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% The Project Gutenberg EBook of The Theory of the Relativity of Motion, by 
% Richard Chace Tolman                                                    %
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% 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    %
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% Title: The Theory of the Relativity of Motion                           %
%                                                                         %
% Author: Richard Chace Tolman                                            %
%                                                                         %
% Release Date: June 17, 2010 [EBook #32857]                              %
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% Language: English                                                       %
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The Project Gutenberg EBook of The Theory of the Relativity of Motion, by 
Richard Chace Tolman

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 the Relativity of Motion

Author: Richard Chace Tolman

Release Date: June 17, 2010 [EBook #32857]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF THE RELATIVITY ***
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Produced by Andrew D. Hwang, Berj Zamanian, Joshua
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from the Cornell University Library: Historical Mathematics
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

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Production Note

Cornell University Library produced
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irreparably deteriorated original.
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Cornell University Library
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BOUGHT WITH THE INCOME OF THE
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1891
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\footnotesize UNIVERSITY OF CALIFORNIA PRESS \\
BERKELEY \\[8pt]
1917
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%%%% Table of contents auto-generated; Scanned ToC commented out. %%%%
%[** TN: Heading below is printed by \contentsname]
THE THEORY OF THE RELATIVITY OF MOTION.
BY
RICHARD C. TOLMAN, PH.D.
TABLE OF CONTENTS.

\textsc{Preface}      1
\textsc{Chapter} I. Historical Development of Ideas as to the Nature of Space and
Time      5
\textit{Part I}. The Space and Time of Galileo and Newton      5
Newtonian Time   6
Newtonian Space   7
The Galileo Transformation Equations   9
\textit{Part II}. The Space and Time of the Ether Theory   10
Rise of the Ether Theory   10
Idea of a Stationary Ether   12
Ether in the Neighborhood of Moving Bodies   12
Ether Entrained in Dielectrics   13
The Lorentz Theory of a Stationary Ether   13
\textit{Part III}. Rise of the Einstein Theory of Relativity   17
The Michelson-Morley Experiment   17
The Postulates of Einstein   18
\textsc{Chapter} II. The Two Postulates of the Einstein Theory of Relativity   20
The First Postulate of Relativity   20
The Second Postulate of the Einstein Theory of Relativity   21
Suggested Alternative to the Postulate of the Independence of the
Velocity of Light and the Velocity of the Source   23
Evidence against Emission Theories of Light   24
Different Forms of Emission Theory   25
Further Postulates of the Theory of Relativity   27
\textsc{Chapter} III. Some Elementary Deductions   28
Measurements of Time in a Moving System   28
Measurements of Length in a Moving System   30
The Setting of Clocks in a Moving System   33
The Composition of Velocities   35
The Mass of a Moving Body   37
The Relation between Mass and Energy   39
\textsc{Chapter} IV. The Einstein Transformation Equations for Space and Time   42
The Lorentz Transformation   42
Deduction of the Fundamental Transformation Equations   43
The Three Conditions to be Fulfilled   44
The Transformation Equations   45
Further Transformation Equations   47
Transformation Equations for Velocity   47
Transformation Equations for the Function $\dfrac{1}{\sqrt{1-\frac{u^2}{c^2}}}$      47
%% -----File: 010.png---Folio vi-------
Transformation Equations for Acceleration   48
Chapter V. Kinematical Applications   49
The Kinematical Shape of a Rigid Body   49
The Kinematical Rate of a Clock   50
The Idea of Simultaneity   51
The Composition of Velocities   52
The Case of Parallel Velocities   52
Composition of Velocities in General   53
Velocities Greater than that of Light   54
Applications to Optical Problems   56
The Doppler Effect   57
The Aberration of Light   59
Velocity of Light in Moving Media   60
Group Velocity   61
Chapter VI. The Dynamics of a Particle   62
The Laws of Motion   62
Difference between Newtonian and Relativity Mechanics   62
The Mass of a Moving Particle   63
Transverse Collision   63
Mass the Same in all Directions   66
Longitudinal Collision   67
Collision of any Type   68
Transformation Equations for Mass   72
The Force Acting on a Moving Particle   73
Transformation Equations for Force   73
The Relation between Force and Acceleration   74
Transverse and Longitudinal Acceleration   76
The Force Exerted by a Moving Charge   77
The Field around a Moving Charge   79
Application to a Specific Problem   80
Work   81
Kinetic Energy   81
Potential Energy   82
The Relation between Mass and Energy   83
Application to a Specific Problem   85
Chapter VII. The Dynamics of a System of Particles   88
On the Nature of a System of Particles   88
The Conservation of Momentum   89
The Equation of Angular Momentum   90
The Function $T$   92
The Modified Lagrangian Function   93
The Principle of Least Action   93
Lagrange's Equations   95
Equations of Motion in the Hamiltonian Form   96
Value of the Function $T'$   97
The Principle of the Conservation of Energy   99
On the Location of Energy in Space   100
%% -----File: 011.png---Folio vii-------
\textsc{Chapter} VIII. The Chaotic Motion of a System of Particles   102
The Equations of Motion   102
Representation in Generalized Space   103
Liouville's Theorem   103
A System of Particles   104
Probability of a Given Statistical State   105
Equilibrium Relations   106
The Energy as a Function of the Momentum   108
The Distribution Law   109
Polar Coördinates   110
The Law of Equipartition   110
Criterion for Equality of Temperature   112
Pressure Exerted by a System of Particles   113
The Relativity Expression for Temperature   114
The Partition of Energy   117
Partition of Energy for Zero Mass   117
Approximate Partition for Particles of any Mass   118
\textsc{Chapter} IX. The Principle of Relativity and the Principle of Least Action.  121
The Principle of Least Action   121
The Equations of Motion in the Lagrangian Form   122
Introduction of the Principle of Relativity   124
Relation between $\int W'dt'$ and $\int Wdt$  124
Relation between $H'$ and $H$  127
\textsc{Chapter} X. The Dynamics of Elastic Bodies   130
On the Impossibility of Absolutely Rigid Bodies   130
\textit{Part I}. Stress and Strain   130
Definition of Strain   130
Definition of Stress   132
Transformation Equations for Strain   133
Variation in the Strain   134
\textit{Part II}. Introduction of the Principle of Least Action   137
The Kinetic Potential for an Elastic Body   137
Lagrange's Equations   138
Transformation Equations for Stress   139
Value of $E°$   139
The Equations of Motion in the Lagrangian Form   140
Density of Momentum   142
Density of Energy   142
Summary of Results from the Principle of Least Action   142
\textit{Part III}. Some Mathematical Relations   143
The Unsymmetrical Stress Tensor $\mathrm{t}$   143
The Symmetrical Tensor $\mathrm{p}$   145
Relation between div $\mathrm{t}$ and $\mathrm{t}_n$   146
The Equations of Motion in the Eulerian Form   147
\textit{Part IV}. Applications of the Results   148
Relation between Energy and Momentum   148
The Conservation of Momentum   149
%% -----File: 012.png---Folio viii-------
The Conservation of Angular Momentum   150
Relation between Angular Momentum and the Unsymmetrical
Stress Tensor   151
The Right-Angled Lever   152
Isolated Systems in a Steady State   154
The Dynamics of a Particle   154
Conclusion   154
\textsc{Chapter} XI. The Dynamics of a Thermodynamic System   156
The Generalized Coördinates and Forces   156
Transformation Equation for Volume   156
Transformation Equation for Entropy   157
Introduction of the Principle of Least Action. The Kinetic
Potential   157
The Lagrangian Equations   158
Transformation Equation for Pressure   159
Transformation Equation for Temperature   159
The Equations of Motion for Quasistationary Adiabatic Acceleration
160
The Energy of a Moving Thermodynamic System   161
The Momentum of a Moving Thermodynamic System   161
The Dynamics of a Hohlraum   162
\textsc{Chapter} XII. Electromagnetic Theory   164
The Form of the Kinetic Potential   164
The Principle of Least Action   165
The Partial Integrations   165
Derivation of the Fundamental Equations of Electromagnetic
Theory   166
The Transformation Equations for $\mathrm{e}$, $\mathrm{h}$ and $\rho$   168
The Invariance of Electric Charge   170
The Relativity of Magnetic and Electric Fields   171
Nature of Electromotive Force   172
Derivation of the Fifth Fundamental Equation   172
Difference between the Ether and the Relativity Theories of Electromagnetics
173
Applications to Electromagnetic Theory   176
The Electric and Magnetic Fields around a Moving Charge   176
The Energy of a Moving Electromagnetic System   178
Relation between Mass and Energy   180
The Theory of Moving Dielectrics   181
Relation between Field Equations for Material Media and
Electron Theory   182
Transformation Equations for Moving Media   183
Theory of the Wilson Experiment   186
\textsc{Chapter} XIII. Four-Dimensional Analysis   188
Idea of a Time Axis   188
Non-Euclidean Character of the Space   189
%% -----File: 013.png---Folio ix-------
Part I. Vector Analysis of the Non-Euclidean Four-Dimensional Manifold
191
Space, Time and Singular Vectors  192
Invariance of $x^2 + y^2 + z^2 - c^2t^2$  192
Inner Product of One-Vectors  193
Non-Euclidean Angle  194
Kinematical Interpretation of Angle in Terms of Velocity  194
Vectors of Higher Dimensions  195
Outer Products  195
Inner Product of Vectors in General  198
The Complement of a Vector  198
The Vector Operator, $\Diamond$ or Quad  199
Tensors  200
The Rotation of Axes  201
Interpretation of the Lorentz Transformation as a Rotation of
Axes  206
Graphical Representation  208
Part II. Applications of the Four-Dimensional Analysis  211
Kinematics  211
Extended Position  211
Extended Velocity  212
Extended Acceleration  213
The Velocity of Light  214
The Dynamics of a Particle  214
Extended Momentum  214
The Conservation Laws  215
The Dynamics of an Elastic Body  216
The Tensor of Extended Stress  216
The Equation of Motion  216
Electromagnetics  217
Extended Current  218
The Electromagnetic Vector $\vc{M}$  217
The Field Equations  217
The Conservation of Electricity  218
The Product $\vc{M} · \vc{q}$  218
The Extended Tensor of Electromagnetic Stress  219
Combined Electrical and Mechanical Systems  221
Appendix I. Symbols for Quantities  222
Scalar Quantities  222
Vector Quantities  223
Appendix II. Vector Notation  224
Three Dimensional Space  224
Non-Euclidean Four Dimensional Space  225
\fi
%%%% End of commented table of contents %%%%
%%
%% -----File: 014.png---Folio x-------
%[Blank Page]
%% -----File: 015.png---Folio 1-------
\mainmatter
\phantomsection\pdfbookmark[-1]{Main Matter}{Main Matter}

\Preface

Thirty or forty years ago, in the field of physical science, there
was a widespread feeling that the days of adventurous discovery had
passed forever, and the conservative physicist was only too happy to
devote his life to the measurement to the sixth decimal place of
quantities whose significance for physical theory was already an old
story. The passage of time, however, has completely upset such
bourgeois ideas as to the state of physical science, through the discovery
of some most extraordinary experimental facts and the development
of very fundamental theories for their explanation.

On the experimental side, the intervening years have seen the
discovery of radioactivity, the exhaustive study of the conduction of
electricity through gases, the accompanying discoveries of cathode,
canal and X-rays, the isolation of the electron, the study of the
distribution of energy in the hohlraum, and the final failure of all
attempts to detect the earth's motion through the supposititious
ether. During this same time, the theoretical physicist has been
working hand in hand with the experimenter endeavoring to correlate
the facts already discovered and to point the way to further research.
The theoretical achievements, which have been found particularly
helpful in performing these functions of explanation and prediction,
have been the development of the modern theory of electrons, the
application of thermodynamic and statistical reasoning to the phenomena
of radiation, and the development of Einstein's brilliant
theory of the relativity of motion.

It has been the endeavor of the following book to present an
introduction to this theory of relativity, which in the decade since
the publication of Einstein's first paper in 1905 (\textit{Annalen der Physik})
has become a necessary part of the theoretical equipment of every
physicist. Even if we regard the Einstein theory of relativity merely
as a convenient tool for the prediction of electromagnetic and optical
phenomena, its importance to the physicist is very great, not only
because its introduction greatly simplifies the deduction of many
%% -----File: 016.png---Folio 2-------
theorems which were already familiar in the older theories based on a
stationary ether, but also because it leads simply and directly to correct
conclusions in the case of such experiments as those of Michelson
and Morley, Trouton and Noble, and Kaufman and Bucherer, which
can be made to agree with the idea of a stationary ether only by the
introduction of complicated and \textit{ad~hoc} assumptions. Regarded from
a more philosophical point of view, an acceptance of the Einstein
theory of relativity shows us the advisability of completely remodelling
some of our most fundamental ideas. In particular we shall now
do well to change our concepts of space and time in such a way as
to give up the old idea of their complete independence, a notion
which we have received as the inheritance of a long ancestral experience
with bodies moving with slow velocities, but which no longer proves
pragmatic when we deal with velocities approaching that of light.

The method of treatment adopted in the following chapters is
to a considerable extent original, partly appearing here for the first
time and partly already published elsewhere.\footnote
  {\textit{Philosophical Magazine}, vol.~18, p.~510 (1909);
  \textit{Physical Review}, vol.~31, p.~26 (1910);
  \textit{Phil.\ Mag.}, vol.~21, p.~296 (1911);
  \textit{ibid}., vol.~22, p.~458 (1911);
  \textit{ibid}., vol.~23, p.~375 (1912);
  \textit{Phys.\ Rev.}, vol.~35, p.~136 (1912);
  \textit{Phil.\ Mag.}, vol.~25, p.~150 (1913);
  \textit{ibid}., vol.~28, p.~572 (1914);
  \textit{ibid}., vol.~28, p.~583 (1914).}
\Chapref{III} follows
a method which was first developed by Lewis and Tolman,\footnote
  {\textit{Phil.\ Mag.}, vol.~18, p.~510 (1909).}
and the
\Chapnumref[XIII]{last chapter} a method developed by Wilson and Lewis.\footnote
  {\textit{Proceedings of the American Academy of Arts and Sciences},
  vol.~48, p.~389 (1912).}
The writer
must also express his special obligations to the works of Einstein,
Planck, Poincaré, Laue, Ishiwara and Laub.

It is hoped that the mode of presentation is one that will be found
well adapted not only to introduce the study of relativity theory to
those previously unfamiliar with the subject but also to provide the
necessary methodological equipment for those who wish to pursue
the theory into its more complicated applications.

After presenting, in the \Chapnumref[I]{first chapter}, a brief outline of the historical
development of ideas as to the nature of the space and time of science,
we consider, in \Chapref{II}, the two main postulates upon which the
theory of relativity rests and discuss the direct experimental evidence
for their truth. The \Chapnumref[III]{third chapter} then presents an elementary and
%% -----File: 017.png---Folio 3-------
non-mathematical deduction of a number of the most important
consequences of the postulates of relativity, and it is hoped that this
chapter will prove especially valuable to readers without unusual
mathematical equipment, since they will there be able to obtain a
real grasp of such important new ideas as the change of mass with
velocity, the non-additivity of velocities, and the relation of mass
and energy, without encountering any mathematics beyond the
elements of analysis and geometry.

In \Chapref{IV} we commence the more analytical treatment of
the theory of relativity by obtaining from the two postulates of
relativity Einstein's transformation equations for space and time as
well as transformation equations for velocities, accelerations, and
for an important function of the velocity. \Chapref{V} presents
various kinematical applications of the theory of relativity following
quite closely Einstein's original method of development. In particular
we may call attention to the ease with which we may handle
the optics of moving media by the methods of the theory of relativity
as compared with the difficulty of treatment on the basis of the ether
theory.

In Chapters \Chapnumref{VI},~\Chapnumref{VII} and~\Chapnumref{VIII} we develop and apply a theory of
the dynamics of a particle which is based on the Einstein transformation
equations for space and time, Newton's three laws of motion,
and the principle of the conservation of mass.

We then examine, in \Chapref{IX}, the relation between the theory
of relativity and the principle of least action, and find it possible to
introduce the requirements of relativity theory at the very start into
this basic principle for physical science. We point out that we
might indeed have used this adapted form of the principle of least
action, for developing the dynamics of a particle, and then proceed
in Chapters \Chapnumref{X},~\Chapnumref{XI} and~\Chapnumref{XII} to develop the dynamics of an elastic
body, the dynamics of a thermodynamic system, and the dynamics
of an electromagnetic system, all on the basis of our adapted form
of the principle of least action.

Finally, in \Chapref{XIII}, we consider a four-dimensional method
of expressing and treating the results of relativity theory. This
chapter contains, in Part~I, an epitome of some of the more important
methods in four-dimensional vector analysis and it is hoped that it
%% -----File: 018.png---Folio 4-------
can also be used in connection with the earlier parts of the book as a
convenient reference for those who are not familiar with ordinary
three-dimensional vector analysis.

In the present book, the writer has confined his considerations to
cases in which there is a \emph{uniform} relative velocity between systems of
coördinates. In the future it may be possible greatly to extend the
applications of the theory of relativity by considering accelerated
systems of coördinates, and in this connection Einstein's latest work
on the relation between gravity and acceleration is of great interest.
It does not seem wise, however, at the present time to include such
considerations in a book which intends to present a survey of accepted
theory.

The author will feel amply repaid for the work involved in the
preparation of the book if, through his efforts, some of the younger
American physicists can be helped to obtain a real knowledge of the
important work of Einstein. He is also glad to have this opportunity
to add his testimony to the growing conviction that the conceptual
space and time of science are not God-given and unalterable, but are
rather in the nature of human constructs devised for use in the description
and correlation of scientific phenomena, and that these
spatial and temporal concepts should be altered whenever the discovery
of new facts makes such a change pragmatic.

The writer wishes to express his indebtedness to Mr.~William~H.
Williams for assisting in the preparation of Chapter~I\@. %[** TN: Not a useful cross-reference]
%% -----File: 019.png---Folio 5-------


\Chapter{I}{Historical Development of Ideas as to the Nature of
Space and Time.}
\SetRunningHeads{Chapter One.}{Historical Development.}

\Paragraph{1.} Since the year 1905, which marked the publication of Einstein's
momentous article on the theory of relativity, the development of
scientific thought has led to a complete revolution in accepted ideas
as to the nature of space and time, and this revolution has in turn
profoundly modified those dependent sciences, in particular mechanics
and electromagnetics, which make use of these two fundamental
concepts in their considerations.

In the following pages it will be our endeavor to present a description
of these new notions as to the nature of space and time,\footnote
  {Throughout this work by ``space'' and ``time'' we shall mean the \emph{conceptual}
  space and time of science.}
and to give a partial account of the consequent modifications which
have been introduced into various fields of science. Before proceeding
to this task, however, we may well review those older ideas
as to space and time which until now appeared quite sufficient for
the correlation of scientific phenomena. We shall first consider the
space and time of Galileo and Newton which were employed in the
development of the classical mechanics, and then the space and time
of the ether theory of light.


\Section[I]{The Space and Time of Galileo and Newton.}

\Paragraph{2.}  The publication in 1687 of Newton's \textit{Principia} laid down so
satisfactory a foundation for further dynamical considerations, that
it seemed as though the ideas of Galileo and Newton as to the nature
of space and time, which were there employed, would certainly remain
forever suitable for the interpretation of natural phenomena. And
indeed upon this basis has been built the whole structure of classical
mechanics which, until our recent familiarity with very high velocities,
has been found completely satisfactory for an extremely large number
of very diverse dynamical considerations.
%% -----File: 020.png---Folio 6-------

An examination of the fundamental laws of mechanics will show
how the concepts of space and time entered into the Newtonian
system of mechanics. Newton's laws of motion, from which the
whole of the classical mechanics could be derived, can best be stated
with the help of the equation
\[
\vc{F} = \frac{d}{dt} (m\vc{u}).
\Tag{1}
\]
This equation defines the force~$\vc{F}$ acting on a particle as equal to the
rate of change in its momentum (\ie, the product of its mass~$m$ and
its velocity~$\vc{u}$), and the whole of Newton's laws of motion may be
summed up in the statement that in the case of two interacting particles
the forces which they mutually exert on each other are equal in
magnitude and opposite in direction.

Since in Newtonian mechanics the mass of a particle is assumed
constant, equation~(1) may be more conveniently written
\[
\vc{F}
  = m \frac{d\vc{u}}{dt}
  = m \frac{d}{dt} \left( \frac{d\vc{r}}{dt} \right),
\]
or
\[
\begin{aligned}
  F_x &= m \frac{d}{dt} \left( \frac{dx}{dt} \right),\\
  F_y &= m \frac{d}{dt} \left( \frac{dy}{dt} \right),\\
  F_z &= m \frac{d}{dt} \left( \frac{dz}{dt} \right),
\end{aligned}
\Tag{2}
\]
and this definition of force, together with the above-stated principle
of the equality of action and reaction, forms the starting-point for
the whole of classical mechanics.

The necessary dependence of this mechanics upon the concepts
of space and time becomes quite evident on an examination of this
fundamental equation~(2), in which the expression for the force acting
on a particle is seen to contain both the variables $x$,~$y$, and~$z$, which
specify the position of the particle in \emph{space}, and the variable~$t$, which
specifies the \emph{time}.

\Subsubsection{3}{Newtonian Time.} To attempt a definite statement as to the
%% -----File: 021.png---Folio 7-------
meaning of so fundamental and underlying a notion as that of time
is a task from which even philosophy may shrink. In a general
way, conceptual time may be thought of as a \emph{one-dimensional}, \emph{unidirectional},
\emph{one-valued} continuum. This continuum is a sort of framework
in which the instants at which actual occurrences take place
find an ordered position. Distances from point to point in the
continuum, that is intervals of time, are measured by the periods of
certain continually recurring cyclic processes such as the daily rotation
of the earth. A unidirectional nature is imposed upon the time
continuum among other things by an acceptance of the second law
of thermodynamics, which requires that actual progression in time
shall be accompanied by an increase in the entropy of the material
world, and this same law requires that the continuum shall be one-valued
since it excludes the possibility that time ever returns upon
itself, either to commence a new cycle or to intersect its former path
even at a single point.

In addition to these characteristics of the time continuum, which
have been in no way modified by the theory of relativity, the \emph{Newtonian
mechanics always assumed a complete independence of time and
the three-dimensional space continuum} which exists along with it.
In dynamical equations time entered as an \emph{entirely independent} variable
in no way connected with the variables whose specification
determines position in space. In the following pages, however, we
shall find that the theory of relativity requires a very definite interrelation
between time and space, and in the Einstein transformation
equations we shall see the exact way in which measurements of time
depend upon the choice of a set of variables for measuring position
in space.

\Subsubsection{4}{Newtonian Space.} An exact description of the concept of
space is perhaps just as difficult as a description of the concept of time.
In a general way we think of space as a \emph{three-dimensional}, \emph{homogeneous},
\emph{isotropic} continuum, and these ideas are common to the
conceptual spaces of Newton, Einstein, and the ether theory of light.
The space of Newton, however, differs on the one hand from that of
Einstein because of a tacit assumption of the complete independence
of space and time measurements; and differs on the other hand from
that of the ether theory of light by the fact that ``free'' space was
%% -----File: 022.png---Folio 8-------
assumed completely empty instead of filled with an all-pervading
quasi-material medium---the ether. A more definite idea of the particularly
important characteristics of the Newtonian concept of space
may be obtained by considering somewhat in detail the actual methods
of space measurement.

Positions in space are in general measured with respect to some
arbitrarily fixed system of reference which must be threefold in
character corresponding to the three dimensions of space. In particular
we may make use of a set of Cartesian axes and determine,
for example, the position of a particle by specifying its three Cartesian
coördinates $x$,~$y$ and~$z$.

In Newtonian mechanics the particular set of axes chosen for
specifying position in space has in general been determined in the
first instance by considerations of convenience. For example, it is
found by experience that, if we take as a reference system lines drawn
upon the surface of the earth, the equations of motion based on Newton's
laws give us a simple description of nearly all dynamical phenomena
which are merely terrestrial. When, however, we try to
interpret with these same axes the motion of the heavenly bodies, we
meet difficulties, and the problem is simplified, so far as planetary
motions are concerned, by taking a new reference system determined
by the sun and the fixed stars. But this system, in its turn, becomes
somewhat unsatisfactory when we take account of the observed
motions of the stars themselves, and it is finally convenient to take a
reference system relative to which the sun is moving with a velocity
of twelve miles per second in the direction of the constellation Hercules.
This system of axes is so chosen that the great majority of stars have
on the average no motion with respect to it, and the actual motion
of any particular star with respect to these coördinates is called the
peculiar motion of the star.

Suppose, now, we have a number of such systems of axes in uniform
relative motion; we are confronted by the problem of finding
some method of transposing the description of a given kinematical
occurrence from the variables of one of these sets of axes to those of
another. For example, if we have chosen a system of axes~$S$ and
have found an equation in $x$,~$y$,~$z$, and~$t$ which accurately describes the
motion of a given point, what substitutions for the quantities involved
%% -----File: 023.png---Folio 9-------
can be made so that the new equation thereby obtained will again
correctly describe the same phenomena when we measure the displacements
of the point relative to a new system of reference~$S'$
which is in uniform motion with respect to~$S$? The assumption of
Galileo and Newton that ``free'' space is entirely empty, and the
further tacit assumption of the complete independence of space and
time, led them to propose a very simple solution of the problem, and
the transformation equations which they used are generally called
the Galileo Transformation Equations to distinguish them from the
Einstein Transformation Equations which we shall later consider.

\Subsubsection{5}{The Galileo Transformation Equations.} Consider two systems
of right-angled coördinates, $S$~and~$S'$, which are in relative motion in
the $X$~direction with the velocity~$V$; for convenience let the $X$~axes,
$OX$~and~$O'X'$, of the two systems coincide in direction, and for further
simplification let us take as our zero point for time measurements the
instant when the two origins $O$~and~$O'$ coincide. Consider now a
point which at the time~$t$ has the coördinates $x$,~$y$ and~$z$ measured in
system~$S$. Then, according to the space and time considerations of
Galileo and Newton, the coördinates of the point with reference to
system~$S'$ are given by the following transformation equations:
\begin{align*}
x' &= x-Vt,  \Tag{3}\displaybreak[0] \\
y' &= y,     \Tag{4}\displaybreak[0] \\
z' &= z,     \Tag{5}\displaybreak[0] \\
t' &= t.     \Tag{6}
\end{align*}
These equations are fundamental for Newtonian mechanics, and may
appear to the casual observer to be self-evident and bound up with
necessary ideas as to the nature of space and time. Nevertheless,
the truth of the first and the last of these equations is absolutely
dependent on the unsupported assumption of the complete independence
of space and time measurements, and since in the Einstein
theory we shall find a very definite relation between space and time
measurements we shall be led to quite a different set of transformation
equations. Relations (3),~(4),~(5) and~(6) will be found, however, to
be the limiting form which the correct transformation equations assume
when the velocity between the systems~$V$ becomes small compared
%% -----File: 024.png---Folio 10-------
with that of light. Since until very recent times the human
race in its entire past history has been familiar only with velocities
that are small compared with that of light, it need not cause surprise
that the above equations, which are true merely at the limit, should
appear so self-evident.

\Paragraph{6.} Before leaving the discussion of the space and time system of
Newton and Galileo we must call attention to an important characteristic
which it has in common with the system of Einstein but
which is not a feature of that assumed by the ether theory. If we
have two systems of axes such as those we have just been considering,
we may with equal right consider either one of them at rest and the
other moving past it. All we can say is that the two systems are in
relative motion; it is meaningless to speak of either one as in any
sense ``\textit{absolutely}'' at rest. The equation $x' = x - Vt$ which we
use in transforming the description of a kinematical event from the
variables of system $S$ to those of system $S'$ is perfectly symmetrical
with the equation $x = x' + Vt'$ which we should use for a transformation
in the reverse direction. Of all possible systems no particular
set of axes holds a unique position among the others. We
shall later find that this important principle of the relativity of motion
is permanently incorporated into our system of physical science as
the \textit{first postulate of relativity}. This principle, common both to the
space of Newton and to that of Einstein, is not characteristic of the
space assumed by the classical theory of light. The space of this
theory was supposed to be filled with a stationary medium, the
luminiferous ether, and a system of axes stationary with respect to
this ether would hold a unique position among the other systems
and be the one peculiarly adapted for use as the ultimate system of
reference for the measurement of motions.

We may now briefly sketch the rise of the ether theory of light and
point out the permanent contribution which it has made to physical
science, a contribution which is now codified as the second postulate
of relativity.


\Section[II]{The Space and Time of the Ether Theory.}

\Subsubsection{7}{Rise of the Ether Theory.} Twelve years before the appearance
of the \textit{Principia}, Römer, a Danish astronomer, observed that an
%% -----File: 025.png---Folio 11-------
eclipse of one of the satellites of Jupiter occurred some ten minutes
later than the time predicted for the event from the known period
of the satellite and the time of the preceding eclipse. He explained
this delay by the hypothesis that it took light twenty-two minutes
to travel across the earth's orbit. Previous to Römer's discovery,
light was generally supposed to travel with infinite velocity. Indeed
Galileo had endeavored to find the speed of light by direct experiments
over distances of a few miles and had failed to detect any lapse of
time between the emission of a light flash from a source and its observation
by a distant observer. Römer's hypothesis has been repeatedly
verified and the speed of light measured by different methods
with considerable exactness. The mean of the later determinations
is $2.9986 × 10^{10}$ cm.~per second.

\Paragraph{8.} At the time of Römer's discovery there was much discussion
as to the nature of light. Newton's theory that it consisted of particles
or corpuscles thrown out by a luminous body was attacked by
Hooke and later by Huygens, who advanced the view that it was
something in the nature of wave motions in a supposed space-filling
medium or ether. By this theory Huygens was able to explain
reflection and refraction and the phenomena of color, but assuming
\emph{longitudinal} vibrations he was unable to account for polarization.
Diffraction had not yet been observed and Newton contested the
Hooke-Huygens theory chiefly on the grounds that it was contradicted
by the fact of rectilinear propagation and the formation of
shadows. The scientific prestige of Newton was so great that the
emission or corpuscular theory continued to hold its ground for a
hundred and fifty years. Even the masterly researches of Thomas
Young at the beginning of the nineteenth century were unable to
dislodge the old theory, and it was not until the French physicist,
Fresnel, about 1815, was independently led to an undulatory theory
and added to Young's arguments the weight of his more searching
mathematical analysis, that the balance began to turn. From this
time on the wave theory grew in power and for a period of eighty
years was not seriously questioned. This theory has for its essential
postulate the existence of an all-pervading medium, the ether, in
which wave disturbances can be set up and propagated. And the
physical properties of this medium became an enticing field of inquiry
and speculation.
%% -----File: 026.png---Folio 12-------

\Subsubsection{9}{Idea of a Stationary Ether.} Of all the various properties with
which the physicist found it necessary to endow the ether, for us the
most important is the fact that it must apparently remain stationary,
unaffected by the motion of matter through it. This conclusion was
finally reached through several lines of investigation. We may first
consider whether the ether would be dragged along by the motion of
nearby masses of matter, and, second, whether the ether enclosed in a
moving medium such as water or glass would partake in the latter's
motion.

\Subsubsection{10}{Ether in the Neighborhood of Moving Bodies.} About the
year 1725 the astronomer Bradley, in his efforts to measure the
parallax of certain fixed stars, discovered that the apparent position
of a star continually changes in such a way as to trace annually a
small ellipse in the sky, the apparent position always lying in the
plane determined by the line from the earth to the center of the
ellipse and by the direction of the earth's motion. On the corpuscular
theory of light this admits of ready explanation as Bradley himself
discovered, since we should expect the earth's motion to produce an
apparent change in the direction of the oncoming light, in just the
same way that the motion of a railway train makes the falling drops
of rain take a slanting path across the window pane. If $\DPtypo{\vc{c}}{c}$~be the
velocity of a light particle and $\DPtypo{\vc{v}}{v}$~the earth's velocity, the apparent or
relative velocity would be $\DPtypo{\vc{c - v}}{c - v}$ and the tangent of the angle of
aberration would be~$\dfrac{v}{c}$.

Upon the wave theory, it is obvious that we should \emph{also} expect a
similar aberration of light, provided only that the ether shall be
quite stationary and unaffected by the motion of the earth through it,
and this is one of the important reasons that most ether theories have
assumed a \emph{stationary ether unaffected by the motion of neighboring
matter}.\footnote
  {The most notable exception is the theory of Stokes, which did assume that
  the ether moved along with the earth and then tried to account for aberration with
  the help of a velocity potential, but this led to difficulties, as was shown by Lorentz.}

In more recent years further experimental evidence for assuming
that the ether is not dragged along by the neighboring motion of
large masses of matter was found by Sir Oliver Lodge. His final
experiments were performed with a large rotating spheroid of iron
%% -----File: 027.png---Folio 13-------
with a narrow groove around its equator, which was made the path
for two rays of light, one travelling in the direction of rotation and
the other in the opposite direction. Since by interference methods
no difference could be detected in the velocities of the two rays, here
also the conclusion was reached that \emph{the ether was not appreciably
dragged along by the rotating metal}.

\Subsubsection{11}{Ether Entrained in Dielectrics.} With regard to the action of
a moving medium on the ether which might be entrained within it,
experimental evidence and theoretical consideration here too finally
led to the supposition that the ether itself must remain perfectly
stationary. The earlier view first expressed by Fresnel, in a letter
written to Arago in 1818, was that the entrained ether did receive a
fraction of the total velocity of the moving medium. Fresnel gave
to this fraction the value~$\dfrac{\mu^2-1}{\mu^2}$, where $\mu$~is the index of refraction of
the substance forming the medium. On this supposition, Fresnel
was able to account for the fact that Arago's experiments upon the
reflection and refraction of stellar rays show no influence whatever
of the earth's motion, and for the fact that Airy found the same angle
of aberration with a telescope filled with water as with air. Moreover,
the later work of Fizeau and the accurate determinations of
Michelson and Morley on the velocity of light in a moving stream
of water did show that the speed was changed by an amount corresponding
to Fresnel's fraction. The fuller theoretical investigations
of Lorentz, however, did not lead scientists to look upon this increased
velocity of light in a moving medium as an evidence that the ether
is pulled along by the stream of water, and we may now briefly sketch
the developments which culminated in the Lorentz theory of a completely
stationary ether.

\Subsubsection{12}{The Lorentz Theory of a Stationary Ether.} The considerations
of Lorentz as to the velocity of light in moving media became
possible only after it was evident that optics itself is a branch of the
wider science of electromagnetics, and it became possible to treat
transparent media as a special case of dielectrics in general. In 1873,
in his \textit{Treatise on Electricity and Magnetism}, Maxwell first advanced
the theory that electromagnetic phenomena also have their seat in
the luminiferous ether and further that light itself is merely an electromagnetic
%% -----File: 028.png---Folio 14-------
disturbance in that medium, and Maxwell's theory was
confirmed by the actual discovery of electromagnetic waves in 1888
by Hertz.

The attack upon the problem of the relative motion of matter and
ether was now renewed with great vigor both theoretically and experimentally
from the electromagnetic side. Maxwell in his treatise had
confined himself to phenomena in stationary media. Hertz, however,
extended Maxwell's considerations to moving matter on the assumption
that the entrained ether is carried bodily along by it. It is evident,
however, that in the field of optical theory such an assumption
could not be expected to account for the Fizeau experiment, which
had already been explained on the assumption that the ether receives
only a fraction of the velocity of the moving medium; while in the
field of electromagnetic theory it was found that Hertz's assumptions
would lead us to expect \emph{no} production of a magnetic field in the
neighborhood of a rotating electric condenser providing the plates of
the condenser and the dielectric move together with the same speed
and this was decisively disproved by the experiment of Eichenwald.
The conclusions of the Hertz theory were also out of agreement with
the important experiments of H.~A.~Wilson on moving dielectrics.
It remained for Lorentz to develop a general theory for moving
dielectrics which was consistent with the facts.

The theory of Lorentz developed from that of Maxwell by the
addition of the idea of the \emph{electron}, as the atom of electricity, and his
treatment is often called the ``electron theory.'' This atomistic
conception of electricity was foreshadowed by Faraday's discovery
of the quantitative relations between the amount of electricity associated
with chemical reactions in electrolytes and the weight of
substance involved, a relation which indicates that the atoms act as
carriers of electricity and that the quantity of electricity carried by a
single particle, whatever its nature, is always some small multiple of a
definite quantum of electricity, the electron. Since Faraday's time,
the study of the phenomena accompanying the conduction of electricity
through gases, the study of radioactivity, and finally indeed
the isolation and exact measurement of these atoms of electrical
charge, have led us to a very definite knowledge of many of the
properties of the electron.
%% -----File: 029.png---Folio 15-------

While the experimental physicists were at work obtaining this
more or less first-hand acquaintance with the electron, the theoretical
physicists and in particular Lorentz were increasingly successful in
explaining the electrical and optical properties of matter in general
on the basis of the behavior of the electrons which it contains, the
properties of conductors being accounted for by the presence of movable
electrons, either free as in the case of metals or combined with
atoms to form ions as in electrolytes, while the electrical and optical
properties of dielectrics were ascribed to the presence of electrons
more or less bound by quasi-elastic forces to positions of equilibrium.
This Lorentz electron theory of matter has been developed in great
mathematical detail by Lorentz and has been substantiated by numerous
quantitative experiments. Perhaps the greatest significance
of the Lorentz theory is that such properties of matter as electrical
conductivity, magnetic permeability and dielectric inductivity, which
occupied the position of rather accidental experimental constants in
Maxwell's original theory, are now explainable as the statistical result
of the behavior of the individual electrons.

With regard now to our original question as to the behavior of
\emph{moving} optical and dielectric media, the Lorentz theory was found
capable of accounting quantitatively for all known phenomena, including
Airy's experiment on aberration, Arago's experiments on the
reflection and refraction of stellar rays, Fresnel's coefficient for the
velocity of light in moving media, and the electromagnetic experiments
upon moving dielectrics made by Röntgen, Eichenwald, H.~A.~Wilson,
and others. For us the particular significance of the Lorentz
method of explaining these phenomena is that he does \emph{not} assume, as
did Fresnel, that the ether is partially dragged along by moving
matter. His investigations show rather that the ether must remain
perfectly stationary, and that such phenomena as the changed velocity
of light in moving media are to be accounted for by the modifying
influence which the electrons in the moving matter have upon the
propagation of electromagnetic disturbances, rather than by a dragging
along of the ether itself.

Although it would not be proper in this place to present the
mathematical details of Lorentz's treatment of moving media, we
may obtain a clearer idea of what is meant in the Lorentz theory by a
%% -----File: 030.png---Folio 16-------
stationary ether if we look for a moment at the five fundamental
equations upon which the theory rests. These familiar equations, of
which the first four are merely Maxwell's four field equations, modified
by the introduction of the idea of the electron, may be written
\begin{align*}
\curl \vc{h}
  &= \frac{1}{c}\, \frac{\partial \vc{e}}{\partial t}
     + \rho\, \frac{\vc{u}}{c},\\
\curl \vc{e}
  &= -\frac{1}{c}\, \frac{\partial \vc{h}}{\partial t},\\
\divg \vc{e} &= \rho,\\
\divg \vc{h} &= 0,\\
\vc{f} &= \rho\left\{
  \vc{e} + \left[\frac{\vc{u}}{c} × \vc{h}\right]^*\right\}
\end{align*}
in which the letters have their usual significance. (See \Chapref{XII}\@.)
Now the whole of the Lorentz theory, including of course his treatment
of moving media, is derivable from these five equations, and
the fact that the idea of a stationary ether does lie at the basis of
his theory is most clearly shown by the first and last of these equations,
which contain the velocity~$\vc{u}$ with which the charge in question
is moving, and \emph{for Lorentz this velocity is to be measured with respect
to the assumed stationary ether}.

We have devoted this space to the Lorentz theory, since his work
marks the culmination of the ether theory of light and electromagnetism,
and for us the particularly significant fact is that by this
line of attack science was \emph{inevitably led to the idea of an absolutely
immovable and stationary ether}.

\Paragraph{13.} We have thus briefly traced the development of the ether
theory of light and electromagnetism. We have seen that the space
continuum assumed by this theory is not empty as was the space of
Newton and Galileo but is assumed filled with a stationary medium,
the ether, and in conclusion should further point out that the \emph{time
continuum} assumed by the ether theory was apparently the same as
that of Newton and Galileo, and in particular that the \emph{old ideas as to
the absolute independence of space and time were all retained}.
%% -----File: 031.png---Folio 17-------


\Section[III]{Rise of the Einstein Theory of Relativity.}

\Subsubsection{14}{The Michelson-Morley Experiment.} In spite of all the brilliant
achievements of the theory of a stationary ether, we must now
call attention to an experiment, performed at the very time when
the success of the ether theory seemed most complete, whose result
was in direct contradiction to its predictions. This is the celebrated
Michelson-Morley experiment, and to the masterful interpretation of
its consequences at the hands of Einstein we owe the whole theory of
relativity, a theory which will nevermore permit us to assume that
space and time are independent.

If the theory of a stationary ether were true we should find, contrary
to the expectations of Newton, that systems of coördinates in
relative motion are not symmetrical, a system of axes fixed relatively
to the ether would hold a unique position among all other systems
moving relative to it and would be peculiarly adapted for the measurement
of displacements and velocities. Bodies at rest with respect
to this system of axes fixed in the ether would be spoken of as ``absolutely''
at rest and bodies in motion through the ether would be
said to have ``absolute'' motion. From the point of view of the
ether theory one of the most important physical problems would be
to determine the velocity of various bodies, for example that of the
earth, through the ether.

Now the Michelson-Morley experiment was devised for the very
purpose of determining the relative motion of the earth and the ether.
The experiment consists essentially in a comparison of the velocities
of light parallel and perpendicular to the earth's motion in its orbit.
A ray of light from the source~$S$ falls on the half silvered mirror~$A$,
where it is divided into two rays, one of which travels to the mirror~$B$
and the other to the mirror~$C$, where they are totally reflected. The
rays are recombined and produce a set of interference fringes at~$\DPtypo{0}{O}$.
(See \Figref{1}.)

We may now think of the apparatus as set so that one of the
divided paths is parallel to the earth's motion and the other perpendicular
to it. On the basis of the stationary ether theory, the
velocity of the light with reference to the apparatus would evidently
be different over the two paths, and hence on rotating the apparatus
%% -----File: 032.png---Folio 18-------
through an angle of ninety degrees we should expect a shift in the
position of the fringes. Knowing the magnitude of the earth's
velocity in its orbit and the dimensions of the apparatus, it is quite
possible to calculate the magnitude of the expected shift, a quantity
\begin{figure}[hbt]
  \begin{center}
    \Fig{1}
    \Input[3in]{032}
  \end{center}
\end{figure}
entirely susceptible of experimental determination. Nevertheless the
most careful experiments made at different times of day and at
different seasons of the year entirely failed to show any such shift
at all.

This result is in direct contradiction to the theory of a stationary
ether and could be reconciled with that theory only by very arbitrary
assumptions. Instead of making such assumptions, the Einstein
theory of relativity finds it preferable to return in part to the older
ideas of Newton and Galileo.

\Subsubsection{15}{The Postulates of Einstein.} In fact, in accordance with the
results of this work of Michelson-Morley and other confirmatory
experiments, the Einstein theory takes as its \emph{first postulate} the idea
familiar to Newton of the relativity of all motion. It states that
there is nothing out in space in the nature of an ether or of a fixed
set of coördinates with regard to which motion can be measured,
that there is no such thing as absolute motion, and that all we can
speak of is the relative motion of one body with respect to another.
%% -----File: 033.png---Folio 19-------

Although we thus see that the Einstein theory of relativity has
returned in part to the ideas of Newton and Galileo as to the nature
of space, it is not to be supposed that the ether theory of light and
electromagnetism has made no lasting contribution to physical science.
Quite on the contrary, not only must the ideas as to the periodic and
polarizable nature of the light disturbance, which were first appreciated
and understood with the help of the ether theory, always
remain incorporated in every optical theory, but in particular the
Einstein theory of relativity takes as the basis for its \emph{second postulate}
a principle that has long been familiar to the ether theory, namely
that the velocity of light is independent of the velocity of the source.
We shall see in following chapters that it is the combination of this
principle with the first postulate of relativity that leads to the whole
theory of relativity and to our new ideas as to the nature and interrelation
of space and time.
%% -----File: 034.png---Folio 20-------



\Chapter{II}{The Two Postulates of the Einstein Theory of
Relativity.}
\SetRunningHeads{Chapter Two.}{The Two Postulates.}

\Paragraph{16.} There are two general methods of evaluating the theoretical
development of any branch of science. One of these methods is to
test by direct experiment the fundamental postulates upon which
the theory rests. If these postulates are found to agree with the facts,
we may feel justified in assuming that the whole theoretical structure
is a valid one, providing false logic or unsuspected and incorrect
assumptions have not later crept in to vitiate the conclusions. The
other method of testing a theory is to develop its interlacing chain of
propositions and theorems and examine the results both for their
internal coherence and for their objective validity. If we find that
the conclusions drawn from the theory are neither self-contradictory
nor contradictory of each other, and furthermore that they agree
with the facts of the external world, we may again feel that our theory
has achieved a measure of success. In the present chapter we shall
present the two main postulates of the theory of relativity, and indicate
the direct experimental evidence in favor of their truth. In following
chapters we shall develop the consequences of these postulates, show
that the system of consequences stands the test of internal coherence,
and wherever possible compare the predictions of the theory with
experimental facts.


\Subsection{The First Postulate of Relativity.}

\Paragraph{17.} The first postulate of relativity as originally stated by Newton
was that it is impossible to measure or detect absolute translatory
motion through space. No objections have ever been made to this
statement of the postulate in its original form. In the development
of the theory of relativity, the postulate has been modified to include
the impossibility of detecting translatory motion through any medium
or ether which might be assumed to pervade space.

In support of the principle is the general fact that no effects due
to the motion of the earth or other body through the supposed ether
%% -----File: 035.png---Folio 21-------
have ever been observed. Of the many unsuccessful attempts to
detect the earth's motion through the ether we may call attention to
the experiments on the refraction of light made by Arago, Respighi,
Hoek, Ketteler and Mascart, the interference experiments of Ketteler
and Mascart, the work of Klinkerfuess and Haga on the position of
the absorption bands of sodium, the experiment of Nordmeyer on the
intensity of radiation, the experiments of Fizeau, Brace and Strasser
on the rotation of the plane of polarized light by transmission through
glass plates, the experiments of Mascart and of Rayleigh on the
rotation of the plane of polarized light in naturally active substances,
the electromagnetic experiments of Röntgen, Des Coudres, J.~Koenigsberger,
Trouton, Trouton and Noble, and Trouton and Rankine, and
finally the Michelson and Morley experiment, with the further work
of Morley and Miller. For details as to the nature of these experiments
the reader may refer to the original articles or to an excellent
discussion by Laub of the experimental basis of the theory of relativity.\footnote
  {\textit{Jahrbuch der Radioaktivität}, vol.~7, p.~405 (1910).}

In none of the above investigations was it possible to detect any
effect attributable to the earth's motion through the ether. Nevertheless
a number of these experiments \emph{are} in accord with the final
form given to the ether theory by Lorentz, especially since his work
satisfactorily accounts for the Fresnel coefficient for the changed
velocity of light in moving media. Others of the experiments mentioned,
however, could be made to accord with the Lorentz theory
only by very arbitrary assumptions, in particular those of Michelson
and Morley, Mascart and Rayleigh, and Trouton and Noble. For
the purposes of our discussion we shall accept the principle of the
relativity of motion as an experimental fact.


\Subsection{The Second Postulate of the Einstein Theory of Relativity.}

\Paragraph{18.} The second postulate of relativity states that \emph{the velocity of
light in free space appears the same to all observers regardless of the
relative motion of the source of light and the observer}. This postulate
may be obtained by combining the first postulate of relativity with a
principle which has long been familiar to the ether theory of light.
This principle states that the velocity of light is unaffected by a
motion of the emitting source, in other words, that the velocity with
%% -----File: 036.png---Folio 22-------
which light travels past any observer is not increased by a motion
of the source of light towards the observer. The first postulate of
relativity adds the idea that a motion of the source of light towards
the observer is identical with a motion of the observer towards the
source. The second postulate of relativity is seen to be merely a
combination of these two principles, since it states that the velocity
of light in free space appears the same to all observers regardless \emph{both}
of the motion of the source of light and of the observer.

\Paragraph{19.} It should be pointed out that the two principles whose combination
thus leads to the second postulate of Einstein have come
from very different sources. The first postulate of relativity practically
denies the existence of any stationary ether through which
the earth, for instance, might be moving. On the other hand, the
principle that the velocity of light is unaffected by a motion of the
source was originally derived from the idea that light is transmitted
by a stationary medium which does not partake in the motion of the
source. This combination of two principles, which from a historical
point of view seem somewhat contradictory in nature, has given to
the second postulate of relativity a very extraordinary content.
Indeed it should be particularly emphasized that the remarkable
conclusions as to the nature of space and time forced upon science
by the theory of relativity are the special product of the second
postulate of relativity.

A simple example of the conclusions which can be drawn from
this postulate will make its extraordinary nature evident.
\begin{figure}[hbt]
  \begin{center}
    \Fig{2}
    \Input{036}
  \end{center}
\end{figure}

$S$~is a source of light and $A$~and~$B$ two moving systems. $A$~is
moving \emph{towards} the source~$S$, and $B$~\emph{away} from it. Observers on the
systems mark off equal distances $aa'$~and~$bb'$ along the path of the light
and determine the time taken for light to pass from $a$~to~$a'$ and $b$~to~$b'$
respectively. Contrary to what seem the simple conclusions of
common sense, the second postulate requires that the time taken
%% -----File: 037.png---Folio 23-------
for the light to pass from $a$~to~$a'$ shall measure the same as the time
for the light to go from $b$~to~$b'$. Hence if the second postulate of
relativity is correct it is not surprising that science is forced in general
to new ideas as to the nature of space and time, ideas which are in
direct opposition to the requirements of so-called common sense.


\Subsection{Suggested Alternative to the Postulate of the Independence of the
Velocity of Light and the Velocity of the Source.}

\Paragraph{20.} Because of the extraordinary conclusions derived by combining
the principle of the relativity of motion with the postulate
that the velocity of light is independent of the velocity of its source,
a number of attempts have been made to develop so-called \emph{emission}
theories of relativity based on the principle of the relativity of motion
and the further postulate that the velocity of light and the velocity
of its source are additive.

Before examining the available evidence for deciding between the
rival principles as to the velocity of light, we may point out that
this proposed postulate, of the additivity of the velocity of source
and light, would as a matter of fact lead to a very simple kind of
relativity theory without requiring any changes in our notions of
space and time. For if light or other electromagnetic disturbance
which is being emitted from a source did partake in the motion of
that source in such a way that the velocity of the source is added to
the velocity of emission, it is evident that a system consisting of the
source and its surrounding disturbances would act as a whole and
suffer no \emph{permanent} change in configuration if the velocity of the
source were changed. This result would of course be in direct agreement
with the idea of the relativity of motion which merely requires
that the physical properties of a system shall be independent of its
velocity through space.

As a particular example of the simplicity of emission theories we
may show, for instance, how easily they would account for the negative
\begin{wrapfigure}{l}{2in}%[** TN: Width-dependent line break]
  \Fig{3}
  \Input[2in]{038}
\end{wrapfigure}
result of the Michelson-Morley experiment. If~$O$, \Figref{3}, is a
source of light and $A$~and~$B$ are mirrors placed a meter away from~$O$, the
Michelson-Morley experiment shows that the time taken for light to
travel to~$A$ and back is the same as for the light to travel to~$B$ and
back, in spite of the fact that the whole apparatus is moving through
space in the direction $O - B$, due to the earth's motion around the sun.
%% -----File: 038.png---Folio 24-------
The basic assumption of emission theories, however, would require
exactly this result, since it says that light travels out from~$O$ with a
constant velocity in all directions with
respect to~$O$, and not with respect to
some ether through which $O$~is supposed
to be moving.

The problem now before us is to
decide between the two rival principles
as to the velocity of light, and we shall
find that the bulk of the evidence is all
in favor of the principle which has led
to the Einstein theory of relativity with
its complete revolution in our ideas as to space and time, and against
the principle which has led to the superficially simple emission theories
of relativity.

\Subsubsection{21}{Evidence Against Emission Theories of Light.} All emission
theories agree in assuming that light from a moving source has a
velocity equal to the vector sum of the velocity of light from a stationary
source and the velocity of the source itself at the instant of
emission. And without first considering the special assumptions
which distinguish one emission theory from another we may first
present certain astronomical evidence which apparently stands in
contradiction to this basic assumption of all forms of emission
theory. This evidence was pointed out by Comstock\footnote
  {\textit{Phys.\ Rev}., vol.~30, p.~291 (1910).}
and later by
de Sitter.\footnote
  {\textit{Phys.\ Zeitschr}., vol.~14, pp.~429, 1267 (1913).}

Consider the rotation of a binary star as it would appear to an
observer situated at a considerable distance from the star and in its
plane of rotation. (See \Figref{4}.) If an emission theory of light
be true, the velocity of light from the star in position~$A$ will be $c + u$,
where $u$~is the velocity of the star in its orbit, while in the position~$B$
the velocity will be $c - u$. Hence the star will be observed to arrive
in position~$A$, $\dfrac{l}{c+u}$~seconds after the event has actually occurred, and
in position~$B$, $\dfrac{l}{c-u}$~seconds after the event has occurred. This will
%% -----File: 039.png---Folio 25-------
make the period of half rotation from $A$~to~$B$ appear to be
\[
\Delta t - \frac{l}{c+u} + \frac{l}{c-u} = \Delta t + \frac{2ul}{c^2},
\]
where $\Delta t$~is the actual time of a half rotation in the orbit, which for
\begin{figure}[hbt]
  \begin{center}
    \Fig{4}
    \Input[3.25in]{039}
  \end{center}
\end{figure}
simplicity may be taken as circular. On the other hand, the period
of the next half rotation from $B$ back to~$A$ would appear to be
\[
\Delta t - \frac{2ul}{c^2}.
\]

Now in the case of most spectroscopic binaries the quantity~$\dfrac{2ul}{c^2}$
is not only of the same order of magnitude as~$\Delta t$ but oftentimes probably
even larger. Hence, if an emission theory of light were true,
we could hardly expect without correcting for the variable velocity
of light to find that these orbits obey Kepler's laws, as is actually
the case. This is certainly very strong evidence against any form
of emission theory. It may not be out of place, however, to state
briefly the different forms of emission theory which have been tried.

\Subsubsection{22}{Different Forms of Emission Theory.} As we have seen, emission
theories all agree in assuming that light from a moving source
%% -----File: 040.png---Folio 26-------
has a velocity equal to the vector sum of the velocity of light from a
stationary source and the velocity of the source itself at the instant
of emission. Emission theories differ, however, in their assumptions
as to the velocity of light after its reflection from a mirror. The three
assumptions which up to this time have been particularly considered
are (1)~that the excited portion of the reflecting mirror acts as a new
source of light and that the reflected light has the same velocity~$c$
with respect to the mirror as has original light with respect to its source;
(2)~that light reflected from a mirror acquires a component of velocity
equal to the velocity of the mirror image of the original source, and
hence has the velocity~$c$ with respect to this mirror image; and (3)~that
light retains throughout its whole path the component of velocity
which it obtained from its original moving source, and hence after
reflection spreads out with velocity~$c$ in a spherical form around a
center which moves with the same speed as the original source.

Of these possible assumptions as to the velocity of reflected light,
the first seems to be the most natural and was early considered by the
author but shown to be incompatible, not only with an experiment
which he performed on the velocity of light from the two limbs of
the sun,\footnote
  {\textit{Phys.\ Rev}., vol.~31, p.~26 (1910).}
but also with measurements of the Stark effect in canal
rays.\footnote
  {\textit{Phys.\ Rev}., vol.~35, p.~136 (1912).}
The second assumption as to the velocity of light was made
by Stewart,\footnote
  {\textit{Phys.\ Rev}., vol.~32, p.~418 (1911).}
but has also been shown\footnotemark[2] %[** TN: Repeated footnote here, below]
to be incompatible with
measurements of the Stark effect in canal rays. Making use of the
third assumption as to the velocity of reflected light, a somewhat
complete emission theory has been developed by Ritz,\footnote
  {\textit{Ann.\ de chim.\ et phys}., vol.~13, p.~145 (1908);
  \textit{Arch.\ de Génève} vol.~26, p.~232
  (1908); \textit{Scientia}, vol.\ 5 (1909).}
and unfortunately
optical experiments for deciding between the Einstein
and Ritz relativity theories have never been performed, although
such experiments are entirely possible of performance.\footnotemark[2] Against the
Ritz theory, however, we have of course the general astronomical
evidence of Comstock and de Sitter which we have already described
above.

