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THE

UNIVERSITY

OF ILLINOIS
LIBRA~y

ELECTROMAGNETIC
AND

THE MECHANICAL

RADIATION
REACTIONS

ARISING FROM IT
BEING AN ADAMS PRIZE ESSAY IN THE
UNIVERSITY OF CAMBRIDGE

by
G. A. SCHOTT, B.A., D.Sc.
Professor of Applied Mathematics in the University College of Wales, Aberystwyth
Formerly Scholar of Trinity College, Cambridge

Cambridge:
at the University Press
1912

t fandhritvge:
PRINTED BY JOHN CLAY,

h.

AT THE UNIVERSITY PRESS

ELECTROMAGNETIC

RADIATION

CAMBRIDGE UNIVERSITY PRESS
,Lon ron: FETTER LANE, E.C.
C. F. CLAY, MANAGEB

Obinbrgb: 100, PRINCES STREET

i8ertin : A. ASHER AND CO.

Lipig : F. A. EROCKHAUS

)

j~ebo r# : G. P. PUTNAM'S SONS
I8ambap anb

Oalcutta

:

MACMILLAN AND CO.,

All rights reserved

LTD.

PREFACE

THE

subject proposed for the Adams Prize in 1908 was "The Radiation
from Electric Systems or Ions in Accelerated Motion and the Mechanical
Reactions on their Motion which arise from it." This also happened to be
the subject-matter of several investigations which I had carried out from
time to time as preliminaries to a number of papers published in the
Philosophical lMagazine and other journals; advantage was taken of the
opportunity then afforded to collect these investigations together and to
extend them. The time available for this purpose however proved to be
too short for a full treatment of the whole subject; while Radiation was
discussed pretty fully, the Mechanical Reactions had to be dismissed very
briefly. During the interval which has elapsed since the award of the prize
in 1909, the treatment of the latter part of the subject has been extended so
as to complete the original program; I venture to hope that there has been
sufficient improvement in the essay to justify the delay in publication which
has arisen.
In order to avoid any great change in the form of the essay most of the
additional matter has been introduced in seven Appendices, and the alterations
in the original text have been confined to extensions of two problems already
included in it, but somewhat scantily treated. The first problem deals with
the motion of a /-particle in a uniform electric field, Ch. XI, % 151-154,
and the second with the electromagnetic field generated by this motion,
Ch. V, 43-60. Room has been made for the additional matter by slightly
curtailing the treatment of two problems dealing with the field generated by
a uniformly accelerated electric charge, Ch. V, § 25-42. These changes
have been deemed advisable because the second problem admits of a simple
solution in a finite form and thus affords an exceptionally good illustration of
the method of the point potentials. All these additions to the text, as well
as a few others of slight extent, have been enclosed in square brackets in
order to distinguish them from the older portions of the essay.
In order to avoid misunderstandings it will be well to say a few words as
to the scope and method of the present investigation. In consequence of
the discoveries of the last few years in the fields of radioactivity, vacuumtube phenomena, magnetism and radiation, a great need has arisen for a

vi

PREFACE

comprehensive Electron Theory of Matter, which shall systematize the results
already achieved, as well as serve as a guide in future researches. A
beginning has been made by J. J. Thomson in his well known paper on the
Structure of the Atom and in his books on Electricity and Matter and the
Corpuscular Theory of Matter, but these investigations have been carried out
under the restriction, no doubt introduced for the sake of simplicity, that the
negative electrons in the atomic model move with velocities so small compared
with the velocity of light, that they can be treated like the particles of
ordinary mechanics. Moving electric charges however do not behave like the
particles of ordinary mechanics: they do not obey the Law of Action and
Reaction, their mass varies with their speed, and they generate a magnetic
field which reacts on their motion in various ways. When the speeds of the
charges are small compared with that of light, all these effects are small, but
we have no right to regard them as negligible even in that case. For
example, Ritz's well known theory of the production of Spectrum Series rests
on the effect of a magnetic field in the atom on the motion of negative
electrons; a theory of this type is not provided for by an Electron Theory of
Matter, which assimilates the electron to the particle of ordinary mechanics.
Moreover it is by no means certain that all the electrons inside the atom
are moving with speeds small compared with that of light; we know that
3-particles are expelled from comparatively stable atoms, like that of Radium,
with speeds differing from that of light by only 2 0/.
The kinetic energy of
such a -particle amounts to three millionths of an erg, which is five times
the mutual electrostatic energy of two negative electrons in contact. It is
not easy to imagine an arrangement of negative and positive charges in
equilibrium, or in slow stationary motion, which shall be sufficiently permanent
and stable to serve as a model of the Radium atom, and at the same time be
capable of setting free sufficient potential energy to supply the kinetic energy
of a 1-particle and also overcome the attraction of the positive charges. If
on the other hand we suppose the 1-particle to be already moving inside the
Radium atom with a speed comparable with that of light, this difficulty does
not arise. A supposition of this kind no doubt has difficulties of its own, but
it would be unwise to ignore it altogether.
For these reasons it is desirable, before attempting to frame any comprehensive Electron Theory of Matter, to develope the theory of moving
electric charges with as few restrictions respecting their structure and motion
as possible. When expressions for the electromagnetic field due to them have
been obtained, the radiation from them and the mechanical reaction on them
can be calculated; the former depends on the field at a great distance, the
latter on that inside them. The object of this essay is to carry out these
preliminary investigations, to provide, as it were, prolegomena to any future
Electron Theory of Matter.
Accordingly the essay is in method deductive and mathematical, rather

vii

PREFACE

than constructive and physical;- but since there is considerable danger, in a
purely mathematical investigation, of losing touch with reality, the methods
developed have been freely illustrated by concrete examples of their application. The solutions of the problems dealt with in this way have not as a rule
been carried to the point of numerical calculation; this could not have been
done without increasing the size of the book unduly, even if there had been
sufficient time available. Many of the computations would have meant a
great expenditure of labour, as a large number of the integrals and series have
been very little studied. In this connection it is worth noting that the series
principally involved .in the expressions found for the radiation and the
mechanical reaction are of the type known as Kapteyn Series of Bessel
Functions; a simple example is afforded by Bessel's well known solution of
Kepler's equation. The study of the convergence and methods of summation
of these series offers many problems to the pure mathematician and is very
necessary for progress in the Electron Theory of Matter.
Let us now consider the fundamental assumptions on which the present
investigation is based, and also their relation to the Theory of Relativity and
to the Unitary Theory of Radiation (Quantentheorie), which have arisen since
it was begun and have been very widely discussed in the interval. In
choosing the fundamental assumptions I have throughout aimed at securing
the greatest generality consistent with thoroughly well established experimental results. Additional assumptions have only been introduced when
further progress seemed impossible without them, or when a comparison of
results already obtained with experiment clearly indicated that further
restrictions were desirable. For these reasons I have refrained from making
any use, either of the Postulate of Relativity, or of the Aether Hypothesis,
which by some are regarded as inconsistent with each other. Some of the
results obtained in this essay are consistent with the Postulate of Relativity;
others cannot be reconciled with it, at least when it is used in the strictest
possible sense, and find their natural explanation in terms of the aether.
All however are deduced quite independently of either hypothesis. It does
not appear to me that either of the new theories is so well established and so
generally accepted yet as to be properly made the basis of an investigation in
which the utmost generality is aimed at.
The following are the assumptions which have been made:(1) The electromagnetic equations employed are those of Maxwell and
Hertz, in the form used by Larmor and Lorentz. These equations already
imply the existence of a system of axes to which they are referred; it is
immaterial for our purpose whether these axes be regarded as fixed in space,
relative to a fixed aether, or as only fixed relative to the observer.
(2) The expressions for the potentials and the electric and magnetic
forces deduced from the electromagnetic equations are assumed to be continuous functions of the time and coordinates in general. They may indeed,
s.

b

vil

PREFACE

at any given instant, be discontinuous at certain surfaces, or, at any given
point, they may undergo sudden changes at certain times; these discontinuities however will be supposed to be exceptional.
(3) The electric charges, whether positive or negative, are assumed to be
distributed throughout finite, though small regions of space, separated from
each other by regions free from charge. It will be found that, for the purpose
of calculating the radiation by means of the expressions relating to the
distant field, a knowledge of the size and structure of the electric charges
is not needed; only the motion of each charge as a whole must be given.
Hence for this purpose they may be treated as point charges moving in a
prescribed way.
(4) The mechanical force on an element of electric charge, so far as it
depends on the electromagnetic field, is assumed to be given by the formula
of Maxwell for a current-element, as adapted by Larmor and Lorentz to the
case of a moving element of charge. This formula gives the mechanical force
in terms of the velocity of the element and of the electric and magnetic forces
at the place where it is for the moment, whether the field be due to other
elements of the same charge, or to external charges, or to both.
(5) For the sake of generality we shall admit the possible presence of a
mechanical force of non-electromagnetic origin, but we shall suppose that the
resultant of all these forces for the several elements of an electric charge
vanishes identically, or at any rate is proportional to the acceleration of the
charge as a whole. The last alternative amounts to postulating that each
element of charge has associated with it, is as it were loaded with a certain
amount of mass of non-electromagnetic origin; we shall accordingly make
this last assumption as well as the first.
(6) The notion of each separate element of charge is to be found from
the mechanical forces (4) and (5) by applying Newton's Second Law of Motion.
In virtue of (5) the motion of the charge as a whole only depends upon the
resultant of all the electromagnetic forces, internal as well as external, which
act on the several elements, and on the total non-electromagnetic mass, if
there be any.
Little need be said as to assumptions (1) and (4); they are generally
accepted as representing the results of experiments on matter in bulk. Their
extension to the case of an isolated charge is of course hypothetical, but is
justified by the agreement of the conclusions obtained with experiment;
for instance the path calculated for a single Lorentz electron moving in a
given electromagnetic field agrees with that found by experiment for a thin
pencil of /-rays. These assumptions also form the basis of the Theory of
Relativity.
Assumption (2) appears to be a natural, if not a necessary consequence of
(1); nevertheless it may lead to difficulties if some forms of the Unitary
Theory of Radiation be adopted. It is well known that one form of this

PREFACE

ix

theory postulates an atomic structure of radiant energy in order to explain
the production of secondary P-rays from X-rays. In order to obtain the
necessary concentration of energy, it is assumed that the space occupied by a
unit of energy is exceedingly small, so that the distribution of energy in the
field is practically discontinuous. In the case of periodic disturbances we
can represent a discontinuous field of the required kind by means of series of
Fourier's type, built up of periodic solutions of the electromagnetic equations,
and thus are able to explain interference and polarisation. But on this view
a truly homogeneous radiation would be impossible, because the fundamental
vibration would always be accompanied by upper partials, whose intensities
would be comparable with its own. This difficulty does not occur in a later
form of the Unitary Theory, in which the emission of energy by the source
takes place discontinuously, but the radiant energy itself is not discontinuous,
not concentrated in isolated regions of the field. A theory of this later type
appears to be sufficient for the statistical theory of radiation and does not
conflict with our assumption (2).
Assumption (3) is fundamental in the treatment of the mechanical
reactions adopted in the present essay. The only alternative is to assume
that each charge is concentrated in a mathematical point, that is, that it is
a mere centre of force. The two assumptions are equivalent as regards the
determination of the field produced by the moving charge; for since the
point charge is a singular point of the field, we may exclude it by means of
a small closed surface surrounding it, and treat it as equivalent, at all outside
points, to a suitable distribution of charge on the surface. This amounts to
regarding.it as an extended charge for all purposes which do not require us
to enquire into its internal structure.
But when we desire to determine the motion of the charge under the
action of a given external field, the difference between the two assumptions
becomes apparent. For a point charge we must, in the first place, assign a
formula expressing the connection between the mechanical force acting upon
it and the electric and magnetic forces of the field, for instance the formula
of Larmor and Lorentz given by assumption (4). Secondly, we must endow
the point charge with a proper amount of energy and mass, and assume
formulae expressing the variation of these quantities with the speed, for
example those given by the Postulate of Relativity. Lastly, we must
determine the reaction on the charge produced by its radiation, for instance
by the method used by Abraham in 15 of his Elelctromagnetische Theorie
der Strahlung, where he deduces it from the expressions for the momentum
and energy radiated from the moving charge. Thus the intrinsic energy
and the mass are irreducible properties for a point charge, but there is nothing
in its nature to determine the laws according to which they depend upon its
speed; these laws have to be determined by considerations having nothing to
do with the character of the charge itself.
b2

PREFACE

For the extended charge on the other hand we must assume in the first
place a definite size and structure when it is at rest, and secondly a formula
expressing the mechanical force on each element of the charge, for instance
the formula of Larmor and Lorentz adopted in assumption (4). The motion
of each element having been assumed provisionally, we are able to determine
the energy, the momentum, the mass and the mechanical reaction of the
radiation, all by one and the same process. No assumption is needed which
does not follow naturally from the notion of an extended charge; in this
respect this hypothesis is more satisfactory than the last.
In making the assumption of an extended charge we however encounter
a problem which compels our attention. When we assume that a charge has
parts which repel one another, we are bound to explain how it happens that
the charge can exist in spite of the mutual repulsions of its elements. This
problem is meaningless for a point charge, which is ipso facto indivisible.
Its solution for an extended charge requires us to postulate the existence of
forces of non-electromagnetic origin, in accordance with assumption (5).
These, together with the mechanical forces due to the electromagnetic field,
act upon each element of the charge and produce the actual state of rest,
or of motion, as the case may be, in accordance with some assigned law,
such as Newton's Second Law of Motion, adopted in assumption (6). We
see that in this way assumptions (5) and (6) are necessary corollaries of
assumption (3).
Thus there are two alternative methods of finding the mechanical
reactions on an electric charge: one based on the hypothesis of the extended,
and the other on that of the point charge. So far as can be judged a priori,
each, by the help of suitable subsidiary assumptions, can be made to give a
consistent account of the phenomena to be explained, and there seems to be
no decisive reason compelling us to choose one rather than the other. The
choice is largely a matter of personal preference; I have selected the method
of the extended charge because it affords more scope for illustration by means
of mechanical models.
The scheme of the essay may be summarized as follows:Ch. I gives a brief discussion of the funmdamental equations of the electron
theory in the form established by Maxwell and Hertz, and further developed
by Larmor and Lorentz; it involves nothing materially new.
Chs. II and III give the transformation of the Lorentz expressions for the
retarded potentials into integrals, which constitute an extension of Fourier's
type. They are shown to lead to Sommerfeld's integrals and to the point
potentials of Lidnard and Wiechert.
In Chs. IV-VI the point potentials are discussed fully and applied to
the determination of the electromagnetic field in special cases of the motion
of a point charge. The most important problem here considered concerns
the field generated by a Lorentz electron, which moves parallel to the lines

PREFACE

xi

of force of a uniform electrostatic field. When the very small reaction due to
the radiation is neglected, this problem admits of an elegant solution in
a finite form; it is, I believe, the only case of an accelerated motion for
which the field has been completely determined. Attention is also drawn to
the discrepancy between the result obtained in Problem 3 of Ch. V and that
found by Descoudres for the uniform motion of a point charge with a speed
greater than that of light. This discrepancy indicates that there is a
difference in kind between the field due to a uniform motion which has been
acquired from rest, and one which has existed for all past time.
The next four chapters deal with the field generated by motions of electric
charges with any speeds not exceeding that of light, the treatment throughout being based upon the integral solutions obtained in Ch. II.
Chs. VII and VIII are devoted to the consideration of the electromagnetic
fields due to periodic motions of systems of electric charges. Ch. VII gives
the case of a monoperiodic motion, a motion involving only one period. The
results are applied to the uniform motion of a circular ring of charges and to
the calculation of the radiation from it.
Ch. VIII deals with the fields generated by polyperiodic motions, that is
motions involving several incommensurable periods. General expressions are
obtained for the electric and magnetic forces at a great distance from the
system of charges producing the field and are applied to the cases of a
uniform circular motion, a simple harmonic rectilinear motion, elliptic motion
about the centre, disturbed circular motion, motion in an epitrochoid and the
precessional motion of a vibrating system of charges. The last problem is of
some importance for the theory of the Zeeman effect developed by Ritz;
the results show that precession of suitable amount will account for resolutions of a line into as many as nine components, symmetrical as to position
but not necessarily symmetrical as to intensity.
Ch. IX deals with motions which are not strictly periodic, or are
aperiodic, for instance damped vibrations. Motions of this kind involve
discontinuities occurring at one or more periods of their existence; it is shown
that the effect of one of these discontinuities is confined to a greater or less
interval of time near the instant at which the discontinuity occurred, and
that at times other than this the potentials and forces of the field can be expanded in Power Series of Lagrange's type.
Ch. X concerns the electromagnetic field at points on or close to the
orbit of a moving charge, but distant from the charge itself by several
diameters at least. The expressions obtained for the electric and magnetic
forces are applied to the case of a circular ring of electrons in uniform
motion.
Ch. XI gives a brief account of the equations of motion of a single
charge, with applications to uniform circular motion, the same motion when

xii

PREFACE

slightly disturbed, and the motion of a Lorentz electron in a uniform
electrostatic field. The last problem has been already referred to in Ch. V,
Problems 4 and 5.
Ch. XII deals with the motion of a group of electric charges, particularly
with the steady motion of a circular ring of electrons. Owing to the loss of
energy by radiation, a strictly uniform motion is impossible if the charges be
absolutely invariable, unless the loss be supplied from some external source,
which produces the tangential force needed to keep the speed uniform.
Moreover, a system of charges of this kind is essentially indeterminate in
structure, because there is but one equation of motion, the radial one, while
there are two independent variables, the speed and the radius of the orbit.
One of the two variables may have its value arbitrarily assigned, and then
the other is completely determined by the equation of motion; there is
however no condition which shall enable us to fix upon any particular value
of the first variable, and consequently the structure of the ring is only
determined within certain limits necessary to secure stability.
The determinateness of structure of the ring, as well as a high degree of
permanence in spite of radiation, can be secured, if the mass of each charge
be allowed to diminish at an exceedingly slow rate, which would occur in
consequence of a very slow expansion of the charge. Then the loss of energy
due to radiation takes place at the expense of the internal electric energy of
the charges; the condition that this compensation may occur gives a second
equation of motion, the tangential one, and thus we have the two equations
necessary to fix the values of the speed and radius of the ring. They are not
absolutely constant, but are subject to small secular changes; the ring is not
absolutely permanent, but it changes very much more slowly than it would
do if its radiation were not compensated. It has definite periods of vibration,
but they undergo small secular changes.
It is possible that the same result may be secured in another way,
namely by means of an asymmetric charge. It is shown in Appendix D
that a tangential force component is required, as well as the radial component, in order to keep an asymmetric charge moving in a circle; it may
happen that the drag due to radiation just supplies the force that is
needed.
These twelve chapters constitute the essay practically as it was submitted
to the examiners; it has been stated already that the treatment of the
motion of electric charges given in Chs. XI and XII is altogether inadequate,
owing to the lack of time to complete the essay in accordance with the
original plan. The rest of the book, with the exception of Appendix A, is
devoted to remedying this defect.
Appendix A gives an application of the results of Chs. VIII and IX to
the theory of the Doppler effect.

PREFACE

lii

Appendix B is devoted to an investigation of the disturbed motion of a
circular ring of charges. In Problem 1 the methods of Ch. X are employed
to find the electric and magnetic forces produced by an electric charge, when
slightly disturbed from its uniform motion in a circle, at points close to the
circle but at a distance from the charge itself. In Problem 2 the field due to
a ring of charges is calculated, and the mechanical force on one of them due
to all the rest is found. In Problem 3 the methods of Chs. XI and XII and
the results of the two preceding problems are applied to find the equations
of motion of a circular ring of charges when it is disturbed from its steady
motion.
Appendix C deals with the electromagnetic field close to a moving
point charge. The Lagrangian Series obtained for the potentials in Ch. IX
are transformed into expansions proceeding according to the accelerations of
increasing order. These expansions are required in the investigations of
Appendices D and E.
The object of Appendix D is to supply a proof from first principles of the
equations of motion of a single charge, which have been already given in
Ch. XI. The resultant of all the mechanical forces exerted on one element
of the charge by all the rest is calculated, and the mechanical force exerted
by the charge on itself is found by summation over all its elements. On
account of the failure of Newton's Third Law of Motion in the case of electric
charges, the total mechanical force on a charge generally differs from zero;
an expression is found for it, which leads to definitions of the electromagnetic
momentum and mass of the charge, as well as of the reaction on it due to its
radiation; this we already know from the investigations of Abraham,
Sommerfeld and others. For the extended electron the electromagnetic
momentum and mass depend upon its size and structure; the reaction due
to radiation however does not, but has the same value as for an equal point
charge. It is here shown that the electromagnetic momentum is not in the
direction of the mean motion of the charge unless the charge be symmetrical
with respect to a plane perpendicular to the direction of the mean motion,
or to two planes parallel to the direction of the mean motion and perpendicular
to each other.
concerning an objection raised by Abraham to the Theory of Relativity in
the second edition of his Elektromagnetische Theorie der Strahlung, § 49,
which came to my notice too late to be considered in the text. Abraham
points out that, according to the Theory of Relativity, the mass, mno, and the
electric energy, Wo, of a slowly moving electron satisfy the relation
mo = Wo/c2, while the mass of the Lorentz electron is 4/3 of the amount
determined by this equation. The equations (350), § 229 of the text, show

that even for a symmetrical electron m0 always exceeds Wo/c 2 , whatever the

xiv

PREFACE

configuration of the electron may be when it is at rest. Thus Abraham's
objection appears to be valid, namely that the electromagnetic dynamics of the
electron is inconsistent with the Postulate of Relativity; at any rate this is
so if the hypothesis of the extended electron be accepted.
The results of this Appendix are applied to the particular cases of the
electrons of Abraham, Bucherer and Lorentz, and also to that of an
asymmetrical electron derived from the Lorentz electron by a small deformation symmetrical about an axis which is inclined to the direction of the mean
motion.
Appendix E deals with the mechanical explanation of the electron, a
problem which compels our attention since we have accepted the hypothesis
of the extended electron, as we have already pointed out. Poincard's
explanation of the Lorentz electron by means of a uniform surface pressure
is here found to have a much wider application. It is shown that the
hypothesis that the electron is subjected to a uniform surface pressure,
whatever its motion may be, is of itself sufficient to lead to the Lorentz
mass-formula, and to a configuration of the electron in which the surfaces of
equal density are Heaviside ellipsoids, so long as the speed remains constant.
When the speed changes, small redistributions of the charge inside the
electron occur, as well as slight deformations of its bounding surface. By
means of these changes an observer moving with the electron could in theory
detect the acceleration, so that the Postulate of Relativity cannot hold for
accelerated motion.
Moreover it is found that an extended electron, whatever the distribution
of its charge may be, cannot exist unless it is subjected to a suitable pressure
on its outer surface. Hence we must either postulate the existence of some
external medium which shall produce the required surface pressure, or admit
that the elements of charge of the electron exert on each other actions at a
distance, which are not electromagnetic and follow quite different laws. The
first hypothesis, that of an external medium, appears to be the more
reasonable of the two and amounts to admitting the existence of the electromagnetic aether.
Thus it appears that the acceptance of the Postulate of Relativity in its
strictest form almost necessitates the adoption of the hypothesis of the
point charge, while the hypothesis of the extended charge leads naturally to
the adoption of the aether hypothesis.
Appendix F gives a brief sketch of the mechanics of the Lorentz electron;
its object is to show that the adoption of the Lorentz mass-formula leads to
a workable system of equations of motion, not indeed as simple as those of
ordinary mechanics, yet simple enough to allow of our obtaining complete
solutions of many important problems. It is found possible to write down
the equations of Lagrange and Hamilton in the most general case of the

PREFACE

Xv

motion of a Lorentz electron under the influence of a variable electromagnetic
field. When the external field is steady, an energy integral exists; when it
is symmetrical about an axis, an integral of angular momentum about that
axis can be found. These integrals can be used as in ordinary mechanics'to
reduce the order of the system of differential equations of motion; all the
necessary eliminations can be actually performed owing to the simple form
of the Lorentz mass-formula. It is possible to deduce Lagrangian and
Hamiltonian equations of motion for other mass-formulae also, at any rate in
theory, but in practice they are useless, because they are too unwieldy on
account of the complications introduced by the forms of the mass-formulae.
In this respect the Lorentz mass-formula possesses an overwhelming advan-

tage over all others.
Appendix G gives applications of the methods of Appendix F to particular
cases, namely the motion of a Lorentz electron in a steady and uniform
electromagnetic field, including the theory of the experiments of Kauffmann
and Bucherer, and its motion in a steady field, in which the electric force is
central and a function of the radius alone, while the magnetic force is
uniform and of small intensity. The last problem is important for the
precessional theory of the Zeeman effect, which is due to Ritz and has already
been considered in Problem 6, Ch. VIII. It is here found that the angular
velocity of precession, / in Problem 6, is equal to the usual value, eH/2 m,
assumed for the Zeeman effect, but the mass of the electron, m, is that
appropriate to its speed at the moment considered. The angular velocity of
the precessing system about its own axis, n in Problem 6, depends on the
constitution of the system under consideration; in the particular example
studied in the present Appendix the angular velocity n is variable, and has
no connection with the angular velocity of precession, , so that Runge's rule
remains unexplained.
It is obvious that in a long-continued investigation like the present one,
there must be many points of contact with the work of other writers, and
accordingly many of the results here obtained have been anticipated. In
cases of this kind I have not made any change in the text, because the proofs
here given are as a rule different from those used by my predecessors in
publication and may very well serve as verifications of their methods. I hope
that I have taken note of most cases of anticipation of this kind, but no
doubt there are others which have escaped my notice altogether, or have
been seen too late to receive due mention. I cannot however close this
preface without expressing the very great debt I owe to Sir J. Larmor and
to Prof. H. A. Lorentz, whose writings have furnished the theoretical
foundations of this essay, and particularly to my former teacher, Sir J. J.
Thomson, to whose paper on Cathode Rays this investigation owes its
inception.

xvi

PREFACE

I have to thank my friend and colleague, Mr J. S. G. Thomas, for his
kindness in reading the proofs of the essay and for his valuable criticisms and
suggestions.
My thanks are also due to the officials of the University Press for the
exactness of their printing and the care shown in the reproduction of the
diagrams, as well as for the obliging manner in which they have received
many corrections.
G. A. SCHOTT.
ABERYSTWYTH,

March 30, 1912.

CONTENTS
PAGE

CHAPTER

I.

FUNDAMENTAL

EQUATIONS

THEORY.

OF

THE

ELECTRON
1.

...

The equations of the electromagnetic field
The potentials
.

.

..

The complementary functions
The equation of energy.
.

.

.

.

.

.

.
.

.

.

.

.
.

2
3
4

..

.

.

..

The resultant mechanical forcive.

9

CHAPTER II.

TRANSFORMATION OF THE POTENTIALS
Transformation of the retarded potentials .
Schott's solutions.
.
The electric and magnetic forces

CHAPTER III.

.

.

*

*

.

.

.

*

.

*21-

.

.

*

*

22

24

.

PHYSICAL INTERPRETATION OF THE SOLUTIONS

OBTAINED

*

...

28

*...

Physical interpretation of the characteristic equation and the point laws.

Graphic representation of the potentials
Geometrical constructions
CHAPTER V.

20

20

.

Point law for velocities greater than that of light.
CHAPTER IV.

17
18

.

*

Sommerfeld's solutions.
.
.
.
New solution for the sphere
The point laws of Lienard and Wiechert

14
14

*

OTHER TYPES OF SOLUTION.

.

.

.

.

ILLUSTRATIVE EXAMPLES

Uniform rectilinear motion

.

.

* 37

.

.

*

.".

*

*

37

*

*

Uniformly accelerated rectilinear motion
Geometrical considerations
Characteristic curve
Potentials
*
*

29'
* 33
* 34

.

*

40

"

.

*
*

44
46
. 48

.

*

Uniform motion generated from rest by a uniform acceleration
General considerations respecting the last two problems
"
Field due to a Lorentz electron moving along the lines of force of a tuniform
*

electrostatic field

*

*"

.

*

."

Characteristic curve, potentials and forces ..
Geometrical representation
Case when the Lorentz electron is subjected to the action of the exte nrnal

*

53

*

*

'31

*O63
*

*"

for a finite time

"

X-rays

Application to the pulse theory of
"
Field due to a point charge moving in a circle with uniform speed
Characteristic curve
Geometrical representation

*

.

*

*

field

*

*

*

.

65
66

69
77
8384
86

xiiiCONTENTS

xviil

PAGE

CHAPTER VI. REMARKS ON THE SOLUTIONS OBTAINED AND ON
THE METHODS OF CALCULATING THE POTENTIALS IN
GENERAL
."
Difference in character of motions with speeds less and greater than that of light.
Question whether charges exist which move with speeds greater than that of light
Comparison of the various methods of calculating potentials
"

88
88
89
91
92

CHAPTER VII.

95

Various types of characteristic curves

.

."

PERIODIC MOTIONS

Monoperiodic motions
.
Field at a great distance

"

.

.

*

*

Definition of a group of charges
.
Circular ring of equidistant charges in uniform motion
Distant field
Radiation

95
99
102
103
106
109

"
.

"
.

"
."

.

."

ON THE DISTANT FIELD DUE TO A MOVING

CHAPTER VIII.
CHARGE

.

.

0

.

.

General expressions for the potentials and forces

"

Polyperiodic motions

111
112
113
116
119
120
122
126
1-29
133
140
143
145

"

..

rn

s

Expansion of the potentials and forces in Fourier series
Character of the distant field
.
*.

Uniform circular

motion-verification

.

Simple harmonic rectilinear motion
Elliptic motion about the centre ,."
Electron disturbed from uniform circular motion

"

.

Orbital and axial oscillations
Group of electrons and their radiation
Epitrochoidal motion

.

"

"

.

"

Precessional motion of a system of vibrating charges.
Application to Ritz's theory of the Zeeman effect
CHAPR IX.

.

PSEUDO-PERIODIC AND APERIODIC MOTIONS

Treatment of discontinuous motions

.

*

.

.

Series of Lagrange for the potentials

149
149
154
156
159
161

.

.

Distant field due to monoperiodic motion of variable amplitude
Temporary

effe~ct

of a discontinuity
.
Damped rectilinear vibration
.& .It

CHAPTER X.

.

*

.

.

.

111

.

ON THE FIELD NEAR THE ORBIT OF A MOVING

CHARGE OR GROUP

......

The potentials and forces
.
Point charge moving in a circle with uniform velocity
Fourier series for the potentials and forces
Group of charges moving in a circle
.

.

*

.

165

.

.
*
*.
*.
*

165
168
170
171

CONTENTS

xix

x

PAGE

THE MECHANICAL FORCES ACTING ON ELECTRIC

CHAPTER XI.

173

CHARGES IN MOTION

Internal and external mechanical forces
Definitions of electromagnetic momentum and mass and of radiation pressure
.174

Equations of motion and energy.176
.179
Uniform circular motion
.
Uniform circular motion slightly disturbed
in a uniform electrostatic field
Motion of a
.
Error due to neglecting radiation

175

179

.

-particle

181

.

183

CHAPTER XII. THE MOTION OF GROUPS OF ELECTRIC CHARGES.
Groups of electrons in uniform circular motion. Mechanical force on one electron
due to the rest

186

.

Equations of motion of the ring of electrons
Conditions for steady motion

.188

.190

ON THE DOPPLER EFFECT

APPENDIX A.

.193

Radiating system moving with a small uniform velocity
Expressions for the forces

.194

Intensity of the radiation

196

.

.196

OF A RING OF

ON THE DISTURBED MOTION

B.

ELECTRONS

193

.

Change of frequency of the vibrations of the system

APPENDIX

185

.200

Point charge slightly disturbed from uniform circular motion
.205
Electric and magnetic force-components

.200

Group of electrons in uniform circular motion slightly disturbed by a wave
travelling round the circle
Expansion of the forces in series.210
Kapteyn series of Bessel functions

Maxwell's trigonometrical series
.
Expressions for the forces

.206

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.
.

.

.

.

.

.

21911

.

.

.

.

221

Disturbed motion of a circular ring of electrons.
The disturbing forces

The

.

.

.

.

equations of disturbed motion

.
..

.

.

.

.

.

MOTION

"

.

.

.

.

Development of the Lagrangian series for the potentials.
The function VT(k)
.

expansion

.

-

.

.

-

for the potentials.

.4

.

.

.

227

228
.

.

230

.

.

.

.

.

.

232

.

.

-.232-

.9

.

.

.
.

.

.

.

.

Second expansion for the potentials

APPENDIX D.

217

ON THE FIELD CLOSE TO A POINT CHARGE IN

APPENDIX C.

First

212
215
217

.

.

.

THE MECHANICAL FORCE EXERTED BY AN ELEC-

TRON ON ITSELF

.

.

.

.

.

.

.

.

.

.

.

235

CONTENTS

xx

xx
CONTENTS
.243
The relative motion of the electron
The resultant internal force on the electron.245
The electromagnetic momentum

246
249

.

.

The electromagnetic mass
The energy relations of the electron

".

.
.251

255
255

The Abraham electron.
The Bucherer electron.

The Lorentz electron

.257

The deformed Lorentz electron

.258

OF THE ELECTRON

APPENDIX E. THE MECHANICAL EXPLANATION
The equation for the pressure

262

.263

Uniform rectilinear translation.265

Uniform and invariable surface pressure gives the Lorentz
Motion

of the

most general type

mass-formla

268

.

269

.

The increase of pressure due to the acceleration

272
274
278

.

Additional con~ditions for a uniform and invariable surface pressure
The surface pressures and the electromagnetic mass

.

.

The solid Lorentz electron.279

General conclusions.

APPENDIX F.

280

THE MECHANICS OF THE LORENTZ ELECTRON

282

.

The Newtonian equations of motion.

283

The Lagrangian equations.

284

External field symmetrical about an axis.

Ignoration of the angular coordinate

The Hamiltonian equations
The equation of Hamilton and Jacobi
Equations of transformation free from the time
Steady external field. The energy integral

APPENDIX 0.

.

Steady and uniform

290

..

291

.

.

.

.

.

.

.

.

Steady and uniform electromagnetic
.

.

.

field-electric
..

New

295

.

forces

electromagnetic field-electric and magnetic

Experimental applications : Kauffmann's experiment.
c/rn and verifying the Lorentz mass-formula
.

290

.

PROBLEMS ILL'USTRATI VE OF THE MOTION OF THE

LORENTZ ELECTRON

angles

286
287

.

parallel

.

.

.

299

.

and magnetic forces at right

.

.

301

.

Electric force less than the magnetic force--Bucherer's experiment.
Electric force greater than the magnetic_ force
..
.

.

Steady and uniform electromagnetic
at any angle

.

field-electric

296

method of finding

.
.

.

and magnetic forces inclined
.

.

Electric force radial, magnetic force uniform and small
,
.
.
The plane of the orbit precesses about a line of magnetic force through the
centre of electric

force

with angular velocity

- ell/2cm.

Direct deduction of the two integrals of angular momentum

INDEX

.

302
306

.
.

.

.
.

.

308

313

316
317

327

CORRIGENDA
Page 5, line 33
for (XIII) read (XII).
6, ~ 29
7, lines 9, 11, 13
~
,,

~

9, line 16, for (344), § 227 read (354), § 231.
38, ,,
4, for t=vr-x read O=V)T-X.

~

38, Fig. 9, the Roman numerals 1 and II are to be interchanged.

,,

43, line 29,

before

continues insert and.

45, at the foot, for C read c.
,,

47, line

2, for (c) read (b).

,,

48,
49,

2 from the foot, for (502) read (504).

,,

,,

,,
10, for [a] read a.
49, last line, for t=e/f read z=c 2 /f 2.

,,

57,

Fig.

the letter C has been used to denote the position of the
it is used
charge at the time U by inadvertence,
elsewhere to denote the initial wave

19

ig.s13,

F081

although

14,19

GDG'.

,,

4, for § 43 read § 44.
7, for R2 read 112
,, 27, for Fig. 25 read Fig. 24.
,, 11, for § 43 read § 44.
,, 33, forl 3= w/c read 3= 'v/c.
read lR=fA.
8, for RA
,,
f.
lines 14, 15, 38, 39, the values of 4) and a must be divided
line 19, for latitude read longitude.
,,
18, after (146) insert (159) and (160).
equal to S.
25, after as before insert except that sr is
,,
3, eq. (g), the sign of the second member must be changed.
,,
15, for (m - 43)2 read (rn -432)2.
,,
,,
16, for II read

,,

70, line

,

74,
74,

,,

78,

~

83,
84,

,,

~ 85,

109,
130,
~132,

~137,
137,
145,
146,

,,

by

now

ML.

23, for A

,,

=B read A

or B be zero.

Pages 147 and 148, for 0 read e throughout.
Page 147, line 24, for D read D2 .
151, at the foot, for 8, read d.
eq. (179).
154, line 12, delete the second 2
read (178).
155, lines 1 and 12, for
165, line 4, for Problem 2, Appendix A read Appendix C.
170, ,, 15, after sin j+ insert ,j.

from

(177)

,,

180,

,,

30, for --

w

read

+

(
w

xxii
CORRIGENDA

xxii

Page 185,
187,
194,
201,
202,
209,
,,

line 4 from the foot, after generally insert continue to.
at the foot, the lower limit of the sum 'V should be 0.
line 23, before (r1v') insert
lines 5 and 6, interchange (x+8x, y+ci, z+8z) and (
line 13, P U is the forward tangent to the circle at P.
,, 18, for

a

F=

in the first term read

e

+

C

c

a
-.

0.

218, ,, 30, for
Q read F'Z -Q.
223, ,,
9, for b read
223, lines 12 and 19, for hP read - h,.
223, line 22, for

~

225,

~

230,

2:31,

2

D+ e )o
D+

read

,, 17, delete f13, f31.
,, 30, for (- v/c)8 read (v/e)8.
,,
6, for x' read $.

264, at the foot,

for

i1 readC!

~ 265, line 4, non-electromnagnetic momentum is really produced at the

per unit volume, but the term
neglected.
268, add footnote,

Liapounoff

8

rate p

(~u

is of the order zero and therefore is to be

has p~roved that the sphere is the only stable

equilibrium figure for homogeneous gravitating liquid at rest (cf. Poincard,
Figures d'1'quilibre dune Masse Fluide, pp. 15 and 22, 1903).
282, add footnote, cf. McLaren, Phil. itag. [6], xxi. p. 15, xxii. p. 66, 1911.
Wassrnuth, Ber. d. Deuttsc/h. t
Ges., p. 76, 1912.
286, line 17, after (459) insert and (460).

P
hys.

CHAPTER I
FUNDAMENTAL EQUATIONS OF THE ELECTRON THEORY
THE most satisfactory foundation for the following investigation at the
present day is furnished by the electromagnetic equations for the free aether,
in the form developed by Maxwell and Hertz, together with the additional
equations representing the effect of electric charges (electrons, ions) due in
the main to Larmor and Lorentz. The former set of equations is sufficiently
firmly established as a result of the experiments of Hertz, as well as of all
subsequent experience. The latter set rests, not only on the fact of the
existence of discrete electric charges capable of motion relative to the aether,
or if we prefer, relative to a system of axes fixed with respect to the observer,
but also on certain assumptions, expressed or implied, as to the connection
between the aether and the electric charges.
In the first place it is assumed that the connection is a mechanical one,
in so far as the equations expressing its effect are developed by an application
of the principle of Least Action. Secondly it is assumed that electric charges
move freely through the aether, without disturbing it or altering its properties
to any appreciable extent.
That these two assumptions are mutually consistent is a distinct postulate, in favour of which not much evidence has yet been offered, and which is
made mainly for the sake of simplicity.
Any disturbing effect of the electric charge on the aether may be expected
to be limited to its immediate vicinity, and probably the details of structure
of the charge will have no appreciable effect on the nature of the disturbance
produced by it in the aether at a great distance away : this will practically
be that due to a point charge moving with the mean velocity of the actual
charge. Thus the electron theory in the form developed by Larmor and
Lorentz affords a sufficient basis for the calculation of the radiation due to
electric charges moving in prescribed ways.
But when we wish to determine the motion of an electric charge under
the influence of a prescribed system of electric and magnetic forces the bases
of the calculation are not so certain. The mechanical forces acting on the
8.
1
1.

