi}{c} \int \ssfK_{\nu}\, d\nu,
\]
which agrees with \Eq{(22)}~and~\Eq{(24)}.

\Section{177.} Let us perform the spectral resolution of the intensity~$J$
according to \Sec{174}; namely,
\[
J = \int \ssfI_{\nu}\, d\nu.
\]
%% -----File: 215.png---Folio 199-------
Then, by comparison with~\Eq{(318)}, we find for the intensity of a
definite frequency~$\nu$ contained in the exciting vibration the value
\[
\ssfI = \frac{4\pi}{c} \int \sin^{2} \theta\, d\Omega\, (\ssfK_{\nu} \cos^{2} \psi + \ssfK_{\nu}' \sin^{2} \psi).
\Tag{(320)}
\]
For radiation which is unpolarized and uniform in all directions
we obtain again, in agreement with~\Eq{(160)},
\[
\ssfI = \frac{32\pi^{2}}{3c} \ssfK.
\]

\Section{178.} With the value~\Eq{(320)} obtained for~$\ssfI$ the total energy
absorbed by the oscillator in an element of time~$dt$ from the
radiation falling on it is found from~\Eq{(249)} to be
\[
\frac{\pi\, dt}{cL} \int \sin^{2} \theta\, d\Omega\, (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi).
\]
Hence the oscillator absorbs in the time~$dt$ from the pencil striking
it within the conical element~$d\Omega$ an amount of energy equal to
\[
\frac{\pi\, dt}{cL} \sin^{2} \theta (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\, d\Omega.
\Tag{(321)}
\]
%% -----File: 216.png---Folio 200-------

\Chapter{III}{A System of Oscillators}

\Section{179.} Let us suppose that a large number~$N$ of similar oscillators
with parallel axes, acting quite independently of one another, are
distributed irregularly in a volume-element of the field of radiation,
the dimensions of which are so small that within it the intensities
of radiation~$\ssfK$ do not vary appreciably.  We shall investigate
the mutual action between the oscillators and the radiation
which is propagated freely in space.

As before, the state of the field of radiation may be given by
the magnitude and the azimuth of vibration~$\psi$ of the principal
intensities $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ of the pencils which strike the system of
oscillators, where $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ depend in an arbitrary way on the
direction angles $\theta$~and~$\phi$. On the other hand, let the state of the
system of oscillators be given by the densities of distribution
$w_{1}$,~$w_{2}$, $w_{3},~\dots$~\Eq{(166)}, with which the oscillators are distributed
among the different region elements, $w_{1}$,~$w_{2}$, $w_{3},~\dots$
being any proper fractions whose sum is~$1$. Herein, as always,
the $n$th~region element is supposed to contain the oscillators
with energies between $(n - 1)h\nu$ and~$nh\nu$.

The energy absorbed by the system in the time~$dt$ within the
conical element~$d\Omega$ is, according to~\Eq{(321)},
\[
\frac{\pi N\, dt}{cL} \sin^{2} \theta (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\, d\Omega.
\Tag{(322)}
\]
Let us now calculate also the energy emitted within the same
conical element.

\Section{180.} The total energy emitted in the time element~$dt$ by all $N$
oscillators is found from the consideration that a single oscillator,
according to~\Eq{(249)}, takes up an energy element~$h\nu$ during the time
\[
\frac{4h\nu L}{\ssfI} = \tau,
\Tag{(323)}
\]
%% -----File: 217.png---Folio 201-------
and hence has a chance to emit once, the probability being~$\eta$.
We shall assume that the intensity~$\ssfI$ of the exciting vibration
does not change appreciably in the time~$\tau$. Of the $Nw_{n}$~oscillators
which at the time~$t$ are in the $n$th~region element a number
$Nw_{n}\eta$ will emit during the time~$\tau$, the energy emitted by each
being~$nh\nu$. From~\Eq{(323)} we see that the energy emitted by all
oscillators during the time element~$dt$ is
\[
\sum Nw_{n}\, \eta\, nh\nu \frac{dt}{\tau} = \frac{N\eta \ssfI\, dt}{4L} \sum nw_{n},
\]
or, according to~\Eq{(265)},
\[
\frac{N(1 - \eta)\, dt}{4pL} \sum nw_{n}.
\Tag{(324)}
\]

From this the energy emitted within the conical element~$d\Omega$
may be calculated by considering that, in the state of thermodynamic
equilibrium, the energy emitted in every conical element
is equal to the energy absorbed and that, in the general case, the
energy emitted in a certain direction is independent of the energy
simultaneously absorbed. For the stationary state we have
from \Eq{(160)}~and~\Eq{(265)}
\[
\ssfK = \ssfK'
  = \frac{3c}{32\pi^{2}} \ssfI
  = \frac{3c}{32\pi^{2}}\, \frac{1 - \eta}{p\eta}
\Tag{(325)}
\]
and further from \Eq{(271)}~and~\Eq{(265)}
\[
w_{n} = \frac{1}{p \ssfI} \left(\frac{p \ssfI}{1 + p \ssfI}\right)^{n}
  = \eta(1 - \eta)^{n-1},
\Tag{(326)}
\]
and hence
\[
\sum nw_{n} = \eta \sum n(1 - \eta)^{n-1} = \frac{1}{\eta}.
\Tag{(327)}
\]
Thus the energy emitted~\Eq{(324)} becomes
\[
\frac{N(1 - \eta)\, dt}{4Lp\eta}.
\Tag{(328)}
\]
This is, in fact, equal to the total energy absorbed, as may be
found by integrating the expression~\Eq{(322)} over all conical elements~$d\Omega$
and taking account of~\Eq{(325)}.
%% -----File: 218.png---Folio 202-------

Within the conical element~$d\Omega$ the energy emitted or absorbed
will then be
\[
\frac{\pi N\, dt}{c} \sin^{2} \theta\, \ssfK\, d\Omega,
\]
or, from \Eq{(325)},~\Eq{(327)} and~\Eq{(268)},
\[
\frac{\pi h\nu^{3} (1 - \eta)N}{c^{3}L} \sum nw_{n} \sin^{2} \theta\, d\Omega\, dt,
\Tag{(329)}
\]
and this is the general expression for the energy emitted by the
system of oscillators in the time element~$dt$ within the conical
element~$d\Omega$, as is seen by comparison with~\Eq{(324)}.

\Section{181.} Let us now, as a preparation for the following deductions,
consider more closely the properties of the different pencils
passing the system of oscillators. From all directions rays
strike the volume-element that contains the oscillators; if we
again consider those which come toward it in the direction
$(\theta, \phi)$ within the conical element~$d\Omega$, the vertex of which lies in
the volume-element, we may in the first place think of them as
being resolved into their monochromatic constituents, and then
we need consider further only that one of these constituents which
corresponds to the frequency~$\nu$ of the oscillators; for all other rays
simply pass the oscillators without influencing them or being
influenced by them. The specific intensity of a monochromatic
ray of frequency~$\nu$ is
\[
\ssfK + \ssfK'
\]
where $\ssfK$~and~$\ssfK'$ represent the principal intensities which we
assume as non-coherent. This ray is now resolved into two components
according to the directions of its principal planes of
vibration (\Sec{176}).