For the present, the observations described above, comprise the
whole of the direct experimental evidence against emission theories
%% -----File: 041.png---Folio 27-------
of light and in favor of the principle which has led to the second
postulate of the Einstein theory. One of the consequences of the
Einstein theory, however, has been the deduction of an expression
for the mass of a moving body which has been closely verified by the
Kaufmann-Bucherer experiment. Now it is very interesting to note,
that starting with what has thus become an \emph{experimental} expression
for the mass of a moving body, it is possible to work backwards to a
derivation of the second postulate of relativity. For the details of
the proof we must refer the reader to the original article.\footnote
  {\textit{Phys.\ Rev}., vol.\ 31, p.\ 26 (1910).}


\Subsection{Further Postulates of the Theory of Relativity.}

\Paragraph{23.} In the development of the theory of relativity to which we
shall now proceed we shall of course make use of many postulates.
The two which we have just considered, however, are the only ones,
so far as we are aware, which are essentially different from those
common to the usual theoretical developments of physical science.
In particular in our further work we shall assume without examination
all such general principles as the homogeneity and isotropism of the
space continuum, and the unidirectional, one-valued, one-dimensional
character of the time continuum. In our treatment of the dynamics
of a particle we shall also assume Newton's laws of motion, and the
principle of the conservation of mass, although we shall find, of course,
that the Einstein ideas as to the connection between space and time
will lead us to a non-Newtonian mechanics. We shall also make
extensive use of the principle of least action, which we shall find a
powerful principle in all the fields of dynamics.
%% -----File: 042.png---Folio 28-------


\Chapter{III}{Some Elementary Deductions.}
\SetRunningHeads{Chapter Three.}{Some Elementary Deductions.}

\Paragraph{24.} In order gradually to familiarize the reader with the consequences
of the theory of relativity we shall now develop by very
elementary methods a few of the more important relations. In this
preliminary consideration we shall lay no stress on mathematical
elegance or logical exactness. It is believed, however, that the
chapter will present a substantially correct account of some of the
more important conclusions of the theory of relativity, in a form
which can be understood even by readers without mathematical
equipment.


\Subsection{Measurements of Time in a Moving System.}

\Paragraph{25.} We may first derive from the postulates of relativity a relation
connecting measurements of time intervals as made by observers in
systems moving with different velocities. Consider a system~$S$
(\Figref[Fig.]{5}) provided with a plane mirror~$a\, a$, and an observer~$A$, who
\begin{figure}[hbt]
  \begin{center}
    \Fig{5}
    \Input[3.25in]{042}
  \end{center}
\end{figure}
has a clock so that he can determine the time taken for a beam of
light to travel up to the mirror and back along the path~$A\, m\, A$.
Consider also another similar system~$S'$, provided with a mirror~$b\, b$,
and an observer~$B$, who also has a clock for measuring the time it
takes for light to go up to his mirror and back. System~$S'$ is moving
past~$S$ with the velocity~$V$, the direction of motion being parallel
to the mirrors $a\, a$~and~$b\, b$, the two systems being arranged, moreover,
%% -----File: 043.png---Folio 29-------
so that when they pass one another the two mirrors $a\,a$~and~$b\,b$
will coincide, and the two observers $A$~and~$B$ will also come into
coincidence.

$A$,~considering his system at rest and the other in motion, measures
the time taken for a beam of light to pass to his mirror and return,
over the path~$A\, m\, A$, and compares the time interval thus obtained
with that necessary for the performance of a similar experiment
by~$B$, in which the light has to pass over a longer path such as~$B\, n\, B'$,
shown in \Figref{6}, where $B\, B'$~is the distance through which the
\begin{figure}[hbt]
  \begin{center}
    \Fig{6}
    \Input{043}
  \end{center}
\end{figure}
observer~$B$ has moved during the time taken for the passage of the
light up to the mirror and back.

Since, in accordance with the second postulate of relativity, the
velocity of light is independent of the velocity of its source, it is
evident that the ratio of these two time intervals will be proportional
to the ratio of the two paths $A\, m\, A$~and~$B\, n\, B'$, and this can easily
be calculated in terms of the velocity of light~$c$ and the velocity~$V$
of the system~$S'$.

From \Figref{6} we have
\[
(A\, m)^2 = (p\, n)^2 = (B\, n)^2 - (B\, p)^2.
\]
Dividing by $(B\, n)^2$,
\[
\frac{(A\, m)^2}{(B\, n)^2} = 1 - \frac{(B\, p)^2}{(B\, n)^2}.
\]
But the distance $B\, p$ is to $B\, n$ as $V$ is to~$c$, giving us
\[
\frac{A\, m }{ B\, n} =  \sqrt{1 - \frac{V^2}{c^2}},
\]
%% -----File: 044.png---Folio 30-------
and hence $A$~will find, either by calculation or by direct measurement
if he has arranged clocks at $B$~and~$B'$, that it takes a longer time for
the performance of $B$'s~experiment than for the performance of his
own in the ratio $1: \sqrt{ 1 - \dfrac{V^2}{c^2}}$.

It is evident from the first postulate of relativity, however, that
$B$~himself must find exactly the same length of time for the light to
pass up to his mirror and come back as did~$A$ in his experiment,
because the two systems are, as a matter of fact, entirely symmetrical
and we could with equal right consider $B$'s~system to be the one at
rest and $A$'s~the one in motion.

\emph{We thus find that two observers, $A$~and~$B$, who are in relative motion
will not in general agree in their measurements of the time interval necessary
for a given event to take place}, the event in this particular case,
for example, having been the performance of $B$'s~experiment; indeed,
making use of the ratio obtained in a preceding paragraph, we may
go further and make the quantitative statement that measurements of
\emph{time intervals made with a moving clock must be multiplied by the quantity
$\dfrac{1}{\sqrt{ 1 - \smfrac{V^2}{c^2}}}$ in order to agree with measurements made with a stationary
system of clocks}.

It is sometimes more convenient to state this principle in the
form: A stationary observer using a set of stationary clocks will
obtain a greater measurement in the ratio $1: \sqrt{ 1 - \dfrac{V^2}{c^2}}$ for a given
time interval than an observer who uses a clock moving with the
velocity~$V$.


\Subsection{Measurements of Length in a Moving System.}

\Paragraph{26.} We may now extend our considerations, to obtain a relation
between measurements of \emph{length} made in stationary and moving
systems.

As to measurements of length \emph{perpendicular} to the line of motion
of the two systems $S$~and~$S'$, a little consideration will make it at once
evident that both $A$~and~$B$ must obtain identical results. This is
true because the possibility is always present of making a direct comparison
%% -----File: 045.png---Folio 31-------
of the meter sticks which $A$~and~$B$ use for such measurements
by holding them perpendicular to the line of motion. When the
relative motion of the two systems brings such meter sticks into
juxtaposition, it is evident from the first postulate of relativity that
$A$'s~meter and $B$'s~meter must coincide in length. Any difference in
length could be due only to the different velocity of the two systems
through space, and such an occurrence is ruled out by our first postulate.
\emph{Hence measurements made with a moving meter stick held perpendicular
to its line of motion will agree with those made with stationary meter
sticks.}

\Paragraph{27.} With regard to measurements of length \emph{parallel} to the line of
motion of the systems, the affair is much more complicated. Any
direct comparison of the lengths of meter sticks in the two systems
would be very difficult to carry out; the consideration, however, of a
simple experiment on the velocity of light parallel to the motion of
the systems will lead to the desired relation.

Let us again consider two systems $S$~and~$S'$ (\Figref[fig.]{7}), $S'$~moving
past~$S$ with the velocity~$V$.
\begin{figure}[hbt]
  \begin{center}
    \Fig{7}
    \Input[3.5in]{045}
  \end{center}
\end{figure}

$A$ and $B$ are observers on these systems provided with clocks and
meter sticks. The two observers lay off, each on his own system,
paths for measuring the velocity of light. $A$~lays off a distance of
one meter,~$A\, m$, so that he can measure the time for light to travel
to the mirror~$m$ and return, and $B$, using a meter stick which has
the same length as~$A$'s when they are both at rest, lays off the distance~$B\, n$.

Each observer measures the length of time it takes for light to
travel to his mirror and return, and will evidently have to find the
same length of time, since the postulates of relativity require that the
velocity of light shall be the same for all observers.
%% -----File: 046.png---Folio 32-------

Now the observer~$A$, taking himself as at rest, finds that $B$'s~light
travels over a path~$B\, n'\, B'$ (\Figref[fig.]{8}), where $n\, n'$~is the distance
\begin{figure}[hbt]
  \begin{center}
    \Fig{8}
    \Input[3in]{046}
  \end{center}
\end{figure}
through which the mirror~$n$ moves while the light is travelling up to
it, and $B\, B'$~is the distance through which the source travels before
the light gets back. It is easy to calculate the length of this path.

We have
\[
\frac{n\,n'}{B\,n'} = \frac{V}{c}
\]
and
\[
\frac{B\,B'}{B\,n'\,B'} = \frac{V}{c}.
\]
Also, from the figure,
\begin{align*}
B\,n' &= B\,n + n\,n',\\
B\,n'\,B' &= B\,n\,B + 2\,n\,n'- B\, B'.
\end{align*}
Combining, we obtain
\[
\frac{B\,n'\,B'}{B\,n\,B} = \frac{1}{1 - \smfrac{V^2}{c^2}}.
\]

Thus $A$ finds that the path traversed by $B$'s~light, instead of being
exactly two meters as was his own, will be longer in the ratio of
$1:\left(1 - \dfrac{V^2}{c^2}\right)$. For this reason $A$~is rather surprised that $B$~does
not report a longer time interval for the passage of the light than he
himself found. He remembers, however, that he has already found
that measurements of time made with a moving clock must be multiplied
by the quantity $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$ in order to agree with his own, and
sees that this will account for part of the discrepancy between the
expected and observed results. To account for the remaining discrepancy
the further conclusion is now obtained \emph{that measurements of
%% -----File: 047.png---Folio 33-------
length made with a moving meter stick, parallel to its motion, must be
multiplied by the quantity $\sqrt{1 - \dfrac{V^2}{c^2}}$ in order to agree with those made
in a stationary system}.

In accordance with this principle, a stationary observer will
obtain a smaller measurement for the length of a moving body than
will an observer moving along with the object. This has been called
the Lorentz shortening, the shortening occurring in the ratio
\[
\sqrt{1 - \frac{V^2}{c^2}}:1
\]
in the line of motion.


\Subsection{The Setting of Clocks in a Moving System.}

\Paragraph{28.} It will be noticed that in our considerations up to this point
we have considered cases where only a \emph{single} moving clock was needed
in performing the desired experiment, and this was done purposely,
since we shall find, not only that a given time interval measures
shorter on a moving clock than on a system of stationary clocks,
but that a system of moving clocks which have been set in synchronism
by an observer moving along with them will not be set in synchronism
for a stationary observer.

Consider again two systems $S$~and~$S'$ in relative motion with the
velocity~$V$. An observer~$A$ on system~$S$ places two carefully compared
clocks, unit distance apart, in the line of motion, and has the
time on each clock read when a given point on the other system
passes it. An observer~$B$ on system~$S'$ performs a similar experiment.
The time interval obtained in the two sets of readings must be the
same, since the first postulate of relativity obviously requires that the
relative velocity of the two systems $V$~shall have the same value for
both observers.

The observer~$A$, however, taking himself as at rest, and familiar
with the change in the measurements of length and time in the moving
system which have already been deduced, expects that the velocity
as measured by~$B$ will be greater than the value that he himself
obtains in the ratio $\dfrac{1}{1 - \smfrac{V^2}{c^2}}$, since any particular one of $B$'s~clocks
%% -----File: 048.png---Folio 34-------
gives a shorter value for a given time interval than his own, while
$B$'s~measurements of the length of a moving object are greater than
his own, each by the factor $\sqrt{1 - \dfrac{V^2}{c^2}}$. In order to explain the actual
result of $B$'s~experiment he now has to conclude that the clocks which
for $B$ are set synchronously are not set in synchronism for himself.

From what has preceded it is easy to see that in the moving system,
from the point of view of the stationary observer, clocks must be set
further and further ahead as we proceed towards the rear of the
system, since otherwise $B$~would not obtain a great enough difference
in the readings of the clocks as they come opposite the given point
on the other system. Indeed, if two clocks are situated in the moving
system,~$S'$, one in front of the other by the distance $l'$, as measured
by~$B$, then for $A$ it will appear as though $B$~had set his rear clock ahead
by the amount~$\dfrac{l'V}{c^2}$.

\Paragraph{29.} We have now obtained all the information which we shall
need in this chapter as to measurements of time and length in systems
moving with different velocities. We may point out, however, before
proceeding to the application of these considerations, that our choice
of $A$'s system as the one which we should call stationary was of course
entirely arbitrary and immaterial. We can at any time equally well
take $B$'s~system as the one to which we shall ultimately refer all our
measurements, and indeed all that we shall mean when we call one of
our systems stationary is that for reasons of convenience we have
picked out that particular system as the one with reference to which
we particularly wish to make our measurements. We may also
point out that of course $B$~has to subject $A$'s~measurements of time
and length to just the same multiplications by the factor $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$
as did~$A$ in order to make them agree with his own.

These conclusions as to measurements of space and time are of course
very startling when first encountered. The mere fact, however, that
they appear strange to so-called ``common sense'' need cause us
no difficulty, since the older ideas of space and time were obtained
from an ancestral experience which never included experiments with
%% -----File: 049.png---Folio 35-------
high relative velocities, and it is only when the ratio $\dfrac{V^2}{c^2}$ becomes
appreciable that we obtain unexpected results. To those scientists
who do not wish to give up their ``common sense'' ideas of space
and time we can merely say that if they accept the two postulates
of relativity then they will also have to accept the consequences
which can be deduced therefrom. The remarkable nature of these
consequences merely indicates the very imperfect nature of our older
conceptions of space and time.


\Subsection{The Composition of Velocities.}

\Paragraph{30.} Our conclusions as to the setting of clocks make it possible
to obtain an important expression for the composition of velocities.
Suppose we have a system~$S$, which we shall take as stationary, and
on the system an observer~$A$. Moving past~$S$ with the velocity~$V$
is another system~$S'$ with an observer~$B$, and finally moving past~$S'$
in the same direction is a body whose velocity is~$u'$ as measured by
observer~$B$. What will be the velocity~$u$ of this body as measured
by~$A$?

Our older ideas led us to believe in the simple additivity of velocities
and we should have calculated~$u$ in accordance with the simple
expression
\[
u = V + u'.
\]
We must now allow, however, for the fact that $u'$~is measured with
clocks which to~$A$ appear to be set in a peculiar fashion and running
at a different rate from his own, and with meter sticks which give
longer measurements than those used in the stationary system.

The determination of~$u'$ by observer~$B$ would be obtained by
measuring the time interval necessary for the body in question to
move a given distance~$l'$ along the system~$S'$. If $t'$~is the difference
in the respective clock readings when the body reaches the ends of
the line~$l'$, we have
\[
u' = \frac{l'}{t'}.
\]
We have already seen, however, that the two clocks are for~$A$ set $\dfrac{l'V}{c^2}$~units
apart and hence for clocks set together the time interval would
%% -----File: 050.png---Folio 36-------
have measured $t' + \dfrac{l'V}{c^2}$. Furthermore these moving clocks give
time measurements which are shorter in the ratio $\sqrt{1 - \dfrac{V^2}{c^2}}:1$ than
those obtained by~$A$, so that for~$A$ the time interval for the body to
move from one end of~$l'$ to the other would measure
\[
\frac{t' + \smfrac{l'V}{c^2}}{\sqrt{1 - \smfrac{V^2}{c^2}}};
\]
furthermore, owing to the difference in measurements of length, this
line~$l'$ has for~$A$ the length $l'\sqrt{1 - \dfrac{V^2}{c^2}}$. Hence $A$~finds that the
body is moving past~$S'$ with the velocity,
\[
\frac{\ l'\sqrt{1 - \smfrac{V^2}{c^2}}\ }
     {\smfrac{t' + \smfrac{l'V}{c^2}}{\sqrt{1 - \smfrac{V^2}{c^2}}}}
  = \frac{\smfrac{l'}{t'} \left(1 - \smfrac{V^2}{c^2}\right)}
         {1 + \smfrac{l'}{t'}\, \smfrac{V}{c^2}}
  = \frac{u' \left(1 - \smfrac{V^2}{c^2}\right)}
         {1+ \smfrac{u'V}{c^2}}.
\]
This makes the total velocity of the body past~$S$ equal to the sum
\[
u = V + \frac{u' \left(1 - \smfrac{V^2}{c^2}\right)}{1 + \smfrac{u'V}{c^2}},
\]
or
\[
u = \frac{V + u'}{1 + \smfrac{u'V}{c^2}}.
\]

This new expression for the composition of velocities is extremely
important. When the velocities $u'$~and~$V$ are small compared with
the velocity of light~$c$, we observe that the formula reduces to the simple
additivity principle which we know by common experience to be true
%% -----File: 051.png---Folio 37-------
for all ordinary velocities. Until very recently the human race has
had practically no experience with high velocities and we now see
that for velocities in the neighborhood of that of light, the simple
additivity principle is nowhere near true.

In particular it should be noticed that by the composition of
velocities which are themselves less than that of light we can never
obtain any velocity greater than that of light. As an extreme case,
suppose for example that the system~$S'$ were moving past $S$~itself
with the velocity of light (\ie, $V = c$) and that in the system~$S'$ a
particle should itself be given the velocity of light in the same direction
(\ie, $u' = c$); we find on substitution that the particle still has
only the velocity of light with respect to~$S$. We have
\[
u = \frac{c + c}{1 + \smfrac{c^2}{c^2}} = \frac{2c}{2} = c.
\]

By the consideration of such conclusions as these the reader will
appreciate more and more the necessity of abandoning his older
naïve ideas of space and time which are the inheritance of a long
human experience with physical systems in which only slow velocities
were encountered.


\Subsection{The Mass of a Moving Body.}

\Paragraph{31.} We may now obtain an important relation for the mass of a
moving body. Consider again two similar systems, $S$~at rest and $S'$~moving
past with the velocity~$V$. The observer~$A$ on system~$S$ has a
sphere made from some rigid elastic material, having a mass of $m$~grams,
and the observer~$B$ on system~$S'$ is also provided with a similar
sphere. The two spheres are made so that they are exactly alike
when both are at rest; thus $B$'s~sphere, since it is at rest with respect
to him, looks to him just the same as the other sphere does to~$A$.
As the two systems pass each other (\Figref[fig.]{9}) each of these clever experimenters
rolls his sphere towards the other system with a velocity of
$u$~cm.~per second, so that they will just collide and rebound in a line
perpendicular to the direction of motion. Now, from the first postulate
of relativity, system~$S'$ appears to~$B$ just the same as system $S$~appears
to~$A$, and $B$'s~ball appears to him to go through the same
evolutions that $A$~finds for his ball. $A$~finds that his ball on collision
%% -----File: 052.png---Folio 38-------
undergoes the algebraic change of velocity~$2u$, $B$~finds the same change
in velocity~$2u$ for his ball. $B$~reports this fact to~$A$, and $A$~knowing
that $B$'s~measurements of length agree with his own in this transverse
\begin{figure}[hbt]
  \begin{center}
    \Fig{9}
    \Input{052}
  \end{center}
\end{figure}
direction, but that his clock gives time intervals that are shorter than
his own in the ratio $\sqrt{1 - \dfrac{V^2}{c^2}}:1$, calculates that the change in velocity
of $B$'s~ball must be~$2u\sqrt{1 - \dfrac{V^2}{c^2}}$.

From the principle of the conservation of momentum, however,
$A$~knows that the change in momentum of $B$'s~ball must be the same
as that of his own and hence can write the equation
\[
m_au = m_bu\sqrt{1 - \frac{V^2}{c^2}},
\]
where $m_a$~is the mass of $A$'s~ball and $m_b$~is the mass of $B$'s~ball. Solving
we have
\[
m_b = \frac{m_a}{\sqrt{1 - \smfrac{V^2}{c^2}}}.
\]

In other words, $B$'s~ball, which had the same mass~$m_a$ as~$A$'s when
%% -----File: 053.png---Folio 39-------
both were at rest, is found to have the larger mass $\dfrac{m_a}{\sqrt{1 - \smfrac{v^2}{c^2}}}$ when
placed in a system that is moving with the velocity~$V$.\footnote
  {In carrying out this experiment the transverse velocities of the balls should
  be made negligibly small in comparison with the relative velocity of the systems~$V$.}

The theory of relativity thus leads to the general expression
\[
m = \frac{m_0}{\sqrt{1 - \smfrac{v^2}{c^2}}}
\]
for the mass of a body moving with the velocity~$u$ and having the
mass~$m_0$ when at rest.

Since we have very few velocities comparable with that of light
it is obvious that the quantity $\sqrt{1 - \dfrac{v^2}{c^2}}$ seldom differs much from
unity, which makes the experimental verification of this expression
difficult. In the case of electrons, however, which are shot off from
radioactive substances, or indeed in the case of cathode rays produced
with high potentials, we do have particles moving with velocities
comparable to that of light, and the experimental work of Kaufmann,
Bucherer, Hupka and others in this field provides one of the most
striking triumphs of the theory of relativity.


\Subsection{The Relation Between Mass and Energy.}

\Paragraph{32.} The theory of relativity has led to very important conclusions
as to the nature of mass and energy. In fact, we shall see that matter
and energy are apparently different names for the same fundamental
entity.

When we set a body in motion it is evident from the previous
section that we increase both its mass as well as its energy. Now
we can show that there is a definite ratio between the amount of
energy that we give to the body and the amount of mass that we
give to it.

If the force~$f$ acts on a particle which is free to move, its increase in
kinetic energy is evidently
\[
\Delta E = \int f\, dl.
\]
But the force acting\DPtypo{, is}{ is,} by definition, equal to the rate of increase in
%% -----File: 054.png---Folio 40-------
the momentum of the particle
\[
f =
\frac{d}{dt}(mu).
\]
Substituting we have
\[
\Delta E
  = \int \frac{d(mu)}{dt}\, dl
  = \int \frac{dl}{dt}\, d(mu)
  = \int u\, d(mu).
\]
We have, however, from the previous section,
\[
m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\]
which, solved for~$u$, gives us
\[
u = c \sqrt{1 - \frac{{m_0}^2}{m^2}}.
\]
Substituting this value of~$u$ in our equation for~$\Delta E$ we obtain, after
simplification,
\[
\Delta E = \int c^2\, dm = c^2\, \Delta m.
\]

This says that the increase of the kinetic energy of the particle,
in ergs, is equal to the increase in mass, in grams, multiplied by the
square of the velocity of light. If now we bring the particle to rest
it will give up both its kinetic energy and its excess mass. Accepting
the principles of the conservation of mass and energy, we know, however,
that neither this energy nor the mass has been destroyed; they
have merely been passed on to other bodies. There is, moreover,
every reason to believe that this mass and energy, which were associated
together when the body was in motion and left the body when
it was brought to rest, still remain always associated together. For
example, if the body should be brought to rest by setting another
body into motion, it is of course a necessary consequence of our considerations
that the kinetic energy and the excess mass both pass
on together to the new body which is set in motion. A similar conclusion
would be true if the body is brought to rest by frictional forces,
since the heat produced by the friction means an increase in the kinetic
energies of ultimate particles.
%% -----File: 055.png---Folio 41-------

In general we shall find it pragmatic to consider that matter and
energy are merely different names for the same fundamental entity.
One gram of matter is equal to $10^{21}$~ergs of energy.
\[
c^2 = (2.9986 × 10^{10})^2 = \text{approx.\ }10^{21}.
\]

This apparently extraordinary conclusion is in reality one which
produces the greatest simplification in science. Not to mention
numerous special applications where this principle is useful, we may
call attention to the fact that the great laws of the conservation of
mass and of energy have now become identical. We may also point
out that those opposing camps of philosophic materialists who defend
matter on the one hand or energy on the other as the fundamental
entity of the universe may now forever cease their unimportant bickerings.
%% -----File: 056.png---Folio 42-------


\Chapter{IV}{The Einstein Transformation Equations for Space
and Time.}
\SetRunningHeads{Chapter Four.}{Transformation Equations for Space and Time.}

\Subsection{The Lorentz Transformation.}

\Paragraph{33.} We may now proceed to a systematic study of the consequences
of the theory of relativity.

The fundamental problem that first arises in considering
spatial and temporal measurements is that of transforming the
description of a given kinematical occurrence from the variables of
one system of coördinates to those of another system which is in
motion relative to the first.

Consider two systems of right-angled Cartesian coördinates $S$~and~$S'$
(\Figref[fig.]{10}) in relative motion in the $X$~direction with the velocity~$V$.
\begin{figure}[hbt]
  \begin{center}
    \Fig{10}
    \Input{056}
  \end{center}
\end{figure}
The \emph{position} of any given point in space can be determined by specifying
its coördinates $x$,~$y$, and~$z$ with respect to system~$S$ or its coördinates
$x'$,~$y'$ and~$z'$ with respect to system~$S'$. Furthermore, for the
purpose of determining the \emph{time} at which any event takes place, we
may think of each system of coördinates as provided with a whole
series of clocks placed at convenient intervals throughout the system,
the clocks of each series being set and regulated\footnote
  {We may think of the clocks as being set in any of the ways that are usual
  in practice. Perhaps the simplest is to consider the clocks as mechanisms which
  have been found to ``keep time'' when they are all together where they can be
  examined by one individual observer. The assumption can then be made, in accordance
  with our ideas of the homogeneity of space, that they will continue to
  ``keep time'' after they have been distributed throughout the system.}
by observers in the
%% -----File: 057.png---Folio 43-------
corresponding system. The time at which the event in question
takes place may be denoted by~$t$ if determined by the clocks belonging
to system~$S$ and by~$t'$ if determined by the clocks of system~$S'$.

For convenience the two systems $S$~and~$S'$ are chosen so that the
axes $OX$~and~$O'X'$ lie in the same line, and for further simplification
we choose, as our starting-point for time measurements, $t$~and~$t'$ both
equal to zero when the two origins come into coincidence.

The specific problem now before us is as follows: If a given kinematical
occurrence has been observed and described in terms of the
variables $x'$,~$y'$,~$z'$ and~$t'$, what substitutions must we make for the
values of these variables in order to obtain a correct description of the
\emph{same} kinematical event in terms of the variables $x$,~$y$,~$z$ and~$t$? In
other words, we want to obtain a set of transformation equations
from the variables of system~$S'$ to those of system~$S$. The equations
which we shall present were first obtained by Lorentz, and the process
of changing from one set of variables to the other has generally been
called the Lorentz transformation. The significance of these equations
from the point of view of the theory of relativity was first appreciated
by Einstein.


\Subsection{Deduction of the Fundamental Transformation Equations.}

\Paragraph{34.} It is evident that these transformation equations are going
to depend on the relative velocity $V$ of the two systems, so that we
may write for them the expressions
\begin{align*}
x' &= F_1(V, x, y, z, t), \displaybreak[0] \\
y' &= F_2(V, x, y, z, t), \displaybreak[0] \\
z' &= F_3(V, x, y, z, t), \displaybreak[0] \\
t' &= F_4(V, x, y, z, t),
\end{align*}
where $F_1$,~$F_2$,~etc., are the unknown functions whose form we wish
to determine.

It is possible at the outset, however, greatly to simplify these
relations. If we accept the idea of the homogeneity of space it is
evident that any other line parallel to~$OXX'$ might just as well have
been chosen as our line of $X$-axes, and hence our two equations for
$x'$~and~$t'$ must be independent of $y$~and~$z$. Moreover, as to the equations
%% -----File: 058.png---Folio 44-------
for $y'$~and~$z'$ it is at once evident that the only possible solutions
are $y' = y$ and $z' = z$. This is obvious because a meter stick held
in the system~$S'$ perpendicular to the line of relative motion,~$OX'$,
of the system can be directly compared with meter sticks held similarly
in system~$S$, and in accordance with the first postulate of relativity
they must agree in length in order that the systems may be entirely
symmetrical. We may now rewrite our transformation equations
in the simplified form
\begin{align*}
x' &= F_1(V, t, x), \\
y' &= y, \\
z' &= z, \\
t' &= F_2(V, t, x),
\end{align*}
and have only two functions, $F_1$~and~$F_2$, whose form has to be determined.

To complete the solution of the problem we may make use of three
further conditions which must govern the transformation equations.

\Subsubsection{35}{Three Conditions to be Fulfilled.} In the first place, when the
velocity~$V$ between the systems is small, it is evident that the transformation
equations must reduce to the form that they had in Newtonian
mechanics, since we know both from measurements and from
everyday experience that the Newtonian concepts of space and time
are correct as long as we deal with slow velocities. Hence the limiting
form of the equations as $V$~approaches zero will be (cf.~\Chapref{I},
equations \DPchg{3--4--5--6}{(3),~(4), (5),~(6)})
\begin{align*}
x' &= x - Vt,\\
y' &= y, \\
z' &= z, \\
t' &= t.
\end{align*}

\Paragraph{36.} A second condition is imposed upon the form of the functions
$F_1$~and~$F_2$ by the first postulate of relativity, which requires that the
two systems $S$~and~$S'$ shall be entirely symmetrical. Hence the
transformation equations for changing from the variables of system~$S$
to those of system~$S'$ must be of exactly the same form as those used
in the reverse transformation, containing, however, $-V$~wherever
$+V$~occurs in the latter equations. Expressing this requirement in
%% -----File: 059.png---Folio 45-------
mathematical form, we may write as true equations
\begin{align*}
x &= F_1(-V, t', x'), \\
t &= F_2(-V, t', x'),
\end{align*}
where $F_1$~and~$F_2$ must have the same form as above.

\Paragraph{37.} A final condition is imposed upon the form of $F_1$~and~$F_2$ by
the second postulate of relativity, which states that the velocity of a
beam of light appears the same to all observers regardless of the
motion of the source of light or of the observer. Hence our transformation
equations must be of such a form that a given beam of
light has the same velocity,~$c$, when measured in the variables of either
system. Let us suppose, for example, that at the instant $t = t' = 0$,
when the two origins come into coincidence, a light impulse is started
from the common point occupied by $O$~and~$O'$. Then, measured in
the coördinates of either system, the optical disturbance which is
generated must spread out from the origin in a spherical form with
the velocity~$c$. Hence, using the variables of system~$S$, the coördinates
of any point on the surface of the disturbance will be given by the
expression
\[
x^2 + y^2 + z^2 = c^2t^2,
\Tag{7}
\]
while using the variables of system~$S'$ we should have the similar
expression
\[
x'^2 + y'^2 + z'^2 = c^2t'^2.
\Tag{8}
\]
Thus we have a particular kinematical occurrence, the spreading out
of a light disturbance, whose description is known in the variables
of either system, and our transformation equations must be of such
a form that their substitution will change equation (8) to equation (7).
In other words, the expression $x^2 + y^2 + z^2 - c^2t^2$ is to be an invariant
for the Lorentz transformation.

\Subsubsection{38}{The Transformation Equations.} The three sets of conditions
which, as we have seen in the last three paragraphs, are imposed upon
the form of $F_1$~and~$F_2$ are sufficient to determine the solution of the
problem. The natural method of solution is obviously that of trial,
%% -----File: 060.png---Folio 46-------
and we may suggest the solution:
\begin{align*}
x' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, (x - Vt)
    = \kappa(x - Vt), \Tag{9}\\
y' &= y, \Tag{10}\\
z' &= z, \Tag{11}\\
t' &= \frac{1}{\sqrt{1- \smfrac{V^2}{c^2}}} \left(t - \frac{V}{c^2}\, x\right)
    = \kappa \left(t - \frac{V}{c^2}\, x\right), \Tag{12}
\end{align*}
where we have placed~$\kappa$ to represent the important and continually
recurring quantity $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$.

It will be found as a matter of fact by examination that these\rule{0em}{1.8em}
solutions do fit all three requirements which we have stated. Thus,
when $V$~becomes small compared with the velocity of light,~$c$, the
equations do reduce to those of Galileo and Newton. Secondly, if
the equations are solved for the unprimed quantities in terms of the
primed, the resulting expressions have an unchanged form except for
the introduction of~$-V$ in place of~$+V$, thus fulfilling the requirements
of symmetry imposed by the first postulate of relativity. And
finally, if we substitute the expressions for $x'$,~$y'$,~$z'$ and~$t'$ in the polynomial
$x'^2 + y'^2 + z'^2 = c^2t'^2$, we shall obtain the expression $x^2 + y^2
+ z^2 - c^2t^2$ and have thus secured the invariance of $x^2 + y^2 + z^2 - c^2t^2$
which is required by the second postulate of relativity.

We may further point out that the whole series of possible Lorentz
transformations form a group such that the result of two successive
transformations could itself be represented by a single transformation
provided we picked out suitable magnitudes and directions for the
velocities between the various systems.

It is also to be noted that the transformation becomes imaginary
for cases where $V > c$, and we shall find that this agrees with ideas
obtained in other ways as to the speed of light being an upper limit
for the magnitude of all velocities.
%% -----File: 061.png---Folio 47-------


\Subsection{Further Transformation Equations.}

\Paragraph{39.} Before making any applications of our equations we shall find
it desirable to obtain by simple substitutions and differentiations a
series of further transformation equations which will be of great value
in our future work.

By the simple differentiation of equation~(12) we can obtain
\[
\frac{dt'}{dt} = \kappa\left(1 - \frac{\dot{x}V}{c^2}\right),
\Tag{13}
\]
where we have put~$\dot{x}$ for~$\dfrac{dx}{dt}$.

\Subsubsection{40}{Transformation Equations for Velocity.} By differentiation of
the equations for $x'$,~$y'$ and~$z'$, nos.\ (9),~(10) and~(11), and substitution
of the value just found for~$\dfrac{dt'}{dt}$ we may obtain the following transformation
equations for velocity:
\begin{alignat*}{3}
\dot{x}' &= \frac{\dot{x} - V}{1 - \smfrac{\dot{x}V}{c^2}}
  &&\qquad\text{or}\qquad&
    u'_x &= \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}}, \Tag{14} \\
%
\dot{y}' &= \frac{\dot{y}\kappa^{-1}}{1 - \smfrac{\dot{y}V}{c^2}} &&&
    u'_y &= \frac{u_y\kappa^{-1}}{1 - \smfrac{u_xV}{c^2}}, \Tag{15} \\
%
\dot{z}' &= \frac{\dot{z}\kappa^{-1}}{1 - \smfrac{\dot{z}V}{c^2}} &&&
    u'_z &= \frac{u_z\kappa^{-1}}{1 - \smfrac{u_xV}{c^2}}, \Tag{16}
\end{alignat*}
where the placing of a dot has the familiar significance of differentiation
with respect to time, $\dfrac{dx}{dt}$~being represented by~$\dot{x}$ and $\dfrac{dx'}{dt'}$ by~$\dot{x}'$.

The significance of these equations for the transformation of
velocities is as follows: If for an observer in system~$S$ a point appears
to be moving with the uniform velocity $(\dot{x}, \dot{y}, \dot{z})$ its velocity $(\dot{x}', \dot{y}', \dot{z}')$,
as measured by an observer in system~$S'$, is given by these expressions
(14),~(15) and~(16).

\Subsubsection{41}{Transformation Equations for the Function $\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}$.} These
%% -----File: 062.png---Folio 48-------
three transformation equations for the velocity components of a point\DPtypo{,}{}
permit us to obtain a further transformation equation for an important
function of the velocity which we shall find continually recurring in
our later work. This is the function $\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, where we have indicated
the total velocity of the point by~$u$, according to the expression
$u^2 = \dot{x}^2 + \dot{y}^2 + \dot{z}^2$. By the substitution of equations (14),~(15) and~(16)
we obtain the transformation equation
\[
\frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  = \frac{\left(1 - \smfrac{u_xV}{c^2}\right) \kappa}
         {\sqrt{1 - \smfrac{u^2}{c^2}}}.
\Tag{17}
\]

\Subsubsection{42}{Transformation Equations for Acceleration.} By further differentiating
equations (14),~(15) and~(16) and simplifying, we easily
obtain three new equations for transforming measurements of \emph{acceleration}
from system $S'$~to~$S$, viz.:
\begin{align*}
\ddot{x}' &= \left(1 -
  \frac{\dot{x}V}{c^2}\right)^{-3}\kappa^{-3}\ddot{x},
\Tag{18} \\
%
\ddot{y}' &= \left(1 - \frac{\dot{x}V}{c^2}\right)^{-2} \kappa^{-2}\ddot{y}
  + \dot{y}\, \frac{V}{c^2}
    \left(1 - \frac{\dot{x}V}{c^2}\right)^{-3} \kappa^{-2}\ddot{x},
\Tag{19} \\
%
\ddot{z}' &= \left(1 - \frac{\dot{x}V}{c^2}\right)^{-2} \kappa^{-2}\ddot{z}
  + \dot{z} \frac{V}{c^2}
    \left(1 - \frac{\dot{x}V}{c^2}\right)^{-3} \kappa^{-2}\ddot{x},
\Tag{20}
\intertext{or}
{\dot{u}_x}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-3}\ddot{u}_x,
\Tag{18} \\
%
{\dot{u}_y}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-2} \kappa^{-2}\ddot{u}_y
  + u_y\, \frac{V}{c^2}
    \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-2}\dot{u}_x,
\Tag{19} \\
%
{\dot{u}_z}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-2} \kappa^{-2}\ddot{u}_z
  + u_z\, \frac{V}{c^2}
    \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-2}\dot{u}_x.
\Tag{20}
\end{align*}
%% -----File: 063.png---Folio 49-------


\Chapter{V}{Kinematical Applications.}
\SetRunningHeads{Chapter Five.}{Kinematical Applications.}

\Paragraph{43.} The various transformation equations for spatial and temporal
measurements which were derived in the \Chapnumref[IV]{previous chapter} may now be
used for the treatment of a number of kinematical problems. In
particular it will be shown in the latter part of the chapter that a
number of optical problems can be handled with extraordinary facility
by the methods now at our disposal.


\Subsection{The Kinematical Shape of a Rigid Body.}

\Paragraph{44.} We may first point out that the conclusions of relativity theory
lead us to quite new ideas as to what is meant by the shape of a rigid
body. We shall find that the shape of a rigid body will depend entirely
upon the relative motion of the body and the observer who is making
measurements on it.

Consider a rigid body which is at rest with respect to system~$S'$.
Let $x_1'$,~$y_1'$,~$z_1'$ and $x_2'$,~$y_2'$,~$z_2'$ be the coördinates in system~$S'$ of two
points in the body. The coördinates of the same points as measured
in system~$S$ can be found from transformation equations (9),~(10)
and~(11), and by subtraction we can obtain the following expressions
\begin{gather*}
(x_2 - x_1) = \sqrt{1 - \frac{V^2}{c^2}}\, ({x_2}' - {x_1}'),
\Tag{21} \\
(y_2 - y_1) = (y_2' - y_1'),
\Tag{22} \\
(z_2 - \DPtypo{y_2}{z_1}) = (z_2' - z_1'),
\Tag{23}
\end{gather*}
connecting the distances between the pair of points as viewed in the
two systems. In making this subtraction terms containing~$t$ have
been cancelled out since we are interested in the \emph{simultaneous} positions
of the points. In accordance with these equations we may distinguish
then between the \emph{geometrical shape} of a body, which is the shape that
it has when measured on a system of axes which are at rest relative
to it, and its \emph{kinematical shape}, which is given by the coördinates which
%% -----File: 064.png---Folio 50-------
express the \emph{simultaneous} positions of its various points when it is in
motion with respect to the axes of reference. We see that the kinematical
shape of a rigid body differs from its geometrical shape by a
shortening of all its dimensions in the line of motion in the ratio
$\sqrt{1 - \dfrac{V^2}{c^2}}:1$; thus a sphere, for example, becomes a Heaviside ellipsoid.

In order to avoid incorrectness of speech we must be very careful
not to give the idea that the kinematical shape of a body is in
any sense either more or less real than its geometrical shape. We
must merely learn to realize that the shape of a body is entirely dependent
on the particular set of coördinates chosen for the making
of space measurements.


\Subsection{The Kinematical Rate of a Clock.}

\Paragraph{45.} Just as we have seen that the shape of a body depends upon
our choice of a system of coördinates, so we shall find that the rate of
a given clock depends upon the relative motion of the clock and its
observer. Consider a clock or any mechanism which is performing
a periodic action. Let the clock be at rest with respect to system~$S'$
and let a given period commence at~${t_1}'$ and end at~${t_2}'$, the length of
the interval thus being $\Delta t' = {t_2}' - {t_1}'$.

From transformation equation~(12) we may obtain
\begin{align*}
t_1' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}
        \left(t_1 - \frac{V}{c^2}\, x_1\right), \\
t_2' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}
        \left(t_2 - \frac{V}{c^2}\, x_2\right),
\end{align*}
and by subtraction, since $x_2 - x_1$ is obviously equal to~$Vt$, we have
\begin{align*}%[** TN: Not aligned in orig.]
t_2 - t_1 &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, ({t_2}' - {t_1}'), \\
\Delta t  &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, \Delta t'.
\end{align*}
%% -----File: 065.png---Folio 51-------
Thus an observer who is moving past a clock finds a longer period for
the clock in the ratio $1 : \sqrt{1 - \dfrac{V^2}{c^2}}$ than an observer who is stationary
with respect to it. Suppose, for example, we have a particle which
is turning alternately red and blue. For an observer who is moving
past the particle the periods for which it remains a given color measure
longer in the ratio  $1 : \sqrt{1  -\dfrac{V^2}{c^2}}$ than they do to an observer who is
stationary with respect to the particle.

\Paragraph{46.} A possible opportunity for testing this interesting conclusion
of the theory of relativity is presented by the phenomena of canal
rays. We may regard the atoms which are moving in these rays as
little clocks, the frequency of the light which they emit corresponding
to the period of the clock. If now we should make spectroscopic
observations on canal rays of high velocity, the frequency of the
emitted light ought to be less than that of light from stationary atoms
of the same kind if our considerations are correct. It would of course
be necessary to view the canal rays at right angles to their direction
of motion, to prevent a confusion of the expected shift in the spectrum
with that produced by the ordinary Doppler effect (see \Secref{54}).


\Subsection{The Idea of Simultaneity.}

\Paragraph{47.}  We may now also point out that the idea of the \emph{absolute} simultaneity
of two events must henceforth be given up. Suppose, for
example, an observer in the system~$S$ is interested in two events
which take place simultaneously at the time~$t$. Suppose one of these
events occurs at a point having the $X$~coördinate~$x_1$ and the other
at a point having the coördinate~$x_2$; then by transformation equation~(12)
it is evident that to an observer in system~$S'$, which is moving
relative to~$S$ with the velocity~$V$, the two events would take place
respectively at the times
\begin{align*}
{t_1}' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}
  \left(t - \frac{V}{c^2}\, x_1\right) \\
\intertext{and}
{t_2}' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}
  \left(t - \frac{V}{c^2}\, x_2\right)
\end{align*}
%% -----File: 066.png---Folio 52-------
or the difference in time between the occurrence of the events would
appear to this other observer to be
\[
{t_2}' - {t_1}'
  = \frac{V}{c^2\, \sqrt{1 - \smfrac{V^2}{c^2}}}\, (x_1 - x_2).
\Tag{25}
\]


\Subsection{The Composition of Velocities.}

\Subsubsection{48}{The Case of Parallel Velocities.} We may now present one of
the most important characteristics of Einstein's space and time,
which can be best appreciated by considering transformation equation~(14),
or more simply its analogue for the transformation in the reverse
direction
\[
u_x = \frac{{u_x}' + V}{1 + \smfrac{{u_x}'V}{c^2}}.
\Tag{26}
\]

Consider now the significance of the above equation. If we
have a particle which is moving in the $X$~direction with the velocity~$u_x'$
as measured in system~$S'$, its velocity~$u_x$ with respect to system~$S$
is to be obtained by adding the relative velocity of the two systems~$V$
\emph{and dividing the sum of the two velocities by} $1 + \dfrac{{u_x}'V}{c^2}$. Thus we see
that we must completely throw overboard our old naïve ideas of the
direct additivity of velocities. Of course, in the case of very slow
velocities, when $u_x'$~and~$V$ are both small compared with the velocity
of light, the quantity~$\dfrac{{u_x}'V}{c^2}$ is very nearly zero and the direct addition
of velocities is a close approximation to the truth. In the case of
velocities, however, which are in the neighborhood of the speed of
light, the direct addition of velocities would lead to extremely erroneous
results.

\Paragraph{49.}  In particular it should be noticed that by the composition of
velocities which are themselves less than that of light we can never
obtain any velocity greater than that of light. Suppose, for example,
that the system~$S'$ were moving past~$S$ with the velocity of light
(\ie, $V = c$), and that in the system~$S'$ a particle should itself be
given the velocity of light in the $X$~direction (\ie, $u_x' = c$); we find
on substitution that the particle still has only the velocity of light
%% -----File: 067.png---Folio 53-------
with respect to~$S$. We have
\[
u_x = \frac{c + c}{1 + \smfrac{c^2}{c^2}} = \frac{2c}{2} = c.
\]

If the relative velocity between the systems should be one half
the velocity of light,~$\dfrac{c}{2}$, and an experimenter on~$S'$ should shoot off a
particle in the $X$~direction with half the velocity of light, the total
velocity with respect to~$S$ would be
\[
u_x = \frac{\frac{1}{2}c + \frac{1}{2}c}{1 + \smfrac{\frac{1}{4}c^2}{c^2}}
  = \frac{4}{5}\, c.
\]

\Subsubsection{50}{Composition of Velocities in General.} In the case of particles
which have components of velocity in other than the $X$~direction it
is obvious that our transformation equations will here also provide
methods of calculation to supersede the simple addition of velocities.
If we place
\begin{align*}
u^2    &= {u_x}^2 + {u_y}^2 + {u_z}^2 ,\\
{u'}^2 &= {{u_x}'}^2 + {{u_y}'}^2 + {{u_z}'}^2 ,
\end{align*}
we may obtain by the substitution of equations (14),~(15) and~(16)
\[
u = \frac{\left({u'}^2 + V^2 + 2u'V \cos\alpha
          - \smfrac{{u'}^2V^2 \sin^2\alpha}{c^2}\right)^{1/2}}
         {1 + \smfrac{u'V \cos\alpha}{c^2}},
\Tag{27}
\]
where $\alpha$ is the angle in the system~$S'$ between the $X'$~axis and the
velocity of the particle~$u'$. For the particular case that $V$~and~$u'$
are in the same direction, the equation obviously reduces to the
simpler form
\[
u = \frac{u' + V}{1 + \smfrac{u'V}{c^2}},
\]
which we have already considered.

\Paragraph{51.} We may also call attention at this point to an interesting characteristic
of the equations for the transformation of velocities. It will
%% -----File: 068.png---Folio 54-------
be noted from an examination of these equations that if to any observer
a particle appears to have a constant velocity, \ie, to be
unacted on by any force, it will also appear to have a \emph{uniform} although
of course different velocity to any observer who is himself in uniform
motion with respect to the first. An examination, however, of the
transformation equations for acceleration (18),~(19),~(20) will show
that here a different state of affairs is true, since it will be seen that a
point which has \emph{uniform acceleration} $(\ddot{x}, \ddot{y}, \ddot{z})$ with respect to an observer
in system~$S$ will not in general have a uniform acceleration in
another system~$S'$, since the acceleration in system~$S'$ depends not
only on the constant acceleration but also on the velocity in system~$S$,
which is necessarily varying.


\Subsection{Velocities Greater than that of Light.}

\Paragraph{52.} In the preceding section we have called attention to the fact
that the mere composition of velocities which are not themselves
greater than that of light will never lead to a speed that is greater
than that of light. The question naturally arises whether velocities
which are greater than that of light could ever possibly be obtained
in any way.

This problem can be attacked in an extremely interesting manner.
Consider two points $A$~and~$B$ on the $X$~axis of the system~$S$, and
suppose that some impulse originates at~$A$, travels to~$B$ with the
velocity~$u$ and at~$B$ produces some observable phenomenon, the starting
of the impulse at~$A$ and the resulting phenomenon at~$B$ thus
being connected by the relation of \emph{cause and effect}.

The time elapsing between the cause and its effect as measured
in the units of system~$S$ will evidently be
\[
\Delta t = t_B - t_A = \frac{x_B - x_A}{u},
\Tag{28}
\]
where $x_A$~and~$x_B$ are the coördinates of the two points $A$~and~$B$.

Now in another system~$S'$, which has the velocity~$V$ with respect
to~$S$, the time elapsing between cause and effect would evidently be
\[
\Delta t' = {t'}_B - {t'}_A
  = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t_B - \frac{V}{c^2}\, x_B\right)
  - \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t_A - \frac{V}{c^2}\, x_A\right),
\]
%% -----File: 069.png---Folio 55-------
where we have substituted for $t'_B$~and~$t'_A$ in accordance with equation~(12).
Simplifying and introducing equation~(28) we obtain
\[
\Delta t'
  = \frac{1 - \smfrac{uV}{ c^2}}
         {\sqrt{1 - \smfrac{V^2}{c^2}}}\, \Delta t.
\Tag{29}
\]
Let us suppose now that there are no limits to the possible magnitude
of the velocities $u$~and~$V$, and in particular that the causal impulse
can travel from~$A$ to~$B$ with a velocity~$u$ greater than that of
light. It is evident that we could then take a velocity~$u$ great enough
so that $\dfrac{uV}{c^2}$~would be greater than unity and $\Delta t'$~would become negative.
In other words, for an observer in system~$S'$ the effect which
occurs at~$B$ would \emph{precede} in time its cause which originates at~$A$.
Such a condition of affairs might not be a logical impossibility; nevertheless
its extraordinary nature might incline us to believe that no
causal impulse can travel with a velocity greater than that of light.

We may point out in passing, however, that in the case of kinematic
occurrences in which there is no causal connection there is no
reason for supposing that the velocity must be less than that of light.
Consider, for example, a set of blocks arranged side by side in a long
row. For each block there could be an \emph{independent} time mechanism
like an alarm clock which would go off at just the right instant so
that the blocks would fall down one after another along the line.
The velocity with which the phenomenon would travel along the
line of blocks could be arranged to have any value. In fact, the
blocks could evidently all be fixed to fall just at the same instant,
which would correspond to an infinite velocity. It is to be noticed
here, however, that there is no causal connection between the falling
of one block and that of the next, and no transfer of energy.

%[** TN: ToC entry reads "Applications to Optical Problems"]
\Subsection{Application of the Principles of Kinematics to Certain Optical Problems.}

\Paragraph{53.} Let us now apply our kinematical considerations to some
problems in the field of optics. We may consider a beam of light
as a periodic electromagnetic disturbance which is propagated through
a vacuum with the velocity~$c$. At any point in the path of a beam of
%% -----File: 070.png---Folio 56-------
light the intensity of the electric and magnetic fields will be undergoing
periodic changes in magnitude. Since the intensities of both the
electric and the magnetic fields vary together, the statement of a
single vector is sufficient to determine the instantaneous condition
at any point in the path of a beam of light. It is customary to call
this vector (which might be either the strength of the electric or of
the magnetic field) the light vector.

For the case of a simple plane wave (\ie, a beam of monochromatic
light from a distant source) the light vector at any point in the path
of the light may be put proportional to
\[
\sin\omega \left(t - \frac{lx + my + nz}{c}\right),
\Tag{30}
\]
where $x$,~$y$ and~$z$ are the coördinates of the point under observation,
$t$~is the time, $l$,~$m$ and~$n$ are the cosines of the angles $\alpha$,~$\beta$ and~$\gamma$ which
determine the direction of the beam of light with reference to our
system, and $\omega$~is a constant which determines the period of the light.

If now this same beam of light were examined by an observer in
system~$S'$ which is moving past the original system in the $X$~direction
with the velocity~$V$, we could write the light vector proportional to
\[
\sin\omega' \left(t' - \frac{l'x' + m'y' + n'z'}{c}\right),
\Tag{31}
\]
It is not difficult to show that the transformation equations which
we have already developed must lead to the following relations between
the measurements in the two systems\footnote
  {Methods for deriving the relation between the accented and unaccented
  quantities will be obvious to the reader. For example, consider the relation between
  $\omega$~and~$\omega'$. At the origin of coördinates $x = y = z = 0$ in system~$S$, we shall have
  in accordance with expression~(30) the light vector proportional to $\sin \omega t$, and hence
  similarly at the point~$O'$, which is the origin of coördinates in system~$S'$, we shall
  have the light vector proportional to $\sin \omega' t'$. But the point~$O'$ as observed from
  system~$S$ moves with the velocity~$V$ along the $X$\DPchg{-}{~}axis and at any instant has the
  position $x = Vt$; hence substituting in expression~(30) we have the light vector at
  the point~$O'$ as measured in system~$S$ proportional to
  \[
  \sin\omega t \left(1 - l\, \frac{V}{c}\right),
  \Tag{36}
  \]
  while as measured in system~$S'$ the intensity is proportional to
  \[
  \sin\omega' t'.
  \Tag{37}
  \]
  We have already obtained, however, a transformation equation for~$t'$, namely,
  \[
  t' = \kappa \left(t - \frac{V}{c^2}\, x\right),
  \]
  and further may place $x = Vt$. Making these substitutions and comparing expressions
  (36)~and~(37) we see that we must have the relation
  \[
  \omega' = \omega \kappa \left(1 - l\, \frac{V}{c}\right).
  \]
  Methods of obtaining the relation between the cosines $l$,~$m$ and~$n$ and the corresponding
  cosines $l'$,~$m'$, and~$n'$ as measured in system~$S'$ may be left to the reader.}
%% -----File: 071.png---Folio 57-------
\begin{align*}%[* TN: Aligning; centered in original]
\omega' &= \omega\kappa \left(1 - l\, \smfrac{V}{c}\right), \Tag{32} \\
l' &= \frac{l - \smfrac{V}{c}}{1 - l\, \smfrac{V}{c}},      \Tag{33} \\
m' &= \frac{m}{\kappa\left(1 - l\smfrac{V}{c}\right)},      \Tag{34} \\
n' &= \frac{n}{\kappa\left(1 - l\smfrac{V}{c}\right)}.      \Tag{35}
\end{align*}

With the help of these equations we may now treat some important
optical problems.

\Subsubsection{54}{The Doppler Effect.} At the origin of coördinates, $x = y = z
= 0$, in system~$S$ we shall evidently have from expression~(30) the
light vector proportional to $\sin \omega t$. That means that the vector
becomes zero whenever $\omega t = 2N \pi$, where $N$~is any integer; in other
words, the period of the light is $p = \dfrac{2\pi}{\omega}$ or the frequency
\[
\nu = \frac{\omega}{2\pi}.
\]
Similarly the frequency of the light as measured by an observer in
system~$S'$ would be
\[
\nu' = \frac{\omega'}{2\pi}.
\]
%% -----File: 072.png---Folio 58-------
Combining these two equations and substituting the equation connecting
$\omega$~and~$\omega'$ we have
\[
\nu = \frac{\nu'}{\kappa \left(1 - l\smfrac{V}{c}\right)}.
\]
This is the relation between the frequencies of a given beam of light
as it appears to observers who are in relative motion.

If we consider a source of light at rest with respect to system~$S'$
and at a considerable distance from the observer in system~$S$, we
may substitute for~$\nu'$ the frequency of the source itself,~$\nu_0$, and for~$l$
we may write~$\cos\phi$, where $\phi$~is the angle between the line connecting
source and observer and the direction of motion of the source, leading
to the expression
\[
\nu = \frac{\nu_0}{\kappa \left(1 - \cos\phi\, \smfrac{V}{c}\right)}.
\Tag{38}
\]

This is the most general equation for the \emph{Doppler effect}. When
the source of light is moving directly in the line connecting source
and observer, we have $\cos\phi = 1$, and the equation reduces to
\[
\nu = \frac{\nu_0}{\kappa \left(1 - \smfrac{V}{c}\right)},
\Tag{39}
\]
which except for second order terms is identical with the older expressions
for the Doppler effect, and hence agrees with experimental
determinations.