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

[R

[CII.

charge due to the impressed field, and the mechanical reaction on it due to
its own radiation, both depend on the nature of the connection assumed to
exist between the charge and the aether [or on the law assumed for the
action at a distance between it and other charges, if we prefer to discard the
idea of the aether]. Any uncertainty as to the mutual consistency of the
two fundamental hypotheses of the electron theory will affect the conclusions
arrived at concerning the motion of the electric charge and the reaction due
to its radiation.
Experiments on the variation of the mass of a moving charge with its
velocity however lead to results in substantial agreement with calculations
based on the electron theory. We conclude that the equations developed by
Lariiior and Lorentz represent the
of the connection between the charge
and the aether, whatever it be, sufficiently closely to serve as a basis for the
calculation of its motion, even though there may be some uncertainty as to
the assumptions from which those eqluations have been derived.

effect

2. The equations of the field. For the sake of uniformity we shall
use the notation of H. A. Lorentz, employed by him in his [Theory of
Electrons, which is in better agreement with the practice of English writers
than that used in his] article in the Encylclopcdie der Matheinatischen
Iissensc/wften, " Elektronentheorie," Vol. v. Sect. 14, except that for the sake
of brevity we shall use electrostatic units in place of rational (Heaviside)
units for electric quantities, and magnetic units for magnetic quantities.
Scalar quantities will be denoted by plain letters, vectors by Latin letters
of Clarendon type, their tensors by plain letters, and their components by
plain letters with suffixes attached. Scalar products will be enclosed in
round brackets ... ) when necessary to avoid misunderstanding, vector
products in square ones [..]
The equations of the Maxwell-Hertz theory arc:

(

dliv. d
div.

=

cacurldi=v.

curl

div.

(free

aether)............

. .......

d = 47rp (electric charge) .................

at
c

0

d

p

(Ia~"),
~(I a),

4vnp.a.......".................,.I),

h=0......f
"h...............(V

f"......ff...........

......

...... a..(V).

These equations arc either based on experience, or are equations of

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

This expresses the mechanical force per unit charge due to the assumed
connection between charge and aether. It rests on the assumption that there
is no resultant mechanical force acting on a finite charge, due to its connection
with the aether, of any kind, elastic or hydrodynamic, whatsoever it be, other
than that directly due to the electric and magnetic force in the free aether.
[It does not require that there be no stresses between neighbouring elements
of charge, or between elements of aether to which they are attached, provided
that these stresses balance for an isolated charge, or electron. The question
of the existence of such stresses will be considered in Appendix E below.]
In these equations p denotes the volume density of the electric charge at
the point (x, y, z) and at time t, and v its velocity relative to the stagnant
aether, or, if we prefer, relative to a system of axes fixed with respect to the
observer, f is the mechanical force per unit charge, d and h are the electric
and magnetic forces in the free aether, and c is the velocity of light in the
free aether.

(

3. Potentials. Equations (IV) and (V) enable us to express the electric
and vector potential a;
and magnetic forces in terms of a scalar potential
this is generally the best course, as the potentials are more easily determined
than the forces themselves. In virtue of (V) we may put
h = curl a ............................... (VII).
Substituting this expression in (IV) we get
curl (d + /c) = 0,
whence

d = - V

- a/c...........................(VIII).

Obviously only two of the three equations for h, hy, hz, derived from
(VII), are independent. Thus a is not completely determined, and a third
condition may be imposed upon it. Thus a singly infinite series of different
functions a are possible, but only two have been used.
(1) Suppose that we impose the condition div. a =0.
Then (I a) gives

V2- = - 4 rp.

And (III) gives
V2a= - (a + 47rpv)/c.
Hence we find in the usual way
# =fpd/R,
ca= f(pv + d/4 ) dfG/R,
where R is the distance of the fieldpoint (Aufpunkt), that is of the point
(x, y, z) of the field, where the potentials and forces are required, from the
position (f, 4, ') of the volume element d(f at the time t. The volume
integration is to be extended over all space, the density p being zero at points
where there is no charge at the time t.
These expressions for the potentials have been used by Maxwell* and
Larmort. They are convenient in so far as the quantities p and v are to be
* Electricity and Magnetism, Vol. ii. p. 237.

t Aether and Matter, pp. 92, 111.
1-2

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

[R

[OH.

taken for the actual time t, and therefore are known explicitly. When there
is motion this convenience is more than counterbalanced by the fact that the
expression for a involves the total current, that is, the displacement current
d/4r as well as the convection current pv. Generally the quantities p and
v are prescribed functions of t and ( ,
'), but d itself remains to be
determined. If however we content ourselves with a first approximation, in
which v is neglected in comparison with c, the term involving d may be
neglected. This is done by Larmor in calculating the field due to a vibrating
doublet (loc. cit. p. 223), bit the resulting expression is incomplete. This
approximation is insufficient for our purpose.
(2) Suppose that we impose the condition
div. a + k/c = 0..
..............
(IX).
Then (I a) gives
V20 - 0P/c2 = - -rp.
And (III) gives
V'a - i/c2 = - 4wpv/c.
Lorentz* gives the following particular integrals of these equations:
Q~=f
$[p3
df2/R
a= ~f[p] d2/.................(X),
The brackets [...] here mean that the values of p and v are to be taken
for the time t - /c, and not for the actual time t; that is, the values of p
and v correspond to the time of emission of a disturbance, which leaves the
element df2 at ( ,
) so as to reach the fieldpoint (x, y, z) at time t. Thus
the quantities [p] and [pv] are prescribed functions of t - R/c and ( ,
).
They involve ( , 9, ') explicitly in the parameters of the functions, and also
implicitly through
they also involve (, y, z) implicitly through R. The
seven variables t, ~, , ' x, y, z in these integrals are all independent.
The integrals (X) and (XI) satisfy the condition (IX) in virtue of

n,

...

7,
R;

',

(II).

4. Complementary furnctions. The Lorentz integrals (X) and (XI)
generally do not satisfy the~ prescribed boundary and initial conditions, and it
is necessary to add complementary functions,
and a 0 , in order to complete
the solution. We wish to determine the nature of these complementary
functions.
They obviously satisfy equations of the form V, J*- +ce - 0.
a

0(

theorem due to

Kirchhoff t

47

where f (t)

-

-

By

we have

______-)Ric)~t
f (t -R/c)

M,

I is an element of the boundary at

(

,

it',its

I]

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

This expression for J shows that the complementary functions merely
represent the effect of charges on and outside the boundary, so that the
Lorentz potentials represent the required effect of the inside charges
completely. Thus complementary functions are unnecessary for our purpose.
Since the integrals (X) and (XI) give the complete solution of our
problem, and moreover only involve the prescribed quantities p and v, we
shall make them the bases of our investigation.
5. Equation of energy. We get the equation of energy in the usual
way: multiply (III) scalarly by d/4r, (IV) by - h/4ir, add, use (VI) and the
identity (h curl d) - (d curl h) = div. [d. h]. Multiply the resulting equation
by the volume element d12 and integrate throughout the space enclosed by
any fixed surface S; then we get

(f.v) pd +

2d + c [d.hndS-= o ...... (XII).
0

The first term represents the rate at which the field does work in moving
the charges inside S; the second is generally interpreted to mean the rate
at which the electromagnetic energy of the nether inside increases; hence
the third term represents the rate at which electromagnetic energy flows out
across the surface. Since the surface is quite arbitrary, the Poynting vector
a = c [d. h]/47r ........................... (XIII)
must necessarily be identified with the rate of flow of electromagnetic energy
per unit area.
This interpretation however is not free from difficulty if we accept a
mechanical theory of the aether. In deriving the electromagnetic equation
(VI) by means of the Principle of Least Action it has been assumed
generally, whether necessarily or not we will not determine here, that the
potential energy of the free aether is to be taken as d-/87r per unit volume,
but the kinetic energy as (a {d + 47rpv})/8rc, and not as h2/87r, per unit
volume (Lorentz*, Larmort, Macdonald+). Macdonald§ however points out
that the two expressions for the kinetic energy cannot really be identified;
indeed we find by means of (VII) and (III)
h' = (h curl a) = div. [a. h] + (a {d + 4rpv})/c.
When this value is substituted in the equation of energy we get an additional
surface integral on the left-hand side of (XIII), namely

J[a.

h]n dS/87r,

which requires interpretation.
*

Enc. Mat. v. 13, p. 165.

t Aether and Matter, p. 94.
+ Electric Waves, p. 160.

[Theory of Electrons, p. 23.]

§

Loc. cit. p. 32.

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

[cg.

We see at once that in all optical applications, where we wish to find the
mean rate of flow of energy across an element of surface, the value of the
vector [a. h] is stationary on the average for long periods of time. Hence
the additional surface integral contributes nothing to the average radiation
from the system of moving charges, so that we may identify the radiation
vector with the Poynting flux without fear of error.
But when we wish to determine the radiation from a system of moving
charges which are not in stationary motion, such as a stream of /3-particles
and the like, the difficulty signalised above subsists.
[In order to get a clear understanding of these energy relations let us
consider the case of a single unit charge moving in any manner. We shall
take as our ,pace
the space enclosed between two infinitely close fixed
spheres S and S', which coincide with the instantaneous positions at time t
of two consecutive spherical waves emitted by the charge at the times r and
r', when it was in the positions E and E'.

11

The radii, R and R', of these spheres are given by the characteristic
equation (28), § 13, infra, so that R = c (t - 7), and R'= c (t - r').

The field

due to the charge is given by equations such as (26) and (27), or (30)-(33),
§ 13. By choosing R and R' very large, we gain three advantages: (1) we
can treat the charge as if it were concentrated in one point; (2) in the
expressions for the forces, (32) and (33), we can neglect all but the lowest
powers of 1/R; and (3), in differentiating these and similar expressions with
respect to the time we can treat R as constant, because its variation only
gives rise to terms of higher order.

We must bear in mind that
)/c
=
where, by (29), K = 1 -(vR 1)/c; and that
To this approximation (32) and (33),
-

h = [Rjd],d -

= 1/K,

§ 13, give
(R-

v/c) (+R

1)

Thus we have (Rlh) = (Rld) = 0, d2 = h2, [dh] = Rld 2.

Let us begin by verifying equation (XIII) for the shell
enclosed
between the two spheres S and S'.
The first integral vanishes identically, because the electric volume
density p vanishes everywhere inside the shell.
As for the second, d12 is a fixed element of volume; hence the integral is
equal to

-(d2

+ h2).

Now the thickness of the shell is easily seen to be c ('-r)K;

dI2 =c(T-'-)KdS. Also a-

-

r'

hence

and d2 = h2, and ' has the same value

IFUNDAMENTAL

I]

EQUATIONS OF THE ELECTRON THEORY

for every point of the sphere S, since it coincides with the wave emitted at
the time -r. Thus the volume integral reduces to

a (cd2fdS ot
47r ,

to (71'-T)

to-

cd'2 dS'
4or

C(-

r)

d2, that is,

fds

ccl2dS

Lastly, the surface integral has to be taken over the two spheres S and
S'. For the sphere S the outward normal n is along R, so that [dh] =d 2;
for the sphere S' the outward normal n' is along - R', so that [d'h']
Hence the surface integral gives, for both spheres together,
-d'

f cd dS

fcd'dS'

2

that is, it just cancels the second volume integral, so that (XIII) is verified.
Again, let us consider the modified energy equation. In the second
volume integral of (XIII) k2 is to be replaced by (a {d + 4wpvl)/c, that
(ad)/c in the present case, where p = 0. That is, we must add a volume

is,by

-f

integral

{i

d&7 to the left-hand side of (XIII).

-

We have already found that we must add an additional surface integral
D

C[ah],

the sum,

Thus the modified equation will be verified

dS

S

f

1 (ad) -

if

we prove that

hJdf1±f[ah],, . dS, is zero, or independent of t.

In the first place, we have to our approximation in the shell
=

[a [R1 d]]R1

(ad).

a

and [ah],,

=

d

(ad) - (ad); and from (27) we

Now (ad) _

v

, h2

find
vv(+R 1)

('R 1) _d

_R,

cK 3 R

a Kt _cK'2R +c 2 K3 R
Hence, since (R1 d)= 0, we have (a.d)

=-

(ad) 1 ,

a
Hence the volume

f (a'd') dS' - f (ad) dS.

(ad)

C

As before we have
_a

cd2. Therefore

at

%c

(ad),

Kad
a=rdfl -c (T --r) KdS.
reduces

integral

to

(T'-

F TdS, that

T)j

is, to

As before, this just cancels the surface integral

I the same wayfo"nnca
n
we

n

riy

e

1

om fthne ee
C0ithr

equationa

~FUNDAMENTAL OF
EQUATIONS

S

8

[ci

THE ELECTRON THEORY

{CH,

from the case of the infinitely thin shell, provided only that the radius of the
inner wave is sufficiently great. We must now consider the values of the
several terms in the equation.

In

fl

the first place, the electromagnetic energy of the thin shell
according
to the usual definition is equal to f (d'±+h2) df/8?r, that is, just as above, to
2
c (7' -- T) fd(lK dS/4wr.

0)

Take polar coordinates (R, 9,

with origin at E, polar axis along
Writing,8 for v/c as usual,
and a for the angle between v and v, we have K
1 -Rcos 9, and

ER' (v), and initial meridian plane through v.
(vR,)/c =1 - K, (R

Also

1)=

tlS

=

=

Iv I(cos a cos9+sinasin0cos44

R2 sin Odd &;

hence using the approximate value of d found above, we get for the energy

S,(.
'

()

n

(+R)

(1-

4
cK

K

I sin 9d d b.

Substitute for (+R) its value and integrate with respect to 0, bearing in
mind that
f(Ri)d=-2
and

-d=
d)

Q(

7

rfi

cos acos 0,

27r+i(costa cost9+ j sine a sine 9).

Since (r)= I) v cos a, we get
I
(Y-

7)i
T

2c3

JCr
Jo

(1I- $ 2 ) sin 8in

1

.)(vi)2

(r'-.

± 20
Remembering that K

variable from 0to
9

iK,

fIr,12 cos0

fol

-T

(l--

2

cuiP

=1I-j

3 (c2 .v2)2

The coefficient of r'

dO

2K5

f

cos

)(3cos20-1)snd9
2

9, we get, most

{

is precisely the

)

2v Ki

2+T 1
2

in8

8

easily by changing

the

.

expression

found

by Li~nard for the

rate of loss of energy from the moving charge owing to radiation. Thus
his
definition of the radiation rests on the assumption that the

energy of the nether is equal to (d2 + h2)/8wr per unit volume.
Again, we found that on the mechanical

electromagnetic

theory of the aether it isnesar

to replace the term h2/87r by (ad)/8w7c, that is, we must add tolthe ene ry^o

I]
FUNDAMENTAL EQUATIONS OF THE ELECTRON THEORY9

I]

9

Now by (27), § 13, we have a= v/cKR. Using the approximate
of d as above we find that the supplementary term is equal to
G3
i 1
(-K -uK32)(R) J
(i)
(I (vv) +It[21r
(
the T-T7aC

value

-

In the same way as before we find that this expression reduces to

a

r)a

S(T'

2c(vv)
3(0

V)

Hence, when we adopt a mechanical theory of the aether, and accordingly
reckon its kinetic energy as (a t4{± 47rpv )/87rc per .unit volume, we must
supplement Lidnard's expression for the radiation from a moving unit charge
by adding the term-a

2c (vV)

where

is the time of emission of the

V2)2'

aT3(c-;.2-

radiation, and v and v are the velocity and acceleration of the charge at

time

T.

In Chapter XI and Appendix D we shall deduce the equations of motion
and energy of a small extended charge on the basis of the equations (I)-(YI)
of ~ 2, without making any assumptions other than those implied in these
equations. We shall find that the resulting energy equation, (211), § 145,
227, involves not merely the Lienard radiation, but also the
and (344),
supplementary term just found.]
6. Resultant mechanical forcive. The resultant mechanical force
acting on all the charges insile a fixed surface S due to the field is on
account of (VI) given by
F = f[d + [v. h]/c p dfl.
Using (I a), (III) and (IV), and
F

where

F1

integaigrating by parts

we

get*

= F 1 +F
2

=f

t2d,1d + 2hth -- (d" + h2) n1} dS/87r .........

ata-fsdl/c

(XIV).

2

Here n~ denotes a unit vector in the direction of the outward normal to the
as before.
surface element dS, and s is the Poynting

flux

et m ay
m
y

F 2 isn usuallyweinterpreted ,tin ethe sl
ai
or he c Fl
s
ali as ep rate t of tdecreasel e
b
r e ea asr d ai r np a st electrom
e
s
of rr .a t o agneticmo
It sce ry
du
tord
calculated as if the surface

S

were

subject

In calculating F, and F., for a

Lorentz,

Enc.

to the

Maxwell stress.

finite system of charges it is however

at. v. 14, p. 161.

hory of

Electrons, p. 26.]

10

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

[CH.

the shape of the surface. Otherwise we could not use the terms electromagnetic momentum and radiation pressure for the system at all. Now in
any case where charges are moving with acceleration, the parts of the forces
d and h,which depend on the acceleration, are ultimately of the order 1/R,
and therefore the quantities under the signs of integration in (XIV) are
ultimately of the order 1/R. On the other hand the element of surface dS
is ultimately of the order R2 dw, and that of volume dfl of the order R2dRdw,
where dco is an element of solid angle. Hence it may happen that the parts
of space at a great distance may contribute largely to the values of the two
integrals; therefore it is not legitimate to ignore their influence without
examination.
Apart altogether from these considerations it seems a somewhat circuitous
method to evaluate the forcive acting on a finite system of charges from the
actions that take place in infinite space surrounding them, so far as that
forcive is due to the system itself It is surely more appropriate to express
it by means of integrals extending to the regions occupied by the charges
themselves. This can be done easily by making use of the potentials.
Substitute from (VII) and (VIII) in (VI); we get
f= - V -

-+ [v. curl a]/c .................. (XV)

where all differential coefficients are partial.

Now

[v. curl a] = [v [Va]] = Va (va) - (vV) a,

where Va operates only on a, not on v.
When we are dealing with an extended distribution of charge we can
always regard it as continuous, by using the artifice of transition layers. Then
the velocity of elements such as de at (x, z) varies continuously from point
y,
to point; that is, it is a continuous function of (x, y, z) as well as of t. It is
convenient to let the operation V refer to all the variable quantities which
follow it.

If we do this we must write

(va) - a
-

so on, in place of the components of Va (va).
ia =

0

'

"

x'

C1

-a

y

-az

-,and

Let
x,

=12
-

, etc ......(XVI).

a-~, o-2, a,3 are the velocities of elongation in the directions of the three
and
axes; 23 + -, C31 + a-, and a'12 + . are the velocities of shear parallel to the

coordinate planes; and I (a,- a 3 J ( -,3),

) and

ponents of molecular rotation about the three axes.
ment of the point (x, y, z) increases at the rate

(a-.1

0-12)

are the com-

The resulting displace-

(o-zx + o-12y + o'z,us.x + a-,y + o'z, asx + o y + ca'z).

I

I]

~FUNDAMENTAL

1

EQUATIONS OF THE ELECTRON THEORY

11

a

We shall write this vectorially as r, so that o denotes the operator of the
rotational strain. We may also write

or = xr + [or].........

....... (XVII),
where x is the operator of pure strain, and w the molecular rotation. We
shall denote the conjugate strain by &, so that

-r = xr - [wr]

...............

(XVII').

With this notation we get
Va (va) = V (va) - oa

=

V (va)

[v curl a] = V (va)

whence

Again, the differential coefficient

0,a

aa

-

in the

a + [wa],

-

(vV) a.

expression

for f denotes the

da

local rate of incrQase of a.

Let da denote its substantial rate of increase,

that is, the rate of increase at a given element of charge de, as noted by.an
observer moving

with it. Then we have

da=a+

dt

at

(vV) a.

Substituting in the equation for f we get

_

(L

(va)l

dada

'cf

f=-VL

cdt

c..........XVI)

da

xa+[ya<

(va)t

c

cdt

c

c)

This is the mechanical force on a unit charge at (x, y, z) which moves
with velocity v, and is undergoing strain at the rate a-r.
[It is worthy of note that the expressions (XV) and (XVIII) also hold with
but slight modification when the origin is moving with the velocity w, and
the axes are
angular velocity 0 about themselves. In this case

rotating with

the

symbols

areal

- i

v, - ,~
n
is to be replaced by n
, ' ,w

-

to be taken relative

to

the

moving

(wa)/calaxes,
- (Ora)/c, and both 0 and
a are to

be calculated from the absolute velocity, relative to the aether, or if we prefer,
relative to the observer.]
In order to find the total mechanical force

IF

on a system of charges we

must multiply either of the expressions (XV) or (XVIII) for f by the element
of charge de and integrate over the whole system. In order to get the total
couple N we must multiply vectorially by rdle and integrate.
get
F'=ffde,
=frf de.

N

That is, we

12

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

charge is commutative with the substantial differentiation,
local differentiation,

dt
dt

d

[CHi.

, not with the

Thus, we have

~

=

dtJ

a de, but
a de =
de bu J at e dtJ

(vV)
ade.

The symbol Ja de has .no meaning when there is relative motion of the
parts of the system, and the operations atand fde are not commutative.

For

this reason it is convenient to use the expression (XVIII) for f, rather than
(XV). Hence we get

d

F =- d a de/c - IV - (va)/c +a/c}de......(XIX).
This expression can be transformed as follows. In the second term write
de = pdf2, where df2 is an element of volume and p the corresponding volume
density of electric charge.
Integrating the first term by parts we get
fV {0 - (va)/c} pd2 =f - (va)/c} pdS - f - (va)/c} Vpd,
where dS denotes a vector element of surface with the positive normal outwards. Hence we get
d
F =- d fa de/c - fbe - (va)/c} p dS
+ f({ - (va)/c} V log p - va/c) pdf ...........
(XX).
The first term in the first line on the right represents rate of decrease of
momentum, not necessarily the whole; for the second line may, and generally
does, contribute a term of the same form.
The second term in the first line includes not merely a part due to the
boundary of the system, but also parts due to surfaces of discontinuity, whenever such exist. But if we regard these surfaces as limits of thin layers of
transition in the usual way, their effect will be included in the terms in the
second line, and need not be considered any further. Then the term in
question represents the resultant of a distribution of pressure on the
boundary, amounting to
- (va)/c} p per unit area, and we may regard
the corresponding part of the mechanical force as due to a hydrostatic
pressure throughout the charged parts of the system.
[This resolution of the mechanical force is of importance in connection
with the possibility of a mechanical explanation of the electron to be considered in Appendix E below. The volume integral in the second line of
(XX) generally does not vanish, nor reduce to terms of the type of those in
the first line. For example, the condition that it should vanish is that
V log p = Fa/cp - (va)}; but in virtue of the equation of continuity for p,
(II), this leads to a condition limiting the character of the motion.

o

I]

FUNDAMENTAL

EQUATIONS OF THE ELECTRON THEORY

13

Just as we derived the expression (XX) for the resultant mechanical
force F, so we may derive the following expression for the resultant couple:

to

d
N = - d I[ra] de/c - f - (va)/c} p [rdS] + f[va] de/c
+ f[r ({' - (va)/c} V log p - va/c)] pdf .............. (XXI).
The first term in the first line represents rate of decrease of angular
momentum, and the second the moment of the surface pressure, but the
third has no analogue in (XX). The second line, like that in (XX), only
vanishes for particular types of motion. We shall make no use of (XXI);
it is only added here for the sake of completeness.]

CHAPTER II
TRANSFORMATION OF THE POTENTIALS
7. IN order to coordinate the facts of our experience we may make the
following assumptions:
(1) Electric charges of two kinds exist, negative and positive. Their
properties and modes of occurrence are so different that it is best to treat
them as separate entities.
(2) Electrification of either kind by itself is indestructible. Although
negative and positive charges occasionally perhaps occupy the same space
and neutralize each other's action to a greater or less extent, yet we are able
to separate them by the application of a sufficiently powerful external field.
Thus they do not annul one another.
(3) Neither kind of electrification is distributed continuously throughout all space; each is confined to limited regions, separated from each other.
These aggregations of charge are capable of an individual existence and can
move about in space; they constitute the electrons and ions. Whether their
inertia is entirely electromagnetic, so that they may be regarded as free from
matter, as in the case of the negative electrons, or not, as for positive ions, is
immaterial for our purpose.
In our problem of determining the radiation due to electric charges in
motion, we are given the position and configuration of each electron or ion
as functions of the time. But the integrals (X) and (XI), § 3, involve the
values of the density p and velocity v of the electric charge which is at each
point ( , , ) of space at the time of emission t - R/c. It is desirable to
transform them so as to put in evidence the data actually given, that is to
say, the coordinates and velocities of every element of charge at the time t.
Every such element de can be identified by means of three parameters, which
may be its initial coordinates, or any three independent functions of them;
and its coordinates ( , q, ) at time t will be certain prescribed functions of
these parameters and of the time t. Our transformation corresponds in some
sort to the passage from Eulerian to Lagrangian coordinates in hydrodynamics.

CH.

1

CH.
TRANSFORMATION OF THE POTENTIALS
ii]

ir]

Y5

point

Regarded as functions of t, the density p and current pv for a given
of space may in the limit be discontinuous, when the boundary of an electron
crosses that point; or they may be infinite, when a charged surface passes.
But from the nature of the case the number of such discontinuities or
infinities occurring during a finite interval of time is necessarily finite and
for each infinity f [p] dr is finite. Hence the quantities [p] and [pv], which
occur in the integrals (X) and (XI), can always be expressed as Fourier
Integrals. If we suppose the values of the density and current at the point
to be given for all values of the time, we may write
o,

fE gyp,(t-RCT) pdd

[P=[pv] =

)pv

Here p and pv are prescribed functions of

while
.. =

(y

17,

......... )2 +

and is a function of x, y, z, ,

.)2

(1)

. . . . . . .

dTrdL................. (2 ).

involving

,

7,
'

as parameters,

+(..............(3),

but not of T. It must be particularly noted

the

root, since the time of
that the positive sign is always to be taken for
emission of the disturbance from ( , j, ), namely t - B/c, is always anterior
to the time t.
If however we prescribe the values of the density and current only for
times anterior to the actual time t, we must use the Cosine and Sine Integrals,
and write

[p] = - fo
2

"

7r

[p]

or

=

f

tt

-

cos/p (t - B/c) cos -. p dd
0o

2-fJ

sin

(t

B/c) sin T. pddj,

-

with similar expressions for pv.

a-

In this case it is simpler to change the variable from T to = t - T, so
that a- is the time reckoned back from t to all past time. Thus we may
write
Srt+0R
dp ........... (4),
ta-.
010 cos
[P] =

pt,

-cos

2

[pv]

or
~

~

=

[~=
~

-I

7

00ft+00
I

R

cos

0 Jo

cos

sn-Ci

0o t +o.R
or[P]

sin

2i
=

Th perlmtfo
indpenentof - ad ~ itis

sin

-k-

-iswitn7.

0
/h .

a

.dog

(pv)t

Q.

it........(5),

d-d

.d-~....()

gka-. (pt-Q.
o

d-d

niat

..........

ht

(7).

atog

otaabouecnatutafcinof.

16

TRANSFORMATION OF THE POTENTIALS

[CH.

The importance of this fact was pointed out by Lindemann* in his criticism
of Sommerfeld's calculation of the mass of the electron.
8. Thus we have the choice of three methods of representation, either
by means of (1) and (2), or (4) and (5), or (6) and (7). The last two
equations, as we shall see later, lead to Sommerfeld's well known expressions
for the potentials of a spherical electron. They have the disadvantage of
failing at the limit o = 0, that is, r = t; but this is of importance only
when R = 0, that is, for points close to the element of charge de. This
however does not matter very much, since the elements of charge close to
the fieldpoint (x, y, z) contribute only a vanishing amount to the potentials,
as in the ordinary theory of the potential. Nor does it affect the values of
the forces due to volume distributions for the same reason; but it may very
well lead to error in estimating the forces due to surface distributions, where
the elements of charge contribute largely to the forces in their immediate
neighbourhoods.
It is interesting to note that Sommerfeldt comes to the conclusion that
while the motion of a spherical volume charge with velocity exceeding that
of light is possible, the motion of a spherical surface charge with such a
velocity is impossible, because it involves an infinitely great mechanical
force. It would be interesting to enquire, although beyond the scope of the
present investigation, whether this difference may not after all be due to the
failure of the integrals (6) and (7) to represent the motion completely at the
time

= t.

Again, the deduction of the electric and magnetic forces from the
potentials involves differentiations with respect to t, x, y, z; we must
therefore enquire how far it is allowable to differentiate the Fourier integrals
which we have obtained.
As regards (1) and (2) differentiation is allowable provided p and pv
vanish at both limits, conditions which are generally satisfied in all physical
applications. These conditions however are not sufficient for the cosine and
sine integrals (4)-(7), which involve t in the upper limit, as well as in the
functions pt-, and (pv)t_. The difficulties arising from these causes have
been completely discussed by Lindemann in the memoir already referred to,
but they may be completely evaded by the use of the integrals (1) and (2).
At first sight it might appear that these latter integrals require more
data to be given than the others, since they involve the values of p and v
for future as well as for past time. This difficulty however is only apparent;
for we may choose the values of p and v quite arbitrarily so far as future
time is concerned. But when these values have been so chosen, the values
* Abhandlungen der K. Bayer. Akademie der Wiss. n. KI. xxm. Bd. xi. Abt. p. 320, 1907.
t GittingerNachrichten, 1904, p. 387.

~TRANSFORMATION
OF THE POTENTIALS

II]

II]

1
17

is

of [p] and [pv] are quite determinate for all time, and this
the important
point-thcy have the proper values for all past time, whatever values of

and v be selected for the future. If experience should show that at some
future time the values of [p] and [pv] ceased to be
represented
the integrals (1) and (2), because the charges had not moved in the way that
was anticipated, that would only be in accordance with our limitedpower of
foretelling the course of future events. For these reasons the integrals (1)
and (2) constitute the best foundation for our investigation. They lead to
the expressions for the potentials given by Schott*.

correctly

9.

(1)

and

by

Schott's solutions. Substitute the values of [p] and [pv] from
(2), § 7, in (X) and (XI), § 3; we get

1 as rsL,(t-R/e-r)nddJ
00~CoR(
0**0,0
~-Jd1J
E
-J
...........
27rf

=

I
Here

cf
L7

ex

ao

V/{(x -

d

r f

+ (y -

a

°+

- - -

e
f'~(

(- t - / - T)

it is

(z -

_

_ .

.

.

()
.

.

( 9)

function of the six independent variables x, y, z, , ,
but not of T. p and pv are prescribed
functions of ,
anl T. d172 = d-d 'and the integration is for all parts
of space for which p is different from zero. Thus the variables ()
are analogous to Eulerian coordinates in hydrodynamics.
It has already been pointed out that the data usually available are the
coordinates of each clement of charge for every value of the time T. Any
B =

a

71,

element of charge de pursues an individual existence, and therefore can'be

7,

identified by means of its initial coordinates (E,10,

standard time r0. Its coordinates
equations of the type

(,

0, o, 4 n --g ( r. , o, ),'4 = It(
where the form of each of the functions f, g, Ii is known.
fbi

(, ,

The set of variables
coordinates of the element.

o,

) for any selected

} at the time T are given by
o,fbi

'T..,(10))

'r) constitute, as it were, Lagrangian

Since the charge is indestructible we have

e=pd~dnd =pd~od od~o.......1)
where Po is the density at time 'r, and therefore independent of T.
may in our integrals choose the element of volume d51 arbitrarily for

We

each time r, since this only amnounts to a rearrangemnent of the
triple sure which

is

forphysical

reasons known

to be

termus

of a

absolutely convergent.

[n

~TRANSFORMATION
OF THE POTENTIALS

18

18

[cg

r

same for all times r; hence we can, if we please, change the order of
integration and sum with respect to
first.
Thus we get from,(8) and (9)
1Ifde 1 1
0
27r JJB _-mJ
1

a=

2wrc J

de
a.i

°°

E(t-i/C-T) dd,12)
L1

0f0f

q(t-12/C-Ti)V dTd/.L

fJ0R

J

.

E.9.1

Here 1? is a function of w,y, z and 7, as wvell as fteprmtr
,n,~
briefly of the charge dc-in virtue of the equations (10). The expressions
(12) and (13) are essentially different in character from (8) and (9); for
whereas the latter, regarded as functions of T, are Fourier integrals, because
1 does not involve T, the former are so no longer, since in virtue of the
transformation (10) R has become a function of T.
The remaining integrals (4)-(7), § 7, lead to analogous expressions:
2ff
t+a
R
ddd,
decos
to- B.....(14),
2 ffooft±ml:
ttR
C
vdo-d
-cos

;

a=

Jo

0

jde J

doB...

cos--cos

d
d- e

.

from (4) and (5); and
2

f

ft+00
rr i,
de

2

a

.oodd

sin t+°°

!fdfeff

R

sin a.

CCR

.
va-dIc

sin

sin a'Vd.d........(17),

from (6) and (7).

97,

Here o- is time measured backwards

from the variable instant t, and

a-.

1?.,v are therefore functions of t -The solutions (16) and (17)
will be found later to lead to Sommerfeld's solutions for a spherical electron.
10. Electric and magnetic forces. The equations (VII) and (VIII);
§3, at once lead to the expressions for these forces. Remembering that
VR=R 1 ,
where R1 denotes the unit vector in the direction of R, we get
(12) and
(13), 9,
,dg
q(t -R/c - ) [VR 1] drTd,u.
2 Ii
2

from

7rc

at

jjj

0

R

[VR]dT do
*EL(t-Re-T)

+def?,
A

2

vn1c7tde

E

L.(t./-T),-I(R.

-

V

/cdd

I,]

TRANSFORMATION OF THIIE POTENTIALS

19

The first integral in each case, being of order 1/R, is effective at large
distances from the charge, the second at small ones. It is easily seen that
for a finite system of charges, and at a fieldpoint at a distance so large
compared with the dimensions of the system, that quantities of order 1/R2
may be neglected and R, treated as a constant vector, we have h = [R 1 . d]
and 0 = (R,. d). That is, at great distances h and d are at right angles to
each other and to the radius vector, just as in the case of a Hertz vibrator.
It is to be noted that in (18) and (19) the differentiationa is partial,
(x, y, z) being kept constant. This is of importance in calculating the
mechanical force exerted by the system on a moving charge.
In the same way other expressions may be deduced from (14)-(17), § 9,
but as they are more complicated and will not be required in our investigation, we refrain from writing them down.

2-2

CHAPTER III
OTHER TYPES OF SOLUTION

11. WE shall now compare our solutions with other expressions which
have been given by various authors. It will be found that these can all be
deduced from the integrals (12)-(17), § 9. The proofs may serve as a
verification of them.
Sommerfeld's solUtions*. Sommerfeld has given expressions for the
potentials due to spherical electrons carrying uniform surface or volume
charges. These may be deduced as follows from the expressions (16) and
(17). Consider first the case of an electron of radius a charged with e units
of electricity uniformly distributed over its surface.
Take the centre as origin of polar coordinates; let the fieldpoint be
at (r, 9, 0) and the element de at (a, 9', /'). We may write
R-=V,(r3 ± a2 2ra cosy), de = e sin 9' d6' d4'
47r

-

where
Write

.

=

cos 'y = cos 9 cos 9' + sin 9 sin 9' cos(bos ; we get from (16)
=ec

27r2)

Now sin Rs
J J(R s)

=

((s.
Josin O'd9'd4'

V(jirRs)

ft+00

sin csa- dsda-

inRs.

J j (Rs), and by a theorem due to Gegenbauerf

a ~ ,= (n
R

-=

f

+ j) J

,,,+I(r

)

Jk + j

(as) P n(cos

y).

. .( 0,41

in the usual notation of Bessel Functions and Zonal Harmonics.
Substituting this in the expression for
of the sphere, we

find

0

and integrating over the surface

that, owing to the presence of the zonal harmonic,

every term vanishes except the term n = 0, and we get
2ec "°°t+°°
.
dsdor
I1 sin as. sin cso-.sin rS......(21),
Iraq'0
o
s

107-110.

*

Gott. Nad. 1904, pp.

t

Gray and Mathews, Bessel' Functions, p.

239,

Ex. 50.

CH.

2

OH.
ii]OTHER TYPES OF SOLUTION

III

21.

differences

which is identical with Sommerfeld's expression (16), but for slight
of notation. The chief difference is that the upper limit for the T integration
is expressed as t ± oo, and not simply as ooin accordance with the contention
of Lindemann already referred to, that the upper limit must be treated as
variable in differentiating to derive the electric force. The reader who is
desirous of following up the controversy between these two authors will find
their papers in the Abhandl ungen and in the Sitzungsberichte der K. Bayer.
Alcademie for 1907 and 1908, and a paper by Schott in the Annalen der
Physik, 1908, 25, p. 63.
Sommerfeld's expression for a volume charge is got at once from (21) by
putting r' for a, 3er'2dr'/a3 for e, and integrating with respect to r' from,0 to a.
We get
Gec 1 +C sin sa - sa cos sa
(t
dsdosin csoa . sin sr ---(sa)2--7 j

S=

rrrar

o

frar
0u

.(22),

(sa)2

agreeing with Sommerfeld's expression (18).
The expressions for the vector potential are got from (17) in precisely the
same way.

12. New solution for the sphere. Proceeding by the method just
used we can get a new solution from the integrals (14) and (15), § 9. We
have an expansion corresponding to (20),
cos

Rs =

V(J-irIs). Y

(R.S)

irR
far),,-_O

)

(n+ )

h+4 (rs)

.+j (s)

P

(cos y)...(23),

(r

<a)

provided r <a. If r > a, r and a must be interchanged.
Using this in (14) we get at once for a surface charge
=YI-sj
.

ec
As before

we get
Sec

0 = 7r

ec

dr j

s

cos i(t

-)-

ds

alsinSr-..,

zds

rfdrfcos s {c (t T)-a }sin sa , (r>aj

{2)
..

for a volume charge

:10f,

i(sin

In these expressions r

I

LC~~){

is

rs

t

s(1 + tsat)ea1 3

)d

.....
(5)
.(25).rSp

the distance of the fixed point (x, y, z) from

the

22

OTHER

TYPES

OF

[ca.

SOLUTION

13. The point laws of Lienard* and Wiechertf. These writers
have given the following expressions for the potentials due to a unit electric
charge of small dimensions at a fieldpoint so distant that the charge may be
treated as if concentrated at a point:

a= [K..............(26),
[KR]..............
c[KR]
The square brackets mean that the enclosed function of the time is to be
taken for a time r given in terms of the actual time t by the equation
t= + R/c...................(28).
In other words the disturbance which reaches the fieldpoint at time t
was emitted by the unit charge at time 7. [We shall call (28) the characteristic equation of the motion.]
K is the so-called Doppler factor; it is given by
K =

at
aT

1 aR
1t-

- (v.R,)/c..................(29).

c a7

There has been some controversy as Lo the proof and the proper expression
for the point law". We shall now show that, provided the velocity of the
electron be less than that of light, the expressions (26) and (27) can be
deduced without ambiguity from the general expressions (12) and (13), § 9,
so that there can be little doubt as to their correctness, at any rate so long as
the velocity of the point charge is less than that of light. The proof however
can be extended so as to give the proper expressions when the velocity of the
charge exceeds that of light.
Change the variable in (12), § 9, from T to t', where
t'

We get

=

fde

7

R

= + - ................................. (28').
0C

d

feLt-t')

dt

where K' and R' are the values of K and R when t' is substituted
for T by
means of (28'). The limits remain to be determined.
In the first place suppose the velocity of the electron to remain constantly
less than that of light.
Since K'= 1 - (v. R,)/c, and the tensor of v < c, while that of R, is unity,
it follows that K', that is, d, is always positive.