The first component,
\[
\ssfK \sin^{2} \psi + \ssfK' \cos^{2} \psi,
\]
passes by the oscillators and emerges on the other side with no
change whatever. Hence it gives a plane polarized ray, which
starts from the system of oscillators in the direction $(\theta, \phi)$ within
the solid angle~$d\Omega$ and whose vibrations are perpendicular to the
axis of the oscillators and whose intensity is
\[
\ssfK \sin^{2} \psi + \ssfK' \cos^{2} \psi = K''.
\Tag{(330)}
\]
%% -----File: 219.png---Folio 203-------
The second component,
\[
\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi,
\]
polarized at right angles to the first consists again, according to
\Sec{176}, of two parts
\begin{alignat*}{2}
&(\ssfK \cos^{2} \psi &&+ \ssfK' \sin^{2} \psi) \cos^{2} \theta
\Tag{(331)} \\
\LeftText{and}
&(\ssfK \cos^{2} \psi &&+ \ssfK' \sin^{2} \psi) \sin^{2} \theta,
\Tag{(332)}
\end{alignat*}
of which the first passes by the system without any change, since
its direction of vibration is at right angles to the axes of the oscillators,
while the second is weakened by absorption, say by the
small fraction~$\beta$. Hence on emergence this component has only
the intensity
\[
(1 - \beta) (\ssfK \cos^{2} \psi + K' \sin^{2} \psi) \sin^{2} \theta.
\Tag{(333)}
\]
It is, however, strengthened by the radiation emitted by the system
of oscillators~\Eq{(329)}, which has the value
\[
\beta' (1 - \eta) \sum nw_{n} \sin^{2} \theta,
\Tag{(334)}
\]
where $\beta'$~denotes a certain other constant, which depends only
on the nature of the system and whose value is obtained at once
from the condition that, in the state of thermodynamic equilibrium,
the loss is just compensated by the gain.

For this purpose we make use of the relations \Eq{(325)}~and~\Eq{(327)}
corresponding to the stationary state, and thus find that the sum
of the expressions \Eq{(333)}~and~\Eq{(334)} becomes just equal to~\Eq{(332)};
and thus for the constant~$\beta'$ the following value is found:
\[
\beta'
  = \beta \frac{3c}{32\pi^{2} p}
  = \beta \frac{h\nu^{3}}{c^{2}}.
\]
Then by addition of \Eq{(331)},~\Eq{(333)} and~\Eq{(334)} the total specific
intensity of the radiation which emanates from the system of
oscillators within the conical element~$d\Omega$, and whose plane of
vibration is parallel to the axes of the oscillators, is found to be
\[
%[** TN: Set on two lines in the original]
\ssfK''' = \ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi +
\beta \sin^{2} \theta \bigl(\ssfK_{e} - (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\bigr)
\Tag{(335)}
\]
where for the sake of brevity the term referring to the emission
is written
\[
\frac{h\nu^{3}}{c^{2}}(1 - \eta) \sum nw_{n} = \ssfK_{e}.
\Tag{(336)}
\]
%% -----File: 220.png---Folio 204-------

Thus we finally have a ray starting from the system of oscillators
in the direction $(\theta, \phi)$ within the conical element~$d\Omega$ and
consisting of two components $\ssfK''$~and~$\ssfK'''$ polarized perpendicularly
to each other, the first component vibrating at right angles
to the axes of the oscillators.

In the state of thermodynamic equilibrium
\[
\ssfK = \ssfK' = \ssfK'' = \ssfK''' = \ssfK_{e},
\]
a result which follows in several ways from the last equations.

\Section{182.} The constant~$\beta$ introduced above, a small positive number,
is determined by the spacial and spectral limits of the radiation
influenced by the system of oscillators. If $q$~denotes the
cross-section at right angles to the direction of the ray, $\Delta\nu$~the
spectral width of the pencil cut out of the total incident radiation
by the system, the energy which is capable of absorption and
which is brought to the system of oscillators within the conical
element~$d\Omega$ in the time~$dt$ is, according to \Eq{(332)}~and~\Eq{(11)},
\[
q\, \Delta\nu\, (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi) \sin^{2} \theta\, d\Omega\, dt.
\Tag{(337)}
\]
Hence the energy actually absorbed is the fraction~$\beta$ of this value.
Comparing this with~\Eq{(322)} we get
\[
\beta = \frac{\pi N}{q · \Delta\nu · cL}.
\Tag{(338)}
\]
%% -----File: 221.png---Folio 205-------

\Chapter[Conservation of Energy and Increase Of Entropy]
{IV}{Conservation of Energy and Increase Of
Entropy. Conclusion}

\Section{183.} It is now easy to state the relation of the two principles of
thermodynamics to the irreversible processes here considered.
Let us consider first the \emph{conservation of energy}. If there is no
oscillator in the field, every one of the elementary pencils, infinite
in number, retains, during its rectilinear propagation, both its
specific intensity~$\ssfK$ and its energy without change, even though it
be reflected at the surface, assumed as plane and reflecting, which
bounds the field (\Sec{166}). The system of oscillators, on the
other hand, produces a change in the incident pencils and hence
also a change in the energy of the radiation propagated in the
field. To calculate this we need consider only those monochromatic
rays which lie close to the natural frequency~$\nu$ of the
oscillators, since the rest are not altered at all by the system.

The system is struck in the direction $(\theta, \phi)$ within the conical
element~$d\Omega$ which converges toward the system of oscillators by
a pencil polarized in some arbitrary way, the intensity of which
is given by the sum of the two principal intensities $\ssfK$~and~$\ssfK$'.
This pencil, according to \Sec{182}, conveys the energy
\[
q\, \Delta\nu\, (\ssfK + \ssfK')\, d\Omega\, dt
\]
to the system in the time~$dt$; hence this energy is taken from the
field of radiation on the side of the rays arriving within~$d\Omega$. As
a compensation there emerges from the system on the other side
in the same direction $(\theta, \phi)$ a pencil polarized in some definite
way, the intensity of which is given by the sum of the two components
$\ssfK''$~and~$\ssfK'''$. By it an amount of energy
\[
q\, \Delta\nu\, (\ssfK'' + \ssfK''')\, d\Omega\, dt,
\]
is added to the field of radiation. Hence, all told, the change in
energy of the field of radiation in the time~$dt$ is obtained by subtracting
%% -----File: 222.png---Folio 206-------
the first expression from the second and by integrating
with respect to~$d\Omega$. Thus we get
\[
dt\, \Delta\nu \int (\ssfK'' + \ssfK''' - \ssfK - \ssfK')q\, d\Omega,
\]
or by taking account of \Eq{(330)},~\Eq{(335)}, and~\Eq{(338)}
\[
\frac{\pi N\, dt}{cL} \int d\Omega \sin^{2} \theta \bigl(\ssfK_{e} - (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\bigr).
\Tag{(339)}
\]