We must also observe, however, that even when the source of
light moves at right angles to the line connecting source and observer
there still remains a second-order effect on the observed frequency,
in contradiction to the predictions of older theories. We have in this
case $\cos\phi = 0$,
\[
\nu = \nu_0\, \sqrt{1 - \frac{V^2}{c^2}}.
\Tag{40}
\]
This is the change in frequency which we have already considered
when we discussed the rate of a moving clock. The possibilities of
%% -----File: 073.png---Folio 59-------
direct experimental verification should not be overlooked (see \Secref[section]{46}).

\Subsubsection{55}{The Aberration of Light.} Returning now to our transformation
equations, we see that equation~(33) provides an expression for
calculating the \emph{aberration of light}. Let us consider that the source
of light is stationary with respect to system~$S$, and let there be an
observer situated at the origin of \DPchg{coordinates}{coördinates} of system~$S'$ and thus
moving past the source with the velocity~$V$ in the $X$~direction. Let $\phi$~be
the angle between the $X$\DPchg{-}{~}axis and the line connecting source of
light and observer and let $\phi'$~be the same angle as it appears to the
moving observer; then we can obviously substitute in equation~(33),
$\cos\phi = l$, $\cos\phi' = l'$, giving us
\[
\cos\phi' = \frac{\cos\phi - \smfrac{V}{c}}{1 - \cos\phi\, \smfrac{V}{c}}.
\Tag{41}
\]
This is a general equation for the aberration of light.

For the particular case that the direction of the beam of light is
perpendicular to the motion of the observer we have $\cos\phi = 0$
\[
\cos\phi' = - \frac{V}{c},
\Tag{42}
\]
which, except for second-order differences, is identical with the familiar
expression which makes the tangent of the angle of aberration numerically
equal to~$V/c$.\DPnote{** Slant fractions start here} The experimental verification of the formula
by astronomical measurements is familiar.

\Subsubsection{56}{Velocity of Light in Moving Media.} It is also possible to treat
very simply by kinematic methods the problem of the velocity of
light in moving media. We shall confine ourselves to the particular
case of a beam of light in a medium which is itself moving parallel
to the light.

Let the medium be moving with the velocity~$V$ in the $X$~direction,
and let us consider the system of coördinates~$S'$ as stationary with
respect to the medium. Now since the medium appears to be stationary
with respect to observers in~$S'$ it is evident that the velocity
of the light with respect to~$S'$ will be~$c/\mu$, where $\mu$~is index of refraction
%% -----File: 074.png---Folio 60-------
for the medium. If now we use our equation~(26) for the addition of
velocities we shall obtain for the velocity of light, as measured by
observers in~$S$,
\[
u = \frac{\smfrac{c}{\mu} + V}{1 + \smfrac{V\, \smfrac{c}{\mu}}{c^2}}.
\Tag{43}
\]
Carrying out the division and neglecting terms of higher order we
obtain
\[
u = \frac{c}{\mu} + \left(\frac{\mu^2 - 1}{\mu^2}\right) V.
\Tag{44}
\]
The equation thus obtained is identical with that of Fresnel, the
quantity $\left(\dfrac{\mu^2 - 1}{\mu^2}\right)$ being the well-known Fresnel coefficient. The
empirical verification of this equation by the experiments of Fizeau
and of Michelson and Morley is too well known to need further
mention.

For the case of a dispersive medium we should obviously have to
substitute in equation~(44) the value of~$\mu$ corresponding to the particular
frequency,~$\nu'$, which the light has in system~$S'$. It should be
noticed in this connection that the frequencies $\nu'$~and~$\nu$ which the
light has respectively in system~$S$ and system~$S'$, although nearly
enough the same for the practical use of equation~(44), are in reality
connected by an expression which can easily be shown (see \Secref[section]{54})
to have the form
\[
\nu' = \kappa \left(1 - \frac{V}{c}\right)\nu.
\Tag{45}
\]

\Subsubsection{57}{Group Velocity.} In an entirely similar way we may treat the
problem of group velocity and obtain the equation
\[
G = \frac{G' + V}{1 + \smfrac{G'V}{c^2}},
\Tag{46}
\]
where $G'$ is the group velocity as it appears to an observer who is
%% -----File: 075.png---Folio 61-------
stationary with respect to the medium. $G'$~is, of course, an experimental
quantity, connected with frequency and the properties of the
medium, in a way to be determined by experiments on the stationary
medium.

In conclusion we wish to call particular attention to the extraordinary
simplicity of this method of handling the optics of moving
media as compared with those that had to be employed before the
introduction of the principle of relativity.
%% -----File: 076.png---Folio 62-------


\Chapter{VI}{The Dynamics of a Particle.}
\SetRunningHeads{Chapter Six.}{Dynamics of a Particle.}

\Paragraph{58.} In this chapter and the two following, we shall present a
system of ``relativity mechanics'' based on Newton's three laws of
motion, the Einstein transformation equations for space and time,
and the principle of the conservation of mass.


\Subsection{The Laws of Motion.}

Newton's laws of motion may be stated in the following form:

I\@. Every particle continues in its state of rest or of uniform motion
in a straight line, unless it is acted upon by an external force.

II\@. The rate of change of the momentum of the particle is equal
to the force acting and is in the same direction.

III\@. For the action of every force there is an equal force acting
in the opposite direction.

Of these laws the first two merely serve to define the concept of
force, and their content may be expressed in mathematical form by
the following equation of definition
\[
\vc{F}
  = \frac{d}{dt}(m\vc{u})
  = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u},
\Tag{47}
\]
where $\vc{F}$ is the force acting on a particle of mass~$m$ which has the
velocity~$\vc{u}$, and hence the momentum~$m\vc{u}$.

Quite different in its nature from the first two laws, which merely
give us a definition of force, the third law states a very definite physical
postulate, since it requires for every change in the momentum of a
body an equal and opposite change in the momentum of some other
body. The truth of this postulate will of course be tested by comparing
with experiment the results of the theory of mechanics which
we base upon its assumption.


\Subsection{Difference between Newtonian and Relativity Mechanics.}

\Paragraph{59.} Before proceeding we may point out the particular difference
between the older Newtonian mechanics, which were based on the
laws of motion and the \emph{Galilean} transformation equations for space
%% -----File: 077.png---Folio 63-------
and time, and our new system of relativity mechanics based on
those same laws of motion and the \emph{Einstein} transformation equations.

In the older mechanics there was no reason for supposing that the
mass of a body varied in any way with its velocity, and hence force
could be defined interchangeably as the rate of change of momentum
or as mass times acceleration, since the two were identical. In relativity
mechanics, however, we shall be forced to conclude that the
mass of a body increases in a perfectly definite way with its velocity,
and hence in our new mechanics we must define force as equal to the
total rate of change of momentum
\[
\frac{d(m\vc{u})}{dt}
  = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u}
\]
instead of merely as mass times acceleration $m\, \dfrac{d\vc{u}}{dt}$. If we should try
to define force in ``relativity mechanics'' as merely equal to mass
times acceleration, we should find that the application of Newton's
third law of motion would then lead to very peculiar results, which
would make the mass of a body different in different directions and
force us to give up the idea of the conservation of mass.


\Subsection{The Mass of a Moving Particle.}

\Paragraph{60.} In \Secref{31} we have already obtained in an elementary way
an expression for the mass of a moving particle, by considering a
collision between elastic particles and calculating how the resulting
changes in velocity would appear to different observers who are
themselves in relative motion. Since we now have at our command
general formulæ for the transformation of velocities, we are now in
a position to handle this problem much more generally, and \emph{in particular
to show that the expression obtained for the mass of a moving particle
is entirely independent of the consideration of any particular type of
collision}.

\Subsubsection{61}{Transverse Collision.} Let us first treat the case of a so-called
``transverse'' collision. Consider a system of coördinates and two
\begin{wrapfigure}{l}{3in}%[** TN: Width-dependent break]
  \Fig{11}
  \Input[3in]{078}
\end{wrapfigure}
exactly similar elastic particles, each having the mass~$m_0$ when at
rest, one moving in the $X$~direction with the velocity~$+u$ and the
other with the velocity~$-u$. (See \Figref{11}.) Besides the large
components of velocity $+u$~and~$-u$ which they have in the $X$~direction
%% -----File: 078.png---Folio 64-------
let them also have small components of velocity in the $Y$~direction,
$+v$~and~$-v$. The experiment is so arranged that the particles
will just undergo a glancing collision as they pass each other and
rebound with components
of velocity in the $Y$~direction
of the same magnitude,~$v$,
which they originally had,
but in the reverse direction.
(It is evident from the symmetry of the arrangement that the experiment
would actually occur as we have stated.)

We shall now be interested in the way this experiment would appear
to an observer who is in motion in the $X$ direction with the velocity~$V$
relative to our original system of coördinates.

From equation~(14) for the transformation of velocities, it can
be seen that this \emph{new observer} would find for the $X$~component velocities
of the two particles the values
\[
u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}} \qquad\text{and}\qquad
u_2 = \frac{-u - V}{1 + \smfrac{uV}{c^2}}
\Tag{48}
\]
and from equation~(15) for the $Y$~component velocities would find the
values
\[
v_1 = \pm \frac{v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}}
\qquad\text{and}\qquad
v_2 = \mp \frac{v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}},
\Tag{49}
\]
the signs depending on whether the velocities are measured before or
after the collision.

Now from Newton's third law of motion (\ie, the principle of
the equality of action and reaction) it is evident that on collision
the two particles must undergo the same numerical change in momentum.

For the experiment that we have chosen the only change in momentum
is in the $Y$~direction, and the observer whose measurements
we are considering finds that one particle undergoes the total change
%% -----File: 079.png---Folio 65-------
in velocity
\begin{align*}
2v_1 &= \frac{2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} \\
\intertext{and that the other particle undergoes the change in velocity}
2v_2 &= \frac{2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 + \smfrac{uV}{c^2}}.
\end{align*}

Since these changes in the velocities of the particles are not equal,
it is evident that their masses must also be unequal if the principle
of the equality of action and reaction is true for all observers, as we
have assumed. This difference in the mass of the particles, each of
which has the mass~$m_0$ when at rest, arises from the fact that the mass
of a particle is a function of its velocity and for the observer in question
the two particles are not moving with the same velocity.

Using the symbols $m_1$~and~$m_2$ for the masses of the particles, we
may now write as a mathematical expression of the requirements of
the third law of motion
\[
\frac{2m_1v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} =
\frac{2m_2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 + \smfrac{uV}{c^2}}.
\]

Simplifying, we obtain by direct algebraic transformation
%[** TN: Setting innermost denominator fractions textstyle for clarity]
\[
\frac{m_1}{m_2}
  = \frac{1 - \smfrac{uV}{c^2}}{1 + \smfrac{uV}{c^2}}
  = \frac{\sqrt{
    1 - \smfrac{\Biggl(\smfrac{-u - V}{1 + \tfrac{uV}{c^2}}\Biggr)^2}{c^2}}}
         {\sqrt{
    1 - \smfrac{\Biggl(\smfrac{ u - V}{1 - \tfrac{uV}{c^2}}\Biggr)^2}{c^2}}},
\]
%% -----File: 080.png---Folio 66-------
which on the substitution of equations~(48) gives us
\[
\frac{m_1}{m_2}
  = \frac{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}}
         {\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}.
\Tag{50}
\]

This equation thus shows that the mass of a particle moving with
the velocity~$u$\footnote
  {For simplicity of calculation we consider the case where the components of
  velocity in the $Y$~direction are small enough to be negligible in their effect on the
  mass of the particles compared with the large components of velocity $u_1$~and~$u_2$ in
  the $X$~direction.}
is inversely proportional to $\sqrt{1 - \dfrac{u^2}{c^2}}$, and, denoting
the mass of the particle at rest by~$m_0$, we may write as a \emph{general expression
for the mass of a moving particle}
\[
m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}.
\Tag{51}
\]

\Subsubsection{62}{Mass the Same in All Directions.} The method of derivation
that we have just used to obtain this expression for the mass of a
moving particle is based on the consideration of a so-called ``transverse
collision,'' and in fact the expression obtained has often been
spoken of as that for the \emph{transverse mass} of a moving particle, while
a different expression, $\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$, has been used for the so-called
\emph{longitudinal mass} of the particle. These expressions $\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ and
$\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$ are, as a matter of fact, the values of the electric force
necessary to give a charged particle unit acceleration respectively
at right angles and in the same direction as its original velocity, and
hence such expressions would be proper for the mass of a moving particle
if we should define force as mass times acceleration. As already
%% -----File: 081.png---Folio 67-------
stated, however, it has seemed preferable to retain, for force, Newton's
original definition which makes it equal to the rate of change of
momentum, and we shall presently see that this more suitable definition
is in perfect accord with the idea that the mass of a particle is
the same in all directions.

Aside from the unnecessary complexity which would be introduced,
the particular reason making it unfortunate to have different
expressions for mass in different directions is that under such conditions
it would be impossible to retain or interpret the principle of
the conservation of mass. And we shall now proceed to show that
by introducing the principle of the conservation of mass, the consideration
of a ``longitudinal collision'' will also lead to exactly the
same expression, $\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, for the mass of a moving particle as we
have already obtained from the consideration of a transverse collision.

\Subsubsection{63}{Longitudinal Collision.} Consider a system of coördinates and
two elastic particles moving in the $X$~direction with the velocities
$+u$~and~$-u$ so that a ``longitudinal'' (\ie, head-on) collision will
occur. Let the particles be exactly alike, each of them having the
mass~$m_0$ when at rest. On collision the particles will evidently come
to rest, and then under the action of the elastic forces developed start
up and move back over their original paths with the respective velocities
$-u$~and~$+u$ of the same magnitude as before.

Let us now consider how this collision would appear to an observer
who is moving past the original system of coördinates with the velocity~$V$
in the $X$~direction. Let $u_1$~and~$u_2$ be the velocities of the particles
as they appear to this new observer before the collision has taken
place. Then, from our formula for the transformation of velocities~(14),
it is evident that we shall have
\[
u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}}\qquad\text{and}\qquad
u_2 = \frac{-u - V}{1 + \smfrac{uV}{c^2}}.
\Tag{52}
\]

Since these velocities $u_1$~and~$u_2$ are not of the same magnitude,
the two particles which have the same mass when at rest do not have
the same mass for this observer. Let us call the masses before collision
$m_1$~and~$m_2$.
%% -----File: 082.png---Folio 68-------

Now during the collision the velocities of the particles will all the
time be changing, but from the principle of the conservation of mass
the sum of the two masses must all the time be equal to $m_1 + m_2$.
When in the course of the collision the particles have come to relative
rest, they will be moving past our observer with the velocity~$-V$,
and their momentum will be $-(m_1 + m_2)V$. But, from the principle
of the equality of action and reaction, it is evident that this momentum
must be equal to the original momentum before collision occurred.
This gives us the equation $-(m_1 + m_2)V = m_1 u_1 + m_2 u_2$. Substituting
our values~(52) for $u_1$~and~$u_2$ we have
\[
\frac{m_1}{\left(1 - \smfrac{uV}{c^2}\right)} =
\frac{m_2}{\left(1 + \smfrac{uV}{c^2}\right)},
\]
and by direct algebraic transformation, as in the previous proof,
this can be shown to be identical with
\[
\frac{m_1}{m_2}
  = \frac{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}}
         {\sqrt{1 - \smfrac{{u_1}^2}{c^2}}},
\]
leading to the same expression that we obtained before for the mass
of a moving particle, viz.:
\[
m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}.
\]

\Subsubsection{64}{Collision of Any Type.} We have derived this formula for the
mass of a moving particle first from the consideration of a transverse
and then of a longitudinal collision between particles which are elastic
and have the same mass when at rest. It seems to be desirable to
show, however, that the consideration of any type of collision between
particles of any mass leads to the same formula for the mass of a
moving particle.

For the mass~$m$ of a particle moving with the velocity~$u$ let us
write the equation $m = m_0 F(u^2)$, where $F(\:)$~is the function whose
form we wish to determine. The mass is written as a function of
%% -----File: 083.png---Folio 69-------
the square of the velocity, since from the homogeneity of space the
mass will be independent of the direction of the velocity, and the
mass is made proportional to the mass at rest, since a moving body
may evidently be thought of as divided into parts without change in
mass. It may be further remarked that the form of the function~$F(\:)$
must be such that its value approaches unity as the variable
approaches zero.

Let us now consider two particles having respectively the masses
$m_0$~and~$n_0$ when at rest, moving with the velocities $u$~and~$w$ before
collision, and with the velocities $U$~and~$W$ after a collision has taken
place.

From the principle of the conservation of mass we have
\begin{multline*}
m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2) +
n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2) \\
= m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)
+ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2),
\Tag{53}
\end{multline*}
and from the principle of the equality of action and reaction (\ie,
Newton's third law of motion)
\begin{multline*}
m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_x +
n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_x \\
= m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_x
+ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_x,
\Tag{54}
\end{multline*}
\begin{multline*}
m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_y +
n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_y \\
= m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_y
+ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_y,
\Tag{55}
\end{multline*}
\begin{multline*}
m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_z +
n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_z \\
= m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_z
+ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_z.
\Tag{56}
\end{multline*}

These velocities, $u_x$,~$u_y$,~$u_x$, $w_x$,~$w_y$,~$w_z$, $U_x$,~etc., are measured, of
course, with respect to some definite system of ``space-time'' coördinates.
An observer moving past this system of coördinates with the
velocity~$V$ in the $X$~direction would find for the corresponding component
velocities the values
\[
\frac{u_x - V}{1 - \smfrac{u_xV}{c^2}},\quad
\frac{\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}}\, u_y,\quad
\frac{\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}}\, u_z,\quad
\frac{w_x - V}{1 - \smfrac{w_xV}{c^2}},\quad\text{etc.},
\]
as given by our transformation equations for velocity \DPchg{(14, 15, 16)}{(14),~(15),~(16)}.
%% -----File: 084.png---Folio 70-------

Since the law of the conservation of mass and Newton's third
law of motion must also hold for the measurements of the new observer,
we may write the following new relations corresponding to
equations \DPchg{53~to~56}{(53)~to~(56)}:

{\footnotesize%
\[
\begin{aligned}
m_0 F&\left\{
  \left(\frac{u_x - V}{1 - \smfrac{u_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_y}{1 - \smfrac{u_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_z}{1 - \smfrac{u_x V}{c^2}}\right)^2
\right\} \\
+ n_0F&\left\{
\left(\frac{w_x - V}{1 - \smfrac{w_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_y}{1 - \smfrac{w_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_z}{1 - \smfrac{w_x V}{c^2}}\right)^2
\right\} \\
= m_0F&\left\{
\left(\frac{U_x - V}{1 - \smfrac{U_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_y}{1 - \smfrac{U_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_z}{1 - \smfrac{U_x V}{c^2}}\right)^2
\right\} \\
+ n_0F&\left\{
\left(\frac{W_x - V}{1 - \smfrac{W_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_y}{1 - \smfrac{W_x V}{c^2}}\right)^2
+ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_z}{1 - \smfrac{W_x V}{c^2}}\right)^2
\right\},
\end{aligned}
\Tag{53\textit{a}}
\]}%
\[
\begin{aligned}
&m_0F\{u_x\cdots\}\, \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}} +
n_0F\{w_x\cdots\}\, \frac{w_x - V}{1 - \smfrac{w_xV}{c^2}} \\
&\qquad=
m_0F\{U_x\cdots\}\, \frac{U_x - V}{1 - \smfrac{U_xV}{c^2}} +
n_0F\{W_x\cdots\}\, \frac{W_x - V}{1 - \smfrac{W_xV}{c^2}},
\end{aligned}
\Tag{54\textit{a}}
\]
{\small%
\[
\begin{aligned}
&m_0F\{u_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_y}{1 - \smfrac{u_xV}{c^2}} +
n_0F\{w_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_y}{1 - \smfrac{w_xV}{c^2}} \\
&\qquad=
m_0F\{U_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_y}{1 - \smfrac{U_xV}{c^2}} +
n_0F\{W_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_y}{1 - \smfrac{W_xV}{c^2}},
\end{aligned}
\Tag{55\textit{a}}
\]}%
%% -----File: 085.png---Folio 71-------
\[
\begin{aligned}
&m_0F{u_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_x}{1 - \smfrac{u_xV}{c^2}} +
n_0F{w_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_x}{1 - \smfrac{w_xV}{c^2}} \\
&\qquad=
m_0F{U_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_x}{1 - \smfrac{U_xV}{c^2}} +
n_0F{W_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_x}{1 - \smfrac{W_xV}{c^2}}.
\end{aligned}
\Tag{56\textit{a}}
\]

It is evident that these equations \DPchg{(53\textit{a}--56\textit{a})}{(53\textit{a})--(56\textit{a})} must be true no
matter what the velocity between the new observer and the original
system of coördinates, that is, true for all values of~$V$. The velocities
$u_x$,~$u_y$,~$u_z$, $w_x$,~etc., are, however, perfectly definite quantities, measured
with reference to a definite system of coördinates and entirely independent
of~$V$. If these equations are to be true for perfectly definite
values of $u_x$,~$u_y$,~$u_z$, $w_x$,~etc., and for all values of~$V$, it is evident that
the function~$F(\:\,)$ must be of such a form that the equations are
identities in~$V$. As a matter of fact, it is found by trial that $V$~can
be cancelled from all the equations if we make $F(\:\,)$ of the form
$\dfrac{1}{\sqrt{1 - \smfrac{(\:)}{c^2}}}$; and we see that the expected relation is a solution of the
equations, although perhaps not necessarily a unique solution.

Before proceeding to use our formula for the mass of a moving
particle for the further development of our system of mechanics,
we may call attention in passing to the fact that the experiments of
Kaufmann, Bucherer, and Hupka have in reality shown that the mass
of the electron increases with its velocity according to the formula
which we have just obtained. We shall consider the dynamics of the
electron more in detail in the chapter devoted to \Chapnumref[XII]{electromagnetic
theory}. We wish to point out now, however, that in this derivation
we have made no reference to any electrical charge which might be
carried by the particle whose mass is to be determined. Hence we
may reject the possibility of explaining the Kaufmann experiment
by assuming that the charge of the electron decreases with its velocity,
since the increase in mass is alone sufficient to account for the results
of the measurement.
%% -----File: 086.png---Folio 72-------


\Subsection{Transformation Equations for Mass.}

\Paragraph{65.} Since the velocity of a particle depends on the particular
system of coördinates chosen for the measurement, it is evident that
the mass of the particle will also depend on our reference system of
coördinates. For the further development of our system of dynamics,
we shall find it desirable to obtain transformation equations for mass
similar to those already obtained for velocity, acceleration, etc.

We have
\[
m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\]
where the velocity~$u$ is measured with respect to some definite system
of coördinates,~$S$. Similarly with respect to a system of coördinates~$S'$
which is moving relatively to~$S$ with the velocity~$V$ in the $X$~direction
we shall have
\[
m' = \frac{m_0}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}.
\]
We have already obtained, however, a transformation equation~(17)
for the function of the velocity occurring in these equations and on
substitution we obtain the desired transformation equation
\[
m' = \left(1 - \frac{u_x V}{c^2}\right) \kappa m,
\Tag{57}
\]
where $\kappa$ has the customary significance
\[
\kappa = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}.
\]

By differentiation of~(57) with respect to the time and simplification,
we obtain the following transformation equation for the
\emph{rate at which the mass of a particle is changing} owing to change in
velocity
\[
\dot{m}' = \dot{m} - \frac{mV}{c^2}
  \left(1 - \frac{u_xV}{c^2}\right)^{-1} \frac{du_x}{dt}.
\Tag{58}
\]
%% -----File: 087.png---Folio 73-------

%[** TN: ToC entry reads "The Force Acting on a Moving Particle"]
\Subsection{Equation for the Force Acting on a Moving Particle.}

\Paragraph{66.} We are now in a position to return to our development of the
dynamics of a particle. In the first place, the equation which we
have now obtained for the mass of a moving particle will permit
us to rewrite the original equation by which we defined force, in a
number of ways which will be useful for future reference.

We have our equation of definition~(47)
\[
\vc{F} = \frac{d}{dt}(m\vc{u})
  = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u},
\]
which, on substitution of the expression for~$m$, gives us
\[
\vc{F}
  = \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u}\Biggr]
  = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}
  + \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr] \vc{u}
\Tag{59}
\]
or, carrying out the indicated differentiation,
\[
\vc{F}
  = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}
  + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\,
    \frac{u}{c^2}\, \frac{du}{dt}\, \vc{u}.
\Tag{60}
\]


\Subsection{Transformation Equations for Force.}

\Paragraph{67.} We are also in position to obtain transformation equations for
force. We have
\[
\vc{F} = \frac{d}{dt}(m\vc{u}) = m\vc{\dot{u}} + \dot{m}\vc{u}
\]
or
\begin{align*}
F_x &= m\dot{u}_x + \dot{m}u_x, \\
F_y &= m\dot{u}_y + \dot{m}u_y, \\
F_z &= m\dot{u}_z + \dot{m}u_z.
\end{align*}
We have transformation equations, however, for all the quantities
on the right-hand side of these equations. For the velocities we
have equations (14),~(15) and~(16), for the accelerations (18),~(19)
and~(20), for mass, equation~(57) and for rate of change of mass,
equation~(58). Substituting above we obtain as our \emph{transformation
%% -----File: 088.png---Folio 74-------
equations for force}
\begin{align*}
F_x' &= \frac{F_x - \dot{m}V}{1 - \smfrac{u_xV}{c^2}}
      = F_x - \frac{u_y V}{c^2 - u_x V}\, F_y
            - \frac{u_z V}{c^2 - u_x V}\, F_z,            \Tag{61} \\
F_y' &= \frac{\kappa^{-1}}{1 - \smfrac{u_x V}{c^2}}\, F_y, \Tag{62}\\
F_z' &= \frac{\kappa^{-1}}{1 - \smfrac{u_x V}{c^2}}\, F_z. \Tag{63}
\end{align*}

We may now consider a few applications of the principles governing
the dynamics of a particle.


\Subsection{The Relation between Force and Acceleration.}

\Paragraph{68.} If we examine our equation~(59) for the force acting on a
particle
\[
F = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}
  + \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr] \vc{u},
\Tag{59}
\]
we see that the force is equal to the sum of two vectors, one of which
is in the direction of the acceleration $\dfrac{d\vc{u}}{dt}$ and the other in the direction
of
\begin{wrapfigure}[17]{l}{2.5in}%[** TN: Width-dependent break]
  \Fig{12}
  \Input[2.5in]{088}
\end{wrapfigure}
the existing velocity~$\vc{u}$, so that \emph{in general force and the acceleration
it produces are not in the same direction}.
We shall find it interesting
to see, however, that if the force
which does produce acceleration in
a given direction be resolved perpendicular
and parallel to the acceleration,
the two components will
be connected by a definite relation.

Consider a particle (\Figref[fig.]{12}) in
plane space moving with the velocity
\[
\vc{u} = {u_x}\vc{i} + {u_y}\vc{j}.
\]
%% -----File: 089.png---Folio 75-------
Let it be accelerated in the $X$~direction by the action of the component
forces $F_x$~and~$F_y$.

From our general equation~(59) for the force acting on a particle
we have for these component forces
\begin{align*}
F_x &= \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{du_x}{dt}
  + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]u_x,
\Tag{64} \\
F_y &= \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{du_y}{dt}
  + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]u_y.
\Tag{65}
\end{align*}

Introducing the condition that all the acceleration is to be in the $Y$~direction,
which makes $\dfrac{du_x}{dt} = 0$, and further noting that $u^2 = u_x^2 + u_y^2$,
by the division of equation~(64) by~(65), we obtain
\begin{align*}
\frac{F_x}{F_y} &= \frac{u_x u_y}{c^2 - {u_x}^2}, \\
F_x &= \frac{u_x u_y}{c^2 - {u_x}^2}\, F_y.
\Tag{66}
\end{align*}

\emph{Hence, in order to accelerate a particle in a given direction, we may
apply any force~$F_y$ in the desired direction, but must at the same time
apply at right angles another force~$F_x$ whose magnitude is given by
equation~\upshape{(66)}.}

Although at first sight this state of affairs might seem rather
unexpected, a simple qualitative consideration will show the necessity
of a component of force perpendicular to the desired acceleration.
Refer again to \Figref{12}; since the particle is being accelerated in the $Y$~direction,
its total velocity and hence its mass are increasing. This
increasing mass is accompanied by increasing momentum in the $X$~direction
even when the velocity in that direction remains constant.
The component force~$F_x$ is necessary for the production of this increase
in $X$-momentum.

In a later paragraph we shall show an application of equation~(66)
in electrical theory.
%% -----File: 090.png---Folio 76-------


\Subsection{Transverse and Longitudinal Acceleration.}

\Paragraph{69.} An examination of equation~(66) shows that there are two
special cases in which the component force~$F_x$ disappears and the
force and acceleration are in the same direction. $F_x$~will disappear
when either $u_x$~or~$u_y$ is equal to zero, so that force and acceleration
will be in the same direction when the force acts exactly at right
angles to the line of motion of the particle, or in the direction of the
motion (or of course also when $u_x$~and~$u_y$ are both equal to zero and
the particle is at rest). It is instructive to obtain simplified expressions
for force for these two cases of transverse and longitudinal
acceleration.

Let us again examine our equation~(60) for the force acting on a
particle
\[
\vc{F}
  = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}
  + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\,
    \frac{u}{c^2}\, \frac{du}{dt} \vc{u}.
\Tag{60}%[** TN: [sic] Repeated equation]
\]

For the case of a \emph{transverse acceleration} there is no component of
force in the direction of the velocity~$\vc{u}$ and the second term of the
equation is equal to zero, giving us
\[
\vc{F} = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}.
\Tag{67}
\]

For the case of \emph{longitudinal acceleration}, the velocity~$\vc{u}$ and the
acceleration~$\dfrac{d\vc{u}}{dt}$ are in the same direction, so that we may rewrite the
second term of~(60), giving us
\[
\vc{F}
  = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}
  + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\,
    \frac{u^2}{c^2}\, \frac{d\vc{u}}{dt},
\]
and on simplification this becomes
\[
\vc{F}
  = \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, \frac{d\vc{u}}{dt}.
\Tag{68}
\]
%% -----File: 091.png---Folio 77-------
An examination of this expression shows the reason why $\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$
is sometimes spoken of as the expression for the \emph{longitudinal mass} of a
particle.


\Subsection{The Force Exerted by a Moving Charge.}

\Paragraph{70.} In a \Chapnumref[XII]{later chapter} we shall present a consistent development
of the fundamentals of electromagnetic theory based on the Einstein
transformation equations for space and time and the four field equations.
At this point, however, it may not be amiss to point out that
the principles of mechanics themselves may sometimes be employed
to obtain a simple and direct solution of electrical problems.

Suppose, for example, we wish to calculate the force with which a
\emph{point charge in uniform motion} acts on any other point charge. We
can solve this problem by considering a system of coördinates which
move with the same velocity as the charge itself. An observer
making use of the new system of coördinates could evidently calculate
the force exerted by the charge in question by Coulomb's familiar
inverse square law for static charges, and the magnitude of the force
as measured in the original system of coördinates can then be determined
from our transformation equations for force. Let us proceed
to the specific solution of the problem.

Consider a system of coördinates~$S$, and a charge~$e$ in uniform
motion along the $X$~axis with the velocity~$V$. We desire to know
the force acting at the time~$t$ on any other charge~$e_1$ which has any
desired coördinates $x$,~$y$, and~$z$ and any desired velocity $u_x$,~$u_y$ and~$u_z$.

Assume a system of coördinates,~$S'$, moving with the same velocity
as the charge~$e$ which is taken coincident with the origin. To an
observer moving with the system~$S'$, the charge~$e$ appears to be
always at rest and surrounded by a pure electrostatic field. Hence
in system~$S'$ the force with which $e$~acts on~$e_1$ will be, in accordance
with Coulomb's law\footnote
  {It should be noted that in its original form Coulomb's law merely stated
  that the force between two stationary charges was proportional to the product of
  the charges and inversely to the distance between them. In the present derivation
  we have extended this law to apply to the instantaneous force exerted by a stationary
  charge upon any other charge.

  The fact that a charge of electricity appears the same to observers in all systems
  is obviously also necessary for the setting up of equations (69),~(70),~(71). That
  such is the case, however, is an evident consequence of the atomic nature of electricity.
  The charge~$e$ would appear of the same magnitude to observers both in
  system~$S$ and system~$S'$, since they would both count the same number of electrons
  on the charge. (See \Secref{157}.)}
\[
\vc{F'} = \frac{e e_1 \vc{r'}}{{r'}^3}
\]
%% -----File: 092.png---Folio 78-------
or
\begin{align*}
F_x' &= \frac{ee_1x'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{69} \\
F_y' &= \frac{ee_1x'}{({y'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{70} \\
F_z' &= \frac{ee_1x'}{({z'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{71}
\end{align*}
where $x'$,~$y'$, and~$z'$ are the coördinates of the charge~$e_1$ at the time~$t'$.
For simplicity let us consider the force at the time $t' = 0$; then from
transformation equations (9),~(10), (11),~(12) we shall have
\[
x' = \kappa^{-1} x,\qquad y' = y, \qquad z'= z.
\]
Substituting in (69),~(70),~(71) and also using our transformation
equations for force (61),~(62),~(63), we obtain the following equations
for the force acting on~$e_1$, as it appears to an observer in system~$S$:
\begin{align*}%[** TN: Aligning on "="s]
F_x &= \frac{ee_1x'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}
       \left(x + \frac{V}{c^2}\, \kappa^2(yu_y + zu_z)\right),
\Tag{72} \\
F_y &= \frac{ee_1\left(1 - \smfrac{u_xV}{c^2}\right) \kappa y}
            {({\kappa^{-2}x'}^2 + {y'}^2 + {z'}^2)^{3/2}},
\Tag{73} \\
F_z &= \frac{ee_1\left(1 - \smfrac{u_xV}{c^2}\right) \kappa z}
            {({\kappa^{-2}x'}^2 + {y'}^2 + {z'}^2)^{3/2}}.
\Tag{74}
\end{align*}

These equations give the force acting on~$e_1$ at the time~$t$. From
transformation equation~(12) we have $t = \dfrac{V}{c^2}\, x$, since $t' = 0$. At this
time the charge~$e$, which is moving with the uniform velocity~$V$ along
%% -----File: 093.png---Folio 79-------
the $X$~axis, will evidently have the position
\[
x_e = \frac{V^2}{c^2}\, x,\qquad
y_e = 0, \qquad
z_e = 0.
\]

For convenience we may now refer our results to a system of
coördinates whose origin coincides with the position of the charge~$e$
at the instant under consideration. If $X$,~$Y$ and~$Z$ are the coördinates
of~$e_1$ with respect to this new system, we shall evidently have
the relations
\begin{gather*}
X = x - \frac{V^2 }{c^2}\, x = \kappa^{-2} x,\qquad Y = y,\qquad Z = z,\\
U_x = u_x, \qquad U_y = u_y, \qquad U_z = u_z.
\end{gather*}
Substituting into (72),~(73),~(74) we obtain
\begin{align*}%[** TN: Aligning on "="s]
F_x &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right)
       \left(X + \frac{V}{c^2}\, (YU_y + ZU_z)\right),
\Tag{75} \\
F_y &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right)
       \left(1 - \frac{U_xV}{c^2})\right) Y,
\Tag{76} \\
F_z &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right)
       \left(1 - \frac{U_xV}{c^2})\right) Z,
\Tag{77}
\end{align*}
where for simplicity we have placed
\[
s = \sqrt{X^2 + \left(1 - \frac{V^2}{c^2}\right)(Y^2 + Z^2)}.
\]

These are the same equations which would be obtained by substituting
the well-known formulæ for the strength of the electric and
magnetic field around a moving point charge into the fifth fundamental
equation of the Maxwell-Lorentz theory, $\vc{f} = \rho \left(\vc{e} + \dfrac{1}{c}\, [\vc{u} × \vc{h}]^*\right)$.
They are really obtained in this way more easily, however, and are
seen to come directly from Coulomb's law.

%[** TN: Unnumbered, but has a ToC entry]
\Subsubsection{}{The Field around a Moving Charge.} Evidently we may also use
these considerations to obtain an expression for the electric field
produced by a moving charge~$e$, if we consider the particular case
that the charge~$e_1$ is stationary (\ie, $U_x = U_y = U_z = 0$) and equal
%% -----File: 094.png---Folio 80-------
to unity. Making these substitutions in (75),~(76),~(77) we obtain
the well-known expression for the electrical field in the neighborhood
of a moving point charge
\[
\vc{F} = e = \frac{\vc{e}}{s^3} \left(1 - \frac{V^2}{c^2}\right)\vc{r},
\Tag{78}
\]
where
\[
\vc{r} = X\vc{i} + Y\vc{j} + Z\vc{k}.
\]

\Subsubsection{71}{Application to a Specific Problem.} Equations (75), (76), (77)
can also be applied in the solution of a
rather interesting specific problem.

Consider a charge~$e$ constrained to
move in the $X$~direction with the velocity~$V$
and at the instant under consideration
let it coincide with the origin
of a system of stationary coördinates
$YeX$ (\Figref[fig.]{13}). Suppose now a second
charge~$e_1$, situated at the point $X = 0$,
$Y = Y$ and moving in the $X$~direction
with the same velocity~$V$ as the charge~$e$,
and also having a component velocity
in the $Y$~direction~$U_y$. Let us
%[** TN: Move down past page break; width-dependent line break]
\begin{wrapfigure}{l}{2.25in}
  \Fig{13}
  \Input[2.25in]{094}
\end{wrapfigure}
predict
the nature of its motion under the influence
of the charge~$e$, it being otherwise
unconstrained.

From the simple qualitative considerations placed at our disposal
by the theory of relativity, it seems evident that the charge~$e_1$ ought
merely to increase its component of velocity in the $Y$~direction and
retain unchanged its component in the $X$~direction, since from the
point of view of an observer moving along with~$e$ the phenomenon is
merely one of ordinary \emph{electrostatic} repulsion.

Let us see whether our equations for the force exerted by a moving
charge actually lead to this result. By making the obvious substitutions
in equations (75)~and~(76) we obtain for the component
forces on~$e_1$
\begin{align*}%[** TN: Aligning on "="s]
F_x &= \frac{e e_1}{s^3}\left(1 - \frac{V^2}{c^2}\right)
       \frac{V}{c^2}\, Y U_y,
\Tag{79} \\
F_x &= \frac{e e_1}{s^3}\left(1 - \frac{V^2}{c^2}\right)^2 Y.
\Tag{80}
\end{align*}
%% -----File: 095.png---Folio 81-------

Now under the action of the component force~$F_x$ we might at
first sight expect the charge~$e_1$ to obtain an acceleration in the $X$~direction,
in contradiction to the simple qualitative prediction that
we have just made on the basis of the theory of relativity. We
remember, however, that equation~(66) prescribes a definite ratio
between the component forces $F_x$~and~$F_y$ if the acceleration is to be
in the $Y$~direction, and dividing~(79) by~(80) we actually obtain the
necessary relation
\[
\frac{F_x}{F_y} = \frac{V U_y}{c^2 - V^2}.
\]

Other applications of the new principles of dynamics to electrical,
magnetic and gravitational problems will be evident to the reader.


\Subsection{Work.}

\Paragraph{72.} Before proceeding with the further development of our theory
of dynamics we shall find it desirable to define the quantities work,
kinetic, and potential energy.

We have already obtained an expression for the force acting on a
particle and shall define the work done on the particle as the integral
of the force times the distance through which the particle is displaced.
Thus
\[
W = \int \vc{F} · d\vc{r},
\Tag{81}
\]
where $\vc{r}$ is the radius vector determining the position of the particle.


\Subsection{Kinetic Energy.}

\Paragraph{73.} When a particle is brought from a state of rest to the velocity~$\vc{u}$
by the action of an unbalanced force~$\vc{F}$, we shall define its kinetic
energy as numerically equal to the work done in producing the velocity.
Thus
\[
K = W = \int \vc{F} · d\vc{r}.
\]

Since, however, the kinetic energy of a particle turns out to be
entirely independent of the particular choice of forces used in producing
the final velocity, it is much more useful to have an expression
for kinetic energy in terms of the mass and velocity of the particle.

We have
\[
K = \int \vc{F} · d\vc{r}
  = \int \vc{F} · \frac{d\vc{r}}{dt}\, dt
  = \int \vc{F} · \vc{u}\, dt.
\]
%% -----File: 096.png---Folio 82-------
Substituting the value of~$\vc{F}$ given by the equation of definition~(47)
we obtain
\begin{align*}
K &= \int m\, \frac{d\vc{u}}{dt} · \vc{u}\, dt
   + \int \frac{dm}{dt}\, \vc{u} · \vc{u}dt \\
  &= \int m\, \vc{u} · d\vc{u} + \int \vc{u} · \vc{u}\, dm \\
  &= \int mu\, du + \int u^2\, dm.
\end{align*}
Introducing the expression~(51) for the mass of a moving particle
$m = \dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, we obtain
\[
K = \int m_0\, \frac{u}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, du
  + \int\frac{m_0}{c^2}\, \frac{u^3}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, du
\]
and on integrating and evaluating the constant of integration by
placing the kinetic energy equal to zero when the velocity is zero,
we easily obtain the desired expression for the kinetic energy of a
particle:
\begin{align*}
K &= m_0 c^2 \Biggl[\frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1\Biggr],
\Tag{82} \\
  &= c^2(m - m_0).
\Tag{83}
\end{align*}

It should be noticed, as was stated above, that the kinetic energy
of a particle \emph{does} depend merely on its mass and final velocity and is
entirely independent of the particular choice of forces which happened
to be used in producing the state of motion.

It will also be noticed, on expansion into a series, that our expression~(82)
for the kinetic energy of a particle approaches at low
velocities the form familiar in the older Newtonian mechanics,
\[
K = \tfrac{1}{2} m_0 u^2.
\]


\Subsection{Potential Energy.}

\Paragraph{74.} When a moving particle is brought to rest by the action of a
%% -----File: 097.png---Folio 83-------
\emph{conservative}\footnote
  {A conservative force is one such that any work done by displacing a system
  against it would be completely regained if the motion of the system should be reversed.

  Since we believe that the forces which act on the ultimate particles and constituents
  of matter are in reality all of them conservative, we shall accept the general
  principle of the conservation of energy just as in Newtonian mechanics. (For a
  logical deduction of the principle of the conservation of energy in a system of particles,
  see the next chapter, \Secref[section]{89}.)} %[** TN: Not a useful chapter cross-ref]
force we say that its kinetic energy has been transformed
into potential energy. The increase in the potential energy
of the particle is equal to the kinetic energy which has been destroyed
and hence equal to the work done by the particle against the force,
giving us the equation
\[
\Delta U = -W = -\int \vc{F} · d\vc{r}.
\Tag{84}
\]


\Subsection{The Relation between Mass and Energy.}

\Paragraph{75.} We may now consider a very important relation between the
mass and energy of a particle, which was first pointed out in our
chapter on ``\Chapnumref[III]{Some Elementary Deductions}.''

When an isolated particle is set in motion, both its mass and
energy are increased. For the increase in mass we may write
\[
\Delta m = m - m_0,
\]
and for the increase in energy we have the expression for kinetic energy
given in equation~(83), giving us
\[
\Delta E = c^2(m-m_0),
\]
or, combining with the previous equation,
\[
\Delta E = c^2 \Delta m.
\Tag{85}
\]

Thus the increase in the kinetic energy of a particle always bears
the same definite ratio (the square of the velocity of light) to its
increase in mass. Furthermore, when a moving particle is brought
to rest and thus loses both its kinetic energy and its extra (``kinetic'')
mass, there seems to be every reason for believing that this mass
and energy which are associated together when the particle is in
motion and leave the particle when it is brought to rest will still
remain always associated together. For example, if the particle is
brought to rest by collision with another particle, it is an evident
%% -----File: 098.png---Folio 84-------
consequence of our considerations that the energy and the mass
corresponding to it do remain associated together since they are both
passed on to the new particle. On the other hand, if the particle
is brought to rest by the action of a conservative force, say for example
that exerted by an elastic spring, the kinetic energy which leaves the
particle will be transformed into the potential energy of the stretched
spring, and since the mass which has undoubtedly left the particle
must still be in existence, we shall believe that this mass is now associated
with the potential energy of the stretched spring.

\Paragraph{76.} Such considerations have led us to believe that matter and
energy may be best regarded as different names for the same fundamental
entity: \emph{matter}, the name which has been applied when we
have been interested in the property of mass or inertia possessed
by the entity, and \emph{energy}, the name applied when we have been
interested in the part taken by the entity in the production of motion
and other changes in the physical universe. We shall find these
ideas as to the relations between matter, energy and mass very fruitful
in the simplification of physical reasoning, not only because it
identifies the two laws of the conservation of mass and the conservation
of energy, but also for its frequent application in the solution
of specific problems.

\Paragraph{77.} We must call attention to the great difference in size between
the two units, the gram and the erg, both of which are used for the
measurement of the one fundamental entity, call it matter or energy
as we please. Equation~(85) gives us the relation
\[
E = c^2 m,
\Tag{86}
\]
where $E$~is expressed in ergs and $m$~in grams; hence, taking the velocity
of light as $3 × 10^{10}$~centimeters per second, we shall have
\[
1\text{ gram} = 9 × 10^{20}\text{ ergs}.
\Tag{87}
\]
The enormous number of ergs necessary for increasing the mass of
a system to the amount of a single gram makes it evident that experimental
proofs of the relation between mass and energy will be hard to
find, and outside of the experimental work on electrons at high velocities,
already mentioned in \Secref{64} and the well-known relations
%% -----File: 099.png---Folio 85-------
between the energy and momentum of a beam of light, such evidence
has not yet been forthcoming.

As to the possibility of obtaining further direct experimental
evidence of the relation between mass and energy, we certainly cannot
look towards thermal experiments with any degree of confidence,
since even on cooling a body down to the absolute zero of temperature
it loses but an inappreciable fraction of its mass at ordinary temperatures.\footnote
  {It should be noticed that our theory points to the presence of enormous
  stores of interatomic energy which are still left in substances cooled to the absolute
  zero.}
In the case of some radioactive processes, however, we may
find a transfer of energy large enough to bring about measurable
differences in mass. And making use of this point of view we might
account for the lack of exact relations between the atomic weights of
the successive products of radioactive decomposition.\footnote
  {See, for example, Comstock, \textit{Philosophical Magazine}, vol.~15, p.~1 (1908).}

\Subsubsection{78}{Application to a Specific Problem.} We may show an interesting
application of our ideas as to the relation between mass and
energy, in the treatment of a specific problem. Consider, just as in
\Secref{63}, two elastic particles both of which have the mass~$m_0$ at
rest, one moving in the $X$~direction with the velocity~$+u$ and the
other with the velocity~$-u$, in such a way that a head-on collision
between the particles will occur and they will rebound over their
original paths with the respective velocities $-u$~and~$+u$ of the
same magnitude as before.

Let us now consider how this collision would appear to an observer
who is moving past the original system of coördinates with the velocity~$V$
in the $X$~direction. To this new observer the particles will be
moving before the collision with the respective velocities
\[
u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}}\qquad\text{and}\qquad
u_2 = \frac{-u - V}{1 - \smfrac{uV}{c^2}},
\Tag{88}
\]
as given by equation~(14) for the transformation of velocities. Furthermore,
when in the course of the collision the particles have come
to relative rest they will obviously be moving past our observer with
the velocity~$-V$.
%% -----File: 100.png---Folio 86-------

Let us see what the masses of the particles will be both before and
during the collision. Before the collision, the mass of the first particle
will be
\[
\frac{m_0}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}} =
\frac{m_0}{\sqrt{1 - \smfrac{\Biggl[\smfrac{ u - V}{1 - \tfrac{uV}{c^2}}\Biggr]^2}{c^2}}} =
\frac{m_0 \left(1 - \smfrac{uV}{c^2}\right)}
     {\sqrt{\left(1 - \smfrac{V^2}{c^2}\right)\left(1 - \smfrac{u^2}{c^2}\right)}}
\]
and the mass of the second particle will be
\[
\frac{m_0}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} =
\frac{m_0}{\sqrt{1 - \smfrac{\Biggl[\smfrac{-u - V}{1 + \tfrac{uV}{c^2}}\Biggr]^2}{c^2}}} =
\frac{m_0 \left(1 + \smfrac{uV}{c^2}\right)}
     {\sqrt{\left(1 - \smfrac{V^2}{c^2}\right)\left(1 - \smfrac{u^2}{c^2}\right)}}.
\]
Adding these two expressions, we obtain for the sum of the masses of
the two particles before collision,
\[
\frac{2m_0}{\sqrt{\left(1 - \smfrac{V^2}{c^2}\right)
                  \left(1 - \smfrac{u^2}{c^2}\right)}}.
\]

Now during the collision, when the two particles have come to
relative rest, they will evidently both be moving past our observer
with the velocity~$-V$ and hence the sum of their masses at the
instant of relative rest would appear to be
\[
\frac{2m_0}{\sqrt{1 - \smfrac{V^2}{c^2}}},
\]
a quantity which is smaller than that which we have just found for
the sum of the two masses before the collision occurred. This apparent
discrepancy between the total mass of the system before and during
the collision, is removed, however, if we realize that when the particles
%% -----File: 101.png---Folio 87-------
have come to relative rest an amount of potential energy of
elastic deformation has been produced, which is just sufficient to restore
them to their original velocities, and the mass corresponding to
this potential energy will evidently be just sufficient to make the
total mass of the system the same as before collision.

In the following chapter on the dynamics of a system of particles
we shall make further use of our ideas as to the mass corresponding
to potential energy.
%% -----File: 102.png---Folio 88-------


\Chapter{VII}{The Dynamics of a System of Particles.}
\SetRunningHeads{Chapter Seven.}{Dynamics of a System of Particles.}

\Paragraph{79.}  In the \Chapnumref[VI]{preceding chapter} we discussed the laws of motion
of a particle. With the help of those laws we shall now derive some
useful general dynamical principles which describe the motions of a
system of particles, and in the \Chapnumref[VIII]{following chapter} shall consider an
application of some of these principles to the kinetic theory of gases.

The general dynamical principles which we shall present in this
chapter will be similar \emph{in form} to principles which are already familiar
in the classical Newtonian mechanics. Thus we shall deduce principles
corresponding to the principles of the conservation of momentum,
of the conservation of moment of momentum, of least action and of
\textit{vis~viva}, as well as the equations of motion in the Lagrangian and
Hamiltonian (canonical) forms. For cases where the velocities of all
the particles involved are slow compared with that of light, we shall
find, moreover, that our principles become identical in content, as
well as in form, with the corresponding principles of the classical
mechanics. Where high velocities are involved, however, our new
principles will differ from those of Newtonian mechanics. In particular
we shall find among other differences that in the case of high
velocities it will no longer be possible to define the Lagrangian function
as the difference between the kinetic and potential energies of the
system, nor to define the generalized momenta used in the Hamiltonian
equations as the partial differential of the kinetic energy with
respect to the generalized velocity.


\Subsection{On the Nature of a System of Particles.}

\Paragraph{80.} Our purpose in this chapter is to treat dynamical systems
consisting of a finite number of particles, each obeying the equation
of motion which we have already written in the forms,
\begin{gather*}
\vc{F}
  = \frac{d}{dt}(m\vc{u})
  = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u},
\Tag{47} \displaybreak[0] \\
\vc{F}
  = \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\,\vc{u}\Biggr]
  = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\, \frac{d\vc{u}}{dt}
  + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\Biggr] \vc{u}.
\Tag{59}
\end{gather*}
%% -----File: 103.png---Folio 89-------

It is not to be supposed, however, that the total mass of such a
system can be taken as located solely in these particles. It is evident
rather, since potential energy has mass, that there will in general be
mass distributed more or less continuously throughout the space in
the neighborhood of the particles. Indeed we have shown at the
end of the preceding chapter\DPnote{** TN: Not a useful cross-ref} (\Secref{78}) that unless we take account
of the mass corresponding to potential energy we can not maintain
the principle of the conservation of mass, and we should also find it
impossible to retain the principle of the conservation of momentum
unless we included the momentum corresponding to potential energy.

For a continuous distribution of mass we may write for the force
acting at any point on the material in a small volume,~$\delta V$,
\[
\vc{f}\, \delta V = \frac{d}{dt} (\vc{g}\, \delta V),
\Tag{47\textit{A}}
\]
where $\vc{f}$ is the force per unit volume and $\vc{g}$~is the density of momentum.
This equation is of course merely an equation of definition for the
intensity of force at a point. We shall assume, however, that Newton's
third law, that is, the principle of the equality of action and
reaction, holds for forces of this type as well as for those acting on
particles. In later chapters we shall investigate the way in which $\vc{g}$~depends
on velocity, state of strain, etc., but for the purposes of this
chapter we shall not need any further information as to the nature
of the distributed momentum.

Let us proceed to the solution of our specific problems.


\Subsection{The Conservation of Momentum.}

\Paragraph{81.} We may first show from Newton's third law of motion that
the momentum of an isolated system of particles remains constant.

Considering a system of particles of masses $m_1$,~$m_2$, $m_3$,~etc., we
may write in accordance with equation~\DPtypo{47}{(47)},
\[
\begin{aligned}
\vc{F}_1 + \vc{I}_1 &= \frac{d}{dt} (m_1 \vc{u}_1), \\
\vc{F}_2 + \vc{I}_2 &= \frac{d}{dt} (m_2 \vc{u}_2), \\
\text{etc.,}\quad  &
\end{aligned}
\Tag{89}
\]
%% -----File: 104.png---Folio 90-------
where $\vc{F}_1$,~$\vc{F}_2$,~etc., are the external forces impressed on the individual
particles from outside the system and $\vc{I}_1$,~$\vc{I}_2$,~etc., are the internal
forces arising from mutual reactions within the interior of the system.
Considering the distributed mass in the system, we may also write,
in accordance with~(47\textit{A}) the further equation
\[
(\vc{f} + \vc{i})\, \delta V = \frac{d}{dt}(\vc{g}\, \delta V),
\Tag{90}
\]
where $\vc{f}$~and~$\vc{i}$ are respectively the external and internal forces acting
\emph{per unit volume} of the distributed mass. Integrating throughout the
whole volume of the system~$V$ we have
\[
\int (\vc{f} + \vc{i})\, dV = \frac{d\vc{G}}{dt},
\Tag{91}
\]
where $\vc{G}$ is the total distributed momentum in the system. Adding
this to our previous equations~(89) for the forces acting on the individual
particles, we have
\[
%[** TN: \textstyle \sum in original]
\Sum \vc{F}_1 + \Sum  \vc{I}_1 + \int \vc{f}\, dV + \int \vc{i}\, dV
  = \frac{d}{dt} \Sum m_1 u_1 + \frac{d\vc{G}}{dt}.
\]

But from Newton's third law of motion (\ie, the principle of the
equality of action and reaction) it is evident that the sum of the
internal forces, $\Sum \vc{I}_1 + \int \vc{i}\, dV$, which arise from mutual reactions within
the system must be equal to zero, which leads to the desired equation
of momentum
\[
\Sum \vc{F}_1 + \int \vc{f}\, dv = \frac{d}{dt}(\Sum m_1 u_1 + \vc{G}).
\Tag{92}
\]

In words this equation states that at any given instant the vector
sum of the external forces acting on the system is equal to the rate
at which the total momentum of the system is changing.

For the particular case of an isolated system there are no external
forces and our equation becomes a statement of the principle of the
\emph{conservation of momentum}.


\Subsection{The Equation of Angular Momentum.}

\Paragraph{82.} We may next obtain an equation for the moment of momentum
of a system about a point.
%% -----File: 105.png---Folio 91-------
Consider a particle of mass~$m_1$ and velocity~$u_1$. Let $\vc{r}_1$~be the
radius vector from any given point of reference to the particle. Then
for the moment of momentum of the particle about the point we may
write
\[
\vc{M}_1 = \vc{r}_1 × m_1\vc{u}_1,
\]
and summing up for all the particles of the system we may write
\[
\Sum \vc{M}_1 = \Sum (\vc{r}_1 × m_1\vc{u}_1).
\Tag{93}
\]
Similarly, for the moment of momentum of the \emph{distributed mass} we
may write
\[
\vc{M}_{\text{dist.}} = \int (\vc{r} × \vc{g})\, dV,
\Tag{94}
\]
where $\vc{r}$ is the radius vector from our chosen point of reference to a
point in space where the density of momentum is~$\vc{g}$ and the integration
is to be taken throughout the whole volume,~$V$, of the system.