Hence as r increases from

* L'l'clairageElectrique, July, 1898.
t Archives vierlamlaises, 1900, p. 549.
+ De la Rive, Archives de Genave, 1907, p. 433.

OTHER

III

rII00 to +

-

TYPES

OF

SOLUTION

23

00, t' constantlyOTHER TYPES OF SOLUTON2 limits.
increases between the same

limits for cfare also

-

00cand

Thus the-new

+ oc .

But the double integral for 0, namely
"d

c

F wc~
(t-

t-)

dt'/KR

is a Fourier integral, and has the value 27r/[KR]. For the equation (28')
reduces to (28), since the function I/K' R' is to
taken for t' = t. Hence

be

_=

fR]
de

......

.

(30).

In the same way (13), § 9, leads to
[y de
a = c[KR].........(3)
a
These equations express the point laws in another form; in fact when R
is large compared with the linear dimensions of the charge, so that the latter
may be regarded as concentrated at one point, and when the total charge is
unity, (30) and (31) reduce to (26) and (27) respectively. When however the
charge is extended we must use the former equations.
[Similarly we can get expressions for the electric and magnetic forces by
differentiating (30) and (31), or, for a point charge, (26) and (27). We may
also deduce them from (18) and (19), § 10, as we deduced (30) from (12), § 9.
The latter method gives for a unit point charge

at

a[[vR
a

C2K

1]

+

CR

VR
[[vR

1

FRi-v/c]L Ri

d=a

We must bear in mind that the quantities in the square brackets are
functions of Tr, where T = t - R/c, by (28), as well as of (x, y, z) explicitly.
By (29) we have a = 1/K ; hence we find
RiR)± c2 v2}/c2K,

=1-_

and
Using these results we

aplR

(1-K)-- v/c}/KR.

find
Ltri

1

.r

<
.J.....%.2)

24

OTHER

TYPES

OF

[CR.

SOLUTION

These equations agree with those found by Lidnard (lee. cit. p. 4). They
1
show that the terms involving the acceleration v are of the order R- , while
those involving the velocity only are of the order R - . The former preponderate at great distances, the latter near the charge.
We see that h = [Rid], that is, the magnetic force is perpendicular to the
radius vector and the electric force everywhere. Also (Rld)= (c2 - v2)/c 2K 2R';
thus the electric force is not transverse to the radius vector, except when the
velocity of the charge equals that of light, but the deviation from transversality
becomes smaller and smaller as the distance increases.]
14. Point law for velocities greater than that of light. The
preceding investigation obviously applies so long as K', that is dt'/dT, does not
vanish. Now by (29) we have
K' = 1 - p' cos 0',
where 8' = v'/c and 0' is the angle between the directions of the velocity and
the radius vector drawn from the point charge to the fieldpoint, all for the
time t'.

It may very well happen that even when
'> 1, yet /3'cos 0' < 1,
particularly in directions approximately transverse to the direction of motion,
for example, in the case where the path is a plane curve of small dimensions,
and the fieldpoint is in a direction nearly perpendicular to its plane and far
away. In this case (30)-(33) still hold.
This ceases to be the case when K' vanishes at any time; this certainly
occurs when /3' > 1 and the fieldpoint lies in the direction of motion, so that
0'= 0. Thus (30)-(33) fail for some points of the field, whenever the velocity
exceeds that of light.
Now we have t'= + R/c, while R is essentially positive; hence t'= oo
when r = oo, whether R be finite or not, so that for large positive values of
r K' is certainly positive. Its behaviour for other values is best seen from a
diagram.

7

Plot a graph of t' against T*.

The

line AB is given by t'= r, and the graph
lies above it everywhere.
Let P be the minimum correspond_
ing to the greatest value of 7 which
makes K' vanish, Q the next maximumM
(if it exist), R the next minimum (if it
exist), and so on; and let OK, OL, OM,...
be the corresponding abscissae. In the
A
Kco we have K' positive;
interval
denote this interval by a suffix 1, so that K'> 0.

L K

Fig. 1.

In the interval LK, K'

*[We shall call the curve PQR the characteristic curve of the motion.]

r

OTHER

III

TYPES

OF

SOLUTION

25

Ill]negative;
is

OTHER TYPES OF SOLUTIN2
denoting this interval by suffix 2 we have 14'< 0. Similarly in
ML we have K3' > 0, and so OiL. It is obvious that for the last interval
extending to T = - oo, K' is positive or negative according as t' is or
+ oo. In the former case the number of intervals is odd, and the number of
maxima and minima together even, being one less; in the latter case the
reverse is true.

Denote OK, OL, OM, ... by 12r, T23
,..., and KP, LQ, Ml, ... by t'12, t'2,
t'34,..., the pair of indices referring to the index numbers of the adjacent
intervals. Then we have obviously
T12

<

t'12

>

T23

t'231

>

T34

t'23

>

>

...

,

t'34;

.

whilst t' 12 may be greater or less than t 4 according to circumstances.

-1,1).

Let the number of intervals be p, so that the last pair of critical values
are Tp-1, , and t
When p is even we have a minimum, and Ip'<O;
when it is odd, we have a maximum, and Kp' > 0.
We may write (12), (§ 9), in the form

fJcl

de j

disyj+j+J

)

+...±j

where the limits for the several integrals have been put in evidence
explicitly.
Transform from T to t'; we must write dt = dt'/K' and arrange each
integral so that the new variable t' increases throughout its range. Thus
we must put

Jder

=0fdt

T17t1

r

.T

frI2dL.

f

=

-t

j t23dt'

3T

'34

1

T33t12

t34

1

7

and so on, and finally,
dt'0or

p-1,pp1,P

-

fp=iP dr=JK/K/,
the first or second alternative being chosen in the last case, according as K'
is positive or negative, that is, according as p is odd or even.
Hence we get
1Lfdeae

t C(7

+

3
t'f2

Each of
in

tegrsonthe

t'-23or

pf)1Et-tdt'

-the

34right-hndsieisa

o

0
r

ptgrlo

h

26

[OH.

OTHER TYPES OF SOLUTION

denote that in the function KR the variable T is to be eliminated and t
substituted by means of the characteristic equation (28), § 13.
Thus we get
=

fe (

]

[K,R][KKRR]] 3 ]3R

±[KR]4). ]
-+[K

The suffixes have been added to indicate the various intervals for t. It
is to be understood that each term is present when t lies within the interval
belonging to it; whether it lies within other intervals at the same time or
not is immaterial. When t lies outside any interval the corresponding term
is absent. Since K is positive in the first, third,..., in fact in all odd intervals,
and negative in the even ones, it is obvious that every term of (34) is
essentially positive, and that no two terms can ever annul one another.
A precisely similar expression is got for a, viz.
a

__

[v,

/]

[v,/c]

a [KR,]
de

+

[K,,]

[v3/c]

[K,]

[vp/c]

"' [R]......(35).

15. For example, suppose that p = 2, so that we have but one stationary
value of t, which is necessarily a minimum, since t and r become + oo together. We get from (34),

O

= de ([

-

[Kde(
Ra]

[KR1-)]
[KR,]

when t > t',

= 0, when t < t'1
,.

In other words, until t = t'1, there is no disturbance at the given fieldpoint
at all, not even an electrostatic one. The character of the field changes once,
at the time t' 1 ; the change is discontinuous in this sense, that the form of
2
the potentials changes suddenly. The values of the potentials and of their
differential coefficients up to a certain order are continuous, but the
differential coefficients from some finite order onwards are discontinuous
when t =t'1 ; otherwise the potentials would always be zero, for t > t'1 , as
2
2
well as for t < t'1 .
2
Since the critical value t'12 is determined as the single real root less than
t of the equation K'= 0, regarded as an equation in t', it follows that when
t = t'1 , each of the two terms in the expression for 0 becomes infinite. As
2
we have already pointed out, K is positive and K 2 negative, whilst R1
and R2 must both be taken positively. Hence both terms are essentially
positive, and cannot balance each other. Apparently t becomes infinite
for t =t' .
2
This however is not really the case; the infinity affects only one element
of charge de at a time, and this element contributes only an infinitesimal

iii]

OTHER TYPES OF SOLUTION

27

amount to the value of 4 in its neighbourhood or elsewhere. Hence the
value of 0, and similarly that of a, due to an extended distribution of charge,
remains finite and continuous everywhere, although that due to a finite charge
concentrated in a point, if that were physically possible, would become
infinitely great, not only at the charge itself, but at all points for which t
had the critical value t'1 .
2
The case just considered occurs for a charge moving with uniform velocity
in a straight line when its velocity exceeds that of light, as we shall see
later.

CHAPTER IV
PHYSICAL INTERPRETATION

OF THE SOLUTIONS OBTAINED

16. WE have obtained two types of solution of the problem of finding
the potentials due to an electric charge moving in a prescribed way. The
first type, given by the equations (12)-(17), § 9, requires the evaluation of
double integrals, which may be regarded as generalisations of Fourier's
integrals. The second, given by the equations (30)-(35), §§ 13, 14, constitutes a generalisation of the ordinary " point-law" expression for the electrostatic potential. It requires the solution of the characteristic equation,
t=T + R/c, where R is given as an explicit function of r. The value of T
found must be substituted in the function 1/KR occurring in the integrals.

The solution of the equation in finite terms is rarely possible; moreover its
solution by means of infinite series, by the use of Lagrange's Theorem or
some equivalent expansion, has only been effected in the case when the
characteristic equation has only a single real root less than t, that is, when
the velocity of the charge has never at any time exceeded the velocity of
light.
It is true that from a physical point of view this restriction, at any rate
at present, is of no moment, since no experiment has yet been made which
indicates the occurrence of such large velocities; whilst useful physical
theories such as the "Relatiftheorie" of Lorentz and Einstein are incompatible
with their existence. Nevertheless from the standpoint of complete mathematical generality this limitation is undesirable. In so far as the use of the
integrals (12)--(17) does not require it, these integral solutions are preferable.
Even when velocities exceeding that of light are excluded and the solution
of the characteristic equation can be effected by means of infinite series,
these series can often be obtained more directly from the integrals. But the
integrals can also be used directly for the calculation of the field due to the
system of charges and of the mechanical forcive acting on the system. In
fact they have been so applied by Sommerfeld* and by Lindemannt to the
* Loc. cit. G6tt. Nach. 1904, pp. 99, 363; 1905, p. 201.
+ Abh. der K Bay. Akad. n. K1. 1907, p. 235 and p. 339.
Sitzungasberichte der K. Bay. Akad. i. 1(1. 1907.

See also the discussion in the

CH. IV]

PHYSICAL INTERPRETATION

29

OF THE SOLUTIONS OBTAINED

problem of the uniform rectilinear motion of a spherical charge and also to
that of its quasi-stationary motion. Even in these simple cases the investigation is very exacting and liable to error, because the integrals are discontinuous.
For this reason the safest, though perhaps a circuitous, method is to develope
the integrals in series and determine the electric and magnetic forces from
them, and to calculate the mechanical reaction on the system by direct
integration over it. This is the method we shall follow later.
17. Although the point laws are unsuitable for the actual calculation of
the forces of the field, they give us the best picture of the physical processes
involved in the propagation of the disturbance from the moving charge to the
observer. For this reason we shall consider them a little in detail.
The characteristic equation, t = r + R/c, may be interpreted as follows:
R is the distance from the fieldpoint (x, y, z) to the point (,
), where
the element of charge was at 7, the time of emission of the disturbance which
reaches (x, y, z) at time t. The equation simply expresses the fact that the
disturbance traverses this distance with the velocity c.
If (x, y, z) be regarded as a variable point, then the equation represents a
The sphere
sphere, whose centre is at ( , q, ') and whose radius is c (t- -T).
is the position at time t of that wave which was emitted by the moving
charge when it occupied the position (,, , (), at the time T.
If t be regarded as a variable parameter, the equation determines a
family of concentric spheres, namely the successive positions of the particular
wave emitted at the time T.
If T be regarded as a variable parameter, it determines a family of
spheres, whose centres lie on the path of the charge, namely the positions at
any time t of all the waves emitted up to that time.
So long as the velocity of the charge is less than that of light the oldest
wave contains all the subsequent ones, and every wave equally contains all

'I,

Fig.

2.

Fig. 3.

30

PHYSICAL INTERPRETATION OF TIHE SOLUTIONS

OBTAINED

[CR.

the later ones, and therefore also the whole of that portion of the path
described subsequent to its time of emission, as shown in Fig. 2. In this
case no two waves intersect.
When the velocity of the charge exceeds that of light, so that the charge
outruns the waves emitted by it, some of the waves cut each other as shown
in Fig. 3. Thus the family of spheres has an envelope PEQ, with a cusp
at E, the position of the charge at time t. The envelope may have several
sheets, but only one is shown.
If both t and r be regarded as variable parameters, the characteristic
equation represents a doubly infinite system of spheres, namely all the
positions of all the waves emitted by the charge. We get this system by
supposing either family of spheres just considered to take up all possible
positions; for instance, the second family may be supposed to expand, each
sphere with the velocity c, while new spheres are being constantly generated
as the charge moves along, and the envelope is constantly being added to at
the cusp and is moving outwards elsewhere.
All these families of spheres can be constructed quite uniquely when the
path of the charge and its law of description are given; and the construction
thus supplies a complete solution of the problem of determining the disturbance at every point of the field. [See note at the end of this chapter.]
18. In practice however the problem is different; we fix upon a definite
fieldpoint (x, y, z) and a definite time t, and wish to determine the values of
the potentials for these particular values without having to solve the whole
problem at the same time.
So long as the velocity is less than that of light there is no difficulty,
beyond the labour of computation. If the fieldpoint be outside the oldest
wave emitted by the moving charge, there is no disturbance there, for
as we
have seen, this wave includes the whole region of disturbance.
If it be inside it is disturbed, but only by a single wave at a time; for
no
two waves, corresponding to the same value of t, but to different values
of
ever intersect. Thus t being given, T is determined uniquely; the characteristic equation gives but one real value of less than t. The corresponding
values of K and R can be calculated, if not directly, at all events by
means of
Lagrange's Theorem, and the values of the potentials are then given
by (30)
and (31), § 13.
When the velocity exceeds that of light, difficulties arise for
points
inside the envelope corresponding to the time t. Every such point
is the
intersection of at least two waves emitted at different times; for instance, in
Fig. 3 the point C is the intersection of the spherical waves whose centres are
at the points A and B. For points on the envelope the two waves touch,
and their centres coalesce. For points outside it the waves are imaginary.

7,

T

PHYSICAL INTERPRETATION OF THE SOLUTIONS OBTAINED

IV]

In this case the characteristic equation, regarded as an equation giving T
when x, y, z and t are prescribed, has two or m6re real roots less than t, one
corresponding to each of the waves which reach the point (x, y, z) at the
time t. When (x, y, z) passes the envelope, two of these roots become equal
and then imaginary. Thus the envelope is the locus of points for which
two roots of the characteristic equation are equal and is given by the equation
at
1 aR
= 7++
.K
1
= 0........................(29).
aT

-c

aT

The roots of this equation are just the critical values T12, T, T34, ... used in
§ 14; and the corresponding values of t, given by the characteristic equation,
are identical with the maximum and minimum values of t', denoted by t'12,
t' 23 , t%, ... , both sets of quantities being taken for the prescribed values of
They are perfectly determinate functions of (x, y, z); therefore
an equation, such as t'12 = t, represents a family of surfaces, namely the
(x, y, z).

successive positions of one of the sheets of the envelope, such as PEQ in
Fig. 3. When the characteristic equation leads to p - 1 double roots 12 ,
equations
723, ... 7T_, p, the set ofp -1
t'12= t, t'2;= t, ... t'p ,p = t ..................... 36)
determines p - 1 families of surfaces, each of which represents the successive
positions of one of the p - 1 sheets of the envelope. These sheets separate
space into regions, differing as regards the number of waves which reach a
point simultaneously. As we pass from one region to the next, this number
changes by two, either because two waves coalesce and become imaginary, or
because two new waves appear.
The interpretation of expressions (34) and (35), § 14, is obvious; fixing
our attention on the particular fieldpoint (x, y, z), let us draw the (t, r) curve
corresponding to it, as in Fig. 1, § 14. Two cases occur: in the one t is - c
when is - ao, and the curve has an even number of maxima and minima
when r = - o, and their number is odd.
of t; in the other t = +

T

co

Fig. 4.

32

PHYSICAL INTERPRETATION OF THE SOLUTIONS OBTAINED

[CH.

In the first case, to begin with there is one wave, until the time corresponding to the least of all the minima is reached, say at A. Until this
moment each of the expressions (34) and (35) has but one term.
At the time A the first sheet of the envelope reaches (x, y, z), and thereafter this point is disturbed by three waves simultaneously, corresponding to
the three points in which the horizontal BCD cuts the characteristic curve.
Each of the expressions (34) and (35) now has three terms, one representing
the effect of each wave.
This continues until another sheet of the envelope reaches the point
(x, y, z); if it correspond to a maximum, as at E, the oldest wave coalesces
with one of the later pair, and only one wave remains; accordingly the
expressions (34) and (35) are again reduced to single terms. But if the
sheet of the envelope correspond to a second minimum, as at F, two new
waves appear, so that (x, y, z) is disturbed by five waves at once, and the
expressions (34) and (35) each have five terms. It is easy to understand
what occurs in other possible cases: whenever the passing envelope sheet
corresponds to a minimum value of t, two new waves appear, but for a
maximum two waves already present coalesce and disappear. The number of
waves disturbing (x, y, z) simultaneously is always odd. (Fig. 4.)

C
B

Fig. 5.

In the second case there is no disturbance whatever to begin with. The
first envelope sheet to reach (w, y, z) corresponds to a minimum, as at A,
and brings with it two waves; accordingly the expressions (34) and (35) have
two terms. The second envelope sheet again corresponds to a minimum, as
at B, bringing two new waves, so that now the point (x, y, z) is disturbed by
four waves at once, and the expressions (34) and (35) have four terms. If
the next envelope sheet correspond to a maximum, as at U, two out of the four
waves coalesce and disappear, otherwise we may get six. In every case we
have an even number of waves. (Fig. 5.)

IV]

PHYSICAL INTERPRETATION

OF THE SOLUTIONS OBTAINED

33

An example of the first case is, as we shall see presently, afforded by the
problem of the uniform rectilinear motion of a charge when its velocity is
less than that of light; and one of the second by the same problem when the
velocity exceeds that of light.
When the moving charge starts from rest, or when its motion is suddenly
changed by some impressed force, these results are modified to this extent,
that the disturbance at the point (x, y, z) only begins when the first spherical
wave emitted by the charge reaches it, or that the character of the disturbance changes suddenly on the arrival of the wave emitted by the charge when
its motion was suddenly changed.
19. Graphic representation of the potentials. The characteristic
curve for the fieldpoint (x, y, z) affords a graphic representation of the
potentials, which is sufficiently accurate to allow us to draw conclusions as
to the relative importance of the terms due to the several disturbing waves.
The diagram represents a characteristic curve with two minima and one
maximum. The horizontal ABCDK is shown cutting it in four points

A

"

K

M
M

Fig. 6.

A, B, C and D, corresponding to four disturbing waves. Let it cut the line
t = T in the point K, and draw the ordinate EGKHF, cutting the tangents
to the curve at A, B, C and D respectively in E, F, G and H (the intersection E is not shown in the diagram).
Consider the effect of one of the waves, say that of B. The term due to
it in the potential 4 is by (34)

fde

where it is to be remembered that the square bracket denotes that the value
of 7 for B is to be substituted in K and R. The term is numbered 3 because
B lies in the third interval of

, reckoning fiom

'

=+ c .

The value of K3

is positive, because B is on an ascending branch of the (t, r) curve, and
K- . Thus [K 3] = tan KBJIF.

PHYSICAL INTERPRETATION OF THE SOLUTIONS OBTAINED

34

[H

[OH.

Again, R3 = c (t - T) by definition (characteristic equation); thus [Rd] = c
times the intercept MB. But MB = BK; hence
[1C3R 3]

c. BK tan KBF=c. KF.

In precisely the same way we have

[KR,]-c.KH, [K2R]=-c. KG, [K 4 1 4 ]=-c.KE,
where KE, ... denote distances without regard to sign.
Now for the present case (34) gives
r
[K

[1

Hence we get

P=fde {JJE 1

Similarly

If
a= -C"
jd

i

,

, herc VA,
rmission

7

...

[K 4R4]}

KH

........... 37).

VA
Vs
V
VD
l + K1'+K +KJ

.........

(38),

denote the velocities of the element of charge at the times of
of the waves A, ....

A, ...

most important waves are
those which have been omitted most recently, e.g. the wave D. But two
These expressions show that generally the

waves rise into exceptional importance as they approach to coalescence,
since in that case the inclinations of the corresponding tangents of the curve
to the horizontal approach the limit zero, and their intercepts on the ordinate
tend to vanish.
20. [Note. Geometrical constructions. The relations between the
path of the charge, the waves emitted by it and their envelope are rendered
much clearer

by the following geometrical constructions.

timier

We sawv on p. 29 that the spherical wave emitted at
from the point
(,v, ) is represented in its position at time t by the characteristic equation

t=TR /c ........................ (28),

(a)
where

R-I

V{(x -

) + (y

-

?)+

(z

-

)21

The equation of the wave-envelope is got by eliminatingT between this
equation and its first differential
(b)
R' =-c............................(39),
where Ai' is the total
coefficient with respect to T,(, ,) being
treated as given functions of T. We shall denote the resulting eliminant by
(abp).

differential

PHYSICAL INTERPRETATION OF THE SOLUTIONS OBTAINED
]

iv]

3

35

equal
E, and El be allowed to coincide the hyperboloid reduces to 10e(T2asymptotic
to the
of revolution with E1 and E2 as foci and real semiaxis

Tr')If

-

cone, and if its semivertical angle be 0, we obviously have cos 9= clv, where
v is the velocity of the charge at the time of emission 7. The front half of
this cone belongs to times t greater than T. Now we have

identically

R'

cs 9......."(40).

Thus (39) is the equation of the front half of the asymptotic cone.
The semicone (39) obviously cuts the spherical wave and the wave
envelope perpendicularly. The circle of intersection is a line of curvature of
the envelope, the vertex is the corresponding centre of curvature, and the
path of the charge represents the corresponding sheet of the surface of
centres.
The condition that three successive waves (a) touch is
(c)

R"

T

_=.

.

j.

involving

Eliminating
between (b) and (c) we get an equation (bc),
neither t nor T, but only (tr, y, z). It represents a
surface along which
three successive waves ((t), and therefore also two sheets of the wave-envelope
(ab) touch. Thus the wave-envelope has a cuspidal ridge, which as it moves
generates the surface (bc).

fixed

But (c) also represents the condition that two successive semicones
(b) touch, and therefore the surface (be) is also the envelope of the family
wave,.and to
of semicones. Since each cone is normal to the
the envelope, the surface (bc) is also the second sheet of the surface of centres
of the family of envelopes, which are parallel surfaces.

corresponding

Again, differentiating

Let
angle

f

(40) we get

be the resultant acceleration of the charge

it makes with the radius vector

at

time Tr, and ~rthe

Then using (40) and (41) we

with (:39) and
we given theipth of the charge andl

This equation, toge'ther

cusplocus (be), when

R.

ltrc

get

(40).('natbleu~s to construct the
it.

mode of

36

PHYSICAL INTERPRETATION OF THE SOLUTIONS OBTAINED

[CH. IV

In Fig. 7 E is the position of the charge at the time T, ET the direction
in which it is moving, and EF a length
equal to (v2 -c2)/f, drawn in the direction

of the resultant acceleration. The plane of
the paper is the osculating plane at E.
PEP' is the trace of the semicone (b), so
that PET = P'ET = 8 = cos- (c/v).
PFP'is perpendicular to EF.
The dotted curve PP' is the conic in
which the cone (b) is cut by the plane (42),
which passes through PP' and is perpendicular to the osculating plane TEF. The
cusplocus is generated by this conic, and is
indicated by its traces at P and P'. These
traces touch PE and P'E respectively.
Again, to construct the envelope, we
get from (39), (40), and (28)
0
R cosO=- RR
v

Fig. 7.

_-

(t-

v

v

..................

(43).

Make ET =c2 (t- T)/v, and draw K'TK perpendicular to ET. The plane
through KK' perpendicular to the osculating plane is given by (43), and cuts
the cone (39) in the circle KK'. The envelope corresponding to E in its
position at time t is generated by this circle, and is indicated by its traces at
K and K', which are perpendicular to EP and EP' respectively.
As the charge moves, K describes a line of curvature of the envelope and
P its evolute, both of which are generally tortuous curves. The line of
curvature is limited, either by its intersection with the path, or with its
evolute. In the former case all the lines of curvature of the same family cut
the path in the same point, which represents the limit of the circular lines
of curvature of the second family, is a conical point of the envelope, and
corresponds to the position of the charge at the time t. In the latter case
the intersection of line of curvature and evolute is the point of contact of
three successive spherical waves, and therefore forms the starting-point of a
line of curvature belonging to a new sheet of the envelope.]

CHAPTER V
ILLUSTRATIVE EXAMPLES

21. WE shall now investigate some problems in illustration of the
methods developed in the last chapter. For this purpose it will be
to confine ourselves to the case of a single point charge, for the field due to
several such charges is merely the geometric sum of those due to each taken
singly; in particular that due to an extended charge can be obtained by
integration. As we have seen, the potentials, and therefore also the electric
and magnetic forces, due to a point charge moving with a velocity greater
than that of light, become infinitely great at points on the envelope of the
spherical waves, as well as at the charge itself. It has already been
out that this difficulty does not exist in the case of extended distributions,
whether on surfaces or in space, provided that the surface, or volume, density,
our
will be met with
A good
as the case may be, remains finite.
velocity in
first problem, that of a point charge moving with uniform
line; the difficulty has in this case been completely resolved by ommerfeld.
The general case may be treated in a similar way.

sufficient

pointed
in

example

This problem
several writers, particularly by Sommerfeld O, but as

Problem 1.

22.

by

been treated

a straight

Uniform rectilinear motion.

has
the

method.
solution is simple, it will afford a good illustration of the general m
T.ake the line of emotion as the x-axis of a systemi of cylindrical coordinates
of (x,

,

fieldpoint

);

ymmtetry about this axis, we need only consider a

since there is s

in the plane

~=

0, say the point (x,

The position of the charge is given by
=v'r, i= =o from T=Hence we

'a).

00to

T~=+ ~

PR V( T- x)u +

get

ICE.=l+v(r-)/cI?......4)
*n~1

90
)4,

p.

w3

7.

~ILLUSTRATIVE
EXAMPLES

38

38

[H

[OH.

We see from (442) that the characteristic curve is the upper branch of the

hyperbola
2

c

(t -

T)2 -

(VT

-

x) 2

=2

The lines t = ' and t = vT -- x are conjugate diameters.

The lines

± (VT-x)

c(t-T)=

are asymptotes, and obviously furnish the characteristic curve for a point on
the line of motion, for which
= 0. The characteristic curves for the cases
of a velocity less than that of light, and one greater, are essentially distinct
and are shown in Fig. 8 and Fig. 9 respectively.
i

E
E

g

D

%

1

/2

1

Sp
IT

''

II

C-

C

D
AjA

Fig. 8.

For v= c, x=c, w=c.

Fig. 9.

For v=2c, x=2c,

=2c.

In the first case the curve ascends continually, and a line t = constant cuts
it once and only once. In the second case the curve has a minimum at P,
and a line t = constant cuts it either twice or not at all. The minimum value
of t is t12, which is equal to KP, arid is given by vt12 = x +
-1);
thus
t= 2.73 in Fig. 9.
The corresponding critical value of T is T12, which is
equal. to OK arid is given by VT = .x - W/V(V2/c2 -1); thus T = 0.42 in Fig. 9.
12
12
W hen t < t12 , the characteristic equation has no roots ; when t > t12 , it has two,
T and T2, of which T 1 > T12 and 2 < T12" One corresponds to the branch
1
(1), FE, the other to the branch (2), DP. When t = t12 we have T1 = T2 = T2
-V(V2/e2

In the present problem the roots of the characteristic equation are easily
found analytically. Rationalizing (442) we get
(v 2 -c 2)T 2 - 2 (Vx -- c2 t) T +x2 +'2- c2 t2 -=0,
T= vx- c 2 t±+Vo2(vt-x2_(v2-2)a21

whexnce
rhi

lhsgives

C2V
2-

___V

-

_-

(vt

- .:) +1/
__.___
+

.
v"

2

fC02(vt _l_

-c

ax) 2
v -C2

.___

[ ,

2

-(V2

""""""""""""(45s).
-

C ') W}
__

_

(46__

).
(46.f"

v]

ILLUSTRATIVE

23.

(a)

39

EXAMPLES

Velocity less than that of light: v < c, Fig. 8.

The square root in (45) and (46) never vanishes, and is always greater
than c (vt - x) in

absolute

value, therefore still greater than v (vt - x).

(46) shows that the upper sign must be taken in order to make t > T.
This corresponds to the point in which the horizontal line at level t cuts
the upper branch DE of the hyperbola in Fig. 8. The second non-available
root corresponds to its intersection with the lower branch. Thus there is
always one, and never more than one solution; (444) gives, with (46), and
(30), § 13,
[KR] = V{(vt - x)2 + (1 - v2/C2) 2},
1
= (vt -

)2 + (1 -

=c /{(Vt -x) 2 + (1 -

2/)

}

(47)

......

2/ 2
)2}

which is the well-known solution for this case.
24.

(b)

Velocity greater than that of light: v > c, Fig. 9.

The square root in (45) and (46) is real or imaginary according as
(vt - x) 2

(v1C2

1) 2.

The equation (vt - x) 2

=

(v2/c

-

1)

2

represents

a

cone, whose vertex is at the charge and which moves through space with
the charge. Its semivertical angle is equal to sin - 1 (c/v). Inside it the roots
of (45) are real, outside imaginary.
When the square root is real it is obviously less than c (vt - x) in absolute
value, therefore still less than v (vt - x). It follows from (46) that both signs
make t > r according as vt -x > 0. Thus the characteristic equation (442)
has two solutions, or none at all, according as vt - w >
/(v 2/ 2 - 1). The
equation
represents the back half of the cone just considered.
semicone the two roots, T and

T2,

At the surface of this

are both equal to the critical value T12, which

is less than the corresponding minimum, t12 , of the characteristic curve. Inside,
behind the semicone, they are both real and less than t; outside it, between
it and the front half of the complete cone, they are imaginary; and inside, in
front of the front half they are both real but greater than t, and therefore not
available. For a given fieldpoint (x, w), the semicone (48) corresponds to the
time t = t12, where as before vt 2 = x + V(v2/c 2 - 1); that is, t is the time at
12
which the moving semicone reaches the given fieldpoint. When t < t12, that
is, before the semicone reaches the point, there are no solutions, and there is
no disturbance. When t > t12, that is, after it has passed the point, there are
two solutions, and the disturbance is due to two waves at once. The semicone
corresponds to the minimum P of the characteristic curve DPE of Fig. 9.

40

ILLUSTRATIVE

[cH.

EXAMPLES

Fig. 10 gives a meridian section of the semicone for the case v = 2c.
RQEQ' is the trace of the semicone, and
R
E is the charge, both in their positions
for time t. P is the fieldpoint (x, vr),
inside the semicone. E 1 and E 2 are the
centres of the two spherical waves which

The roots of (45),

disturb P at time t.

E,

E

II

0

x

1

and T7, are the times at which the charge

passed through E 1 and E 2. As P moves
up to the semicone, E1 and E2 obviously approach and ultimately coincide,
and so also do Q and R. Thus the semicone is the envelope, in its position
at time t, of all the waves emitted by the charge at earlier times. Since
E 1 Q = c (t - T1) and E 1 E = v (t - 7T), we have
sin E 1 EQ = c/v.

The semicone at any time divides space into two regions: (I) in front of
it, where there is no disturbance; and (II) behind it, where the disturbance
is due to two waves. These regions are indicated in Figs. 9 and 10.
The potentials are easily found. In region (I) they are of course zero.
In region (II) they are given by (34) and (35),§ 14, together with (444), (45)
and (46). The value of T7 corresponds to the upper sign in (45); hence we
get
[K 1R 1] = + /{(vt -

2

x) -

(V2/C2

)

2},

[K 2R2] =

With these values we find, for (II),
2
2
2
2 - (v 2
{(tx)
(vt - _
which is the well-known result for this case.

-

'{(vt-

x)2-

2v
2
)2- (v/-

(2/C2

1)

2

)

2}.

(49),

We notice that as the envelope passes the point, the potentials due to
each of the two waves become infinite, equal and of the same sign, so that the
resultant potentials themselves are infinite. It has already been pointed out
that this infinity is merely due to the supposed concentration of a finite charge
at a point, but does not exist in any actually realizable case, as Sommerfeld
has exhaustively proved in the investigation already referred to.
Uniformly accelerated rectilinear motion.
This problem leads to biquadratic equations and thus can be completely
solved in theory, though in practice the expressions obtained are too
unwieldly to work with.
25.

Problem 2.

Using cylindrical coordinates (x, wr) we write for the coordinates of the
moving charge
=

=

=0 from T=- -frT2, to T=+ oO.

The charge moves in the negative directi n with uniform retardation

v]

ILLUSTRATIVE

41

EXAMPLES

from an infinite distance up to the origin, reaches it at time zero, reverses its
motion there and moves away again with uniform acceleration. Thus the
motion for positive times is exactly the reverse of that for negative ones;
nevertheless the field produced is not reversed, nor is it symmetrical with
respect to the time zero.
We get

R =

V(fr2 - X)2 +

2

V(U72-w)2+&I

t= 7+

= R +fT (fT 2 --)/c

KR
Rationalizing (502) we get

)
4 (fx + C2

ct 2) = 0 ...... (51).
Only those roots of (51) which are less than t give solutions of our
problem. Putting
= t, and r=-oo, in succession, we see that their
number is always even.
F()

-

2

+ 8t

+ 4(r

2

-

Using the same notation as before we denote the roots of (51) by
T1

when all are real.

>

72 >

7T
3

>

74,

Now (51) may obviously be written in the form
c2 (t -

7)2

R2 =

0.

Thus it represents, not only the characteristic equation t = 7 + R/c, whose
roots, when real, are less than t, but also the derived equation t = 7 - R/c,
whose roots, when real, are greater than t.
Hence when the four real roots of (51) are distributed between the two
equations, we must take 73 and 74 as the roots of the characteristic, and rT

and 73 as the roots of the derived equation.
When the equation (51) has two real roots we must, on account of the
principle of continuity, choose them to be T3 and 74, when they belong to the
characteristic, and Tr and 72, when they belong to the derived equation.
In our study of the problem we shall begin by determining the conditions
under which the characteristic equation may have four, or two, or no real
roots. This amounts to determining the critical values of T, which are
double roots of (51) less than t and determine the minima and maxima of
the characteristic curve, as explained in Oh. III, 14. We can then determine
the position of the various sheets of the envelope at the time t, and so obtain
a general idea of the nature of the field.
We shall then consider the characteristic curve, and explain how to
calculate the roots of the characteristic equation and obtain expressions for
the potentials in the various parts of the field. In this way we shall get
some idea of the way in which the field developes itself, both for velocities

~ILLUSTRATIVE
EXAMPLES[o.

42

42

[CH.

less and greater than that of light, in the ease of the most general motion in
a straight line; for the present problem obviously gives a second approximation to every motion of this kind.
26. Critical values. The critical values of T are found by eliminating
t between the characteristic equation and its first differential with respect
to T. We may use (51) in place of the characteristic equation, provided we
bear in mind that values of T less than t are alone relevant to our problem.
In this way we find

(fT2 -

2x) 2 (f

2T2 -

c2)

4C2w

-

2

=

0

(52).

............

Using (502) and (503) we get
2
t = T +fT (x - f7 2)/c

. . . . . . . . . . . . (53).

The only available roots of (52) are those which are real and positive, and
at the same time make t > T. That is, we must choose the sign of T, which
is at our disposal, so that T and x - 2 f 2 may have the same sign. This
gives us three solutions. The remaining three roots of (52) correspond to
the derived equation. Thus we get the following expressions:
(a)

2fx - c2 < 0 - one real root.

(b)

ft 34 = - \/{C2 + s (c2 - 2fx) sinh2 01
sinh 2 30 = 2C 2 f2 ~2(C2
2x 3 . . . .
27c2f22 > (2fx - C2)3 > 0 - one real root.
-

_--

f34
2

cosh 30

V{e 2

27c 2f

=

(C) (2fx - c2)3 > 27C2 f22
fT34 =

+ 3(2.tx

2/(2f

2

-

.

Ccosh20.

.

. .

..

.

5)

(55),

C2

-

three real roots.
ce± (2fx- c2) sin 2
-

-

1723=_

+

fr13

+V{C

\VIc

2

+

(2fx

-

2+ 3
(2fx -

sin 2

C2

(7/3

+

&)}

01

2
C) sine (7r/3

5)
-

0)1

2
sine 30 = 27Ic 2f 2v 2/(2fx - C)3, 0 < 0 < 7r/6)
It is easy to see that the values of T34 given by these expressions are
continuous.
The notation has been chosen to fit our conventions as to the roots of (51).

27. Wave-envelope.
The several sheets of the wave-envelope are
found by putting t = t34, t = t23 and t = t12, where t34, ... are minimum and
maximum values of t given by (53) with T34, ... in place of T.
Both t and
(t 34)

(7-T

are real when, and only when, T is real.

exists for all points, but the two sheets

points for which T13 and
paraboloid

T23

(t 1 3)

Hence the sheet

and (t23) exist only for

are real, that is, by (56), inside the semicubical

27C2f

[This is the cusplocus of p. 35.]

2
2vT

=

(2fa

-

C23...................(57).

xr]

v]

~ILLUSTRATIVE
EXAMPLES

4

43

From (53) we see that the ratio t/7 has the sign of the quantity
c +fx - ff.
Thus t12/Tr 2 and t23/7 23 are positive on account of (56), and,
as T12 and T23 are also positive, t12 and t23 are both positive. Again, t34/7 34 is
positive (negative), and, as T34 is negative, t34 is negative (positive) according
as the fieldpoint (x, Z) is inside (outside) the paraboloid
2

C2)

(58).
................
f 2 2 = C2 (2fX +
t34 vanishes at all points of this paraboloid, which is therefore the envelope
sheet (t34) in its position at time zero.

In order to be able to construct the wave-envelope we require to know its
intersections with the x-axis and with the cusplocus.

0;

On the x-axis we have Z =
this gives 0 0 in cases (a) and (c), but
cannot occur in case (b). We find, by (53)-(56)
(a) outside the cusplocus (57): x < c2/2f,
fi 3 4 =
c, ft34 = - (2fx + C2)/2C,

..

(59),

x- ft 3
ft34 = V(27x),
ft 2 3 = (2fx + C2)/2c, X = Ct23 -- c2/2f.
2
ft12 + V(2fx),
x
1

(60).