\Section{184.} Let us now calculate the change in energy of the system
of oscillators which has taken place in the same time~$dt$. According
to~\Eq{(219)}, this energy at the time~$t$ is
\[
E = Nh\nu \sum_{1}^{\infty} (n - \DPchg{\frac{1}{2}}{\tfrac{1}{2}})w_{n},
\]
where the quantities~$w_{n}$ whose total sum is equal to~$1$ represent
the densities of distribution characteristic of the state. Hence
the energy change in the time~$dt$ is
\[
dE = Nh\nu \sum_{1}^{\infty} (n - \DPchg{\frac{1}{2}}{\tfrac{1}{2}})\, dw_{n}
   = Nh\nu \sum_{1}^{\infty} n\, dw_{n}.
\Tag{(340)}
\]
To calculate~$dw_{n}$ we consider the $n$th~region element. All of
the oscillators which lie in this region at the time~$t$ have, after
the lapse of time~$\tau$, given by~\Eq{(323)}, left this region; they have
either passed into the $(n + 1)$st~region, or they have performed
an emission at the boundary of the two regions. In compensation
there have entered $(1 - \eta)Nw_{n-1}$ oscillators during the
time~$\tau$, that is, all oscillators which, at the time~$t$, were in the
$(n - 1)$st~region element, excepting such as have lost their energy
by emission. Thus we obtain for the required change in the
time~$dt$
\[
N\, dw_{n} = \frac{dt}{\tau} N \bigl((1 - \eta)w_{n-1} - w_{n}\bigr).
\Tag{(341)}
\]
A separate discussion is required for the first region element $n = 1$.
For into this region there enter in the time~$\tau$ all those oscillators
which have performed an emission in this time. Their number
is
\[
\eta (w_{1} + w_{2} + w_{3} + \dots)N = \eta N.
\]
%% -----File: 223.png---Folio 207-------
Hence we have
\[
N\, dw_{1} = \frac{dt}{\tau} N(\eta - w_{1}).
\]
We may include this equation in the general one~\Eq{(341)} if we
introduce as a new expression
\[
w_{0} = \frac{\eta}{1 - \eta}.
\Tag{(342)}
\]
Then~\Eq{(341)} gives, substituting $\tau$ from~\Eq{(323)},
\[
dw_{n} = \frac{\ssfI\, dt}{4h\nu L} \bigl((1 - \eta) w_{n-1} - w_{n}\bigr),
\Tag{(343)}
\]
and the energy change~\Eq{(340)} of the system of oscillators becomes
\[
dE = \frac{N\ssfI\, dt}{4L} \sum_{1}^{\infty} n \bigl((1 - \eta)w_{n-1} - w_{n}\bigr).
\]
The sum $\sum$ may be simplified by recalling that
\begin{align*}
\sum_{1}^{\infty} nw_{n-1}
  &= \sum_{1}^{\infty} (n - 1)w_{n-1} + \sum_{1}^{\infty} w_{n-1}\\
  &= \sum_{1}^{\infty} nw_{n} + w_{0} + 1
   = \sum_{1}^{\infty} nw_{n} + \frac{1}{1 - \eta}.
\end{align*}
Then we have
\[
dE = \frac{N\ssfI\, dt}{4L} (1 - \eta \sum_{1}^{\infty} nw_{n}).
\Tag{(344)}
\]
This expression may be obtained more readily by considering that
$dE$~is the difference of the total energy absorbed and the total
energy emitted. The former is found from~\Eq{(250)}, the latter from~\Eq{(324)},
by taking account of~\Eq{(265)}.

The principle of the conservation of energy demands that the
sum of the energy change~\Eq{(339)} of the field of radiation and the
energy change~\Eq{(344)} of the system of oscillators shall be zero,
which, in fact, is quite generally the case, as is seen from the relations
\Eq{(320)}~and~\Eq{(336)}.

\Section{185.} We now turn to the discussion of the second principle, the
principle of the \emph{increase of entropy}, and follow closely the above
discussion regarding the energy. When there is no oscillator in
the field, every one of the elementary pencils, infinite in number,
%% -----File: 224.png---Folio 208-------
retains during rectilinear propagation both its specific intensity
and its entropy without change, even when reflected at the surface,
assumed as plane and reflecting, which bounds the field.
The system of oscillators, however, produces a change in the
incident pencils and hence also a change in the entropy of the
radiation propagated in the field. For the calculation of this
change we need to investigate only those monochromatic rays
which lie close to the natural frequency~$\nu$ of the oscillators, since
the rest are not altered at all by the system.

The system of oscillators is struck in the direction $(\theta, \phi)$ within
the conical element~$d\Omega$ converging toward the system by a pencil
polarized in some arbitrary way, the spectral intensity of which
is given by the sum of the two principal intensities $\ssfK$~and~$\ssfK'$ with
the azimuth of vibration~$\psi$ and $\dfrac{\pi}{2} + \psi$ respectively, which are
assumed to be non-coherent. According to~\Eq{(141)} and \Sec{182}
this pencil conveys the entropy
\[
q\, \Delta\nu\, [\ssfL(\ssfK) + \ssfL(\ssfK')]\, d\Omega\, dt
\Tag{(345)}
\]
to the system of oscillators in the time~$dt$, where the function~$\ssfL(\ssfK)$
is given by~\Eq{(278)}. Hence this amount of entropy is taken
from the field of radiation on the side of the rays arriving within~$d\Omega$.
In compensation a pencil starts from the system on the
other side in the same direction $(\theta, \phi)$ within~$d\Omega$ having the
components $\ssfK''$~and~$\ssfK'''$ with the azimuth of vibration $\dfrac{\pi}{2}$ and $0$
respectively, but its entropy radiation is not represented by
$\ssfL(\ssfK'') + \ssfL(\ssfK''')$, since $\ssfK''$~and~$\ssfK'''$ are not non-coherent, but by
\[
\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}')
\Tag{(346)}
\]
where $\ssfK_{0}$~and~$\ssfK_{0}'$ represent the principal intensities of the pencil.

For the calculation of $\ssfK_{0}$~and~$\ssfK_{0}'$ we make use of the fact that,
according to \Eq{(330)}~and~\Eq{(335)}, the radiation $\ssfK''$~and~$\ssfK'''$, of which
the component~$\ssfK'''$ vibrates in the azimuth~$0$, consists of the
following three components, non-coherent with one another:
% [** TN: Next two equations reformatted slightly from the original]
\[
\ssfK_{1}
  = \ssfK \sin^{2} \psi + \ssfK \cos^{2} \psi(1 - \beta \sin^{2} \theta)
  = \ssfK(1 - \beta \sin^{2} \theta \cos^{2} \psi)
\]
with the azimuth of vibration $\DPchg{tg}{\tg}^{2}\psi_{1} = \dfrac{\DPchg{tg}{\tg}^{2}\psi}{1 - \beta \sin^{2} \theta}$,
%% -----File: 225.png---Folio 209-------
\[
\ssfK_{2}
  = \ssfK' \cos^{2} \psi + \ssfK' \sin^{2} \psi (1 - \beta \sin^{2} \theta)
  = \ssfK'(1 - \beta \sin^{2} \theta \sin^{2} \psi)
\]
with the azimuth of vibration $\tg^{2} \psi_{2} = \dfrac{\cot^{2} \psi}{1 - \beta \sin^{2} \theta}$,
and,
\[
\ssfK_{3} = \beta \sin^{2} \theta\, \ssfK_{e}
\]
with the azimuth of vibration $\tg\psi_{3} = 0$.

According to~\Eq{(147)} these values give the principal intensities
$\ssfK_{0}$~and~$\ssfK_{0}'$ required and hence the entropy radiation~\Eq{(346)}.
Thereby the amount of entropy
\[
q\, \Delta\nu\, [\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}')]\, d\Omega\, dt
\Tag{(347)}
\]
is added to the field of radiation in the time~$dt$. All told, the entropy
change of the field of radiation in the time~$dt$, as given by
subtraction of the expression~\Eq{(345)} from~\Eq{(347)} and integration
with respect to~$d\Omega$, is
\[
dt\, \Delta\nu \int q\, d\Omega\, [\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}') - \ssfL(\ssfK) - \ssfL(\ssfK')].
\Tag{(348)}
\]

Let us now calculate the entropy change of the system of
oscillators which has taken place in the same time~$dt$. According
to~\Eq{(173)} the entropy at the time~$t$ is
\[
S = -kN \sum_{1}^{\infty} w_{n} \log w_{n}.
\]
Hence the entropy change in the time~$dt$ is
\[
dS = -kN \sum_{1}^{\infty} \log w_{n}\, dw_{n},
\]
and, by taking account of~\Eq{(343)}, we have:
\[
dS = \frac{Nk \ssfI\, dt}{4h\nu L} \sum_{1}^{\infty}
  \bigl(w_{n} - (1 - \eta) w_{n-1}\bigr) \log w_{n}.
\Tag{(349)}
\]

\Section{186.} The principle of increase of entropy requires that the sum
of the entropy change~\Eq{(348)} of the field of radiation and the
entropy change~\Eq{(349)} of the system of oscillators be always
positive, or zero in the limiting case. That this condition is in
fact satisfied we shall prove only for the special case when all rays
falling on the oscillators are unpolarized, \ie, when $\ssfK' = \ssfK$.
%% -----File: 226.png---Folio 210-------