Adding these two equations (93)~and~(94), we obtain for the total
amount of momentum of the system about our chosen point
\[
\vc{M} = \Sum(\vc{r}_1 × m_1\vc{u}_1) + \int (\vc{r} × \vc{g})\, dV;
\]
and differentiating with respect to the time we have, for the rate of
change of the moment of momentum,
\begin{multline*}
\frac{d\vc{M}}{dt}
  = \Sum \left\{\vc{r}_1 × \frac{d}{dt}(m_1\vc{u}_1)\right\}
  + \Sum \left(\frac{d\vc{r}_1}{dt} × m_1\vc{u}_1\right) \\
  + \int \left(\vc{r} × \frac{d\vc{g}}{dt} \right) dV
  + \int \left(\frac{d\vc{r}}{dt} × \vc{g} \right) dV;
\end{multline*}
or, making the substitutions given by equations (89)~and~(90), and
writing $\dfrac{d\vc{r}_1}{dt} = \vc{u}_1$, etc.\DPtypo{}{,} we have
\begin{multline*}
\frac{d\vc{M}}{dt}
  = \Sum (\vc{r}_1 × \vc{F}_1) + \Sum (\vc{r}_1 × \vc{I}_1)
  + \Sum (\vc{u}_1 × m_1\vc{u}_1) \\
  + \int (\vc{r} × \vc{f})\, dV + \int (\vc{r} × \vc{i})\, dV
  + \int (\vc{u} × \vc{g})\, dV.
\end{multline*}
To simplify this equation we may note that the third term is equal to
zero because it contains the outer product of a vector by itself. Furthermore,
if we accept the principle of the equality of action and
%% -----File: 106.png---Folio 92-------
reaction, together with the further requirement that forces are not
only equal and opposite but that their points of application be in the
same straight line, we may put the moment of all the internal forces
equal to zero and thus eliminate the second and fifth terms. We
obtain as the equation of angular momentum
\[
\frac{d\vc{M}}{dt} = \Sum(\vc{r}_1 × \vc{F}_1)
  + \int (\vc{r} × \vc{f})\, dV + \int (\vc{u} × \vc{g})\, dV.
\Tag{95}
\]

We may call attention to the inclusion in this equation of the
interesting term $\int(\vc{u} × \vc{g})\, dV$. If density of momentum and velocity
should always be in the same direction this term would vanish, since
the outer product of a vector by itself is equal to zero. In our consideration
of the ``Dynamics of Elastic Bodies,'' however, we shall
find bodies with a component of momentum at right angles to their
direction of motion and hence must include this term in a general
treatment. For a completely isolated system it can be shown, however,
that this term vanishes along with the external forces and we
then have the principle of the \emph{conservation of moment of momentum.}


\Subsection{The Function $T$.}

\Paragraph{83.} We may now proceed to the definition of a function which
will be needed in our treatment of the principle of least action.

One of the most valuable properties of the Newtonian expression,
$\frac{1}{2}m_0u^2$, for kinetic energy was the fact that its derivative with respect
to velocity is evidently the Newtonian expression for momentum,~$m_0u$.
It is not true, however, that the derivative of our new expression
for kinetic energy (see \Secref{73}), $m_0c^2 \Biggl[\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1\Biggr]$, with respect
to velocity is equal to momentum, and for that reason in our non-Newtonian
mechanics we shall find it desirable to define a new function,~$T$,
by the equation,
\[
T = m_0c^2\left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right).
\Tag{96}
\]

For slow velocities (\ie, small values of~$u$) this reduces to the
Newtonian expression for kinetic energy and at all velocities we have
%% -----File: 107.png---Folio 93-------
the relation,
\[
\frac{dT}{du}
  = -m_0 c^2\, \frac{d}{du} \sqrt{1 - \frac{u^2}{c^2}}
  = \frac{m_0u}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  = mu,
\Tag{97}
\]
showing that the differential of~$T$ with respect to velocity is momentum.

For a system of particles we shall define~$T$ as the summation of
the values for the individual particles:
\[
T = \Sum m_0 c^2 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right).
\Tag{98}
\]


\Subsection{The Modified Lagrangian Function.}

\Paragraph{84.} In the older mechanics the Lagrangian function for a system
of particles was defined as the difference between the kinetic and
potential energies of the system. The value of the definition rested,
however, on the fact that the differential of the kinetic energy with
respect to velocity was equal to momentum, so that we shall now
find it advisable to define the Lagrangian function with the help of
our new function~$T$ in accordance with the equation
\[
L = T - U.
\Tag{99}
\]


\Subsection{The Principle of Least Action.}

\Paragraph{85.} We are now in a position to derive a principle corresponding
to that of least action in the older mechanics. Consider the path
by which our dynamical system actually moves from state~(1) to
state~(2). The motion of any particle in the system of mass $m$ will
be governed by the equation
\[
\vc{F} = \frac{d}{dt} (m\vc{u}).
\Tag{100}
\]

Let us now compare the actual path by which the system moves
from state~(1) to state~(2) with a slightly displaced path in which the
laws of motion are not obeyed, and let the displacement of the particle
at the instant in question be~$\delta \vc{r}$.

Let us take the inner product of both sides of equation~(100) with~$\delta \vc{r}$;
%% -----File: 108.png---Folio 94-------
we have
\begin{gather*}
\begin{aligned}
\vc{F} ·\delta\vc{r}
  &= \frac{d}{dt}(m \vc{u}) · \delta \vc{r} \\
  &= \frac{d}{dt}(m\vc{u} · \delta\vc{r})
   - m\vc{u} · \frac{d\, \delta\vc{r}}{dt} \\
  &= \frac{d}{dt}(m\vc{u} · \delta\vc{r}) - m\vc{u} · \delta\vc{u})
\end{aligned} \\
(m\vc{u} · \delta\vc{u} + \vc{F} · \delta\vc{r})\, dt
  = d(m\vc{u} · \delta\vc{r}).
\end{gather*}

Summing up for all the particles of the system and integrating
between the limits $t_1$~and~$t_2$, we have
\[
\int_{t_1}^{t_2} \left(\Sum m\vc{u}· \delta\vc{u} + \Sum \vc{F} · \delta\vc{r}\right) dt
  = \left[\Sum m\vc{u} · \delta\vc{r} \right]_{t_1}^{t_2}.
\]
Since $t_1$~and~$t_2$ are the times when the actual and displaced motions
coincide, we have at these times $\delta\vc{r} = 0$; furthermore we also have
$\vc{u} · \delta\vc{u} = u\, \delta u$, so that we may write
\[
\int_{t_1}^{t_2}\left(\Sum mu\, \delta u + \vc{F} · \delta\vc{r}\right) dt = 0.
\]
With the help of equation~(97), however, we see that $\Sum mu\, \delta u = \delta T$,
giving us
\[
\int_{t_1}^{t_2} (\delta T + \vc{F} · \delta r)\, dt = 0.
\Tag{101}
\]
\emph{If the forces~$F$ are conservative}, we may write $\vc{F} · \delta r = -\delta U$, where
$\delta U$~is the difference between the potential energies of the displaced
and the actual configurations. This gives us
\[
\delta \int_{t_1}^{t_2} (T - U)\, dt = 0
\]
or
\[
\delta \int_{t_1}^{t_2} L\, dt = 0,
\Tag{102}
\]
which is the modified principle of least action. The principle evidently
requires that for the actual path by which the system goes
%% -----File: 109.png---Folio 95-------
from state~(1) to state~(2), the quantity $\ds\int_{t_1}^{t_2} L\, dt$ shall be a minimum (or
maximum).


\Subsection{Lagrange's Equations.}

\Paragraph{86.} We may now derive the Lagrangian equations of motion from
the above principle of least action. Let us suppose that the position
of each particle of the system under consideration is completely determined
by $n$~\emph{independent} generalized coördinates $\phi_{1}$,~$\phi_{2}$, $\phi_{3} \cdots \phi_{n}$ and
hence that $L$~is some function of $\phi_{1}$,~$\phi_{2}$, $\phi_{3} \cdots \phi_{n}$, $\dot{\phi}_{1}$,~$\dot{\phi}_{2}$, $\dot{\phi}_{3} \cdots \dot{\phi}_{n}$,
where for simplicity we have put $\dot{\phi}_{1} = \dfrac{d\phi_1}{dt}$, $\dot{\phi}_{2} = \dfrac{d\phi_2}{dt}$,~etc.

%%%% Use of "1" as a subscript in the original starts here %%%%
From equation~(102) we have
\[
\int_{t_1}^{t_2} (\delta L)\, dt = \int_{t_1}^{t_2} \left(
    \Sum_1^n \frac{\partial L}{\partial\phi_{\1}}\, \delta\phi_{\1}
  + \Sum_1^n \frac{\partial L}{\partial\dot{\phi}_{\1}}\, \delta\dot{\phi}_{\1}
  \right)dt = 0.
\Tag{103}
\]
But
\[
\delta\dot{\phi}_{\1} = \frac{d}{dt}(\delta\phi_{\1})\DPchg{}{,}
\]
which gives us
\begin{align*}
\int_{t_1}^{t_2} \frac{\partial L}{\partial\dot{\phi}_{\1}}\,
     \delta\dot{\phi}_{\1}\, dt
  &= \int_{t_1}^{t_2} \frac{\partial L}{\partial\dot{\phi}_{\1}}\,
       \frac{d}{dt}(\delta\phi_{\1})\, dt \\
  &= \left[\frac{\partial L}{\partial\dot{\phi}_{\1}}\, \delta\phi_{\1}\right]_{t_1}^{t_2}
  - \int_{t_1}^{t_2} \delta\phi_{\1}\,
      \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right) dt
\end{align*}
or, since at times $t_{1}$~and~$t_{2}$, $\delta \phi_{\1}$~is zero, the first term in this expression
disappears and on substituting in equation~(103) we obtain
\[
\int_{t_1}^{t2} \left[\Sum_{1}^{n} \delta \phi_{\1}
  \left\{ \frac{\partial L}{\partial\phi_{\1}}
        - \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right)
  \right\} \right] dt = 0.
\]
Since, however, the limits $t_{1}$~and~$t_{2}$ are entirely at our disposal we must
have at every instant
\[
\Sum_{1}^{n} \delta \phi_{\1}
  \left\{ \frac{\partial L}{\partial\phi_{\1}}
        - \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right)
  \right\} = 0.
\]
Finally, moreover, since the $\phi$'s~are independent parameters, we can
assign perfectly arbitrary values to $\delta\phi_{1}$,~$\delta\phi_{2}$,~etc., and hence must have
%% -----File: 110.png---Folio 96-------
the series of equations
\[
\begin{aligned}
&\frac{d}{dt} \left( \frac{\partial L}{\partial\dot{\phi}_1} \right)
                   - \frac{\partial L}{\partial\phi_1} = 0, \\
&\frac{d}{dt} \left( \frac{\partial L}{\partial\dot{\phi}_2} \right)
                   - \frac{\partial L}{\partial\phi_2} = 0, \\
&\text{etc.}
\end{aligned}
\Tag{104}
\]
These correspond to Lagrange's equations in the older mechanics,
differing only in the definition of~$L$.


\Subsection{Equations of Motion in the Hamiltonian Form.}

\Paragraph{87.} We shall also find it desirable to obtain equations of motion
in the Hamiltonian or canonical form.

Let us define the \emph{generalized momentum}~$\psi_{\1}$ corresponding to the
coördinate~$\phi_{\1}$ by the equation,
\[
\psi_{\1} = \frac{\partial T}{\partial\dot{\phi}_{\1}}.
\Tag{105}
\]

It should be noted that the generalized momentum is not as in
ordinary mechanics the derivative of the kinetic energy with respect
to the generalized velocity but approaches that value at low velocities.

Consider now a function~$T'$ defined by the equation
\[
T' = \psi_1\dot{\phi}_1 + \psi_{2}\dot{\phi}_2 + \cdots - T.
\Tag{106}
\]
Differentiating we have
\begin{align*}
dT' &= \psi_1\, d\dot{\phi}_1 + \psi_2\, d\dot{\phi}_2 + \cdots \\
  &\quad+ \dot{\phi}_1\, d\psi_{1} + \dot{\phi}_2\, d\psi_{2} + \cdots \\
  &\quad- \frac{\partial T}{\partial\phi_1}\, d\phi_1
        - \frac{\partial T}{\partial\phi_2}\, d\phi_2 - \cdots \\
  &\quad- \frac{\partial T}{\partial\dot{\phi}_1}\, d\dot{\phi}_1
        - \frac{\partial T}{\partial\dot{\phi}_2}\, d\dot{\phi}_2 - \cdots,
\end{align*}
and this, by the introduction of~(105), becomes
\[
dT' = \dot{\phi}_1\, d\psi_1 + \dot{\phi}_2\, d\psi_{2} + \cdots
  - \frac{\partial T}{\partial\phi_1}\, d\phi_1
  - \frac{\partial T}{\partial\phi_2}\, d\phi_2 - \cdots.
\Tag{107}
\]
%% -----File: 111.png---Folio 97-------
Examining this equation we have
\begin{align*}
\frac{\partial T'}{\partial\phi_{\1}}
  &= - \frac{\partial T}{\partial\phi_{\1}},
\Tag{108} \\
\frac{\partial T'}{\partial\psi_{\1}}
  & = \dot{\phi}_{\1}.
\Tag{109}
\end{align*}
In Lagrange's equations we have
\[
\frac{d}{dt}\left\{ \frac{\partial}{\partial\dot{\phi}_{\1}}(T - U)\right\}
  - \frac{\partial}{\partial\phi_{\1}}(T - U) = 0.
\]
But since $U$ is independent of~$\psi_{\1}$ we may write
\[
\frac{\partial(T - U)}{\partial\dot{\phi}_{\1}}
  = \frac{\partial T}{\partial\dot{\phi}_{\1}} = \psi_{\1},
\]
and furthermore by~(108),
\[
\frac{\partial T}{\partial\phi_{\1}} = -\frac{\partial T'}{\partial\phi_{\1}}.
\]
Substituting these two expressions in Lagrange's equations we obtain
\[
\frac{d\psi_{\1}}{dt} = -\frac{\partial(T' + U)}{\partial\phi_{\1}}
\]
or, writing $T' + U = E$, we have
\[
\frac{d\psi_{\1}}{dt} = -\frac{\partial E}{\partial\phi_{\1}}
\Tag{110}
\]
and since $U$~is independent of~$\psi_{\1}$ we may rewrite equation~(109) in
the form
\[
\frac{d\phi_{\1}}{dt} = \frac{\partial E}{\partial\psi_{\1}}.
\Tag{111}
\]

The set of equations corresponding to (110)~and~(111) for all the
coördinates $\phi_{1}$,~$\phi_{2}$, $\phi_{3}, \cdots \phi_{n}$  and the momenta $\psi_{1}$,~$\psi_{2}$, $\psi_{3}, \cdots \psi_{n}$ are
the desired equations of motion in the canonical form.

\Subsubsection{88}{Value of the Function $T'$.} We have given the symbol~$E$ to
the quantity $T' + U$, since $T'$~actually turns out to be identical with
%% -----File: 112.png---Folio 98-------
the expression by which we defined kinetic energy, thus making
$E = T' + U$ the sum of the kinetic and potential energies of the
system.

To show that $T'$~is equal to~$K$, the kinetic energy, we have by the
equation of definition~(106)
\begin{align*}
T' &= \phi_1\psi_1 + \phi_2\psi_2 + \cdots - T, \\
   &= \phi_1\, \frac{\partial T}{\partial\dot{\phi}_1}
    + \phi_2\, \frac{\partial T}{\partial\dot{\phi}_2} + \cdots - T.
\end{align*}
But $T$ by definition, equation~(98), is
\begin{align*}
T &= \Sum c^2m_0 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right), \\
\intertext{which gives us}
\frac{\partial T}{\partial\dot{\phi}_{\1}}
  &= \Sum m_0 \left(1 - \frac{u^2}{c^2}\right)^{-1/2}
     u\, \frac{\partial u}{\partial\dot{\phi}_{\1}} \\
  &= \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_{\1}}
\end{align*}
and substituting we obtain
\[
\begin{aligned}
T' &= \dot{\phi}_1 \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_1}
    + \dot{\phi}_2 \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_2}
    + \cdots - T \\
   &= \Sum mu \left\{
        \dot{\phi}_1\, \frac{\partial u}{\partial\dot{\phi}_1}
      + \dot{\phi}_2\, \frac{\partial u}{\partial\dot{\phi}_2}
      + \cdots \right\} - T.
\end{aligned}
\Tag{112}
\]
We can show, however, that the term in parenthesis is equal to~$u$.
If the \DPchg{coordinates}{coördinates} $x$,~$y$,~$z$ determine the position of the particle in
question, we have,
\begin{align*}
x &= f(\phi_1\phi_2\phi_3 \cdots \phi_n), \\
\dot{x} = \frac{dx}{dt}
  &= \dot{\phi}_1\, \frac{\partial f(\:)}{\partial\phi_1}
   + \dot{\phi}_2\, \frac{\partial f(\:)}{\partial\phi_2}
   + \dot{\phi}_3\, \frac{\partial f(\:)}{\partial\phi_3} + \cdots
\end{align*}
and differentiating with respect to the~$\dot{\phi}$'s, we obtain,
\[
\frac{\partial\dot{x}}{\partial\dot{\phi}_1}
  = \frac{\partial f(\:)}{\partial\phi_1}
  = \frac{\partial x}{\partial\phi_1}, \quad
\frac{\partial\dot{x}}{\partial\dot{\phi}_2}
  = \frac{\partial x}{\partial\phi_2}, \quad
\frac{\partial\dot{x}}{\partial\dot{\phi}_3}
  = \frac{\partial x}{\partial\phi_3}, \quad \text{etc.}\DPtypo{,}{}
\]
%% -----File: 113.png---Folio 99-------
Similarly
\begin{alignat*}{3}
\frac{\partial\dot{y}}{\partial\dot{\phi}_1}
  &= \frac{\partial y}{\partial\phi_1},
&\qquad
\frac{\partial\dot{y}}{\partial\dot{\phi}_2}
  &= \frac{\partial y}{\partial\phi_2}, &\qquad \text{etc.}, \\
\frac{\partial\dot{z}}{\partial\dot{\phi}_1}
  &= \frac{\partial z}{\partial\phi_1},
&\qquad
\frac{\partial\dot{z}}{\partial\dot{\phi}_2}
  &= \frac{\partial z}{\partial\phi_2}, &\qquad \text{etc.},
\end{alignat*}
Let us write now
\begin{align*}
u &= \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}, \\
\frac{\partial u}{\partial\dot{\phi}_{\1}}
  &= \frac{1}{\sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}}
     \left(\dot{x}\, \frac{\partial\dot{x}}{\partial\dot{\phi}_{\1}}
         + \dot{y}\, \frac{\partial\dot{y}}{\partial\dot{\phi}_{\1}}
         + \dot{z}\, \frac{\partial\dot{z}}{\partial\dot{\phi}_{\1}}\right),
\end{align*}
or making the substitutions for $\dfrac{\partial\dot{x}}{\partial\dot{\phi}_{\1}}$, $\dfrac{\partial\dot{y}}{\partial\dot{\phi}_{\1}}$, etc., given above, we have,
\[
\frac{\partial u}{\partial\dot{\phi}_{\1}}
  = \frac{1}{u}
    \left(\dot{x}\, \frac{\partial x}{\partial\phi_{\1}}
        + \dot{y}\, \frac{\partial y}{\partial\phi_{\1}}
        + \dot{z}\, \frac{\partial z}{\partial\phi_{\1}}\right).
\]
%%%% Use of "1" as a subscript in the original ends here %%%%
Substituting now in~(112) we shall obtain,
{\footnotesize%
\begin{align*}
T'& = \Sum mu
\begin{aligned}[t]
\Biggl\{\frac{\dot{x}}{u}
  \left(\phi_1\, \frac{\partial x}{\partial\phi_1}
      + \phi_2\, \frac{\partial x}{\partial\phi_2} + \cdots \right)
    &+ \frac{\dot{y}}{u}
  \left(\phi_1\, \frac{\partial y}{\partial\phi_1}
      + \phi_2\, \frac{\partial y}{\partial\phi_2} + \cdots \right) \\
    &+ \frac{\dot{z}}{u}
  \left(\phi_1\, \frac{\partial z}{\partial\phi_1}
      + \phi_2\, \frac{\partial z}{\partial\phi_2} + \cdots \right)
        \Biggr\} - T
\end{aligned} \\
  &= \Sum  mu^2 - T
\end{align*}}%
or, introducing the value of~$T$ given by equation~(98), we have
\begin{align*}
T' &= \Sum \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  \left\{u^2 - c^2 \sqrt{1 - \frac{u^2}{c^2}}
         + c^2 \left(1 - \frac{u^2}{c^2}\right)\right\} \\
   &= \Sum c^2(m - m_0),
\end{align*}
which is the expression~(83) for kinetic energy.

Hence we see that the Hamiltonian function $E = T' + U$ is the
sum of the kinetic and potential energies of the system as in Newtonian
mechanics.


\Subsection{The Principle of the Conservation of Energy.}

\Paragraph{89.} We may now make use of our equations of motion in the
canonical form to show that the total energy of a system of interacting
%% -----File: 114.png---Folio 100-------
particles remains constant. If such were not the case it is obvious
that our definitions of potential and kinetic energy would not be
very useful.

Since $E = T' + U$ is a function of $\phi_1$,~$\phi_2$, $\phi_3, \cdots$ $\psi_1$,~$\psi_2$, $\psi_3, \cdots$, we
may write
\begin{align*}
\frac{dE}{dt}
  &= \frac{\partial E}{\partial\phi_1}\, \dot{\phi}_1
   + \frac{\partial E}{\partial\phi_2}\, \dot{\phi}_2 + \cdots \\
  &\quad
   + \frac{\partial E}{\partial\psi_1}\, \dot{\psi}_1
   + \frac{\partial E}{\partial\psi_2}\, \dot{\psi}_2 + \cdots.
\end{align*}
Substituting the values of $\dfrac{\partial E}{\partial\phi_1}$, $\dfrac{\partial E}{\partial\psi_1}$, etc., given by the canonical
equations of motion (110)~and~(111), we have
\begin{align*}
\frac{dE}{dt}
  &= -\dot{\psi}_1\dot{\phi}_1 - \dot{\psi}_2\dot{\phi}_2 - \cdots \\
  &\quad
   +  \dot{\psi}_1\dot{\phi}_1 + \dot{\psi}_2\dot{\phi}_2 + \cdots \\
  &= 0,
\end{align*}
which gives us the desired proof that just as in the older Newtonian
mechanics the total energy of an isolated system of particles is a
conservative quantity.


\Subsection{On the Location of Energy in Space.}

\Paragraph{90.} This proof of the conservation of energy in a system of interacting
particles justifies us in the belief that the concept of energy
will not fail to retain in the newer mechanics the position of great
importance which it gradually acquired in the older systems of physical
theory. Indeed, our newer considerations have augmented the
important rôle of energy by adding to its properties the attribute of
mass or inertia, and thus leading to the further belief that matter
and energy are in reality different names for the same fundamental
entity.

The importance of this entity, energy, makes it very interesting
to consider the possibility of ascribing a definite location in space to
any given quantity of energy. In the older mechanics we had a
hazy notion that the kinetic energy of a moving body was probably
located in some way in the moving body itself, and possibly a vague
%% -----File: 115.png---Folio 101-------
idea that the potential energy of a raised weight might be located in
the space between the weight and the earth. Our discovery of the
relation between mass and energy has made it possible, however, to
give a much more definite, although not a complete, answer to inquiries
of this kind.

In our discussions of the dynamics of a particle (Chapter~VI, %[** TN: Not a useful cross-reference]
\Secref{61}) we saw that an acceptance of Newton's principle of the
equality of action and reaction forced us to ascribe an increased mass
to a moving particle over that which it has at rest. This increase in
the mass of the moving particle is necessarily located either in the
particle itself or distributed in the surrounding space in such a way
that its center of mass always coincides with the position of the
particle, and since the kinetic energy of the particle is the energy
corresponding to this increased mass we may say that \emph{the kinetic energy
of a moving particle is so distributed in space that its center of mass
always coincides with the position of the particle}.

If now we consider the transformation of kinetic energy into
potential energy we can also draw somewhat definite conclusions as to
the location of potential energy. By the principle of the conservation
of mass we shall be able to say that the mass of any potential
energy formed is just equal to the ``kinetic'' mass which has disappeared,
and by the principle of the conservation of momentum we
can say that the velocity of this potential energy is just that necessary
to keep the total momentum of the system constant. Such considerations
will often permit us to reach a good idea as to the location
of potential energy.

Consider, for example, a pair of similar attracting particles which
are moving apart from each other with the velocities $+u$~and~$-u$
and are gradually coming to rest under the action of their mutual
attraction, their kinetic energy thus being gradually changed into
potential energy. Since the total momentum of the system must
always remain zero, we may think of the potential energy which is
formed as left stationary in the space between the two particles.
%% -----File: 116.png---Folio 102-------


\Chapter{VIII}{The Chaotic Motion of a System of Particles.}
\SetRunningHeads{Chapter Eight.}{Chaotic Motion of a System of Particles.}

The discussions of the \Chapnumref[VII]{previous chapter} have placed at our disposal
generalized equations of motion for a system of particles similar in
form to those familiar in the classical mechanics, and differing only
in the definition of the Lagrangian function. With the help of these
equations it is possible to carry out investigations parallel to those
already developed in the classical mechanics, and in the present
chapter we shall discuss the chaotic motion of a system of particles.
This problem has received much attention in the classical mechanics
because of the close relations between the theoretical behavior of
such an ideal system of particles and the actual behavior of a monatomic
gas. We shall find no more difficulty in handling the problem
than was experienced in the older mechanics, and our results will of
course reduce to those of Newtonian mechanics in the case of slow
velocities. Thus we shall find a distribution law for momenta which
reduces to that of Maxwell for slow velocities, and an equipartition
law for the average value of a function which at low velocities becomes
identical with the kinetic energy of the particles.

\Subsubsection{91}{The Equations of Motion.} It has been shown that the Hamiltonian
equations of motion
\[
\begin{aligned}
&\frac{\partial E}{\partial\phi_1} = -\frac{d\psi_1}{dt} = -\dot{\psi}_1, \\
&\frac{\partial E}{\partial\psi_1} =  \frac{d\phi_1}{dt} =  \dot{\phi}_1, \\
&\text{etc.},
\end{aligned}
\Tag{113}
\]
will hold in relativity mechanics provided we define the generalized
momenta $\psi_1$,~$\psi_2$,~etc., \emph{not} as the differential of the kinetic energy
with respect to the generalized velocities $\dot{\phi}_1$,~$\dot{\phi}_2$,~etc., but as the differential
with respect to $\dot{\phi}_1$,~$\dot{\phi}_2$,~etc., of a function
\[
T = \Sum m_0c^2 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right),
\]
%% -----File: 117.png---Folio 103-------
where $m_0$~is the mass of a particle having the velocity~$u$ and the summation~$\Sum$
extends over all the particles of the system.

\Subsubsection{92}{Representation in Generalized Space.} Consider now a system
defined by the $n$~generalized coördinates $\phi_1$,~$\phi_2$, $\phi_3, \cdots, \phi_n$, and the
corresponding momenta $\psi_1$,~$\psi_2$, $\psi_3, \cdots, \psi_n$. Employing the methods
so successfully used by Jeans,\footnote
  {Jeans, \textit{The Dynamical Theory of Gases}, Cambridge, 1916.}
we may think of the state of the
system at any instant as determined by the position of a point plotted
in a $2n$-dimensional space. Suppose now we had a large number of
systems of the same structure but differing in state, then for each
system we should have at any instant a corresponding point in our
$2n$-dimensional space, and as the systems changed their state, in the
manner required by the laws of motion, the points would describe
stream lines in this space.

\Subsubsection{93}{Liouville's Theorem.} Suppose now that the points were
originally distributed in the generalized space with the uniform
density~$\rho$. Then it can be shown by familiar methods that, just as
in the classical mechanics, the density of distribution remains uniform.

Take, for example, some particular cubical element of our generalized
space $d\phi_1\, d\phi_2\, d\phi_3 \dots d\psi_1\, d\psi_2\, d\psi_3\dots$. The density of distribution
will evidently remain uniform if the number of points
entering any such cube per second is equal to the number leaving.
Consider now the two parallel bounding surfaces of the cube which
are perpendicular to the $\phi_1$~axis, one cutting the axis at the point~$\phi_1$
and the other at the point~$\phi_1 + d\phi_1$. The area of each of these
surfaces is $d\phi_2\, d\phi_3\dots d\psi_1\, d\psi_2\, d\psi_3\dots$, and hence, if $\dot{\phi}_1$~is the component
of velocity which the points have parallel to the $\phi_1$~axis, and $\dfrac{\partial\dot{\phi}_1}{\partial\phi_1}$~is
the rate at which this component is changing as we move along the
axis, we may obviously write the following expression for the difference
between the number of points leaving and entering per second
through these two parallel surfaces
\[
\rho\left[\left(\frac{\partial\dot{\phi}_1}{\partial\phi_1}\right) d\phi_1\right]
d\phi_2\, d\phi_3\, \dots d\psi_1\, d\psi_2\, d\psi_3 \cdots
  = \rho\, \frac{\partial\dot{\phi}_1}{\partial\phi_1}\, dV.
\]

Finally, considering all the pairs of parallel bounding surfaces, we
%% -----File: 118.png---Folio 104-------
find for the total decrease per second in the contents of the element
\[
\rho\left(
    \frac{\partial\dot{\phi}_1}{\partial\phi_1}
  + \frac{\partial\dot{\phi}_2}{\partial\phi_2}
  + \frac{\partial\dot{\phi}_3}{\partial\phi_3} + \cdots
  + \frac{\partial\dot{\psi}_1}{\partial\psi_1}
  + \frac{\partial\dot{\psi}_2}{\partial\psi_2}
  + \frac{\partial\dot{\psi}_3}{\partial\psi_3} + \cdots\right) dV.
\]
But the motions of the points are necessarily governed by the Hamiltonian
equations~(113) given above, and these obviously lead to the
relations
\begin{align*}
& \frac{\partial\dot{\phi}_1}{\partial\phi_1}
+ \frac{\partial\dot{\psi}_1}{\partial\psi_1} = 0, \\
& \frac{\partial\dot{\phi}_2}{\partial\phi_2}
+ \frac{\partial\dot{\psi}_2}{\partial\psi_2} = 0\DPtypo{.}{,} \\
& \text{etc.}
\end{align*}
So that our expression for the change per second in the number of
points in the cube becomes equal to zero, the necessary requirement
for preserving uniform density.

This maintenance of a uniform distribution means that there is
no tendency for the points to crowd into any particular region of the
generalized space, and hence if we start some one system going and
plot its state in our generalized space, we may \emph{assume} that, after an
indefinite lapse of time, the point is equally likely to be in any one of
the little elements~$dV$. \emph{In other words, the different states of a system,
which we can specify by stating the region $d\phi_1\, d\phi_2\, d\phi_3 \dots d\psi_1\, d\psi_2\, d\psi_3 \dots$
in which the values of the \DPchg{coordinates}{coördinates} and momenta of the system fall,
are all equally likely to occur.}\footnote
  {The criterion here used for determining whether or not the states are equally
  liable to occur is obviously a necessary requirement, although it is not so evident
  that it is a sufficient requirement for equal probability.}

\Subsubsection{94}{A System of Particles.} Consider now a system containing $N_a$~particles
which have the mass~$m_a$ when at rest, $N_b$~particles which
have the mass~$m_b$, $N_c$~particles which have the mass~$m_c$, etc. If at
any given instant we specify the particular differential element
$dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ which contains the coördinates $x$,~$y$,~$z$, and the
corresponding momenta $\psi_x$,~$\psi_y$,~$\psi_z$ for \emph{each} particle, we shall thereby
completely determine what Planck\footnote
  {Planck, \textit{Wärmestrahlung}, Leipzig, 1913.}
has well called the \emph{microscopic}
state of the system, and by the previous paragraph any microscopic
%% -----File: 119.png---Folio 105-------
state of the system in which we thus specify the six-dimensional
position of each particle is just as likely to occur as any other microscopic
state.

It must be noticed, however, that many of the possible microscopic
states which are determined by specifying the six-dimensional
position of each individual particle are in reality completely identical,
since if all the particles having a given mass~$m_a$ are alike among themselves,
it makes no difference which particular one of the various
available identical particles we pick out to put into a specified range
$dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$.

For this reason we shall usually be interested in specifying the
\emph{statistical} state\footnote
  {What we have here defined as the \emph{statistical} state is what Planck calls the
  \emph{macroscopic} state of the system. The word macroscopic is unfortunate, however, in
  implying a less minute observation as to the size of the elements $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$
in which the representative points are found.}
of the system, for which purpose we shall merely
state the number of particles of a given kind which have coördinates
falling in a given range $dx\, dy\,dz\, d\psi_x\, d\psi_y\, d\psi_z$. We see that corresponding
to any given statistical state there will be in general a
large number of microscopic states.

\Subsubsection{95}{Probability of a Given Statistical State.} We shall now be
particularly interested in the probability that the system of particles
will actually be in some specified \emph{statistical} state, and since Liouville's
theorem has justified our belief that all \emph{microscopic} states are
equally likely to occur, we see that the probability of a given statistical
state will be proportional to the number of microscopic states
which correspond to it.

For the system under consideration let a particular statistical
state be specified by stating that ${N_a}'$,~${N_a}''$, ${N_a}''', \cdots$, ${N_b}'$,~${N_b}''$, ${N_b}''', \cdots$,~etc., are the number of particles of the corresponding masses
$m_a$,~$m_b$,~etc., which fall in the specified elementary regions $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, Nos.\ $1a$,~$2a$, $3a, \cdots$, $1b$,~$2b$, $3b, \cdots$,~etc. By familiar
methods of calculation it is evident that the number of arrangements
by which the particular distribution of particles can be effected,
that is, in other words, the number of microscopic states,~$W$, which
correspond to the given statistical state, is given by the expression
\[
%[** TN: Modernized factorial notation]
W = \frac{N_a!\, N_b!\, N_c! \cdots}
         {{N_a}'!\, {N_a}''!\, {N_a}'''! \cdots
          {N_b}'!\, {N_b}''!\, {N_b}'''! \cdots}
\]
%% -----File: 120.png---Folio 106-------
and this number~$W$ is proportional to the probability that the system
will be found in the particular statistical state considered.

If now we assume that each of the regions
\[
dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z,\
\text{Nos.}\ 1a,\ 2a,\ 3a,\ \cdots,\ 1b,\ 2b,\ 3b,\ \cdots\ \text{etc.}
\]
is great enough to contain a large number of particles,\footnote
  {The idea of successive orders of infinitesimals which permit the differential
  region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, to contain a large number of particles is a familiar one in
  mathematics.}
we may
apply the Stirling formula
\[
N! = \sqrt{2\pi\, N} \left(\frac{N}{\epsilon}\right)^N
\]
for evaluating $N_a!$,~$N_b!$,~etc., and omitting negligible terms, shall
obtain for~$\log W$ the result
\begin{align*}
\log W &= -N_a \left(
    \frac{{N_a}'}{N_a} \log\frac{{N_a}'}{N_a}
  + \frac{{N_a}''}{N_a} \log\frac{{N_a}''}{N_a}
  + \frac{{N_a}'''}{N_a} \log\frac{{N_a}'''}{N_a} + \cdots\right) \\
  &\quad -N_b\left(
    \frac{{N_b}'}{N_b} \log\frac{{N_b}'}{N_b}
  + \frac{{N_b}''}{N_b} \log\frac{{N_b}''}{N_b}
  + \frac{{N_b}'''}{N_b} \log\frac{{N_b}'''}{N_b} + \cdots\right),\\
  &\quad\text{etc.}
\end{align*}

For simplicity let us denote the ratios $\dfrac{{N_a}'}{N_a}$, $\dfrac{{N_a}''}{N_a}$,~etc., by the
symbols ${w_a}'$, ${w_a}''$,~etc. These quantities ${w_a}'$, ${w_a}''$,~etc., are evidently
the probabilities, in the case of this particular statistical state,
that any given particle~$m_a$ will be found in the respective regions
Nos.\ $1a$,~$2a$,~etc.

We may now write
\[
\log W = -N_a\Sum w_a\log w_a - N_b\Sum w_b\log w_b -{}, \text{ etc.},
\]
where the summation extends over all the regions Nos.\ $1a$,~$2a$, $\cdots
1b$, $2b$,~etc.

\Subsubsection{96}{Equilibrium Relations.} Let us now suppose that the system
of particles is contained in an enclosed space and has the definite
energy content~$E$. Let us find the most probable distribution of the
particles. For this the necessary condition will be
\begin{multline*}
\delta\log W = -N_a\Sum (\log w_a + 1)\, \delta w_a \\
               -N_b\Sum (\log w_b + 1)\, \delta w_b \cdots = 0.
\Tag{114}
\end{multline*}
In carrying out our variation, however, the number of particles of
%% -----File: 121.png---Folio 107-------
each kind must remain constant so that we have the added relations
\[
\Sum \delta w_a=0, \qquad
\Sum \delta w_b=0, \qquad \text{etc.}
\Tag{115}
\]
Finally, since the energy is to have a definite value~$E$, it must also
remain constant in the variation, which will provide still a further
relation. Since the energy of a particle will be a definite function of
its position and momentum,\footnote
  {We thus exclude from our considerations systems in which the potential energy
  depends appreciably on the \emph{relative} positions of the independent particles.}
let us write the energy of the system
in the form
\[
E = N_a \Sum w_a E_a + N_b \Sum w_b E_b + \cdots,
\]
where $E_a$ is the energy of a particle in the region $1a$,~etc.

Since in carrying out our variation the energy is to remain constant,
we have the relation
\[
E = N_a \Sum E_a\, \delta w_a
  + N_b \Sum E_b\, \delta w_b + \cdots = 0.
\Tag{116}
\]

Solving the simultaneous equations (114),~(115),~(116) by familiar
methods we obtain
\begin{align*}
&\log w_a + 1 + \lambda E_a + \mu_b = 0, \\
&\log w_b + 1 + \lambda E_b + \mu_b = 0, \\
&\text{etc.},
\end{align*}
where $\lambda$,~$\mu_a$, $\mu_b$,~etc., are undetermined constants. (It should be
specially noticed that $\lambda$~is the same constant in each of the series of
equations.)

Transforming we have
\[
\begin{aligned}
& w_a = \alpha_a\, e^{-hE_a}, \\
& w_b = \alpha_b\, e^{-hE_b}, \\
& \text{etc.},
\end{aligned}
\Tag{117}
\]
as the expressions which determine the chance that a given particle
of mass $m_a$,~$m_b$,~etc., will fall in a given region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$,
when we have the distribution of maximum probability. It should
be noticed that~$h$, which corresponds to the~$\lambda$ of the preceding equations,
is the same constant in all of the equations, while $\alpha_a$,~$\alpha_b$,~etc.,
are different constants, depending on the mass of the particles $m_a$,~$m_b$,~etc.
%% -----File: 122.png---Folio 108-------

\Subsubsection{97}{The Energy as a Function of the Momentum.} $E_a$,~$E_b$,~etc.,
are of course functions of $x$,~$y$,~$z$, $\psi_x$,~$\psi_y$,~$\psi_z$. Let us now obtain an
expression for~$E_a$ in terms of these quantities. If there is no external
field of force acting, the energy of a particle~$E_a$ will be independent
of $x$,~$y$, and~$z$, and will be determined entirely by its velocity and
mass. In accordance with the theory of relativity we shall have\footnote
  {This expression is that for the total energy of the particle, including that internal energy~$m_0 c^2$
  which, according to relativity theory, the particle has when it is at rest. (See \Secref{75}.)
  It would be just as correct to substitute for~$E_a$ in equation~(117) the value of the kinetic energy
  $m_a c^2 \Biggl(\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}- 1 \Biggr)$
  instead of the total energy $\dfrac{m_ac^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$,
  since the two differ merely by a constant~$m_a c^2$ which would be taken care of by assigning a suitable value to~$\alpha_a$.}
\[
E_a = \frac{m_ac^2}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\Tag{118}
\]
where $m_a$ is the mass of the particle at rest.

Let us now express $E_a$ as a function of $\psi_x$,~$\psi_y$,~$\psi_z$.

We have from our equations (105)~and~(98), which were used for
defining momentum
\begin{align*}
\psi_x
  &= \frac{\partial}{\partial\dot{x}}\, m_a
     \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right)\displaybreak[0]  \\
  &= \frac{\partial}{\partial\dot{x}}\, m_a
     \left(1 - \sqrt{1 - \frac{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}{c^2}}\right)\displaybreak[0] \\
  &= \frac{m_0\dot{x}}{\sqrt{1 - \smfrac{u^2}{c^2}}}.
\end{align*}
Constructing the similar expressions for $\psi_y$~and~$\psi_z$ we may write the
relation
\[
\psi^2 = \psi_x^2 + \psi_y^2 + \psi_z^2
  = \frac{m^2_a (\dot{x}^2 + \dot{y}^2 + \dot{z}^2)}{1 - \smfrac{u^2}{c^2}}
  = \frac{m^2_au^2}{1 - \smfrac{u^2}{c^2}},
\Tag{119}
\]
which also defines~$\psi^2$.
%% -----File: 123.png---Folio 109-------

By simple transformations and the introduction of equation~(118)
we obtain the desired relation
\[
E_a = c\sqrt{\psi^2 + {m_a}^2c^2}.
\Tag{120}
\]

\Subsubsection{98}{The Distribution Law.} We may now rewrite equations~(117)
in the form
\[
\begin{aligned}
& w_a = \alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}, \\
& w_b = \alpha_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}}, \\
& \text{etc.}
\end{aligned}
\Tag{121}
\]

These expressions determine the probability that a given particle
of mass $m_a$,~$m_b$,~etc.\DPtypo{}{,} will fall in a given region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, and
correspond to Maxwell's distribution law in ordinary mechanics. We
see that these probabilities are independent of the position $x$,~$y$,~$z$\footnote
  {This is true only when, as assumed, no external field of force is acting.}
but dependent on the momentum.

$\alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}$ is the probability that a given particle will fall in a
particular six-dimensional cube of volume $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$. Let us
now introduce, for convenience, a new quantity $a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}$ which
will be the probability per \emph{unit} volume that a given particle will have
the six dimensional location in question, the constants $\alpha_a$~and~$a_a$
standing in the same ratio as the volumes $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ and unity.

We may then write
\begin{alignat*}{2}
w_a &= \alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}} &
    &=      a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,
            dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z \\
w_b &= \alpha_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}} &
    &=      a_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}}\,
            dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z
\end{alignat*}
etc.

Since every particle must have components of momentum lying
between minus and plus infinity, and lie somewhere in the whole
volume~$V$ occupied by the mixture, we have the relation
\[
V \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}
  a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z = 1.
\Tag{122}
\]

It is further evident that the average value of any quantity~$A$
which depends on the momentum of the particles is given by the
%% -----File: 124.png---Folio 110-------
expression
\[
[A]_{\text{av.}}
  = V \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}
  a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,A \, d\psi_x\, d\psi_y\, d\psi_z,
\Tag{123}
\]
where $A$ is some function of $\psi_x$,~$\psi_y$, and~$\psi_z$.

\Subsubsection{99}{Polar Coördinates.} We may express relations corresponding
to (122)~and~(123) more simply if we make use of polar coördinates.
Consider instead of the elementary volume $d\psi_x\, d\psi_y\, d\psi_z$ the volume
$\psi^2\sin\theta\, d\theta\, d\phi\, d\psi$ expressed in polar coördinates, where
\[
\psi^2 = {\psi_x}^2 + {\psi_y}^2 + {\psi_z}^2.
\]

The probability that a particle~$m_a$ will fall in the region
\[%[** TN: Displaying to avoid bad line break]
dx\, dy\, dz\, \psi^2 \sin\theta\, d\theta\, d\phi\, d\psi
\]
will be
\[
a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,
  dx\, dy\, dz\, \psi^2 \sin\theta\, d\theta\, d\phi\, d\psi,
\]
and since each particle must fall somewhere in the space $x\:y\:z\: \psi_x\: \psi_y\: \psi_z$
we shall have corresponding to~(122) the relation
\[
\begin{gathered}
V \int_{0}^{\pi}\int_{0}^{2\pi}\int_{0}^{\infty}
  a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2 \sin\theta\,
  d\theta\, d\phi\, d\psi = 1, \\
4\pi V \int_{0}^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2\, d\psi = 1.
\end{gathered}
\Tag{124}
\]
Corresponding to equation~(123), we also see that the average value
of any quantity~$A$, which is dependent on the momentum of the
molecules of mass~$m_a$, will be given by the expression
\[
[A]_{\text{av.}}
  = 4\pi V \int_{0}^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,
    A\, \psi^2\, d\psi.
\Tag{125}
\]

\Subsubsection{100}{The Law of Equipartition.} We may now obtain a law which
corresponds to that of the equipartition of \textit{vis~viva} in the classical
mechanics. Considering equation~(124) let us integrate by parts, we
obtain
\begin{multline*}
\left[ 4\pi V a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \frac{\psi^3}{3}\right]_{\psi=0}^{\psi=\infty} \\
-4\pi V\int_0^{\infty} \frac{\psi^3}{3}\, a_a\,
  e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,(-hc)\,
  \frac{\psi}{\sqrt{\psi^2 + {m_a}^2c^2}}\, d\psi = 1.
\end{multline*}
%% -----File: 125.png---Folio 111-------
Substituting the limits into the first term we find that it becomes
zero and may write
\[
4\pi V\int_0^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,
  \frac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2\, d\psi = \frac{3}{h}.
\]

But by equation~(125) the left-hand side of this relation is the
average value of $\dfrac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}$ for the particles of mass~$m_a$. We have
\[
\left[ \frac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}\right]_{\text{av.}}
  = \frac{3}{h}.
\]
Introducing equation~(119) which defines~$\psi^2$, we may transform this
expression into
\[
\Biggl[\frac{m_au^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggl]_{\text{av.}}
  = \frac{3}{h}.
\Tag{126}
\]

Since we have shown that $h$~is independent of the mass of the
particles, \emph{we see that the average value of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is the same for particles
of all different masses}. This is the principle in relativity mechanics
that corresponds to the law of the equipartition of \textit{vis~viva} in the
classical mechanics. Indeed, for low velocities the above expression
reduces to~$m_0 u^2$, the \textit{vis~viva} of Newtonian mechanics, a fact which
affords an illustration of the general principle that the laws of Newtonian
mechanics are always the limiting form assumed at low velocities
by the more exact formulations of relativity mechanics.

We may now call attention in passing to the fact that this quantity
$\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, whose value is the same for particles of different masses, is
not the relativity expression for kinetic energy, which is given rather
by the formula $c^2\Biggl[\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0\Biggr]$. So that in relativity mechanics
%% -----File: 126.png---Folio 112-------
the principle of the equipartition of energy is merely an approximation.
We shall later return to this subject.

\Subsubsection{101}{Criterion for Equality of Temperature.} For a system of particles
of masses $m_a$,~$m_b$,~etc., enclosed in the volume~$V$, and having the
definite energy content~$E$, we have shown that
\[
4\pi V\, a_a\, e^{-hc\sqrt{\psi^2+{m_a}^2c^2}}\, \psi^2\, d\psi
\]
and
\[
4\pi V\, a_b\, e^{-hc\sqrt{\psi^2+{m_b}^2c^2}}\, \psi^2\, d\psi
\]
are the respective probabilities that given particles of mass~$m_a$ or
mass~$m_b$ will have momenta between $\psi$~and~$\psi + d\psi$. Suppose now
we consider a differently arranged system in which we have $N_a$~particles
of mass~$m_a$ by themselves in a space of volume~$V_a$ and $N_b$~particles
of mass~$m_b$ in a contiguous space of volume~$V_b$, separated
from~$V_a$ by a partition which permits a transfer of energy, and let
the total energy of the double system be, as before, a definite quantity~$E$
(the energy content of the partition being taken as negligible).
Then, by reasoning entirely similar to that just employed, we can
obviously show that
\[
4\pi V_a\, a_a\, e^{-hc\sqrt{\psi^2+{m_a}^2c^2}}\, \psi^2\, d\psi
\]
and
\[
4\pi V_b\, a_b\, e^{-hc\sqrt{\psi^2+{m_b}^2c^2}}\, \psi^2\, d\psi
\]
are now the respective probabilities that given particles of mass~$m_a$
or mass~$m_b$ will have momenta between $\psi$~and~$\psi + d\psi$, the only
changes in the expressions being the substitution of the volumes
$V_a$~and~$V_b$ in the place of the one volume~$V$. Furthermore, this
distribution law will evidently lead as before to the equality of the
average values of
\[
\frac{m_au^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\qquad \text{and}\qquad
\frac{m_bu^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}.
\]
Since, however, the spaces containing the two kinds of particles are in
thermal contact, their temperature is the same. Hence we find that
\emph{the equality of the average values of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is the necessary condition for
equality of temperature}.
%% -----File: 127.png---Folio 113-------

\emph{The above distribution law also leads to the important corollary that
for any given system of particles at a definite temperature the momenta
and hence the total energy content is independent of the volume.}

We may now proceed to the derivation of relations which will
permit us to show that the important quantity $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is directly
proportional to the temperature as measured on the absolute thermodynamic
temperature scale.

\Subsubsection{102}{Pressure Exerted by a System of Particles.} We first need
to obtain an expression for the pressure exerted by a system of $N$~particles
enclosed in the volume~$V$. Consider an element of surface~$dS$
perpendicular to the $X$~axis, and let the pressure acting on it be~$p$.
The total force which the element~$dS$ exerts on the particles that
impinge will be~$p\, dS$, and this will be equal to the rate of change of
the momenta in the $X$~direction of these particles.\footnote
  {The system is considered dilute enough for the mutual attractions of the
  particles to be negligible in their effect on the external pressure.}

Now by equation~(122) the total number of particles having
momenta between $\psi_x$~and~$\psi_x + d\psi_x$ in the \emph{positive} direction is
\[
NV \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}
  a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z.
\]
But $\dot{x}\, dS$ gives us the volume which contains the number of particles
having momenta between $\psi_x$~and~$\psi_x + d\psi_x$ which will reach~$dS$ in a
second. Hence the number of such particles which impinge per
second will be
\[
NV\, \frac{\dot{x}\, dS}{V}
  \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}
  a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z.
\]
and their change in momentum, allowing for the effect of the rebound,
will be
\[
2N\, dS \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}
  a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi_x\, \dot{x}\, d\psi_x\, d\psi_y\, d\psi_z.
\]
Finally, the total change in momentum per second for all particles
can be found by integrating for all possible positive values of~$\psi_x$.
%% -----File: 128.png---Folio 114-------
Equating this to the total force~$p\, dS$ we have
\[
p\, dS = 2N\, dS
  \int_{0}^{\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}
  a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi_x\, \dot{x}\, d\psi_x\, d\psi_y\, d\psi_z.
\]
Cancelling~$dS$, multiplying both sides of the equation by the volume~$V$,
changing the limits of integration and substituting $\dfrac{m_0\dot{x}}{\sqrt{1 + \smfrac{u^2}{c^2}}}$ for~$\psi_x$,
we have
\[
pV = NV
  \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}
  a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\,
  \frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, d\psi_x\, d\psi_y\, d\psi_z.
\]
But this by equation~(123) reduces to
\[
pV = N \Biggl[\frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}}
\]
or, since
\[
\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  = \frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  + \frac{m_0\dot{y}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  + \frac{m_0\dot{z}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\]
we have from symmetry
\[
pV = \frac{N}{3}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Bigg]_{\text{av.}}.
\Tag{127}
\]
Since at a given temperature we have seen that the term in parenthesis
is independent of the volume and the nature of the particles, we see
that the laws of Boyle and Avogadro hold also in relativity mechanics
for a system of particles.

For slow velocities equation~(127) reduces to the familiar expression
$pV = \dfrac{N}{3}\, (m_0u^2)_{\text{av.}}$.

\Subsubsection{103}{The Relativity Expression for Temperature.} We are now in
a position to derive the relativity expression for temperature. The
thermodynamic scale of temperature may be defined in terms of the
efficiency of a heat engine. Consider a four-step cycle performed
with a working substance contained in a cylinder provided with a
piston. In the first step let the substance expand isothermally and
%% -----File: 129.png---Folio 115-------
reversibly, absorbing the heat~$Q_2$ from a reservoir at temperature~$T_2$;
in the second step cool the cylinder down at constant volume to~$T_1$;
in the third step compress to the original volume, giving out the
heat~$Q_1$ at temperature~$T_1$, and in the fourth step heat to the original
temperature. Now if the working substance is of such a nature that
the heat given out in the second step could be used for the \emph{reversible}
heating of the cylinder in the fourth step, we may define the absolute
temperatures $T_2$~and~$T_1$ by the relation $\dfrac{T_2}{T_1} = \dfrac{Q_2}{Q_1}$.\footnote
  {We have used this cycle for defining the thermodynamic temperature scale
  instead of the familiar Carnot cycle, since it avoids the necessity of obtaining an
  expression for the relation between pressure and volume in an adiabatic expansion.}

Consider now such a cycle performed on a cylinder which contains
one of our systems of particles. Since we have shown (\Secref{101})
that at a definite temperature the energy content of such a
system is independent of the volume, it is evident that our working
substance fulfils the requirement that the heat given out in the second
step shall be sufficient for the reversible heating in the last step.
Hence, in accordance with the thermodynamic scale, we may measure
the temperatures of the two heat reservoirs by the relation $\dfrac{T_2}{T_1} = \dfrac{Q_2}{Q_1}$
and may proceed to obtain expressions for $Q_2$~and~$Q_1$.

In order to obtain these expressions we may again make use of the
principle that the energy content at a definite temperature is independent
of the volume. This being true, we see that $Q_2$~and~$Q_1$
must be equal to the work done in the changes of volume that take
place respectively at $T_2$~and~$T_1$, and we may write the relations
\begin{align*}
Q_2 &= \int_V^{V'} p\, dV\quad \text{(at $T_2$)}, \\
Q_1 &= \int_V^{V'} p\, dV\quad \text{(at $T_1$)}.
\end{align*}
But equation~(127) provides an expression for~$p$ in terms of~$V$, leading
on integration to the relations
\begin{align*}
Q_2 &= \frac{N}{3}\Biggl[
    \frac{m_0{u_2}^2}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}}
  \Biggr]_{\text{av.}} \log\frac{V'}{V}, \\
%% -----File: 130.png---Folio 116-------
Q_1 &= \frac{N}{3}\Biggl[
    \frac{m_0{u_1}^2}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}
  \Biggl]_{\text{av.}} \log\frac{V'}{V},
\end{align*}
which gives us on division
\[
\frac{T_2}{T_1} = \frac{Q_2}{Q_1}
  = \frac{\Biggl[\smfrac{m_0{u_2}^2}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}}\Biggr]_{\text{av.}}}
         {\Biggl[\smfrac{m_0{u_1}^2}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}\Biggr]_{\text{av.}}}.
\]

\emph{We see that the absolute temperature measured on the thermodynamic
scale is proportional to the average value of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$.}

We may finally express our temperature in the same units customarily
employed by comparing equation~(127)
\[
pV = \frac{N}{3}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}},
\]
with the ordinary form of the gas law
\[
pV = nRT,
\]
where $n$~is the number of mols of gas present.

We evidently obtain
\[
\begin{aligned}
nRT &= \frac{N}{3} \Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}},\\
T  &= \frac{N}{3nR}\Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}}
  = \frac{1}{3k}   \Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}},
\end{aligned}
\Tag{128}
\]
where the quantity $\dfrac{nR}{N}$, which may be called the gas constant for a
single molecule, has been denoted, as is customary, by the letter~$k$.
%% -----File: 131.png---Folio 117-------
Remembering the relation $\Biggl[\dfrac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}} = \dfrac{3}{h}$, we have
\[
kT = \frac{1}{h}.
\Tag{129}
\]

\Subsubsection{104}{The Partition of Energy.} We have seen that our new equipartition
law precludes the possibility of an exact equipartition of
energy. It becomes very important to see what the average energy
of a particle of a given mass does become at any temperature.