-

xr-Ct

34 -c2/

2

(c) inside the cusplocus: x > c2/2f,
fT34 = fT 23 =

V(2fw),

+ C,

fT12 = + V(2f'),

-

-ft

J

Putting t = t34, and remembering that the x-coordinate of the charge is 2ft2
at time t, we see that the vertex of the envelope sheet (t34) coincides with the
charge until t = - c/f, that is, so long as the charge is moving towards the
origin with velocity greater than that of light. It leaves it just as the
velocity becomes equal to that of light, and ever afterwards moves ahead of

it with that velocity, so that at time t its distance from the charge is equal
to f(t +c/f) 2
.
Putting t = t23 we see that the envelope sheet (t23) arises at the time t c/f,
and at that time its vertex coincides with the charge, which is moving with
the velocity f~light in the positive direction. The vertex at once falls behind
the charge,>continues to travel with the velocity of light in the positive
direction, so that at time t its distance from the charge is equal to
2 f (t - c/f)2.
-

Putting t = t12 we see that the envelope sheet (t12) arises at the time t
and that its vertex coincides with the charge ever afterwards.

=

c/f,

The intersection of the envelope with the cusplocus is got by making
in (56); the value of T~34 is of no interest, but for the other critical
0
values we get
2
f'r 23 =f7T12 =±+
V3(fx+ C ),

=,7r/6

44

ILLUSTRATIVE

[OH.

EXAMPLES

whence, using (53) and (57), we find for this intersection
x

fr

2

2 2

. . . ...

(61).
If s be the are of the semicubical parabola (57), measured from its vertex
ds
dt
2
=

2/f

=(f

-

c2/c

to the intersection with the envelope, we easily find

dT=

3f

.....

T2/C

=

C

,

whence s = c (t - c/f). This result verifies that the intersection travels
along the semicubical paraboloid with the velocity of light away from the
vertex.
Again, we find that /{(x -

frT)2

+ ~2} = c (t - 7); this verifies that

is

the time of emission of the three coincident spherical waves, which touch the
sheets (t12) and (t23) along the circle (61).
P

H

II
t 4

A

D

B

Fig. 11.

G

For t = F 2c/f.

[28. Construction of the wave-envelope and its cusplocus for the
case of uniformly accelerated motion. In the present problem there is
symmetry about Ox, so that it is sufficient to construct the traces of the
various surfaces on a meridian plane of (x,
In Fig. 11 C is the point
(c2/2f, 0), through which the charge passes at the times t = + c/f, when its
velocity is just equal to i c. E is the position of the charge at times
- 2c/f, when its velocity is equal to - 2c, so that OE = 2c2/f The envelopes
for these two times are shown in the figure.
PEP' is the semicone (b) for both times + 2c/f (cf. Fig. 7, § 20).
PNP' represents the plane PFP' for both times; EF reduces to EN,
because f is always directed along Ox, to the right. The intersection of the
semicone and plane of course reduces to a circle indicated by PP' (P' is
the intersection of HEP' with PNP' and is below the diagram).

w).

v]

ILLUSTRATIVE

45

EXAMPLES

The locus of the circle PP' is the semicubical paraboloid (57), of which
the trace is PBCB'P'; it is the envelope of the semicone PEP', and the
cusplocus of the wave-envelope. BB' indicates the cuspidal ridge for the
time t = + 2c/f.
EP touches the arc CBP at P; if t' be the time at which the cuspidal
ridge of the wave-envelope passes through P, we find, by (61), EP = c (t' - t).
This merely expresses the fact that the disturbance emitted at time t from
E in the direction EP reaches P at the time t'. Further, arc CB = c (t - c/f),
and arc CP = c (t' - c/f), since the cuspidal ridge starts from C at time t = + c/f,
and travels along the cusplocus with the velocity of light. Hence EP = arc BP.
(Parenthetically we may notice a simple construction for P; we have
OC=c2/2f, OE = ft 2, so that CE= ft 2 - c2/2f. Again, by (61),
ON = ft 2 -

so that CN = 2ft

2

- 3c2/2f.

2/f,

Hence we have CN= 3CE.)

Again, we found, in (602), that OA = ct - c2/2f, so that
CA = c (t - c/f) = are CB.

Lastly, we found in (59), that OF=ct + c2/2f, so that CF=c (t + c/f).
Thus FA = 2ct.

Since the trace of the wave-envelope is an involute of the trace of the
cusplocus (§ 20) we can construct the envelope as follows:
From E, the position of the charge at times ± t, draw EP to touch the
3CE, and drawing NP
semicubical parabola CBP at P (by making
vertically).
Unwrap PE from the infinite arc beyond P; the end E describes the
arc EG',which generates the envelope sheet (t34), that is, the complete envelope

ON=

for time - t.

Wrap PE on to the finite arc PB; the end E describes the arc EB
belonging to the envelope sheet (t 2 ) for time + t.
Unwrap a length CB from the arc CB; the end B describes the arc BA
belonging to the envelope sheet (t23) for time + t.
Lastly, choose a length greater than PE by the length 2ct, and unwrap it
from the complete arc GBP; the free end describes the arc FH' belonging to
the envelope sheet (t34) for time + t.
It is evident from the construction that the envelope sheet GEG' for time
- t, and the sheet (t12) of the envelope for time + t, are continuous with each
other at E, and have equal and opposite conical points there, provided t > c/f.
The semivertical angle at E is sin - (c/v); as E approaches the point C,
and its velocity v approaches to equality with the velocity of light N, the

46

ILLUSTRATIVE

[CH.

EXAMPLES

semivertical angle approaches to 90 ° , and the conical point disappears. The
envelope GEG' then assumes the shape HFH', and retains it for all times
later than - c/f At the time zero it takes up the position JDJ', which
represents the trace of the paraboloid (58).]
29. Characteristic curve. The biquadratic (51), § 25, may be represented by a quartic curve with two infinite branches. The upper branch,
shown as a thick full line in Figs. 12-15, is the characteristic curve, the
lower, shown as a broken line, represents the derived equation got by changing
the sign of R. Thus the line t = always bisects the vertical distance between
the two branches.
In addition, in the present problem R is an even function of T, so that the
derived is got from the characteristic equation by changing the sign both of
t and of T7. Hence the two branches of the quartic are symmetrical with
respect to the origin.

T

There are four cases: (a) fieldpoint inside the paraboloid (58), but outside
the cusplocus (57), Fig. 12; (b) fieldpoint inside both surfaces, Fig. 13;

Fig. 12.

For x = 2c

2

/f,

w = 2c 2 /f.

Fig. 13.

2

For x=7c /2f,

-

2

=2c /f.

(c) fieldpoint outside both surfaces, Fig. 14; (d) fieldpoint outside the paraboloid (58), but inside the cusplocus (57), Fig. 15.
In cases (a) and (b) we have t34 < 0, and both branches cut the axis t = 0;
in (c) and (d) we have t34 > 0, and neither branch cuts the axis t=0.

v]

ILLUSTRATIVE

47

EXAMPLES

Vii In cases (a) and (c) theILLUSTRATIVE EXAMPES4
critical value j3 alone exists, and the characteristic

equation has at most two real roots ; in (~ and (d) T12 and T23 are real as well
as T , and the characteristic equation may have as many as four real roots.
34

Fig. 14.

For x=-

2
C1f,

z=2c /f.
2

Fig. 15.

2
For x=-5c2 /f, z =: 4c /f.

All the curves show the minimum 1?, which corresponds to the enve'lope
sheet (t34) ; but only the curves in Figs. 13 and 15 have the second minimum
P and maximum Q, corresponding respectively to the envelope sheets (t12)
and (t23).

The dotted curve is the parabola t

= T-

+

j

fr2/2c,

which is a curvilinear

asymptote.

=/c

When x >0, the curve has a double tangent FG, which is parallel to the
above it, and touches the curve where
line t = T at a vertical height
T= ± V1(2x/f) and is below it elsewhere.
30.

Progress of events in the field.

A study of Figs.

11-15

leads

to the following results :
At all times previous to + c/f the envelope has but one sheet (t34), which
has the conical shape GEG' when t < - c/f, and the paraboloidal shape HFH'
when - c/f < t < + c/f (cf. Fig. 11). This sheet comes from the right, expanding with the velocity of light, and reaches the fieldpoint when t t34. This
-

48

ILLUSTRATIVE

EXAMPLES

[oH.

time is positive (negative) according as the point is outside (inside) the
paraboloid (58) (JDJ'in Fig. 11). The two cases are illustrated by Figs. 14
and 15 (12 and 13) respectively. Times earlier than t34 correspond to levels
below the minimum R in these diagrams, where the horizontal line t does not
cut the characteristic curve CED, although it may or may not cut the
symmetrical curve. Equation (51), § 25, may, or may not, have real roots, but
if it has, they are greater than t, and do not belong to the characteristic
equation. Thus no waves, properly speaking no diverging waves, reach the
fieldpoint, and it is undisturbed. This region of no disturbance, outside the
envelope sheet (t34), we call region (I).
For times immediately after the time t4, when the sheet (t34) has already
passed the fieldpoint, the horizontal line t cuts the characteristic curve CED
twice, and may also cut the symmetrical curve twice, but not more than
twice. The characteristic equation has two real roots, but (51) may have
four, in which case two of them are greater than t and belong to the derived
equation. The fieldpoint is now disturbed by two divergent waves simultaneously, which according to our notation correspond to roots 7 and 74 of
3
is
(51), of which 74 is negative, while 37 negative (positive) according as
t > r/c. This region of two-wave disturbance (34) we call region (II). For
points outside the cusplocus it lasts for ever, as shown in Figs. 12 and 14.
For points inside the cusplocus however, where the second minimum (t12)
and the maximum (t23) exist, this type of disturbance lasts only until t = t12,
that is, until the conical envelope sheet (tl2), BEB' in Fig. 11, reaches the
point. Immediately afterwards the horizontal line t cuts the characteristic
curve CED four times, when it lies between P and Q in Figs. 13 and 15.
> 73 > 73> r4, and the point
The characteristic equation has four real roots,
is disturbed by four divergent waves simultaneously. The region of fourwave disturbance, lying between the envelope sheets (t1 ) and (t23), BEB'A in
2
Fig. 11, we call region (III).
The four-wave disturbance is only temporary, and disappears as soon as
t = t23, that is, when the envelope sheet (t23), BAB', passes the fieldpoint.
The two intermediate roots, T2 and 73, of (51) become imaginary, while the
extreme roots, Ti and T4 , remain real. The point is again disturbed by two
waves simultaneously, but these are the waves (T1) and (74). This type of
disturbance lasts for ever, but is not continuous with the type (34), existing
on the other side of the cusplocus. The region, where it exists, lying between
the envelope sheet (t23) and the cusplocus, we call region (IV).
It remains to consider the roots of the characteristic equation in detail,
and to find expressions for the potentials of the several types of disturbance.

T1

31. The potentials. Their values are given by (34) and (35), § 14.
, § 25, by
The values of [KB] for the various waves can be got from (
substituting the values of the roots v3, 'r, 73 and 74, as the case may be.

v]

v]

Since R

=

c (t -

Putting t =

~ILLUSTRATIVE
EXAMPLES

4

49

(51), § 25, KR

7) we easily find by help of

=

F' (T)/8c.

1, ... we get

[KR]]
1

=f 2 (7- -

72)

(7- -

73)

(7--4)/8...........(62),

with similar expressions for the other roots.
Again, since the coordinates of the charge are given by t

we find for the velocity at time

Tl

[v1 ]=fmr, [v1y]

= [viz]

=

0

-

'fT

2

i7-=0,

.............. (63),

with similar expressions for the other roots.

t,

It is of course obvious that the y and z components of the vector potential

[st] vanish identically, so that we have only to deal with the components
a1 , ... , besides the scalar potential 0 , ....
1

It is difficult to express the potentials explicitly in terms of t, x and w-,
but it is easy to find expressions in terms of the roots of the resolving cubic
of (51).
Putting in the usual way
71

U+V-W,

72 =U-V+W,

73=- -+V+w,

T

4

-

-v-W...(6

4

),

we satisfy the necessary condition 71 + 72 + 73 + T4= 0, and find that u2, v2, w2
are the roots of the cubic
f 4 z 3 - 2 (fX + ) f 2z2 + {(fx + c2)2 -f 2 (r2 - cft2)} Z - c:t2 = 0..
provided only that
uvw = c2t/f2
.................... (66).
Our object is to study the variations of the potentials at a given fieldpoint
as the time changes; hence we must determine the roots of (51) and (65) as
functions of t when x and
are given.

w

When we plot t as a function of z by means of (65) we obtain a graph of
the type shown in Fig. 16, for the case where the fieldpoint lies inside both
the cusplocus (57) and the paraboloid (58).

its asmtt

5.Fig./16.

n

fteoa

FF,

hc

oce

h

xsz=0a

50

[CE.

EXAMPLES

ILLUSTRATIVE

the origin. Since (65) involves only the square of t, the graph is symmetrical
with respect to the axis t = 0. It has maxima and minima for t at B, C, F
and B', C', F', corresponding respectively to Q, P, R on the characteristic,
and to their symmetrical points on the symmetrical dotted curve of Fig. 13.
The points B (t = t23), C (t = t12), and F (t = t4) are, the first two on the
positive, and the third on the negative side, because t23 and t12 are positive,
and, in the present case, t, is negative.
The graph for Fig. 12 differs in so far as the folds BC and B'C' are
absent, and those for Figs. 14 and 15 differ in so far as the oval EFOF' lies
below the axis z = 0 instead of lying above it, as in the former figures.
The intersections of the ordinate t with the infinite branch give real
and positive roots of (65); those with the oval give real roots, which are
positive (negative) according as the oval lies above (below) the axis t = 0,
that is, according as t3 > 0. This is easily verified by a study of (65), but
follows directly from our previous results. We shall denote the roots of (65)
by a, R and 7y.

,

When the roots are all real we shall choose them so that a > > 7.
When two are conjugate imaginaries, they will be denoted by a and /, or by
as may be necessary for continuity.
and

,

ry,

When a, / and 7 are all real and positive, u, v and w are real, and so also
are the roots T1,. T2, T3 and ,4of (51). When / and 7 are real and negative,
v and w are pure imaginaries and unequal, so that T1, T, T, and T are all
4
imaginary. When / and 7 are conjugate imaginaries, v and w are so
also, and two of the roots T, Te, T3 and T4 are real, and the remaining two
imaginary.
The proper expressions for the roots r, 2 , T3 and T4 are best got by
a study of Fig. 16, and the corresponding graphs for the other cases; but
we cannot in this way determine which of them are less and which greater
than t, that is to say, which belong to the characteristic equation of our
problem and which do not. This is best determined by a study of
Figs. 12-15.

32.

For example let us study the case corresponding to Figs. 13 and 16.

When t lies between + t3, the ordinate t cuts the graph of Fig. 16 between
are
FJ and F'J', therefore in three points. Thus a, 8 and 7ry all real and
positive, and the roots T, 2, T3 and T4 are all real. The horizontal line
t cuts the characteristic curve of Fig. 13 twice, above R, and the symmetrical,
broken curve also twice. We take the first two intersections to correspond
to T4 and T3 , the others to T, and

r.

This choice makes

74 ,

and

than t, so that they belong to the characteristic equation, and

T3

less

and r1

v])

ILLUSTRATIVE

greater than t, so that they belong to the derived equation.
choose u = +
v = + /Band w = + /ry, so that (64) gives

/a,

T= a+V/ -y,

51

EXAMPLES

Ta=
-/N +V7l,
T3 = - /a+ /3 + /

Further we

T4=-/a-V//-/...(67).

8>

This choice makes T > 72> 3
T >T3> in virtue of the inequalities a >
y
provided that 1/a and 1/B are taken to mean the absolute values of the roots.
In order to satisfy (66) we must choose the positive (negative) sign for /7,
according'as t 0.

As t increases, the ordinate in Fig. 16 moves to the right; when t = - t3
(which is positive in the present case) the ordinate reaches F'J',and / = y,
becoming conjugate imaginaries immediately afterwards. By (67) T1 and
become imaginary, while T7and T7remain real. This agrees with Fig. 13,
where the horizontal line t just touches the broken curve at its highest point,
and thereafter ceases to cut it.
So long as t < t12, the ordinate in Fig. 16 lies to the left of OH and cuts
the graph only once, so that the characteristic equation has only the two real
roots 7Tand 74. Up to this time the disturbance is of the two-wave type (34)
and the fieldpoint lies in the region (II) of § 30.
When t = t12 the roots / and y of (65) are again equal, and become real.
Between t12 and t, the ordinate in Fig. 16 cuts the graph three times, between
OH and BG,and the roots T1, 2, T3 and 74 are again all real. But now they
are all less than t, for the horizontal line t in Fig. 13 cuts the characteristic
curve four times, between P and Q, and does not cut the symmetrical curve
at all. The disturbance is of the four-wave type, and the fieldpoint is in
region (III).
When t = t23, the ordinate in Fig. 16 coincides with BG, the roots a and /
are equal and become conjugate imaginaries, while y remains real and
positive.

From (67) we see that T1 and

74

remain real, while T2 and 7T3

become imaginary. The horizontal line t in Fig. 13 touches the characteristic
curve at Q, and thereafter cuts it twice, on the arcs RO and PD. The
fieldpoint lies in region (IV).
Thus we see that with our conventions the root a is represented by the
BO
arc B'AB of the graph in Fig. 16, the root / by the arcs FEF', and B'O',
and the root y by FOF', CD and O'D'. This is in agreement with the fact
that /a and 1/ are always positive, while /y changes sign with t, and
therefore vanishes when t = 0, at 0.
33. The remaining cases can be treated in precisely the same way,
and the expressions (67), with the conventions made, are quite general.
Substituting the values of T1, ... in (62) and (63) and using (34) and (35), § 14,
we easily get the following expressions for the potentials.
4--2

52

ILLUSTRATIVE

(I)

Ti,

2, T3 and 7T4all

[CHO

EXAMPLES

imaginary, or greater than t.

Here /3 and ry are negative, as in Figs. 14 and 15 for t between ± t34 ;
otherwise the horizontal line t cuts the symmetrical curve alone.
. = az =.0
...

..................... (68).

(II) T1 and T2 are imaginary, or greater than t, while T3 and 4 are real
and less than t.
Here / and y are real and positive, or are conjugate imaginaries; the
horizontal line t cuts the characteristic curve twice and only twice, and the
symmetrical curve twice, or not at all.
2

a-'8

c

+

2)

2 (/3 + -+/3)V

f(a - /3) (a (III)

T1, ~,,

a/3

y)

................

(69).

(/+V)

r3 and 74 are all real and less than t.

Here and y are real and positive, and besides the horizontal line t cuts
the characteristic curve four times, and the symmetrical curve not at all.
4c

.f
(IV)

V/9

_/_./()(70).

71 and 74 are real and less than t, while

2

and3 are imaginary.

Here a and /3 are conjugate imaginaries, and the horizontal line t cuts the
characteristic curve twice, and the symmetrical curve not at all.

j(

2c (ry+

f (a - y) (/3
+

/a/3)

ry) (./a +V/3)(
-,2+(aa+7)
-

....

(71).

A comparison of (71) with (69) shows that the disturbance (IV), of twowave type (14), differs from the disturbance (II), of two-wave type (34),
that is by a change of notation. The
merely by the interchange of a and
expressions for the potentials in fact involve the single real root, a or y, as the
case may be, in precisely the same way.

y,

The solution of our problem is now completed, so far as this is possible
when the position of the fieldpoint and the time are not actually specified.
It represents the solution of the most general problem of accelerated motion
with unlimited velocities in all its essentials, for the development of the
envelope and of the various possible types of disturbance will take place in a
similar manner.

v]

ILLUSTRATIVE

53

EXAMPLES

34. Problem 3. A point charge, initially at rest, is subject to a
constant acceleration for a given interval of time, and thereafter
continues moving uniformly in a straight line with the acquired
velocity. Required the resulting electromagnetic field.
This problem illustrates the mode of combining the solutions of two known
problems, so as to give the solution of a new one. Moreover it has a special
interest in so far as it illustrates, crudely it may be, the process of the generation of X-rays, or of ry-rays, according to the commonly accepted views as to
their nature.
(= 0, v = 0 from =- 00 to = 0
We have

f=

fr2, v =fT from T = 0 to r = T..........(72).
= fT (T - IT), vz=fT from T= T toT=
Further y = z = 0, v,= vz= 0, a = az = 0 always.
There are two cases, according as the final velocity is less or greater than
that of light.
(a) Final velocity less than that of light: T < c/f.
The equations (54)-(56), § 26, show that the critical value T is negative
in every case. But in our present problem negative values of T, the time
of emission, correspond to times when the charge was at rest, and its field
everywhere electrostatic. Hence the critical value T. does not enter into
consideration at all. This means that the field is everywhere electrostatic
until the first wave of disturbance reaches the fieldpoint. This is obviously
always true, so that we can leave the envelope sheet (t,) out of account,
whether the final velocity be less than that of light or greater.
35.

The discussion is much simplified by a study of the forms and positions of
the several critical surfaces separating the various regions of space in which
the potentials are represented by expressions of different form.
The diagram, Fig. 17, shows the traces of these surfaces on the meridian
plane.

O is the origin, where the charge
was at rest until t = 0, T the point
(ifT 2, 0), where the uniform motion
began, E the position of the charge
at the time t. The circle DGD' is'
the initial spherical wave, emitted
at the moment of starting at O, and
HKH' is the spherical wave (T),

I

/
n
;K
H'
ATEC
Fig. 17.

H

D

x

For T=2c/3f, and t=c/f.

emitted at time T, when the motion changed, both in their positions at the
actual time t. The dotted curve AB is the position of the (absent) envelope
sheet (t,); it is absent because it is completely outside the initial wave,

54

ILLUSTRATIVE

[Cg.

EXAMPLES

corresponding as it does to negative times of emission. The several surfaces
move normally outward with the velocity of light, c, as indicated by the
arrows, while the charge moves onward along Ox. This diagram holds for all
time after the steady state has been reached.
36. The sequence of events at a given fieldpoint is best discussed with
the help of the characteristic curve, whose equation is

t= + R/c.................................(28).
From (72) we see that, since in the present problem
R = V {(x- )2+ 21,
R has three different expressions according to the interval for
used, say RB R 2 and R 3.
1,

r which is

Then we have
2
R=R 1 =V{x +W- 2 }=r,from r=-oo to

7=0

2 2
.. (73).
0
) + , from = Oto T = T
R = R 2 /{x-fr
2
from = T to = co '
R = R 3 = V/{(x + fT2 -fTT) +2},
There are three corresponding characteristic curves, which together make
R and the velocity v are all continuous
up the complete curve. Now
throughout the whole range; hence the three sections of the curve must form
a continuous line, without corners, though in general with sudden changes of
curvature. A break in the curve would imply a sudden jump of the charge
from one position to another, a corner a sudden change in its velocity, neither
of which is physically possible. The only difficulty that can conceivably occur

e,

is when

a

=

0; but this, as we have seen in Ch. IV, requires the velocity

to be greater than that of light, a case which will be discussed presently.

Fig. 18.

For x=0,

=c 2/2f.

v]

ILLUSTRATIVE

EXAMPLES

55

The first section of the curve, CD, is given by t = T + r/c.
It is a straight line parallel to the line t= T, AB, and at a height r/c
above it.
The section, DE, is an arc of the curve of the type shown in Fig. 14, § 29,
for the case of uniformly accelerated motion, but extending only from 7 = 0 to
7= T. It touches the straight line CD at D.
The last section, EF,is an are of the hyperbola, of the type shown in
Fig. 8, § 22, for the case of uniform motion with a velocity less than that of
light.
The remaining portions of these curves are indicated by dotted lines. It
is obvious that these portions must be ignored. In fact t is a uniquely
determined function of 7 and the coordinates of the fieldpoint (x, r); for the
charge emits but one wave at a time, and this has no double points. Each
wave corresponds to a determinate ordinate in the (t, r) diagram, which
ordinate can cut the characteristic curve only in a single point. Since the
points of the full line curve do give actual waves, they are the only ones that
do so.
It should be noted that on the present view the state of rest, corresponding
to the section CD, is to be regarded as the limiting case of an infinitely slow
motion, during which waves are being constantly emitted, but from positions
of the charge which are infinitely close together, so that the variations of R
become infinitely small.
37. The sequence of events at the selected fieldpoint is now seen to be
as follows:
(I) Until t = r/c, corresponding to r =0, the state is that corresponding
to the line CD; the graphic method of representing potentials given in § 19
shows that during this interval we have an electrostatic field given by

to

(II) At the time t = r/c, corresponding
the point D, the initial spherical
wave arrives, that is, the wave DGD' of Fig. 17. The form of the solution
changes to that belonging to region II of the last problem; but the root T4,
being less than the critical value r8 of T, which gives the minimum of the
curve II in Fig. 18, and is negative, is itself negative, so that the corresponding wave does not exist. Thus we have merely the one wave corresponding
to the root r, given by the intersection of the horizontal line t = constant

56

[CH.

EXAMPLES

ILLUSTRATIVE

Thus we have, by (62), § 31,

with the are DE of the characteristic curve.
8c
1
[K 3,R]

[v 3 ]

c [KR,]

f2 (T

-

(T3 -

4)

8T3

_
-f(

T3)

3 ) (72-

-

r)(72

-

T)

(T

-

4)

is
Tl and 72 are the two imaginary roots, 37 the real positive and 74 the
real negative root of (51), § 25.
(III) The last expressions hold until t reaches the value corresponding to
the point E on the characteristic curve, and the spherical wave (T), HKH'
of Fig. 17, reaches the fieldpoint. The equation of this wave is
where

R 2 = c (t - T),

where in B2 7 is to be put equal to T. Hence we have for E
t = T + V/{(e - fT 2 )2
}/c
At this moment the forms of the potentials change to those corresponding
to rectilinear motion with the uniform velocity v =fT, which the electron had
at the time T, when the wave was emitted.
The potentials are henceforth given by (47), § 23; thus,
1
{(x + fT 2 - fTt) + (1 - f 2 T2 /c2 ) 2} ......... (76).
c

V{(x + fT 2 - ft)

fT
2

+ (1

f2T2/c22

}

It has been remarked already that the continuity of motion of the charge,
and the uniqueness of the disturbance due to it, require the potentials to be
continuous in value, though discontinuous in form, at each time of transition;
but this continuity does not in general extend to the electric and magnetic
forces. These are given by (VII) and (VIII), § 3, and involve differentiations
with respect to coordinates and time. Since 7 is an implicit function of both,
in virtue of (28), § 13, these differentiations involve differentiation with
respect to 7, and therefore give rise to terms involving the acceleration,
which is discontinuous.
38. (b) Final velocity greater than that of light: T > c/f.
So long as the velocity of the charge is still less than that of light,
that is for t < c/f, the circumstances are much the same as in case (a). The
field outside the initial spherical wave is electrostatic and given by (74), § 37;
the field inside is due to the accelerated motion and given by (75), § 37.
There is yet no inner boundary, since the state of uniform motion has not
been reached.
What occurs when the velocity of the charge exceeds that of light will be
best understood by reference to Fig. 19, which represents the state of affairs
at the time 3c/f, for the case where
2c/f, that is, where the final velocity
is twice that of light.

T=

V]

vILLUSTRATIVE

EXAMPLES

57

II
0i

ILLUSTRATIVE EXAMPLES

58

[CII.

§ 27,

of which the equation is

B'CBP is the semicubical parabola (57),
(2fx - c)3= 27c2fr 2

.

Thus OC= c2/2f.

T is the point reached by the charge

at time T, when the motion becomes uniform. Thus OT= fT 2.
As the charge passes through C, where its velocity is equal to that of
light, the envelope sheets (t12) and (t23) first appear (vide § 28 and Fig. 11).
At time T they occupy the positions bTb' and bab' respectively. The semivertical angle at T is equal to a, where sin a = clv, and v =fT is the velocity
in the uniform motion. The two sheets touch each other along the circle bb',
and cut the paraboloid (57) perpendicularly. The line TP touches the semicubical parabola at P, which is the position of the cusp at some later time
t= U. This time is given by (61), § 27, so that
U =f T3 /c2 ...

. .. . ... .. . ... .. .. .

(77).
The charge then is at G, the last vestiges of the envelope sheet BKK'J'J
have disappeared, and the envelope is now represented by the semicone PGP'
belonging to uniform motion. The spherical wave (T) has expanded into
the sphere PM, and extends far beyond the point C, for its radius TP is
greater than TN, which is itself equal to twice CT. Thus the region, where
the disturbance is due to acceleration, is now limited to the region (II) of the
last problem, which lies between the cone PGP' and sphere PM and the
initial wave. The disturbance here is of the same one-wave type (75), and
has acquired as it were a uniform character. Hence the quantity U may be
taken as a measure of the duration of the variable state during which the
uniform motion is established.
The field during the variable state is of no great interest; its character
will be sufficiently understood by reference to the last problem, when it is
stated that it is of type (III) in the ring-shaped region BKJ, between the
envelope sheets (t12) and (t23) and the spherical wave (T), of the type (IV) in
the ring-shaped region BJI, between the sheet (t23), the spherical wave (T)
and the cusplocus (57), and of the type (II) everywhere else between the
initial wave GDG', the spherical wave (T) and the semicone KEK',
39. Since the charge acquires a velocity greater than that of light,
whilst the initial wave only expands with the velocity of light, the charge
will sooner or later catch it up and penetrate into the region outside. If
T < 2c/f, this happens when t = jfT 2/(fT - c), after the state of uniform
motion has been reached; but if T> 2c/f, it happens when t = 2c/f during
the state of accelerated motion. If T< (1/3 +1)c/2f, it happens at a time
t > U, that is, after the variable state has ceased, otherwise before. Until
the charge pierces the initial wave the disturbance is entirely confined to the
space inside it; afterwards the disturbed region is bounded only partially by
the initial wave, the rest of the boundary being formed by the portion of the
conical envelope sheet which projects beyond the initial wave.

v]

59

ILLUSTRATIVE EXAMPLES

Fig. 20 gives the positions at time t = 49c/8f of the two spherical waves
and the conical envelope sheet for the case where T = 7c/4f, after the variable
regime is over and when the charge E has already pierced the initial
wave GLD.
/
P

I

---,-

-

C-T'

H'

Fig. 20.

IV

N

III

II
H

7,
aE

x

K
HD
For T=7c/4f, and t=49c/8f.

As before HKH' is the wave (T), and KLEL' is the semiconical envelope
sheet belonging to the steady motion.
The intersection of the wave (T) with the semicone is a circle through K,
which travels along the cone KTK'; the intersection of the initial wave GLD
with the semicone is a circle through L, which travels along a surface of
revolution generated by the parabolic arc ML.
40. It would lead us too far to examine in detail all the cases which may
occur; in order to understand the changes that take place in the field it will
be sufficient if we study the progress of events in one particular case, for
instance, for a fieldpoint lying within, in front of, the surface of revolution
LML'. For this purpose we use the characteristic curve as a guide; it is
represented in Fig. 21.
i.

F\

Fig. 21.

(30
ILUSTRATVE

60

Just asin the case of Fig.

setins.

[H

.18

[OH.

the characteristic curve consists of three

, is the straight line t = 'r±/c which is

The first sect,

initial

EXAPLES

electrostatic field.

The

Fig. 14, 29,
Te-,
type of

, L', is

second sectio

thirdt

due to

,R

E

curve

", of the type of

accelerated m.otion.
F
of the hyperbolic curve,
k pi!'is an
the

arc

secton

Fig. 9,

an arc of the

2, due to the uniform

tion.

EPF,of the

Its minimum, P, as before

corresponds to the time .when the conical envelope sheet KEKpasesthe
fieldpoint.

41.

A consideration

of Figs.:20

and 21

leads to the following

con

clusions:
So long as the conical env elope sheet KEK' has not yet reached the

(I)

the
The field is electrostatic, and the potentials are given by (74),

fildpoint, the .horizontal line t cuts the characteristic curve below ,:on
section CD.

point is in region (I).
(11) As soon as the envelope sheet has passed across the fieldpoint, it is
disturbed simultaneously by two waves emitted during theuifor motion.
The .horizontal line t cuts the characteristic curve three ties, namely the
straight line CD once, and the hyperbola EPFtwice.
potentials consist
of three termss; one due to the electrstatic field and given by(74) the others
due to:the uniform motion and given by (49), § 24. Hence we et
37.

The

The

r

r
=C

N

2

(vt

A.tvt

-

-

,

-UT

(111) When the initial wave,
static

-

field: disappears and is replaced

motion.

fT) 2 . -

2)2 -

(V2/02

).~2
LD, reaches the fieldpoint, the electro.
(V2/C2

:by the

disturbance due to the accelerated.

The horizontal line t cuts the characteristic curve three times,

arc

namely the
.D:;, due to the accelerated motion, once, and the hyperbola,
EPF, due to the uniform, motion,. twice.
again consist of
terms, and the only difference from the last case is that the :electro-

The'potentials
static term is replaced by"a term due to the wave (T3),which belongs to the
arc
by
three

ascending

R.DE E", and is got
8c

-

means of (62),

§

31.

efind

v]

61

EXAMPLES

ILLUSTRATIVE

the uniform motion disappear, and only a single wave, due to the uniform
motion, reaches the fieldpoint. The horizontal line t cuts the characteristic
curve only once, on the ascending hyperbolic arc PF. The potentials have
only one term each, due to a single wave, and these are obviously just one half
of the values given in (49), § 24. Hence we get

-

V4{(vt
(vt

--

SV{(vt

x --

x

--

)

2

12fT2

--

T22

(v/2

-

(22_

_

2

-

1)

(V2/C2-1

72

}..............(80).

2

42. General considerations. These results complete all that we shall
give for the present problem; the other cases lead to analogous expressions, and
the differences which occur only concern matters of detail. It only remains to
draw attention to some important conclusions which follow from them. These
problems, though in themselves very special ones, are typical of large classes
of problems, and therefore difficulties which arise in them may be expected to
occur generally. The reason why they have been selected for treatment is
because they appear to be the only cases which can be treated by comparatively simple methods; others require the solution of algebraic equations of
higher degree than the fourth, or lead to transcendental equations.
Attention must be called to a discrepancy between the results of problems
1 and 3 for the case of a velocity greater than that of light.
The usual treatment of the problem of uniform motion due to Heaviside,
Descoudres, Sommerfeld and others is that adopted in problem 1. When
the velocity exceeds that of light it leads to the expressions (49), § 24.
These expressions hold inside the conical envelope sheet, that is, inside the
shadow of the motion, to use Sommerfeld's term. Outside the potentials are
zero; at its surface they are infinite (for a point charge).
The result of problem 3 is however quite different. Inside the shadow
of motion, at some distance behind the charge, in region (IV), we have the
expressions (80), § 41.
The coordinates must be changed for purposes of comparison with
problem 1, so that the new origin is the point T, where the uniform motion
just begins, and the time is reckoned from the instant at which it begins.
Expressions (80) are just one half of (49), because in the present case only
one wave is effective, whilst in problem 1 there are two.
Inside the shadow of motion, just behind the charge, in region (III), we
have the expressions (79), § 41.
These differ from (49) by the addition of the first term, due to the
accelerated motion; thus here there are three effective waves in place
of two.

62

ILLUSTRATIVE

EXAMPLES

[CH.

Outside the shadow of motion, in the region (I), we have the expressions

(74), § 37.
Thus the field is electrostatic, not zero as in problem 1.
At the surface separating regions (III) and (IV), which is the limiting
spherical wave emitted at the instant when the uniform motion begins, we
pass from (79) to (80); the potentials, though finite, are discontinuous.
At the surface separating regions (I) and (II), which is the conical sheet of
the wave-envelope, we pass from (74) to (78); here the potentials are infinite.
At the surface separating regions (II) and (III) the potentials are finite
and continuous.
The first discontinuity is absent from problem 1; the second is common to
both problems.
At first sight the discrepancy between the two sets of results would
appear to be due to errors of calculation, but a very little examination shows
that it is fundamental, and that the two problems are essentially different in
kind.
In problem 1 we have assumed that the motion has been going on in
precisely the same way for ever, and that there never has been any electrostatic field at all. It is therefore not surprising that outside the shadow of
the motion there should be no field.
When however we ask ourselves what is the effect of the initial conditions, we implicitly assume that the motion has started from rest. (In
cases where the motion has started from some previously existing motion of
a different kind similar considerations obviously apply.) Hence we are
dealing with a new problem altogether, in fact problem 3. We cannot
without further enquiry apply the results obtained for problem 1 to this new
problem, but must begin the investigation ab initio.
The result of the investigation proves that the initial circumstances of
the motion do have a determining influence for all future time, in the sense
that there is a fundamental difference between aperiodic motions which have
been going on for ever and which necessarily take place in unclosed orbits
extending both ways to infinity, and motions which have started from rest.
The latter are necessarily discontinuous, in the sense that the form of the
analytical expressions for the coordinates of the moving charge as functions
of the time is not the same for all time. The former may be continuous.
[I have shown elsewhere * that the discontinuity of form has no effect on
the expressions for the transverse and longitudinal masses of an electron
calculated from the field due to its motion, and that its effect is confined to
the terms of higher order in the reaction due to the radiation.]
* Schott, Ann. der Phys. 25, p. 63, 1908.

v]

63

ILLUSTRATIVE EXAMPLES

43. Problem 4. An electron moves in a uniform electric field
in a direction parallel to the lines of force. Required the resulting
electromagnetic field.
This problem is of interest for two reasons: apart from the case of
uniform rectilinear motion, it is the only case which has been solved in
finite terms; moreover the motion is almost exactly realisable.
We make two assumptions:
(1) The mass of the electron varies according to the Lorentz mass
formula, that is, m, = cm/V(c2 - v2), where m is the mass for zero velocity,
and m, that for velocity v. This means that the momentum is equal to
cm. v/V(c 2 -_ v), and changes in accordance with Newton's Second Law.
(2) In calculating the mechanical force acting on the electron we neglect
the reaction due to radiation, that is, we assume that the motion is quasistationary. Our solution nevertheless holds so long as the velocity differs
appreciably from the velocity of light, in fact far beyond the range of velocity
actually realised with /-particles.
We shall treat the electron as a point charge, and as before calculate the
potentials per unit charge.
44. We choose the line of motion as x-axis of cylindrical coordinates as
before. With the assumptions made the motion of the charge is given by
the equations*
=

V(k 2 + C)T2)

v
cT/(k2+c2T2)
where

from T = - o to

T=+ 00.(81

k = cm/eX

X is the electric force, and e the charge, both measured in electrostatic
units.

When T = 0 we have
= k, v= 0. The charge starts from positive
infinity with the velocity of light, moves against the field, is brought to
rest by it, and moves away again to positive infinity, ultimately acquiring
the velocity of light. Actually, on account of radiation, it would be brought
to rest somewhat sooner, and in the reversed motion would not regain its
initial velocity. Of course only a portion of the motion can be realised,
so that any real motion is necessarily discontinuous; but the method of the
last problem enables us to extend our results to any case of this kind,
e.g. the case of problem 5 below.
* See the problem, Ch. XI, § 153 below.

EXAMPLES
~ILLUSTRATIVE

64

64

[H

ACA.

From (81) we deduce the following equations of our problem:
/(k 2 ± C2T2)
R = / {(V
t= +
K=

)2

-

+

W2}

B/c

at

{1/(k2±07>IT.......8)
2

B V/(k

+

C2Tr2)

C7X

= ct -

KR

,/(k±2 + C2T )
2

Rationalizing the second equation and using the first we get a quadratic
for T, which gives
CT =

V(k2 +

C2T2)

-x

R=
s=

where

=

{ct (k

2

+ ,Z2+

x2 -

c2t2)

+ .}/2

(x2

c2 t2)

-

2
(kc + W2 + x2 - Ct 2) ± Cts}/2 ( 2 - C t2 )
olt (k2 + w 2 - x2 + C2t2) ± xs}/2 (x 2 - C212)
........................................
2 (a2-c.2t2)t}
2
2t)24k
/{(k + & +

X

2
Vt (k2 + W-

2

V{(c +

+

c2 t2)2
4W2 (x2
22
22
2 - C t - k ) + 4k 2&}
x2 +

-

(83).

02t2)}

We must choose the sign of s, the square root in which we take as
essentially positive, so that B and /(k 2 + 02T2) are both positive, the former
because we must have less than t as before, the latter because the electron
moves on the positive half of the xo-axis.