In this case we have from~\Eq{(147)} and \Sec{185}.
\[
\left.
\begin{aligned}
&\ssfK_{0} \\
&\ssfK_{0}'
\end{aligned}
\right\} = \tfrac{1}{2} \{2\ssfK + \beta \sin^{2} \theta(\ssfK_{e} - \ssfK) ± \beta \sin^{2} \theta(\ssfK_{e} - \ssfK)\},
\]
and hence
\[
\ssfK_{0} = \ssfK + \beta \sin^{2} \theta (\ssfK_{e} - \ssfK),\quad
\ssfK_{0}' = \ssfK.
\]
The entropy change~\Eq{(348)} of the field of radiation becomes
\[
dt\, \Delta\nu \int q\, d\Omega\, \{\ssfL(\ssfK_{o}) - \ssfL(\ssfK)\}
  = dt\, \Delta\nu \int q\, d\Omega\, \beta \sin^{2} \theta(\ssfK_{e} - \ssfK)\, \frac{d\ssfL(\ssfK)}{d\ssfK}
\]
or, by \Eq{(338)}~and~\Eq{(278)},
\[
= \frac{\pi kN\, dt}{hc\nu L} \int d\Omega \sin^{2} \theta(\ssfK_{e} - \ssfK) \log\left(1 + \frac{h\nu^{3}}{c^{2} \ssfK}\right).
\]

On adding to this the entropy change~\Eq{(349)} of the system of
oscillators and taking account of~\Eq{(320)}, the total increase in entropy
in the time~$dt$ is found to be equal to the expression
\[
\Squeeze[0.95]{$\displaystyle\frac{\pi kN\, dt}{ch\nu L} \int d\Omega \sin^{2} \theta
  \biggl\{
  \ssfK \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n}
  + (\ssfK_{e} - \ssfK) \log\biggl(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\biggr)
\biggr\}$}
\]
where
\[
\zeta = 1 - \eta.
\Tag{(350)}
\]

We now must prove that the expression
\begin{multline*}
F = \int d\Omega \sin^{2} \theta \biggl\{
  \ssfK \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n} \\
  + (\ssfK_{e} - \ssfK) \log\left(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\right)
\biggr\}
\Tag{(351)}
\end{multline*}
is always positive and for that purpose we set down once more the
meaning of the quantities involved. $\ssfK$~is an arbitrary positive
function of the polar angles $\theta$~and~$\phi$. The positive proper fraction~$\zeta$
is according to \Eq{(350)},~\Eq{(265)}, and~\Eq{(320)} given by
\[
\frac{\zeta}{1 - \zeta} = \frac{3c^{2}}{8\pi h\nu^{3}} \int \ssfK \sin^{2} \theta\, d\Omega.
\Tag{(352)}
\]
The quantities $w_{1}$,~$w_{2}$, $w_{3},~\dots$ are any positive proper
%% -----File: 227.png---Folio 211-------
fractions whatever which, according to~\Eq{(167)}, satisfy the condition
\[
\sum_{\DPtypo{I}{1}}^{\infty} w_{n} = 1
\Tag{(353)}
\]
while, according to~\Eq{(342)},
\[
w_{0} = \frac{1 - \zeta}{\zeta}.
\Tag{(354)}
\]
Finally we have from~\Eq{(336)}
\[
\ssfK_{e} = \frac{h\nu^{3} \zeta}{c^{2}} \sum_{1}^{\infty} nw_{n}.
\Tag{(355)}
\]

\Section{187.} To give the proof required we shall show that the least
value which the function~$F$ can assume is positive or zero. For
this purpose we consider first that positive function,~$\ssfK$, of $\theta$~and~$\phi$,
which, with fixed values of $\zeta$, $w_{1}$,~$w_{2}$, $w_{3},~\dots$ and~$\ssfK_{e}$, will
make $F$ a minimum. The necessary condition for this is $\delta F = 0$,
where according to~\Eq{(352)}
\[
\int \delta\ssfK \sin^{2} \theta\, d\Omega = 0.
\]
This gives, by considering that the quantities $w$~and~$\zeta$ do not
depend on $\theta$~and~$\phi$, as a necessary condition for the minimum,
\[
\delta F = 0 = \int d\Omega \sin^{2} \theta\, \delta\ssfK \left\{
  -\log\left(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\right)
  - \frac{\ssfK_{e} - \ssfK}{\dfrac{c^{2}\ssfK}{h\nu^{3}} + 1} · \frac{1}{\ssfK}
\right\}
\]
and it follows, therefore, that the quantity in brackets, and hence
also $\ssfK$~itself is independent of $\theta$~and~$\phi$. That in this case $F$~really
has a minimum value is readily seen by forming the second variation
\[
\delta^{2} F = \int d\Omega \sin^{2} \theta\, \delta\ssfK\,
  \delta \left\{-\log\left(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\right)
    - \frac{\ssfK_{e} - \ssfK}{\dfrac{c^{2}\ssfK}{h\nu^{3}} + 1} · \frac{1}{\ssfK}
\right\}
\]
which may by direct computation be seen to be positive under all
circumstances.

In order to form the minimum value of~$F$ we calculate the value
of~$\ssfK$, which, from~\Eq{(352)}, is independent of $\theta$~and~$\phi$. Then it
follows, by taking account of~\Eq{(319a)}, that
\[
\ssfK = \frac{h\nu^{3}}{c^{2}}\, \frac{\zeta}{1 - \zeta}
\]
%% -----File: 228.png---Folio 212-------
and, by also substituting~$\ssfK_{e}$ from~\Eq{(355)},
\[
F = \frac{8\pi h\nu^{3}}{3c^{2}}\, \frac{\zeta}{1 - \zeta}
  \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n}
    - [(1 - \zeta) n - 1] w_{n} \log \zeta.
\]

\Section{188.} It now remains to prove that the sum
\[
\Phi = \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n} - [(1 - \zeta) n - 1] w_{n} \log\zeta,
\Tag{(356)}
\]
where the quantities~$w_{n}$ are subject only to the restrictions that
\Eq{(353)}~and~\Eq{(354)} can never become negative. For this purpose
we determine that system of values of the~$w$'s which, with a fixed
value of~$\zeta$, makes the sum~$\Phi$ a minimum. In this case $\delta\Phi = 0$, or
\begin{multline*}
\sum_{1}^{\infty} (\delta w_{n} - \zeta\, \delta w_{n-1}) \log w_{n}
  + (w_{n} - \zeta w_{n-1})\, \frac{\delta w_{n}}{w_{n}}
\Tag{(357)} \\ %[** Attn tag on first line of broken display]
  - [(1 - \zeta) n - 1]\, \delta w_{n} \log\zeta = 0,
\end{multline*}
where, according to \Eq{(353)}~and~\Eq{(354)},
\[
\sum_{1}^{\infty} \delta w_{n} = 0 \quad\text{and}\quad \delta w_{0} = 0.
\Tag{(358)}
\]
If we suppose all the separate terms of the sum to be written out,
the equation may be put into the following form:
\[
\sum_{1}^{\infty} \delta w_{n} \{\log w_{n} - \zeta \log w_{n+1}
    + \frac{w_{n} - \zeta w_{n-1}}{w_{n}} - [(1 - \zeta) n - 1] \log\zeta\} = 0.
\Tag{(359)}
\]
From this, by taking account of~\Eq{(358)}, we get as the condition
for a minimum, that
\[
\log w_{n} - \zeta \log w_{n+1} + \frac{w_{n} - \zeta w_{n-1}}{w_{n}} - [(1 - \zeta) n - 1] \log\zeta
\Tag{(360)}
\]
must be independent of~$n$.