Equation~(125) provides a general expression for the average value
of any property of the particles. For the average value of the energy
$c\sqrt{\psi^2 + {m_0}^2c^2}$ of particles of mass~$m_0$ (see equation~120) we shall have
\[
[E]_{\text{av.}}
  = 4\pi V \int_0^{\infty} a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\,
    c\, \sqrt{\psi^2 + {m_0}^2c^2}\, \psi^2\, d\psi.
\]
The unknown constant~$a$ may be eliminated with the help of the
relation~(124)
\[
4\pi V \int_0^{\infty} a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi^2\, d\psi = 1
\]
and for~$h$ we may substitute the value given by~(129), which gives us
the desired equation
\[
[E]_{\text{av.}}
  = \frac{\ds\int_0^{\infty} e^{-(c/kT)\sqrt{\psi^2 + {m_0}^2c^2}}\, c\, \sqrt{\psi^2 + {m_0}^2c^2}\, \psi^2\, d\psi}
         {\ds\int_0^{\infty} e^{-(c/kT)\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi^2\, d\psi}.
\Tag{130}
\]

\Subsubsection{105}{Partition of Energy for Zero Mass.} Unfortunately, no general
method for the evaluation of this expression seems to be available.
For the particular case that the mass~$m_0$ of the particles approaches
zero compared to the momentum, the expression reduces to
\[
[E]_{\text{av.}}
  = \frac{c\ds\int_0^{\infty} e^{-(c\psi /kT)}\, \psi^3\, d\psi}
         { \ds\int_0^{\infty} e^{-(c\psi /kT)}\, \psi^2\, d\psi}
\]
%% -----File: 132.png---Folio 118-------
in terms of integrals whose values are known. Evaluating, we obtain
\[
[E]_{\text{av.}} = 3kT.
\]
For the total energy of $N$ such particles we obtain
\[
E = 3NkT,
\]
and introducing the relation $k = \dfrac{nR}{N}$ by which we defined~$k$ we have
\[
E = 3nRT
\Tag{131}
\]
as the expression for the energy of $n$~mols of particles if their value of~$m_0$
is small compared with their momentum.

It is instructive to compare this with the ordinary expression of
Newtonian mechanics
\[
E = \frac{3}{2}\, nRT,
\]
which undoubtedly holds when the masses are so large and the velocities
so small that no appreciable deviations from the laws of Newtonian
mechanics are to be expected. We see that for particles of
very small mass the average kinetic energy at any temperature is
twice as large as that for large particles at the same temperature.
It is also interesting to note that in accordance with equation~(131)
a mol of particles which approach zero mass at the absolute zero,
would have a mass of
\[
\frac{3 × 8.31 × 10^{7} × 300}{10^{21}} = 7.47 × 10^{-11}
\]
grams at room temperature ($300°$~absolute). This suggests a field
of fascinating if profitless speculation.

%[** TN: ToC entry reads "Approximate Partition for Particles of any Mass"]
\Subsubsection{106}{Approximate Partition of Energy for Particles of any Desired
Mass.} For particles of any desired mass we may obtain an approximate
idea of the relation between energy and temperature by expanding
the expression for kinetic energy into a series. For the average
kinetic energy of a particle we have
\[
[K]_{\text{av.}}
  = c^2\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0\Biggr]_{\text{av.}}.
\]
%% -----File: 133.png---Folio 119-------
Expanding into a series we obtain for the total kinetic energy of $N$~particles
\[
K = Nm_0\left(
   \frac{1}{2}\, \vc{u}^2 + \frac{3}{8}\, \frac{\vc{u}^4}{c^2}
 + \frac{15}{48}\, \frac{\vc{u}^6}{c^4}
 + \frac{105}{384}\, \frac{\vc{u}^8}{c^6} + \cdots\right),
\Tag{132}
\]
where $\vc{u}^2$, $\vc{u}^4$,~etc., are the average values of $u^2$,~$u^4$,~etc., for the individual
particles.

To determine approximately how the value of~$K$ varies with the
temperature we may also expand our expression~(128) for temperature,
\[
T = \frac{1}{3k}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{Av.}},
\]
into a series; we obtain
{\small%
\[
\frac{3}{2}\, kNT = \frac{3}{2}\, nRT
  = Nm_0 \left(
   \frac{1}{2}\, \vc{u}^2 + \frac{1}{4}\, \frac{\vc{u}^4}{c^2}
 + \frac{3}{16}\, \frac{\vc{u}^6}{c^4}
 + \frac{15}{96}\, \frac{\vc{u}^8}{c^6} + \cdots\right).
\Tag{133}
\]}%
Combining expressions (132)~and~(133) by subtraction and transposition,
we obtain
\[
K = \frac{3}{2}\, nRT
  + Nm_0 \left(
    \frac{1}{8}\, \frac{\vc{u}^4}{c^2}
  + \frac{1}{8}\, \frac{\vc{u}^6}{c^4}
  + \frac{15}{128}\, \frac{\vc{u}^8}{c^6} + \cdots\right).
\Tag{134}
\]
For the case of velocities low enough so that $\vc{u}^4$~and higher powers
can be neglected, this reduces to the familiar expression of Newtonian
mechanics,~$K = \dfrac{3}{2}\, nRT$.

In case we neglect in expression~(134) powers higher than~$\vc{u}^4$ we
have the approximate relation
\[
\frac{Nm_0 \vc{u}^4}{8c^2}
  = \frac{1}{2Nm_0c^2} \left(\frac{Nm_0\vc{u}^2}{2}\right)^2,
\]
the left-hand term really being the larger, since the average square of a
quantity is greater than the square of its average. Since $\left(\dfrac{Nm_0\vc{u}^2}{2}\right)^2$
is approximately equal to $\left(\dfrac{3}{2}\, nRT\right)^2$, we may write the approximation
%% -----File: 134.png---Folio 120-------
\[
K = \frac{3}{2}\, nRT + \frac{1}{2Nm_0c^2} \left(\frac{3}{2}\, nRT\right)^2,
\]
or, noting that $N m_0 = M$, the total mass of the system at the absolute
zero, we have
\[
K = \frac{3}{2}\, nRT + \frac{9}{8}\, \frac{n^2R^2}{Mc^2}\, T^2.
\]
If we use the erg as our unit of energy, $R$~will be~$8.31 × 10^7$; expressing
velocities in centimeters per second, $c^2$~will be~$10^{21}$, and $M$~will be the
mass of the system in grams.

For one mol of a monatomic gas we should have in ergs
\[
K = 12.4 × 10^7T + \frac{7.77}{M}\, 10^{-6}\, T^2.
\]

In the case of the electron $M$~may be taken as approximately
$1/1800$. At room temperature the second term of our equation would
be entirely negligible, being only $3.5 × 10^{-6}$~per cent of the first, and
still be only $3.5 × 10^{-4}$~per cent in a fixed star having a temperature of
$30,000°$. Hence at all ordinary temperatures we may expect the
law of the equipartition of energy to be substantially exact for particles
of mass as small as the electron.

Our purpose in carrying through the calculations of this chapter
has been to show that a very important and interesting problem in
the classical mechanics can be handled just as easily in the newer
mechanics, and also to point out the nature of the modifications in
existing theory which will have to be introduced if the later developments
of physics should force us to consider equilibrium relations for
particles of mass much smaller than that of the electron.

We may also call attention to the fact that we have here considered
a system whose equations of motion agree with the principles
of dynamics and yet do not lead to the equipartition of energy. This
is of particular interest at a time when many scientists have thought
that the failure of equipartition in the hohlraum stood in necessary
conflict with the principles of dynamics.
%% -----File: 135.png---Folio 121-------


\Chapter{IX}{The Principle of Relativity and the Principle of
Least Action.}
\SetRunningHeads{Chapter Nine.}{Relativity and the Principle of Least Action.}

It has been shown by the work of Helmholtz, J.~J. Thomson,
Planck and others that the principle of least action is applicable in
the most diverse fields of physical science, and is perhaps the most
general dynamical principle at our disposal. Indeed, for any system
whose future behavior is determined by the instantaneous values of a
number of \DPchg{coordinates}{coördinates} and their time rate of change, it seems possible
to throw the equations describing the behavior of the system into
the form prescribed by the principle of least action. This generality
of the principle of least action makes it very desirable to develop the
relation between it and the principle of relativity, and we shall obtain
in this way the most important and most general method for deriving
the consequences of the theory of relativity. We have already
developed in \Chapref{VII} the particular application of the principle
of least action in the case of a system of particles, and with the help
of the more general development which we are about to present, we
shall be able to apply the principle of relativity to the theories of
elasticity, of thermodynamics and of electricity and magnetism.

\Subsubsection{107}{The Principle of Least Action.} For our purposes the principle
of least action may be most simply stated by the equation
\[
\int_{t_1}^{t_2}(\delta H + W)\, dt = 0.
\Tag{135}
\]
This equation applies to any system whose behavior is determined
by the values of a number of independent coördinates $\phi_1\phi_2\phi_3\cdots$
and their rate of change with the time $\dot{\phi}_1\dot{\phi}_2\dot{\phi}_3\cdots$, and the equation
describes the path by which the system travels from its configuration
at any time~$t_1$ to its configuration at any subsequent time~$t_2$.

$H$~is the so-called kinetic potential of the system and is a function
of the coördinates and their generalized velocities:
\[
H = F(\phi_1\phi_2\phi_3\cdots \dot{\phi}_1\dot{\phi}_2\dot{\phi}_3\cdots).
\Tag{136}
\]
%% -----File: 136.png---Folio 122-------
$\delta H$~is the variation of~$H$ at any instant corresponding to a slightly
displaced path by which the system might travel from the same
initial to the same final state in the same time interval, and $W$~is the
external work corresponding to the variation~$\delta$ which would be done
on the system by the external forces if at the instant in question the
system should be displaced from its actual configuration to its configuration
on the displaced path. Thus
\[
W = \Phi_1\, \delta\phi_1
  + \Phi_2\, \delta\phi_2
  + \Phi_3\, \delta\phi_3 + \cdots,
\Tag{137}
\]
where $\Phi_1$, $\Phi_2$,~etc., are the so-called generalized external forces which
act in such a direction as to increase the values of the corresponding
coördinates.

The form of the function which determines the kinetic potential~$H$
depends on the particular nature of the system to which the principle
of least action is being applied, and it is one of the chief tasks of
general physics to discover the form of the function in the various
fields of mechanical, electrical and thermodynamic investigation.
As soon as we have found out experimentally what the form of~$H$ is
for any particular field of investigation, the principle of least action,
as expressed by equation~(135), becomes the basic equation for the
mathematical development of the field in question, a development
which can then be carried out by well-known methods.

The special task for the theory of relativity will be to find a general
relation applicable to any kind of a system, which shall connect the
value of the kinetic potential~$H$ as measured with respect to a set of
coördinates~$S$ with its value~$H'$ as measured with reference to another
set of coördinates~$S'$ which is in motion relative to~$S$. This relation
will of course be of such a nature as to agree with the principle of the
relativity of motion, and in this way we shall introduce the principle
of relativity at the very start into the fundamental equation for all
fields of dynamics.

Before proceeding to the solution of that problem we may put
the principle of least action into another form which is sometimes
more convenient, by obtaining the equations for the motion of a
system in the so-called Lagrangian form.

\Subsubsection{108}{The Equations of Motion in the Lagrangian Form.} To obtain
the equations of motion in the Lagrangian form we may evidently
%% -----File: 137.png---Folio 123-------
rewrite our fundamental equation~(135) in the form
\[
\begin{aligned}
\int_{t_1}^{t_2} \biggl(
    \frac{\partial H}{\partial\phi_1}\, \delta\phi_1
  + \frac{\partial H}{\partial\phi_2}\, \delta\phi_2 + \cdots
 &+ \frac{\partial H}{\partial\dot{\phi}_1}\, \delta\dot{\phi}_1
  + \frac{\partial H}{\partial\dot{\phi}_2}\, \delta\dot{\phi}_2 + \cdots \\
 &+ \Phi_1\, \delta\phi_1 + \Phi_2\, \delta\phi_2 + \cdots\biggr) dt = 0
\end{aligned}
\Tag{138}
\]

We have now, however,
\[
\delta\dot\phi_1 = \frac{d}{dt}(\delta\phi_1), \qquad
\delta\dot\phi_2 = \frac{d}{dt}(\delta\phi_2), \qquad \text{etc.,}
\]
which gives us
\begin{align*}
\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, \delta\dot\phi_1\, dt
  &= \int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\,
       \frac{d}{dt}(\delta\phi_1)\, dt \\
  &= \left[\frac{\partial H}{\partial\dot\phi_1}\, \delta\phi_1\right]_{t_1}^{t_2}
   - \int_{t_1}^{t_2} \delta\phi_1\,
       \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_1}\right) dt,
\end{align*}
or, since $\delta\phi_1$, $\delta\phi_2$,~etc., are by hypothesis zero at times $t_1$~and~$t_2$, we
obtain
\begin{align*}
&\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, \delta\dot\phi_1
  = - \int_{t_1}^{t_2} \frac{d}{dt}
        \left(\frac{\partial H}{\partial\dot\phi_1}\right) \delta\phi_1\, dt, \\
&\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_2}\, \delta\dot\phi_2
  = - \int_{t_1}^{t_2} \frac{d}{dt}
        \left(\frac{\partial H}{\partial\dot\phi_2}\right) \delta\phi_2\, dt, \\
&\text{etc.}
\end{align*}
On substituting these expressions in~(138) we obtain
\begin{multline*}
 \int_{t_1}^{t_2} \left[
  \left(\frac{\partial H}{\partial\phi_1}
      - \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_1}\right)
      + \Phi_1\right) \right. \delta\phi_1 \\
  \left. + \left(\frac{\partial H}{\partial\phi_2}
      - \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_2}\right)
      + \Phi_2\right) \delta\phi_2 + \cdots \right] dt = 0\DPtypo{}{,}
\end{multline*}
and since the variations of $\phi_1$, $\phi_2$,~etc., are entirely independent and
the limits of integration $t_1$~and~$t_2$ are entirely at our disposal, this
equation will be true only when each of the following equations is
true. And these are the equations of motion in the desired Lagrangian
%% -----File: 138.png---Folio 124-------
form,
\[
\begin{aligned}
&\frac{d}{dt}\, \frac{\partial H}{\partial\dot{\phi}_1}
  - \frac{\partial H}{\partial \phi_1} = \Phi_1, \\
&\frac{d}{dt}\, \frac{\partial H}{\partial\dot{\phi}_2}
  - \frac{\partial H}{\partial \phi_2} = \Phi_2, \\
&\text{etc.}
\end{aligned}
\Tag{139}
\]

In these equations $H$ is the kinetic potential of a system whose
state is determined by the generalized coördinates $\phi_1$,~$\phi_2$,~etc., and
their time derivatives $\dot{\phi}_1$,~$\dot{\phi}_2$~etc., where $\Phi_1$,~$\Phi_2$,~etc., are the generalized
external forces acting on the system in such a sense as to tend
to \emph{increase} the values of the corresponding generalized coördinates.

\Subsubsection{109}{Introduction of the Principle of Relativity.} Let us now investigate
the relation between our dynamical principle and the principle
of the relativity of motion. To do this we must derive an equation
for transforming the kinetic potential~$H$ for a given system
from one set of \DPchg{coordinates}{coördinates} to another. In other words, if $S$~and~$S'$
are two sets of reference axes, $S'$~moving past~$S$ in the $X$\DPchg{-}{~}direction
with the velocity~$V$, what will be the relation between $H$~and~$H'$,
the values for the kinetic potential of a given system as measured
with reference to $S$~and~$S'$?

It is evident from the theory of relativity that our fundamental
equation~(135) must hold for the behavior of a given system using
either set of \DPchg{coordinates}{coördinates} $S$~or~$S'$, so that both of the equations
\[
\int_{t_1}^{t_2} (\delta H + W)\, dt = 0\qquad\text{and}\qquad
\int_{{t_1}'}^{{t_2}'} (\delta H' + W')\, dt' = 0\DPtypo{}{,} \Tag{140}
\]
or
\[
\int_{t_1}^{t_2} (\delta H + W)\, dt
  = \int_{{t_1}'}^{{t_2}'} (\delta H' + W')\, dt' = 0\DPtypo{}{,}
\]
must hold for a given process, where it will be necessary, of course,
to choose the limits of integration $t_1$~and~$t_2$, ${t_1}'$~and~${t_2}'$ wide enough
apart so that for both sets of coördinates the varied motion will be
completed within the time interval. Since we shall find it possible
now to show that in general $\ds\int W\, dt = \int W'\, dt'$, we shall be able to
obtain from the above equations a simple relation between $H$~and~$H'$.

%[** TN: Bold symbols in original]
\Subsubsection{110}{Relation between $\int W\, dt$ and $\int W'\, dt'$.} To obtain the desired
%% -----File: 139.png---Folio 125-------
proof we must call attention in the first place to the fact that all
kinds of force which can act at a given point must be governed by
the same transformation equations when changing from system~$S$ to
system~$S'$. This arises because when two forces of a different nature
are of such a magnitude as to exactly balance each other and produce
no acceleration for measurements made with one set of coördinates
they must evidently do so for any set of coördinates (see Chapter~IV, %[** TN: Not a useful cross-reference]
\Secref{42}). Since we have already found transformation equations
for the force acting at a point, in our consideration of the dynamics
of a particle, we may now use these expressions in general for the
evaluation~$\int W'\, dt'$.

$W'$ is the work which would be done by the external forces if at
any instant~$t'$ we should displace our system from its actual configuration
to the simultaneous configuration on the displaced path.
Hence it is evident that $\int W'\, dt\DPtypo{}{'}$~will be equal to a sum of terms of the
type
\[
%[** TN: Subscripts y and z misprinted (not as subscripts) in original]
\int ({F_x}'\, \delta x' + {F_y}'\, \delta y' + {F_z}'\, \delta z')\, dt',
\]
where ${F_x}'$,~${F_y}'$,~${F_z}'$, is the force acting at a given point of the system
and $\delta x'$,~$\delta y'$,~$\delta z'$ are the displacements necessary to reach the corresponding
point on the displaced path, all these quantities being
measured with respect to~$S'$.

Into this expression we may substitute, however, in accordance
with equations (61),~(62),~(63) and~(13), the values
\[
\begin{aligned}
{F_x}' &= F_x - \frac{\dot{y}V}{c^2}\, \frac{1}{1 - \smfrac{\dot{x}V}{c^2}}\, F_y
              - \frac{\dot{z}V}{c^2}\, \frac{1}{1 - \smfrac{\dot{x}V}{c^2}}\, F_z, \\
{F_y}' &= \frac{F_y\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}, \\
{F_z}' &= \frac{F_z\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}, \\
dt'    &= \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) dt.
\end{aligned}
\Tag{141}
\]
%% -----File: 140.png---Folio 126-------

We may also make substitutions for $\delta x'$,~$\delta y'$ and~$\delta z'$ in terms of
$\delta x$,~$\delta y$ and~$\delta z$, but to obtain transformation equations for these quantities
is somewhat complicated owing to the fact that positions on the
actual and displaced path, which are simultaneous when measured
with respect to~$S'$, will not be simultaneous with respect to~$S$. We
have denoted by~$t'$ the time in system~$S'$ when the point on the \emph{actual}
path has the position $x'$,~$y'$,~$z'$ and simultaneously the point on the
\emph{displaced} path has the position $(x'+ \delta x')$, $(y' + \delta y')$, $(z' + \delta z')$,
when measured in system~$S'$, or by our fundamental transformation
equations (9),~(10) and~(11) the positions $\kappa (x' + Vt')$,~$y'$,~$z'$ and
$\kappa \bigl([x' + \delta x'] + Vt'\bigr)$, $(y'+ \delta y')$, $(z'+\delta z')$ when measured in system~$S$.
If now we denote by $t_A$~and~$t_D$ the corresponding times in system~$S$
we shall have, by our fundamental transformation equation~(12),
\begin{align*}
t_A &= \kappa \left(t' + \frac{Vx'}{c^2}\right), \\
t_D &= \kappa \left(t' + \frac{V}{c^2}[x' + \delta x']\right),
\end{align*}
and we see that in system~$S$ the point has reached the displaced
position at a time later than that of the actual position by the amount
\[
t_D - t_A = \frac{\kappa V}{c^2}\, \delta x',
\]
and, since during this time-interval the displaced point would have
moved, neglecting higher-order terms, the distances
\[
\dot{x}\, \frac{\kappa V}{c^2}\, \delta x', \qquad
\dot{y}\, \frac{\kappa V}{c^2}\, \delta x', \qquad
\dot{z}\, \frac{\kappa V}{c^2}\, \delta x',
\]
these quantities must be subtracted from the coördinates of the
displaced point in order to obtain a position on the displaced path
which will be simultaneous with~$t_A$ as measured in system~$S$. We
obtain for the simultaneous position on the displaced path
\begin{gather*}
\kappa \bigl([x' + \delta x'] + Vt'\bigr)
  - \kappa\, \frac{\dot{x}V}{c^2}\, \delta x', \qquad
y' + \delta y' - \kappa\, \frac{\dot{x}V}{c^2}\, x', \\
z' + \delta z' - \kappa\, \frac{\dot{z}V}{c^2}\, \delta x',
\end{gather*}
%% -----File: 141.png---Folio 127-------
and for the corresponding position on the actual path
\[
\kappa (x' + Vt'), \quad y', \quad z',
\]
and obtain by subtraction
\[
\begin{aligned}
\delta x &= \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) \delta x', \\
\delta y &= \delta y' - \kappa \frac{\dot{y}V}{c^2}\, \delta x', \\
\delta z &= \delta z' - \kappa \frac{\dot{z}V}{c^2}\, \delta x'.
\end{aligned}
\Tag{142}
\]
Substituting now these equations, together with the other transformation
equations~(141), in our expression we obtain
\[
\begin{aligned}
\int ({F_x}'\, \delta x' &+ {F_y}'\, \delta y' + {F_z}'\, \delta z')\, dt' \\
  &= \int \Biggl(\Biggl[
    F_x - \frac{\dot{y}V}{c^2}\, \frac{F_y}{1 - \smfrac{\dot{x}V}{c^2}}
        - \frac{\dot{z}V}{c^2}\, \frac{F_z}{1 - \smfrac{\dot{x}V}{c^2}}
    \Biggr]
%[** TN: \rlap hack to get equation number centered]
    \rlap{$\ds \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x$} \\
  &+ \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, F_y \Biggl[
     \delta y + \frac{\dot{y}V/c^2}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x
    \Biggr] \\
  &+ \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, F_z \Biggl[
     \delta z + \frac{\dot{z}V/c^2}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x
    \Biggr]\Biggr) \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) \\
  &= \int(F_x\, \delta x + F_y\, \delta y + F_z\, \delta z)\, dt'.
\end{aligned}
\Tag{143}
\]
We thus see that we must always have the general equality
\[
\int W'\, dt' = \int W\, dt.
\Tag{144}
\]

\Subsubsection{111}{Relation between $H'$~and~$H$.} Introducing this equation into
our earlier expression~(140) we obtain as a general relation between
$H'$~and~$H$
\[
\int \delta H'\, dt' = \int \delta H\, dt.
\Tag{145}
\]

Restricting ourselves to systems of such a nature that we can
%% -----File: 142.png---Folio 128-------
assign them a definite velocity $u = \dot{x}\vc{i} + \dot{y}\vc{j} + \dot{z}\vc{k}$, we can rewrite
this expression in the following form, where by $H_D$~and~$H_A$ we denote
the values of the kinetic potential respectively on the displaced and
actual paths
\begin{align*}
\int \delta H'\, dt' = \int {H_D}'\, dt'
  &- \int {H_A}'\, dt'
   = \int {H_D}' \kappa
          \left(1 - \frac{(\dot{x} + \delta \dot{x})V}{c^2}\right) dt \\
  &- \int {H_A}' \kappa
          \left(1 - \frac{\dot{x}V}{c^2}\right) dt
   = \int H_D\, dt - \int H_A\, dt,
\end{align*}
and hence obtain for such systems the simple expression
\[
H' = \frac{H}{\kappa \left(1 - \smfrac{\dot{x}V}{c^2}\right)}.
\]
Noting the relation between $\sqrt{1 - \dfrac{{u'}^2}{c^2}}$ and $\sqrt{1 - \dfrac{u^2}{c^2}}$ given in equation~(17),
this can be rewritten
\[
\frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}
  = \frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\Tag{146}
\]
and this is the expression which we shall find most useful for our
future development of the consequences of the theory of relativity.
Expressing the requirement of the equation in words we may say
that the theory of relativity requires an invariance of $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ in the
Lorentz transformation.

\Paragraph{112.} As indicated above, the use of this equation is obviously
restricted to systems moving with some perfectly definite velocity~$\vc{u}$.
Systems satisfying this condition would include particles, infinitesimal
portions of continuous systems, and larger systems in a steady state.

\Paragraph{113.} Our general method of procedure in different fields of investigation
will now be to examine the expression for kinetic potential
which is known to hold for the field in question, provided the velocities
involved are low and by making slight alterations when necessary,
%% -----File: 143.png---Folio 129-------
see if this expression can be made to agree with the requirements of
equation~(146) without changing its value for low velocities. Thus
it is well known, for example, that, in the case of low velocities, for a
single particle acted on by external forces the kinetic potential may
be taken as the kinetic energy~$\frac{1}{2}m_0u^2$. For relativity mechanics, as
will be seen from the developments of \Chapref{VII}, we may take for
the kinetic potential, $-m_0c^2 \sqrt{1 - \dfrac{u^2}{c^2}}$, an expression which, except for
an additive constant, becomes identical with~$\frac{1}{2}m_0u^2$ at low velocities,
and which at all velocities agrees with equation~(146).
%% -----File: 144.png---Folio 130-------


\Chapter{X}{The Dynamics of Elastic Bodies.}
\SetRunningHeads{Chapter Ten.}{Dynamics of Elastic Bodies.}

We shall now treat with the help of the principle of least action
the rather complicated problem of the dynamics of continuous elastic
media. Our considerations will \emph{extend} the appreciation of the intimate
relation between mass and energy which we found in our treatment
of the dynamics of a particle. We shall also be able to show
that the dynamics of a particle may be regarded as a special case
of the dynamics of a continuous elastic medium, and to apply our
considerations to a number of other important problems.

\Subsubsection{114}{On the Impossibility of Absolutely Rigid Bodies.} In the
older treatises on mechanics, after considering the dynamics of a
particle it was customary to proceed to a discussion of the dynamics
of rigid bodies. These rigid bodies were endowed with definite and
\DPtypo{nu}{un}changeable size and shape and hence were assigned five degrees
of freedom, since it was necessary to state the values of five variables
completely to specify their position in space. As pointed out by
Laue, however, our newer ideas as to the velocity of light as a limiting
value will no longer permit us to conceive of a continuous body as
having only a finite number of degrees of freedom. This is evident
since it is obvious that we could start disturbances simultaneously
at an indefinite number of points in a continuous body, and as these
disturbances cannot spread with infinite velocity it will be necessary
to give the values of an infinite number of variables in order completely
to specify the succeeding states of the system. For our newer
mechanics the nearest approach to an absolutely rigid body would
of course be one in which disturbances are transmitted with the
velocity of light. Since, then, the theory of relativity does not
permit rigid bodies we may proceed at once to the general theory of
deformable bodies.


\Section[I]{Stress and Strain.}

\Subsubsection{115}{Definition of Strain.} In the more familiar developments of
the theory of elasticity it is customary to limit the considerations to
%% -----File: 145.png---Folio 131-------
the case of strains small enough so that higher powers of the displacements
can be neglected, and this introduces considerable simplification
into a science which under any circumstances is necessarily
one of great complication. Unfortunately for our purposes, we
cannot in general introduce such a simplification if we wish to apply
the theory of relativity, since in consequence of the Lorentz shortening
a body which appears unstrained to one observer may appear tremendously
compressed or elongated to an observer moving with a
different velocity. The best that we can do will be arbitrarily to
choose our state of zero deformation such that the strains will be
small when measured in the particular system of coördinates $S$ in
which we are specially interested.

A theory of strains of any magnitude was first attempted by
Saint-Venant and has been amplified and excellently presented by
Love in his \textit{Treatise on the Theory of Elasticity}, Appendix to Chapter~I.
In accordance with this theory, the strain at any point in a body is
completely determined by six component strains which can be defined
by the following equations, wherein $(u, v, w)$~is the displacement of a
point having the unstrained position $(x, y, z)$:
%[** TN: Setting as two groups, both numbered (148), to permit a page break]
\begin{align*}
&\begin{aligned}
\epsilon_{xx} &= \frac{\partial u}{\partial x}
  + \tfrac{1}{2} \left\{
      \left(\frac{\partial u}{\partial x}\right)^2
    + \left(\frac{\partial v}{\partial x}\right)^2
    + \left(\frac{\partial w}{\partial x}\right)^2 \right\}, \\
%
\epsilon_{yy} &= \frac{\partial y}{\partial v}
  + \tfrac{1}{2} \left\{
      \left(\frac{\partial u}{\partial y}\right)^2
    + \left(\frac{\partial v}{\partial y}\right)^2
    + \left(\frac{\partial w}{\partial y}\right)^2 \right\}, \\
%
\epsilon_{zz} &= \frac{\partial w}{\partial z}
  + \tfrac{1}{2} \left\{
      \left(\frac{\partial u}{\partial z}\right)^2
    + \left(\frac{\partial v}{\partial z}\right)^2
    + \left(\frac{\partial w}{\partial z}\right)^2 \right\},
\end{aligned}
\Tag{148}
\displaybreak[0] \\
%
&\begin{aligned}
\epsilon_{yz} &= \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}
  + \frac{\partial u}{\partial y}\, \frac{\partial u}{\partial z}
  + \frac{\partial v}{\partial y}\, \frac{\partial v}{\partial z}
  + \frac{\partial w}{\partial y}\, \frac{\partial w}{\partial z}, \\
%
\epsilon_{xz} &= \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}
  + \frac{\partial u}{\partial x}\, \frac{\partial u}{\partial z}
  + \frac{\partial v}{\partial x}\, \frac{\partial v}{\partial z}
  + \frac{\partial w}{\partial x}\, \frac{\partial w}{\partial z}, \\
%
\epsilon_{xy} &= \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}
  + \frac{\partial u}{\partial x}\, \frac{\partial u}{\partial y}
  + \frac{\partial v}{\partial x}\, \frac{\partial v}{\partial y}
  + \frac{\partial w}{\partial x}\, \frac{\partial w}{\partial y}.
\end{aligned}
\Tag{148}
\end{align*}

It will be seen that these expressions for strain reduce to those
familiar in the theory of small strains if such second-order quantities as
$\left(\dfrac{\partial u}{\partial x}\right)^2$ or $\dfrac{\partial u}{\partial y}\,\dfrac{\partial u}{\partial z}$ can be neglected.
%% -----File: 146.png---Folio 132-------

\Paragraph{116.} A physical significance for these strain components will be
obtained if we note that it can be shown from geometrical considerations
that lines which are originally parallel to the axes have, when
strained, the elongations
\[
\begin{aligned}
e_x &= \sqrt{1 + 2\epsilon_{xx}} - 1, \\
e_y &= \sqrt{1 + 2\epsilon_{yy}} - 1, \\
e_z &= \sqrt{1 + 2\epsilon_{zz}} - 1,
\end{aligned}
\Tag{149}
\]
and that the angles between lines originally parallel to the axes are
given in the strained condition by the expressions
\[
\begin{aligned}
\cos \theta_{yz}
  &= \frac{\epsilon_{yz}}
          {\sqrt{1 + 2\epsilon_{yy}}\, \sqrt{1 + 2\epsilon_{zz}}}, \\
\cos \theta_{xz}
  &= \frac{\epsilon_{xz}}
          {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1 + 2\epsilon_{zz}}}, \\
\cos \theta_{xy}
  &= \frac{\epsilon_{xy}}
          {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1 + 2\epsilon_{yy}}},
\end{aligned}
\Tag{150}
\]

Geometrical considerations are also sufficient to show that in
case the strain is a simple elongation of amount~$e$ the following equation
will be true:
\[
\frac{\epsilon_{xx}}{l^2} =
\frac{\epsilon_{yy}}{m^2} =
\frac{\epsilon_{zz}}{n^2} =
\frac{\epsilon_{yz}}{2mn} =
\frac{\epsilon_{xz}}{2ln} =
\frac{\epsilon_{xy}}{2lm} = e + \tfrac{1}{2}e^2,
\Tag{151}
\]
where $l$,~$m$,~$n$ are the cosines which determine the direction of the
elongation.

\Subsubsection{117}{Definition of Stress.} We have just considered the expressions
for the strain at a given point in an elastic medium; we may
now define stress in terms of the work done in changing from one
state of strain to another. Considering the material contained in
\emph{unit volume when the body is unstrained}, we may write, for the work
done by this material on its surroundings when a change in strain
takes place,
%% -----File: 147.png---Folio 133-------
\[
\begin{aligned}
\delta W = -\delta E
  = t_{xx}\, \delta\epsilon_{xx}
  + t_{yy}\, \delta\epsilon_{yy}
 &+ t_{zz}\, \delta\epsilon_{zz} \\
 &+ t_{yz}\, \delta\epsilon_{yz}
  + t_{xz}\, \delta\epsilon_{xz}
  + t_{xy}\, \delta\epsilon_{xy},
\end{aligned}
\Tag{152}
\]
and this equation serves to define the stresses $t_{xx}$,~$t_{yy}$,~etc. In case
the strain varies from point to point we must consider of course the
work done \textit{per}~unit volume of the unstrained material. In case the
strains are small it will be noticed that the stresses thus defined are
identical with those used in the familiar theories of elasticity.

\Subsubsection{118}{Transformation Equations for Strain.} We must now prepare
for the introduction of the theory of relativity into our considerations,
by determining the way the strain at a given point~$P$ appears to observers
moving with different velocities. Let the point~$P$ in question
be moving with the velocity $\vc{u} = x\vc{i} + y\vc{j} + z\vc{k}$ as measured in system~$S$.
Since the state of zero deformation from which to measure
strains can be chosen perfectly arbitrarily, let us for convenience
take the strain as zero as measured in system~$S$, giving us
\[
\epsilon_{xx} =
\epsilon_{yy} =
\epsilon_{zz} =
\epsilon_{yz} =
\epsilon_{xz} =
\epsilon_{xy} = 0.
\Tag{153}
\]
What now will be the strains as measured by an observer moving
along with the point~$P$ in question? Let us call the system of coördinates
used by this observer~$S°$. It is evident now from our considerations
as to the shape of moving systems presented in \Chapref{V} that
in system~$S°$ the material in the neighborhood of the point in question
will appear to have been elongated in the direction of motion in the
ratio of $1 : \sqrt{1 - \dfrac{u^2}{c^2}}$. Hence in system~$S°$ the strain will be an elongation
\[
e = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1
\Tag{154}
\]
in the line determined by the direction cosines
\[
l = \frac{\dot{x}}{u},\qquad
m = \frac{\dot{y}}{u},\qquad
\DPtypo{u}{n}
  = \frac{\dot{z}}{u}.
\Tag{155}
\]

We may now calculate from this elongation the components of
strain by using equation~(151). We obtain
%% -----File: 148.png---Folio 134-------
{\small%[** TN: Setting on two lines, not six]
\[
\begin{aligned}
%[** TN: \llap coaxes equation to the left without crowding the tag]
\llap{$\epsilon°$}_{xx}
  &= \frac{\dot{x}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],&
{\epsilon°}_{yy}
  &= \frac{\dot{y}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],&
{\epsilon°}_{zz}
  &= \frac{\dot{z}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],\\
%
\llap{$\epsilon°$}_{yz}%[** See above]
  &= \ \frac{\dot{y}\dot{z}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],&
{\epsilon°}_{xz}
  &= \ \frac{\dot{x}\dot{z}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],&
{\epsilon°}_{xy}
  &= \ \frac{\dot{x}\dot{y}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],
\end{aligned}
\Tag{156}
\]}%
and these are the desired equations for the strains at the point~$P$,
the accent~$°$ indicating that they are measured with reference to a
system of coördinates~$S°$ moving along with the point itself.

\Subsubsection{119}{Variation in the Strain.} We shall be particularly interested
in the variation in the strain as measured in~$S°$ when the velocity
experiences a small variation~$\delta\vc{u}$, the strains remaining zero as measured
in~$S$. For the sake of simplicity let us choose our coördinates
in such a way that the $X$\DPchg{-}{~}axis is parallel to the original velocity, so
that our change in velocity will be from $\vc{u} = \dot{x}\vc{i}$ to
\[
\vc{u} + \delta\vc{u}
  = (\dot{x} + \delta\dot{x})\, \vc{i}
  + \delta\dot{y}\, \vc{j} + \delta\dot{z}\, \vc{k}.
\]
Taking $\delta\vc{u}$~small enough so that higher orders can be neglected, and
noting that $\dot{y} = \dot{z} = 0$, we shall then have, from equations~(156),
%% -----File: 149.png---Folio 135-------
\[
\begin{aligned}
\delta{\epsilon°}_{xx}
  &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\,
     \frac{\dot{x}}{c^2}\, \delta\dot{x},&
\delta{\epsilon°}_{yy} &= 0, \\
\delta{\epsilon°}_{zz} &= 0, &
\delta{\epsilon°}_{yz} &= 0, \\
\delta{\epsilon°}_{xz}
  &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\,
     \frac{\dot{x}}{c^2}\, \delta\dot{z},&\qquad
\delta{\epsilon°}_{xy}
  &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\,
     \frac{\dot{x}}{c^2}\, \delta\dot{y}.
\end{aligned}
\Tag{157}
\]

We shall also be interested in the variation in the strain as measured
in~$S°$ produced by a variation in the strain as measured in~$S$. Considering
again for simplicity that the $X$\DPchg{-}{~}axis is parallel to the motion
of the point, we must calculate the variation produced in ${\epsilon°}_{xx}$,~${\epsilon°}_{yy}$,~etc.,
by changing the values of $\epsilon_{xx}$,~$\epsilon_{yy}$,~etc., from zero to $\delta\epsilon_{xx}$,~$\delta\epsilon_{yy}$,~etc.

The variation~$\delta\epsilon_{xx}$ will produce a variation in~${\epsilon°}_{xx}$ whose amount
can be calculated as follows: By equations~(149) a line which has unit
length and is parallel to the $X$\DPchg{-}{~}axis in the unstrained condition will
have when strained the length $\sqrt{1 + 2\epsilon_{xx}}$ when measured in system~$S$
and $\sqrt{1 + 2{\epsilon°}_{xx}}$ when measured in system~$S°$. Since the strain in
system~$S$ is small, the line remains sensibly parallel to the $X$\DPchg{-}{~}axis,
which is also the direction of motion, and these quantities will be
connected in accordance with the Lorentz shortening by the equation
\[
\sqrt{1 + 2\epsilon_{xx}}
  = \sqrt{1 - \frac{u^2}{c^2}}\,
    \sqrt{1 + 2{\epsilon°}_{xx}}.
\Tag{158}
\]
Carrying out now our variation~$\delta\epsilon_{xx}$, neglecting~$\epsilon_{xx}$ in comparison
with larger quantities and noting that except for second order quantities,
\[
\sqrt{1 + 2{\epsilon°}_{xx}} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}
\Tag{159}
\]
we obtain
\[
\delta{\epsilon°}_{xx}
  = \frac{\delta\epsilon_{xx}}{\left(1 - \smfrac{u^2}{c^2}\right)}.
\Tag{160}
\]

Since the variations $\delta\epsilon_{yy}$,~$\delta\epsilon_{zz}$,~$\delta\epsilon_{yz}$ affect only lines which are at
right angles to the direction of motion, we may evidently write
\[
\delta{\epsilon°}_{yy} = \delta\epsilon_{yy}, \qquad
\delta{\epsilon°}_{zz} = \delta\epsilon_{zz}, \qquad
\delta{\epsilon°}_{yz} = \delta\epsilon_{yz}.
\Tag{161}
\]
%% -----File: 150.png---Folio 136-------
To calculate $\delta{\epsilon°}_{xz}$ we may note that in accordance with equations~(150)
we must have
\begin{align*}
\cos \theta_{xz}
  &= \frac{\epsilon_{xz}}
          {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1+2\epsilon_{zz}}},\\
\cos {\theta°}_{xz}
  &= \frac{{\epsilon°}_{xz}}
          {\sqrt{1 + 2{\epsilon°}_{xx}}\, \sqrt{1 + 2{\epsilon°}_{zz}}},
\end{align*}
where $\theta_{xz}$~is the angle between lines which in the unstrained condition
are parallel to the $X$~and~$Z$ axes respectively. In accordance with
the Lorentz shortening, however, we shall have
\[
\cos \theta_{xz} = \sqrt{1 - \frac{u^2}{c^2}} \cos {\theta°}_{xz}.
\]
Introducing this relation, remembering that $\epsilon_{xx} = {\epsilon°}_{zz} = 0$, and
noting equation~(159), we obtain
\begin{align*}
\delta {\epsilon°}_{xz}
  &= \frac{\delta\epsilon_{xz}}{\left(1 - \smfrac{u^2}{x^2}\right)},
\Tag{162} \\
\intertext{and similarly}
\delta {\epsilon°}_{xy}
  &= \frac{\delta\epsilon_{xy}}{\left(1 - \smfrac{u^2}{x^2}\right)}.
\Tag{163}
\end{align*}

We may now combine these equations (160),~(161),~(162) and~(163)
with those for the variation in strain with velocity and obtain
the final set which we desire:
\[
\begin{aligned}
\delta {\epsilon°}_{xx}
  &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\,
     \frac{\dot{x}}{c^2}\, \delta \dot{x}
   + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xx}, \\
\delta {\epsilon°}_{yy} &= \delta\epsilon_{yy}, \\
\delta {\epsilon°}_{zz} &= \delta\epsilon_{zz}, \\
\delta {\epsilon°}_{yz} &= \delta\epsilon_{yz}, \\
\delta {\epsilon°}_{xz}
  &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\,
     \frac{\dot{x}}{c^2}\, \delta \dot{z}
   + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xz}, \\
\delta {\epsilon°}_{xy}
  &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\,
     \frac{\dot{x}}{c^2}\, \delta \dot{y}
   + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xy}.
\end{aligned}
\Tag{164}
\]
%% -----File: 151.png---Folio 137-------

These equations give the variation in the strain measured in
system~$S°$ at a point~$P$ moving in the $X$~direction with velocity~$u$,
provided the strains are negligibly small as measured in~$S$.


\Section[II]{Introduction of the Principle of Least Action.}

\Subsubsection{120}{The Kinetic Potential for an Elastic Body.} We are now in
a position to develop the mechanics of an elastic body with the help
of the principle of least action. In Newtonian mechanics, as is well
known, the kinetic potential for unit volume of material at a given
point~$P$ in an elastic body may be put equal to the density of kinetic
energy minus the density of potential energy, and it is obvious that
our choice for kinetic potential must reduce to that value at low
velocities. Our choice of an expression for kinetic potential is furthermore
limited by the fundamental transformation equation for kinetic
potential which we found in the last chapter
\[
\frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  = \frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}.
\Tag{146}
\]

Taking these requirements into consideration, we may write for
the kinetic potential per unit volume of the material at a point~$P$
moving with the velocity~$\vc{u}$ the expression
\[
H = -E° \sqrt{1 - \frac{u^2}{c^2}},
\]
where $E°$~is the energy as measured in system~$S°$ of the amount of
material which in the unstrained condition (\ie, as measured in
system~$S$) is contained in unit volume.

The above expression obviously satisfies our fundamental transformation
equation~(146) and at low velocities reduces in accordance
with the requirements of Newtonian mechanics to
\[
H = \tfrac{1}{2} m° u^2 - E°,
\]
provided we introduce the substitution made familiar by our previous
work, $m° = \dfrac{E°}{c^2}$.
%% -----File: 152.png---Folio 138-------

\Subsubsection{121}{Lagrange's Equations.} Making use of this expression for the
kinetic potential in an elastic body, we may now obtain the equations
of motion and stress for an elastic body by substituting into Lagrange's
equations~(139) Chapter~IX\@. %[** TN: Not a useful cross-reference.]

Considering the material at the point~$P$ contained in unit volume
in the unstrained condition, we may choose as our generalized coördinates
the six component strains $\epsilon_{xx}$,~$\epsilon_{yy}$,~etc., with the corresponding
stresses $-t_{xx}$,~$-t_{yy}$,~etc., as generalized forces, and the
three coördinates $x$,~$y$,~$z$ which give the position of the point with the
corresponding forces $F_x$,~$F_y$ and~$F_z$.

It is evident that the kinetic potential will be independent of
the time derivatives of the strains, and if we consider cases in which
$E°$~is independent of position, the kinetic potential will also be independent
of the absolute magnitudes of the coördinates $x$,~$y$ and~$z$.
Substituting in Lagrange's equations~(139), we then obtain
\[
\left.
\begin{aligned}
-\frac{\partial}{\partial \epsilon_{xx}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xx}, \\
-\frac{\partial}{\partial \epsilon_{yy}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{yy}, \\
-\frac{\partial}{\partial \epsilon_{zz}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{zz}, \\
-\frac{\partial}{\partial \epsilon_{yz}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{yz}, \\
-\frac{\partial}{\partial \epsilon_{xz}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xz}, \\
-\frac{\partial}{\partial \epsilon_{xy}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xy},
\end{aligned}
\right\}
\Tag{165}
\]
\[
\left.
\begin{aligned}
\frac{d}{dt}\, \frac{\partial}{\partial \dot{x}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_x, \\
\frac{d}{dt}\, \frac{\partial}{\partial \dot{y}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_y, \\
\frac{d}{dt}\, \frac{\partial}{\partial \dot{z}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_z.
\end{aligned}
\right\}
\Tag{166}
\]
%% -----File: 153.png---Folio 139-------

We may simplify these equations, however; by performing the
indicated differentiations and making suitable substitutions, we have
\[
\frac{\partial {E°}_{xx}}{\partial \epsilon_{xx}}
  = \frac{\partial {E°}_{xx}}{\partial {\epsilon°}_{xx}}\,
    \frac{\partial {\epsilon°}_{xx}}{\partial \epsilon_{xx}}.
\]
But in accordance with equation~(152) we may write
\[
\frac{\partial {E°}_{xx}}{\partial {\epsilon°}_{xx}} = -{t°}_{xx}
\]
and from equations~(164) we may put
\[
\frac{\partial {\epsilon°}_{xx}}{\partial \epsilon_{xx}}
  = \frac{1}{1 - \smfrac{u^2}{c^2}}.
\]
Making the substitutions in the first of the Lagrangian equations we
obtain
\[
t_{xx} = -\frac{\partial}{\partial\epsilon_{xx}}
    \left(E° \sqrt{1 - \frac{u^2}{c^2}}\;\right)
  = {t°}_{xx} \frac{1}{1 - \smfrac{u^2}{c^2}} \sqrt{1 - \frac{u^2}{c^2}}
  = \frac{{t°}_{xx}}{\sqrt{1 - \frac{u^2}{c^2}}}.
\]

\Subsubsection{122}{Transformation Equations for Stress.} Similar substitutions
can be made in all the equations of stress, and we obtain as our set
of transformation equations
\[
\begin{aligned}
t_{xx} &= \frac{{t°}_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, &
t_{yy} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{yy}, &
t_{zz} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{zz}, \\
t_{yx} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{yx}, &
t_{xz} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, &
t_{xy} &= \frac{{t°}_{xy}}{\sqrt{1 - \smfrac{u^2}{c^2}}}.
\end{aligned}
\Tag{167}
\]

%[** TN: Bold symbol in original]
\Subsubsection{123}{Value of $E^\circ$.} With the help of these transformation equations
for stress we may calculate the value of~$E°$, the energy content, as
measured in system~$S°$, of material which in the unstrained condition
is contained in unit volume.

Consider unit volume of the material in the unstrained condition
and call its energy content~$w°°$. Give it now the velocity $u = \dot{x}$,
keeping its state of strain unchanged in system~$S$. Since the \emph{strain}
%% -----File: 154.png---Folio 140-------
is not changing in system~$S$, the stresses $t_{xx}$,~etc., will also be constant
in system~$S$. In system~$S°$, however, the component strain will
change in accordance with equations~(156) from zero to
\[
{\epsilon°}_{xx}
  = \frac{\dot{x}^2}{2c^2}\, \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)},
\]
and the corresponding stress will be given at any instant by the
expression just derived,
\[
{t°}_{xx} = t_{xx} \sqrt{1 - \frac{u^2}{c^2}},
\]
$t_{xx}$ being, as we have just seen, a constant. We may then write for~$E°$
the expression
\[
E° = w°° - t_{xx} \int_0^w \sqrt{1 - \frac{u^2}{c^2}}\,
  d\Biggl[\frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\,
  \frac{\dot{x}}{2c^2}\Biggr].
\]
Noting that $u = \dot{x}$ we obtain on integration,
\[
E° = w°° + t_{xx} - \frac{t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}
\Tag{168}
\]
as the desired expression for the energy as measured in system~$S°$
contained in the material which in system~$S$ is unstrained and has
unit volume.

\Subsubsection{124}{The Equations of Motion in the Lagrangian Form.} We are
now in a position to simplify the three Lagrangian equations~(166)
for $F_x$,~$F_y$ and~$F_z$. Carrying out the indicated differentiation we have
\[
F_x = \frac{d}{dt}\, \frac{\partial}{\partial \dot{x}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right)
  = \frac{d}{dt} \Biggl[
      \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
      \frac{\dot{x}}{c^2}
    - \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial\dot{x}}
  \Biggr],
\]
and introducing the value of~$E°$ given by equation~(168) we obtain
\[
F_x = \frac{d}{dt} \Biggl[
  \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\dot{x}}{c^2}\Biggr].
\Tag{169}
\]
%% -----File: 155.png---Folio 141-------
Simple calculations will also give us values for $F_y$~and~$F_z$. We have
from~(166)
\[
F_y = \frac{d}{dt}\, \frac{\partial}{\partial \dot{y}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right)
  = \frac{d}{dt} \Biggl[
      \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
      \frac{\dot{y}}{c^2}
    - \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial\dot{y}}
  \Biggr].
\]
But since we have adapted our considerations to cases in which the
direction of motion is along the $X$\DPchg{-}{~}axis, we have $\dot{y} = 0$; furthermore
we may substitute, in accordance with equations (152),~(157) and~(167),
\[
\frac{\partial E°}{\partial \dot{y}}
  = \frac{\partial E°}{\partial {\epsilon°}_{xy}}\,
    \frac{\partial {\epsilon°}_{xy}}{\partial \dot{y}}
  = -{t°}_{xy}\, \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\,
    \frac{\dot{x}}{c^2}
  = \frac{-t_{xy}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}.
\]
We thus obtain as our three equations of motion
\[
\begin{aligned}
F_x &= \frac{d}{dt}\Biggl[
  \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\dot{x}}{c^2}
  \Biggr], \\
F_y &= \frac{d}{dt} \left(t_{xy}\, \frac{\dot{x}}{c^2}\right), \\
F_z &= \frac{d}{dt} \left(t_{xz}\, \frac{\dot{x}}{c^2}\right).
\end{aligned}
\Tag{170}
\]
In these equations the quantities $F_x$,~$F_y$ and~$F_z$ are the components
of force acting on a particular system, namely that quantity of material
which at the instant in question has unit volume. Since the volume
of this material will in general be changing, $F_x$,~$F_y$ and~$F_z$ do not give
us the force per unit volume as usually defined. If we represent,
however, by $f_x$,~$f_y$ and~$f_z$  the components of force per unit volume,
we may rewrite these equations in the form
\[
\begin{aligned}
F_x\, \delta V &= \frac{d}{dt}\Biggl[
  \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\dot{x}}{c^2}\, \delta V\Biggr], \\
F_y\, \delta V &= \frac{d}{dt}\left(t_{xy}\, \frac{\dot{x}}{c^2}\, \delta V\right),\\
F_z\, \delta V &= \frac{d}{dt}\left(t_{xz}\, \frac{\dot{x}}{c^2}\, \delta V\right),
\end{aligned}
 \Tag{171}
\]
%% -----File: 156.png---Folio 142-------
where by $\delta V$ we mean a small element of volume at the point in
question.

\Subsubsection{125}{Density of Momentum.} Since we customarily define force as
equal to the time rate of change of momentum, we may now write for
the density of momentum~$\vc{g}$ at a point in an elastic body which is
moving in the $X$~direction with the velocity $u = \dot{x}$
\[
\vc{g}_x = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\dot{x}}{c^2}, \qquad
\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}, \qquad
\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}.
\Tag{172}
\]

It is interesting to point out that there are components of momentum
in the $Y$~and~$Z$ directions in spite of the fact that the material
at the point in question is moving in the $X$~direction. We shall
later see the important significance of this discovery.

\Subsubsection{126}{Density of Energy.} It will be remembered that the forces
whose equations we have just obtained are those acting on unit
volume of the material as measured in system~$S$, and hence we are
now in a position to calculate the energy density of our material.
Let us start out with unit volume of our material at rest, with the
energy content~$w°°$ and determine the work necessary to give it the
velocity $u = \dot{x}$ without change in stress or strain. Since the only
component of force which suffers displacement is~$F_x$, we have
\[
\begin{aligned}
%[** TN: Commas present in original, arguably serve a grammatical purpose]
w &= w°° + \int_0^u \frac{d}{dt} \Biggl[
  \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\dot{x}}{c^2}\Biggr] \dot{x}\, dt, \\
  &= w°° + (w°° + t_{xx})
  \int_0^u \dot{x}\, d\Biggl[
  \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}\Biggr], \\
  &= \Biggl\{\frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}
    - t_{xx}\Biggr\}
\end{aligned}
\Tag{173}
\]
as an expression for the energy density of the elastic material.

\Subsubsection{127}{Summary of Results Obtained from the Principle of Least
Action.} We may now tabulate for future reference the results obtained
from the principle of least action.
%% -----File: 157.png---Folio 143-------

At a given point in an elastic medium which is moving in the $X$~direction
with the velocity $u = \dot{x}$, we have for the components of
stress
\[
\begin{aligned}
t_{xx} &= \frac{{t°}_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, &
t_{yy} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{yy}, &
t_{zz} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{zz}, \\
t_{yz} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{yz}, &
t_{xz} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, &
t_{xy} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\end{aligned}
\Tag{167}
\]
For the density of energy at the point in question we have
\[
w = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} - t_{xx}.
\Tag{173}
\]
For the density of momentum we have
\[
\vc{g}_{x}
  = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
    \frac{\dot{x}}{c^2}, \qquad
\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}, \qquad
\vc{g}_z = t_{xz}\, \frac{\dot{x}}{c^2}.
\Tag{172}
\]


\Section[III]{Some Mathematical Relations.}

Before proceeding to the applications of these results which we
have obtained from the principle of least action, we shall find it desirable
to present a number of mathematical relations which will
later prove useful.

\Subsubsection{128}{The Unsymmetrical Stress Tensor $\vc{t}$.} We have defined the
components of stress acting at a point by equation~(152)
\[
\delta W
  = t_{xx}\, \delta\epsilon_{xx}
  + t_{yy}\, \delta\epsilon_{yy}
  + t_{zz}\, \delta\epsilon_{zz}
  + t_{yz}\, \delta\epsilon_{yz}
  + t_{xz}\, \delta\epsilon_{xz}
  + t_{xy}\, \delta\epsilon_{xy},
\]
where $\delta W$~is the work which accompanies a change in strain and is
performed on the surroundings by the amount of material which was
contained in unit volume in the unstrained state. Since for convenience
we have taken as our state of zero strain the condition of
the body as measured in system~$S$, it is evident that the components
$t_{xx}$,~$t_{yy}$,~etc., may be taken as the forces acting on the faces of a unit
cube of material at the point in question, the first letter of the subscript
%% -----File: 158.png---Folio 144-------
indicating the direction of the force and the second subscript
the direction of the normal to the face in question.