T

From the last two equations we get at once
to62
2

to

-

2 2

t

(k2 +

+ c2t 2)2

Z2 - x2

00

+
(k2 + w,.2 2 -

2 22

t)

-

2
Ct 2s2

-

C2t2)

{(k2 +

02

=(x2e -

0212)

{(k 2 +

2
w2 + xo - c2t2)2 +

-(X2

-

x

2

2
+ Ct2)2 + 4w 2x2},

4k2c2t2}.

Wx

2
2
2
+ &t2),
Thus we see that tos is numerically greater (less) than ot (k + 2
2 - C t2) is numerically greater (less) than cts, according as to
+x
and to (k2 +
is numerically greater (less) than ct.

W2

In the first case the lower sign alone makes B positive, and both signs
make V/(1k 2 + C2T 2 ) positive (negative), according as to is positive (negative).

2
In the second case the lower sign alone makes V\(k + C272) positive,
positive (negative), according as t is positive
and both signs make
(negative).

B

Thus we must use the lower sign alone in every case.
Moreover the disturbance is confined to the space on the positive side
of the plane to + ct = 0.

~ILLUSTRATIVE
EXAMPLES

v]

V]

6

6.5

45. [Characteristic curve. These conclusions can be verified from
the characteristic curve, which is of the
D'
type shown in Fig. 22. It has two/t
asymptotes, namely the lines t =

/c,

-

and t =27-- Ce/, and is everywhereB
ascending. The broken line as before
is the curve for B negative.]
46. Potentials. The potentials are
given by (26) and (27), § 13, where in
accordance with the above we must
substitute from (83) using the negative
sign.

c
-

sign.

We get by (82),
vU
c

A

Fig.

2.For

x

=2k,

v

=2k.

44,

CT
ct (k±2 2
+
-c 2 t2)-xs
2
/(k2 + C2 )x(I2+-2 + x2c2t2)cts

KR=ct-

c

=2
x(kc +&+x2_C2t2)cts

KR is always positive, as it should be.

We get

2
2
+ X - C2 t )___
22
(x _ C t ) S

f2
+~
2

et (k 2 ±
O
0

a=a=

Z

+

2

-

C2t2)-

(84).

-.

2

The values of 0 and ax are apparently infinite when x 2 = c2t2, but this
is really only the case for xe + ct = 0.
For instance, 0 may be written in the form
{(kc2 + wT2 + x2 c2t 2)2 + 4k2c2t2}/is {x (k2 + w2 + x2 _ c2t 2) + cts},
and this reduces to
2
{(k2 + W2)2 + 4k2c2t 2}/(x + Ct) (/c + Zy2)2
when x2 = c2t2 ; in the same way ax reduces to
{4k2c 2 t2 (k2 ± #.2)2}/(x + ct) (k 2 + w.2)2.
Thus both 0 and ax become infinite at the plane xe + ct = 0, which, as we
have seen, bounds the field on the negative side, but they remain finite at
the plane xe - ct = 0.
-

-

Since the characteristic curve CED, Fig. 22 (§ 45), has the line xe + ct = 0
for an asymptote at Tr = - 00, it is. evident that the boundary is a plane of
concentration of all the disturbances emitted at times infinitely long past,
and thus does not exist in any realisable problem. The infinity there is due
to the supposepd cncerntratin of a finite charga a point, andthe boundary

66

[CH.

ILLUSTRATIVE EXAMPLES

really represents a sheet of relatively intense disturbance, the thickness of
which is of the order of the diameter of the electron, somewhat like the
conical envelope sheet due to motion with velocity greater than that of
light (§ 17).
At infinity the potentials become infinitely small of the order 1/x, for s
obviously becomes infinitely great of order r 2.

w

Near the charge, and also near its image in the plane x = 0, they again
become infinitely great. In fact, write w = ±+ + p cos 0, = p sin 0, where
as before 4 = /(k 2 + c2t2) is the coordinate of the charge, and p is the distance
of the fieldpoint from it. Neglecting higher powers of p we find by (83), § 44,
s = 2 p /( 2 cos2 0 + k2 sin2 8) = 2 p4 /(1 - 32 sin2 ),
where = v/c, so that k = : /(1 - /2) by (81). Hence
1
p /(1 - /2 sin2) '

8

Thus near the charge (4, 0) the potentials are the same as those due to an
equal charge moving with
velocity equal to the instantaneous velocity
actually possessed by the charge, as was to be expected. In addition, near its
image (- :, 0) in the plane x = 0, they are the same as those due to an equal
charge of opposite sign moving with the same uniform velocity. The image,
however, is outside the field, beyond the boundary x + ct = 0.
The potentials become infinite of order l/p, and the equipotential surfaces
are similar Heaviside ellipsoids.

uniform

47. Electric and magnetic forces.
in (VII) and (VIII) (§ 3), we find
dX-

_4k2 (x2-2

2)
s3

,

d, =

Substituting from (83) and (84)
8k 2x

,

h. =

8k 2 ct

...... (85).

The remaining force components vanish.
At the boundary x + ct = 0 we get
U
8k2xr
4k
x
(k2 ± 2 2 '
d=h
(k 2 + .2)3
Thus the forces, unlike the potentials, remain finite at the boundary, but are
discontinuous there.
At infinity the electric and magnetic forces become infinitely small, of the
orders 1/r4 and 1/r5 respectively, although the potentials only vanish to the order
1/x. Hence the difficulty signalised in § 15 does not exist in this problem.
48. [Geometrical representation of the field.
expressed very simply by means of ring coordinates.
Write

x=

4sinh
cosh .- cosv'

a = cosh

sinx

The forces can be

- cosv'

v]
ILLUSTRATIVE EXAMPLES

v]

67

where as before
is the distance of the charge from the origin at time t, so
that by (81), § 44,
2 ±c
k= +(k2t2),

f = v /c
t/,
(The circles = constant form a pencil of circles through the charge and its
imiage in the equatorial plane x = 0, and the circles r = constant form the
family of orthogonal circles with the charge and its image for
points.
The part of the axis between these points is given by x-7,
the remainder
x = 0: the equatorial plane is given by 4Fr = 0, and the charge and its
by
= ± c respectively.

limiting

by
image

Using (83) and (85) we easily deduce the following results:
(1)

The electric and
(1

'2)

magnetic furces are given by
(cosh

-- cos

h=/3sin-.d........(86)
Ns.ear the charge, an near its image, these reduce to the well-known
Heaviide expressions fucr positive, and negative, unit charges respectively,
both moving with the instantaneous velocity v in the positive direction.
(2)

The lines of electric for ce are the circles

x = constant.

The lines of magnetic force are the parallels of latitude.
The electric force acts from the charge to the image.
The magnetic force acts right-handedly with respect to the simultaneous
velocity of the charge.

con-

(3) Let the electric induction across the segment of the sphere4
stant, from the positive x-axis up to the parallel of latitude x = constant, be 1
and let the nagnetic induction across the part of the meridian plane, which
lies between the circles (x) and (i), and the axes x = 0, * = 0, be J. Then
we get

I=2w

f,___cos

x

J

J}

2
AV(
13 sin 2 X)
The total electric induction is given by x = wr, and is equal to 47r, as it
ought to be for unit charge.
Trhe total magnetic induction is infinite, owing to the supposed, but nut
realisable, concentration of a finite charge at a point ; practically the whole is
contributed by the region immediately surrounding the charge, where
±
The image lies beyond the boundary, +ct = 0, that is, it is virtual and
contributes nothing.

x

ILJLUSTRATIVE EXAMPLES

[CH.
[H

The flow of energy vanishes at the axis of x ; it is away from it between the
charge and its image, and towards it beyond them, when the applied electric
force aids the motion, but is reversed when it opposes the motion.
Fig. 23 is a meridian section for the case where the velocity is four-fifths
of that of light and positive, that is in the direction of the applied electric
force.

E

is the position of the charge at time t, and F is its image in x = 0.
Thus OE= OF= ,.
A is the turning-point, so that QA = le.
B) is the trace of the boundary, so that OB = ct.
is the line of electric force through P, so that x -z EPF.
GPJ is the line of flow of energy through P.
K and H are the centres of these circles, so that
CIUDF

OK =

cot

x,

Y.

cosec

KE =

OH

x,

GH

-4coth

=

cosech

f.

D

8/i

Q4

4F5/10

17 6, -2.2,

-- 132, -

8,-'44,

Fig. 23.

44,

"88,

et= -,

2.2,

k

,

1076,
1.32

P=*.

It is easily proved that I = 2wr(1 -- OKrLAK).
With the aid of this relation the lines of
the values

1

=4w

tuies

1/10, 2/10, :3/10),

Thustheeletri
inucton hrogh

force have been constructed for

.and

achofth

O4

ies1

,3

0.44ones enerated

,5
by th

v]

ILLUSTRATIVE

69

EXAMPLES

chosen so that one of the circles bounding these areas, and given by
just touches the boundary BD.

= - 2'20,

We have generally Jrr log, (FP/PE),so that OG = tanh (f/2). Hence
=
as the circle (f) approaches E, the magnetic induction through the enclosed
area becomes logarithmically infinite. This is due to the supposed, but
physically unrealisable, concentration of unit charge at the point E. Thus we
must exclude a very small space around E from consideration; this excluded
space represents the volume occupied by the electron.
In order to secure agreement with our fundamental assumptions we must
suppose the surface of the electron to be an oblate spheroid of revolution about
Ox with its centre at E, its eccentricity equal to 8, and its equatorial radius
equal to a constant, say a. At this spheroid we have
V( 1 - 32 sin2 )/a
to the lowest order in the small quantity a/. The x-momentum per unit
volume is sx/c 2; hence we find by means of (86) that the x-momentum
contained in the part of the field, which lies between the spheroid and the

cosh

*

=

boundary x + ct = 0, is, to the lowest order, equal to 2$/3ca

1/(1

- 32), which

is the value given by the Lorentz mass formula for unit surface charge, as it
ought to be.]
49. The electric and magnetic forces are discontinuous at the boundary
x + ct= 0, for they are finite inside and vanish outside. No difficulty
arises on this account, for in the first place the boundary is really a layer
of transition, of thickness comparable with the radius of the electron, in
which the forces vary rapidly, but continuously, from their zero values outside
to the values given by (86) inside. Secondly, this boundary is never realised;
it requires the motion of the electron to have continued for an infinitely long
time according to the same law, whilst the necessary uniform external electric
field can in reality only be of finite duration.
Thus it becomes necessary to consider cases of discontinuous motion; for
instance, the case where the uniform field is confined to the space between
two planes perpendicular to the x-axis, and the electron is projected along the
axis with a given initial velocity.
50. [Problem 5. An electron, moving with uniform velocity in
a straight line, is subjected to a parallel uniform electric force for
a certain interval. Required to find the electromagnetic field due
to the motion.
After what has been said above we shall content ourselves with a very
brief statement as to the generation of the pulse, and shall then give an
account of its ultimate nature; for it is this alone which is of importance for
the theory of X- and y-rays.

70

ILLUSTRATIVE

[cg.

EXAMPLES

In addition to the initial velocity and the strength of the applied electric
field, we must be given either the final velocity generated, or the time during
which the field is applied, or the space through which it extends, it is quite
immaterial which; the equations (81), § 43, in any case determine the
remaining quantities. To fix the ideas we shall suppose that the electron
moves with uniform velocity v1, until it reaches the point G (g, 0) at time 4.
The uniform electric field X then commences to act, and accelerates the
electron according to (81), until the point H (h, 0) is reached at time t2. Then
the field ceases to act, and the electron continues to move with the acquired
velocity v2. The origins of space and time are supposed to be taken as in
problem 4, and must be determined by means of (81).
The disturbance emitted during the initial and final uniform motions is
determined by (47), § 23, but we must bear in mind that the origin is the
position occupied by the charge at time t in the particular uniform motion
under consideration.
For the initial motion the origin is a point El, of which the x-coordinate
is given* by 1 = g + v1 (t - t1 ). Hence in (47) we must replace v by v1 , and
vt - x by vt - x + g - vt.
This disturbance is confined to a region (I), which is unlimited externally,
but is bounded internally by a sphere (t), the position at time t of the wave
emitted at time t1. Its centre is G, and its radius is given by R 1 = c (t - t).
It cuts Ox in two points B and B', whose x-coordinates are g ± R1 .
For the final motion the origin is E2, for which
in (47) we replace v by v2, and vt - x by vet - x + h

-

~2 = h

+ v2 (t - t), so that

v2t2 .

This disturbance is confined to a region (II), which is bounded externally
by a sphere (t2), the position at time t of the wave emitted at time t2. Its
centre is H, and its radius is given by R 2 =c (t- t2). It cuts Ox in two points
C and C', whose x-coordinates are h ± R 2.
The disturbance emitted during the variable motion is determined by
(84)-(86) of problem 4. We use the same notation; in particular we denote
the position at time t of the electron in the variable motion by E, its
x-coordinate by , and its velocity by v, where
=

2
Q/(k1c),
+ c2t2

and v

=

c2t//
V(kc2 + c2t2).

This disturbance is confined to a region (III)-the pulse-bounded by
the spheres (t1) and (t2).
The electron coincides with El for t < t1 , with E for t1 < t < t2, and with E,
for t > t2 ; we shall however find it necessary to use these points for values of
t outside these intervals, in which case they may be regarded as virtual
electrons.

~ILLUSTRATIVE
EXAMPLES

v]

v]

7

71

61. We shall for convenience collect together below all the data and
equations which we shall require later ; they all follow readily from (81).
We write
v/c = $,
v 1 /c = i3,
v2/C = /2
Then

V/(1

32)=

-

/3( ct/c,
=

2/

/$2 =

t/(1

2)= h

t/(1-

ctl/g,
1V1-112/1t= ct
2

(

Rh=c(t-t1)=/3

-

1>

32)

$32
/k

.(87).
2)$

kI

R2 =c(t-t 2 )=/3 -

-$13g,

h

2

1=g~1(t-1) g+
11,
2 =h+v2 (t-t 2)=h+/3 Rj
2
The time T, for which the field is applied, is given by

T

(l

=

-

1

/(22)

v

-182)

N

The distance between the points G and H, the space over which the
applied field extends, is given by
c

L,

-

V(1

=

1

--

+ v2

-

9'

2)

1c~

V -

1

Either T, or L, may be given in addition to v1 in place of v2.
The thickness of the pulse varies; at the positive pole it is BC (Fig. 24
below), which is given by
BC =g+R1-h--R2=

? =(1

-A1))

-

1

h

-$2

/

1-

l1

12_

!32

2

y
k

At the negative pole it is B'C', which is given by
B'

V'

=

h -

R2

-

g

+

RI

=

(1

+

$2)

h

-

(1

+

j3 g

32_

+13

Which of these thicknesses is the greater depends on the signs of /31
and /32, but is of no great moment. It is important to note that both are
independent of t, and of order ic,in all practical cases at all events. Now we
have, by (81),

k

=

clr/eX ; with e/cm

=

-P77.

10' we

find

k

k

=

1700/X.

Thus

with fields such as can be reached in the open air, may amount to several
ems. ; but in the case of X-rays, diffraction experiments seem to show that
is at most of the order 10- cm. Hence in experiments on X-rays the thickness of the pulse is always exceedingly small compared with distances, such as
R1 and B2, at which observations are made. In these cases it is the ultimate
nature of the pulse that is important.
Fig. 24 illustrates the construction of the lines of force, the notation being
that used above.
PQBE. is a line of electric force, consisting of three sections :
(1) The part QP of the radius vector E 2QP; (2) the
QR of the
circle (x), ERiQF, through E and its image
in the equatorial plane x = 0,
whose centre is at K (0, 4 cot
and (3) the radius vector E211.

X);

F

k

arc

72

72

EXAMPLES[c.
~ILLUSTRATIVE

[CH.

The flow of energy at Q is along QS1 in region (I), and along KQS 3 in
region (III); at R it is along RS2 in region (II), and along KRS3 in region
(III), where QS and RS 2 are respectively perpendicular to E1Q and E 2R.
1

F''00

Fig.F24.

G

5
=A,

For e=k,

H-SE~0

P

88

It should be noticed that both the electric force and flow of energy are
discontinuous at the spheres (t1) and (t2), and the same is true of the magnetic
force.
It is proved below that KG is perpendicular to E Q, and KH to
these facts afford a simple construction for the electric line of force, when the
part of it in any one of the three regions is given.

ER;

52. We shall find it convenient to transform our coordinates by referring
everything to the spherical wave which passes throughP
the fieldpoint P at the time t. This wave was emitted at
some time t', when the electron was at a point E', and
R
moving with velocity v' along E'T. Write R for the
E'
v'T
vector E'P, 0 for the angle it makes with E'T or v'.
Let v'/c = /3', and as usual write K = 1 - /' cos 0.
___

The point laws are

Fig. 25.

1

v

~K-R)

a =cKIW.

They give by means of (VII) and (VIII), § 3, as in § 13, (32) and (33),
y
v___ +R
d -c2K2R +

1 -vy'/c

1'

1

,,

For the uniform motions we must put
ndr

v'-()0

2
+

(v'R)

8~'2f'h[R

$/3,

h=[I

d]

or X3, as the case may be,
2

~ILLUSTRATIVE
EXAMPLES

v]

V]

7

73

For the variable motion we have v' = c2 t'/V(k2 + c2t'2), so that
k 2/(k2

+ C2t'2) - (1
')k
Of course in any case v' and v' are along Ox. Hence we find for the
'U'/C2 =

1

Uniform motions:

dr

112(1

de =h

cos

/3'(1 R12

Variable motion:

dr =

(1

1

sin

8'2)

(88).

/3' CoO)3

-1co

(1

de

18/
0)2

-

-

2-

+

d =(-h3')2sin
6

k

cos 1-

0)3j
0

In the last equation we have made use of (87), which give for the
x-coordinate of E' the value e' = l/(1 - /32). They also give 11=
-3
'
a relation which allows us to replace the factor 1+-

in this equation by

By help of (88) we get the following results:
(1)

At the wave (t,), where /' = $1, we have
dr2 = dri,
d02: d61 =

k:

/

=

tan a:tana,
tan a :tana,

h83: h61 = /3 : /3~ =
where the suffixes refer to the regions (I) and (III), and
tan a

=

/3 /k

=

/3//(1 -

32),

tan a,

=,i,1V(1

-

012)"

Thus the normal electric force is continuous, but the tangential electric force

and the magnetic force are discontinuous. The lines of electric induction are
refracted according to the tangent law, as they would be from a dielectric of
specific inductive capacity tan a to one of capacity tan a2.
(2) Further, we have, for the normal Poynting vector,
sn:s2=

tan2a : tan 2 a .
1

This inequality implies no infraction of the Principle of the Conservation of
Energy. For since the wave (t1) is expanding with velocity c, region (I) gains
energy at the rate s1, per unit area of surface of the wave on account of the
flow of energy, but loses it at the rate cE2 per unit area on account of the
expansion, where E1 is the energy per unit volume of region (I) close to the
wave (t1). Using (88) we find for the total rate of gain of energy of region
(I) per unit area of the wave (t1) the value s1, - cE = - Cdrui/S7t-. In the
same way we find the equal, but opposite, value cdrii/8wr for the rate of gain
of energy of region (III) per unit area of the wave (t,). Similar results hold
for the wave (t2), so that the principle is satisfied for each surface element of
ach-Y
ave1TL
searaey
/.A

~ILLUSTRATIVE
EXAMPLES

74

74

(3)

If

[n
[CH.

Q be the point (x, r) the equation of the circe (x) through
2wfS

cotX=

Zr-

X2+

Also Fig. 24 gives
2

+

0Q-2

R
B2

+ g

2

it gives

2

+2g(x-g);

and from (87) we get
R 1 +/81g, and A/(1 - $X
2
2
2 = B + g + 2g/3 1 B .
1

/'=
whence

Hence we get
x2 +

and

W2
-

=

2

g (x - g -

$1R)=2g(x-

f cut x =g (x - 1)mar or OK = OG cot QE x

=

OG cot GE 1I.

Thus the triangles KOG and E1MG are similar, and
Similarly, N is a right angle.

! is a right angle.

Hence the construction given above for the line of force has been
verified.
(4) As time increases, , B, and B2 become very nearly equal and infinitely large compared with g, h, 4 and , while /3 approaches the limit unity.
Hence for all points of the pulse de ultimately is infinitely large compared
with dr, except just at the poles, and the electric force is ultimately tangential, though quite near to the poles it is radial, from the charge E, or its image
F. The limiting value of the electric force is the same as that of the magnetic

k,

force and equal to (1 -

f'2)3

sin

6icR (1 -

variations inside the pulse below.

8'

cos 6)3.

We shall study its

53. The behaviour of the pulse in the several possible cases will be
evident from Figs.
They are drawn for two typical cases, the first

26-29.

pair for the stoppage of a moderately fast cathode ray particle, the second for
the starting of a very fast /3-particle. The electric lines of force are drawn
by the construction of Fig. 25, so as to correspond to 10 equal increments of
induction, as in Fig. 23, §
the orthogonal lines of flow of energy are omitted,
as they are of no particular interest, and in any case can be easily supplied.

48;

Betarded motion.
In Fig. 26 we have
11005k,

e2=

k-,4=

4.12k,

'

=

060k,

"

=k

E2 and E" coincide with A exactly, while El is practically indistinguishable from it. The waves (t1) and (t2) are very nearly concentric, and the pulse
is very nearly of uniform thickness.
Fig. 27 represents the same pulse after an infinite time, practically for any
time
greaterthan sa 100kn/.
On the1^
scale-used A,"
and E" are-al
1E

"I

v]

ILLUSTRATIVE

75

EXAMPLES

indistinguishable from the origin, the pulse is infinitely thin, OE'= - OB/10,
and E practically coincides with B, and F with B'.

Fig. 26.

For X1= -'10,

Fig. 27.

For fpi= -

pX2=0, ct=4k.

Frontc
E--

-10,

P2=0,

t=c

.

Accelerated motion.
In Fig. 28 we have
5= 1"67k,
E and E' coincide with A.
1=k,

=4"12k,

'=k,

"=3"80k.

Fig. 29 represents the same pulse after an infinite time.
On the scale used A, E1, E2 and E' are all indistinguishable from the origin,
the pulse is infinitely thin, OE"= 80B/10, and E practically coincides with B,
and F with B'.

76

ILLUSTRATIVE

EXAMPLES

[CH.

Owing to the overlapping of the lines of force in the pulse, some care is
necessary in associating the corresponding lines in the regions (I) and (III).
For instance, the second line in Fig. 29, corresponding to I= 87r/10, is
E"PQR, and the curved portion PQ of this line overlaps the lines 3, 4, 5 and
6 in region (III).

Fig. 28.

For

3= 0, /02='80,

ct=4k.

xo

Fig. 29.

For

(31=0,

02='80,

t=00 .

More complicated cases, where neither the initial nor the final velocity
vanishes, can be illustrated by a combination of the two types of diagram.
For instance, if ja = - 10, X2 = "80, we must superpose Fig. 26 on Fig. 28, and
Fig. 27 on Fig. 29, so that region (I) in the first figure represents region (I),
and region (III) in the second represents region (III) finally.

ILLUSTRATIVE

V]

54. The ultimate nature of the pulse.
electron at the time t', so that OE'= '.
Let BPB' be the wave emitted at
time t' in its position at the time t,
and P a point on it, so that
/ PE'x = 0, and
E'P = R = c(t t') ct - ' ',
=
_.
by (87).

From 0 as centre describe

77

EXAMPLES

C'

a sphere CQC'of radius ct, draw OQ
parallel to E'P, and PR perpendicular to it.

Let E' be the position of the

M
R

B'

O

Fig.
Then QR =

CB

A E'
30.

' (cos

0-

3')= n

say. Thus n is the distance of P from the sphere CQC'measured outwards
along the radius OQ. By (87) we get
n=

')
ck (1-')..................(89).
(cos 0 -

As time increases and the pulse expands, the distances QR and PR remain
constant, while OQ and E'P increase indefinitely. Hence ultimately the
points P, Q and R may be regarded as coincident, and the radii OP and OQ as
parallel and equal to R, or ct, indifferently, for all purposes for which we may
neglect the quantity k/R. We know from experiments on the diffraction of
X-rays that the thickness of the pulse is of the order 10- 9 cm., and this is
also the order of the length k, except in very particular cases. Thus our
approximation is sufficient, even when R is as small as ten or a hundred times
the radius of an atom.
From (88) and (89) we see that the radial electric force may be neglected,
and the electric and magnetic forces regarded as equal and given by one or
other of the following equivalent expressions:
............ (90),
=)sin ek2sin0
_
kR (1- 'cos 0)3 R 2 + k2 sin2 0)2
where the charge of the generating electron is taken as e electrostatic units
instead of unity as before.
d= h=

The first expression shows how the force varies in different parts of the
constant, the second how it varies along the same
same wave, given by
radius of the pulse, given by 0 = constant.

3'=

The electric and magnetic forces are ultimately perpendicular to the radius
vector, as well as to each other. The electricforce is along the meridian, from
the positive to the negative pole, that is, always in the reverse direction to the
acceleration of the generating electron at the instant of emission of the wave.
The magnetic force is along the parallel, always right-handed with reference
to the accelerationof the generating electron at the instant of emission. Their
directions are reversed only when the acceleration of the generating electron

78

[CH.

ILLUSTRATIVE EXAMPLES

is reversed. Thus when a pulse is produced by a sudden change in the
motion of an electron, the electric and magnetic forces have the same direction
throughout the pulse.
The pulse is completely polarised at every point in a plane perpendicular
to the meridian, except at the poles, where however the forces vanish.
55. Small velocities. When the velocities ' throughout the pulse
are so small that their squares may be neglected, we find by (89) that the
thickness of the pulse is uniform and given by d = k (12- 13), where
and
32 correspond respectively to the outer and inner surfaces of the pulse. The
electric force which must be applied to the electron to produce this pulse is,

a1

,
by (81), § 43, given by X = c2nm/el = c2m (2 - ()/ed.
Taking e/cm = 177.107
we get X = 1700 (321,)/d. The force at the surface of an electron is of the

order 1016; taking this as an upper limit to the electric force X, we see that
the least thickness is given by d= 1"7.10-13 (1-2-1). Since 13 and 13 are
supposed to be small compared with unity, this is well below the maximum,
10- 9 cm., assigned by the diffraction of X-rays, even allowing for the probability that the force X is much less than the upper limit which we have
assigned. When 3, and 32 are not small, the thickness of the pulse is
variable, but its mean value is given by d = k

1-32

1

Even

when ,31and 3, are of opposite signs and as great as "97-the value given by
Kaufmann for his fastest -particles-d is only eight times k, and about
10- 12 cm., still leaving a sufficiently wide margin between it and the maximum
assigned by diffraction experiments.
As regards the distribution of force inside the pulse in the particular case
of small velocities, we find, by (90), d = h = e sin e/kR.

This law is the same as that in the wave emitted by a Hertzian vibrator,
apart from the periodic factor present in that case. This was to be expected,
seeing that the Hertzian vibrator is equivalent to an electron vibrating with
very small velocity. The force is symmetrical, as regards magnitude, with
respect to the equatorial plane, has a maximum there and vanishes at the
poles. Its maximum, e/lcR, is R/lc times as great as the electrostatic force at
the same distance R from an equal charge at rest. Owing to the smallness
of kc, it is enormously greater for all but extremely small distances, showing
that the pulse is a region of exceptionally intense force.
56. General case. We see from (90) that as the velocity 1' of a wave
increases, the wave loses its symmetry with respect to the equatorial plane.
The maximum of the force is given by cos 0 = {\/(1 + 243 '2) - 1}/43' or, what
is the same thing, by 3' = cos 6/(1 + 2 sin 2 6), and its value is easily found to

6

be e (5 + 4 sine 6) /27kR sine 0. Thus is less, or greater, than wr, according
as 3' is positive, or negative, and the maximum increases as 1' increases

v]

ILLUSTRATIVE

79

EXAMPLES

numerically. For example, when /'= 5, the maximum is given by 0= 36 ° ,
and its value is 1'8e/kR; when /3'= 97, it is given by 9= 6° 23', and its
value is 20e/kR. Hence as 3' approaches unity, the maximum approaches
very closely to either pole, and becomes very high and steep.
Since the extreme values of
are the values /1 and /32, the greatest of
all the maxima is found at one or other of the two surfaces of the pulse.

8'

57. Variation of the force across the pulse. The second expression
(90) shows that as we pass across the pulse along a radius vector 0 = constant,
the force increases to a maximum and then diminishes again. The value of the
maximum is e/kR sin 2 , and it occurs when n = 0, that is, at the point Q on
the circle CQC' of Fig. 30, § 54. The law of variation of the force is the same
for every radius vector, when the normal distance n is expressed in terms of
the length k sin 9 as unit. But we must remember that the pulse is limited
by the values n = n1 and n = n 2 , given by putting 3' = $3 and /3'= /2 respectively in (89), so that only a part of the whole range of the second expression
(90) is available for the pulse. When cos 0 = /3, the maximum of the force
just falls on the outer surface, and when cos 0 = /2 on the inner. When 0
lies between the two limits thus assigned, the maximum falls inside the pulse,
otherwise outside. In the latter case the extreme values of the force are
found at the two surfaces.
58. The energy of the pulse. The energy per unit area of a layer of
thickness dn is equal to (d 2 + h2) dn/8r. By (89) we find
dn = - k (1 - p' cos 6) d$'/(1 -

'2)2;

hence by (90) we get for the energy density per unit area of the pulse
e2 sin29

72

(1

32)2

-

kR
13

=32 7 kR2 sin 0
with

tan

=(l -

-

d/'

/3' cos 6)O
--

2 sin 2 +

1 sin 40}

...... (91).

/3') sin 6

cos d -

'

The indices denote that the values 0 and 02, given by /3'= /3 and /3'= /2,
are to be put for 0 in succession in the bracket, and the first result subtracted from the second.

When /3' is small, tan 4= tan
value of the bracket then is 8 (/32E

=

(/2-

(1 + /3' sec 0), or
/3)

31)

sin,
2
e sin2

4 = 0 +/3' sin 0. The

0, so that

0/4kR2.

Thus for small velocities the energy is distributed in the same way as it is
for a Hertzian vibrator, as was to be expected, symmetrically with respect to
the equatorial plane, with a maximum there and with zeros at the poles.

80

[CH.

EXAMPLES

ILLUSTRATIVE

3'

As
increases, the energy density behaves like the force. Its maximum
approaches nearer to either pole, and increases in height and steepness as
approaches unity.
The total energy of the pulse is found by multiplying E by 27rR 2 sin OdO,
and integrating with respect to 0 from 0 to r,most easily by performing the
0-integration first. In this way we find for the total energy,
2e2
/'
)2

/

V

3k

1

(1

-

...............

)2)1

.... (92).

This expression is easily got by direct integration of the well-known
expression for the radiation from an electron moving with acceleration which
is due to Lidnard and Abraham.
Since the mean thickness of the pulse is given by d= k(3'/V(1-'2),
(92) may also be written
3d

-

1(1-,C3)

/3 - "97, and /32 = + "97 as extreme
values, we find that V = -94.10-8 erg, which may be regarded as an upper
limit for the thickness assumed.
Taking e = 4"7 .10

10, d = 10-9 cm.,

The increase in the kinetic energy of the generating electron according
to the Lorentz formula is given by
K = c2
With m = 8"9.10-28 gram., and
-9

In this case V= 2"4. 10
When

/3=0 and

/2

(1131=

1

0,/32 = "97, we get K = 24. 10 - 6 erg.

erg, and thus is one thousand times smaller.

is very small, the ratio V/K approaches the limit

4e2/3c2md; for d = 10- 9 cm. its value is 1/2700.

Hence for a pulse as thick as 10-9 cm., our original assumption, that the
reaction due to radiation could be neglected in comparison with the force
generating the motion of the electron, is amply justified even for the fastest
/3-particles known. But it is clear that if the pulse were much thinner the
reaction would have to be taken into account.
59. The strength of the pulse. Both the force and the energy of the
pulse involve the constant k,or if we prefer, the mean thickness d, as well
as the terminal velocities /31 and /32, and to this extent depend on the field
which generated the pulse. For many purposes, however, what is required is
not so much the force, nor the energy, but the integral of the force across the
pulse, and this we shall find to be independent of k,or d. Remembering
that by (89) dn = - k (1 - /3' cos 0) d/3'/(1 - /3'2), we get by (90)

fdd

e sin

of9

d,3'

-R , (1A-2'.cos)

e
2

8'sin0

1-/'cos

8

2

.

V]

ILLUSTRATIVE

81

EXAMPLES

We may call I the strength of the pulse at the particular place.
As the pulse expands and passes across a fixed point the time-integral of
the force there is obviously equal to I/c.

d
Fig. 31. Variation of time-integral of field in the pulse.
The numbers attached to the curves are the values of the velocity jS'.

Fig. 31 gives curves for I as a function of 8 for a pulse which is bounded
by waves /3' and /3'=0. If the pulse be bounded by $1 and /32, we need only
and
take the algebraical difference of the two corresponding curves 3' =
'= 13.
The curves are given for /3' = -1, "2, ... , "9, 10; those for negative values are
got by putting 7r -6 in place of 9. The curve for 1'= 0 is the horizontal
axis, that for 3'= 1 is I = cot 10. The unit throughout is e/R.

,31

The maximum is given by cos
I= cot 8
s

= /3', and its value by
= 8'//(1

-

/3'2)
6

82

ILLUSTRATIVE

EXAMPLES

[en.

When 8' is very small the maximum lies on the equator; as 3' increases
it approaches either pole, and at the same time increases in height and
steepness. It always lies at M, the, parallel in which the particular wave
(/3') cuts the sphere CMC' of Fig. 30, § 54.
An important result follows from the fact that I does not involve the
constant k, that is, does not directly involve the strength of the generating
field X, but only the initial and final velocities of the electron.
Let D = fd. dn and H = fh . dn, so that D and H are the integrals across

the pulse of the electric and magnetic forces regarded as vectors.
The vector D is perpendicular to the vector R,, a unit vector along the
radius, and to v', the velocity of the electron, and lies in the meridian plane;
the vector H is perpendicular to the meridian plane; and the magnitude of
each vector is I. Hence we get from (93)
e

[v'

.

:
2

R,]

V
D=c

........................ (94).

:v'

,

Since these equations involve merely the terminal velocities and the angles
they make with the radius vector R, but do not involve the generating field
directly, they may be applied to any pulse whatever, provided only that the
distance through which the generating electron is moved in the process is
negligible in comparison with R.
For any pulse, however complex, may be supposed to be broken up into
a succession of elementary pulses, to each of which (94) may be applied in
succession; the form of the expressions shows that the final terms alone
appear in the result.
The expressions (94) could have been obtained by direct integration of the
general expressions for the forces given on p. 23, and derived from (VII)
and (VIII), but the present process is instructive, and will serve as a verification of the work.
60. Effect of the pulse. Let us suppose the pulse to pass across a
second electron in static, or kinetic, equilibrium under the action of a given
system of forces. We shall suppose the time of passage of the pulse to be so
small that the effect on the given system of forces of the displacement and
change of velocity of the electron due to the pulse may be neglected. This
approximation suffices whenever the time of passage is small compared with
the free periods of the electron.
On this assumption the increase in the momentum of the electron due to
the pulse is given by
A (my)

=

e

{d + [v. h]/c} dt

=e

{D + [v. H]/c}.

V]

83

EXAMPLES

ILLUSTRATIVE

It follows, by (94), that the increase in velocity, Av, is of the order e2/cmR,
which with the usual value for e/cm, and for R = 1 cm., is '008 cm./sec.
The maximum velocity of an electron, necessary to account for the radiation in a spectrum line, is of the order 106 cm./sec.; this follows readily from
Wien's measurement of the energy radiated by a single canal ray ion in the
blue line of hydrogen, namely 10- 6 erg/sec., on the assumption that only one
electron takes part in its production. If all the electrons of the hydrogen
atom shared equally, which isa most improbable assumption, the velocity
would still be as great as 2. 104 cm./sec.
The comparative smallness of the velocity produced by the pulse indicates
that the ionisation produced by X- and y-rays in stable substances like
hydrogen is to be accounted for on the aether-pulse theory by the cumulative
effect of a large number of successive pulses, and that something in the
nature of resonance is involved in the process. The results of Barkla
respecting the selective absorption of secondary rays support this view,
but it would lead us too far to enquire into the matter here. Such an
enquiry presupposes an investigation into the motion of systems of
electrons, which can only be given later.
Note. Since the above was written, Sommerfeld (Phys. Zeitsch. 10,
p. 969, 1909) has given a theory of aether pulses, which agrees with what
has been given above.]
61. Problem 6. A point-charge moves in a circle with uniform
velocity; to find the resulting field.
We shall content ourselves with investigating the general character of
the field, for the calculation of the potentials requires the solution of a
transcendental equation, which cannot be effected in finite terms.
Take the centre of the circle as origin and its axis as axis of x of a system
of cylindrical coordinates of (x, w, 0).
Let the coordinates of the charge be given by
=pcoswOr,

= p sinwr,

'=0,

from

Tr=-

to

0

T=+300.

The radius of the orbit is p, the angular velocity w, the period of
revolution 27r-/w, the linear velocity of the charge v- cop. As usual write
= o/c. We get at once
R = V{x 2 +

+
2

p2

- 2 wp

+

/{x

t

t=

2

cos ( r 2

1-....(95).

KR = R -4+

+

W + p -

sin (cor

-

)

)}

p cos (M -

)}/

~ILLUSTRATIVE
EXAMPLES

84

84

[H

[cii.

It is convenient to write

J = 2(Ot-0+'r),

x =

2

wT(-b+7),

f=/{w +( +p) j, g=1{x2+( p)2},
k = 1/(f 2 g 2 )/f, k' =g/f, A = (1-ksinx), y =wf/2c.
Z
2

Thus f and g are respectively the greatest and least distances of the
fieldpoint (x, w,

0)

from the orbit.

We get from (95)
R=A,

K=1 -rk2sinXcosX/A.

The multiple roots of (952) are given by K
= 0//(1

A2)

-

=

0, that is by

(A2-'2)

This equation gives on solution
A=

V{(l + k')

2

- l/ye} ±+
IV{(1-k')2 -1/y}.......(96).

The roots are real when

y > 1/(I

-

k').

They are equal when

y

k').

They are imaginary when

Since

ry

=

1/(1

-

<

l/(1

- k').

ry

= wf/2c, the limiting case is given by

f-g

= 2c/w = 2p//3

.................

(97).

This equation represents a hyperboloid of revolution of one sheet. Its
focal line is the orbit, its real semi-axis is p/,8, and its eccentricity is /3.
Thus the hyperboloid is real when, and only when, / > 1, that is, when the
velocity of the charge is greater than that of light.
Inside the hyperboloid, that is, on the same side of it as the orbit, the
roots of (96) are real, outside it they are imaginary.
62. The characteristic curve.
teristic equation (952) becomes

In the present notation the charac-

This equation is more convenient to deal with than the original one,
because the coordinates of the fieldpoint enter into it only through f and g,
that is, through

ry

and k.

The least value of A~ is k', and it occurs for
is 1, and it occurs for x = 0.

x = 2ir.

Its greatest value

These values repeat whenever x increases by 'pr, so that iJ is a periodic
function of x of period ir.
The chieftpf

the cFharat

risi crve are the follwing:

V]

ILLUSTRATIVE

85

EXAMPLES

(a) Velocity less than that of light: v < c.
The maxima of A, and of R, correspond to points such as D, and the
minima to points such as C. At both sets of
points the curve is parallel to AB.
Obviously, for every value of ~, and also
of t, there is but one value of x, and therefore
also of 7, that is, but one wave.
D
Since the curve only involves the two
x
-2
parameters k' = g/f, and 7 = wf/2c, the same
curve holds for all points on the anchor ring
g/f= 707, provided that the appropriate
value of w be selected.
Fig. 32.