The solution of this functional equation is
\[
w_{n} = (1 - \zeta) \zeta^{n-1}
\Tag{(361)}
\]
for it satisfies~\Eq{(360)} as well as \Eq{(353)}~and~\Eq{(354)}. With this value
\Eq{(356)}~becomes
\[
\Phi = 0.
\Tag{(362)}
\]
%% -----File: 229.png---Folio 213-------

\Section{189.} In order to show finally that the value~\Eq{(362)} of~$\Phi$ is really
the minimum value, we form from~\Eq{(357)} the second variation
\[
\delta^{2} \Phi = \sum_{1}^{\infty} (\delta w_{n} - \zeta\, \delta w_{n-1}) \frac{\delta w_{n}}{w_{n}}
  - \frac{\zeta\, \delta w_{n-1}}{w_{n}}\, \delta w_{n}
  + \frac{\zeta w_{n-1}}{w_{n}^{2}}\, \delta w_{n}^{2},
\]
where all terms containing the second variation~$\delta^{2}w_{n}$ have been
omitted since their coefficients are, by~\Eq{(360)}, independent of~$n$
and since
\[
\sum_{1}^{\infty} \delta^{2} w_{n} = 0.
\]

This gives, taking account of~\Eq{(361)},
\[
\delta^{2} \Phi
  = \sum_{1}^{\infty} \frac{2\delta w_{n}^{2}}{(1 - \zeta) \zeta^{n-1}}
    - \frac{2\zeta\, \delta w_{n-1}\, \delta w_{n}}{(1 - \zeta) \zeta^{n-1}}
\]
or
\[
\delta^{2} \Phi = \frac{2\zeta}{1 - \zeta}
  \sum_{1}^{\infty} \frac{\delta w_{n}^{2}}{\zeta^{n}}
    - \frac{\delta w_{n-1}\, \delta w_{n}}{\zeta^{n-1}}.
\]
That the sum which occurs here, namely,
\[
\frac{\delta w_{1}^{2}}{\zeta} - \frac{\delta w_{1}\, \delta w_{2}}{\zeta}
  + \frac{\delta w_{2}^{2}}{\zeta^{2}} - \frac{\delta w_{2}\, \delta w_{3}}{\zeta^{2}}
  + \frac{\delta w_{3}^{2}}{\zeta^{3}} - \frac{\delta w_{3}\, \delta w_{4}}{\zeta^{3}} + \dots
\Tag{(363)}
\]
is essentially positive may be seen by resolving it into a sum of
squares. For this purpose we write it in the form
\[
\sum_{1}^{\infty} \frac{1 - \alpha_{n}}{\zeta^{n}}\, \delta w_{n}^{2}
  - \frac{\delta w_{n}\, \delta w_{n+1}}{\zeta^{n}}
  + \frac{\alpha_{n+1}}{\zeta^{n+1}}\, \delta w_{n+1}^{2},
\]
which is identical with~\Eq{(363)} provided $\alpha_{1} = 0$. Now the $\alpha$'s
may be so determined that every term of the last sum is a perfect
square, \ie, that
\[
4 · \frac{1 - \alpha_{n}}{\zeta^{n}} · \frac{\alpha_{n+1}}{\zeta^{n+1}} = \left(\frac{1}{\zeta^{n}}\right)^{2}
\]
or
\[
\alpha_{n+1} = \frac{\zeta}{4(1 - \alpha_{n})}.
\Tag{(364)}
\]
By means of this formula the $\alpha$'s may be readily calculated. The
first values are:
\[
\alpha_{1} = 0,\quad
\alpha_{2} = \frac{\zeta}{4},\quad
\alpha_{3} = \frac{\zeta}{4 - \zeta},\ \dots.
\]
%% -----File: 230.png---Folio 214-------
{\Loosen Continuing the procedure $\alpha_{n}$~remains always positive and less
than $\alpha' = \DPchg{\dfrac{1}{2}}{\frac{1}{2}} \left(1 - \sqrt{1 - \zeta}\right)$.} To prove the correctness of this statement
we show that, if it holds for~$\alpha_{n}$, it holds also for~$\alpha_{n+1}$.
We assume, therefore, that $\alpha_{n}$~is positive and~$< \alpha'$. Then from~\Eq{(364)}
$\alpha_{n+1}$~is positive and $< \dfrac{\zeta}{4(1 - \alpha')}$. But $\dfrac{\zeta}{4(1 - \alpha')} = \alpha'$.
Hence $\alpha_{n+1} < \alpha'$. Now, since the assumption made does actually
hold for $n = 1$, it holds in general. The sum~\Eq{(363)} is thus
essentially positive and hence the value~\Eq{(362)} of~$\Phi$ really is a
minimum, so that the increase of entropy is proven generally.

The limiting case~\Eq{(361)}, in which the increase of entropy
vanishes, corresponds, of course, to the case of thermodynamic
equilibrium between radiation and oscillators, as may also be
seen directly by comparison of~\Eq{(361)} with \Eq{(271)},~\Eq{(265)}, and~\Eq{(360)}.

\Section[190.]{190. Conclusion.}---The theory of irreversible radiation processes
here developed explains how, with an arbitrarily assumed
initial state, a stationary state is, in the course of time, established
in a cavity through which radiation passes and which contains
oscillators of all kinds of natural vibrations, by the intensities
and polarizations of all rays equalizing one another as regards
magnitude and direction. But the theory is still incomplete in
an important respect. For it deals only with the mutual actions
of rays and vibrations of oscillators of the same period. For a
definite frequency the increase of entropy in every time element
until the maximum value is attained, as demanded by the second
principle of thermodynamics, has been proven directly. But, for
all frequencies taken together, the maximum thus attained does
not yet represent the absolute maximum of the entropy of the
system and the corresponding state of radiation does not, in general,
represent the absolutely stable equilibrium (compare \Sec{27}).
For the theory gives no information as to the way in which
the intensities of radiation corresponding to different frequencies
equalize one another, that is to say, how from any arbitrary
initial spectral distribution of energy the normal energy distribution
corresponding to black radiation is, in the course of time,
developed. For the oscillators on which the consideration was
based influence only the intensities of rays which correspond
%% -----File: 231.png---Folio 215-------
to their natural vibration, but they are not capable of changing
their frequencies, so long as they exert or suffer no other action
than emitting or absorbing radiant energy.\footnote
  {Compare \Name{P.~Ehrenfest}, Wien.\ Ber.\ \textbf{114}~[2a], p.~1301, 1905. Ann.\ d.\ Phys.\ \textbf{36}, p.~91,
  1911. \Name{H.~A. Lorentz}, Phys.\ Zeitschr.\ \textbf{11}, p.~1244, 1910. \Name{H.~Poincaré}, Journ.\ de Phys.~(5)
  \textbf{2}, p.~5, p.~347, 1912.}

To get an insight into those processes by which the exchange of
energy between rays of different frequencies takes place in nature
would require also an investigation of the influence which the
motion of the oscillators and of the electrons flying back and
forth between them exerts on the radiation phenomena. For, if
the oscillators and electrons are in motion, there will be impacts
between them, and, at every impact, actions must come into play
which influence the energy of vibration of the oscillators in a
quite different and much more radical way than the simple emission
and absorption of radiant energy. It is true that the final
result of all such impact actions may be anticipated by the aid
of the probability considerations discussed in the third section,
but to show in detail how and in what time intervals this result
is arrived at will be the problem of a future theory. It is certain
that, from such a theory, further information may be expected
as to the nature of the oscillators which really exist in nature,
for the very reason that it must give a closer explanation of
the physical significance of the universal elementary quantity of
action, a significance which is certainly not second in importance
to that of the elementary quantity of electricity.
%% -----File: 232.png---Folio 216-------

\BackMatter

\Bibliography

List of the papers published by the author on heat radiation and the hypothesis
of quanta, with references to the sections of this book where the
same subject is treated.
\bigskip

\selectlanguage{german}
Absorption und Emission elektrischer Wellen durch Resonanz. Sitzungsber.\
d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~21. März~1895, p.~289--301.
\textsc{Wied.}\ Ann.\ \textbf{57}, p.~1--14, 1896.