Interpreting the components of stress in this fashion, we may
now add three further components and obtain a complete tensor
\[
\vc{t} = \left\{
\begin{matrix}
t_{xx} & t_{xy} & t_{xz} \\
t_{yx} & t_{yy} & t_{yz} \\
t_{zx} & t_{zy} & t_{zz}
\end{matrix}
\right.
\Tag{174}
\]

The three new components $t_{yx}$,~$t_{zx}$,~$t_{zy}$ are forces acting on the
unit cube, in the directions and on the faces indicated by the subscripts.
A knowledge of their value was not necessary for our developments
of the consequences of the principle of least action, since it was
possible to obtain an expression for the work accompanying a change
in strain without their introduction. We shall find them quite important
for our later considerations, however, and may proceed to
determine their value.

$t_{yz}$ is the force acting in the $Y$~direction tangentially to a face of
the cube perpendicular to the $X$\DPchg{-}{~}axis, and measured with a system
of coördinates~$S$. Using a system of \DPchg{coordinates}{coördinates}~$S°$ which is stationary
with respect to the point in question, we should obtain, for the measurement
of this force,
\[
{t°}_{yx} = \frac{t_{yx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}
\]
in accordance with our transformation equation for force~(62), Chapter~VI\@. %[** TN: Not a useful cross-reference]
Similarly we shall have the relation
\[
{t°}_{xy} = t_{xy}.
\]
In accordance with the elementary theory of elasticity, however, the
forces ${t°}_{yx}$~and~${t°}_{xy}$ which are measured by an observer moving with
the body will be connected by the relation
\[
{t°}_{xy} = \frac{{t°}_{yx}}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\]
${t°}_{xy}$ being larger than~${t°}_{yx}$ in the ratio of the areas of face upon which
they act. Combining these three equations, and using similar methods
%% -----File: 159.png---Folio 145-------
for the other quantities, we can obtain the desired relations
\[
t_{yx} = \left(1 - \frac{u^2}{c^2}\right) t_{xy}, \qquad
t_{zx} = \left(1 - \frac{u^2}{c^2}\right) t_{xz}, \qquad
t_{zy} = t_{yz}.
\Tag{175}
\]
We see that $\vc{t}$~is an unsymmetrical tensor.

\Subsubsection{129}{The Symmetrical Tensor $\vc{p}$.} Besides this unsymmetrical tensor~$\vc{t}$
we shall find it desirable to define a further tensor~$\vc{p}$ by the
equation
\[
\vc{p} = \vc{t} + \vc{gu}.
\Tag{176}
\]

We shall call $\vc{gu}$ the tensor product of $\vc{g}$~and~$\vc{u}$ and may indicate
tensor products in general by a simple juxtaposition of vectors. $\vc{gu}$~is
itself a tensor with components as indicated below:
\[
\vc{gu} = \left\{
\begin{matrix}
g_xu_x & g_xu_y & g_xu_z, \\
g_yu_x & g_yu_y & g_yu_z, \\
g_zu_x & g_zu_y & g_zu_z.
\end{matrix}
\right.
\Tag{177}
\]

Unlike $\vc{t}$, $\vc{p}$~will be a symmetrical tensor, since we may show, by
substitution of the values for $\vc{g}$~and~$\vc{u}$ already obtained, that
\[
p_{yx} = p_{xy}, \qquad
p_{zx} = p_{xz}, \qquad
p_{zy} = p_{yz}.
\Tag{178}
\]
Consider for example the value of~$p_{yx}$; we have from our definition
\[
p_{yx} = t_{yx} + g_{y}u_{x},
\]
and by equations (175)~and~(172) we have
\[
t_{xy} = \left(1 - \frac{u^2}{c^2}\right) t_{xy},\qquad
g_y = t_{xy}\, \frac{u_x}{c^2},
\]
and hence by substitution obtain
\[
p_{yx} = t_{xy}.
\]
We also have, however, by definition
\[
p_{xy} = t_{xy} + g_xu_y,
\]
and since for the case we are considering $u_y = 0$, we arrive at the
equality
\[
p_{xy} = p_{yx}.
\]
The other equalities may be shown in a similar way.
%% -----File: 160.png---Folio 146-------

\Subsubsection{130}{Relation between $\divg\vc{t}$~and~$\vc{t}_n$.} At a given point~$P$ in our
elastic body we shall define the divergence of the tensor~$\vc{t}$ by the equation
\[
\begin{aligned}
\divg\vc{t}
  &= \left(\frac{\partial t_{xx}}{\partial x}
         + \frac{\partial t_{xy}}{\partial y}
         + \frac{\partial t_{xz}}{\partial z}\right) \vc{i} \\
  &+ \left(\frac{\partial t_{yx}}{\partial x}
         + \frac{\partial t_{yy}}{\partial y}
         + \frac{\partial t_{yz}}{\partial z}\right) \vc{j} \\
  &+ \left(\frac{\partial t_{zx}}{\partial x}
         + \frac{\partial t_{zy}}{\partial y}
         + \frac{\partial t_{zz}}{\partial z}\right) \vc{k},
\end{aligned}
 \Tag{179}
\]
where $\vc{i}$,~$\vc{j}$ and~$\vc{k}$ are unit vectors parallel to the axes, $\divg\vc{t}$~thus being
an ordinary vector. It will be seen that $\divg\vc{t}$~is the elastic force
acting per unit volume of material at the point~$P$.

Considering an element of surface~$dS$, we shall define a further
vector~$\vc{t}_n$ by the equation
\[
\begin{aligned}
\vc{t}_n
  &= (t_{xx}\cos\alpha + t_{xy}\cos\beta + t_{xz}\cos\gamma)\, \vc{i} \\
  &+ (t_{yx}\cos\alpha + t_{yy}\cos\beta + t_{yz}\cos\gamma)\, \vc{j} \\
  &+ (t_{zx}\cos\alpha + t_{zy}\cos\beta + t_{zz}\cos\gamma)\, \vc{k},
\end{aligned}
\Tag{180}
\]
where $\cos \alpha$,~$\cos \beta$ and~$\cos \gamma$ are the direction cosines of the inward-pointing
normal to the element of surface~$dS$.

Considering now a definite volume~$V$ enclosed by the surface~$S$
it is evident that $\divg\vc{t}$~and~$\vc{t}_n$ will be connected by the relation
\[
-\int \divg\vc{t}\, dV = \int_0 \vc{t}_n\, dS,
\Tag{181}
\]
where the symbol~$0$ indicates that the integration is to be taken over
the whole surface which encloses the volume~$V$. This equation is
of course merely a direct application of Gauss's formula, which states
in general the equality
{\small%[** TN: Not breaking]
\[
-\int \left(
    \frac{\partial P}{\partial x}
  + \frac{\partial Q}{\partial y}
  + \frac{\partial R}{\partial z}\right) dV
  = \int_0 (P\cos \alpha + Q\cos \beta + R\cos \gamma)\, dS,
\Tag{182}
\]}%
where $P$,~$Q$ and~$R$ may be any functions of $x$,~$y$ and~$z$.
%% -----File: 161.png---Folio 147-------

We shall also find use for a further relation between $\divg\vc{t}$~and~$\vc{t}_n$.
Consider a given point of reference~$O$, and let $\vc{r}$~be the radius vector
to any point~$P$ in the elastic body; we can then show with the help
of Gauss's Formula~(182) that
\begin{multline*}
-\int (\vc{r} × \divg\vc{t})\, dV = \int_0 (\vc{r} × \vc{t}_n)\, dS \\
-\int \bigl[(t_{yz}-t_{zy})\vc{jk}
          + (t_{xz}-t_{zx})\vc{ik}
          + (t_{xy}-t_{yx})\vc{ij}\bigr]\, dV,
\end{multline*}
where $×$~signifies as usual the outer product. Taking account of
equations (172)~and~(175) this can be rewritten
\[
-\int (\vc{r} × \divg\vc{t})\, dV
  = \int_0 (\vc{r} × \vc{t}_n)\, dS - \int(\vc{u} × \vc{g})\, dV.
\Tag{183}
\]

\Subsubsection{131}{The Equations of Motion in the Eulerian Form.} We saw in
\DPchg{sections (\Secnumref{124})~and~(\Secnumref{125})}{Sections \Secnumref{124}~and~\Secnumref{125}} that the equations of motion in the Lagrangian
form might be written
\[
\vc{f}\, \delta V = \frac{d}{dt} (\vc{g}\, \delta V),
\]
where $\vc{f}$~is the density of force acting at any point and $\vc{g}$~is the density
of momentum.

Provided that there are no external forces acting and $\vc{f}$~is produced
solely by the elastic forces, our definition of the divergence of a
tensor will now permit us to put
\[
\vc{f} = - \divg\vc{t},
\]
and write for our equation of motion
\[
(-\divg\vc{t})\, \delta V
  = \frac{d}{dt} (\vc{g}\, \delta V)
  = \delta V\, \frac{d\vc{g}}{dt} + \vc{g}\, \frac{d(\delta V)}{dt}.
\]
Expressing $\dfrac{d\vc{g}}{dt}$ in terms of partial differentials, and putting
\[
\frac{d(\delta V)}{dt} = \delta V \divg\vc{u}
\]
we obtain
\[
-\divg \vc{t} = \left(\frac{\partial \vc{g}}{\partial t}
  + u_x\, \frac{\partial \vc{g}}{\partial x}
  + u_y\, \frac{\partial \vc{g}}{\partial y}
  + u_z\, \frac{\partial \vc{g}}{\partial z}\right)
  + \vc{g} \divg\vc{u}.
\]
%% -----File: 162.png---Folio 148-------
Our symmetrical tensor~$\vc{p}$, however, was defined by the equation~(176)
\[
\vc{p} = \vc{t} + \vc{gu},
\]
and hence we may now write our equations of motion in the very
beautiful Eulerian form
\[
-\divg\vc{p} = \frac{\partial \vc{g}}{\partial t}.
\Tag{184}
\]

We shall find this simple form for the equations of motion very
interesting in connection with our considerations in the last chapter.


\Section[IV]{Applications of the Results.}

We may now use the results which we have obtained from the
principle of least action to elucidate various problems concerning
the behavior of elastic bodies.

\Subsubsection{132}{Relation between Energy and Momentum.} In our work on
the dynamics of a particle we found that the mass of a particle was
equal to its energy divided by the square of the velocity of light, and
hence have come to expect in general a necessary relation between
the existence of momentum in any particular direction and the transfer
of energy in that same direction. We find, however, in the case
of elastically stressed bodies a somewhat more complicated state of
affairs than in the case of particles, since besides the energy which is
transported bodily by the motion of the medium an additional quantity
of energy may be transferred through the medium by the action
of the forces which hold it in its state of strain. Thus, for example,
in the case of a longitudinally compressed rod moving parallel to its
length, the forces holding it in its state of longitudinal compression
will be doing work at the rear end of the rod and delivering an equal
quantity of energy at the front end, and this additional transfer of
energy must be included in the calculation of the momentum of the
bar.

As a matter of fact, an examination of the expressions for momentum
which we obtained from the principle of least action will show
the justice of these considerations. For the density of momentum
in the $X$~direction we obtained the expression
\[
g_x = (w + t_{xx})\, \frac{\dot{x}}{c^2},
\]
%% -----File: 163.png---Folio 149-------
and we see that in order to calculate the momentum in the $X$~direction
we must consider not merely the energy~$w$ which is being bodily
carried along in that direction with the velocity~$\dot{x}$, but also must take
into account the additional flow of energy which arises from the
stress~$t_{xx}$. As we have already seen in \Secref{128}, this stress~$t_{xx}$ can
be thought of as resulting from forces which act on the front and
rear faces of a centimeter cube of our material. Since the cube is
moving with the velocity~$\dot{x}$, the force on the rear face will do the
work $t_{xx}\dot{x}$~per second and this will be given up at the forward face.
We thus have an additional density of energy-flow in the $X$~direction
of the magnitude~$t_{xx}\dot{x}$ and hence a corresponding density of momentum~$\dfrac{t_{xx}\dot{x}}{c^2}$.

Similar considerations explain the interesting occurrence of components
of momentum in the $Y$~and~$Z$ directions,
\[
g_y = t_{xy}\, \frac{\dot{x}}{c^2},\qquad
g_z = t_{xz}\, \frac{\dot{x}}{c^2},
\]
in spite of the fact that the material involved is moving in the $X$~direction.
The stress~$t_{xy}$, for example, can be thought of as resulting
from forces which act tangentially in the $X$~direction on the pair of
faces of our unit cube which are perpendicular to the $Y$~axis. Since
the cube is moving in the $X$~direction with the velocity~$\dot{x}$, we shall
have the work~$t_{xy}\dot{x}$, done at one surface per second and transferred to
the other, and the resulting flow of energy in the $X$~direction is accompanied
by the corresponding momentum~$\dfrac{t_{xy}\dot{x}}{c^2}$.

\Subsubsection{133}{The Conservation of Momentum.} It is evident from our
previous discussions that we may write the equation of motion for
an elastic medium in the form
\[
\vc{f}\, \delta V = \frac{d(\vc{g}\, \delta V)}{dt},
\]
where $\vc{g}$~is the density of momentum at any given point and $\vc{f}$~is the
force acting per unit volume of material. We have already obtained,
from the principle of least action, expressions~(172) which permit
the calculation of~$\vc{g}$ in terms of the energy density, stress and velocity
at the point in question, and our present problem is to discuss somewhat
further the nature of the force~$\vc{f}$.
%% -----File: 164.png---Folio 150-------

We shall find it convenient to analyze the total force per unit
volume of material~$\vc{f}$ into those external forces~$\vc{f}_{\text{ext.}}$ like gravity, which
are produced by agencies outside of the elastic body and the internal
force~$\vc{f}_{\text{int.}}$ which arises from the elastic interaction of the parts of the
strained body itself. It is evident from the way in which we have
defined the divergence of a tensor~(179) that for this latter we may
write
\[
\vc{f}_{\text{int.}} = -\divg\vc{t}.
\Tag{185}
\]
Our equation of motion then becomes
\[
(\vc{f}_{\text{ext.}} - \divg\vc{t})\, \delta V
  = \frac{d(\vc{g}\, \delta V)}{dt},
\Tag{186}
\]
or, integrating over the total volume of the elastic body,
\[
\int \vc{f}_{\text{ext.}}\, dV - \int \divg\vc{t}\, dV
  = \frac{d}{dt} \int \vc{g}\, dV
  = \frac{d\vc{G}}{dt},
\Tag{187}
\]
where $\vc{G}$ is the total momentum of the body. With the help of the
purely analytical relation~(181) we may transform the above equation
into
\[
\int \vc{f}_{\text{ext.}}\, dV + \int \vc{t}_n\, dS = \frac{d\vc{G}}{dt},
\Tag{188}
\]
where $\vc{t}_n$~is defined in accordance with~(180) so that the integral
$\ds\int_{0} \vc{t}_n\, dS$ becomes the force exerted by the surroundings on the surface
of the elastic body.

In the case of an isolated system both $\vc{f}_{\text{ext.}}$~and~$\vc{t}_n$ would evidently
be equal to zero and we have the principle of the conservation of
momentum.

\Subsubsection{134}{The Conservation of Angular Momentum.} Consider the
%[** TN: O and P in next line are boldface in the original]
radius vector~$\vc{r}$ from a point of reference~$O$ to any point~$P$ in an elastic
body; then the angular momentum of the body about~$O$ will be
\[
\vc{M} = \int (\vc{r} × \vc{g})\, dV,
\]
and its rate of change will be
\[
\frac{d\vc{M}}{dt}
  = \int \left(\vc{r} × \frac{d\vc{g}}{dt}\right) dV
  + \int \left(\frac{d\vc{r}}{dt} × \vc{g}\right) dV.
\Tag{189}
\]
%% -----File: 165.png---Folio 151-------
Substituting equation~(186), this may be written
\[
\frac{d\vc{M}}{dt}
  = \int (\vc{r} × \vc{f}_{\text{ext.}})\, dV
  - \int (\vc{r} × \divg\vc{t})\, dV + \int (\vc{u} × \vc{g})\, dV,
\]
or, introducing the purely mathematical relation~(183) we have,
\[
\frac{d\vc{M}}{dt}
  = \int (\vc{r} × \vc{f}_{\text{ext.}})\, dV
  + \int_{0} (\vc{r} × \vc{t}_n)\, dS.
\Tag{190}
\]
We see from this equation that the rate of change of the angular
momentum of an elastic body is equal to the moment of the external
forces acting on the body plus the moment of the surface forces.

In the case of an isolated system this reduces to the important
principle of the conservation of angular momentum.

\Subsubsection{135}{Relation between Angular Momentum and the Unsymmetrical
Stress Tensor.} The fact that at a point in a strained elastic medium
there may be components of momentum at right angles to the motion
of the point itself, leads to the interesting conclusion that even in a
state of steady motion the angular momentum of a strained body
will in general be changing.

This is evident from equation~(189), in the preceding section,
which may be written
\[
\frac{d\vc{M}}{dt}
  = \int \left(\vc{r} × \frac{d\vc{g}}{dt}\right) dV
  + \int(\vc{u} × \vc{g})\, dV.
\Tag{191}
\]
In the older mechanics velocity~$\vc{u}$ and momentum~$\vc{g}$ were always in
the same direction so that the last term of this equation became zero.
%[** TN: Awkward grammar/repeated verb in the original.]
In our newer mechanics, however, we have found~(172) components
of momentum at right angles to the velocity and \emph{hence even for a body
moving in a straight line with unchanging stresses and velocity we find
that the angular momentum is increasing at the rate
\[
\frac{d\vc{M}}{dt} = \int (\vc{u} × \vc{g})\, dV,
\Tag{192}
\]
and in order to maintain the body in its state of uniform motion we
must apply external forces with a turning moment of this same amount}.

The presence of this increasing angular momentum in a strained
body arises from the unsymmetrical nature of the stress tensor, the integral
$\int (\vc{u} × \vc{g})\, dV$ being as a matter of fact exactly equal to the integral
%% -----File: 166.png---Folio 152-------
over the same volume of the turning moments of the unsymmetrical
components of the stress. Thus, for example, if we have a body moving
in the $X$~direction with the velocity $\vc{u} = \dot{x}\vc{i}$ we can easily see from
equations (172)~and~(175) the truth of the equality
\[
(\vc{u} × \vc{g})
  = \bigl[(t_{yz} - t_{zy})\, \vc{jk}
        + (t_{xz} - t_{zx})\, \vc{ik}
        + (t_{xy} - t_{yx})\, \vc{ij}\bigr].
\]

\Subsubsection{136}{The Right-Angled Lever.} An interesting example of the
\begin{wrapfigure}{l}{2.125in}
  \Fig{14}
  \Input[2in]{166}
\end{wrapfigure}
principle that in general a turning
moment is needed for the uniform
translatory motion of a strained body
is seen in the apparently paradoxical
case of the right-angled lever.

Consider the right-angled lever
shown in \Figref{14}. This lever is stationary
with respect to a system of
coördinates~$S°$. Referred to~$S°$ the
two lever arms are equal in length:
\[
{l_1}° = {l_2}°,
\]
and the lever is in equilibrium under the action of the equal forces
\[
{F_1}° = {F_2}°.
\]

Let us now consider the equilibrium as it appears, using a system
of coördinates~$S$ with reference to which the lever is moving in $X$~direction
with the velocity~$V$. Referred to this new system of coördinates
the length~$l_1$ of the arm which lies in the $Y$~direction will be
the same as in system~$S°$, giving us
\[
l_1 = {l_1}°.
\]
But for the other arm which lies in the direction of motion we shall
have, in accordance with the Lorentz shortening,
\[
l_2 = {l_2}° \sqrt{1 - \frac{V^2}{c^2}}.
\]
For the forces $F_1$~and~$F_2$ we shall have, in accordance with our equations
%% -----File: 167.png---Folio 153-------
for the transformation of force (61)~and~(62),
\begin{align*}
F_1 &= {F_1}°, \\
F_2 &= {F_2}° \sqrt{1 - \frac{V^2}{c^2}}.
\end{align*}
We thus obtain for the moment of the forces around the pivot~$B$
\[
F_1l_1 - F_2l_2
  = {F_1}° {l_1}°
  - {F_2}° {l_2}° \left(1 - \frac{V^2}{c^2}\right)
  = {F_1}°{l_1}°\, \frac{V^2}{c^2},
  = F_1l_1\, \frac{V^2}{c^2},
\]
and are led to the remarkable conclusion that such a moving lever
will be in equilibrium only if the external forces have a definite turning
moment of the magnitude given above.

The explanation of this apparent paradox is obvious, however,
in the light of our previous discussion. In spite of the fact that the
lever is in uniform motion in a straight line, its angular momentum
is continually increasing owing to the fact that it is elastically strained,
and it can be shown by carrying out the integration indicated in
equation~(192) that the rate of change of angular momentum is as a
matter of fact just equal to the turning moment $F_1l_1\, \dfrac{V^2}{c^2}$.

This necessity for a turning moment $F_1l_1\, \dfrac{V^2}{c^2}$ can also be shown
directly from a consideration of the energy flow in the lever. Since
the force~$F_1$ is doing the work $F_1V$~per second at the point~$A$, a stream
of energy of this amount is continually flowing through the lever
from~$A$ to the pivot~$B$. In accordance with our ideas as to the relation
between energy and mass, this new energy which enters at~$A$ each
second has the mass~$\dfrac{F_1V}{c^2}$, and hence each second the angular momentum
of the system around the point~$B$ is increased by the amount
\[
\frac{F_1V}{c^2}\, Vl_1 = F_1l_1\, \frac{V^2}{c^2}.
\]
We have already found, however, exactly this same expression for
the moment of the forces around the pivot~$B$ and hence see that they
are of just the magnitude necessary to keep the lever from turning,
thus solving completely our apparent paradox.
%% -----File: 168.png---Folio 154-------

\Subsubsection{137}{Isolated Systems in a Steady State.} Our considerations have
shown that the density of momentum is equal to the density of energy
flow divided by the square of the velocity of light. If we have a
system which is in a steady internal state, and is either isolated or
merely subjected to an external pressure with no components of force
tangential to the bounding surface, it is evident that the resultant
flow of energy for the whole body must be in the direction of motion,
and hence for these systems momentum and velocity will be in the
same direction without the complications introduced by a transverse
energy flow.

Thus for an \emph{isolated} system in a steady \emph{internal} state we may
write,
\[
\vc{G} = \frac{E}{c^2}\, \vc{u}
  = \frac{\smfrac{E°}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u}.
\Tag{193}
\]

\Subsubsection{138}{The Dynamics of a Particle.} It is important to note that
particles are interesting examples of systems in which there will
obviously be no transverse component of energy flow since their
infinitesimal size precludes the action of tangential surface forces.
We thus see that the dynamics of a particle may be regarded as a
special case of the more general dynamics which we have developed
in this chapter, the equation of motion for a particle being
\[
\vc{F} = \frac{d}{dt} \left[
  \frac{\smfrac{E°}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,\vc{u}
  \right]
  = \frac{d}{dt} \Biggl[
  \frac{m°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\vc{u}\Biggr],
\]
in agreement with the work of \Chapref{VI}.

\Subsubsection{139}{Conclusion.} We may now point out in conclusion the chief
results of this chapter. With the help of Einstein's equations for
spatial and temporal considerations, we have developed a set of
transformation equations for the strain in an elastic body. Using the
components of strain and velocity as generalized coördinates, we then
introduced the principle of least action, choosing a form of function
%% -----File: 169.png---Folio 155-------
for kinetic potential which agrees at low velocities with the choice
made in the older theories of elasticity and at all velocities agrees
with the requirements of the principle of relativity. Using the
Lagrangian equations, we were then able to develop all that is necessary
for a complete theory of elasticity.

The most important consequence of these considerations is an
extension in our ideas as to the relation between momentum and
energy. We find that the density of momentum in any direction
must be placed equal to the total density of energy flow in that same
direction divided by the square of the velocity of light; and we find
that we must include in our density of energy flow that transferred
through the elastic body by the forces which hold it in its state of
strain and suffer displacement as the body moves. This involves in
general a flow of energy and hence momentum at right angles to the
motion of the body itself.

At present we have no experiments of sufficient accuracy so that
we can investigate the differences between this new theory of elasticity
and the older ones, and hence of course have found no experimental
contradiction to the new theory. It will be seen, however, from the
expressions for momentum that even at low velocities the consequences
of this new theory will become important as soon as we
run across elastic systems in which very large stresses are involved.
It is also important to show that a theory of elasticity can be developed
which agrees with the requirements of the theory of relativity.
In fairness, it must, however, be pointed out in conclusion that since
our expression for kinetic potential was not absolutely uniquely determined
there may also be other theories of elasticity which will agree
with the principle of relativity and with all the facts as now known.
%% -----File: 170.png---Folio 156-------


\Chapter{XI}{The Dynamics of a Thermodynamic System.}
\SetRunningHeads{Chapter Eleven.}{Dynamics of a Thermodynamic System.}

We may now use our conclusions as to the relation between the
principle of least action and the theory of relativity to obtain information
as to the behavior of thermodynamic systems in motion.

\Subsubsection{140}{The Generalized Coördinates and Forces.} Let us consider a
thermodynamic system whose state is defined by the \emph{generalized
coördinates} volume~$v$, entropy~$S$ and the values of $x$,~$y$ and~$z$ which
determine its position. Corresponding to these coördinates we shall
have the generalized external forces, the negative of the pressure,~$-p$,
temperature,~$T$, and the components of force, $F_x$,~$F_y$ and~$F_z$.
These generalized coördinates and forces are related to the energy
change~$\delta E$ accompanying a small displacement~$\delta$, in accordance with
the equation
\[
\delta E = -\delta W
  = -p\, \delta v + T\, \delta S
  + F_x\, \delta x + F_y\, \delta y + F_z\, \delta z.
\Tag{194}
\]

\Subsubsection{141}{Transformation Equation for Volume.} Before we can apply
the principle of least action we shall need to have transformation
equations for the generalized coördinates, volume and entropy.

In accordance with the Lorentz shortening, we may write the
following expression for the volume~$v$ of the system in terms of~$v°$ as
measured with a set of axes~$S°$ with respect to which the system is
stationary:
\[
v = v° \sqrt{1 - \frac{u^2}{c^2}}
  = v° \sqrt{1 - \frac{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}{c^2}},
\]
where $u$ is the velocity of the system.

By differentiation we may obtain expressions which we shall find
useful,
\begin{align*}
\frac{\partial v°}{\partial v}
  &= \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\Tag{195}\displaybreak[0] \\
\frac{\partial v°}{\partial \dot{x}}
  &= \frac{v}{\left(1 - \smfrac{u^2}{c^2}\right)^{\frac{3}{2}}}\,
     \frac{\dot{x}}{c^2}
   = \frac{v°}{\left(1 - \smfrac{u^2}{c^2}\right)}\,
     \frac{\dot{x}}{c^2}.
\Tag{196}
\end{align*}
%% -----File: 171.png---Folio 157-------

\Subsubsection{142}{Transformation Equation for Entropy.} As for the entropy
of a thermodynamic system, this is a quantity which must appear
the same to all observers regardless of their motion. This invariance
of entropy is a direct consequence of the close relation between the
entropy of a system in a given state and the probability of that state.
Let us write, in accordance with the Boltzmann-Planck ideas as to
the interdependence of these quantities,
\[
S = k\log W,
\]
where $S$ is the entropy of the system in the state in question, $k$~is a
universal constant, and $W$~the probability of having a microscopic
arrangement of molecules or other elementary constituent parts which
corresponds to the desired thermodynamic state. Since this probability
is evidently independent of the relative motion of the observer
and the system we see that the entropy of a system~$S$ must be an
invariant and may write
\[
S = S°.
\Tag{197}
\]

\Subsubsection{143}{Introduction of the Principle of Least Action. The Kinetic
Potential.} We are now in a position to introduce the principle of
least action into our considerations by choosing a form of function
for the kinetic potential which will agree at low velocities with the
familiar principles of thermodynamics and will agree at all velocities
with the requirements of the theory of relativity.

If we use volume and entropy as our generalized coördinates, these
conditions are met by taking for kinetic potential the expression
\[
H = -E° \sqrt{1 - \frac{u^2}{c^2}}.
\Tag{198}
\]

This expression agrees with the requirements of the theory of
relativity that $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ shall be an invariant (see \Secref{111}) and
at low velocities reduces to $H = -E$, which with our choice of
coördinates is the familiar form for the kinetic potential of a thermodynamic
system.
%% -----File: 172.png---Folio 158-------

It should be noted that this expression for the kinetic potential
of a thermodynamic system applies of course only provided we pick
out volume~$v$ and entropy~$S$ as generalized coördinates. If, following
Helmholtz, we should think it more rational to take $v$ as one coördinate
and a quantity~$\theta$ whose time derivative is equal to temperature,
$\dot{\theta} = T$, as the other coördinate, we should obtain of course a different
expression for the kinetic potential; in fact should have under those
circumstances
\[
H = (E° - T° S°) \sqrt{1 - \frac{u^2}{c^2}}.
\]
Using this value of kinetic potential, however, with the corresponding
coördinates we should obtain results exactly the same as those which
we are now going to work out with the help of the other set of coördinates.

\Subsubsection{144}{The Lagrangian Equations.} Having chosen a form for the
kinetic potential we may now substitute into the Lagrangian equations~(139)
and obtain the desired information with regard to the
behavior of thermodynamic systems.

Since we shall consider cases in which the energy of the system is
independent of the position in space, the kinetic potential will be
independent of the coördinates $x$,~$y$ and~$z$, depending only on their
time derivatives. Noting also that the kinetic potential is independent
of the time derivatives of volume and entropy, we shall
obtain the Lagrangian equations in the simple form
\[
\begin{aligned}
-\frac{\partial}{\partial v}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -p, \\
-\frac{\partial}{\partial S}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= T, \\
\frac{d}{dt}\, \frac{\partial}{\partial\dot{x}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_x, \\
\frac{d}{dt}\, \frac{\partial}{\partial\dot{y}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_y, \\
\frac{d}{dt}\, \frac{\partial}{\partial\dot{z}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_z.
\end{aligned}
\Tag{199}
\]
%% -----File: 173.png---Folio 159-------

\Subsubsection{145}{Transformation Equation for Pressure.} We may use the first
of these equations to show that the pressure is a quantity which
appears the same to all observers regardless of their relative motion.
We have
\[
p = \frac{\partial}{\partial v} \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right)
  = -\sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial v}
  = -\sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial v°}\,
     \frac{\partial v°}{\partial v}.
\]
But, in accordance with equation~(194), $p° = -\dfrac{\partial E°}{\partial v°}$, and in accordance
with equation~(195),
\[
\frac{\partial v°}{\partial v} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\]
which gives us the desired relation
\[
p = p°.
\Tag{200}
\]

Defining pressure as force per unit area, this result will be seen
to be identical with that which is obtained from the transformation
equations for force and area which result from our earliest considerations.

\Subsubsection{146}{Transformation Equation for Temperature.} The second of
the Lagrangian equations~(199) will provide us information as to
measurements of temperature made by observers moving with different
velocities. We have
\[
T = \frac{\partial}{\partial S}
    \left(E° \sqrt{1 - \frac{u^2}{c^2}}\right)
  = \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial S°}\,
    \frac{\partial S°}{\partial S}.
\]
But, in accordance with equation~(194), $\dfrac{\partial E°}{\partial S°} = T°$ and in accordance
with~(197) $\dfrac{\partial S°}{\partial S} = 1$. We obtain as our transformation equation,
\[
T = T° \sqrt{1 - \frac{u^2}{c^2}},
\Tag{201}
\]
and see that the quantity $\dfrac{T}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is an invariant for the Lorentz
transformation\DPchg{}{.}
%% -----File: 174.png---Folio 160-------

\Subsubsection{147}{The Equations of Motion for Quasistationary Adiabatic
Acceleration.} Let us now turn our attention to the last three of the
Lagrangian equations. These are the equations for the motion of a
thermodynamic system under the action of external force. It is
evident, however, that these equations will necessarily apply only
to cases of quasistationary acceleration, since our development of
the principle of least action gave us an equation for kinetic potential
which was true only for systems of infinitesimal extent or large systems
in a steady internal state. It is also evident that we must confine our
considerations to cases of adiabatic acceleration, since otherwise the
value of~$E°$ which occurs in the expression for kinetic potential might
be varying in a perfectly unknown manner.

The Lagrangian equations for force may be advantageously transformed.
We have
\begin{align*}
F_x &= \frac{d}{dt}\, \frac{\partial}{\partial \dot{x}}
  \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right)
  = \frac{d}{dt} \Biggl[
    \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}
    - \sqrt{1 - \frac{u^2}{c^2}}\,
      \frac{\partial E°}{\partial \dot{x}}\Biggr] \\
  &= \frac{d}{dt}\Biggl\{
    \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}
    - \sqrt{1 - \frac{u^2}{c^2}}
      \left(\frac{\partial E°}{\partial v°}\,
            \frac{\partial v°}{\partial \dot{x}}
          + \frac{\partial E°}{\partial S°}\,
            \frac{\partial S°}{\partial \dot{x}}\right)\Biggr\}.
\end{align*}
But by equations (194),~(196) and~(197) we have
\[
\frac{\partial E°}{\partial v°} = -p°, \qquad
\frac{\partial v°}{\partial \dot{x}}
  = \frac{v°}{\left(1 - \smfrac{u^2}{c^2}\right)}\,
    \frac{\dot{x}}{c^2}, \qquad\text{and}\qquad
\frac{\partial S°}{\partial \dot{x}} = 0.
\]
We obtain
\[
F_x = \frac{d}{dt}\Biggl\{
  \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\dot{x}}{c^2}\Biggr\}.
\Tag{202}
\]

Similar equations may be obtained for the components of force in
the $Y$~and~$Z$ directions and these combined to give the vector equation
\[
\vc{F} = \frac{d}{dt} \Biggl\{
  \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\vc{u}}{c^2}\Biggr\}.
\Tag{203}
\]
%% -----File: 175.png---Folio 161-------

This is the fundamental equation of motion for the dynamics of a
thermodynamic system.

\Subsubsection{148}{The Energy of a Moving Thermodynamic System.} We may
use this equation to obtain an expression for the energy of a moving
thermodynamic system. If we adiabatically accelerate a thermodynamic
system in the direction of its motion, its energy will increase
both because of the work done by the force
\[
\vc{F} = \frac{d}{dt} \Biggl\{
  \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\vc{u}}{c^2}\Biggr\}
\]
which produces the acceleration and because of the work done by the
pressure $p = p°$ which acts on a volume which is continually diminishing
as the velocity~$u$ increases, in accordance with the expression
$v = v° \sqrt{1 - \dfrac{u^2}{c^2}}$. Hence we may write for the total energy
\begin{align*}
E &= E° + \int_0^u \frac{d}{dt}\Biggl\{
     \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
     \frac{\vc{u}}{c^2}\Biggr\} \vc{u}\, dt
  + p° v° \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\:\right)\DPchg{}{,} \\
E &= \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  - p° v° \sqrt{1 - \frac{u^2}{c^2}}
  = \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}} - pv.
\Tag{204}
\end{align*}

\Subsubsection{149}{The Momentum of a Moving Thermodynamic System.} We
may compare this expression for the energy of a thermodynamic
system with the following expression for momentum which is evident
from the equation~(203) for force:
\[
\vc{G} = \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\vc{u}}{c^2}.
\Tag{205}
\]

We find again, as in our treatment of \Chapnumref[X]{elastic bodies} presented
in the last chapter, that the momentum of a moving system may be
calculated by taking the \emph{total} flow of energy in the desired direction
%% -----File: 176.png---Folio 162-------
and dividing by~$c^2$. Thus, comparing equations (204)~and~(205),
we have
\[
\vc{G} = \frac{E}{c^2}\, \vc{u} + \frac{pv}{c^2}\, \vc{u},
\Tag{206}
\]
where the term $\dfrac{E}{c^2}\, \vc{u}$ takes care of the energy transported bodily along
by the system and the term $\dfrac{pv}{c^2}\, \vc{u}$ takes care of the energy transferred
in the $\vc{u}$~direction by the action of the external pressure on the rear
and front end of the moving system.

\Subsubsection{150}{The Dynamics of a Hohlraum.} As an application of our considerations
we may consider the dynamics of a hohlraum, since a
hohlraum in thermodynamic equilibrium is of course merely a special
example of the general dynamics which we have just developed. The
simplicity of the hohlraum and its importance from a theoretical
point of view make it interesting to obtain by the present method the
same expression for momentum that can be obtained directly but
with less ease of calculation from electromagnetic considerations.

As is well known from the work of Stefan and Boltzmann, the
energy content~$E°$ and pressure~$p°$ of a hohlraum at rest and in thermodynamic
equilibrium are completely determined by the temperature~$T°$
and volume~$v°$ in accordance with the equations
\begin{align*}
E° &= av° {T°}^4, \\
p° &= \frac{a}{3}\, {T°}^4,
\end{align*}
where $a$~is the so-called Stefan's constant.

Substituting these values of $E°$~and~$p°$ in the equation for the
motion of a thermodynamic system~(203), we obtain
\[
\vc{F} = \frac{d}{dt}\Biggl[
  \frac{4}{3}\, \frac{av° {T°}^4}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\vc{u}}{c^2}\Biggr]
  = \frac{d}{dt}\Biggl[
  \frac{4}{3}\, \frac{avT^4}{\left(1 - \smfrac{u^2}{c^2}\right)^3}\,
  \frac{\vc{u}}{c^2}\Biggr]
\Tag{207}
\]
as the equation for the quasistationary adiabatic acceleration of a
%% -----File: 177.png---Folio 163-------
hohlraum. In view of this equation we may write for the momentum
of a hohlraum the expression
\[
\vc{G} = \frac{4}{3}\, \frac{av°{T°}^4}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,
  \frac{\vc{u}}{c^2}.
\Tag{208}
\]

It is a fact of significance that our dynamics leads to a result for
the momentum of a hohlraum which had been adopted on the ground
of electromagnetic considerations even without the express introduction
of relativity theory.
%% -----File: 178.png---Folio 164-------


\Chapter{XII}{Electromagnetic Theory.}
\SetRunningHeads{Chapter Twelve.}{Electromagnetic Theory.}

The Einstein theory of relativity proves to be of the greatest
significance for electromagnetics. On the one hand, the new electromagnetic
theory based on the first postulate of relativity obviously
accounts in a direct and straightforward manner for the results of the
Michelson-Morley experiment and other unsuccessful attempts to
detect an ether drift, and on the other hand also accounts just as
simply for the phenomena of moving dielectrics as did the older
theory of a stationary ether. Furthermore, the theory of relativity
provides considerably simplified methods for deriving a great many
theorems which were already known on the basis of the ether theory,
and gives us in general a clarified insight into the nature of electromagnetic
action.

\Subsubsection{151}{The Form of the Kinetic Potential.} In \Chapref{IX} we investigated
the general relation between the principle of least action
and the theory of the relativity of motion. We saw that the development
of any branch of dynamics would agree with the requirements
of relativity provided only that the kinetic potential~$H$ has such a form
that the quantity $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is an invariant for the Lorentz transformation.
Making use of this discovery we have seen the possibility
of developing the dynamics of a particle, the dynamics of an elastic
body, and the dynamics of a thermodynamic system, all of them in
forms which agree with the theory of relativity by merely introducing
slight modifications into the older expressions for kinetic potential in
such a way as to obtain the necessary invariance for $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$.
In the case of electrodynamics, however, on account of the closely
interwoven historical development of the theories of electricity and
relativity, we shall not find it necessary to introduce any modification
%% -----File: 179.png---Folio 165-------
in the form of the kinetic potential, but may take for~$H$ the following
expression, which is known to lead to the familiar equations of the
Lorentz electron theory
\[
H = \int dV \left\{\frac{\vc{e}^2}{2} + \frac{\curl \vc{\phi})^2}{2}
  - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right)
\right\},
\Tag{209}
\]
where the integration is to extend over the whole volume of the
system~$V$, $\vc{e}$~is the intensity of the electric field at the point in question,
$\vc{\phi}$~is the value of the vector potential, $\rho$~the density of charge and $\vc{u}$~its
velocity.\footnote
  {Strictly speaking this expression for kinetic potential is not quite correct,
  since kinetic potential must have the dimensions of energy. To complete the equation
  and give all the terms their correct dimensions, we could multiply the first term
  by the dielectric inductivity of free space~$\epsilon$, and the last two terms by the magnetic
  permeability~$\mu$. Since, however, $\epsilon$~and~$\mu$ have the numerical value unity with the
  usual choice of units, we shall not be led into error in our particular considerations
  if we omit these factors.}

Let us now show that the expression which we have chosen for
kinetic potential does lead to the familiar equations of the electron
theory.

\Subsubsection{152}{The Principle of Least Action.} If now we denote by~$\vc{f}$ the
force per unit volume of material exerted by the electromagnetic
action it is evident that we may write in accordance with the principle
of least action~(135)
\[
\int dt\, dV \left[\delta \left\{
  \frac{\vc{e}^2}{2} + \frac{(\curl \vc{\phi})^2}{2}
    - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right)
  \right\} + \vc{f}· \delta\vc{r} \right] = 0,
\Tag{210}
\]
where $\delta\vc{r}$ is the variation in the radius vector to the particle under
consideration, and where the integration is to be taken over the
whole volume occupied by the system and between two instants of
time $t_1$~and~$t_2$ at which the actual and displaced configurations of the
system coincide.

\Subsubsection{153}{The Partial Integrations.} In order to simplify this equation,
we shall need to make use of two results which can be obtained by
partial integrations with respect to time and space respectively.

Thus we may write
\[
\int_{t_1}^{t_2} dt\, (a\, \dot{\delta b})
  = \int_{t_1}^{t_2} a\, d(\delta b)
  = [a\, \delta b]_{t_1}^{t_2}
  - \int_{t_1}^{t_2} dt \left(\frac{da}{dt}\, \delta b\right),
\]
%% -----File: 180.png---Folio 166-------
or, since the displaced and actual motions coincide at $t_1$~and~$t_2$,
\[
\int dt\, (a\, \dot{\delta b})
  = -\int dt \left(\frac{da}{dt}\, \delta b\right)\DPtypo{}{.}
\Tag{211}
\]
We may also write
\[
\int dV \left(a\, \frac{db}{dx}\right)
  = \int dy\, dz\, (a\, db)
  = \int dy\, dz\, [ab]_{x=-\infty}^{x=+\infty}
  - \int dV \left(b\, \frac{da}{dx}\right),
\]
or, since we are to carry out our integrations over the whole volume
occupied by the system, we shall take our functions as zero at the
limits of integration and may write
\[
\int dV \left(a\, \frac{db}{dx}\right)
  = -\int dV \left(b\, \frac{da}{dx}\right).
\Tag{212}
\]
Since similar considerations apply to derivatives with respect to the
other variables $y$~and~$z$, we can also obtain
\begin{gather*}
\int dV\, a \divg\vc{b} = -\int dV\, \vc{b} · \grad a,
\Tag{213} \\
\int dV\, \vc{a} · \curl\vc{b} = \int dV\, \vc{b} · \curl\vc{a}.
\Tag{214}
\end{gather*}

\Subsubsection{154}{Derivation of the Fundamental Equations of Electromagnetic
Theory.} {\stretchyspace%
Making use of these purely mathematical relationships we
are now in a position to develop our fundamental equation~(210).
Carrying out the indicated variation, noting that $\delta \vc{u} = \dfrac{d(\delta\vc{r})}{dt}$ and
making use of (211)~and~(214) we easily obtain}
\[
\begin{aligned}
\int dt\, dV \Biggl[
  \left\{\vc{e} + \frac{1}{c}\, \frac{\partial\vc{\phi}}{\partial t}\right\}
 · \delta\vc{e}
 &+ \left\{\curl\curl\vc{\phi} - \left(\frac{\dot{\vc{e}}}{c}
         + \rho\, \frac{\vc{u}}{c}\right) \right\} · \delta\vc{\phi} \\
 &\qquad\qquad
  - \frac{\vc{\phi}}{c} · \delta(\rho\vc{u}) + \vc{f}· \delta\vc{r}\Biggr]
  = 0.
\end{aligned}
\Tag{215}
\]

In developing the consequences of this equation, it should be
noted, however, that the variations are not all of them independent;
thus, since we shall define the density of charge by the equation
\[
\rho = \divg\vc{e},
\Tag{216}
\]
it is evident that it will be necessary to preserve the truth of this
equation in any variation that we carry out. This can evidently be
%% -----File: 181.png---Folio 167-------
done if we add to our equation~(215) the expression
\[
\int dt\, dV\, \psi[\delta\rho - \divg\delta\vc{e}] = 0,
\]
where $\psi$~is an undetermined scalar multiplier. We then obtain with
the help of~(213)
{\small%
%[** TN: Re-breaking]
\[
\begin{aligned}
&\int dt\, dV \Biggl[
  \left\{\vc{e} + \frac{1}{c}\, \frac{\partial\vc{\phi}}{\partial t}
       + \grad\psi \right\} · \delta\vc{e} \\
+& \left\{\curl\curl\vc{\phi} - \left(\frac{\dot{\vc{e}}}{c}
        + \rho\, \frac{\vc{u}}{c}\right)\right\} · \delta\vc{\phi}
  - \frac{\vc{\phi}}{c} · \delta(\rho\vc{u}) + \psi\, \delta\rho
  + \vc{f} · \delta\vc{r}\Biggr] = 0,
\end{aligned}
\Tag{217}
\]}%
and may now treat the variations $\delta \vc{e}$~and~$\delta\vc{\phi}$ as entirely independent
of the others; we must then have the following equations true
\begin{gather*}
\vc{e} = -\frac{1}{c}\, \frac{\partial \vc{\phi}}{\partial t} - \grad \psi,
\Tag{218} \\
\curl\curl\vc{\phi} = \frac{\dot{\vc{e}}}{c} + \frac{\rho\vc{u}}{c},
\Tag{219}
\end{gather*}
and have thus derived from the principle of least action the fundamental
equations of modern electron theory. We may put these in
their familiar form by defining the magnetic field strength~$\vc{h}$ by the
equation
\[
\vc{h} = \curl\vc{\phi}\DPtypo{}{.}
\Tag{220}
\]
We then obtain from~(219)
\begin{align*}%[** TN: Next four equations not aligned in original]
\curl\vc{h} &= \frac{1}{c}\, \frac{\partial\vc{e}}{\partial t}
  + \rho\, \frac{\vc{u}}{c},
\Tag{221} \\
\intertext{and, noting the mathematical identity $\curl\grad\psi = 0$, we obtain
from (218)}
\curl\vc{e} &= -\frac{1}{c}\, \frac{\partial\vc{h}}{\partial t}.
\Tag{222} \\
\intertext{We have furthermore by definition~(216)}
\divg\vc{e} &= \rho,
\Tag{223} \\
\intertext{and noting equation~(220) may write the mathematical identity}
\divg\vc{h} &= 0.
\Tag{224}
\end{align*}
%% -----File: 182.png---Folio 168-------

These four equations~\DPchg{(221--4)}{(221)--(224)} are the familiar expressions which
have been made the foundation of modern electron theory. They
differ from Maxwell's original four field equations only by the introduction
in (221)~and~(223) of terms which arise from the density of
charge~$\rho$ of the electrons, and reduce to Maxwell's set in free space.

\Paragraph{155.} We have not yet made use of the last three terms in the
fundamental equation~(217) which results from the principle of least
action. As a matter of fact, it can be shown that these terms can be
transformed into the expression
\[
\int dt\, dV \left[
  \frac{\rho}{c}\, \frac{\partial\vc{\phi}}{\partial t}
  - \frac{\rho}{c}\, [\vc{u} × \curl\vc{\phi}]^*
  + \rho \grad\psi + \vc{f}\right] · \delta\vc{r},
\Tag{225}
\]
and hence lead to the familiar fifth fundamental equation of modern
electron theory,
\begin{align*}
\vc{f} &= \rho \left\{-\frac{\partial\vc{\phi}}{c\partial t}
  - \grad\psi + \left[\frac{\vc{u}}{c} × \curl\vc{\Phi}\right]^*\right\}, \\
\vc{f} &= \rho \left\{\vc{e} + \left[\frac{\vc{u}}{c} × \vc{h}\right]^*\right\}.
\Tag{226}
\end{align*}
The transformation of the last three terms of~(217) into the form
given above~(225) is a complicated one and it has not seemed necessary
to present it here since in a later paragraph we shall show the
possibility of deriving the fifth fundamental equation of the electron
theory~(226) by combining the four field equations~\DPchg{(221--4)}{(221)--(224)} with the
transformation equations for force already obtained from the principle
of relativity. The reader may carry out the transformation himself,
however, if he makes use of the partial integrations which we have
already obtained, notes that in accordance with the principle of the
conservation of electricity we must have $\delta\rho = - \divg\rho\, \delta\vc{r}$ and notes
that $\delta\vc{u} = \dfrac{d(\delta\vc{r})}{dt}$, where the differentiation $\smash{\dfrac{d}{dt}}\rule{0pt}{12pt}$ indicates that we are
following some particular particle in its motion, while the differentiation
$\dfrac{\partial}{\partial t}$ occurring in $\dfrac{\partial\vc{\phi}}{\partial t}$ indicates that we intend the rate of change
at some particular stationary point.

\Subsubsection{156}{The Transformation Equations for $\vc{e}$,~$\vc{h}$ and~$\rho$.} We have thus
shown the possibility of deriving the fundamental equations of modern
%% -----File: 183.png---Folio 169-------
electron theory from the principle of least action. We now wish to
introduce the theory of relativity into our discussions by presenting
a set of equations for transforming measurements of $\vc{e}$,~$\vc{h}$ and~$\rho$ from
one set of space-time coördinates~$S$ to another set~$S'$ moving past~$S$
in the $X$\DPchg{-}{~}direction with the velocity~$V$. This set of equations is as
follows:
\begin{gather*} %[** TN: Set equation groups on one line each]
\begin{alignat*}{3}
{e_x}' &= e_x, \qquad &
{e_y}' &= \kappa \left(e_y - \frac{V}{c}h_z\right),\qquad &
{e_z}' &= \kappa \left(e_z + \frac{V}{c}h_y\right), \Tag{227}\displaybreak[0] \\
{h_x}' &= h_x, &
{h_y}' &= \kappa \left(h_y + \frac{V}{c}e_z\right), &
{h_z}' &= \kappa \left(h_z - \frac{V}{c}e_y\right), \Tag{228}
\end{alignat*} \displaybreak[0] \\
\rho' = \rho\kappa \left(1 - \frac{u_zV}{c^2}\right), \Tag{229}
\end{gather*}
where $\kappa$ has its customary significance $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$.
\bigskip%[** Explicit space]

As a matter of fact, this set of transformation equations fulfills
all the requirements imposed by the theory of relativity. Thus, in
the first place, it will be seen, on development, that these equations
are themselves perfectly symmetrical with respect to the primed and
unprimed quantities except for the necessary change from $+V$~to~$-V$.
In the second place, it will be found that the substitution of
these equations into our five fundamental equations for electromagnetic
theory \DPchg{(221--2--3--4--6)}{(221), (222), (223), (224), (226)} will successfully transform them
into an entirely similar set with primed quantities replacing the
unprimed ones. And finally it can be shown that these equations
agree with the general requirement derived in \Chapref{IX} that the
%% -----File: 184.png---Folio 170-------
quantity $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ shall be an invariant for the Lorentz transformation.

To demonstrate this important invariance of $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ we may
point out that by introducing equations (220),~(221) and~(214), our
original expression for kinetic potential
\[
H = \int dV \left\{
  \frac{\vc{e}^2}{2} + \frac{(\curl\vc{\phi})^2}{2}
     - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right)
  \right\}
\]
can easily be shown equal to
\[
\int dV \left(\frac{\vc{e}^2}{2} - \frac{\vc{h}^2}{2}\right),
\Tag{230}
\]
and, noting that our fundamental equations for space and time provide
us with the relation
\[
\frac{dV}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  = \frac{dV'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}},
\]
we can easily show that our transformation equations for $\vc{e}$~and~$\vc{h}$ do
lead to the equality
\[
\frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  = \frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}.
\]

We thus know that our development of the fundamental equations
for electromagnetic theory from the principle of least action is indeed
in complete accordance with the theory of relativity, since it conforms
with the general requirement which was found in \Chapref{IX} to be
imposed by the theory of relativity on all dynamical considerations.

\Subsubsection{157}{The Invariance of Electric Charge.} As to the significance of
the transformation equations which we have presented for $\vc{e}$,~$\vc{h}$ and~$\rho$,
we may first show, in accordance with the last of these equations,
that a given electric charge will appear the same to all observers no
matter what their relative motion.
%% -----File: 185.png---Folio 171-------

To demonstrate this we merely have to point out that, by introducing
equation~(17), we may write our transformation equation
for~$\rho$~(229) in the form
\[
\frac{\rho'}{\rho}
  = \frac{\sqrt{1 - \smfrac{u^2}{c^2}}}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}},
\]
which shows at once that the two measurements of density of charge
made by $O$~and~$O'$ are in exactly the same ratio as the corresponding
measurements for the Lorentz shortening of the charged body, so
that the total charge will evidently measure the same for the two
observers.

We might express this invariance of electric charge by writing the
equation
\[
Q' = Q.
\Tag{231}
\]

It should be noted in passing that this result is in entire accord
with the whole modern development of electrical theory, which lays
increasing stress on the fundamentality and indivisibility of the
electron as the natural unit quantity of electricity. On this basis
the most direct method of determining the charge on an electrified
body would be to count the number of electrons present and this
number must obviously appear the same both to observer~$O$ and
observer~$O'$.\footnote
  {A similar invariance of electric charge has been made fundamental in the
  author's development of the theory of similitude (\ie, the theory of the relativity
  of size). See for example \textit{Phys.\ Rev}., vol.~3, p.~244 (1914).}

\Subsubsection{158}{The Relativity of Magnetic and Electric Fields.} As to the
significance of equations (227)~and~(228) for transforming the values
of the electric and magnetic field strengths from one system to another,
we see that at a given point in space we may distinguish between the
electric vector $\vc{e} = e_x\, \vc{i} + e_y\, \vc{j} + e_z\, \vc{k}$ as measured by our original
observer~$O$ and the vector $\vc{e}' = {e_x}'\, \vc{i} + {e_y}'\, \vc{j} + {e_z}'\, \vc{k}$ as measured in
units of his own system by an observer~$O'$ who is moving past~$O$ with
the velocity~$V$ in the $X$\DPchg{-}{~}direction. Thus if $O$~finds in an unvarying
electromagnetic field that $Q\vc{e}$~is the force on a small test charge~$Q$
which is stationary with respect to his system, $O'$~will find experimentally
%% -----File: 186.png---Folio 172-------
for a similar test charge that moves along with him a value
for the force~$Q\vc{e}'$, where $\vc{e}'$~can be calculated from with the help of
these equations~(227). Similar remarks would apply to the forces
which would act on magnetic poles.

These considerations show us that we should now use caution in
speaking of a pure electrostatic or pure magnetic field, since the
description of an electromagnetic field is determined by the particular
choice of coördinates with reference to which the field is measured.

\Subsubsection{159}{Nature of Electromotive Force.} We also see that the ``electromotive''
force which acts on a charge moving through a magnetic
field finds its interpretation as an ``electric'' force provided we make
use of a system of coördinates which are themselves stationary with
respect to the charge. Such considerations throw light on such questions,
for example, as to the seat of the ``electromotive'' forces in
``homopolar'' electric dynamos where there is relative motion of a
conductor and a magnetic field.


\Subsection{Derivation of the Fifth Fundamental Equation.}

\Paragraph{160.} We may now make use of this fact that the forces acting on
a moving charge of electricity may be treated as purely electrostatic,
by using a set of coördinates which are themselves moving along with
the charge, to derive the fifth fundamental equation of electromagnetic
theory.

Consider an electromagnetic field having the values $\vc{e}$~and~$\vc{h}$ for
the electric and magnetic field strengths at some particular point.
What will be the value of the electromagnetic force~$\vc{f}$ acting per
unit volume on a charge of density~$\rho$ which is passing through the
point in question with the velocity~$\vc{u}$?