The values of the potentials are

cos
7ykx cos 'y by (26), § 13,

-

2

a=
(n

-y=-785.

1
2
2 sin
sin

Ac

For k=k'= 707, and

-

yk2

n

8;

sin

COS

)

by (27), § 13.

x)

The magnitude of the vector v/c is
its direction is that of the motion
of the charge at the time of emission T, and makes with the initial line the
-.
angle or + rr-,that is, 2 x + (b) Velocity greater than that of light : v > c.
(1) As the velocity increases, while the fieldpoint remains fixed,
7 increases while f, g, k and k' remain constant. In the present case 7 may
increase until it equals the quantity 1/(1 - k'), that is, = 2p/(f- g). The
hyperboloid (97) is real, because /3 > 1, and it expands until it passes through
the given fieldpoint. So long however as the fieldpoint is still on its outside,
that is, on the opposite side to the orbit, the roots of (96) are imaginary,
which means that the characteristic curve is continually ascending, without
maxima or minima, and of the type shown in Fig. 32. The potentials are
given by (26) and (27) as before.
(2) When the velocity is so great that the fieldpoint lies inside the
hyperboloid (97), the roots of (96) are real, and
c
Q
the characteristic curve has maxima (Q) and
E
..
minima (P), infinite in number. For all values
of '4 between KP and LQ there are three
values of x, and therefore three simultaneous
waves. For all other values within the range
C
from C to C, there is but one.

/3

Thepotentials are given by(34)and(35),§ 14,
A-

2
c sin

cos

'

-r/2/

A/
Sv

c (A -

/yk y cos y)'
sin
2

Fig. 33.

For k=k'='707, and

93
y= 3 " .

86

[CH.

ILLUSTRATIVE EXAMPLES

where the sum is to be taken for three values of x, xl, X2 and X3, corresponding respectively to the arcs EP, PQ and QF in Fig. 33.
(:3) As the velocity increases still further, it may happen that the
maxima increase and the minima diminish so much that a maximum Q
is not only above the next minimum P, but above the next but one as well.
In this case horizontal lines can be drawn below Q, but above the second
minimum, and these cut the characteristic curve five times and correspond
to five simultaneous waves. The potentials then have five terms each, of the
same form as before. In the same way, for still greater velocities, and
suitable positions of the fieldpoint, we may have seven simultaneous waves,
and so on. We shall not trouble about these more complicated cases, but
merely note that the number of simultaneous waves is always odd.
63. [Note. Geometrical representation. The present problem
affords an interesting example of the construction of p. 36 for finding the
cusplocus and the envelope. In Fig. 34, E is the position of the electron
at time t, and ET its line of motion. PEP' is the trace of the semi-cone
cos 0 = c/v = 1/l, and CF that of the plane fR cos
= v2- c2 , that is,
1)/2.
= (v2 - C2)/f p
R cos
Thus EF = p sin2 9, and OF = p cos 2 9.

Obviously CF and EP intersect at C, so that OC = p cos , and z C = 90 .
Thus EP touches the circle BCD, of centre
O and radius p cos 9.
The cusplocus is generated by the
J
'
intersection of the plane CF and semic
cone PEP',which is obviously a hyperbola,
and is indicated by a dotted line. ItsC
equation is easily found. For if the angle
made by OE with a fixed radius be cot as
E'
before, the equation of the plane CF is
H
j
p = 32 v~ cos (cot 2

cone is /{x +

W2

0), and that of the semi+ p2 - 2 p cos (cot - 0)} =

Fig. 34.

l

sin (cot -

0=600.

4). Hence eliminating

2
t we find that the equation of the cusplocus is x +2

this easily reduces to f- g = 2 p/$, that is, to (97).
the hyperboloid (97), as was to be expected.

For a3=2,

+ p2 =

$2W2

+ p2/) 32;

Thus the cusplocus is

64. The wave-envelope.
The wave-envelope has the cusplocus and
the orbit for the two sheets of its surface of centres. Its trace on the plane
of the diagram is readily found, for it must be generated by the wrapping of
an inextensible and flexible thread on the circle BCDG, which is the intersection of the cusplocus by the plane of the diagram. Thus it consists of
arcs of involutes of this circle, namely the arc GE, got by wrapping CE on to

v]

ILLUSTRATIVE

EXAMPLES

87

the arc CBG of the circle; the are EJ,got by unwrapping C'E; and lastly,
the arc GH, parallel to the last. At G there is a cusp, at E a conical point.
The remaining parts of the arcs GE and JE, namely, the broken lines EJ' and
EG', obviously belong to the reversed motion.
From Fig. 34 it is easy to find the velocity for which we can get five, seven,...
simultaneous waves. Obviously we cannot get five waves, for a point in the
plane of the diagram, until the traces GE' and GE of the wave-envelope
intersect again between the two circles, that is, until E' and E coincide, and
z BOG =
It is easy to show that this occurs when tan V(/ 2 -1)= /(/ 2 - 1),
2
that is, for the values 2 = 1, 1 + (1"43. wr), 1 + (2"45. r)2, ....
These values

v.

correspond to three, five, seven, ... waves respectively.]

CHAPTER VI
REMARKS ON THE SOLUTIONS OBTAINED, AND ON THE
METHODS OF CALCULATING THE POTENTIALS IN GENERAL

65. PROBLEMS concerning the field due to a moving charge may be
grouped in three classes according to the nature of the relation between the
time of emission of a wave of disturbance, T, and its time of action at a given
fieldpoint, t. This relation is illustrated by the characteristic curve for that
fieldpoint as previously explained.

(1) The curve for the first class is ascending both when T has an infinitely
great negative and an infinitely great positive value. Fig. 8, § 22, for uniform
motion with velocity less than that of light (problem 1 a), Figs. 18, § 36, and
21, § 40, for uniform motion with any velocity generated from rest with
uniform acceleration (problem 3), and Fig. 22, § 45 (problem 4), show curves
of this type.
Their characteristic is that a line t = constant cuts the curve an odd
number of times, so that any fieldpoint can be disturbed by an odd, but never
by an even number of waves simultaneously. Ultimately there is but one
such wave.
(2) The curve for the second class is descending for infinitely great
negative values of T, ascending for infinitely great positive values. Fig. 9, § 22,
for uniform motion with velocity greater than that of light (problem 1 b),
and Figs. 12, 13, 14 and 15, § 29, for uniformly retarded motion turning later
into a uniformly accelerated one (problem 2), show curves of this type.
The line t = constant cuts them an even number of times, or not at all.
Any fieldpoint is disturbed by an even number of waves at once, or not at all.
Ultimately there are two such waves.
(3) The third class includes periodic motions, such as uniform circular
motion (problem 6). The curve is either continually ascending (velocity less
than that of light), or alternately ascending and descending, but of the same
character however great the value of may be, whether positive or negative.
The line t--constant cuts the curve an odd number of times; any fieldpoint
is disturbed by an odd number of simultaneous waves.

r

CH. VI]

REMARKS

ON THE SOLUTIONS OBTAINED,

ETC.

89

It will now be understood why the results obtained for uniform motion
with velocity greater than that of light in problem 3 show no agreement
with those obtained here, as well as by all previous writers, for problem 1 b.
The two problems are essentially different in kind, and rest on entirely
different assumptions. Before we can identify two given motions, which for
a given interval of time appear to be the same, we must enquire into their
previous history; if this has been different, the electromagnetic field will
generally be different. We cannot avoid the consideration of the initial
conditions in a given problem merely by supposing the commencement of the
motion to have been infinitely long past.
Our conclusions are in this respect in agreement with the criticism of
Lindemann* on the work of Sommerfeld, so far as motions with velocities
greater than that of light are concerned. But there seems to be no doubt
as to the correctness of Sommerfeld's results for quasi-stationary motions
with velocities less than that of light. This question has also been discussed
from a different point of view by Schottt with the same result.
66. We must next draw attention to the difference in character of the
field due to motions with velocities less and greater than that of light.
When the velocities are less than that of light the potentials are everywhere finite and continuous, and therefore the forces are everywhere finite,
except at a point-charge, just as for the ordinary electrostatic potential and
force.
In consequence the discrepancy just signalised for velocities greater than
that of light between problems of the type of problems 1 and 2, and those of
the type of problem 3, does not exist in this case.
Moreover, the existence of these discontinuities and infinities of the
potentials has a bearing on the physical question whether charges moving
with velocities greater than that of light really occur in nature.
It may be urged that this question is of no real importance, seeing that
all phenomena hitherto observed can be explained without assuming the
existence of such charges.
We must at once admit this, so far as our knowledge of the properties of
free electric charges goes. Indeed the recent experiments of Bucherer4 are
decidedly in favour of Lorentz's formula for the mass of a $-particle, which
makes the mass vary as 1//c2 - v2, and therefore leads to an imaginary
expression for velocities greater than that of light. If this formula were
universally true, velocities greater than that of light would be impossible.
The experiments however have only been made with -particles having
velocities considerably less than that of light; thus the extension of the
* K. Bay. Akad., II. Ki., xxiii. Bd., II. Abt., 1907, p. 235.

t Ann. der Phys. 25, 1908, p. 63.

+ Phys. Zeitech. 1908, p. 755.

90

REMARKS ON THE SOLUTIONS

OBTAINED,

AND ON THE

[CH.

formula up to the velocity of light involves an extrapolation much beyond
the range of experiment, and can only be justified by theoretical reasons.
The Lorentz mass formula is required by the so-called Lorentz-Einstein
Relatiftheorie*, which has the great advantage of explaining phenomena of
aberration as well as the null effect of the earth's motion on optical and
electrical phenomena to every order of approximation.
Unfortunately however this theory takes no account of the loss of energy
from an electric charge moving with acceleration; this loss is very small so
long as the velocity remains small, but becomes very important when it
approaches near to that of light.
In consequence the idea of mass itself becomes ill-defined for velocities
closely approaching that of light; this has been proved conclusively by
Sommerfeldt, precisely on the basis of the Lorentz theory. It is obviously
difficult to reconcile this result with the results of the Relatiftheorie.
We must therefore be careful not to allow theoretical views such as these
unduly to influence our notions as to the possibility, or otherwise, of velocities
greater than that of light.
67. On the other hand there are a number of phenomena which we
have hitherto been utterly unable to account for on any of the current
mechanical theories, namely those afforded by spectrum series.
When we review the experimental evidence, quite apart from theoretical
considerations of any kind, we can hardly avoid the conclusion that a series
actually contains an infinite number of lines. It may fairly be said that this
is the view of those best qualified to judge, namely the observers themselves.
But when we try to account for these series by means of finite systems
of discrete electric charges we are at once met by the difficulty that the
number of their degrees of freedom is finite.
This difficulty is however only apparent; it is not allowable to treat
electric charges as bodies which obey the ordinary laws of mechanics, for
they influence the surrounding aether and are indissolubly linked with it.
If we treat the aether as a continuum, the system of aether and charges has
an infinite number of degrees of freedom.
Accordingly it has been proved by Herglotz 4 and Sommerfeld §
that a spherical electric charge can execute an infinite number of free
rotational, to and fro, oscillations about an axis, and therefore can emit a series
of an infinite number of lines.
*

Einstein, Ann. der Phys. 17, p. 892, 1905.

t Gott. Nach. 1905, p. 201.
§ GItt. Nach. 1904, p. 431.

Planck, K. Preus. Akad. xxix. p. 542.

. Gott. Nach. 1903, p. 375.

vi]

METHODS OF CALCULATING THE POTENTIALS

IN GENERAL

91

These series it is true are not like the spectrum series to be accounted
for; in particular their wave-lengths are much too small, being of the order
of the dimensions of the charge. But they show conclusively that when we
do not limit ourselves to small velocities, and thus alter the form of the
solution from the commencement, infinite series are possible. Whether
motions of charges can be found, which shall give series like those known
in nature, cannot be decided beforehand; a detailed investigation of the
problem is necessary, and it can only be carried out on the assumption that
velocities of all magnitudes are possible.
68. From this digression it appears that anything which may throw
light on the question of the existence of charges moving with velocities
greater than that of light is of importance. If such charges exist they must
be looked for inside the atom.
We may expect evidence of their presence in two ways: when the atom
is disturbed by outside influences it is no doubt set in vibration and its
charges certainly emit waves, for no accelerated motion is possible without
radiation to some extent. It is generally taken for granted that these waves
constitute light. But so far as I am aware no vibrating system of atomic
dimensions has yet been imagined, which can give a sufficient number of
waves as long as those of the spectrum. Every mechanical or electrical
system, made up of separate particles, such as the Saturnian systems of
Nagaoka and J. J. Thomson, so far constructed, has most of its wave-lengths
very little greater than the diameter of the atom. At present we cannot
gain much information as to the motion of the charges inside the atom from
spectrum observations.
But a second way suggests itself. When the disturbance becomes so
large that the charges inside the atom acquire very great accelerations,
aperiodic waves, or pulses, of electric and magnetic force are emitted. It
is usual to regard these as constituting X-rays and 7-rays. We have seen
in problems 3 and 5 that the duration of passage of the pulse across a fieldpoint is of the order of the time that the acceleration lasts, and its thickness
of the order of the distance moved through by the emitting charge while its
velocity is changing.
In the case of a disintegrating atom the latter distance may be expected
to be of atomic dimensions, let us say of order 10- 8cm.; the thickness of the
pulse will be of the same order.
When a series of such pulses falls on a dispersing or diffracting apparatus,
this apparatus acts as a harmonic analyser, and breaks up the disturbance
into Fourier components. The fundamental, which usually predominates,
has a wave-length comparable with the thickness of the pulse, and is therefore

92

REMARKS ON THE SOLUTIONS

OBTAINED,

AND ON THE

[CH.

of the order 10- 8 cm. All this is sufficiently well known; it is in agreement
with the few experiments made regarding the diffraction of X-rays, for
instance those of Haga and Wind *.
When the charge causing the pulse is moving with a velocity greater
than that of light an additional effect may be expected, due to the discontinuity of the potential.
charge the surfaces of discontinuity, which occur only
at a time, become shells of transition, whose thickness
as the linear dimension of the charge; for a /-particle
10-13 cm.
in these shells vary rapidly and remain finite; the same
The potentials
is true of the electric and magnetic forces. Sommerfeld t finds that in
the shell, which replaces the conical envelope sheet in problem 1, the potential

For an extended
for a single element
is of the same order
this is of the order

is of the order a- , where a is the radius of the charge, supposed spherical.
The electric and magnetic forces will be of order a- . In the rest of the
pulse they are of much smaller order. Thus the shells of transition are
places of exceptionally intense forces, and constitute exceptionally thin and
intense aether pulses.
When they fall on the analysing apparatus we therefore expect to get
a fundamental wave of great intensity, but of exceptionally short wave-length,
only of the order 10- 13 cm.
Thus these pulses may be expected to have special properties, for instance,
to be extremely penetrating. Their existence would prove conclusively the
presence of charges in the atom possessing velocities greater than that of
light, and would require us to very materially alter our views as to its
constitution.
I am not acquainted with any experimental result, which cannot be
explained without them, and the theoretical investigation of their motions
would be extremely difficult.
69. We shall therefore during the remainder of this investigation restrict
ourselves to the case of motions less than that of light, except when special
mention is made to the contrary.
Hitherto we have mentioned two methods of calculating the potentials.
(1) The method of Sommerfeld depending on the direct evaluation of
the integral expressions (21) and (22), § 11.
We have already seen that the use of these integrals offers difficulties on
account of the variability of their upper limits, so that it is doubtful how far
* Wied. Ann. 1899, 68, p. 884.

Also Sommerfeld, Phys. Zeitsch. 1901, p. 55.

u. R. Pohl, Ann. der Phys. 1909, 29, p. 331.]

t Amst. Proc. 1904, p. 362.

[B. Walter

vi]

METHODS OF CALCULATING THE POTENTIALS IN GENERAL

93

differentiation is allowable. They have been the subject of the polemic
between Sommerfeld and Lindemann already referred to on p. 21; this
in itself is sufficient evidence that their evaluation is very difficult and
intricate.
The method has only been used for a spherical charge; it is doubtful how
far it can be adapted to other cases, for instance, the case of a Heaviside
ellipsoid, that is, a spheroid of revolution with its short axis in the line of
motion, and of eccentricity v/c.
On the other hand it is not limited by considerations regarding the
magnitude of the velocity; it is equally applicable to velocities less and
greater than that of light, but of course is more complicated for the latter.
Since however for our purposes it is of importance to have a method,
which shall apply equally well whatever the arrangement of the charge may
be, whether it applies to velocities greater than that of light or not, Sommerfeld's method is inconvenient.
(2) The method of the point potentials, due originally to Lidnard and to
Wiechert.
We have used this method hitherto in our illustrative problems. It is
very convenient and suggestive as long as we do not require explicit and
calculable expressions for the potentials, and only wish to get a general idea
of the nature of the field due to a given motion.
Its main disadvantage is that the point potentials are explicit functions
of the time of emission of the disturbing wave, T, and not of its time of action,
t. In order to express them explicitly as functions of t we must solve an
equation, such as 28, § 13. This solution can only be effected in finite terms
in exceptional cases; in the most interesting cases, like that of uniform
circular motion (problem 6), the equation is transcendental, and a finite
solution unobtainable.
Its solution can be effected in series by means of Lagrange's Theorem
whenever there is but one root, that is to say in all problems of motion with
a velocity less than that of light, and for some cases where the velocity is
greater than that of light, but then only for restricted portions of space.
Since for many purposes, such as the calculation of the radiation from the
moving charge, and of the mechanical reaction of the resulting field on it, we
require the potentials for all points of space, this method may be said to be
restricted to problems where the velocity is less than that of light.
In these cases it gives us a complete solution of the problem so long as
the series remain convergent.
When the equation has more than one root Lagrange's Theorem fails. In
this case there seems to be no known general method of obtaining a solution,

94

REMARKS

ON THE

SOLUTIONS OBTAINED,

ETC.

[CH.
VI

but as it can only occur when the velocity is greater than that of light this
failure is not of much importance for our purpose.
This method of expansion though available is not convenient; for the
series can be obtained more easily directly from the integrals themselves,
especially from (12) and (13), § 9, and that too in the form most convenient
for the problem in hand.
Besides, these integrals for periodic motions lead at once to the appropriate
expansions of the potentials in Fourier series. For instance, in problem 6,
where the equation giving 7 in terms of t is practically Kepler's equation,
the integrals give the series corresponding to Bessel's solution of that
equation.
We shall first treat of the problem of determining the field due to given
motions of charges at distant points, with a view to calculating the radiation
from them.
We shall then consider the problem of determining the field in the
neighbourhood and inside the charge, and thence show how to calculate the
mechanical reaction of the field upon it.
Since our object in this investigation is to discover new results rather
than to establish them with absolute mathematical rigour, we shall make
use of the simplest methods available, for instance of symbolical methods,
leaving the expressions obtained to be justified a posteriori. Such a justification will generally consist in a determination of the range of validity of the
expressions obtained, but this is usually sufficiently obvious in a physical
problem without detailed examination.

CHAPTER VII
PERIODIC MOTIONS
70. IT is convenient to classify the motions to be considered according
to the number of different, that is, incommensurable, periods or frequencies
involved. Thus we may speak of monoperiodic motions, for instance,
uniform circular motion or elliptic motion; of diperiodic motions, for
instance, motion in a nonreentrant epicycloid; and of polyperiodic motions,
for instance, simple harmonic oscillations about uniform circular motion.
We shall begin with the simplest case, where there is only a single
period.
Monoperiodic motion.
and (13),§ 9:
1 f

We start from the integral expressions (12)

fO0

de

a =2 f

t-00

l-R

e
c.........(13),

R

where

In these expressions the vector p, whose components are (y, 1, '), is a
given function of T with the single period T,and v, that is to say p, has the
same period.
Separate the integral with respect to T into the sum of an infinite
number of component integrals, that is write
-

j= -N

(j+1)T

where the integer N is to be taken infinitely large ultimately.

[CR.

MOTIONS

PERIODIC

96

.96 In the type .integral written down write jT + ' for T, thus reducing [H
PERIODIC MOTIONS
the
limits from jT and (j + 1) T to 0 and 7. R and v are unaltered by this
change on account of the periodicity. Thus we get, for

example,

95~

I

Limit
m
L

iI

T

Oz

o

2^
rn

N

=

de

i;

NF
N

c
C(t-jT-R/C--)d

t

-3
-N

I f d f~fT i=N cosplk (t - R/e -- jT)Td,
'ni
301 j-N
B
1
a"
T sin
) iT cos (t-B/c
) dT- p

Limit

-

Y-s

= Limit co

j=

+j
2

-'f de

de

(N+

de f

f -: 0

T

2

cos --'-(t--R/c --

T)

dT

... , ..... ........... (98),

T.

j=4

where I' as usual denotes the sum written but with one-half of the first
term. Similarly
a =i Ja=
Y7
'

2

defcos

(t
--

2

2r

c-'T(t.-dT.................(99)
-- )
.
d

The interchanges of summations and integrations are allowed because we
know that the integrals are convergent for physical reasons. It would in any
case be quite easy to introduce an appropriate factor to ensure convergence
of the integrals.
By expanding the circular functions we may write (98) and (99) in the
forms
Q;
_t2

j=

(Di

((Dj cos?

+

fdef

sin

a=~ 2(,cos
j=0

27rjt7~t2

+r)T

-fdefcosi(

j=

in

l

---

±

+

..........

.(100),

T

A,' sin

\

Ai=4fdefcos.2(

+ T!!2

T
Ai'-4defsin (i±.)..
Theequaios gvetheepninsoTh

.........

otnil

(101).

nReiso

vit]

PERIODIC

97

MOTIONS

vi] When the distance R is large compared with the linear dimensions of the
PERIODIC MOTIONS9

charge, though it may be comparable with those of the orbit, the integration
with respect to the element of charge, de, may be dispensed with, and the
whole charge, e, introduced as a factor.
71. The magnetic and electric forces are easily found by means of the
equations (VII) and (VIII), § 3,
h=curla .......

................

(VII),

............ ..... (VI)

d =- grad.

The series (100) and (101) may be differentiated term by term for the
motion is continuous, being periodic.
In finding grad. 0 and curl a we must remember that (x, y, z) occur only
in the quantity R, which enters into the functions'i ... under the sign of
integration.
Now for any scalar function f(R), we have

grad. f(R)

-

R1 .f'(R),

where R1 is the unit vector in the direction R.
Similarly for a vector function g (R), we get
curl g (R) = [R1 . g' (R)].
Applying (VII) and (VIII) to (100) and (101), we get

d="_2O; cos 2-rjt
O"=-

grad.j-

=

2wj A...'(12)

cJ/+±A
'

=O-grad.

where X

.

cT ; it is the wave-length of the fundamental wave of period T.

Performing the differentiations, we get

O = 27j j e T sin27j (R + T (R 1 -v/c) dT
+ 1f

defT cos

2nj (R

sin 2'rj

)

Rl dr

R

27frdefTco27j

+ 1fdej

±

(R

+

7)

T

(R 1 -y/c) dr ......

R1 dr

(0)

98

[CH.

PERIODIC MOTIONS

In the same way we get, from (VII) and (101),
h- =
Hj=

S2rrJ
-

-

=

d

de

fde

-

H'

Hj cos 2

2

- -1

.

sin

27

sin
[R v] d

R
-

+

(R +

cos 2

7r R

j

cos2

de

-X2

T

+ -jt
Hj'

+

R

d

..........

(104).

dT

T

de

o sin

T

c

+

T

R2

The expressions (102)-(104) give the values of the electric and magnetic
forces expanded in series of harmonic terms.
Thus we see that the monoperiodic motion of an electric charge always
gives rise to an infinite series of simple harmonic waves, whose frequencies
are integral multiples of the frequency of the motion itself.
The amplitudes of the several harmonics are given by the equations
(103) and (104); in general they diminish very rapidly as the order of the
harmonic increases.
It is to be noted that each amplitude involves terms which at large
distances are of different orders. The first line in each equation represents
a vibration, which is of order 1/Xr ultimately, r being as usual the distance
of the fieldpoint from the origin supposed to be taken near the orbit; and
this component is transverse. The second line represents a vibration ultimately of order 1/r2, and therefore negligible in comparison with the first;
for the electric force this term is ultimately radial, for the magnetic force
transverse. The magnetic force in each order is ultimately perpendicular to
the electric force.
72. Hence at large distances from the orbit we may make the following
approximations:
(1)

We replace the integration fde by the factor e.

(2) We retain only terms of order 1/Xr, that is, only the first lines of
(103) and (104).
(3) We replace the quantities R, R 1, where they occur outside the circular
functions, by r, r1.
(4) In the circular functions we cannot replace R by r simply, because
we should then neglect a difference of phase of finite amount. We must
proceed as follows:

vrl]

VI]

99,

MOTIONS

PERIODIC

PERIODIC MOTIONS9

We have
R

/

)2 + (y

..

X(.

where terms of order

) + (z

-

-

4 +)2}7
+

=

and higher, have been

1/r,

z

neglected.

Denoting the vector (x, y, z) by r as before, and its unit vector
and the vector ( r, 5) by p, we thus have

S= r -(r

1

by r ,
1

p)

.

=r-p,

where p is the projection of the radius vector of the charge, p, on the vector
r2 at time T.
With this approximate value of R, we get
cos

sin

27r3i

(R/c +

27rj
(R/c

T) =

cos

+ T)= sin

27i~

-T-

(2

27i

+

-

(i/c + T

-

p/c).

When these expressions are substituted in terms such as the general
70,
term of (100),
27j
2irjt+i

411cos

+7(D

sin

we get an expression such as

2irjfd
cos T
~(t - rc f
r

2 7(T

ecs--To
.

2

/sin
c)fdefsin

73.

2

d)

C
)2

z jp

[y

T

d

-- _)dR.

Field at a great distance. Making the approximations mentioned

above, in (1), (2) and (3), and changing the notation slightly,
following expressions for a distant

_

--

f2j'.

2w:j

1 jfT
A,

Tcos

we get the

(2jp/)ci

cos

( &

.

point:

(t

/(Tr,T--

r /c)

+

A4Y

sin

(t

-r/c))

2

pc)

V.

,1F T . 2wj
Aij=;j sin -17 (T-p/c) dr
fo7-2

.10)

I00

MO

PERIODIC

100By direct

differentiation,
47rr
>j

O =
-

PIONS

[OH.

P'EIOI)IC

MIOTION'SLol
or from (102)-(104), we get
(t
cos
- (t
/c) +f
sin 27
-

(

-7/C)

4<r-A'(107),
_ ,

Pj.r 1 +Aj

=

cos 27h
, T(t

..,('.

-

r/c) -

H1 'si2W(t-i/)

H, = - [r.A'1(108).
Expressions of this type were first given

by Schott*.

74. By means of the expressions (107) and (108) we can easily verify
that, to the approximation used, the electric and magnetic forces are
pendicular to the radius vector r, and to each other.

per-

In the first place, form the scalar product of r, and 0j, we get

I

f T

277-j

r.-)}dT.

( sin -7 (T-P/c).{

Now since we have by definition
p=(r1.p) and p=v,
we geta

_r
at

so that

1

-

cc

(r 1 .v)

1

.v)

c

7

Thus the function under the sign of integration is a perfect differential,
and the integral vanishes when taken between the limits 0 and
T on account
of the periodicity. Hence
(r 1 .ej==0,
and in the same way
thus showing that each component of the electric

the radius vector r

1

t.

The form of HH and Hj' shows that the

to r,.
By forming the vector product of
_0

r, and

force

is perpendicular to

magnetic force is perpendicular
4

we find at o

rnci

.l

_u_=

VII]

101

PERIODIC MOTIONS

75. An examination of the equations (107) shows that each harmonic
component of the electric force is the resultant of two components:
(1) A purely radial component proportional to, but one quarter of
its own period in advance of, the corresponding harmonic of the scalar
potential, 0.
(2) A component proportional to and in the direction of, but one
quarter of its own period later than, the corresponding harmonic of the
vector potential, a. The radial component of this term is just sufficient
to neutralize the first term.
Similarly equations (108) show that each harmonic component of the
magnetic force is proportional to, but one quarter of its own period later
than, the corresponding harmonic of the vector potential, a; and its direction
perpendicular to both the radius vector and the harmonic of the vector
potential.
It follows that each harmonic vibration is elliptically polarized.
that we may understand the nature of this
polarization, let us consider Fig. 35. It is taken
irthe wavefront at the fieldpoint P, looking
B
along the radius vector r, but inwards, just as an
observer would view the incident vibration.
PO and PQ are in the directions in which
the polar distance 9, and the longitude 0 increase.
0
These directions form a right-handed system
with r.

In order

A'H

Q

A

Let PA represent the projection on the wavefront of the vector A-,
PA' that of the vector Ai'. These lines are conjugate radii of the ellipse
described by the radius vector PQ, the projection of the harmonic component
aj of the vector potential on the wavefront.

e';

Oj,

It is obvious that PB' represents the vector
and that PA represents
this follows at once from the geometrical interpretation of (107).

Hence PD, the radius conjugate to PQ and behind it, represents the
corresponding harmonic; and the harmonic component, dj, of the electric
force is represented by 47rej/rX times PD.
It follows that the harmonic component, hi, of the magnetic force is
represented by 47rej/rX times PH, where PH is a right angle in front of
PD, or behind it, according as the rotation of all the vectors is right-handed,
as shown, or left-handed.
Thus we see that a knowledge of the vectors A, and Aj' is sufficient for
a complete description of the distant field.

102

PERIODIC

[CH.

MOTIONS

The Poynting vector
76. The Poynting vector.
equation (XIII), § 5,
c [d.h]

is given

by

8=

47r

It is obvious that in finding the mean value of s for a long time, or for
a whole number of periods of the motion, the terms in s due to products of
different harmonics disappear identically on integration. Hence we may
without loss of generality confine our investigation to terms involving only
a single harmonic, such as
c [dj. hj]
j-

47r

The only terms of this type which contribute to the mean radiation are
those which involve cos

2 T (t - r/c) and sin2

(t

-

r/c); and the mean

value of each of these factors is 2.
Thus, omitting the time factors, we get for the mean radiation vector
due to the harmonic j
j
J c (4wej
_ (27rej 2

c 27rej) (H + H'2) r
2
(2r /
Xr

27r

A;)
{A; sin 2 (r. + Aj'2 sin2 (r. A;')} ri ......... (109).

2\Xr

This equation together with (106), § 73, enables us to determine the total
loss of energy due to radiation from the moving charge.
77. Group of charges. The expressions obtained above can easily be
generalized for the case where we have a set of n similar charges moving in
regular succession round the same orbit in the same way.
Let (i, i, i)be the coordinates of the ith charge; and suppose that for
all values of i from 0 to n - 1 we are given equations of the following forms:

where f, g, h are three periodic functions of the same period T.
A set of charges satisfying these conditions may conveniently be called
a "group" of charges; an example is afforded by n equidistant electrons
describing the same circle with the same uniform velocity. Thus corresponding elements of all the charges describe the same curve in succession,
passing each point at intervals T/n.
Hence the quantities Ri and vi for the ith charge at time T have the
same values as the quantities R and v for the 0th charge at the time
T +iT/n.

vii]

103

MOTIONS

PERIODIC

0

vii]The contribution of the ith charge to 0 is, by (98), § 70,
PERIODIC MOTONS
i=0T
2wj
' - de ]cos-y (t - Ri/c - T) dr/Ri.

YW

Change T into 7 - iT/n; then, by what has just been said, R4 takes the
value of 1 for the time T. The limits become iT/n and T+ iT/n, but may
be changed back to 0 and T on account of the periodicity of the motion.
Thus the contribution of the ith charge becomes

f

j' 2decT

R/c

(t + iT/n -

os 2

-

T)dR

1 '
J=0
Summing for all the charges, that is for i from i = 0 to i = n
(2snt
S
2l
4=n
2 ID, cos
Tsft+
sn.
1

- 1, we get

.o

TnT

s=0

Similarly

a=n
~2
s

=

(An cos

2wsnt
T+

A'8 n sin 2Tsn).......(112).

s=o

Thus all the harmonics disappear except those corresponding to multiples
of n.
Whereas for a single charge the mean radiation vector is given by
jci'

s= C' s,.........(113),
j=o

in the present case of a group it is given by
s=o

2
- n

.. .

..........

(114),

a-0

a

for by (112) each harmonic component which is left is n times as great as
before, and therefore each component of is n2 times as great.
We shall find that the component harmonics j = 0 and s = 0 contribute
nothing to the radiation, while successive harmonics diminish very rapidly
in magnitude. Thus the absence from (114) of the components j = 1, 2,..
n -1, n + 1, n±+ ... 2n -1, ... is much more than sufficient to make up for
2,
one factor n in (114). It fcdlows that the radiation per charge from a group
of charges is much less than that from a single charge.
The first example of interference of this kind was given by J. J. Thomson *.
78. Illustrative Problems. We shall now treat of some problems in
illustration of the methods just developed.
Problem 1. n. equidistant electrons move in a circle with uniform
velocity. To
the electro-magnetic field produced. This problem

find

*

Phil. Mia. [6],Vol. VT p. 681.

104~~
PERIODIC MOTIONS

104

[H

[cii.

has been treated by Schott * for the case of a fieldpoint at a great distance.
The approximate solution for infinitely small velocities was first given by
J. J. Thomsonft.
Let the radius of the circle be p and the angular velocity c. If we confine
ourselves to the case of a fieldpoint not very close to any one of the electrons,
we may treat them as points with charge e.

a

Take the centre of the circle as origin, and its axis as axis of z in system
of cylindrical coordinates (z, w, ). Sometimes
we shall use also polar coordinates (r, 6, 0).
E R
P
In Fig. 36, P is the projection of the fieldpoint
A
(z, w., 0) on the plane of the circle, A and B the
intersections of the meridian plane through it withox
the circle. E is the position of the 0th charge at
B
the time T.
wOP is the angle 0, OP is equal to w, and the
Fig. 36.
height of P above the plane of the diagram is z.
Let the angle xOE be equal to w + 8, and let x = COT+ 8- 4). By
choosing the zero of time at the instant when the electron E crosses OP, we
can make 8 = f; but it is convenient to retain f explicitly in the equations.
This we can do by replacing t wherever it occurs by t + (8 - 0)/w.

o is equal to the angular velocity, so that the period T is equal to

2

w/w.

We have, from Fig. 36,

R =\,/{z2 +

'2

+p

2

cos x}

-2

2

where

C = (z + W2 +

=

p2)/

V2wp (C-- cosX)

2

wp.

On the circle, where z = 0 andw= p, we have C = 1; elsewhere it is
greater than 1; at an infinite distance it is infinitely great, and we have

R = 1/( 2apC)

-

cos x

W
=

r - p sin 9 cos x

very approximately ; that is, in equations (105) and (106),
p

=

p sin 6 cos

§ 73, we must put

X.

(8

When we replace t in equations (100) and (101), § 70, by t + - 4)/wO, and
therefore also T by +(8 - 4)/w, and change the variable from Tr to x, we get
equations such as

T

2
1ijef
ffcos j
2rJ0

Tx + aV(C -cos x)} dx
V2wp (C-cos x)

* 1 c. Ann. der Phys. 1907, 24, p. 641; Phil. Mag. [6], Vol. xi . p. 189.
t Phil. Mag. [6], Vol. vi. p. 681.

vi]

f

105

0
, and replacing x in the second by

PERIODIC MOTIONS

Dividing the integral into two,
27r -

MOTIONS

PERIODIC

vtl]

x,

we get

7
rp I de Fcos j a (
!

N

-

cos x)o

x

2zp(C-cosx)

=7r.JoJ

In this equation a stands for w (2wp)G.

Each of the vectors Aj and A' may be divided into a radial component
Aj and Aj, and a component along the parallel of latitude, A andX0
The components of v/c in these directions are given by
c

where =
of light.
79.

p/c, so that

8

C

is the ratio of the velocity of the electron to that

In this way we get
1=C' tI[osi(wt +632

f'

d icos- jac</(C--

=

r

cosx) cos ix dy
cas

f ci f'~sin j
Jo

<(C - cos
/22p(-

{A(j cos (cot + -

a,='2

K, cos .jot

A;}=3fde
77

f
J0

-it

1(C~

x) cosX dx
cos X)

) + A 'j sinj(t+-)}
cos x) cos

jx

cos x dX

02wp(C--cosx)

e

.= .i -"o

""(15}.

J 2 p (G-cos x)

"r

7J

c) +%'sin j(ot+8--)}

sin ja

V/(Ccos x) S

(1)
.(
..

xsilxcx

Vow(C-cosx)

(1)

sincoa\(C cos x) si)X sifx dX

9

de

\2p(C-cosx)
V

Th raue fth lcrcadmgeisocscnesl
t eri
1b
frm(0And (04)e
gra
itrst
7tsufce osa ht

ewitndw
rminsoastearofo
-hy-aebohrdilanxrnses

~PERIODIC
MOTIONS

106

106

[H

[CH.

The values of the potentials for the group of n electrons are at once given
by equations (111) and (112), that is to say, by equations(115)-(117)}with
terms of orders j = si alone.
The functions occurring in these equations are all of the same type.
When a = 0, that is, c = 0, they reduce to toroidal functions; thus we may
regard them in some- sort as generalized toroidal functions.

We shall not stop to discuss

their properties, but pass on to the consideration of an important particular case, namely that of a distant fieldpoint,

for

which

C is very large.

80. Distant field. We have seen that for very large values of C we
may write very approximately,
Pr-psin Ocos x,

-

--

-

c

sin 8-cosX

9 cos.

p
=pSin

c

Introducing these values into (116) and (117), where a V/(C- cos
and

2

/(

p)V/(C

difficulty

I~

-

cos

x)

proceeding as

B, and

in

73, we

x)

=

coR c,

without

any

CC

I3Acos)(w. t -r/c ±Aj=cos (j,8 sin 9 cos x) cos
=

--

) ±+
A'j
cosjx

sin j(w.t-r/c+3-

dy

(118).

T=rfsin

A'

(j$lsin 8Ocos x) cosx cosjX d)

A~r _=---fsin(j~sin~cosX)sinXsinyXdX
cos (j$3 sinG8 cos x) sin x sin jx

A~j-!

d
.......................
(119).

easily
We get at once from the well-known expansion
81.

The normal functions A,

... are

expressed as Bessel

Functions.

E cos = Y' 2t jJJ (x) cosjy
cx
j=4
the

following integrals which occur in (118) and (119):
7Tf

sn(

csx)csx

cos (x cos
s

cx

X=

,w

x) cosjx d= cos ja.

J"(x),

0,

w]hen j is even,
w.hen j is odd,
'hen j is even,

-sin

j.

J (w),

w hen

wl

j is odd.

vilil
PERIODIC MOTIONS10

viYl

107

Using these results, and also the relations
2j(x)

1(x)

J
and

-

Jj+i

(x)=2J,' (x),

putting xw j3 sin-0, we get the following results:
A'01 COS -j.j(jI3 sinG), jeven
,
~
-j=co
oA
j -odd
A j = 0,
Aj =sin 4 w.Jj' (j sinG0),

Ag=

J)j(jsin,

r

-sin

Awe
,1

= 0'A

8si
sodd

(

j3sinG

'

................... ..
(120).
Substituting in (118) and (119), we get
2e3 j=00
ro
j
r
°°
-2e
,
('Jr sin Z' J(j$
'

ad

sin

0)

(co

.sink

t

"

--

4r

/

=

1(122).
sin G).cosj(w.t-r/c+-4+ir)

82. Electric-and-magnetic forces. It is easy to deduce the expressions
for the forces from equations (107) and (108), ':73. Since we know that the
electric force is equal to the magnetic force, perpendicular to it and to the
radius vector, we have
d,=h, , 4qhh.
Thus it is sufficient to calculate h.
Equations (108) give at once
H , =-A' 1 - cos GA'w ", H',
for the

angle between the directions of 0 and is equal to 0.
co. + 8- 4))+ H' sinj'(w .t rlc + 8 4
4))
IA ,,j (o
w

get___
H~~ "cosj

t

-/c

sinj

cos 0.