Ueber elektrische Schwingungen, welche durch Resonanz erregt und
durch Strahlung gedämpft werden. Sitzungsber.\ d.~k.\ preuss.\ Akad.\ d.~Wissensch.\
vom~20. Februar~1896, p.~151--170. \textsc{Wied.}\ Ann.\ \textbf{60}, p.~577--599,
1897.

Ueber irreversible Strahlungsvorgänge. (Erste Mitteilung.) Sitzungsber.\
d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~4. Februar~1897, p.~57--68.

Ueber irreversible Strahlungsvorgänge. (Zweite Mitteilung.) Sitzungsber.\
d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~8. Juli~1897, p.~715--717.

Ueber irreversible Strahlungsvorgänge. (Dritte Mitteilung.) Sitzungsber.\
d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~16. Dezember~1897, p.~1122--1145.

Ueber irreversible Strahlungsvorgänge. (Vierte Mitteilung.) Sitzungsber.\
d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~7. Juli~1898, p.~449--476.

Ueber irreversible Strahlungsvorgänge. (Fünfte Mitteilung.) Sitzungsber.\
d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~18. Mai~1899, p.~440--480.
(§§\;144 bis 190. §\;164.)

Ueber irreversible Strahlungsvorgänge. Ann.\ d.\ Phys.\ \textbf{1}, p.~69--122, 1900.
(§§\;144--190. §\;164.)

Entropie und Temperatur strahlender Wärme. Ann.\ d.\ Phys.\ \textbf{1}, p.~719
bis 737, 1900. (§\;101. §\;166.)

Ueber eine Verbesserung der \Name{Wien}schen Spektralgleichung. Verhandlungen
der Deutschen Physikalischen Gesellschaft \textbf{2}, p.~202--204, 1900.
(§\;156.)

Ein vermeintlicher Widerspruch des magneto-optischen \Name{Faraday}-Effektes
mit der Thermodynamik. Verhandlungen der Deutschen Physikalischen
Gesellschaft \textbf{2}, p.~206--210, 1900.

Kritikzweier Sätze des Herrn \Name{W.~Wien}. Ann.\ d.\ Phys.\ \textbf{3}, p.~764--766, 1900.

Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum.
Verhandlungen der Deutschen Physikalischen Gesellschaft \textbf{2}, p.~237--245,
1900. (§§\;141--143. §\;156 f. §\;163.)

Ueber das Gesetz der Energieverteilung im Normalspektrum. Ann.\ d.\
Phys.\ \textbf{4}, p.~553--563, 1901. (§§\;141--143. §§\;156--162.)

Ueber die Elementarquanta der Materie und der Elektrizität. Ann.\ d.\
Phys.\ \textbf{4}, p.~564--566, 1901. (§\;163.)

Ueber irreversible Strahlungsvorgänge (Nachtrag). Sitzungsber.\ d.~k.\
preuss.\ Akad.\ d.~Wissensch.\ vom~9. Mai~1901, p.~544--555. Ann.\ d.\
Phys.\ \textbf{6}, p.~818--831, 1901. (§§\;185--189.)

Vereinfachte Ableitung der Schwingungsgesetze eines linearen Resonators
im stationär durchstrahlten Felde. Physikalische Zeitschrift \textbf{2}, p.~530
bis p.~534, 1901.

Ueber die Natur des weissen Lichtes.\ Ann.\ d.\ Phys.\ \textbf{7}, p.~390--400, 1902.
(§§\;107--112. §§\;170--174.)

Ueber die von einem elliptisch schwingenden Ion emittierte und absorbierte
%% -----File: 233.png---Folio 217-------
Energie. Archives Néerlandaises, Jubelband für \Name{H.~A. Lorentz}, 1900,
p.~164--174. Ann.\ d.\ Phys.\ \textbf{9}, p.~619--628, 1902.

Ueber die Verteilung der Energie zwischen Aether und Materie. Archives
Néerlandaises, Jubelband für \Name{J.~Bosscha}, 1901, p.~55--66. Ann.\ d.\ Phys.\
\textbf{9}, p.~629--641, 1902. (§§\;121--132.)

Bemerkung über die Konstante des \Name{Wien}schen Verschiebungsgesetzes.
Verhandlungen der Deutschen Physikalischen Gesellschaft \textbf{8}, p.~695--696,
1906. (§\;161.)

Zur Theorie der Wärmestrahlung. Ann.\ d.\ Phys.\ \textbf{31}, p.~758--768, 1910.
Eine neue Strahlungshypothese. Verhandlungen der Deutschen Physikalischen
Gesellschaft \textbf{13}, p.~138--148, 1911. (§\;147.)

Zur Hypothese der Quantenemission. Sitzungsber.\ d.~k.\ preuss.\ Akad.\
d.~Wissensch.\ vom~13. Juli~1911, p.~723--731. (§§\;150--152.)

{\Loosen Ueber neuere thermodynamische Theorien (\Name{Nernst}sches Wärmetheorem
und Quantenhypothese). Ber.\ d.~Deutschen Chemischen Gesellschaft \textbf{45},
p.~5--23, 1912. Physikalische Zeitschrift \textbf{13}, p.~165--175, 1912. Akademische
Verlagsgesellschaft m.~b.~H., Leipzig 1912. (§§\;120--125.)}

Ueber die Begründung des Gesetzes der schwarzen Strahlung. Ann.\ d.\
Phys.\ \textbf{37}, p.~642--656, 1912. (§§\;145--156.)
\selectlanguage{english}
%% -----File: 234.png---Folio 218-------

\Appendix{I}{On Deductions from Stirling's Formula}

The formula is
\Label{218}% [** TN: Page label]
\[
\Tag{(a)}
\lim_{n=\infty} \frac{n!}{n^{n} e^{-n} \sqrt{2\pi n}} = 1,
\]
or, to an approximation quite sufficient for all practical purposes,
provided that $n$~is larger than~$7$
\[
\Tag{(b)}
n! = \left(\frac{n}{e}\right)^{n} \sqrt{2\pi n}.
\]

For a proof of this relation and a discussion of its limits of
accuracy a treatise on probability must be consulted.

%[** F2: Check par break]
On substitution in~\Eq{(170)} this gives
\[
W = \frac{\left(\dfrac{N}{e}\right)^{N}}{\left(\dfrac{N_{1}}{e}\right)^{N_{1}} · \left(\dfrac{N_{2}}{e}\right)^{N_{2}} \dots}
  · \frac{\sqrt{2\pi N}}{\sqrt{2\pi N_{1}} · \sqrt{2\pi N_{2}} \dots}\Add{.}
\]
On account of~\Eq{(165)} this reduces at once to
\[
\frac{N_{N}}{N_{1}^{N_{1}} N_{2}^{N_{2}} \dots}
  · \frac{\sqrt{2\pi N}}{\sqrt{2\pi N_{1}} · \sqrt{2\pi N_{2}} \dots}\Add{.}
\]
Passing now to the logarithmic expression we get
\begin{multline*}
S = k \log W
  = k [N \log N - N_{1} \log N_{1} - N_{2} \log N_{2} - \dots \\
    + \log\sqrt{2\pi N} - \log\sqrt{2\pi N_{1}} - \log\sqrt{2\pi N_{2}} - \dots],
\end{multline*}
or,
\begin{multline*}
\DPtypo{S = k \log W
  = k [(N \log N - \log\sqrt{2\pi N})
    + (N_{1} \log N_{1} - \log\sqrt{2\pi N_{1}}) +\\
      (N_{2} \log N_{2} - \log\sqrt{2\pi N_{2}}) + \dots].}
{S = k \log W
%[** TN: Re-broken to keep terms of same sign on same line]
  = k [(N \log N + \log\sqrt{2\pi N}) \\
    - (N_{1} \log N_{1} + \log\sqrt{2\pi N_{1}})
    - (N_{2} \log N_{2} + \log\sqrt{2\pi N_{2}}) - \dots].}
\end{multline*}