To solve the problem take a system of coördinates~$S'$ which itself
moves with the same velocity as the charge, for convenience letting
the $X$\DPchg{-}{~}axis coincide with the direction of the motion of the charge.
Since the charge of electricity is stationary with respect to this system,
the force acting on it as measured in units of this system will be by
definition equal to the product of the charge by the strength of the
electric field as it appears to an observer in this system, so that we may
write
\[
\vc{F} = Q'\vc{e}',
\]
%% -----File: 187.png---Folio 173-------
or
\[
{F_x}' = Q'{e_x}', \qquad
{F_y}' = Q'{e_y}', \qquad
{F_z}' = Q'{e_z}'.
\]
For the components of the electrical field ${e_x}'$,~${e_y}'$,~${e_z}'$, we have just
obtained the transformation equations~(227), while in our earlier
dynamical considerations in \Chapref{VI} we obtained transformation
equations (61),~(62), and~(63) for the components of force. Substituting
above and bearing in mind that $u_x = V$, $u_y = u_z = 0$, and
that $Q' = Q$, we obtain on simplification
\begin{align*}
F_x &= Q e_x, \\
F_y &= Q \left(e_y - \frac{u_x}{c}h_z\right), \\
F_z &= Q \left(e_z - \frac{u_x}{c}h_y\right),
\end{align*}
which in vectorial form gives us the equation
\[
\vc{F} = Q \left(\vc{e} - \frac{1}{c}\, [\vc{u} × \vc{h}]^*\right)
\]
or for the force per unit volume
\[
\vc{f} = \rho \left(\vc{e} + \frac{1}{c}\, [\vc{u} × \vc{h}]^*\right).
\Tag{226}
\]

This is the well-known fifth fundamental equation of the Maxwell-Lorentz
theory of electromagnetism. We have already indicated the
method by which it could be derived from the principle of least action.
This derivation, however, from the transformation equations, provided
by the theory of relativity, is particularly simple and attractive.


\Subsection{Difference between the Ether and the Relativity Theories of Electromagnetism.}

\Paragraph{161.} In spite of the fact that we have now found five equations
which can be used as a basis for electromagnetic theory which agree
with the requirements of relativity and also have exactly the same
form as the five fundamental equations used by Lorentz in building
up the stationary ether theory, it must not be supposed that the
relativity and ether theories of electromagnetism are identical. Although
the older equations have exactly the same form as the ones
which we shall henceforth use, they have a different interpretation,
since our equations are true for measurements made with the help
of any non-accelerated set of coördinates, while the equations of
%% -----File: 188.png---Folio 174-------
Lorentz were, in the first instance, supposed to be true only for measurements
which were referred to a set of coördinates which were
stationary with respect to the assumed luminiferous ether. Suppose,
for example, we desire to calculate with the help of equation~(226),
\[
\vc{t} = \rho \left(\vc{e} + \frac{1}{\vc{c}}\, [\vc{u} × \vc{h}]^*\right),
\]
the force acting on a charged body which is moving with the velocity~$\vc{u}$;
we must note that for the stationary ether theory, $\vc{u}$~must be the
velocity of the charged body through the ether, while for us $\vc{u}$~may be
taken as the velocity past any set of unaccelerated coördinates, provided
$\vc{e}$~and~$\vc{h}$ are measured with reference to the same set of coördinates.
It will be readily seen that such an extension in the meaning
of the fundamental equations is an important simplification.

\Paragraph{162.} A word about the development from the theory of a stationary
ether to our present theory will not be out of place. When it was
found that the theory of a stationary ether led to incorrect conclusions
in the case of the Michelson-Morley experiment, the hypothesis
was advanced by Lorentz and Fitzgerald that the failure of that
experiment to show any motion through the ether was due to a contraction
of the apparatus in the direction of its motion through the
ether in the ratio $1 : \sqrt{1 - \dfrac{u^2}{c^2}}$. Lorentz then showed that if all systems
should be thus contracted in the line of their motion through the
ether, and observers moving with such system make use of suitably
contracted meter sticks and clocks adjusted to give what Lorentz
called the ``local time,'' their measurements of electromagnetic
phenomena could be described by a set of equations which have
nearly the same form as the original four field equations which would
be used by a stationary observer. It will be seen that Lorentz was
thus making important progress towards our present idea of the complete
relativity of motion. The final step could not be taken, however,
without abandoning our older ideas of space and time and giving up
the Galilean transformation equations as the basis of kinematics.
It was Einstein who, with clearness and boldness of vision, pointed
out that the failure of the Michelson-Morley experiment, and all
other attempts to detect motion through the ether, is not due to a
%% -----File: 189.png---Folio 175-------
fortuitous compensation of effects but is the expression of an important
general principle, and the new transformation equations for kinematics
to which he was led have not only provided the basis for an \emph{exact}
transformation of the field equations but have so completely revolutionized
our ideas of space and time that hardly a branch of science
remains unaffected.

\Paragraph{163.} With regard to the present status of the ether in scientific
theory, it must be definitely stated that this concept has certainly
lost both its fundamentality and the greater part of its usefulness,
and this has been brought about by a gradual process which has only
found its culmination in the work of Einstein. Since the earliest
days of the luminiferous ether, the attempts of science to increase the
substantiality of this medium have met with little success. Thus
we have had solid elastic ethers of most extreme tenuity, and ethers
with a density of a thousand tons per cubic millimeter; we have had
quasi-material tubes of force and lines of force; we have had vibratory
gyrostatic ethers and perfect gases of zero atomic weight; but after
every debauch of model-making, science has recognized anew that a
correct mathematical description of the actual phenomena of light
propagation is superior to any of these sublimated material media.
Already for Lorentz the ether had been reduced to the bare function
of providing a stationary system of reference for the measurement of
positions and velocities, and now even this function has been taken
from it by the work of Einstein, which has shown that any unaccelerated
system of reference is just as good as any other.

To give up the notion of an ether will be very hard for many
physicists, in particular since the phenomena of the interference and
polarization of light are so easily correlated with familiar experience
with wave motions in material elastic media. Consideration will
show us, however, that by giving up the ether we have done nothing
to destroy the periodic or polarizable nature of a light disturbance.
When a plane polarized beam of light is passing through a given
point in space we merely find that the electric and magnetic fields at
that point lie on perpendiculars to the direction of propagation and
undergo regular periodic changes in magnitude. There is no need of
going beyond these actual experimental facts and introducing any
hypothetical medium. It is just as simple, indeed simpler, to say
%% -----File: 190.png---Folio 176-------
that the electric or magnetic field has a certain intensity at a given
point in space as to speak of a complicated sort of strain at a given
point in an assumed ether.


\Subsection{Applications to Electromagnetic Theory.}

\Paragraph{164.} The significant fact that the fundamental equations of the
new electromagnetic theory have the same form as those of Lorentz
makes it of course possible to retain in the structure of modern electrical
theory nearly all the results of his important researches, care
being taken to give his mathematical equations an interpretation in
accordance with the fundamental ideas of the theory of relativity. It
is, however, entirely beyond our present scope to make any presentation
of electromagnetic theory as a whole, and in the following paragraphs
we shall confine ourselves to the proof of a few theorems which
can be handled with special ease and directness by the methods introduced
by the theory of relativity.

\Subsubsection{165}{The Electric and Magnetic Fields around a Moving Charge.}
Our transformation equations for the electromagnetic field make it
very easy to derive expressions for the field around a point charge in
uniform motion. Consider a point charge~$Q$ moving with the velocity~$V$.
For convenience consider a system of reference~$S$ such that $Q$~is
moving along the $X$\DPchg{-}{~}axis and at the instant in question, $t=0$, let the
charge coincide with the origin of coördinates~$O$. We desire now to
calculate the values of electric field~$\vc{e}$ and the magnetic field~$\vc{h}$ at any
point in space $x$,~$y$,~$z$.

Consider another system of reference,~$S'$, which moves along with
the same velocity as the charge~$Q$, the origin of coördinates~$O'$\DPchg{,}{} and
the charge always coinciding in position. Since the charge is stationary
with respect to their new system of reference, we shall have
the electric field at any point $x'$,~$y'$,~$z'$ in this system given by the
equations
\begin{align*}
{e_x}' &= \frac{Qx'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \\
{e_y}' &= \frac{Qy'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \\
{e_z}' &= \frac{Qz'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}},
\end{align*}
%% -----File: 191.png---Folio 177-------
while the magnetic field will obviously be zero for measurements made
in system~$S'$, giving us
\[%[** TN: Setting on one line]
{h_x}' = 0, \qquad {h_y}' = 0, \qquad {h_z}' = 0.
\]
Introducing our transformation equations (9),~(10) and~(11) for $x'$,~$y'$
and~$z'$ and our transformation equations (227)~and~(228) for the
electric and magnetic fields and substituting $t=0$, we obtain for the
values of $\vc{e}$~and~$\vc{h}$ in system~$S$ at the instant when the charge passes
through the point~$O$,
\begin{align*}
e_x &= \frac{Q\kappa x}{(\kappa^2x^2 + y^2 + z^2)^{3/2}}
     = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) x}
       {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}},
\displaybreak[0] \\
e_y &= \frac{Q\kappa y}{(\kappa^2x^2 + y^2 + z^2)^{3/2}}
     = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) y}
       {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}},
\displaybreak[0] \\
e_z &= \frac{Q\kappa z}{(\kappa^2x^2 + y^2 + z^2)^{3/2}}
     = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) z}
       {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}},
\displaybreak[0] \\
h_x &=  0, \\
h_y &= -\frac{V}{c}\, e_z,\\
h_z &=  \frac{V}{c}\, e_y,
\end{align*}
or, putting $s$ for the important quantity $\sqrt{x^2 + \left(1 - \dfrac{V^2}{c^2}\right)(y^2 + z^2)}$
and writing the equations in the vectorial form where we put
\[
\vc{r} = (x\, \vc{i} + y\, \vc{j} + z\, \vc{k}),
\]
we obtain the familiar equations for the field around a point charge
%% -----File: 192.png---Folio 178-------
in uniform motion with the velocity $u=V$ in the $X$\DPchg{-}{~}direction
\begin{gather*}
\vc{e} = Q\, \frac{\left(1 - \smfrac{u^2}{c^2}\right)\vc{r}}{s^3},
\Tag{232} \\
\vc{h} = \frac{1}{c}\, [\vc{u} × \vc{e}]\DPtypo{.^*}{^*.}
\Tag{233}
\end{gather*}

\Subsubsection{166}{The Energy of a Moving Electromagnetic System.} Our
transformation equations will permit us to obtain a very important
expression for the energy of an isolated electromagnetic system in
terms of the velocity of the system and the energy of the same system
as it appears to an observer who is moving along with it.

Consider a physical system surrounded by a shell which is impermeable
to electromagnetic radiation. This system is to be thought
of as consisting of the various mechanical parts, electric charges and
electromagnetic fields which are inside of the impermeable shell.
The system is free in space, except that it may be acted on by external
electromagnetic fields, and its energy content thus be changed.

Let us now equate the increase in the energy of the system to the
work done by the action of the external field on the electric charges
in the system. Since the force which a magnetic field exerts on a
charge is at right angles to the motion of the charge it does no work
and we need to consider only the work done by the external electric
field and may write for the increase in the energy of the system
\[
\Delta E %[** TN: Textstyle integral in original]
  = \iiiint \rho(e_xu_x + e_yu_y + e_zu_z)\, dx\, dy\, dz\, dt,
\Tag{234}
\]
where the integration is to be taken over the total volume of the
system and over any time interval in which we may be interested.

Let us now transform this expression with the help of our transformation
equations for the electric field~(227) for electric charge~(229),
and for velocities \DPchg{(14--15--16)}{(14), (15), (16)}. Noting that our fundamental
equations for kinematic quantities give us $dx\, dy\, dz\, dt = dx'\, dy'\, dz'\, dt'$,
we obtain
\begin{align*}
\Delta E &= \kappa \iiiint
  \rho'({e_x}'{u_x}' + {e_y}'{u_y}' + {e_z}'{u_z}')\, dx'\, dy'\, dz'\, dt' \\
  &\quad
  + \kappa V \iiiint \rho'\left(
    {e_x}' + \frac{{u_y}'}{c}\, {h_z}' - \frac{{u_z}'}{c}\, {h_y}'
  \right) dx'\, dy'\, dz'\, dt'.
\end{align*}
%% -----File: 193.png---Folio 179-------

Consider now a system which \emph{both at the beginning and end of our
time interval is free from the action of external forces}; we may then
rewrite the above equation for this special case in the form
\[
\Delta E = \kappa \Delta E'
  + \kappa V \int \Sum {F_x}'\, dt',
\]
where, in accordance with our earlier equation~(234), $\Delta E'$~is the increase
in the energy of the system as it appears to observer~$O'$ and $\Sum {F_x}'$
is the total force acting on the system in $X$\DPchg{-}{~}direction as measured
by~$O'$.

The restriction that the system shall be unacted on by external
forces both at the beginning and end of our time interval is necessary
because it is only under those circumstances that an integration
between two values of~$t$ can be considered as an integration between
two definite values of~$t'$, simultaneity in different parts of the system
not being the same for observers $O$~and~$O'$.

We may now apply this equation to a specially interesting case.
Let the system be of such a nature that we can speak of it as being
at rest with respect to~$S'$, meaning thereby that all the mechanical
parts have low velocities with respect to~$S'$ and that their center of
gravity moves permanently along with~$S'$. Under these circumstances
we may evidently put $\int\Sum {F_x}'\, dt' = 0$ and may write the
above equation in the form
\begin{align*}
\Delta E &= \frac{\Delta E_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, \\
\intertext{or}
\frac{\partial \Delta E}{\partial E_0}
  &= \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\end{align*}
where $u$~is the velocity of the system, and $E°$~is its energy as measured
by an observer moving along with it. The energy of a system which
is \emph{unacted on by external forces} is thus a function of two variables, its
energy~$E_0$ as measured by an observer moving along with the system
and its velocity~$u$.
%% -----File: 194.png---Folio 180-------

We may now write
\[
E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, E_0 + \phi(u) + \text{const.},
\]
where $\phi(u)$ represents the energy of the system which depends solely
on the velocity of the system and not on the changes in its $E_0$~values.
$\phi(u)$~will thus evidently be the kinetic energy of the mechanical masses
in the system which we have already found~(82) to have the value
$\dfrac{m_0c^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0c^2$ where $m_0$~is to be taken as the total mass of the
mechanical part of our system when at rest. We may now write
\[
E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, (m_0c^2 + E_0)
  - m_0c^2 + \text{const.}
\]
Or, assuming as before that the constant is equal to~$m_0c^2$, which will
be equivalent to making a system which has zero energy also have
zero mass, we obtain
\[
E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, (m_0c^2 + E_0),
\Tag{235}
\]
which is the desired expression for the energy of an isolated system
which may contain both electrical and mechanical parts.

\Subsubsection{167}{Relation between Mass and Energy.} This expression for the
energy of a system that contains electrical parts permits us to show
that the same relation which we found between mass and energy for
mechanical systems also holds in the case of electromagnetic energy.
Consider a system containing electromagnetic energy and enclosed
by a shell which is impermeable to radiation. Let us apply a force~$\vc{F}$
to the system in such a way as to change the velocity of the system
without changing its $E_0$~value. We can then equate the work done
per second by the force to the rate of increase of the energy of the
system. We have
\[
\vc{F} · \vc{u} = \frac{dE}{dt}.
\]
%% -----File: 195.png---Folio 181-------
But from equation~(235) we can obtain a value for the rate of increase
of energy~$\dfrac{dE}{dt}$, giving us
\[
\vc{F} · \vc{u}
  = F_xu_x + F_yu_y + F_zu_z
  = \left(m_0 + \frac{E_0}{c^2}\right)
    \frac{u\, \smfrac{du}{dt}}{\left(1 - \smfrac{u^2}{c^2}\right)^{\tfrac{3}{2}}},
\]
and solving this equation for~$\vc{F}$ we obtain
\begin{align*}
\vc{F} &= \frac{d}{dt}\left[
  \frac{\left(m_0 + \smfrac{E_0}{c^2}\right)}
       {\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u}
  \right],
\Tag{236} \\
\intertext{which for low velocities assumes the form}
\vc{F} &= \frac{d}{dt}\left[\left(m_0 + \frac{E_0}{c^2}\right) \vc{u}\right].
\Tag{237}
\end{align*}

Examination of these expressions shows that our system which
contains electromagnetic energy behaves like an ordinary mechanical
system with the mass $\left(m_0 + \dfrac{E_0}{c^2}\right)$ at low velocities or $\dfrac{m_0 + \smfrac{E_0}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ at
any desired velocity~$u$. To the energy of the system~$E_0$, part of which
is electromagnetic, we must ascribe the mass~$\dfrac{E_0}{c^2}$ just as we found in
the case of mechanical energy. We realize again that matter and
energy are but different names for the same fundamental entity,
$10^{21}$~ergs of energy having the mass $1$~gram.


\Subsection{The Theory of Moving Dielectrics.}

\Paragraph{168.} The principle of relativity proves to be very useful for the
development of the theory of moving dielectrics.

It was first shown by Maxwell that a theory of electromagnetic
phenomena in material media can be based on a set of field equations
similar in form to those for free space, provided we introduce besides
the electric and magnetic field strengths, $\vc{E}$~and~$\vc{F}$, two new field vectors,
%% -----File: 196.png---Folio 182-------
the dielectric displacement~$\vc{D}$ and the magnetic induction~$\vc{B}$, and
also the density of electric current in the medium~$\vc{i}$. These quantities
are found to be connected by the four following equations similar in
form to the four field equations for free space:
\begin{align*}
\curl \vc{H}
  &= \frac{1}{c} \left(\frac{\partial\vc{D}}{\partial t} + \vc{i}\right),
                                                         \Tag{238} \\
\curl \vc{E}
  &= -\frac{1}{c}\, \frac{\partial\vc{B}}{\partial t},   \Tag{239} \\
\divg \vc{D} &= \rho,
                                                         \Tag{240} \\
\divg \vc{B} &= 0.
                                                         \Tag{241}
\end{align*}

For \emph{stationary homogeneous} media, the dielectric displacement,
magnetic induction and electric current are connected with the
electric and magnetic field strengths by the following equations:
\begin{align*}
\vc{D} &= \epsilon \vc{E},  \Tag{242}\\
\vc{B} &= \mu \vc{H},       \Tag{243}\\
\vc{i} &= \sigma \vc{E},    \Tag{244}
\end{align*}
where $\epsilon$~is the dielectric constant, $\mu$~the magnetic permeability and $\sigma$~the
electrical conductivity of the medium in question.

\Subsubsection{169}{Relation between Field Equations for Material Media and
Electron Theory.} It must not be supposed that the four field equations
\DPchg{(238--241)}{(238)--(241)} for electromagnetic phenomena in \emph{material media} are
in any sense contradictory to the four equations \DPchg{(221--224)}{(221)--(224)} for free
space which we took as the fundamental basis for our development of
electromagnetic theory. As a matter of fact, one of the main achievements
of modern electron theory has been to show that the electromagnetic
behavior of material media can be explained in terms of
the behavior of the individual electrons and ions which they contain,
these electrons and ions acting in accordance with the four fundamental
field equations for free space. Thus our new equations for material
media merely express from a \emph{macroscopic} point of view the statistical
result of the behavior of the individual electrons in the material in
question. $\vc{E}$~and~$\vc{H}$ in these new equations are to be looked upon as
the average values of $\vc{e}$~and~$\vc{h}$ which arise from the action of the
individual electrons in the material, the process of averaging being so
%% -----File: 197.png---Folio 183-------
carried out that the results give the values which a \emph{macroscopic} observer
would actually find for the electric and magnetic forces acting
respectively on a unit charge and a unit pole at the point in question.
These average values, $\vc{E}$~and~$\vc{H}$, will thus pay no attention to the
rapid fluctuations of $\vc{e}$~and~$\vc{h}$ which arise from the action and motion
of the individual electrons, the macroscopic observer using in fact
differentials for time,~$dt$, and space,~$dx$, which would be large from a
microscopic or molecular viewpoint.

Since from a microscopic point of view $\vc{E}$~and~$\vc{H}$ are not really
the instantaneous values of the field strength at an actual point in
space, it has been found necessary to introduce two new vectors,
electric displacement,~$\vc{D}$, and magnetic induction,~$\vc{B}$, whose time
rate of change will determine the curl of $\vc{E}$~and~$\vc{H}$ respectively. It will
evidently be possible, however, to relate $\vc{D}$~and~$\vc{B}$ to the actual electric
and magnetic fields $\vc{e}$~and~$\vc{h}$  produced by the individual electrons,
and this relation has been one of the problems solved by modern
electron theory, and the field equations \DPchg{(238--241)}{(238)--(241)} for material media
have thus been shown to stand in complete agreement with the most
modern views as to the structure of matter and electricity. For
the purposes of the rest of our discussion we shall merely take these
equations as expressing the experimental facts in stationary or in
moving media.

\Subsubsection{170}{Transformation Equations for Moving Media.} Since equations
\DPchg{(238 to 241)}{(238) to (241)} are assumed to give a correct description of electromagnetic
phenomena in media whether stationary or moving with
respect to our reference system~$S$, it is evident that the equations
must be unchanged in form if we refer our measurements to a new
system of coördinates~$S'$ moving past~$S$, say, with the velocity~$V$ in the
$X$\DPchg{-}{~}direction.

As a matter of fact, equations \DPchg{(238 to 241)}{(238) to (241)} can be transformed
into an entirely similar set
\begin{align*}
\curl \vc{H'}
  &= \frac{1}{c} \left(\frac{\partial\vc{D'}}{\partial t'} + \vc{i}'\DPtypo{,}{}\right)\DPtypo{}{,} \\
\curl \vc{E'} &= -\frac{1}{c}\, \frac{\partial\vc{B'}}{\partial t'}, \\
\divg \vc{D'} &= \rho', \\
\divg \vc{B'} &= 0,
\end{align*}
%% -----File: 198.png---Folio 184-------
provided we substitute for $x$,~$y$,~$z$ and~$t$ the values of $x'$,~$y'$,~$z'$ and~$t'$
given by the fundamental transformation equations for space and
time \DPchg{(9~to~12)}{(9)~to~(12)}, and substitute for the other quantities in question the
relations
{\small%
\begin{align*}%[** TN: Re-grouping]
\begin{aligned}
{E_x}' &= E_x, &
{E_y}' &= \kappa \left(E_y - \frac{V}{c} B_z\right), &
{E_z}' &= \kappa \left(E_z + \frac{V}{c} B_y\right), \\
%
{D_x}' &= D_x, &
{D_y}' &= \kappa \left(D_y - \frac{V}{c} H_z\right), &
{D_z}' &= \kappa \left(D_z + \frac{V}{c} H_y\right),
\end{aligned}
\Tag{245}\displaybreak[0] \\[12pt]
\begin{aligned}
{H_x}' &= H_x, &
{H_y}' &= \kappa \left(H_y + \frac{V}{c} D_z\right), &
{H_z}' &= \kappa \left(H_z - \frac{V}{c} D_y\right), \\
{B_x}' &= B_x, &
{B_y}' &= \kappa \left(B_y + \frac{V}{c} E_z\right), &
{B_z}' &= \kappa \left(B_z - \frac{V}{c} E_y\right),
\end{aligned}
\Tag{246}\displaybreak[0] \\[12pt]
\begin{gathered}
\rho' = \kappa \left(\rho - \frac{V}{c^2}\, i_x\right),\qquad
{i_x}' = \kappa(i_x - V_\rho), \qquad
{i_y}' = i_y, \qquad
{i_z}' = i_z.
\end{gathered}
\Tag{247}
\end{align*}}%

It will be noted that for free space these equations will reduce to
the same form as our earlier transformation equations \DPchg{(227~to~229)}{(227)~to~(229)}
since we shall have the simplifications $\vc{D} = \vc{E}$, $\vc{B} = \vc{H}$ and $\vc{i} = \rho \vc{u}$.

We may also call attention at this point to the fact that our fundamental
%% -----File: 199.png---Folio 185-------
equations for electromagnetic phenomena \DPchg{(238--241)}{(238)--(241)} in dielectric
media might have been derived from the principle of least
action, making use of an expression for kinetic potential which could
be shown equal to $H = \ds\int dV \left(\frac{\vc{E·D}}{2} - \frac{\vc{H}·\vc{B}}{2}\right)$, and it will be noticed
that our transformation equations for these quantities are such as to
preserve that necessary invariance for $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ which we found in
\Chapref{IX} to be the general requirement for any dynamical development
which agrees with the theory of relativity.

\Paragraph{171.} We are now in a position to handle the theory of moving
media. Consider a homogeneous medium moving past a system of
coördinates $S$ in the $X$\DPchg{-}{~}direction with the velocity~$V$; our problem is
to discover relations between the various electric and magnetic
vectors in this medium. To do this, consider a new system of coördinates~$S'$
also moving past our original system with the velocity~$V$.
Since the medium is stationary with respect to this new system~$S'$ we
may write for measurements referred to~$S'$ in accordance with equations
\DPchg{(242~to~244)}{(242)~to~(244)} the relations
\begin{align*}
\vc{D'} &= \epsilon \vc{E'},\\
\vc{B'} &= \mu \vc{H'},\\
\vc{i'} &= \sigma \vc{E'},
\end{align*}
which, as we have already pointed out, are known experimentally to
be true in the case of \emph{stationary, homogeneous} media. $\epsilon$,~$\mu$ and~$\sigma$ are
evidently the values of dielectric constant, permeability and conductivity
of the material in question, which would be found by an
experimenter with respect to whom the medium is stationary.

Making use of our transformation equations \DPchg{(245~to~247)}{(245)~to~(247)} we can
obtain by obvious substitutions the following set of relations for
measurements made with respect to the original system of coördinates~$S$:
\begin{align*}
&\begin{aligned}
D_x &= \epsilon E_x, \\
D_y - \frac{V}{c} H_z
  &= \epsilon \left(E_y - \frac{V}{c} B_z\right), \\
%% -----File: 200.png---Folio 186-------
D_z + \frac{V}{c} H_y
  &= \epsilon \left(E_z + \frac{V}{c} B_y\right),
\end{aligned}
\Tag{248} \displaybreak[0] \\[12pt]
&\begin{aligned}
B_x &= \mu H_x, \\
B_y + \frac{V}{c} E_z
  &= \mu\left(H_y + \frac{V}{c} D_z\right), \\
B_z - \frac{V}{c}E_y
  &= \mu\left(H_z - \frac{V}{c} D_y\right),
\end{aligned}
\Tag{249} \displaybreak[0] \\[12pt]
&\begin{aligned}
\kappa (i_x - V_\rho) &= \sigma E_x, \\
i_y &= \sigma\kappa \left(E_y - \frac{V}{c} B_z\right), \\
i_z &= \sigma\kappa \left(E_z + \frac{V}{c} B_y\right).
\end{aligned}
\Tag{250}
\end{align*}

\Subsubsection{172}{Theory of the Wilson Experiment.} The equations which we
have just developed for moving media are, as a matter of fact, in
complete accord with the celebrated experiment of H.~A. Wilson on
moving dielectrics and indeed all other experiments that have been
performed on moving media.

Wilson's experiment consisted in the rotation of a hollow cylinder
of dielectric, in a magnetic field which was parallel to the axis of the
cylinder. The inner and outer surfaces of the cylinder were covered
with a thin metal coating, and arrangements made with the help of
wire brushes so that electrical contact could be made from these
coatings to the pairs of quadrants of an electrometer. By reversing
the magnetic field while the apparatus was in rotation it was possible
to measure with the electrometer the charge produced by the electrical
displacement in the dielectric. We may make use of our equations
to compute the quantitative size of the effect.
\begin{figure}[hbt]
  \begin{center}
    \Fig{15}
    \Input[3.75in]{200}
  \end{center}
\end{figure}
%% -----File: 201.png---Folio 187-------

Let \Figref{15} represent a cross-section of the rotating cylinder.
Consider a section of the dielectric~$AA$ which is moving perpendicularly
to the plane of the paper in the $X$\DPchg{-}{~}direction with the velocity~$V$. Let
the magnetic field be in the $Y$\DPchg{-}{~}direction parallel to the axis of rotation.
The problem is to calculate dielectric displacement~$D_z$ in the $Z$\DPchg{-}{~}direction.

Referring to equations~(248) we have
\begin{align*}
D_z + \frac{V}{c} H_y &= \epsilon \left(E_z + \frac{V}{c} B_y\right), \\
\intertext{and, substituting the value of~$B_y$ given by equations~(249),}
B_y + \frac{V}{c} E_z &= \mu \left(H_y + \frac{V}{c} D_z\right)
\end{align*}
we obtain
\[
\left(1 - \epsilon\mu\, \frac{V^2}{c^2}\right) D_z
  = \epsilon \left(1 - \frac{V^2}{c^2}\right) E_z
  + \frac{V}{c}\, (\epsilon\mu - 1)\, H_y,
\]
or, neglecting terms of orders higher than~$\dfrac{V}{c}$, we have
\[
D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon\mu - 1)\, H_y.
\Tag{251}
\]

For a substance whose permeability is practically unity such as
Wilson actually used the equation reduces to
\[
D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon - 1)\, H_y,
\]
and this was found to fit the experimental facts, since measurements
with the electrometer show the surface charge actually to have the
magnitude $D_z$~per square centimeter in accordance with our equation
$\divg D = \rho$.

It would be a matter of great interest to repeat the Wilson experiment
with a dielectric of high permeability so that we could test the
complete equation~(251). This is of some importance since the
original Lorentz theory led to a different equation,
\[
D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon - 1)\, \mu H_y.
\]
%% -----File: 202.png---Folio 188-------


\Chapter{XIII}{Four-Dimensional Analysis.}
%[** TN: Running head not hyphenated in original]
\SetRunningHeads{Chapter Thirteen.}{Four-Dimensional Analysis.}

\Paragraph{173.} In the present chapter we shall present a four-dimensional
method of expressing the results of the Einstein theory of relativity,
a method which was first introduced by Minkowski, and in the form
which we shall use, principally developed by Wilson and Lewis. The
point of view adopted\DPtypo{,}{} consists essentially in considering the properties
of an assumed four-dimensional space in which intervals of time are
thought of as plotted along an axis perpendicular to the three Cartesian
axes of ordinary space, the science of kinematics thus becoming
the geometry of this new four-dimensional space.

The method often has very great advantages not only because it
sometimes leads to considerable simplification of the mathematical
form in which the results of the theory of relativity are expressed,
but also because the analogies between ordinary geometry and the
geometry of this imaginary space often suggest valuable modes of
attack. On the other hand, in order to carry out actual numerical
calculations and often in order to appreciate the physical significance
of the conclusions arrived at, it is necessary to retranslate the results
obtained by this four-dimensional method into the language of ordinary
kinematics. It must further be noted, moreover, that many important
results of the theory of relativity can be more easily obtained
if we do not try to employ this four-dimensional geometry. The
reader should also be on his guard against the fallacy of thinking that
extension in time is of the same nature as extension in space merely
because intervals of space and time can both be represented by
plotting along axes drawn on the same piece of paper.

\Subsubsection{174}{Idea of a Time Axis.} In order to grasp the method let us
consider a particle constrained to move along a single axis, say~$OX$,
and let us consider a time axis~$OT$ perpendicular to~$OX$. Then the
\emph{position} of the particle at any \emph{instant} of time can be represented by a
point in the $XT$~plane, and its motion as time progresses by a line in
the plane. If, for example, the particle were stationary, its behavior
%% -----File: 203.png---Folio 189-------
in time and space could be represented by a line parallel to the time
axis~$OT$ as shown for example by the line~$ab$ in \Figref{16}. A particle
\begin{figure}[hbt]
  \begin{center}
    \Fig{16}
    \Input[3.5in]{203}
  \end{center}
\end{figure}
moving with the uniform velocity $u = \dfrac{dx}{dt}$ could be represented by a
straight line $ac$ making an angle with the time axes, and the kinematical
behavior of an accelerated particle could be represented by a
curved line.

By conceiving of a \emph{four}-dimensional space we can extend this
method which we have just outlined to include motion parallel to
all three space axes, and in accordance with the nomenclature of
Minkowski might call such a geometrical representation of the space-time
manifold ``the world,'' and speak of the points and lines which
represent the instantaneous positions and the motions of particles as
``world-points'' and ``world-lines.''

\Subsubsection{175}{Non-Euclidean Character of the Space.} It will be at once
evident that the graphical method of representing kinematical events
which is shown by \Figref[Figure]{16} still leaves something to be desired. One
of the most important conclusions drawn from the theory of relativity
was the fact that it is impossible for a particle to move with a velocity
greater than that of light, and it is evident that there is nothing in
our plot to indicate that fact, since we could draw a line making any
desired angle with the time axis, up to perpendicularity, and thus
%% -----File: 204.png---Folio 190-------
represent particles moving with any velocity up to infinity,
\[
u = \frac{\Delta x}{\Delta t} = \infty.
\]
It is also evident that there is nothing in our plot to correspond to
that invariance in the velocity of light which is a cornerstone of the
theory of relativity. Suppose, for example, the line~$OC$, in \Figref{17},
\begin{figure}[hbt]
  \begin{center}
    \Fig{17}
    \Input[3.75in]{204}
  \end{center}
\end{figure}
represents the trajectory of a beam of light with the velocity $\dfrac{\Delta x}{\Delta t} = c$;
there is then nothing so far introduced into our method of plotting
to indicate the fact that we could not equally well make use of another
set of axes~$OX'T'$, inclined to the first and thus giving quite a different
value, $\dfrac{\Delta x'}{\Delta t'}$, to the velocity of the beam of light.

There are a number of methods of meeting this difficulty and
obtaining the invariance for the four-dimensional expression $x^2 + y^2
+ z^2 - c^2t^2$ (see \Chapref{IV}) which must characterize our system of
kinematics. One of these is to conceive of a four-dimensional Euclidean
%% -----File: 205.png---Folio 191-------
space with an imaginary time axis, such that instead of plotting
real instants in time along this axis we should plot the quantity
$l = ict$ where $i = \sqrt{-1}$. In this way we should obtain invariance
for the quantity $x^2 + y^2 + z^2 + l^2 = x^2 + y^2 + z^2 - c^2t^2$, since it may
be regarded as the square of the magnitude of an imaginary four-dimensional
radius vector. This method of treatment has been
especially developed by Minkowski, Laue, and Sommerfeld. Another
method of attack, which has been developed by Wilson and Lewis
and is the one which we shall adopt in this chapter, is to use a real
time axis, for plotting the real quantity~$ct$, but to make use of a non-Euclidean
four-dimensional space in which the quantity $(x^2 + y^2 + z^2
- c^2t^2)$ is itself taken as the square of the magnitude of a radius vector.
This latter method has of course the disadvantages that come from
using a non-Euclidean space; we shall find, however, that these reduce
largely to the introduction of certain rules as to signs. The method
has the considerable advantage of retaining a real time axis which is
of some importance, if we wish to visualize the methods of attack and
to represent them graphically.

We may now proceed to develop an analysis for this non-Euclidean
space. We shall find this to be quite a lengthy process but at its
completion we shall have a very valuable instrument for expressing
in condensed language the results of the theory of relativity. Our
method of treatment will be almost wholly analytical, and the geometrical
analogies may be regarded merely as furnishing convenient
names for useful analytical expressions. A more geometrical method
of attack will be found in the original work of Wilson and Lewis.


\Section[I]{Vector Analysis of the Non-Euclidean Four-Dimensional
Manifold.}

\Paragraph{176.} Consider a four-dimensional manifold in which the position
of a point is determined by a radius vector
\[
\vc{r} = (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4),
\]
where $\vc{k}_1$,~$\vc{k}_2$,~$\vc{k}_3$ and~$\vc{k}_4$ may be regarded as unit vectors along four
mutually perpendicular axes and $x_1$,~$x_2$,~$x_3$, and~$x_4$ as the magnitudes
of the four components of~$\vc{r}$ along these four axes. We may identify
$x_1$,~$x_2$, and~$x_3$ with the three spatial coördinates of a point $x$,~$y$ and~$z$
%% -----File: 206.png---Folio 192-------
with reference to an ordinary set of space axes and consider~$x_4$ as a
coördinate which specifies the time (multiplied by the velocity of
light) when the occurrence in question takes place at the point~$xyz$.
We have
\[
x_1 = x,\qquad
x_2 = y,\qquad
x_3 = z,\qquad
x_4 = ct,
\Tag{252}
\]
and from time to time we shall make these substitutions when we
wish to interpret our results in the language of ordinary kinematics.
We shall retain the symbols $x_1$,~$x_2$,~$x_3$, and~$x_4$ throughout our development,
however, for the sake of symmetry.

\Subsubsection{177}{Space, Time and Singular Vectors.} Our space will differ in
an important way from Euclidean space since we shall consider three
classes of one-vector, space, time and singular vectors. Considering
the coördinates $x_1$,~$x_2$,~$x_3$, and~$x_4$ which determine the end of a radius
vector, \\
\emph{Space or $\gamma$-vectors} will have components such that
\[
({x_1}^2 + {x_2}^2 + {x_3}^2) > {x_4}^2,
\]
and we shall put for their magnitude
\[
s = \sqrt{{x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2}.
\Tag{253}
\]
\emph{Time or $\delta$-vectors} will have components such that
\[
{x_4}^2 > ({x_1}^2 + {x_2}^2 + {x_3}^2),
\]
and we shall put for their magnitude
\[
s = \sqrt{{x_4}^2-{x_1}^2- {x_2}^2 - {x_3}^2}.
\Tag{254}
\]
\emph{Singular or $\alpha$-vectors} will have components such that
\[
({x_1}^2 + {x_2}^2 + {x_3}^2) = {x_4}^2,
\]
and their magnitude will be zero.

\Subsubsection{178}{Invariance of $x^2 + y^2 + z^2 - c^2t^2$.} Since we shall naturally
consider the magnitude of a vector to be independent of any particular
choice of axes we have obtained at once by our definition of magnitude
for any rotation of axes that invariance for the expression
\[
({x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2) = (x^2 + y^2 + z^2 - c^2t^2),
\]
%% -----File: 207.png---Folio 193-------
which is characteristic of the Lorentz transformation, and have thus
evidently set up an imaginary space which will be suitable for plotting
kinematical events in accordance with the requirements of the theory
of the relativity of motion.

\Subsubsection{179}{Inner Product of One-Vectors.} We shall define the inner
product of two one-vectors with the help of the following rules for the
multiplication of unit vectors along the axes
\[
\vc{k}_1 · \vc{k}_1 = \vc{k}_2 · \vc{k}_2 = \vc{k}_3· \vc{k}_3 = 1,\qquad
\vc{k}_4 · \vc{k}_4 = -1,\qquad \vc{k}_n · \vc{k}_m = 0.
\Tag{255}
\]

It should be noted, of course, that there is no particular significance
in picking out the product $\vc{k}_4 · \vc{k}_4$ as the one which is negative;
it would be equally possible to develop a system in which the
products $\vc{k}_1 · \vc{k}_1, \vc{k}_2 · \vc{k}_2$ and $\vc{k}_3 · \vc{k}_3$ should be negative and $\vc{k}_4 · \vc{k}_4$ positive.

The above rules for unit vectors are sufficient to define completely
the inner product provided we include the further requirements that
this product shall obey the \emph{associative law} for a scalar factor and the
\emph{distributive} and \emph{commutative} laws, namely
\[
\begin{aligned}
(n\vc{a}) · \vc{b}  &= n(\vc{a} · \vc{b}) = (\vc{a}· \vc{b})(n), \\
\vc{a} · \vc{(b+c)} &= \vc{a} · \vc{b} + \vc{a} · \vc{c}, \\
\vc{a} · \vc{b}     &= \vc{b} · \vc{a}.
\end{aligned}
\Tag{256}
\]

For the inner product of a one-vector by itself we shall have, in
accordance with these rules,
\begin{multline*}
\vc{r} · \vc{r}
  = (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4)
  · (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4) \\
  = (x_1^2 + x_2^2 + x_3^2 - x_4^2)
\Tag{257}
\end{multline*}
and hence may use the following expressions for the magnitudes of
vectors in terms of inner product
\[
s = \sqrt{ \vc{r} · \vc{r}} \text{ for $\gamma$-vectors},\qquad
s = \sqrt{-\vc{r} · \vc{r}} \text{ for $\delta$-vectors}.
\Tag{258}
\]

For curved lines we shall define interval along the curve by the
equations
\[
\begin{aligned}
\int ds &= \int\sqrt { dr · dr} \text{ for $\gamma$-curves}, \\
\int ds &= \int\sqrt {-dr · dr} \text{ for $\delta$-curves}.
\end{aligned}
\Tag{259}
\]
%% -----File: 208.png---Folio 194-------

Our rules further show us that we may obtain the space components
of any one vector by taking its inner product with a unit vector
along the desired axis and may obtain the time component by taking
the negative of the corresponding product. Thus
\[
\begin{aligned}
\vc{r}·\vc{k}_1
  &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_1 =  x_1,\\
\vc{r}·\vc{k}_2
  &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_2 =  x_2,\\
\vc{r}·\vc{k}_3
  &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_3 =  x_3,\\
\vc{r}·\vc{k}_4
  &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_4 = -x_4.\\
\end{aligned}
\Tag{260}
\]

We see finally moreover in general that the inner product of any
pair of vectors will be numerically equal to the product of the magnitude
of either by the projection of the other upon it, the sign depending
on the nature of the vectors involved.

\Subsubsection{180}{Non-Euclidean Angle.} We shall define the non-Euclidean
angle~$\theta$ between two vectors $\vc{r}_1$~and~$\vc{r}_2$ in terms of their magnitudes
$s_1$~and~$s_2$ by the expressions
\[
\pm \vc{r}_1·\vc{r}_2
  = (s_1 × \text{projection}\ s_2)
  = s_1s_2\cosh\theta,
\Tag{261}
\]
the sign depending on the nature of the vectors in the way indicated
in the preceding section. We note the analogy between this equation
and those familiar in Euclidean vector-analysis, the hyperbolic
\DPtypo{trigonometeric}{trigonometric} functions taking the place of the circular functions
used in the more familiar analysis.

For the angle between unit vectors $\vc{k}$~and~$\vc{k'}$ we shall have
\[
\cosh\theta = \pm \vc{k}·\vc{k'},
\Tag{262}
\]
where the sign must be chosen so as to make $\cosh\theta$ positive, the
plus sign holding if both are $\gamma$-vectors and the minus sign if both are
$\delta$-vectors.

\Subsubsection{181}{Kinematical Interpretation of Angle in Terms of Velocity.}
At this point we may temporarily interrupt the development of our
four-dimensional analysis to consider a kinematical interpretation of
non-Euclidean angles in terms of velocity. It will be evident from
our introduction that the behavior of a moving particle can be represented
in our four-dimensional space by a $\delta$-curve,\footnote
  {It is to be noted that the actual trajectories of particles are all of them represented
  by $\delta$-curves since as we shall see $\gamma$-curves would correspond to velocities
  greater than that of light.}
each point on
%% -----File: 209.png---Folio 195-------
this curve denoting the position of the particle at a given instant of
time, and it is evident that the velocity of the particle will be determined
by the angle which this curve makes with the axes.

Let $\vc{r}$ be the radius vector to a given point on the curve and consider
the derivative of~$\vc{r}$ with respect to the interval $s$ along the curve;
we have
\[
\vc{w} = \frac{d\vc{r}}{ds}
  = \frac{dx_1}{ds}\, \vc{k}_1
  + \frac{dx_2}{ds}\, \vc{k}_2
  + \frac{dx_3}{ds}\, \vc{k}_3
  + \frac{dx_4}{ds}\, \vc{k}_4,
\Tag{263}
\]
and this may be regarded as a unit vector tangent to the curve at the
point in question.

If $\phi$ is the angle between the $\vc{k}_4$~axis and the tangent to the curve
at the point in question, we have by equation~(262)
\[
\cosh\phi = - \vc{w}·\vc{k}_4 = \frac{dx_4}{ds};
\]
making the substitutions for $x_1$,~$x_2$,~$x_3$, and~$x_4$, in terms of $x$,~$y$,~$z$ and~$t$
we may write, however,
\[
ds = \sqrt{\smash[b]{dx_4^2 - dx_1^2 - dx_2^2 - dx_3^2}}
   = \sqrt{1 - \frac{u^2}{c^2}}\, c\, dt,                    \Tag{264}
\]
which gives us
\[
\cosh\phi = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}             \Tag{265}
\]
and by the principles of hyperbolic trigonometry we may write the
further relations
\begin{gather*}
\sinh\phi = \frac{\smfrac{u}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, \Tag{266}
\displaybreak[0] \\
\tanh\phi = \frac{u}{c}.                                      \Tag{267}
\end{gather*}


%[** TN: Heading set like a \Section in original]
\Subsection{Vectors of Higher Dimensions}

\Subsubsection{182}{Outer Products.} We shall define the outer product of two
one-vectors so that it obeys the \emph{associative law} for a scalar factor, the
%% -----File: 210.png---Folio 196-------
\emph{distributive law} and the \emph{anti-commutative law}, namely,
\[
\begin{aligned}
(n\vc{a}) × \vc{b} &= n(\vc{a} × \vc{b}) = \vc{a} × (n\vc{b}),\\
  \vc{a}  × (\vc{b} + \vc{c}) &= \vc{a} × \vc{b} + \vc{a} × \vc{c}\DPchg{}{,}\quad
( \vc{a}  + \vc{b}) × \vc{c} = \vc{a} × \vc{c} + \vc{b} × \vc{c}, \\
  \vc{a}  × \vc{b} &= -\vc{b}× \vc{a}.
\end{aligned}
\Tag{268}
\]

From a geometrical point of view, we shall consider the outer
product of two one-vectors to be itself a \emph{two-vector}, namely the parallelogram,
or more generally, the area which they determine. The
sign of the two-vector may be taken to indicate the direction of progression
clockwise or anti-clockwise around the periphery. In order
to accord with the requirement that the area of a parallelogram determined
by two lines becomes zero when they are rotated into the same
direction, we may complete our definition of outer product by adding
the requirement that the outer product of a vector by itself shall be
zero.
\[
\vc{a} × \vc{a} = 0.
\Tag{269}
\]

We may represent the outer products of unit vectors along the
chosen axes as follows:
\[
\begin{aligned}
\vc{k}_1 × \vc{k}_1 &= \vc{k}_2 × \vc{k}_2 = \vc{k}_3 × \vc{k}_3 = \vc{k}_4 × \vc{k}_4 = 0,\\
\vc{k}_1 × \vc{k}_2 &= -\vc{k}_2 × \vc{k}_1 = \vc{k}_{12} = -\vc{k}_{21},\\
\vc{k}_1 × \vc{k}_3 &= -\vc{k}_3 × \vc{k}_1 = \vc{k}_{13} = -\vc{k}_{31},\quad \text{etc.},\\
\end{aligned}
\Tag{270}
\]
where we may regard~$\vc{k}_{12}$, for example, as a unit parallelogram in the
plane~$X_1OX_2$.

We shall continue to use small letters in Clarendon type for one-vectors
and shall use capital letters in Clarendon type for two-vectors.
The components of a two-vector along the six mutually perpendicular
planes $X_1OX_2$,~$X_1OX_3$,~etc., may be obtained by expressing the one-vectors
involved in terms of their components along the axes and
carrying out the indicated multiplication, thus:
\[
\begin{aligned}
\vc{A} &= \vc{a} × \vc{b}
        = (a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4) \\
 &\quad × (b_1\vc{k}_1 + b_2\vc{k}_2 + b_3\vc{k}_3 + b_4\vc{k}_4) \\
       &= (a_1b_2 - a_2b_1)\vc{k}_{12}
        + (a_1b_3 - a_3b_1)\vc{k}_{13}
        + (a_1b_4 - a_4b_1)\vc{k}_{14} \\
 &\quad + (a_2b_3 - a_3b_2)\vc{k}_{23}
        + (a_2b_4 - a_4b_2)\vc{k}_{24}
        + (a_3b_4 - a_4b_3)\vc{k}_{34},
\end{aligned}
\Tag{271}
\]
%% -----File: 211.png---Folio 197-------
or, calling the quantities $(a_1b_2 - a_2b_1)$,~etc., the component magnitudes
of $\vc{A}$,~$A_{12}$,~etc., we may write
\[
\vc{A} = A_{12}\vc{k}_{12} + A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14}
       + A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}.
\Tag{272}
\]

The concept of outer product may be extended to include the
idea of vectors of higher number of dimensions than two. Thus the
outer product of three one-vectors, or of a one-vector and a two-vector
will be a three-vector which may be regarded as a \emph{directed} parallelopiped
in our four-dimensional space. The outer product of four one-vectors
will lead to a four-dimensional solid which would have direction
only in a space of more than four dimensions and hence in our case
will be called a pseudo-scalar. The outer product of vectors the
sum of whose dimensions is greater than that of the space considered
will vanish.

The results which may be obtained from different types of outer
multiplication are tabulated below, where one-vectors are denoted
by small Clarendon type, two-vectors by capital Clarendon type,
three-vectors by Tudor black capitals, and pseudo-scalars by bold face
Greek letters.
{\small%
\begin{align*} %[** TN: Re-breaking]
&\begin{aligned}
\vc{A}
  &= \vc{a} × \vc{b} = -\vc{b} × \vc{a} \\
  &= (a_1b_2 - a_2b_1)\vc{k}_{12}
   + (a_1b_4 - a_3b_1)\vc{k}_{13}
   + (a_1b_4 - a_4b_1)\vc{k}_{14} \\
  &+ (a_2b_3 - a_3b_2)\vc{k}_{23}
   + (a_2b_4 - a_4b_2)\vc{k}_{21}
   + (a_3b_4 - a_4b_3)\vc{k}_{34},
\end{aligned} \displaybreak[0] \\[12pt]
&\begin{aligned}
\Alpha
  &= \vc{c} × \vc{A} \\
  &= (c_1A_{23} - c_2A_{13} + c_3A_{12})\vc{k}_{123}
   + (c_1A_{24} - c_2A_{14} + c_4A_{12})\vc{k}_{124} \\
  &+ (c_1A_{34} - c_2A_{14} + c_4A_{15})\vc{k}_{134}
   + (c_2A_{34} - c_3A_{24} + c_4A_{23})\vc{k}_{234}
\end{aligned}
\Tag{273} \displaybreak[0] \\[12pt]
&\begin{aligned}
\vc{\alpha}
  &= \vc{d} × \Alpha = -\Alpha × \vc{d} \\
  &= (d_1\Alpha_{234} - d_2\Alpha_{134} +
      d_3\Alpha_{124} - d_4\Alpha_{123})\vc{k}_{1234}, \\
\vc{\alpha}
  &= \vc{A} × \vc{B} \\
  &= (A_{12}B_{34} - A_{13}B_{24} + A_{14}B_{23} + A_{23}B_{14}
    - A_{24}B_{13} + A_{34}B_{12})\vc{k}_{1234}.
\end{aligned}
\end{align*}}%

\emph{The signs in these expressions are determined by the general rule
that the sign of any unit vector~$\vc{\bar{k}}_{nmo}$ will be reversed by each transposition
of the order of a pair of adjacent subscripts, thus}:
\[
k_{abcd} = - k_{bacd} = k_{bcad},\qquad \text{etc.},\ \cdots.
\Tag{274}
\]
%% -----File: 212.png---Folio 198-------

\Subsubsection{183}{Inner Product of Vectors in General.} We have previously
defined the inner product for the special case of a pair of one-vectors,
in order to bring out some of the important characteristics of our
non-Euclidean space. We may now give a general rule for the inner
product of vectors of any number of dimensions.

The inner product of any pair of vectors follows the \emph{associative}
law for scalar factors, and follows the \emph{distributive} and \emph{commutative}
laws.

Since we can express any vector in terms of its components, the
above rules will completely determine the inner product of any pair
of vectors provided that we also have a rule for obtaining the inner
products of the unit vectors determined by the mutually perpendicular
axes. This rule is as follows: Transpose the subscripts of the unit
vectors involved so that the common subscripts occur at the end and
in the same order and cancel these common subscripts. If both the
unit vectors still have subscripts the product is zero; if neither vector
has subscripts the product is unity, and if one of the vectors still has
subscripts that itself will be the product. The sign is to be taken
as that resulting from the transposition of the subscripts (see equation~(274)), unless the subscript~$4$ has been cancelled, when the sign
will be changed.

For example:
\[
\begin{aligned}
\vc{k}_{124} · \vc{k}_{34}  &=  \vc{k}_{12} · \vc{k}_{3} = 0, \\
\vc{k}_{132} · \vc{k}_{123} &= -\vc{k}_{123} · \vc{k}_{123} = -1, \\
\vc{k}_{124} · \vc{k}_{42}  &= -\vc{k}_{124} · \vc{k}_{24} = \vc{k}_{1}.
\end{aligned}
\Tag{275}
\]

It is evident from these rules that we may obtain the magnitude
of any desired component of a vector by taking the inner product of
the vector by the corresponding unit vector, it being noticed, of course,
that when the unit vector involved contains the subscript~$4$ we obtain
the negative of the desired component. For example, we may obtain
the $k_{12}$~component of a two-vector as follows:
\[
\begin{aligned}
A_{12}
  = \vc{A} · \vc{k}_{12}
  = (A_{12}\vc{k}_{12} &+ A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} \\
  &+ A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}) · \vc{k}_{12}.
\end{aligned}
\Tag{276}
\]

\Subsubsection{184}{The Complement of a Vector.} In an $n$-dimensional space
any $m$-dimensional vector will uniquely determine a new vector of
%% -----File: 213.png---Folio 199-------
dimensions $(n-m)$ which may be called the complement of the
original vector. The complement of a vector may be exactly defined
as the inner product of the original vector with the unit pseudo-scalar
$\vc{k}_{123\cdots n}$. In general, we may denote the complement of a vector
by placing an asterisk~$*$ after the symbol. As an example we may
write as the complement of a two-vector~$\vc{A}$ in our non-Euclidean four-dimensional
space:
\[
\begin{aligned}
\vc{A}^* &=
\begin{aligned}[t]
  \vc{A} · \vc{k}_{1234}
  = (A_{12}\vc{k}_{12} &+ A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} \\
  &+
      A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}) · \vc{k}_{1234}
\end{aligned} \\
  &= (A_{12}\vc{k}_{34} - A_{13}\vc{k}_{24} - A_{14}\vc{k}_{23}
    + A_{23}\vc{k}_{14} + A_{24}\vc{k}_{13} - A_{34}\vc{k}_{12}).
\end{aligned}
\Tag{277}
\]

\Subsubsection{185}{The Vector Operator, $\Qop$ or Quad.} Analogous to the familiar
three-dimensional vector-operator del,
\[
\nabla
  = \vc{k}_1\, \frac{\partial}{\partial x_1}
  + \vc{k}_2\, \frac{\partial}{\partial x_2}
  + \vc{k}_3\, \frac{\partial}{\partial x_3},
\Tag{278}
\]
we may define the four-dimensional vector-operator quad,
\[
\Qop
  = \vc{k}_1\, \frac{\partial}{\partial x_1}
  + \vc{k}_2\, \frac{\partial}{\partial x_2}
  + \vc{k}_3\, \frac{\partial}{\partial x_3}
  - \vc{k}_4\, \frac{\partial}{\partial x_4}.
\Tag{279}
\]

If we have a scalar or a vector field we may apply these operators
by regarding them formally as one-vectors and applying the rules
for inner and outer multiplication which we have already given.

Thus if we have a scalar function~$F$ which varies continuously
from point to point we can obtain a one-vector which we may call
the four-dimensional gradient of~$F$ at the point in question by simple
multiplication; we have
\[
\grad F= \Qop F
  = \vc{k}_1\, \frac{\partial F}{\partial x_1}
  + \vc{k}_2\, \frac{\partial F}{\partial x_2}
  + \vc{k}_3\, \frac{\partial F}{\partial x_3}
  - \vc{k}_4\, \frac{\partial F}{\partial x_4}.
\Tag{280}
\]
If we have a one-vector field, with a vector~$\vc{f}$ whose value varies
from point to point we may obtain by inner multiplication a scalar
quantity which we may call the four-dimensional divergence of~$\vc{f}$\DPtypo{ we}{. We}
have
\[
\divg\vc{f} = \Qop · \vc{f}
  = \frac{\partial f_1}{\partial x_1}
  + \frac{\partial f_2}{\partial x_2}
  + \frac{\partial f_3}{\partial x_3}
  + \frac{\partial f_4}{\partial x_4}.
\Tag{280}
\]
Taking the outer product with quad we may obtain a two-vector, the
%% -----File: 214.png---Folio 200-------
four-dimensional curl of~$\vc{f}$,
\[
\begin{aligned}%[** TN: Re-aligning]
\curl \vc{f} = \Qop × \vc{f}
  &= \left(\frac{\partial f_2}{\partial x_1}
         - \frac{\partial f_1}{\partial x_2}\right) \vc{k}_{12}
   + \left(\frac{\partial f_3}{\partial x_1}
         - \frac{\partial f_1}{\partial x_3}\right) \vc{k}_{13} \\
  &+ \left(\frac{\partial f_4}{\partial x_1}
         + \frac{\partial f_1}{\partial x_4}\right) \vc{k}_{14}
   + \left(\frac{\partial f_3}{\partial x_2}
         - \frac{\partial f_2}{\partial x_3}\right) \vc{k}_{23} \\
  &+ \left(\frac{\partial f_4}{\partial x_2}
         + \frac{\partial f_2}{\partial x_4}\right) \vc{k}_{24}
   + \left(\frac{\partial f_4}{\partial x_3}
         + \frac{\partial f_3}{\partial x_4}\right) \vc{k}_{34}.
\end{aligned}
\Tag{282}
\]
By similar methods we could apply quad to a two-vector function~$\vc{F}$
and obtain the one-vector function $\Qop · \vc{F}$ and the three-vector function
$\Qop × \vc{F}$.