-

cos 0. {A,,.cos

.

t

-rv/c

t

[j(v.

-

+ 8-

j(c.

t -v?/C

+ 8

-_)

+

1 cosj (
5

-A'

vc +8-))

.

t

-r/c

±

8-)

-2r

± A'
cos

Hence we

-

31

-

HI."

cosB

H's sinj

.j sin

.j

(co. t

-

t
(wo. v/c + 8--4))
--

?-/C + 8

)

NCH.

MOTIONS

PERIODIC

108

108
PERIODIC MOTIONS
[n
phase of each harmonic by 17r, and d4 --he from a by multiplying by
2rj/X, that is, by j,3/p, and diminishing the phase by -:.

Writing f

for o (t

?-/c) +

-

0

,4

2e,

4 +

0

pr

j-

j=1

Pr

dg

=

h

=-

2e/3"n)"

e3

*.

jJj'(j$ -h 6) cosj r ........
sin

(124).

of n electrons are found as before by replacing
the ith electron and summing for i from i =.0
all ha ,rmonics disappear except those for
by

S

which

it.

Hence for the group

2e co)1t 92.s"

= - he = -

9)sin...........(123),

jpi

2e9

The forces due to the group
the constant 8 by 8 + 2wi/fn for
to i a - 1. The result is that
j = sn, and these are multiplied

we get from (12!) and (122)

j Jj fl sin

1cot
cot

t 7,

s $J,,,(sn13sin 6) sin snfr.....(125),

-0

I

s J"(sn sinO)cossn#.....(126).

These agree with the values given by Schott* (loc. cit.).
83. Character of the field. The forces consist of series of harmonic
terms, all vibrating transversely to the radius vector. The term of order zero
obviously vanishes identically on account of the presence of the factor j in
(123) and (124), and s in (125) and (126). The amplitudes of the successive
harmonics are of the order j Jj (j$ sin 9) or j J?' (j$ sin 6). When 5 is small,
that is the velocity very small compared with that of light, they diminish
with very great rapidity as the order of the harmonic, j, increases. In particular, even the harmonic, s = 2, in (125) and (126) may be quite inappreciable,
particularly near the poles, where the factor sin 9 makes the argument of
the Bessel Function smmall. For by Duhamel's formula we have, when
j is very large -,(eo
tn~)

Jj(jsine)=
This

13 < 1,
sin 6

formula

shows

that the series

(2)

always converge absolutely

that is the velocity less than that of light.

= 1, so that sin 6

=

when

When however $ = 1, and

1, the Bessel Function is ultimately of the orderj'

and the series are no longer convergent.

generally

The polarization of each harmonic vibration is
elliptic ; the
ratio of the amplitudes of the two components in and perpendicular to the

VII

PERIODIC

109

MOTIONS

PERIODIC MOTIONS
0
the equator, where 9 = jwr, the polarization is linear, the electric force
being along the equator and the magnetic force perpendicular to it. Near
the axis, where 0 is very small, the polarization is approximately circular, but
the harmonic j = 1 alone has any appreciable value. The direction of rotation
is that of the electron in its orbit. As 9 increases in passing from the axis
to the equator, the polarization becomes elliptic, the axes of the vibration
ellipse being in. and perpendicular to the meridian; and the perpendicular
axis predominates more and more.
VII] At

84. Problem 2. To calculate the radiation from the ring. Using
(123) and (124), § 82, we find for the mean value of the Poynting vector at
the point (r, 9, ) the expression
Ce2I/2j2

[cot2
c=

0.

{Jj(j$ sin

9)12+132

(128).

{J'(jSsin 9)12].

S2p 2 2
Now we have, by Neumann's Addition Theorem*,
*2

{Jj (x)}2 =

f

Jo(2x sin 0) cos 2j4)do,

whence also
J0 (2x sin

{J! (X)}2 =

4) ycos2-

Putting x =j3 sin 0, we get from (128)
2
J(2j,3 sin sin 0) (/2
= -ce/P~j

.i;

0

cos 2j' d .

cos 2

-

1) cos

Since sj is independent of the latitude, we must multiply this by
2wrr 2 sin 9 d9 and integrate over the whole sphere of radius r, that is from
9=0 to 0= -r.
Now we

have t
fJS rsin9sin9d0=
0

/rJ (r)=sin r

Put r = 2j$3 sin 4); then we get for the rate of radiation of energy on
account of the jtli harmonic
R=fj

.27rr2sin~d9
-2ce2/32j

2

IT (2j(3 sin 4))(2d
sin
'~rp

-

2ce21C3

Jo
j/3sin

L_7

sin

)

(3'cos 24)

it

+t

1) cos 2j4

)4

(2j,8 sin 4)) cos 2j4) sin 4)d4
-(1 -/$2) j2

*

-

f71sin 2j
J

sin cs4)d~

2j sin ~

Gray and Mathews, Bessel Functions, p. 28.
ibid. p.n240, x.55

~

110

PERIODIC

MOTIONS

[cH.

VII

110The first integral is easilyPERIODICbe - 7r J' j (2j3).
seen to MOTIIONS[C.
2

I

The second may be written

fr cos (2jx sin 0) cos 2jo do,
and is therefore equal to 7T J j (2jx) dx.
dx

2

Hence we get
='

j

I

2ce

3 [j/3

j '2
2

'

j2( 1

(2j/)

'-

/3

2x)dx

2

]

The total rate of radiation from the ring of n electrons is found by taking
= sn, multiplying by n2 and summing for s from s =1 to s =n; we get

R

2

2Ce2/3n2

=

2

2 8_

P2

00

8=1

[

n$3n

L

(2sn/3)

82 n 2

(1

-

/32)

.J

2,n

Jo

Ii

(2s nx) dx]...

(129).

j.B)."(

85. The series is easily shown to be convergent when < 1. Its terms
are of necessity positive, since every term such as R, is by (128) the sum of
two squares. Hence
is less than n2 tines the value for the case n
But in this case the series can be summed.

R.

= 1.

In fact we have*
S=°°

J28(2sx)

S=1

S2

2

= Ix

2

.

Using the differential equation for the Bessel Function we get by
differentiation in succession
s=x
=0
x2 (1+x 2 )
sJ,(2sx)=
I
(2sx)=
8

Y

8=1
)J

8=1fo6(1

Putting n

= 1 and

s8=1

2
-/32)3"

substituting in (129), we get for a single electron
1? 2ce2 ft(3)
R 2 (1- /2)2 .. ..........
.. 13

)

n2

Thus the series (129) has a finite sum, certainly less than
times this
amount ; actually it is very much smaller.
The expression (130) agrees with the value of the radiation given by
Lie'nard f.

[In strictness the sum of the Kapteyn series used above has only been
found when x < 0.659..., but the agreement of the resulting expression (130)
with Lienard's expression leaves hardly any doubt as to its correctness up to
the limit x =

1.]

*

Nielsen, Cylinderfunktionen, p. 303, 1904.
± 'I lairage vletriu,
July, 1898, q.n(1).

CHAPTER VIII
ON THE DISTANT FIELD DUE TO A MOVING CHARGE

86. THE number of cases in which tractable expressions can be obtained
for the potentials is exceedingly limited, unless the generality of the problem
is circumscribed. If however we limit ourselves to the calculation of the
field at a distance from the orbit, which is large compared with its dimensions, all the expressions become very much simplified. In this way we may,
for instance, calculate the field for polyperiodic and even for some aperiodic
motions. As it happens the only direct experimental means we have of
investigating the motion of charges in the atom is by means of the spectroscope, which enables us to resolve the distant field due to these charges into
its components and to a certain extent to determine their relative intensities.
From these experimental results it is required to infer the nature of the
motions which produce them. Thus the study of the distant field due to
prescribed motions of charges of the most general character is one of the most
important problems we have to attack.
We commence with the general expressions of ~9,
r--~

.(t-R/f-7)

dd

where we have replaced fde by e, since the linear dimensions of the charge are
exceedingly small compared with the distance R.
We now introduce the approximation of § 72. We write
R=r-p,
where p = (r 1 . p), and is the projection on the radius vector, r, to the fieldpoint,
of the radius vector, p, to the charge. Thus, neglecting p in the denominator R,
but not in the exponential (since it modifies the phase to a finite extent, but
the amplitude inappreciably), we get
S=

~

f

E(t-r/IC+P/c-T) d'd

a.... . .(t-r.c.+-.e-r)
2rcr _-

....

. . .. . . .

(1:31),

v d.rd. ............ (132).

112
ON

112

THE DISTANT

FIELD DUE

TO A MOVING CHARGE

[H

[CH.

87. Using equations (VII) and (VIII), § 3, we get the electric and
magnetic forces. We must remember that we need only
exponential, with respect to t or r, because the differentiation of the factor
1/r leads only to terms of order 1/-2, which we have already neglected. Hence
we may write

differentiate the

grad. = r1 . ~--= curl

d = -grad. 0

-

=

r,.

,

1
fi.
arI -ri,~i

Saa

c_
cat

=
h

=

2?

fe

---

.......
(133),

L~~fhwc?"/c7)(r,--)d'rd

a

v~I,

curl a
c.

rf

Ja~~t

v r,] drcp

rcPcT

......... (134).

It is to be noticed that these expressions for d and h are based merely on
our knowledge that the integrals involved in (131) and (132) are functions
of t and r only in the combination t--r/c. They do nut assume that
differentiation under the sign of integration is allowable, for the
are to be evaluated before the integration is performed.
It is well known that if the integrand is a discontinuous function of T,
differentiation under the sign of integration is not generally allowable *. In
our problem the integrand is a function of
and implicitly, the
latter through p and v. If the motion be continuous in foam, that is to
if the differential coefficients of all orders of the radius vector p be finite and

integrals

explicitly

continuous, as in the case of periodic motions,

of integration is allowable to every order.

differentiation

say,

under the sign

If however the acceleration, or

any differential coefficient of higher order, be

discontinuous, as in

the case of

missible, and the discontinuity produces an effect which must be allowed perfor.
In our case, where the radius vector p and the velocity v are both continuous
of necessity, the integrand itself is always continuous. Hence only
one
differentiation under the sign of integration would appear to be allowable.
a motion starting from rest, unlimited differentiation is generally not

88.

Polyperiodic motions,

We

shall begin with the case

where the

viii]
ON THE

VIII

DISTANT FIELD DUE TO A MOVING CHARGE13

We may write

=

#a, sin

(wir +

113

ai)

19).................(135),

l= bisin (T +
= ci sin (wiT + 7)
where (4:, n
are the coordinates of the charge at the time T, and therefore
the components of the vector p.
Let (1, m, n) be the direction cosines of the unit radius vector r1 drawn in
the direction of the fieldpoint. Then
p

=(rl

.p)

If:

+

mr

+

inC

i

+Si

where
pi cos Si = la cos ai + mbi cos /i + nc cos.ry..(136).
pi sin Sj = lai sin aj + mbi sin /3i + nci sin yi
Also we have for the components of v

S=

wiai cos (wiT + ai)

wiiCS(T+a)................. (137).

=
wibcos (wOiT + i)J
These expressions are to be substituted in equations
§§86,
87, and the resulting terms developed. As absolute mathematical rigour is
not of prime importance we shall employ symbolic methods of development,
leaving it to the future to supply a more rigorous treatment.

(131)-(134),

89.

Potentials.

We have to develop exp. t

(t

r/c + p/c

-

-

T).

By Taylor's Theorem we may write this in the form
ePD exp.

(t -

tiL

r/c -

where D is written for the operator a~ and is to be treated as if it were an
algebraic constant quantity.
Now we have the well-known expansion
8=00
si
=ZJ
-

8 (x)~~

8=- 00

Hence we find, using (136),
ED=1I Ep4D.

sin

(wir+s;) =

8;=

Hi

00

J8, (- tpiD) E&8(c)$r+ 84)

00
where

f2 = s81w s 2 w 2 +
1 +

...

+ s8eoi,

818=+ 8282 +
1

...

+ .si~8...(138).

Substituting this symbolic expression for epD in (131), we get

95e 1100
-7r
00

J 8 (- tpiD)

f

:

0

eL; (t-rc-)+lT
+0j

drd. .

114

114

THE DISTANT FIELD DUE TO A MOVING CHARGE
~ON.

[CH.
[c.

The justification lies in the fact that we may develop the Bessel Function
operator in a series of powers of D, that is of a,

and then interchange the

order of the integration and of the differentiations to any order; for since the
integrand is a continuous function, differentiation to any order under the sign
of integration is permissible as we have already explained.
The integral which occurs in

0

is a Fourier Integral of the usual type,

and its value is 27rESID(t-r/c)+o}.

Hence the differentiation can be performed by the substitution of the
quantity tl /c in place of the symbol D; thus we get

INJ

=2...

8

j(Pi)Et{(t-ic)+o}

dJg

(139).

s,

s2, ... si are each to be
can easily be written in a real
form; for if (si, s2, ... si) denote any set of values of the indices, the sum
The i summations with respect to the i indices

taken from - oc to + oc.

0

The value of

also contains a term (- S1, - 82)... - si), and we may group each such pair of

terms together. The values of 1Z and S for the second tcrm are equal but of
opposite sign to those for the first, that is they are -11 and -. &Any
Bessel Function, such as JS, (Lni/c), in the first term is replaced by J_ 8, (- fpi/c)
in the second; since both the order and the argument are merely changed in
sign, the value of the Bessel Function is unaltered. Thus the factors of the
exponentials in the two terms are equal, and the sum of the two terms has
merely 2 cos {fl (t - r/c) +
in place of the exponential. Hence we may also
write

Al

eY.

r

F ()cos{11(t-r/c)+

-

8,

A .....
(140),.

where for convenience of notation the summations from - oo to +
retained.

Xc

are still

The expression for the vector potential is easily found in the same way.
For instance the component ax involves the factor t under the sign of integration ; but by (137) this may be written
=

Y;

&6

(wiT+ai)

+ E-

t('~i

Hence in. the present case the symbolic operator
operates on the function
Y{

,)ji J,6&{E[(

+wi) ( - r/c)+A

F+
a6+C[(a

- w')(t -r/c) ±+A-

ail

VIII

viii]
ON THE

DISTANT FIELD DUE TO A MOVING CHARGE15

115

Thus we get in place of (139)

00

_ e

$

..

(f+

-00

+

0

o

(+wj)
3

F(1G

181Y"82...N4,

-

(t-r/c) ±O+ai]

E&[(f-w) (t-rC)+

w)

This expression may be simplified.
In the first sum replace si by si - 1.
This does not alter the limits of the summation with respect

still extends from - oo to + 00.
The value of
1+ wi= 1 W 1 + 2 2 +..."+ (Si +1) wi
is changed into
f= slwl + s2w'2 + ... + SO*
The value of A is changed into A -8i

8

ai]}.

to 8i,
which

8

In the function F(f11 + wi) the factor J,, {(1i2 + coi) pi/c} occurs ; it is
changed into J8;- (f~pi/c) ; every other factor, such as J8 , (fl + Wi) p1/c}, is
changed into J81 (fp 1/c), its index being unaltered.
Thus the first sum in the bracket becomes

2,8)J

J

-001

+A+ai-a].

...

CCp)JE[(t-r/c)

Similarly, in the second sum we change si into s, + 1, and thus get in the
same way

The sum of these two terms involves the factor

4 -1
8

')

e''c

+J8

dpi

This by the properties of the Bessel Function reduces to

2csi J8"(LC,i )cos (ai -

S
i)+

2 J7

(S
'

lp i(-

8

)"

Substituting in the expression found for az we get
e
a =-

00
...

$

J
t(-rc}siwiai
(fl) e~lt/)~

cos(ai-)

+ c

caJ (I2pi/c)
J8

By taking together the terms with the set of indices (si, sz ... Si) and
(- sl, i ... - si), and remembering that J_8 Z(- &lpi/c) = J8z(&2pi/c), while
J'-8 , (- fIpi/c) = - J' 8, (&2pi/c), we get
0..iF[lsi
e 00

wiai cos
(Ii/)sin
aT'
eii (S.j

wiaiJ';

(ai

(a,- -

i) cos

- (Si) sin

/c).
0() ! _(r"1

{1(

I

/)+t

11(t - r/c) +

A }]

(11

116
ON

116

[c.

THE DISTANT FIELD DUE TO A MOVING CHARGE

Similar expressions hold for ay and
ai, ai.

a,

with

[CII.

and ci, yi in place of

bi, 83

90. Electric and magnetic forces. The expressions for the electric
forces may be found in the same way from (133) and (134), or more simply
from (140) and (141), by the equations

= cat(1

d

l=

a

.

-

. r1.

cat[r]

Remembering that the components of r1 are
(138), (140) and (141)
d =

~)

1...,F

LSii

(ai cos (aj -

8)-

(1, m, n), we get at once from
lpi} sin {f1(t-r/c)+AJ

+ho J'8 , (f~pi/c) ai sin (a cos {fI(t - r/c) + A }
+
f~/)
j.
0 2J
e

r

0
oo

x[sico {mc 1
+
floc

fl

c2Jsl

cos (yi

-

8i)

c

-

sin

nbi cos (8i -

j

(ryi

.. (142),

(fnipi/c)

8i)}
s

sin {2 (t

(3

r/c) + 0}

-

(t

-Scos{i2

-

r/c) +

c 2 JS(ILp2/c)I

.........
with similar expressions for the remaining components.

z}

(143),

The process employed in deducing these absolutely general expressions
for the field due to any polyperiodic motion whatsoever cannot claim to be
rigorous ; but it is extremely difficult to deduce them in any other way.
There is however no reason to doubt their substantial correctness, so long as
the series remain convergent. We shall verify below that they give the
expressions (123) and (124), § 82, for the forces found by a rigorous method
in problem 1 of Chap. VII.
91. Character of the distant field. We shall now discuss the
character of the field represented by the expressions (142) and (143). We
easily deduce the following conclusions :
(1) Each force component consists of an infinity, of multiplicity i, of
simple harmonic vibrations.
The type of their argument is
sLi=S1WI1±+8
2

fl

(A)2+... +jCO,

(t - r/c) + 0, where
A= S1 81 +S 28 2

±...±+Si~i.

Thus the frequency of each harmonic is a linear function with integral
coefficients of i fundamental

frequencies

(o,,

w9,

...

w,.

The fundamental

viii]

viii]
ON THE

DISTANT FIELD DUE TO A MOVING CHARGE17

117

frequencies are those of the harmonic constituents of the polyperiodic
motion which emits the vibrations; the vibrations are as it were sum and
difference harmonics of the fundamentals.
(2) Each harmonic component vibration, e.g. that of frequency l,
consists of i component vibrations, each of which can be resolved into two
sin
components proportional to n{fl(t - r/c) +
cos

Al.

In dx the amplitude of the sine component is of the type
e

{ai cos (i
cp2

FO)sioi

-

r

i}

-

and that of the cosine component is of the type
e
r

F(S)

1aoi J' ( flp/c) ai sin (ai 2J, (fp/c)
0

Now we easily find from (136), § 88,
pi = lai cos (ai -

Si)

+ rnbi cos (,8i

-

Si) + mbi sin (3i -

0 = lai sin (ai -

S)+ rci cos (yi-S
Si) + nci sin (y,

-

Si).

Hence we see at once that each of the two components is perpendicular
to the radius vector (1, m, in).
In the same way each component of hx is perpendicular to the radius
vector (1, m, n), as well as to the component of the same phase in dx.
Thus each component harmonic vibration is transverse, with electric and
magnetic forces at right angles.
(3) Choose (1', in', n') and (l", in", in"), any two directions at right angles
to each other and to the radius vector (1, in, n) and forming with it a righthanded system. Let the ith components of the electric and magnetic forces
of the harmonic fl be d2',
and
hi" in these two directions.
We easily

find from (142) and (143)
i= h"

dil=-

=

hi'

hi'

di",

2e F (n) sZ"[Ci'sin ff1(t- r/c)+A}
r
epi
+ kiSi' cos
=

2

e

F(fl)

~[Gi"sin {fl
pi,

(t

-r/c)-

Jff
(t

+ k2Si" cos

f{1 (t -r/c) +

= S=
l'aicos (cai - Si), 5 Si'=Sl'a sin
0/' = S l" ai cos (ci - Si),
= S l" ai sin

-cs;J

8 , (flp;/c)

r/c) + A}].. .(144),

+Al}

where

Si"

-

] .(145),

(ai --Si)

(oq
-...

(16)

118
ON

118

THE DISTANT FIELD DUE TO A MOVING CHARGE

[H

[ca.

As before we have

F(fl) =J 81 ~
pi = S la, cos (a. -

fl

8i),

= 8~w) + 82w2 + ... +

J2i p%
(

...

0 = S lai sin (ai siwi,

a0

s"1+
8

p ............ (140),

...
8

i),

8282

+

by (136),
...

+ s8..(138).

S denotes a sum of three terms, one for each coordinate.
The factor 2 is introduced in order to take account of the fact that to each
term of (142), or (143), with the set of indices (s1, s2, ... s2) there corresponds
an equal term with each index equal, but of opposite sign to (s1, s2...si).

There are i terms of the type (144), or (145), one for each of the i
component simple harmonic motions which constitute the motion of the
charge. These together give the harmonic vibration of frequency fl under
consideration.
They have a common factor 2eF(fl)/r, involving the product of i Bessel
Functions given in (140).
Now the wave-length X of the harmonic is equal to 2wc/fl, so that the
arguments of the Bessel Functions are, in order, 27rp1/X, 2wp 2/X, ... 2npi/X.
Since pi, P2i ... pi are of the order of the linear dimensions of the orbit, we
see that if X correspond to any observable line of the spectrum, each of the
arguments is exceedingly small. Thus the harmonic is very weak unless the
indices

81,

s2, ... si are small.

We cannot expect a visible line to be produced unless this be the case.
(4) Eq. (144) shows that each component of the harmonic is an elliptic
vibration. By the ordinary formulae we see that the axes of the ellipse are
equal to
2
2eF(12) sio.,,(C'2+±ki2S! )
(12)

ad2eF

and()

cpj4/

V(EP 2

2"s/

k 2S")

rcpi

}1

......... (147),

and that the first axis is reached from (1', in', n') by rotation towards (1",
through the angle 8i, where
tan
6,
.2
(0/0C,"+ kt2S'Sti")

in", n")

2
tan -//28i
0Z/2 + k, ("2
S,12)............(~148).
in the ellipse is right or
iy
rotation left-handed according
LZThe
as
Ci'-

(5)

C,'>

or <

0.

It is worthy of note that the vibration cannot be linear unless either
Ci

i'

=0,or

Ci"=SZ"=0, or ki=0.

viii]
ON THE

VIII

DISTANT FIELD DUE TO A MOVING CHARGE19

119

The first case gives
1' m'

n'

=

bic sin (/3

yz)

-

ciai sin

sin (ryi - ai)+sin

ai

-

+nn' = 0 only if

which is consistent with the condition II' + mm'
sin (,8i - yi)

sin (a,.

-):aib

(yi-

(

b

-

=0.(149).

c

...

The second case is the same as this, merely with (1",
(1', m', n'), and thus is not essentially distinct.

i",in")

in place of

(149) is the equation of a plane through the origin; for every direction
(1, m, n) lying in this plane the elliptic vibration becomes linear.
This plane of linear polarization is generally different for each component
of the harmonic.
The third case occurs whenever Q2pi/c, that is, 27rpi/X, is one of the roots
of the equation
8=
()..= 0........
.....
..............
(150).
Now by (136) we have
2
1a2 + i 2b.2 + 2c-2+ 2mnbiqc cos (/3i - 'y) + 2n1 ciai cos ('yi - ci)

1

+ 2lmaibi cos (ai

-

8)=pi 2 =

(151)

This is the equation of a quadric cone on which the radius vector (1, m, n)
must lie in order that the root x of (150) may correspond to linear polarization.
Since the least root of J',, (x) = 0 is of the order I (2sf + 1) 7r, the condition in this case cannot be satisfied for light waves, for which 2^rpi/X is
always very small.

Hence this case is of no practical importance.

We shall not consider the general case any further, but proceed to the
study of some illustrative problems.
92. Problem 1. Uniform circular motion.
the last chapter we get by (135) and (136), § 88,

'q=psin(ov+8),

=pcos((or+8),
giving

Also

wo

=cw,

a = b = p,
1

1=sin~cos0,

Hence

c 1 = 0,

'=0,

a1=b+j17r,

m= sin~sin4,

p=iP11,

With the notation of

#1=&S

n=cos9.

"1=6-4+2-r.

Thus there is only a single component vibration.
In the equations

(138)-(140), § 89, we have
S21=seo,

F (12)

=

J 8 (s,89 sin 0),

A=s(6-

+j',r),

kcF (12) =/3 sin

0J4' (s/ sin

0).

120
ON

120

THE DISTANT FIELD DUE TO A MOVING CHARGE

[H

[OH.

0;

We shall choose (1', i', n') in the direction of increasing 0 and (1",
in the direction of increasing
hence
l' =cos0cosc,
m'=cos6sin 4, n' =-sin,
l" = -sin4, m"=
cos b, n"=0.

in",

n")

With these values we get from (146)
C'=pcos9, S'=0, C"=0, S"=-p.
=

Substituting in (144) and (145), writing

w

(t

-

r/c) + b

)+ fi, and

-

summing for s, we get
2e$ cot
pr

SssJs
do

=

2e

- h =

soi
.=1
C2

s=

3

pr

00
sJS' (s/3 sin 9)cos

s=i

which are identical with (123) and (124),
tion of the general formulae.

93.

Problem 2.

(sf3 sin 9) sin s-4r,

§ 82.

si

This will serve as a verifica-

Simple harmonic rectilinear motion.

Suppose Oz to be the line of motion, and let
=a

sin (OT +

With the usual notation we get by (136),

).

§ 88,

p1=acos9, 81=a=y1.
By (138) and (140), § 89, we have
Sil=so, 0=sa, F(1)= J(s$
where
light.

$

=

cos0),

ao/c, so that 3 is the ratio of the maximum velocity to that of

Take (l', mn', n') and (l", in", n") in the directions of increasing
before ; then
1'=cos 6cos 4, in' =cos 6sin 0, n'=-sin96,
l"= - sin4, m" =
cos4,n"0.
By (146)
C'=-asin9, S'=0, C"=0, S"=0.
Hence we get from (144) and (145)
2
e8tan
d 9 =e h 0 =0ar

=

9

8
_

9

/
8\

4) as

}...

F1I10c
L

9 and

(152)

I

Thus the electric force is in the meridian, the magnetic force along the
parallel. The vibration is linearly polarized at right angles to the meridian.
The amplitude vanishes on the axis for every harmonic.
For all harmonics, except the
when s = 1 its value there

is

first, it also vanishes

e12/ra,

and is a maximum.

at the equator ; but

VIII

VIII]
ON THE DISTANT FIELD DUE TO A MOVING CHARGE12

0)

121

Since j9 < 1, J8 (s$ cos
can never vanish, and is constantly positive or
negative according as 0 lies between 0 and 2 w, or between
and 7r,
when s is odd, while it is always positive when s is even. Thus the product
tan 0J3 (s/3 cos 6) is always positive when s is odd, but is positive or negative
according as 9 lies between 0 and
or between 17r and 7r, when s is even.

j 7r

17r,

Hence for even values of s the electric and magnetic forces are of opposite
sign on opposite sides of the equator, the electric force in particular being
towards or away from both poles at the same time. For odd values of s they
are of the same sign, so that the electric force is from pole to pole, in the same
direction on both sides of the equator, but reversing every half-period. The
only node occurs at the equator.
94. Radiation. The average value of the Poynting vector due to the
harmonic s is given by
tan20B

ce22

ce/ ta 2
2 {J8 (s$
cos8)}2r
27ra2r
The radiation due to it is equal to
R=

I {IIs cos 9)12tan29 sin
J (s
a Jo
ce 2 I 2s 2 1{J3 (S, )}2 (1
I'"2)
dpt

OdO

a2
rL 2

It does not appear to be possible to find this integral in finite terms, but
it is possible to find the total radiation 1. We get

Rs
e/

2

f1

8

J 8 (,.)}2)

(

)d......6(153).

We require the series of Bessel Functions.
Nielsen gives the following series*
sr {J
({ '
=i

2

By differentiation we get

S

s=1

=- fx,

s8

[J3 (sx) JS" (sx) + {JS'

(sx)}2] -

44

Now by the differential equation for the Bessel Function we have
(1-

X2) J 8

(sX)

=J

J8(sx)(x

Hence, multiplying by J3 (sx) and summing, we get
l_ 2 s ms s)2=
x
J
JS (sa;) J" (sa;) +
3(X)J s;
sa)J'
-=0

S=

1

8"1

:
*

Cvlinderfunktionen,

p.

307,

eq.

8

(8) (with v =0,

p= 1).

12
ON

122

THE DISTANT FIELD DUE TO A MOVING CHARGE

Again, multiplying by 2sx 2 JS (sx) and

by the second equation above.
summing, we get

=

-

[CH.
[n

)}2

2

=d

d

s=1

J(sx2.

s=1

Multiplying the previous equation by x2, differentiating and adding to
the last, we get
00
2
1d2
{Js(sx)}2 - 2xY.{JS (sx)}2 = x
2(1 -x2dx
S-=
d

s00

_

s_1

by the third equation above.
Integrating the last equation and remembering that the sum vanishes
identically for x = 0, we get

8

1

J (sx) } 2 =

=1

1

1l

-1 ,

The same process gives
16 (1X2)/2

s=

Substituting in (153), we get
R-ce2/34

(1

16a2 J_ 1
2

4

ce 3 (4
12a 2 (1

2

( 4 +$
(1

)(1-L2)d

-

$/2)7/2

3/2)
-

/2)2.................................(154).

find

Remembering that $ = coa/c = 2wa/X, we
that this reduces to
167r4ce2a2/3X4 when /3 is very small. This is the value given by Larmor
for this

case*.

The value (154) may easily be obtained direct from the general formula
of Lienard, which for rectilinear motion gives for the rate of radiation
2ce2i 2

Substituting v/c = /3 cos (cot + ae), v)/C2 = - /32 sin (cot + a)/a, and integrating
with respect to t, we get (154). This will serve as another verification of the
general formulae (144) and

95.
write

Problem 3.

(145).

Elliptic motion about the centre.

=acos(coT+a),

f9=bsin(COT+a),

Hence there is but one period, so that
w 1=wo,
a1 =a, b1 =b, c1 =0,,
*

'=O0.

=ca+jw,

Aether and Matter, p. 226.

/31=az.

We may

VIII]
ON THE DISTANT FIELD DUE TO A MOVING CHARGE12

VIII

123

By (136), § 88, we get, using polar coordinates (r, 9, )
sin 9 (- a cos q sin a +bsin bcosa),
psin3=sin9( acos cosa+bsin4sina).
P, cos 8,=

Write

a cos 4 = p cos f,
p = (a2 cos2

so that

+ b2 sin2 ),

tan#=btan0
a
81=a-fr +fi

p ,=psin 9,
1

Then we get

b sin)=psinf,

By (138), § 89, we have
f2=sw,

Choose (1', i', n') and (1",
then we find by (146)

A=s6=s(a-4r+jw).

n") in the directions of increasing 9 and

in",

-(a2-b2)sincos4

"

C,'=pcos9S,
'=0,

0;

-ab

,op sin J'S(swppsin 9/c)
cJ, (sop sin 9/c)

=

With these values we get from (144) and (145)
d 9 = h=

2eco cot
c

2

sJ (swp sin90/c) sin s [w (t-r/c)+a-+i]...(155),

2eo2ab8m
a
sJ' (sopsin/c) s
cos

- h

dr=

8=00

t

2eo(a2

c2pr

[

t-rc+a-+i]

S=1

0

- b2)sin
cos4
cprsin9

sJ(ssin/1c)sin

_

s

(P(

[co t-rc)+

a-

/

+

17r]

................ .........
(156).
It is easily seen that, when a = b = p, these expressions reduce to those
for the circle, (123) and (124), §
when b = 0 they reduce to (152), § 93,
provided we make allowance for the difference in the coordinates used.

82;

(k)

In the diagram ABA'B' is the ellipse and ACA'C' its eccentric circle.
P is the projection of the fieldpoint (r, 9,
on the plane of the ellipse, and Q is thecP
point for which the eccentric angle is 0.B
Then p is the radius OQ, and f is the
angle

xOQ, which OQ makes with

Ox.

*f;

In (155) and (156) we may if we wish
express everything in terms of
we easily
find

-

ab

A,/(a
2

sin2 *I + bcos*)

sin fcos 4- sin 4,cos 4
p2
ab

A

O
Fig. 37.

124
ON

124

THE DISTANT FIELD DUE TO A MOVING CHARGE

[H

[OH.

We notice that along the axis, where 9 = 0 or r, we have
d = heepsin

[ (t - r/c) + a eo2 ab

=e- h=-

e

4

+

i]

cos [t(t - rlc) + a -

+i

eo2 (a2 - b2) sin k cos c sin [w (t

r/c) + a -'

-

c2 pr

, which would be absurd; but we

Apparently the forces depend on
easily find that
dx= de cosck -

dosinf=

sin[w(t-r/c)+a +ir],

2

2r

2
ew b
2 cos
cr

d,=de sin 4) + docos=-e

+ j?].

[co(t-r/c)+a+4r],

that is to say
= ew

where

2

(t2- r/c)
cr
'

(-

d=

c2r

y

(t - r/c) and rq (t - r/c) denote the coordinates of the charge at the

time of emission, t - r/c.

Thus on the axis the electric force is always in the meridian plane drawn
through the charge in its position at the time of emission. The change of
the electric force is exactly proportional to the change in the radius vector to
the charge.
This explains the apparent difficulty.
On the equator, where 9

do~hoO, d=

do

-

0e

_2eo,

2

=

17r, we get

= _2ew2ab S-s'spcsswt
c4=h
2
=, 8
sJ

2

(a - b ) sin 0 cos 048
2

cp r

K

10

r

cos[

J(swoP) sin s[wt - rc)+
sl

c

]

(t-/)+-+

a-

+k7].

[w(

The vibration is linear and polarized in the meridian.
For every other direction the vibration is elliptic.
96. Group of charges. A number, n, of charges describing the ellipse
in succession form a group, that is, partially absorb each other's radiation,
provided the epoch a differs by the angle
from one charge to the next.
For in this case, on summing for all the charges, all the circular functions in
(155) and (156) disappear, except those for which s is a multiple of n.

27r/n

FIELD DUE TO A MOVING CHARGE12

VIII]
ON THE DISTANT

VIII

125

Thus the forces due to the group of n charges are given by (155) and (156),
provided s be replaced by sn, and the expressions be multiplied by n, just as

in § 82.

r;

WT

Now the angle
+ a is the eccentric angle of the charge at the time
hence n charges moving in an ellipse form a group provided their eccentric
angles are always in arithmetical progression.

97. Radiation. We see from (155) and (156) that the average value
of the Poynting vector due to the harmonic of order s is given by
sSi-

_c(d02+d402z)

47
2 2

s

(a

oete
[(t29+

2'n-cr2 Li

2

2 2

) sing 4)co&t4))fj(8cp sine
2
8
c
+ b Sia2 4))2

-b

4

sin2 9 (a 2 cos 2

{j.' sWp sin 9)}2]r,

2
+ W a2b2

[cot2+

e2 W2 s2f'r

(a2
0

in2
S,=

+

2

02
2
W aa

(a2 coS2
cos

C

2
b sin2

-

4))(\

J,("

's

)

in

b

coS24

\

sne )

}JsPn

sin)2

b2)2Sin24)coS24
(a2

2

4

j2sin 9dedc .. .(157)

2

There seems little hope of evaluating this integral; but we can sum the
series for s from 1 to oc and thus get an integral for the total radiation R.

In § 94, p. 122, we found
we find

2

)}2.-

SIS2J8

(

2)

=1Similarly

=i{J8' (Sr)}2 =-14 + 3x

2

1(1

s=1

Put x

R

=

5/"

)

op sin 9/c and substitute in (157); we get

= I R

f2vrf 2,{p2

3

[

coS

2

2 2
W p Sin2e

F+

0+

(a2

b)2srn20coS2 }0

4+
(i

2W

-

4 3

w2p 2 Sin2

a2 b2

+wher

p2
P2

where

-

a

(

4)+

b

cos

2C80+

2

si2

21120

( 2
O

)

2
2

p

9

2
sin2 )

]

2

Sin 2 9)7/2

sin

9ded4),

12
ON

126

By means of the substitution op cos 9 = tan x /(c2
some transformation

±7w4 a2b2

' _ W2(a2+b2)
Re2

[H

THE DISTANT FIELD DUE TO A MOVING CHARGE

(i-

2
C

( p ), we get after
2

2

c 7 0 2 (a + b2)
c2

c4

c2

w22

-

[cg.

pj1

-

d

__d2

This gives on integration
2
e2w2 4 Go 2
(a

ewC2

w42b2
+ b2) + '3+

(/2

a

12c (i-

(A2

4
(a+C

3

4

2b
3 2

1
i-

3/.

2

.. (158).

c2b)/

When we put a = b = p, wa/c = wb/c = $, this reduces to the value for the
circle, (130), § 85.
When we put b = 0, it reduces to the value given above for the rectilinear
vibration.
Thus there can be no doubt as to the correctness of the result, which
may indeed be deduced directly from Linard's general expression for the
rate of radiation*
4
'2/4
"
+
2
ft= 2ce 2- V4/0
V2/C2)2 (1 - V2/C2)3
_____________

where p is the radius of curvature of the orbit.
Thus we have another verification of the formulae (144) and (145), § 91.
98. Problem 4.
An electron is disturbed from its state of
uniform circular motion. To
the
produced.

find

field

We shall use the notation of problem 1, § 78.
We shall use the method of representation of Maxwell, invented for the
problem of Saturn's Ringsft.

Write

=p (1 - ) cos (wr+ +X),
1= p(l - t)sin (cO'±8±X)
=

pv.

Thus (X, , v) measure the component displacements of the electron from
its position in steady motion, in the direction of the tangent, inward radius
and axis respectively._
*

L'E~clairageelectrique, July, 1898.

t Collected Papers, vol.

i. pp.

See also Searle, Phys. Zeitsch. 9, p. 887, 1908.

321-323.

viii]
ON THE

viii]

DISTANT

FIELD DUE TO A MOVING CHARGE12

127

We shall suppose them to be small and shall neglect their squares and
products in the usual way. Then we get
= p 1(1 = p {(1 = pv.

) cos (wr +8) - X sin (or + 8)},
sin (w7 +8) +Xcos(W + 8)1,

/)

We shall suppose (X, , v) to be sums of periodic functions of T, in
accordance with our general assumptions for the present chapter. Thus we
write
=
~A sin(qTr + a),
= B sin (qT + 3),

A
v =

1,C

sin (q'T + y).