{\Loosen Now, for a large value of~$N_{i}$, the term $N_{i} \log N_{i}$ is very much
larger than $\log\sqrt{2\pi N_{i}}$, as is seen by writing the latter in the form
$\frac{1}{2} \log 2\pi + \frac{1}{2} \log N_{i}$. Hence the last expression will, with a fair
approximation, reduce to}
\[
S = k \log W
  = k [N \log N - N_{1} \log N_{1} - N_{2} \log N_{2} - \dots].
\]
%% -----File: 235.png---Folio 219-------
Introducing now the values of the densities of distribution~$w$
by means of the relation
\[
N_{i} = w_{i}N
\]
we obtain
\[
S = k \log W = kN[\log N - w_{1} \log N_{1} - w_{2} \log N_{2} - \dots],
\]
or, since
\[
w_{1} + w_{2} + w_{3} + \dots = 1,
\]
and hence
\[
(w_{1} + w_{2} + w_{3} + \dots) \log N = \log N,
\]
and
\[
\log N - \log N_{1} = \log \frac{N}{N_{1}} = \log \frac{1}{w_{1}} = -\log w_{1},
\]
we obtain by substitution, after one or two simple transformations
\[
S = k \log W = -kN\sum w_{1} \log w_{1},
\]
a relation which is identical with~\Eq{(173)}.

The statements of \Sec{143} may be proven in a similar manner.
From~\Eq{(232)} we get at once
\[
S = k \log W_{m} = k \log\frac{(N + P - 1)!}{(N - 1)!\, P!}
\]
Now
\[
\log (N - 1)! = \log N! - \log N,
\]
and, for large values of~$N$, $\log N$~is negligible compared with
$\log N!$\Add{.} Applying the same reasoning to the numerator we
may without appreciable error write
\[
S = k \log W_{m} = k \log\frac{(N + P)!}{N!\, P!}\Add{.}
\]
Substituting now for $(N + P)!$, $N!$, and $P!$ their values from~\Eq{(b)}
and omitting, as was previously shown to be approximately
correct, the terms arising from the $\sqrt{2\pi (N + P)}$~etc.,\DPnote{** Italicized in orig} we get,
since the terms containing $e$ cancel out
\begin{align*}
S &= k[(N + P) \log (N + P) - N \log N - P \log P]\\
  &= k[(N + P) \log \frac{N + P}{N} + P \log N - P \log P]\\
  &= kN\left[\left(\frac{P}{N} + 1\right) \log \left(\frac{P}{N} + 1\right) - \frac{P}{N} \log \frac{P}{N}\right].
\end{align*}

This is the relation of \Sec{143}.
%% -----File: 236.png---Folio 220-------

\Appendix{II}{References}

Among general papers treating of the application of the theory
of quanta to different parts of physics are:

1. \Name{A.~Sommerfeld}, Das Planck'sche Wirkungsquantum und
seine allgemeine Bedeutung für die Molekularphysik, Phys.\
Zeitschr., \textbf{12}, p.~1057. Report to the Versammlung Deutscher
Naturforscher und Aerzte. Deals especially with applications to
the theory of specific heats and to the photoelectric effect.
Numerous references are quoted.

2. Meeting of the British Association, Sept., 1913. See
Nature, \textbf{92}, p.~305, Nov.~6, 1913, and Phys.\ Zeitschr., \textbf{14}, p.~1297.
Among the principal speakers were \Name{J.~H. Jeans} and \Name{H.~A. Lorentz}.

(Also American Phys.\ Soc., Chicago Meeting, 1913.\footnote
  {Not yet published (Jan.~26, 1914. Tr.)})

3. \Name{R.~A. Millikan}, Atomic Theories of Radiation, Science,
\textbf{37}, p.~119, Jan.~24, 1913. A non-mathematical discussion.

4. \Name{W.~Wien}, Neuere Probleme der Theoretischen Physik,
1913. (\Name{Wien's} Columbia Lectures, in German.) This is perhaps
the most complete review of the entire theory of quanta.

\Name{H.~A. Lorentz}, Alte und Neue Probleme der Physik, Phys.\
Zeitschr., \textbf{11}, p.~1234. Address to the Versammlung Deutscher
Naturforscher und Aerzte, Königsberg, 1910, contains also some
discussion of the theory of quanta.

Among the papers on radiation are:

\Name{E.~Bauer}, Sur la théorie du rayonnement, Comptes Rendus,
\textbf{153}, p.~1466. Adheres to the quantum theory in the original
form, namely, that emission and absorption both take place in a
discontinuous manner.

\Name{E.~Buckingham}, Calculation of~$c_{2}$ in Planck's equation, Bull.\
Bur.\ Stand.\ \textbf{7}, p.~393.

\Name{E.~Buckingham}, On \Name{Wien's} Displacement Law, Bull.\ Bur.\
Stand.\ \textbf{8}, p.~543. Contains a very simple and clear proof of the
displacement law.
%% -----File: 237.png---Folio 221-------

\Name{P.~Ehrenfest}, Strahlungshypothesen, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{36}, p.~91.

\Name{A.~Joffé}, Theorie der Strahlung, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{36}, p.~534.

Discussions of the method of derivation of the radiation formula
are given in many papers on the subject. In addition to those
quoted elsewhere may be mentioned:

\Name{C.~Benedicks}, Ueber die Herleitung von \Name{Planck's} Energieverteilungsgesetz
Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{42}, p.~133. Derives \Name{Planck's} law
without the help of the quantum theory. The law of equipartition
of energy is avoided by the assumption that solids are not
always monatomic, but that, with decreasing temperature, the
atoms form atomic complexes, thus changing the number of
degrees of freedom. The equipartition principle applies only
to the free atoms.

\Name{P.~Debye}, \Name{Planck's} Strahlungsformel, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{33}, p.~1427.
This method is fully discussed by \Name{Wien} (see~4, above).
It somewhat resembles \Name{Jeans'} method (\Sec{169}) since it avoids
all reference to resonators of any particular kind and merely
establishes the most probable energy distribution. It differs,
however, from \Name{Jeans'} method by the assumption of discrete
energy quanta~$h\nu$. The physical nature of these units is not
discussed at all and it is also left undecided whether it is a
property of matter or of the ether or perhaps a property of the
energy exchange between matter and the ether that causes their
existence. (Compare also some remarks of \Name{Lorentz} in~2.)

\Name{P.~Frank}, Zur Ableitung der Planckschen Strahlungsformel,
Phys.\ Zeitschr., \textbf{13}, p.~506.

\Name{L.~Natanson}, Statistische Theorie der Strahlung, Phys.\ Zeitschr.,
\textbf{12}, p.~659.

\Name{W.~Nernst}, Zur Theorie der \DPtypo{Specifischen}{specifischen} Wärme und über die
Anwendung\DPtypo{,}{} der Lehre von den Energiequanten auf \DPtypo{Physikalisch-chemische}{physikalisch-chemische}
Fragen überhaupt, Zeitschr.\ f.\ \DPtypo{Elektochemie}{Elektrochemie}, \textbf{17}, p.~265.