\Paragraph{186.} Still regarding $\Qop$ as a one-vector we may obtain a number of
important expressions containing~$\Qop$ more than once; we have:
\begin{align*}
\Qop × (\Qop F) &= 0, \quad(283) &
\Qop × (\Qop × \vc{f}) &= 0,\quad (286) \\
%
\Qop · (\Qop · \vc{F}) &= 0, \quad (284) &
\Qop × (\Qop × \vc{F}) &= 0, \quad (287) \\
%
\Qop · (\Qop · \frakF) &= 0, \quad (285) &&
\end{align*}
\begin{align*}
\Qop · (\Qop × \vc{f})
  &= \Qop (\Qop · \vc{f}) - (\Qop · \Qop)\vc{f},
\Tag{288} \\
\Qop · (\Qop × \vc{F})
  &= \Qop × (\Qop · \vc{F}) + (\Qop · \Qop)\vc{F},
\Tag{289}\\
\Qop · (\Qop × \frakF)
  &= \Qop × (\Qop · \frakF) - (\Qop · \Qop)\frakF.
\Tag{290}
\end{align*}

The operator $\Qop · \Qop$ or~$\Qop^2$ has long been known under the name
of the D'Alembertian,
\[
\Qop^2 = \frac{\partial^2}{\partial {x_1}^2}
       + \frac{\partial^2}{\partial {x_2}^2}
       + \frac{\partial^2}{\partial {x_3}^2}
       - \frac{\partial^2}{\partial {x_4}^2}
  = \Delta^2 - \frac{\partial^2}{c^2\, \partial t^2}.
\Tag{291}
\]

From the definition of the complement of a vector given in the
previous section it may be shown by carrying out the proper expansions
that
\[
(\Qop × \phi)^* = \Qop · \phi^*,
\Tag{292}
\]
where $\phi$~is a vector of any number of dimensions.

\Subsubsection{187}{Tensors.} In analogy to three-dimensional tensors we may
define a four-dimensional tensor as a quantity with sixteen components
as given in the following table:
\[
T = \left\{
\begin{matrix}
T_{11} & T_{12} & T_{13} &T_{14}, \\
T_{21} & T_{22} & T_{23} &T_{24}, \\
T_{31} & T_{32} & T_{33} &T_{34}, \\
T_{41} & T_{42} & T_{43} &T_{44},
\end{matrix}
\right.
\Tag{293}
\]
%% -----File: 215.png---Folio 201-------
with the additional requirement that the divergence of the tensor,
defined as follows, shall itself be a one-vector.
\[
\settowidth{\TmpLen}{$\ds\frac{\partial T_{12}}{\partial x_2}
                        +\frac{\partial T_{13}}{\partial x_3}
                        +\frac{\partial T_{14}}{\partial x_4}\,$}%
\begin{aligned}
\divg T &= \left\{
    \frac{\partial T_{11}}{\partial x_1}
  + \frac{\partial T_{12}}{\partial x_2}
  + \frac{\partial T_{13}}{\partial x_3}
  + \frac{\partial T_{14}}{\partial x_4}\right\}\vc{k}_1 \\
 &+ \left\{\frac{\partial T_{21}}{\partial x_1}
  + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_2 \\
 &+ \left\{\frac{\partial T_{31}}{\partial x_1}
  + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_3 \\
 &+ \left\{\frac{\partial T_{41}}{\partial x_1}
  + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_4 \\
\end{aligned}
\Tag{294}
\]

\Subsubsection{188}{The Rotation of Axes.} Before proceeding to the application
of our four-dimensional analysis to the actual problems of relativity
theory we may finally consider the changes in the components of a
vector which would be produced by a rotation of the axes. We have
already pointed out that the quantity $({x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2)$ is an
invariant in our space for any set of rectangular coördinates having
the same origin since it is the square of the magnitude of a radius
vector, and have noted that in this way we have obtained for the
quantity $(x^2 + y^2 + z^2 - c^2t^2)$ the desired invariance which is characteristic
of the Lorentz transformation. In fact we may look upon
the Lorentz transformation as a rotation from a given set of axes to a
new set, with a corresponding re-expression of quantities in terms of
the new components. The particular form of Lorentz transformation,
familiar in preceding chapters, in which the new set of spatial axes
has a velocity component relative to the original set, in the $X$\DPchg{-}{~}direction
alone, will be found to correspond to a rotation of the axes in which
only the directions of the $X_1$~and~$X_4$ axes are changed, the $X_2$~and~$X_3$
axes remaining unchanged in direction.

Let us consider a one-vector
\[
\vc{a}
  = (a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4)
  = ({a_1}'\vc{k_1}' + {a_2}'\vc{k_2}' + {a_3}'\vc{k_3}' + {a_4}'\vc{k_4}'),
\]
where $a_1$,~$a_2$,~$a_3$ and~$a_4$ are the component magnitudes, using a set of
axes which have $\vc{k}_1$,~$\vc{k}_2$,~$\vc{k}_3$ and~$\vc{k}_4$ as unit vectors and ${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$
the corresponding magnitudes using another set of mutually perpendicular
axes with the unit vectors $\vc{k_1}'$,~$\vc{k_2}'$,~$\vc{k_3}'$ and~$\vc{k_4}'$. Our problem,
%% -----File: 216.png---Folio 202-------
now, is to find relations between the magnitudes $a_1$,~$a_2$,~$a_3$ and~$a_4$ and
${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$.

We have already seen\DPtypo{}{,} \DPchg{sections (\Secnumref{179})~and~(\Secnumref{183})}{Sections \Secnumref{179}~and~\Secnumref{183}}, that we may obtain
any desired component magnitude of a vector by taking its inner
product with a unit vector in the desired direction, reversing the
sign if the subscript~$4$ is involved. We may obtain in this way an
expression for~$a_1$ in terms of ${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$. We have
\begin{align*}
a_1 = \vc{a}·\vc{k}_1
  &= ({a_1}'{\vc{k}_1}' + {a_2}'{\vc{k}_2}'
    + {a_3}'{\vc{k}_3}' + {a_4}'{\vc{k}_4}') · {\vc{k}_1} \\
  &= {a_1}'{\vc{k}_1}' · \vc{k}_1 + {a_2}'{\vc{k}_2}' · \vc{k}_1
   + {a_3}'{\vc{k}_3}' · \vc{k}_1 + {a_4}'{\vc{k}_4}' · \vc{k}_1.
\Tag{295}
\end{align*}
By similar multiplications with $\vc{k_2}$,~$\vc{k_3}$ and~$\vc{k_4}$ we may obtain expressions
for $a_2$,~$a_3$ and~$-a_4$. The results can be tabulated in the convenient
form
\[
\begin{array}{c|*{4}{l|}}
    & \Neg{a_1}' & \Neg{a_2}' & \Neg{a_3}' & \Neg{a_4}' \\
\hline
a_1 & \Neg{\vc{k}_1}' · \vc{k}_1 & \Neg{\vc{k}_2}' · \vc{k}_1
    & \Neg{\vc{k}_3}' · \vc{k}_1 & \Neg{\vc{k}_4}' · \vc{k}_1 \\
\hline
a_2 & \Neg{\vc{k}_1}' · \vc{k}_2 & \Neg{\vc{k}_2}' · \vc{k}_2
    & \Neg{\vc{k}_3}' · \vc{k}_2 & \Neg{\vc{k}_4}' · \vc{k}_2 \\
\hline
a_3 & \Neg{\vc{k}_1}' · \vc{k}_3 & \Neg{\vc{k}_2}' · \vc{k}_3
    & \Neg{\vc{k}_3}' · \vc{k}_3 & \Neg{\vc{k}_4}' · \vc{k}_3 \\
\hline
a_4 &    -{\vc{k}_1}' · \vc{k}_4 &    -{\vc{k}_2}' · \vc{k}_4
    &    -{\vc{k}_3}' · \vc{k}_4 &    -{\vc{k}_4}' · \vc{k}_4 \\
\hline
\end{array}
\Tag{296}
\]

Since the square of the magnitude of the vector, $({a_1}^2 + {a_2}^2 + {a_3}^2
- {a_4}^2)$, is a quantity which is to be independent of the choice of axes,
we shall have certain relations holding between the quantities ${\vc{k}_1}'· \vc{k}_1$,
${\vc{k}_1}' · \vc{k}_2$, etc. These relations, which are analogous to the familiar
%% -----File: 217.png---Folio 203-------
conditions of orthogonality in Euclidean space, can easily be shown
to be
\[
\begin{aligned}
({\vc{k}_1}'· \vc{k}_1)^2 + ({\vc{k}_1}'· \vc{k}_2)^2 + ({\vc{k}_1}'· \vc{k}_3)^2 - ({\vc{k}_1}'· \vc{k}_4)^2 &= 1,\\
({\vc{k}_2}'· \vc{k}_1)^2 + ({\vc{k}_2}'· \vc{k}_2)^2 + ({\vc{k}_2}'· \vc{k}_3)^2 - ({\vc{k}_2}'· \vc{k}_4)^2 &= 1,\\
({\vc{k}_3}'· \vc{k}_1)^2 + ({\vc{k}_3}'· \vc{k}_2)^2 + ({\vc{k}_3}'· \vc{k}_3)^2 - ({\vc{k}_3}'· \vc{k}_4)^2 &= 1,\\
({\vc{k}_4}'· \vc{k}_1)^2 + ({\vc{k}_4}'· \vc{k}_2)^2 + ({\vc{k}_4}'· \vc{k}_3)^2 - ({\vc{k}_4}'· \vc{k}_4)^2 &= - 1,
\end{aligned}
\Tag{297}
\]
and
\begin{align*}%[** TN: Re-breaking]
({\vc{k}_1}'· \vc{k}_1)({\vc{k}_2}' · \vc{k}_1)
  &+ ({\vc{k}_1}' · \vc{k}_2)({\vc{k}_2}' · \vc{k}_2) \\
  &+ ({\vc{k}_1}' · \vc{k}_3)({\vc{k}_2}' · \vc{k}_3)
   - ({\vc{k}_1}' · \vc{k}_4)({\vc{k}_2}' · \vc{k}_4) = 0,
\end{align*}
etc., for each of the six pairs of vertical columns in table~(296).

Since we shall often be interested in a simple rotation in which
the directions of the $X_2$~and~$X_3$ axes are not changed, we shall be able
to simplify this table for that particular case by writing
\[
{\vc{k}_2}' = \vc{k}_2,\qquad
{\vc{k}_3}' = \vc{k}_3,
\]
and noting the simplifications thus introduced in the products of the
unit vectors, we shall obtain
\[
\begin{array}{*{5}{c|}}
    & \Neg {a_1}' & {a_2}' & {a_3}' & \Neg {a_4}' \\
\hline
a_1 & \Neg{\vc{k}_1}' · \vc{k}_1  & 0  & 0 & \Neg{\vc{k}_4}' · \vc{k}_1 \\
\hline
a_2 & \Neg 0  &  1  &  0  &  \Neg 0 \\
\hline
a_3 & \Neg 0  &  0  &  1  &  \Neg 0 \\
\hline
a_4 & -{\vc{k}_1}' · \vc{k}_4 & 0 & 0 & -{\vc{k}_4}' · \vc{k}_4 \\
\hline
\end{array}
\Tag{298}
\]
%% -----File: 218.png---Folio 204-------

If now we call~$\phi$ the angle of rotation between the two time axes
${OX_4}'$~and~$OX_4$, we may write, in accordance with equation~(262),
\[
-{\vc{k}_4}' · \vc{k}_4 = \cosh \phi.
\]

Since we must preserve the orthogonal relations~(297) and may
also make use of the well-known expression of hyperbolic trigonometry
\[
\cosh^2 \phi - \sinh^2 \phi = 1,
\]
we may now rewrite our transformation table in the form
\[
\begin{array}{*{5}{c|}}
    & {a_1}' & {a_2}' & {a_3}' & {a_4}' \\
\hline
a_1 & \cosh\phi & 0 &  0  & \sinh \phi \\
\hline
a_2 & 0 & 1 & 0 & 0 \\
\hline
a_3 & 0 & 0 & 1 & 0 \\
\hline
a_4 & \sinh \phi & 0 & 0 & \cosh \phi \\
\hline
\end{array}
\Tag{299}
\]

By a similar process we may obtain transformation tables for the
components of a two-vector~$\vc{A}$. Expressing~$\vc{A}$ in terms of the unit
vectors ${\vc{k}_{12}}'$,~${\vc{k}_{13}}'$, ${\vc{k}_{14}}'$,~etc., and taking successive inner products with
the unit vectors $\vc{k}_{12}$,~$\vc{k}_{13}$, $\vc{k}_{14}$,~etc., we may obtain transformation
equations which can be expressed by the \hyperref[table:300]{tabulation~(300)} shown on
the following page.\DPnote{[** TN: No need for varioref]}
%% -----File: 219.png---Folio 205-------
\begin{sidewaystable}[p]
\phantomsection\label{table:300}%
\renewcommand{\arraystretch}{3}
\[
\begin{array}{c|*{6}{r|}}
   & \multicolumn{1}{c|}{{A_{12}}'} & \multicolumn{1}{c|}{{A_{13}}'}
   & \multicolumn{1}{c|}{{A_{14}}'} & \multicolumn{1}{c|}{{A_{23}}'}
   & \multicolumn{1}{c|}{{A_{24}}'} & \multicolumn{1}{c|}{{A_{34}}'} \\
\hline
A_{12} & {\vc{k}_{12}}' · \vc{k}_{12} & {\vc{k}_{13}}' · \vc{k}_{12}
      & {\vc{k}_{14}}' · \vc{k}_{12} & {\vc{k}_{23}}' · \vc{k}_{12}
      & {\vc{k}_{24}}' · \vc{k}_{12} & {\vc{k}_{34}}' · \vc{k}_{12} \\
\hline
A_{13} & {\vc{k}_{12}}' · \vc{k}_{13} & {\vc{k}_{13}}' · \vc{k}_{13}
      & {\vc{k}_{14}}' · \vc{k}_{13} & {\vc{k}_{23}}' · \vc{k}_{13}
      & {\vc{k}_{24}}' · \vc{k}_{13} & {\vc{k}_{34}}' · \vc{k}_{13} \\
\hline
A_{14} &-{\vc{k}_{12}}' · \vc{k}_{14} & -{\vc{k}_{13}}' · \vc{k}_{14}
      & -{\vc{k}_{14}}' · \vc{k}_{14} & -{\vc{k}_{23}}' · \vc{k}_{14}
      & -{\vc{k}_{24}}' · \vc{k}_{14} & -{\vc{k}_{34}}' · \vc{k}_{14} \\
\hline
A_{23} & {\vc{k}_{12}}' · \vc{k}_{23} & {\vc{k}_{13}}' · \vc{k}_{23}
      & {\vc{k}_{14}}' · \vc{k}_{23} & {\vc{k}_{23}}' · \vc{k}_{23}
      & {\vc{k}_{24}}' · \vc{k}_{23} & {\vc{k}_{34}}' · \vc{k}_{23} \\
\hline
A_{24} & -{\vc{k}_{12}}' · \vc{k}_{24} & -{\vc{k}_{13}}' · \vc{k}_{24}
      & -{\vc{k}_{14}}' · \vc{k}_{24} & -{\vc{k}_{23}}' · \vc{k}_{24}
      & -{\vc{k}_{24}}' · \vc{k}_{24} & -{\vc{k}_{34}}' · \vc{k}_{24} \\
\hline
A_{34} & -{\vc{k}_{12}}' · \vc{k}_{34} & -{\vc{k}_{13}}' · \vc{k}_{34}
      & -{\vc{k}_{14}}' · \vc{k}_{34} & -{\vc{k}_{23}}' · \vc{k}_{34}
      & -{\vc{k}_{24}}' · \vc{k}_{34} & -{\vc{k}_{34}}' · \vc{k}_{34} \\
\hline
\end{array}
\Tag{300}
\]
\end{sidewaystable}

For the particular case of a rotation in which the direction of the
$X_2$~and~$X_3$ axes are not changed we shall have
\[
{\vc{k}_2}' = \vc{k}_2,\qquad
{\vc{k}_3}' = \vc{k}_3,
\]
and very considerable simplification will be introduced. We shall
have, for example,
\begin{alignat*}{4}
&{\vc{k}_{12}}'· \vc{k}_{12}
  &&= ({\vc{k}_1}' × {\vc{k}_2}') · (\vc{k}_1 × \vc{k}_2)
  &&= ({\vc{k}_1}' × \vc{k}_2) · (\vc{k}_1 × \vc{k}_2)
  &&= {\vc{k}_1}' · \vc{k}_1, \\
&{\vc{k}_{13}}' · \vc{k}_{12}
  &&= ({\vc{k}_1}' × {\vc{k}_3}') · (\vc{k}_1 × \vc{k}_2)
  &&= ({\vc{k}_1}' × \vc{k}_3 ) · (\vc{k}_1 × \vc{k}_2)
  &&= 0, \\
&\text{etc.}
\end{alignat*}
Making these and similar substitutions and introducing, as before,
%% -----File: 220.png---Folio 206-------
the relation $-\DPtypo{{\vc{k}'}_4}{{\vc{k}_4}'} · \vc{k}_4 = \cosh \phi$ where $\phi$~is the non-Euclidean angle
between the two time axes, we may write our transformation table
in the form
\[
\begin{array}{*{7}{c|}}
  & \Neg{A_{12}}' & \Neg{A_{13}}' & {A_{14}}' & {A_{23}}' & {A_{24}}' &{A_{34}}' \\
\hline
A_{12} & \Neg\cosh\phi & 0 & 0 & 0 & \sinh\phi & 0 \\
\hline
A_{13} & \Neg0 & \Neg\cosh\phi & 0 & 0 & 0 & \sinh\phi \\
\hline
A_{14} & \Neg0 & \Neg0 & 1 & 0 & 0 & 0 \\
\hline
A_{23} & \Neg0 & \Neg0 & 0 & 1 & 0 & 0 \\
\hline
A_{24} & -\sinh\phi & 0 & 0 & 0 & \cosh\phi & 0 \\
\hline
A_{34} & \Neg0 & -\sinh\phi & 0 & 0 & 0 & \cosh\phi \\
\hline
\end{array}
\Tag{301}
\]

\Subsubsection{189}{Interpretation of the Lorentz Transformation as a Rotation
of Axes.} We may now show that the Lorentz transformation may
be looked upon as a change from a given set of axes to a rotated set.

Since the angle~$\phi$ which occurs in our transformation tables is
that between the $\vc{k}_4$~axis and the new ${\vc{k}_4}'$~axis, we may write, in accordance
with equations (265)~and~(266),
\[
\cosh \phi = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}, \qquad
\sinh \phi = \frac{\smfrac{V}{c}}{\sqrt{1 - \smfrac{V^2}{c^2}}},
\]
where $V$~is the velocity between the two sets of space axes which
correspond to the original and the rotated set of four-dimensional
axes. This will permit us to rewrite our transformation table for the
%% -----File: 221.png---Folio 207-------
components of a one-vector in the forms
\begin{gather*}
\phantomsection\label{table:302}%
\renewcommand{\arraystretch}{2}
\begin{array}{*{5}{c|}}
    & {a_1}' & {a_2}'  & {a_3}' & {a_4}' \\
\hline
a_1 & \dfrac{1}  {\sqrt{1 - \smfrac{V^2}{c^2}}}& 0 & 0
    & \dfrac{V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\
\hline
a_2 &  0  & 1 & 0 & 0  \\
\hline
a_3 & 0  & 0 & 1  & 0 \\
\hline
a_4 & \dfrac{V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} & 0  & 0
    & \dfrac{1}  {\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\
\hline
\end{array} \\
\Tag{302} \\
\renewcommand{\arraystretch}{2}
\begin{array}{*{5}{c|}}
 & a_1 & a_2 & a_3 & a_4 \\
\hline
{a_1}' & \dfrac{1}   {\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0
       & \dfrac{-V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\
\hline
{a_2}' & 0 & 1 & 0 & 0 \\
\hline
{a_3}' & 0 & 0 & 1 & 0 \\
\hline
{a_4}' & \dfrac{-V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0
       & \dfrac{1}   {\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\
\hline
\end{array}
\end{gather*}

Consider now any point $P(x_1, x_2, x_3, x_4)$. The radius vector from
the origin to this point will be $\vc{r} = (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)$, or,
making use of the relations between $x_1$,~$x_2$, $x_3$,~$x_4$ and $x$,~$y$, $z$,~$t$ given
by equations~(252), we may write
\[
\vc{r} = (x\vc{k}_1 + y\vc{k}_2 + z\vc{k}_3 + ct\vc{k}_4).
\]
Applying our transformation table to the components of this one-vector,
we obtain the familiar equations for the Lorentz transformation
\begin{align*}
x' &= \frac{x - Vt}{\sqrt{1 - \smfrac{V^2}{c^2}}}, \\
%% -----File: 222.png---Folio 208-------
y' &= y, \\
z' &= z, \\
t' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t - \frac{V}{c^2}\, x\right).
\end{align*}

We thus see that the Lorentz transformation is to be interpreted
in our four-dimensional analysis as a rotation of axes.

\Subsubsection{190}{Graphical Representation.} Although we have purposely restricted
ourselves in the foregoing treatment to methods of attack
which are almost purely analytical rather than geometrical in nature,
the importance of a graphical representation of our four-dimensional
manifold should not be neglected. The difficulty of representing all
four axes on a single piece of two-dimensional paper is not essentially
different from that encountered in the graphical representation of the
facts of ordinary three-dimensional solid geometry, and these difficulties
can often be solved by considering only one pair of axes at a
time, say $OX_1$~and~$OX_4$, and plotting the occurrences in the $X_1OX_4$
plane. The fact that the geometry of this plane is a non-Euclidean
one presents a more serious complication since the figures that we
draw on our sheet of paper will obviously be Euclidean in nature,
but this difficulty also can be met if we make certain conventions as
to the significance of the lines we draw, conventions which are fundamentally
not so very unlike the conventions by which we interpret as
solid, a figure drawn in ordinary perspective.

Consider for example the diagram shown in \Figref{18}, where we
have drawn a pair of perpendicular axes, $OX_1$,~and~$OX_4$ and the
two unit hyperbolæ given by the equations
\[
\begin{aligned}
{x_1}^2 - {x_4}^2 &= 1, \\
{x_1}^2 - {x_4}^2 &= -1,
\end{aligned}
\Tag{303}
\]
together with their asymptotes, $OA$~and~$OB$, given by the equation
\[
{x_1}^2 - {x_4}^2 = 0.
\Tag{304}
\]
This purely Euclidean figure permits, as a matter of fact, a fairly
satisfactory representation of the non-Euclidean properties of the
manifold with which we have been dealing.
%% -----File: 223.png---Folio 209-------

$OX_1$~and~$OX_4$ may be considered as perpendicular axes in the
non-Euclidean $X_1OX_4$~plane. Radius vectors lying in the quadrant~$AOB$\DPtypo{,}{}
will have a greater component along the~$X_4$ than along the $X_1$~axis
and hence will be $\delta$-vectors with the magnitude $s = \sqrt{{x_4}^2 - {x_1}^2}$,
where $x_1$~and~$x_4$ are the coördinates of the terminal of the vector.
\begin{figure}[hbt]
  \begin{center}
    \Fig{18}
    \Input[4in]{223}
  \end{center}
\end{figure}
$\gamma$-radius-vectors will lie in the quadrant~$BOC$ and will have the magnitude
$s = \sqrt{{x_1}^2 - {x_4}^2}$. Radius vectors lying along the asymptotes
$OA$~and~$OB$ will have zero magnitudes ($s = \sqrt{{x_1}^2 - {x_4}^2} = 0$) and
hence will be singular vectors.

Since the two hyperbolæ have the equations ${x_1}^2 - {x_4}^2 = 1$ and
${x_1}^2 - {x_4}^2 = -1$, rays such as $Oa$,~${Oa}'$, $Ob$,~etc., starting from the
origin and terminating on the hyperbolæ, will all have unit magnitude.
Hence we may consider the hyperbolæ as representing unit pseudo-circles
in our non-Euclidean plane and consider the rays as representing
the radii of these pseudo-circles.

A non-Euclidean rotation of axes will then be represented by
changing from the axes $OX_1$~and~$OX_4$ to ${OX_1}'$~and~${OX_4}'$, and taking
${Oa}'$~and~${Ob}'$ as unit distances along the axes instead of $Oa$~and~$Ob$.
%% -----File: 224.png---Folio 210-------

It is easy to show, as a matter of fact, that such a change of axes
and units does correspond to the Lorentz transformation. Let $x_1$~and~$x_4$
be the coördinates of any point with respect to the original
axes $OX_1$~and~$OX_4$, and ${x_1}''$~and~${x_4}''$ the coördinates of the same point
referred to the oblique axes ${OX_1}'$~and~${OX_4}'$, no change having yet
been made in the actual lengths of the units of measurement. Then,
by familiar equations of analytical geometry, we shall have
\[
\begin{aligned}
x_1 &= {x_1}'' \cos\theta + {x_4}'' \sin\theta, \\
x_4 &= {x_1}'' \sin\theta + {x_4}'' \cos\theta,
\end{aligned}
\Tag{305}
\]
where $\theta$ is the angle~$X_1O{X_1}'$.

We have, moreover, from the properties of the hyperbola,
\[
\frac{{Oa}'}{Oa} = \frac{{Ob}'}{Ob}
  = \frac{1}{\sqrt{\cos^2\theta - \sin^2\theta}},
\]
and hence if we represent by ${x_1}'$~and~${x_4}'$ the coördinates of the point
with respect to the oblique axes and use $O{a}'$~and~$O{b}'$ as unit distances
instead of $Oa$~and~$Ob$, we shall obtain
\begin{align*}
x_1 &= {x_1}'\, \frac{\cos\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}
     + {x_4}'\, \frac{\sin\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}, \\
x_4 &= {x_1}'\, \frac{\sin\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}
     + {x_4}'\, \frac{\cos\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}.
\end{align*}

It is evident, however, that we may write
\[
\frac{\sin\theta}{\cos\theta} = \tan\theta = \frac{dx_1}{ dx_4} = \frac{V}{c},
\]
where $V$ may be regarded as the relative velocity of our two sets of
space axes. Introducing this into the above equations and also
writing $x_1 = x$, $x_4 = ct$, ${x_1}' = x'$, ${x_4}' = ct'$, we may obtain the familiar
equations
\begin{align*}
x &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, (x' + Vt'), \\
t &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} \left(t' + \frac{V}{c^2}\, x'\right).
\end{align*}
%% -----File: 225.png---Folio 211-------
We thus see that our diagrammatic representation of non-Euclidean
rotation in the ${X_1}OX_4$~plane does as a matter of fact correspond to
the Lorentz transformation.

Diagrams of this kind can now be used to study various kinematical
events. $\delta$-curves can be drawn in the quadrant~$AOB$ to represent
the space-time trajectories of particles, their form can be investigated
using different sets of rotated axes, and the equations for
the transformation of velocities and accelerations thus studied.
$\gamma$-lines perpendicular to the particular time axis used can be drawn to
correspond to the instantaneous positions of actual lines in ordinary
space and studies made of the Lorentz shortening. Singular vectors
along the asymptote~$OB$ can be used to represent the trajectory of a
ray of light and it can be shown that our rotation of axes is so devised
as to leave unaltered, the angle between such singular vectors and the
$OX_4$~axis, corresponding to the fact that the velocity of light must
appear the same to all observers. Further development of the possibilities
of graphical representation of the properties of our non-Euclidean
space may be left to the reader.


\Section[II]{Applications of the Four-Dimensional Analysis.}

\Paragraph{191.} We may now apply our four-dimensional methods to a
number of problems in the fields of kinematics, mechanics and electromagnetics.
Our general plan will be to express the laws of the particular
field in question in four-dimensional language, making use of
four-dimensional vector quantities of a kinematical, mechanical, or
electromagnetic nature. Since the components of these vectors
along the three spatial axes and the temporal axis will be closely
related to the ordinary quantities familiar in kinematical, mechanical,
and electrical discussions, there will always be an easy transition from
our four-dimensional language to that ordinarily used in such discussions,
and necessarily used when actual numerical computations
are to be made. We shall find, however, that our four-dimensional
language introduces an extraordinary brevity into the statement of a
number of important laws of physics.


%[** TN: Heading set like a \Section in original]
\Subsection{Kinematics.}

\Subsubsection{192}{Extended Position.} The position of a particle and the particular
instant at which it occupies that position can both be indicated
%% -----File: 226.png---Folio 212-------
by a point in our four-dimensional space. We can call this
the extended position of the particle and determine it by stating the
value of a four-dimensional radius vector
\[
\vc{r} = (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4).
\Tag{306}
\]

\Subsubsection{193}{Extended Velocity.} Since the velocity of a real particle can
never exceed that of light, its changing position in space and time
will be represented by a $\delta$-curve.

The equation for a unit vector tangent to this $\delta$-curve will be
\[
\vc{w} = \frac{d\vc{r}}{ds}
  = \left(\frac{dx_1}{ds}\, \vc{k}_1 + \frac{dx_2}{ds}\, \vc{k}_2
        + \frac{dx_3}{ds}\, \vc{k}_3 + \frac{dx_4}{ds}\, \vc{k}_4\right),
\Tag{307}
\]
where $ds$~indicates interval along the $\delta$-curve; and this important
vector~$\vc{w}$ may be called the extended velocity of the particle.

Remembering that for a $\delta$-curve
\[
ds = \sqrt{d{x_4}^2 - d{x_1}^2 - d{x_2}^2 - d{x_3}^2}
   = c\, dt \sqrt{1 - \frac{u^2}{c^2}},
\Tag{308}
\]
we may rewrite our expression for extended velocity in the form
\[
\vc{w} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\},
\Tag{309}
\]
where $\vc{u}$ is evidently the ordinary three-dimensional velocity of the
particle.

Since $\vc{w}$ is a four-dimensional vector in our imaginary space, we
may use our tables for transforming the components of~$\vc{w}$ from one
set of axes to another. We shall find that we may thus obtain transformation
equations for velocity identical with those already familiar
in \Chapref{IV}.

The four components of $\vc{w}$ are
\[
\frac{\smfrac{u_x}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_1, \qquad
\frac{\smfrac{u_y}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_2, \qquad
\frac{\smfrac{u_z}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_3, \qquad
\frac{\vc{k}_4}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\]
and with the help of \hyperref[table:302]{table~(302)} we may easily obtain, by making
simple algebraic substitutions, the following familiar transformation
%% -----File: 227.png---Folio 213-------
equations:
\begin{gather*}%[** TN: Re-breaking]
  {u_x}' = \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}},\qquad
  {u_y}' = \frac{u_y\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}},\qquad
  {u_z}' = \frac{u_z\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}},\\
\frac{1}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}
  = \frac{1 - \smfrac{u_xV}{c^2}}
         {\sqrt{1 - \smfrac{u^2}{c^2}}\, \sqrt{1 - \smfrac{V^2}{c^2}}}.
\end{gather*}

This is a good example of the ease with which we can derive our
familiar transformation equations with the help of the four-dimensional
method.

\Subsubsection{194}{Extended Acceleration.} We may define the extended acceleration
of a particle as the rate of curvature of the $\delta$-line which determines
its four-dimensional position. We have
\[
c = \frac{d^2\vc{r}}{ds^2} = \frac{d\vc{w}}{ds}
  = \frac{d}{ds}\left[
    \frac{\smfrac{\vc{u}}{c} + \vc{k}_4}
         {\sqrt{1 - \smfrac{u^2}{c^2}}}\right].
\Tag{310}
\]
Or, introducing as before the relation $ds = c\, dt \sqrt{1 - \dfrac{u^2}{c^2}}$, we may write
\begin{multline*}
c = \frac{1}{c^2} \Biggl\{
    \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \frac{d\vc{u}}{dt}
  + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, \frac{u}{c^2}\,
    \frac{du}{dt}\, \vc{u} \\
  + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, \frac{u}{c}\,
    \frac{du}{dt}\, \vc{k}_4\Biggr\},
\Tag{311}
\end{multline*}
%% -----File: 228.png---Folio 214-------
where $\vc{u}$ is evidently the ordinary three-dimensional velocity, and $\dfrac{d\vc{u}}{dt}$
the three-dimensional acceleration; and we might now use our transformation
table to determine the transformation equations for acceleration
which we originally obtained in \Chapref{IV}.

\Subsubsection{195}{The Velocity of Light.} As an interesting illustration of the
application to kinematics of our four-dimensional methods, we may
point out that the trajectory of a ray of light will be represented by a
singular line. Since the magnitude of all singular vectors is zero by
definition, we have for any singular line
\[
{dx_1}^2 + {dx_2}^2 + {dx_3}^2 = {dx_4}^2,
\]
or, since the magnitude will be independent of any particular choice
of axes, we may also write
\[
{{dx_1}'}^2 + {{dx_2}'}^2 + {{dx_3}'}^2 = {{dx_4}'}^2.
\]
Transforming the first of these equations we may write
\[
\frac{{dx_1}^2 + {dx_2}^2 + {dx_3}^2 }{{dx_4}^2}
  = \frac{dx^2 + dy^2 + dz^2 }{c^2\, dt^2} = 1
\]
or
\[
\frac{dl}{dt} = c.
\]
Similarly we could obtain from the second equation
\[
\frac{dl'}{dt'} = c.
\]
We thus see that a singular line does as a matter of fact correspond
to the four-dimensional trajectory of a ray of light having the velocity~$c$,
and that our four-dimensional analysis corresponds to the requirements
of the second postulate of relativity that a ray of light shall
have the same velocity for all reference systems.


%[** TN: Heading set like a \Section in original]
\Subsection{The Dynamics of a Particle.}

\Subsubsection{196}{Extended Momentum.} We may define the extended momentum
of a material particle as equal to the product~$m_0\vc{w}$ of its mass~$m_0$,
measured when at rest, and its extended velocity~$\vc{w}$. In accordance
%% -----File: 229.png---Folio 215-------
with equation~(309) for extended velocity, we may write then, for
the extended momentum,
\[
m_0\vc{w} = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}
  \left(\frac{\vc{u}}{c} + \vc{k}_4\right).
\Tag{312}
\]
Or, if in accordance with our considerations of \Chapref{VI} we put
for the mass of the particle at the velocity~$u$
\[
m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}},
\]
we may write
\[
m_0\vc{w} = m\, \frac{\vc{u}}{c} + m\vc{k}_4.
\Tag{313}
\]
We note that the space component of this vector is ordinary momentum
and the time component has the magnitude of mass, and by
applying our \hyperref[table:302]{transformation table~(302)} we can derive very simply
the transformation equations for mass and momentum already
obtained in \Chapref{VI}.

\Subsubsection{197}{The Conservation Laws.} We may now express the laws for
the dynamics of a system of particles in a very simple form by stating
the principle that the extended momentum of a system of particles is a
quantity which remains constant in all interactions of the particles,
we have then
\[
\Sum m_0\vc{w}
  = \Sum\left(\frac{m\vc{u}}{c} + m\vc{k}_4 \right)
  = \text{ a constant},
\Tag{314}
\]
where the summation $\Sum$ extends over all the particles of the system.

It is evident that this one principle really includes the three
principles of the conservation of momentum, mass, and energy.
This is true because in order for the vector~$\Sum m_0\vc{w}$ to be a constant
quantity, its components along each of the four axes must be constant,
and as will be seen from the above equation this necessitates
the constancy of the momentum~$\Sum m\vc{u}$, of the total mass~$\Sum m$, and of
the total energy~$\Sum \dfrac{m}{c^2}$.
%% -----File: 230.png---Folio 216-------


%[** TN: Heading set like a \Section in original]
\Subsection{The Dynamics of an Elastic Body.}

Our four-dimensional methods may also be used to present the
results of our theory of elasticity in a very compact form.

\Subsubsection{198}{The Tensor of Extended Stress.} In order to do this we shall
first need to define an expression which may be called the four-dimensional
stress in the elastic medium. For this purpose we may take the
symmetrical tensor~$T_m$ defined by the following table:
\[
T_m = \left\{
\begin{matrix}
p_{xx} & p_{xy} & p_{xz} & cg_x, \\
p_{yx} & p_{yy} & p_{yz} & cg_y, \\
p_{zx} & p_{zy} & p_{zz} & cg_z, \\
\dfrac{s_x}{c} & \dfrac{s_y}{c} & \dfrac{s_z}{c} & w,
\end{matrix}
\right.
\Tag{315}
\]
where the spatial components of~$T_m$ are equal to the components of
the symmetrical tensor~$\vc{p}$ which we have already defined in \Chapref{X}
and the time components are related to the density of momentum~$\vc{g}$,
density of energy flow~$\vc{s}$ and energy density~$w$, as shown in the tabulation.

From the symmetry of this tensor we may infer at once the simple
relation between density of momentum and density of energy flow:
\[
\vc{g} = \frac{\vc{s}}{c^2},
\Tag{316}
\]
with which we have already become familiar in \Secref{132}.

\Subsubsection{199}{The Equation of Motion.} We may, moreover, express the
equation of motion for an elastic medium unacted on by external
forces in the very simple form
\[
\divg T_m = 0.
\Tag{317}
\]

It will be seen from our definition of the divergence of a four-dimensional
tensor, \Secref{187}, that this one equation is in reality
equivalent to the two equations
\begin{align*}
\divg\vc{p} + \frac{\partial\vc{g}}{\partial t} &= 0
\Tag{318} \\
\intertext{and}
\divg\vc{s} + \frac{\partial w}{\partial t} &= 0.
\end{align*}
%% -----File: 231.png---Folio 217-------
The first of these equations is identical with~(184) of Chapter~X, %[** TN: Not a useful cross-reference]
which we found to be the equation for the motion of an elastic medium
in the absence of external forces, and the second of these equations
expresses the principle of the conservation of energy.

The elegance and simplicity of this four-dimensional method of
expressing the results of our laborious calculations in \Chapref{X} cannot
fail to be appreciated.


%[** TN: Heading set like a \Section in original]
\Subsection{Electromagnetics.}

We also find it possible to express the laws of the electromagnetic
field very simply in our four-dimensional language.

\Subsubsection{200}{Extended Current.} We may first define the extended current,
a simple but important one-vector, whose value at any point will depend
on the density and velocity of charge at that point. We shall
take as the equation of definition
\[
\vc{q} = \rho_0\vc{w}
  = \rho \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\},
\Tag{319}
\]
where
\[
\rho = \frac{\rho_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}
\]
is the density of charge at the point in question.

\Subsubsection{201}{The Electromagnetic Vector $\vc{M}$.} We may further define a
two-vector~$\vc{M}$ which will be directly related to the familiar vectors
strength of electric field~$\vc{e}$ and strength of magnetic field~$\vc{h}$ by the
equation of definition
\begin{align*}
\vc{M} &= (h_1\vc{k}_{23} + h_2\vc{k}_{31} + h_3\vc{k}_{12}
         - e_1\vc{k}_{14} - e_2\vc{k}_{24} - e_3\vc{k}_{34}) \\
%[** TN: Hack to get equation number vertically centered]
\intertext{or\hfill(320)}
\vc{M^*} &= (e_1\vc{k}_{23} + e_2\vc{k}_{31} + e_3\vc{k}_{12}
           + h_1\vc{k}_{14} + h_2\vc{k}_{24} + h_3\vc{k}_{34}),
\end{align*}
where $e_1$,~$e_2$,~$e_3$, and $h_1$,~$h_2$,~$h_3$ are the components of $\vc{e}$~and~$\vc{h}$.

\Subsubsection{202}{The Field Equations.} We may now state the laws of the
electromagnetic field in the extremely simple form
\begin{align*}
\Qop · \vc{M} &= \vc{q},  \Tag{321} \\
\Qop × \vc{M} &= 0.       \Tag{322}
\end{align*}
%% -----File: 232.png---Folio 218-------

These two simple equations are, as a matter of fact, completely
equivalent to the four field equations which we made fundamental
for our treatment of electromagnetic theory in \Chapref{XII}. Indeed
if we treat~$\Qop$ formally as a one-vector
\[
\left(\vc{k}_1\, \frac{\partial}{\partial x_1}
    + \vc{k}_2\, \frac{\partial}{\partial x_2}
    + \vc{k}_3\, \frac{\partial}{\partial x_3}
    - \vc{k}_4\, \frac{\partial}{\partial x_4}\right)
\]
and apply it to the electromagnetic vector~$\vc{M}$ expressed in the extended
form given in the equation of definition~(320) we shall obtain from~(321)
the two equations
\begin{align*}
\curl \vc{h} - \frac{1}{c}\, \frac{\partial\vc{e}}{\partial t}
  &= \rho\, \frac{\vc{u}}{c}, \\
\divg\vc{e} &= \rho, \\
\intertext{and from (322)}
\divg \vc{h} &= 0,\\
\curl \vc{e} + \frac{1}{c}\, \frac{\partial\vc{h}}{\partial t} &= 0,
\end{align*}
where we have made the substitution $x_4 = ct$. These are of course
the familiar field equations for the Maxwell-Lorentz theory of electromagnetism.

\Subsubsection{203}{The Conservation of Electricity.} We may also obtain very
easily an equation for the conservation of electric charge. In accordance
with equation~(284) we may write as a necessary mathematical
identity
\[
\Qop · (\Qop · \vc{M}) = 0.
\Tag{323}
\]
Noting that $\Qop · \vc{M} = \vc{q}$, this may be expanded to give us the equation
of continuity.
\[
\divg \rho\vc{u} + \frac{\partial\rho}{\partial t} = 0.
\Tag{324}
\]

\Subsubsection{204}{The Product $\vc{M}·\vc{q}$.} We have thus shown the form taken by
the four field equations when they are expressed in four dimensional
language. Let us now consider with the help of our four-dimensional
methods what can be said about the forces which determine the
motion of electricity under the action of the electromagnetic field.

Consider the inner product of the electromagnetic vector and
%% -----File: 233.png---Folio 219-------
the extended current:
\begin{multline*}
\vc{M} · \vc{q}
  = (h_1\vc{k}_{23} + h_2\vc{k}_{31} + h_3\vc{k}_{12}
   - e_1\vc{k}_{14} - e_2\vc{k}_{24} - e_3\vc{k}_{34})
  · \rho \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\} \\
  = \rho \left\{\vc{e} + \frac{[\vc{u} × \vc{h}]^*}{c}\right\}
  + \rho\, \frac{\vc{e} · \vc{h}}{c}\vc{k}_4.
\Tag{325}
\end{multline*}
We see that the space component of this vector is equal to the expression
which we have already found in \Chapref{XII} as the force
acting on the charge contained in unit volume, and the time component
is proportional to the work done by this force on the moving
charge; hence we may write the equation
\[
\vc{M} · \vc{q} = \left\{\vc{f} + \frac{\vc{f} · \vc{u}}{c}\, \vc{k}_4\right\},
\Tag{326}
\]
an expression which contains the same information as that given by
the so-called fifth fundamental equation of electromagnetic theory,
$\vc{f}$~being the force exerted by the electromagnetic field per unit volume
of charged material.

\Subsubsection{205}{The Extended Tensor of Electromagnetic Stress.} We may
now show the possibility of defining a four-dimensional tensor~$T_e$, such
that the important quantity $\vc{M} · \vc{q}$ shall be equal to~$-\divg T_e$. This
will be valuable since we shall then be able to express the equation
of motion for a combined mechanical and electrical system in a very
simple and beautiful form.

Consider the symmetrical tensor
\[
T_e =
\left\{
\begin{matrix}
T_{11} & T_{12} & T_{13} & T_{14}, \\
T_{21} & T_{22} & T_{23} & T_{24}, \\
T_{31} & T_{32} & T_{33} & T_{34}, \\
T_{41} & T_{42} & T_{43} & T_{44},
\end{matrix}
\right.
\Tag{327}
\]
defined by the expression
\[
\begin{aligned}
T_{jk} &= \tfrac{1}{2}
  \{M_{j1}M_{k1} + M_{j2}M_{k2} + M_{j3}M_{k3} - M_{j4}M_{k4} \\
 &\qquad
  + {M_{j1}}^*{M_{k1}}^* + {M_{j2}}^*{M_{k2}}^*
  + {M_{j3}}^*{M_{k3}}^* - {M_{j4}}^*{M_{k4}}^*\},
\end{aligned}
\Tag{328}
\]
where $j$, $k = 1$, $2$, $3$, $4$.
%% -----File: 234.png---Folio 220-------

It can then readily be shown by expansion that
\[
-\divg T_e = \vc{M} · (\Qop · \vc{M}) + \vc{M}^* · (\Qop · \vc{M}^*).
\]
But, in accordance with equations (321),~(326),~(292) and~(322), this
is equivalent to
\[
-\divg T_e = \vc{M} · \vc{q}
%[** TN: Keeping () in numerator, cf. (326) above]
  = \left\{\vc{f} + \frac{(\vc{f} · \vc{u})}{c}\, \vc{k}_4\right\}.
\Tag{329}
\]

Since in free space the value of the force~$\vc{f}$ is zero, we may write
for free space the equation
\[
\divg T_e = 0.
\Tag{330}
\]

This one equation is equivalent, as a matter of fact, to two important
and well-known equations of electromagnetic theory. If we
develop the components $T_{11}$,~$T_{12}$,~etc., of our tensor in accordance
with equations (328)~and~(320) we find that we can write
\[
T_e =
\left\{
\renewcommand{\arraystretch}{2}
\begin{matrix}
\psi_{xx} & \psi_{xy}  & \psi_{xz} & \dfrac{S_x}{c}, \\
\psi_{yx} & \psi_{yy}  & \psi_{yz} & \dfrac{S_y}{c}, \\
\psi_{zx} & \psi_{zxy} & \psi_{zz} & \dfrac{S_z}{c}, \\
\dfrac{s_x}{c}& \dfrac{s_x}{c} & \dfrac{s_x}{c} & w,
\end{matrix}
\right.
\Tag{331}
\]
where we shall have
\[
\begin{aligned}
\psi_{xx}
  &= -\tfrac{1}{2}({e_x}^2 - {e_y}^2 - {e_z}^2 + {h_x}^2 - {h_y}^2 - {h_z}^2), \\
\psi_{xy}
  &= -(e_xh_y + h_xh_y), \\
\text{etc.}& \\
s_x &= c(e_yh_z - e_zh_y), \\
\text{etc.}& \\
w &= \tfrac{1}{2}(e^2 + h^2),
\end{aligned}
\Tag{332}
\]
$\psi$ thus being equivalent to the well-known Maxwell three-dimensional
stress tensor, $s_x$,~$s_y$,~etc., being the components of the Poynting vector
$c\, [\vc{e} × \vc{h}]^*$, and $w$~being the familiar expression for density of electromagnetic
%% -----File: 235.png---Folio 221-------
energy $\dfrac{e^2 + h^2}{s}$. We thus see that equation~(330) is equivalent
to the two equations
\begin{align*}
\divg \psi   + \frac{1}{c^2}\, \frac{\partial s}{\partial t} = 0, \\
\divg \vc{s} + \frac{\partial w}{\partial t} = 0.
\end{align*}
The first of these is the so-called equation of electromagnetic momentum,
and the second, Poynting's equation for the flow of electromagnetic
energy.

\Subsubsection{206}{Combined Electrical and Mechanical Systems.} For a point
not in free space where mechanical and electrical systems are both
involved, taking into account our previous considerations, we may
now write the equation of motion for a combined electrical and
mechanical system in the very simple form
\[
\divg T_m + \divg T_e = 0.
\]
And we may point out in closing that we may reasonably expect all
forces to be of such a nature that our most general equation of motion
for any continuous system can be written in the form
\[
\divg T_1 + \divg T_2 + \cdots = 0.
\]
%% -----File: 236.png---Folio 222-------


\Appendix{I}{Symbols for Quantities.}

\AppSection{Scalar Quantities}{Scalar Quantities. \(Indicated by Italic type.\)}

\begin{longtable}{rl}
$c$&       speed of light.\\
$e$&       electric charge.\\
$E$&       energy.\\
$H$&       kinetic potential.\\
$K$&       kinetic energy.\\
$l$, $m$, $n$&       direction cosines.\\
$L$&       Lagrangian function.\\
$p$&       pressure.\\
$Q$&       quantity of electricity.\\
$S$&       entropy.\\
$t$&       time.\\
$T$&       temperature, function $\ds\Sum m_0c^2 \left(1-\sqrt{1-\frac{u^2}{c^2}}\;\right)$.\\
$U$&       potential energy.\\
$v$&       volume.\\
$V$&       relative speed of coördinate systems, volume.\\
$w$&       energy density.\\
$W$&       work.\\
$\epsilon$&dielectric constant.\\
$\kappa$  &$\dfrac{1}{\sqrt{1-\smfrac{V^2}{c^2}}}$.\\
$\mu$     &index of refraction, magnetic permeability.\\
$\nu $    &frequency.\\
$\rho$    &density of charge.\\
$\sigma$  &electrical conductivity.\\
$\phi$    &non-Euclidean angle between time axes.\\
$\phi_1\phi_2\phi_3 \cdots $& generalized coördinates.\\
$\psi$    &scalar potential.\\
$\psi_1\psi_2\psi_3\cdots$  &  generalized momenta.
\end{longtable}
%% -----File: 237.png---Folio 223-------


\AppSection{Vector Quantities}{Vector Quantities. \(Indicated by Clarendon type.\)}

\begin{longtable}{r l}
$\vc{B}$&    magnetic induction.\\
$\vc{c}$&    extended acceleration.\\
$\vc{D}$&    dielectric displacement.\\
$\vc{e}$&    electric field strength in free space.\\
$\vc{E}$&    electric field strength in a medium.\\
$\vc{f}$&    force per unit volume.\\
$\vc{F}$&    force acting on a particle.\\
$\vc{g}$&    density of momentum.\\
$\vc{h}$&    magnetic field strength in free space.\\
$\vc{H}$&    magnetic field strength in a medium.\\
$\vc{i}$&    density of electric current.\\
$\vc{M}$&    angular momentum, electromagnetic vector.\\
$\vc{p}$&    symmetrical elastic stress tensor.\\
$\vc{q}$&    extended current.\\
$\vc{r}$&    radius vector\DPtypo{}{.}\\
$\vc{s}$&    density of energy flow.\\
$\vc{t}$&    unsymmetrical elastic stress tensor.\\
$\vc{u}$&    velocity.\\
$\vc{w}$&    extended velocity.\\
$\vc{\phi}$& vector potential.
\end{longtable}
%% -----File: 238.png---Folio 224-------


\Appendix{II}{Vector Notation.}

\AppSection{Three Dimensional Space}{Three Dimensional Space.}

%[** TN: No periods after items in this section.]
Unit Vectors, $\vc{i}\ \vc{j}\ \vc{k}$

Radius Vector, $\vc{r} = x\vc{i} + y\vc{j} + z\vc{k}$

Velocity,
\begin{align*}
\vc{u} = \frac{d\vc{r}}{dt}
  &= \dot{x}\vc{i} + \dot{y}\vc{j} + \dot{z}\vc{k} \\
  &= u_x\vc{i} + u_y\vc{j} + u_z\vc{k} \\
\intertext{\indent Acceleration,}
\dot{\vc{u}} = \frac{d^2\vc{r}}{dt^2}
  &= \ddot{x}\vc{i} + \ddot{y}\vc{j} + \ddot{z}\vc{k} \\
  &= \dot{u}_x\vc{i} + \dot{u}_y\vc{j} + \dot{u}_z\vc{k}
\end{align*}

Inner Product,
\[
\vc{a}·\vc{b} = a_xb_x + a_yb_y + a_zb_z
\]

Outer Product,
\[
\vc{a} × \vc{b}
  = (a_xb_y - a_yb_x)\vc{ij}
  + (a_yb_z - a_zb_y)\vc{jk}
  + (a_zb_x - a_xb_z)\vc{ki}
\]

Complement of Outer Product,
\[
[\vc{a} × \vc{b}]^*
  = (a_yb_z - a_zb_y)\vc{i}
  + (a_zb_x - a_xb_z)\vc{j}
  + (a_xb_y - a_yb_x)\vc{k}
\]

The Vector Operator Del or~$\nabla$,
\[
\nabla
  = \vc{i}\, \frac{\partial}{\partial x}
  + \vc{j}\, \frac{\partial}{\partial y}
  + \vc{k}\, \frac{\partial}{\partial z}
\]
\begin{align*}
\grad A &= \nabla A
  = \vc{i}\, \frac{\partial A}{\partial x}
  + \vc{j}\, \frac{\partial A}{\partial y}
  + \vc{k}\, \frac{\partial A}{\partial z} \\
\divg\vc{a} &= \nabla · \vc{a}
  = \frac{\partial a_x}{\partial x}
  + \frac{\partial a_y}{\partial y}
  + \frac{\partial a_z}{\partial z} \\
\curl\vc{a} &= [\nabla × \vc{a}]^* \\
 &= \left(\frac{\partial a_z}{\partial y}
        - \frac{\partial a_y}{\partial z}\right) \vc{i}
  + \left(\frac{\partial a_x}{\partial z}
        - \frac{\partial a_z}{\partial x}\right) \vc{j}
  + \left(\frac{\partial a_y}{\partial x}
        - \frac{\partial a_x}{\partial y}\right) \vc{k}
\end{align*}
%% -----File: 239.png---Folio 225-------

\AppSection{Non-Euclidean Four Dimensional Space.}{Non-Euclidean Four Dimensional Space.}

Unit Vectors, $\vc{k}_1$ $\vc{k}_2$ $\vc{k}_3$ $\vc{k}_4$

Radius Vector,
\begin{align*}
\vc{r} &= x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4 \\
       &= x\vc{i} + y\vc{j} + z\vc{k} + ct\vc{k}_4
\end{align*}

One Vector,
\[
\vc{a} = a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4
\]

Two Vector,
\[
\vc{A} = A_{12}\vc{k}_{12} + A_{13}\vc{k}_{13}
       + A_{14}\vc{k}_{14} + A_{23}\vc{k}_{23}
       + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}
\]

Three Vector,
\[
\Alpha = \frakA_{123}\vc{k}_{123} + \frakA_{124}\vc{k}_{124}
       + \frakA_{134}\vc{k}_{134} + \frakA_{234}\vc{k}_{234}
\]

Pseudo Scalar,
\[
\vc{\alpha} = \alpha\vc{k}_{1234}
\]

Transposition of Subscripts,
\[
\vc{k}_{abc\cdots} = -\vc{k}_{bac\cdots} = \vc{k}_{bca\cdots}
\]

Inner Product of One Vectors,

(\textit{See \Secref{183}}).

Outer Product of One Vectors,
\[
\vc{k}_{ab\cdots} × \vc{k}_{nm\cdots} = \vc{k}_{ab\cdots nm\cdots}
\]

Complement of a Vector,
\[
\vc{\phi}^* = \phi·\vc{k}_{1234}
\]

The Vector Operator Quad or~$\Qop$,
\[
\Qop = \vc{k}_1\frac{\partial}{\partial x_1}
     + \vc{k}_2\frac{\partial}{\partial x_2}
     + \vc{k}_3\frac{\partial}{\partial x_3}
     + \vc{k}_4\frac{\partial}{\partial x_4}
\]

\cleardoublepage
\backmatter

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graphicx.sty    1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
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graphics.sty    2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
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graphics.cfg    2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
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