The experimental facts on which the recent theories of specific
heat (quantum theories) rely, were discovered by \Name{W.~Nernst}
and his fellow workers. The results are published in a large
number of papers that have appeared in different periodicals.
See, \eg, \Name{W.~Nernst}, Der Energieinhalt fester Substanzen, Ann.\
d.\ Phys.\DPchg{,}{}\ \textbf{36}, p.~395, where also numerous other papers are quoted.
(See also references given in~1.) These experimental facts give
very strong support to the heat theorem of \Name{Nernst} (\Sec{120}),
%% -----File: 238.png---Folio 222-------
according to which the entropy approaches a definite limit
(perhaps the value zero, see \Name{Planck's} Thermodynamics, 3.~ed.,
sec.~282, \textit{et~seq.})\ at the absolute zero of temperature, and which
is consistent with the quantum theory. This work is in close
connection with the recent attempts to develop an equation of
state applicable to the solid state of matter. In addition to
the papers by \Name{Nernst} and his school there may be mentioned:

\Name{K.~Eisenmann}, Canonische Zustandsgleichung einatomiger
fester Körper und die Quantentheorie, Verhandlungen der
Deutschen Physikalischen Gesellschaft, \textbf{14}, p.~769.

\Name{W.~H. Keesom}, Entropy and the Equation of State, Konink.\
Akad.\ Wetensch.\ Amsterdam Proc., \textbf{15}, p.~240.

\Name{L.~Natanson}, Energy Content of Bodies, Acad.\ Science Cracovie
Bull.\ Ser.~A, p.~95. In \Name{Einstein's} theory of specific heats
(\Sec{140}) the atoms of actual bodies in nature are apparently
identified with the ideal resonators of \Name{Planck}. In this paper it
is pointed out that this is implying too special features for the
atoms of real bodies, and also, that such far-reaching specializations
do not seem necessary for deriving the laws of specific heat
from the quantum theory.

\Name{L.~S. Ornstein}, Statistical Theory of the Solid State, Konink.\
Akad.\ Wetensch.\ Amsterdam Proc., \textbf{14}, p.~983.

\Name{S.~Ratnowsky}, Die Zustandsgleichung einatomiger fester
Körper und die Quantentheorie, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{38}, p.~637.

Among papers on the law of equipartition of energy (\Sec{169})
are:

\Name{J.~H. Jeans}, Planck's Radiation Theory and \Name{Non-Newtonian}
Mechanics, Phil.\ Mag., \textbf{20}, p.~943.

\Name{S.~B. McLaren}, Partition of Energy between Matter and
Radiation, Phil.\ Mag., \textbf{21}, p.~15.

\Name{S.~B. McLaren}, Complete Radiation, Phil.\ Mag.\Add{,} \textbf{23}, p.~513.
This paper and the one of Jeans deal with the fact that from
%[** Unitalicized names]
Newtonian Mechanics (Hamilton's Principle) the equipartition
principle necessarily follows, and that hence either Planck's law
or the fundamental principles of mechanics need a modification.

For the law of equipartition compare also the discussion at the
meeting of the British Association (see~2).

In many of the papers cited so far deductions from the quantum
%% -----File: 239.png---Folio 223-------
theory are compared with experimental facts. This is also
done by:

\Name{F.~Haber}, Absorptionsspectra fester Körper und die Quantentheorie,
Verhandlungen der Deutschen Physikalischen Gesellschaft,
\textbf{13}, p.~1117.

\Name{J.~Franck} \emph{und} \Name{G.~Hertz}, Quantumhypothese und Ionisation,
Ibid., \textbf{13}, p.~967.

Attempts of giving a concrete physical idea of Planck's constant~$h$
are made by:

\Name{A.~Schidlof}, Zur Aufklärung der universellen electrodynamischen
Bedeutung der Planckschen Strahlungsconstanten~$h$,
Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{35}, p.~96.

\Name{D.~A. Goldhammer}, Ueber die Lichtquantenhypothese, Phys.\
Zeitschr., \textbf{13}, p.~535.

\Name{J.~J. Thomson}, On the Structure of the Atom, Phil.\ Mag., \textbf{26},
p.~792.

\Name{N.~Bohr}, On the Constitution of the Atom, Phil.\ Mag., \textbf{26},
p.~1.

\Name{S.~B. McLaren}, The Magneton and Planck's Universal Constant,
Phil.\ Mag., \textbf{26}, p.~800.

The line of reasoning may be briefly stated thus: Find some
quantity of the same dimension as~$h$, and then construct a model
of an atom where this property plays an important part and can
be made, by a simple hypothesis, to vary by finite amounts instead
of continuously. The simplest of these is \Name{Bohr's}, where $h$~is
interpreted as angular momentum.

The logical reason for the quantum theory is found in the
fact that the \Name{Rayleigh-Jeans} radiation formula does not agree
with experiment. Formerly \Name{Jeans} attempted to reconcile
theory and experiment by the assumption that the equilibrium
of radiation and a black body observed and agreeing with
\Name{Planck's} law rather than his own, was only apparent, and that
the true state of equilibrium which really corresponds to his law
and the equipartition of energy among all variables, is so slowly
reached that it is never actually observed. This standpoint,
which was strongly objected to by authorities on the experimental
side of the question (see, \eg, \Name{E.~Pringsheim} in~2), he
has recently abandoned. \Name{H.~Poincaré}, in a profound mathematical
investigation (\Name{H.~Poincaré}, Sur\DPtypo{.}{} la Théorie des Quanta,
%% -----File: 240.png---Folio 224-------
Journal de Physique~(5), \textbf{2}, p.~1, 1912) reached the conclusion
that whatever the law of radiation may be, it must always, if
the total radiation is assumed as finite, lead to a function presenting
similar \DPtypo{discontinuites}{discontinuities} as the one obtained from the
hypothesis of quanta.

While most authorities have accepted the quantum theory for
good (see \Name{J.~H. Jeans} and \Name{H.~A. Lorentz} in~2), a few still entertain
%[** More non-italic names; maybe italics used only for authors in the appendix?]
doubts as to the general validity of Poincaré's conclusion
(see above \Name{C.~Benedicks} and \Name{R.~A\Add{.} Millikan}~3). Others still
reject the quantum theory on account of the fact that the experimental
evidence in favor of \Name{Planck's} law is not absolutely
conclusive (see \Name{R.~A. Millikan}~3); among these is \Name{A.~E.~H. Love}~(2),
who suggests that \Name{Korn's} (\Name{A.~Korn}, Neue Mechanische
Vorstellungen \DPtypo{uber}{über} die \DPtypo{Schwarze}{schwarze} Strahlung und eine sich aus
denselben ergebende Modification des Planckschen Verteilungsgesetzes,
Phys.\ Zeitschr., \textbf{14}, p.~632) radiation formula fits the
facts as well as that of Planck.

\Name{H.~A. Callendar}, Note on Radiation and Specific Heat, Phil.\
Mag., \textbf{26}, p.~787, has also suggested a radiation formula that fits
the data well. Both Korn's and Callendar's formulæ conform
to \Name{Wien's} displacement law and degenerate for large values of~$\lambda T$
into the Rayleigh-Jeans, and for small values of~$\lambda T$ into
Wien's radiation law. Whether Planck's law or one of these
is the correct law, and whether, if either of the others should
prove to be right, it would eliminate the necessity of the adoption
of the quantum theory, are questions as yet undecided.
Both Korn and Callendar have promised in their papers to follow
them by further ones.
%% -----File: 241.png---Folio 225-------
\iffalse
%[** TN: Errata have been applied to the text body]
ERRATA

Page~77. The last sentence of \Sec{77} should be replaced by: \\
The corresponding additional terms may, however, be omitted
here without appreciable error, since the correction caused by
them would consist merely of the addition to the energy change
here calculated of a comparatively infinitesimal energy change of
the same kind with an external work that is infinitesimal of the
second order.

Page~83. Insert at the end of \Sec{84a}: \\
These laws hold for any original distribution of energy whatever;
hence, \eg, an originally monochromatic radiation remains
monochromatic during the process described, its color changing
in the way stated.
\fi

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