The frequency is relative to the ring, not absolute.
With these values we get

VIA

2+B2 +2ABsin(a-/)} sin {(q+(0) 7+ 8+r+r}
±+rr) +M p
= p sin(on++
2
2
+ p /{A + B - 2AB sin(a-,e)} sin {(q w)r-8+ v -

fl=p sin (co +

8)+

lp

+ Yip
= pC sin (q'T + 7),
where

V{A 2 +13+ 2AB
V{A

2

+B32

-

{(q +c)T+8+}-}

2AB sin (a-18)} sin {(q -(0)

Asina+Bcos$
Scos a - B sin

tan

sin(-)}sin

T-8+ v},

Asina-Bcos/3
tanvA cosa+Bsin

These expressions are of the form (135), § 88, where we must write
w1=w, a1 =b1 =p, c=O, a= 8 +1.
i8=5
o 2=q+ w, a 2 b 2 = ipV{/}A2 +
2AB sin(a fl)}, c 2 =0,
eo=-a
I 2 = 8 ZT'
2 =8+'+jr,
159)
(03q-,as=b =ipVtA2+B
2~i~
I),c
3 =0,

=

w 4 =q', a 4 =b 4 =0, C = pC,
4

B2+

/748+ti

Thus we have to deal with a preponderant uniform circular motion
(index 1), compounded with a number of triplets of small harmonic motions,
in general of incommensurable periods.
Two of the constituent motions of each triplet are in the plane of the
undisturbed motion, and are sum and difference motions of the uniform
circular motion and the given relative disturbance. Their amplitudes and
phases are different, the amplitude B of the radial disturbance appearing
with opposite sign in the two.
The third constituent of the triplet constitutes an independent motion of
the same
as the or'iginal relative disturbance parallel to the axis.

frequency

In general the quantities q and eo are incommensurable, so that the same
is true of the periods of the constituent motions.

128
ON

128

THE DISTANT FIELD DUE TO A MOVING CHARGE

[CH.
[H

By means of (136), § 88, and (159), we find
pl=psin,
p

8

1 =a 1

2 =aasin,

-4

-0, 2k.......(160).

p 3=a 3 sin0, 83= 3
p 4 =C cOS0, 84=7
4
By (138),

§ 89, (159) and (160), we get
L= ( 1 - s2- s 3 ) W ± (82 +83) q+s 4 q+
S=

- (s+

+

+ sc3 +

2-) 83)

sa+83 +84y+(6

}.

(161)

99. Electric and magnetic forces. We must now consider the
values of the electric and magnetic forces given by (144) and (145), § 91.
In the first place we must examine the function

F(11) = J31 (&1p1/c) JS2(p

2/c)

J83 (f p 3 /c) "?4 (fp

4 /c).

The value of J81 (flp1/c), that is, J,, (flp sin 0/c), is for the present to he
regarded as finite, not very small, since we wish to limit ourselves in no wise
as to the possible values of the velocity of the charge.
The values of JS, (f1p2/c), ... on the other hand are small quantities of the
2
2 8 , ..., since p2, ... are all of the order of the amplitude of the disturbance.
Hence we must reject all values of 82, ... greater than unity, and replace
functions like Jo (f2p 2/c) by unity, and functions like J1 (fp 2/c) by f~p2/2c,

ordersp

neglecting higher terms in each case.
We shall for convenience omit the index 1, writing s instead of sl in future.
Thus we must consider terms of two types in (144) and (145):
(1) Terms for which
1 tocc.

82, 83,

...

(fl)

are all zero, while

8l

takes any value from

cap/c

In this case F
= JS (s$3 sin 9), where /3 =
as before. These zero
order terms represent the field due to the undisturbed uniform circular
motion, and have been fully considered in problem 1.

(2) Terms for which one of the indices 82, 83, 84, ... is either plus or minus
unity, all the rest being zero, while sl takes every value from - on to cc.
These terms represent the first order effect of the disturbance, and are the
only ones to be taken into account.

,

82 = ± 1, 83 =84 = ... = 0, and to 83 = ±
0, represent the effect due to orbital oscillations, that is,
oscillations in the plane of the orbit. Those corresponding to 83 = 83 = 0,
84 = + 1, represent the effect due to oscillations parallel to the axis, i.e. axial
oscillations. These are quite independent of the orbital oscillations, so far as
the field due to them is concerned. We shall consider the two types of
disturbance separately.

The terms corresponding to

82 = 84 = ... =

viii]
ON THE DISTANT

viii]

FIELD DUE TO A MOVING CHARGE12

129

100. Orbital oscillations. In equations (142) and (143), § 90, terms
occur for which 82= ± 1 and 81= ± integer, and for which 83= + 1
s,= ± integer. These group themselves in two sets, as may be seen from
the following table, which gives the frequencies 11 and epochs 0 for the
typical terms of each set.
83

+1

+(8-1)

0

+(q+so)

0

-0
0
0

-(8+1)

+1

A

+{s (a-P)-a1+a2

-- (q+sw)

82

s1

-{s(al-4)-a+a2}

-1

+(q+&o)

+{s (al-

)+a1+a3

-s(a1-0)+a1+a3}

+1

(s-1)

0

+(q-sco)

-{s(a1-0)+a1-a2I

-1

-(+)
+(s+1)

0

-- (q -sco)

+ {S(a

0

+(q

0

+(s-.1)

+1
-1

1

-95)+ai-a

2

}

-(q-o)

-S)

The values of 12 and of A are given by (161).
An inspection of this table shows that there are four terms in the sums
(142) and (143) which give the same frequency q + sw, though with different
epochs, namely those corresponding to the upper half of the table. Similarly
there are four terms giving the frequency q - 8( belonging to the lower half
q, but the two
of the table. When 8=0, these terms all give the
halves of the table being identical for this case, we get four terms for this
value also. Hence to each frequency of the form q + 8cc, four terms
correspond for all values of 8, positive, negative or zero.

frequency

Let us consider the terms of

frequency q +

In equations (144) and (145) the terms

8cc.

8i1=8s-1, 8 = 1,

and 8s1=

-

(s

-

1),

1 are taken together, as also the terms s1=8s + 1, 83=1, and 81= - (s + 1),
83 = -1, this in fact being the reason for introducing the factor 2, which does
not appear in (142) and (143).
83= -

Using the notation of (144) and (145) we get the following terms :
d22.er=___

dcp -Fcl)

1
+

0l)s11'11+
82w3C(i0Jf3t
1
p2
si

(8;0 kl S

(t-

, +82w k2S ')cosg2

F

+
d~,-h2 )sleo 1 0Ci"+
=-~f2 Ff
H
r
LCpl

=h8=2e

rc

slw i
l

s
83t

,\

il(J

9A C2) sin~f(-/)A

pep.

SGV2/283')S "cos

1L(

+
r/c)+A

130

These arise from the sets of indices

s=

[OH.

ON THE DISTANT FIELD DUE TO A MOVING CHARGE
s1 =

s - 1,

s2

1,

=

and s1=

-

(s -

1),

There is an exactly similar set of terms arising from the sets of

1.

-

indices s1

= s

+ 1,

s3

1, and

=

s1

= -

(s + 1),

83 =

-

1, which

are got by merely

interchanging the indices 2 and 3, and changing s - 1 into s + 1.

By
(10)

F
L (fI)

(flp 1/c)
n

=Js-1

J1(nflp /c) JO (&lp
2

3 /C)....

To our degree of approximation we may write
J1 (12p 2/c) = Qp2/2c, Jo (f~p 3 /c)

=

1, ..

terms involving squares and higher powers of p2, p3, ... , but not of

ne gle ct ed.

H e nce

Again by

(146)
k

81

1r
nF=

lP

1C(s

-

J's- 1

(f2p1/C)

1) Js-1

pi,

being

1r

/k'-= 1P2

(flp 1/c) '

J 1 ' (flp 2/c)

=

1.

CJ 1 (f Lp2/c)

Equations (159) and (160) give with (138)

~=s(8- + 2'r)-a1
we get

i)

/)+s(

x=1(

Hence writing

+2 1 .

fI(t-r/c)±0=x- a 1+a 2 .

Further, we get by (146),
01 1=pcosO,

C2 =a 2 cos9,
C"-p,=30,

S'=0,
Si"

C"= 0,

p 1 =psin9,

-

29

=C

a,

cW=q+W.
2

Using these results we get
si(-a
p,
h
efI 2 cos 9J

~=2

= -

p 2 = a 2 sin e,

W1k=W,

S2 -0,
S2

2

he --

,

j

C

1

{_ Pl1
W

1

a,

o

a2sn~a+~,a

(- pl)+ (q +

W)

J8_1i (fi)

}a~o(

x-a
~

2

i

a

In precisely the same way we get
d9= h,

d,=-

=

efl 2 cos

-

fact thag "

8

+{1

( flp\)

ell {7J
J'

a 3 sin

(tpl)
1

+a 3,whles+"1=C-

a

(x

+

-

a1

+ a3),

q-W)
q

J+, P
8 1

a3 cos

(x + a1 + a3 ).

viii]
ON THE

VIII

DISTANT FIELD DUE TO A MOVING CHARGE13

131

The terms we require are the sums of these two sets of terms, and the
complete expressions for the forces are obtained by summing with respect to
- oo to oo ; for as we see from the table above we also get a similar
s
set of terms for which 11~= q - sw.

from

Now we get by (159)
1/ 2 +B2
a ip
t{ a2 = +2ABsin
+ +A~i(a-)},

an 2 - l)=A sin a + B cos ,C3
a(al-a)A
cos a- Bsin3

-8 ,

2

Hence
a2

sin (x-al + a )
2

a 2 cos(

a1 +a 2)

-

sin

{A

= 2p

cos (x+$/)},

a) +B

(x+

4.p fA cos(X +a)- Bsin(x +$)}.

Similarly by changing the sign of B we get
a3

sin

± a1 +;a)= jp {A sin (x +a)

B cos(x +/)

-

a3 cos(x +a1 +a3) = p {A cos (x-+-a) +B sin (x +$/)}.
When we substitute these expressions in the values got for d29 and d30, we
find for the factor in d~e + die of

{'(

)±+ J

ofe2cos pJfp

A sin(+a):

_1p c1

exU) cors
9,S

B cos (x +$/)':

2

f2eo r pJ

')},

8+1

\1 ) -s\2+
i?

PC\
1

kj7

0
Similarly we find for the factor in d24 + dam of
Ax o )
Acs~a:

_efZlp [ofip 1 $
2c~r Lc 1

1 p1 +

c

+ (q +
p [w~lp 1

Bsnefl

(o)

J8 -- (1 Pi)

8+1

p

c

(q

8+

use the well known equations
J8

I+

1

)

J8-(1
p l

and write
so that

->)

-C

±(q w)J

We

(q

1
"p1 -_1(8P

1

2c~r LC

Bi(x+-}-):

J"8+

J

8-1p8±

C+f2 1
+
,J

3
1C
=wP/c,

f

2
pi

=

2csJ 8 (f 1p 1 /c)

1

=

2J '

c8

=

{8_
2

+,

'"

8
J

-

1GP)

_

S= q +sw=l1w,

f~p sin 9 =lsn0
cl3snO

2}

J

f P I)

C

132

[CH.

ON THE DISTANT FIELD DUE TO A MOVING CHARGE

Then we easily get, on summation from 8 =

de= h,
0

=

-

o

to

8=x
sl3J,(13sin8).sin(x + a)
00

ecot 8
A
pr

8=

+

00

I

_

e cosO8

+

pr

s °

B

12/32J8 '(1,3 sin 0). cos

f

(x +,8)...(162),

S= 00

d=-he=-

s= -0

12/32J (lsin).cos(x+a)

8(
+sinB
3 -,,sin 0

$--pr

where
and

pr

A

ela~c
1,-i82sin

6) l/3J8 (1/3 sin 0). sin

1,82

fl

(x +,/3)

. .. (163),

(t - r/c)+ s (8 - 0 + 17r) =
fr =(t-r/c)+8-0+-jr.

x=

101. Axial oscillations. The field due to these oscillations is easily
obtained by using (159) and (160) in (144) and (145).
As before we neglect higher powers of C, that is, of p4' than the first, but
not higher powers of pi.
Hence in (144) and (145) we have to consider terms of the two types:
S1=S,

84=+1

S23=0,

The upper sign gives by (138)
Z=so+wo =q+so,
4

8

A=

1 +8

4

=s(8-0+w)+

7

.

The lower sign gives
-W 4=-(q-so),
fl=so

a=s 1-8 4 =s(8-4+ir)-y.
Both may be included by summing for negative, as well as positive, values
of s; thus we need only consider the upper sign.
As before we get from (144) and (145)
F (n~) H

2e!11r8'4d=

C1l +

sin {fl (t-r/)+A

-4W4

(S(J)1_1I {f1
As')cos
r8

d4=2e F

(n) 1)
F8

Cl " + 84(04

4(041k4

(i 4

(t/)+z]
t-rc++

singlfI(t -r/c) + }

012
pl,
The values of W1, 1c1,Ci', Si', C0'", Si" are the same as before. The
given by the following equations, got just as before :
other quantities are

104 =1,

f (t -r/c)+0A=

x ±y,

VIII]
ON THE DISTANT FIELD DUE TO A MOVING CHARGE13

ViII]

133

We easily find
ce

=

d4

= -

j

e
h4

s0

=

h=

efCOS9

sino) 1J

-

8

l/sin) sin(+)...(164).

122J,'(13sin9)cos(+ y)

8= - 00

pr

......

(165.

102. Group of n electrons. To get the forces due to a group of n
electrons describing the same circle we need only write for the ith electron
=pcos

+n+

2'i)
n

2wi)

X=2Asin
(Tr+a-k2
v=Y Csin

(qr+

)

=
=BsinqT+3-k272)

7 _ik!i-).

The first three equations represent the steady motion of the group as in
problem 1. They show that the electrons are equidistant.
The last three equations represent sets of waves of disturbance travelling
round the circle with a velocity qlck relative to the rotating electrons. For if
i be increased by unity, r must be increased by 2wlc/qn in order that X, , v
may be unaltered. In other words the disturbance requires the time 27Tk/qn
to travel from one electron to the next, through the intervening angle 2'n/n.
The substitutions only affect the values of the three arguments
x

+,8,

x+

rY
in

x ± a,

(162)-(165).

x

Remembering that

=

fl

(t

-

r/c) + s (8

-

0-

pr), the substitutions of

S+ 27ri/n for 8, a-k27ri/n for a, /3-k2'ni/n for 8, and y-k2wi/n for y,
change x + a,

x

+,83,
x+

y respectively into

x+a+(s-k)2wri/n, x+83+(s-k)2ri/n, x+ +(s-k)2wi/n.
When the equations are now written down for all the n electrons and
summed from i = 0 to i = n - 1, all the circular functions disappear except
those for which s is of the form k + sn, s being an integer, and these are
multiplied by n.
Write
Then

m =k +sn,
fl =q+ kcosnw,

'4=CO (t -r/c) +8s-s
l =h +sn +q/W,

+P1r.

= q (t -r/c)+ m*.

Then we get finally, from (162)-(165),
de = h,=
+

ne cot 98=°
rA

ne cos 9
pr
ne

~ml3Jz

B I

0

(1,6sin
9)sin {m4r q(t -r/c) +a}

l2/3J (C3sin 9)cosmr+ q (t -r/c) +,}

8=-0

8= 0(m/

+C 1

sin 9-1/C3 sin 9)1$3Jm (1/3sin 9) sin nvqr+q(t -r/c)+ y}
pr ........................
16)

134

ON THE DISTANT

do = - he =

ne

pr

FIELD DUE TO A MOVING

A Z lS2J'(1sin) cos{m

=

ne+B 8=° m - 1
+
sin 0

pr

+ q(t - r/c)+a}

13J, (1/3 sin 9)

2sin

sin {m4 + q (t - r/c) +
ne cos 0
pr

[CH.

CHARGE

122J' (

sin ) cos

3}

+ q (t - r/c) + }

s=- ,

........ (167).
These equations were first given by Schott* in a slightly different form
(there is a misprint in the coefficient of B in do in the first article referred to
The above
- 1/32 sin 0 is to be deleted, as above).
sin
alternative proof has been selected in order to test the formulae (144) and
(145) for the case of a polyperiodic motion.
-the

first /3 in

103. Character of the field.
the form mi + q (t r/c) + a,....
Now

The argument in (166) and (167) is of

9 = W(t-r/c)+8-

+

.

It is the argument of the fundamental wave due to the steady circular
motion, just as in problem 1.
On the other hand the remaining part of the argument, q (t - r/c) + a,..
is the argument of the corresponding component X, ... of the oscillation
relative to the rotating ring which generates the harmonic under discussion.
Every harmonic of the field is therefore a sum or difference vibration of
the fundamental due to the steady motion, and of one of the fundamentals
due to the disturbance, as was to be expected.
It is to be noted that the important harmonics as usual are those for
which the order, m, of the Bessel Function involved is least, that is to say,
one of the two harmonics of order k,or of order n - k, whichever is least.
We can include all the possible disturbances of the ring in our scheme by
giving to k all integral values from 0 to n - 1, or if we prefer, from - n to
+ n. The latter is generally the most convenient arrangement for our
purpose; it makes the orders of the most important harmonics positive or
negative, according as k is positive or negative, that is, according as the wave
of disturbance travels round the ring in a forward, or backward, direction
relative to the direction of rotation of the ring.
The polarization of the harmonic m is as follows:
On the axis, for 0= 0 and 0= wr, forces vanish (except for the case of
the
a single electron).
* Phil. Mag. [6], Vol. xmi. p. 197.

Ann. der Phys. 24, p. 653, 1907.

ON THE DISTANT

VIII]

FIELD DUE TO A MOVING CHARGE

135

At the equator, for 0= r, we get
do =

h =-

pr

=-

(1 - m)l12J(l)sin{m

+q(t-r/c)+y},.

d, = - he=-- A
ne
t12J (l) cos {m + q (t - r/c) + a}
pr ,=-G
ne 9
+ B
(m-132)lIJ(1 )cos{mr+q(t-r/c)+/3}.
pr

,=-O

In other words, at the equator the waves due to the axial disturbance are
completely polarized in the plane of the equator, those due to the orbital
disturbance are completely polarized in the perpendicular plane. Thus the
plane of the equator is a plane of complete polarization for each type of
disturbance separately.
In every other direction each type of disturbance emits waves with elliptic
polarization.
104. Radiation. The calculation of the radiation has been given by
Schott in his paper in the Annalen (1. c. p. 653). [But he only gives it for the
case where none of the waves due to the disturbance interfere with waves
due to the steady motion. On reference to (125) and (126), § 82, and (166)
and (167), § 102, we notice that the arguments of the periodic functions in
the field due to steady motion are of the form sn#, and those in the field due
to the disturbance of the form m# + q (t - r/c) + constant, where m = k + sn,
and ' = c (t - r/c) + S - q + rr. The radiation is given by the integral

R

$
i

o

(dl + d2)r 2 sin Od9dd4,

where de and d2 denote mean values of de2 and d02 for a long interval of
time, or, what comes to the same thing, for a whole period.
The integration with respect to 0, and the averaging with respect to time,
cause all products of different harmonics to disappear, except those for which
both factors have the same coefficients in their arguments both for 0 and for t.
Clearly this cannot happen for harmonics, both of which are due to
steady motion, or both of which are due to disturbance, but it can happen
when one harmonic is due to steady motion and the other to disturbance, but
only when the disturbance is such that q= 0 and k =0. Thus we have to
deal with terms of three types:
(1) Terms due to steady motion alone; these have been calculated in

§ 84.
(2) Terms due to disturbance alone; these will be calculated below.
(3) Terms resulting from interaction between waves due to steady
motion and those due to a disturbance, if any, for which q = 0 and k = 0;
these will also be calculated below.

13
ON

136

THE DISTANT FIELD DUE TO A MOVING CHARGE

[H

[ca.

105. Radiation due to disturbance alone. From what
stated above it follows that this part of the radiation is given by
the form R = SRS, where B3 is the part of the radiation due to the
of order s in (166) and (167), and is the same as if these harmonics
themselves.

has been
a series of
harmonics
existed by

On averaging with respect to t and integrating with respect to 0, we
easily see that the remaining integral with respect to 9 is of such a form
that terms involving the first power of cos 9 as a factor vanish identically;
such are the terms which involve the products A C and BC. The remaining
ones give
ce2n2122[A , B2 2 + 2AB sin (c- 13+ C-14],

8=

h,

where

I

=

12

=

i

... are integrals defined as follows:
[i

2

cote9. {J,, (1 sin

f2[1232

13 =4l
14-

141

f[m.

2f

{m

cos29

(

9

(1 sin

122 tJm' (113 sin

9)12+

9)}2+(si

/

sin 6d9,

-12sin0)2{Jm(1sin{9)121{sin

+ sin91 32 sine]

-1 sin9).{J

9)}2]

(11 sin 9) Jm' (1/3 sin 9)

Jm

sin 8d9,
d9.

sin9)21232cost9{J(sin

We proceed just as in problem 2, 84; for this purpose we require the
following results, which follow without much difficulty from the Addition
Theorem and the known integral expressions for the Bessel Function :
(a)

'.r{J(13sin 6)}2 =

(b)

it- tJm'Q13 sin 9)}2=

rJo (2113 sin 9 sin /)) cos 2mqdq,
J (21,8 sin 9 sin

x(1(c)

7rJm

(113 sin 9) fm~ (1$3 sin 9) =oJo'

(218sin9sin 0) sin 9 d9
J

=

f
3si2

sin2 0)cos 2m4do,

(218 sin 9 sin

f0) 0b cos 2m4/dc/,
sin

sin((21xsinn)01
in(18in1(d)f
sin

°2113

(e)

-

J° (2113 sin 9 sin 0) cost 9 sin 9d9- sin (21/3 sin

fo(21,3

/

sin 0)3

9 s

=

rx
f;os 2w i

x '4

(2113

sin ~
cos
(21,8 sin 4))2

cos (21x sin 0)

(32_

x2) dx,

ON THE

'Till]

DISTANT

'211 i

sn4)d

VIII

(f J

(\

o (1

i

i

cos (21/3 sin 4))
21(3 sin )

cs

4) co

(7/21/3sin0 sn
' 0

2

2l(3 sin ¢)

fI sin (21wc sin4) dx,

d

~

=

f

if
'()

cos

(21wc

=

f

(k)

sin

=

4)

_sin

x)

dx,

fJ , (21wc) dx,

cos 2m4)cr~dwd4) =1J2(1w

4l132J rn'

-

sin (21w sin

fff

(1)

4))

f

-J

cxd.4)
sin4) cos 2m4)d

cos (21wcsin

(21(3 sin 0))
(21,8 sin 0)2
sin (21x sin 4)) (13-

1
21/3 sin

-

(Ia})

137

1

B=

a(l3sn0sn

fU

FIELD' DUE TO A MOVING CHARGE13

xd

(2118) + j/3J~,n (2113) + (mW

4)) sin 4)cos 2m4)idxd4) =

0Jam

-

(21wc) dx,

J-,n (2113),

(21x sin 4)) sin 4)cos 2m4)wdxd4)
J_ (1)

IJm(21/3) +

=

x

We get, by means of (a), (b), (d) and (h),

J J9

(21/3 sin

11=
=(
6/

~'(2G

1/3

0 sin

2..12/32

4)) (12132

-IM

2

f ,,,(21x)

-212/32sin2e4))

x

1

i

(

cos 2-nab sin GdMdO

= 4

Od~d4)

dx.

A gainbyr2Ieans o f (a ), i ),,O (e), (h ) an d2&.)
,7 m
(b (d),
(,
~(

cos 2mo4 sin

( )

4

c

s

1

(1 -/32) 1$3J 2,,' (21/3) + (3

i

+

182)

J

e )

(2t)3)

Again, by means of (c), (f),(g), (kc) and (1),
=

1
-

=~(M

2J

2l3sin

9 sin

- 1/3) Jm(2118) +

Lastly, by means of

4))

{11

(a), (b). (dl),

L1/3+

:

2

(rn+ 1/2) cos29}1 sin

(21v) dx.

(e), (h) and (i),

4) cos 2m4)Od4)

13
ON

1.8

THE DISTANT FIELD DUE TO A MOVING CHARGE

[H

[CH.

all the terms together] we find for the radiation due to dis-

Collecting

tuirbance alone

R

A2

C= t

.E1j3L12$32J«

X1f13

+ J3

' (21,x) -- (rn2 -- 1/3~) lifJJ,(21x) dx]

(1_-/32) 123'' 2
JA

-

2A B

sin (a

-,/3)

~1/3 J

1/32)

(35+

-,11,3 2-~

(m

+ { (1 + /32) (m2 + 1282)
+

(1$) +

+
1/3r -

+02

+{

(1 +

$2)

(n2

4

(1

-

/32) l22J

m

(1 +

if:an (21x) dx]

m(2143)

(m + 1/32)

(21/3) +

+ 12/3)2) 91/32 -

+ /$2)}

(3

43J

18 J2m (21/C3)

2)

S(1

if: J

(21w)

dxl

± 3/2) 127JZ (21$3)

3,8)'f

J

(21x) dxj$..(168).

106. [Radiation due to interaction between steady motion and
disturbance. Putting q =0 and Ik=0 in (166) and- (167) we see that the
parts giving rise to terms of the present type in the radiation reduce to

,= prS2/3-co

A sin a I

s2J3,, (sn./ sin 6) cos sn*

sin /38'I

2e/32ncos6

=0

--- B
_

2J' z(sn/3

S=C
+2e$&-n3cos20
prsi 6Csinl 7s2J
2e/23

d oA

sin

expressions
products

also involve the product sin

sn'4

6) sin

snip'

6) cos z
~,

(sn$ sin 6) sin snJi

3
+2e/n 7 $~in
(

Comparing these
(126), § 82, we see that

(@in$sin

s 2J+,,

Sx

sin a

+"/~co
P

sin

-B

sin ,8

8

2JJ

(sn/3 sin

0) Cos

sn

Jr

ryE82ehn (s12,3 sin 6) sin Sw9r.
with the

corresponding- ones in (125) and
terms involving A and

of the two sets of
cos snij

as a

averaging for the time, but products involving

factor,

B

C

which. disappears

in

also involve either cos 2 snJ

VIII]

ON THE DISTANT FIELD DUE TO A MOVING CHARGE

.139

divided by sn2$, where for 1 and m are put the value sn, to which they
reduce when q and k are zero. Using the value of I given, we find
B sin

R = P

1

s

3(1 -

$2)J2.

(2sn3)(1+
1+

2

J

(2snx)d

s=l

When we put p (1l -B sin 1) in place of p, and.(1- Bsin 3) in placeof 1in expression (129), 84, for the radiation from a ring in steady motion, we
find, on developing in powers of B, that the linear term obtained is precisely
the value just found for R. This must be so; for a disturbance, for which
q and k both vanish, represents either a rotation of the ring as a whole about
its centre proportional to A sin a, or a contraction of the ring as a whole
proportional to B sin 1, or a displacement of the ring as a whole perpendicular
to its plane proportional to C sin-t. Of these the second alone alters the
steady motion radiation, and since q = 0,it takes place without altering the
angular velocity o of this rotation, so that 3 must diminish in proportion
as p diminishes. This being the character of the exceptional disturbance
just considered, it is obviously unnecessary to trouble any further about
it, and we shall therefore confine our discussion to the expression (168)].

1

+

mw) p/c. Now
a
107. We have = p/c and = m + q ; hence 13= (
of the particular type of wave under consideration
q + mco is the frequency
the. correhis
relative to a stationary observer; thus 1 = 27rp/X, where X
sponding wave-length. For spectrum lines it is always small; in fact i
only 1/160 for the extreme ultraviolet lines of Schumann. For uch waves
,
we may put J , (2113) = (1) 2 ,/2d! where L is the absolute value of m.
Using this approximation I have calculated an upper limit to the radition
from a ring for several classes of disturbance, that is for several values ofIk,
namely for Ic = ± 3, + 2, + 1, and 0. [See the paper in the Phil. Ma.
=
referred to in 102, note.] For the classes kIc± 2, the radiation per ring
= 0
=
is about 3.10-- erg/sec. and for the classes kc + I and Ikc of the order
10- 9 erg/sec., all for violet light.
Wien* finds that for the canal ray ion in a Geissler tube and for the
-7
line H the radiation per ion is of the order 10 erg/sec. This result shows
that if spectrum lines can be ascribed to disturbances of rings of electrons
from their steady motion at all, they can only be accounted for by means
These disturbances have
+ I and Ick=0.
of disturbances of the classes
; the former include to andfro
one pair of nodes and no nodes respectively
oscillations in the plane of the ring (orbital), and oscillations about a diameter
(axial), and the latter, periodic expansions and contractions of the ring (orbital),
and to and fro oscillations perpendicular to its plane (axial).
It should be noted that nothing has as yet been proved as to aperiodic
motions involving periodic components, such as damped-oscillations ; for the

k-=

* Ann. der Phys. [4), 23, p. 415, 1907.

14 THE DISTANT
ON

140

TO A MOVING CHARGE

HIELD DUE

method used fails for such motions.
that our results hold for these

[n

ACA.

It will however be proved in Oh. IX

cases also, provided

that the damping be not

too great.
108.

An electron moves in an epitrochoidal orbit:.

Problem 6.

To find the
example of a

field produced. We shall treat
diperiodic motion. We may write

this case briefly as

an

~=acoswr- bcos c'r,
=asin

wT--b5ll~CoT',

Hence, comparing with (135), § 88, we have
a1

b, a, c 1 =0,

i,3=0,

2

a=z,

(00 =CO

0sin

=-b=b,

c 2 0, a 2 = 7, j& 2 -7T.
Hence by (136), § 88, with I=sin 0cos ,m =sin
a2

7r-

p 1 =a sinO8,

Choose (1',

respectively,

n')-

in',

b

P2 =bsin98,

8 .-

lcos

0cos4

,

I"=_,sin4,
By (146),

§ 91,

k2

-

cos

6?,

sin

fib

0sin

,1'

C

0,

§ 89, we have

0,

(fla sin

S2'=0,

0J

8

,'

1'r =

=--sin90,

n"=0.
S~"=-a

-6/c)

£22

=

0,S

21 =

b

(fi~b sin 6/1c)

cs2 J4,(fib sin 8/c)

F(f)
=J
Lastly,

21' =

fa sinG J /

2'=b

't

m"=cos4),

C'= a cos 0,

By (140),

'=cos

we have
=

,

(1", in", n") in the direction of increasing 8? and

and

so that

4, n =cos68,

in

L8(Ib

sin 0)

by (138),§ 89, we have
=S

1W +

~w,

= j (s1

+ 2 )'IT--(S + 2).
38
S

f)

VIII]
ON THE DISTANT FIELD DUE TO A MOVING CHARGE14

VIII

141

where we take all values, positive and negative, both of s1 and s2, and
therefore omit the factor 2 in (144) and (145) We must however bear in
mind that there are two terms giving the same frequency, namely those
whose indices are (si, s,) and (- s , -

respectively.

82)

109. The character of the field is easily understood from (169). On the
axis the forces vanish. This is obvious for all harmonics of order 2 or higher.
The harmonic 81= s2= 0, of order zero, is absent on account of the vanishing
factor
hence the only difficulty is for s1= ± 1, 82=0 and s1 =0, 82= + 1.
The corresponding terms do not vanish. We easily find that for these two
sets together
a2
ebo 2
de cos - d sin4= 2 cos
cerwcosw't- rc
cr

fl;

d,=

co(t-r/c)-

d,=desin4+ cos
d

= cer sin o(t-r/c)2rcr

which of course are independent of 4,.
circularly polarized vibrations.

-sin

They evidently represent

two

At the equator, where 6= fir, we have
de

=0,

=
d

eZ

ef

-h

00/Za
11 [o
aE

J(\ 'f\ J

+ bw'J,(

~
82

(fb\1
c

)J (1
82'

.

cos {&)J

(t -r/c)

+O

which represents a vibration completely polarized in a plane at right angles to
the equator.
In every other direction the vibration is elliptically polarized.
The most intense vibrations are those for which
(1)
motion.

s1 =

± 1,

(2) s = 0,
motion.

82 =

82=-±

0 : the frequency is w, that of the first fundamental
1: the frequency is

wo',

that of the second fundamental

The amplitudes of these vibrations are of the orders
(ewc/r) J1 (coa sin 0/c) Jo (cob sin 0/c),
and
respectively.

(eco/cr) Jo (c'asin 9/c) J1 (co'b sin 9/c),
If wa/c and Wo'b/c be both small, the orders are

e(o2a/c2

r, and

142

ON THE DISTANT

(3)

FIELD DUE TO A MOVING CHARGE

[CH.

s, = ± 1, s2 = ± 1 : frequency o + w', that of the first sum or difference

vibration.
s = + 2, s,= 0: frequency 2w.
s,= 0, s28= ± 2: frequency

2

w'.

The amplitudes are of the order ewo3ab/c r, that is of order higher by unity
than the last.
110. Group of electrons. It is of interest to enquire whether it is
possible to arrange n electrons, moving in the same epitrochoid, so as to
interfere. This we found was always possible in the case of a mono-periodic
motion.
In order that interference may occur it is necessary that the argument of
the circular functions in (169) be of the form fI (t -r/c) + A + kc27i/n for the
ith electron.
This can only be accomplished by adding multiples of 2wi/n to the phases
wc, and w'T, of the two constituent motions; that is we must replace
o (t - ric) by w (t - r/c) + m2ri/n,
w' (t - r/c) by w' (t - r/c) + m'2ri/n,

and

where m and m' are integers.
Thus the argument in question becomes
£t (t -

r/c) +

0 +

(ms 1 +

m's2 ) 27ri/n.

The circular functions then disappear for all values of s, and s2, when the
expressions (169) are summed from i = 0 to i= n - 1, except only those which
satisfy an equation of the form
Inms + nm's jn,
2=
where j is any integer including zero.

inm=m'=

When
1, the value j=0 gives rise to terms s1= ± 1, s2= -1,
which are the second order; these are obviously the terms of lowest order,
the terms of the first order having been destroyed by interference.
When mn= 2, m'= 1, the value j=0

gives terms s= ± 1, s2= - 2, of the

third order, those of the second, as well as of the first order, having been
destroyed by interference.
It is obvious that we may confine ourselves to values of min m' which
and
are less than n; the only restriction is that they must be so chosen that no
two electrons can occupy the same position on the curve. In general large
values give greater interference than small ones; it is an interesting problem
to find which is the best arrangement for a given number of electrons, that is
to say, which makes the smallest value of Is I+!s2 as large as possible.

VIII]

ON THE DISTANT

FIELD DUE TO A MOVING CHARGE

143

111. Radiation. It is easy to write down the expression for the
Poynting vector corresponding to (169), but it is of little use, since it will
involve products of four Bessel Functions. Series of this type have not been
summed, nor does it seem likely that integrals involving such products can
be evaluated. For this reason we refrain from writing down the expression
for the radiation.
112. Problem 6. Precessional motion of a system of vibrating
charges. In conclusion we shall discuss the effect on the waves emitted by
a vibrating system of the type considered in this chapter, when it is given a
precessional motion. There is some reason for supposing that an atom may
be likened to a symmetrical magnetic top, of the type studied by Du Bois;
such an analogy has been employed by Langevin* in his electron theory
of magnetism, and by Ritz t in his paper on Spectrum Series and Atomic
Fields. The problem is of some importance, since it has an obvious bearing
on the theory of the Zeeman effect.
Let us suppose that a charge is executing oscillations relative to axes
(OA, OB, OC), which are in precessional motion relative to the fixed axes

(Ox, Oy, Oz).
With the usual notation of rigid dynamics, let O be the angle between Oz
and OC, Jr that between the planes zOx and zOC, and 0 that between zOC
and AOC; and suppose that

r=

0 = constant,

4 = nr, at the time r.

T7,

Further suppose that the components parallel to the moving axes of the
radius vector to the charge are
ZA sin (qT + a),

1B sin

(qT

+

,),

1C

sin (qr + 7ry).

The well known equations for the direction cosines in this case are
(XA) =
(XB) = -

(XC)

=

(YA) =
(YB) = -

0 cos
cos 0 cos
sin O cos
cos

0
cos 0
cos

sin

tr cos nr - sin tr

sin nTr,

r sin nr - sin r cos nr,
T,

r cos nr + cos pr sin nT,

sin pr sin nTr + cos

Tprcos nr,

(YC) = sin sin r,
(ZA)= - sin0 cosn'r,
(ZB) = sin 0 sin nr,
(ZC) = cos 0,
* Journal de Physique, [4], 4, p. 678, 1905.
t Ann. der Phys. 25, p. 660, 1908.

144
ON

144

[H

THE DISTANT FIELD DUE TO A MOVING CHARGE

[ca.

By their means we easily find, for the components of the radius vector
parallel to the fixed axes, the expressions
[

j(1

(1

-

_I

+

4

} + Esin {(q
)r+
+cos ())(D sin {(q +nu+
{(q + - )T'+ Z - wr+ Esin {(q
(H))(D
ry})]
sineC(sin {(q + tT +y} -sin{((q -

+±

sin

cos

-it

i

-

-

n+

v})

)T+

p)T+

7r})

v+

(170),

...........

±cos 0) (D sin {(q + n±+)7-±+- 17r} + Esin{(q -n - ) + v + 7r})
-+ jr} +Esin {(q-n+t)'7+ v+4w})
+ 4 (1 - cose)(Dsin (q +n-)
S(I.

+1[ () C(Dsin {(q +
sin
±

)

T + ry

+

7r}

+ sin {(q

+ 7})

+

1.)

.(7)

cos 0 C sin (q7- + 'y)]...........................................(172),

where

-,),

D =1{A2±f+J3+2AB sin (a -I3)}, E=1/{A2±+B2 - 2AB sin (a

A sin a- Bcos /3

A sin a +B cos,3

tny A cos a-Bsin/3'

tn

Acos a±BsinI3

We notice that each oscillation, of frequency q, gives rise to eight sum
and difference oscillations, of frequencies
q+n+wi q-n- u, q--n+t, q+n- ~, q+,ct, q-tp, q+n, q-n,
in addition to the original vibration of frequency q.
Using the notation of (135), § 88, we may classify them as follows :

113.

Circular vibrations.
Right-handed about Oz.
1
w 1 =q~n+ ~, a 1 =b,=4 (1±cosH))D, C =0,
a2 -=b 1(1 -cos0)E, C = 0,
wo =q- nt+/Z,
2
2
2
C=0,
b3 =-2sin OC,
cv3}-/~c,
(1)

as=

=

Left-handed about Oz.
a 4 =b 4 -I(1-cos ©)D, C = 0,
coo=q+n-jL,
4
co=q-n- c, a5 =b 5 =4 (1+cos0) E, C =0,
5
Cs= 0,
1.1q-,, as= b6 = sin ()C,

a1 =

8=W

,,

a2 = v+7T,

a3 =ry,

-7,

=v
18 +j2

7 r,

/33=ey -j7r.

(2)

IT

(3)

a

4 =

-- r,I8 =-'T
7
4

a5 =V,

8 =-v+-

a6 -y,

/6 =7fy+

5

7r,

17r.

Linear vibrations parallel to Oz.

W 7 =q+n, a,=b,=0, c 7 = 2

WB~-na=b8=0,c
W9 =q,

sin(HOD,

7 7 -=+1r,

8 =-,sin &E,78v+r

a9 = b9 =0, c9 =cos0 C,

79=7.

The waves of electric and
114. Electric and magnetic forces.
magnetic force emitted by the vibrating charge are determined by (144) and
(145), § 91, as before.
If we suppose that the amplitudes A, B, C are all small, so that their
squares and products may be neglected, we need only consider the fundamental waves for which one of the numbers, s,, is unity, all the others being

viii]
ON THE

VIII]

DISTANT FIELD DUE TO A MOVING CHARGE14

145

zero. In any case these are the most important, and we shall for simplicity
consider them alone.
We have, by (136), § 88,
Si = sin 9 (ai cos 0 cos c + bi sin 0 cos $i) + c, cos 9 cos
sin Si = sin 9 (ai cos 4 sin cai + bi sin 0 sin f3i) + ci cos 9 sin
where we have used polar coordinates (r, 9, 0) as before.
Again we have, by (146), § 91,
Ci = cos 9 {ai cos 4 cos (aj - Si) + bi sin 4) cos (iSi)} - ci sin 9 cos (yi Si' = cos 9 {ai cos 0 sin (cai - Si) + bi sin 4 sin (f3 - S)} - ci sin 9 sin (yi (§1/ = - ai sin 4 cos (ai - i) + b, cos 4 cos (/3i -Si
S= - ai sin 4)sin (cj - Si) + b, cos 4 sin (/3 -Si
where the directions (1', i', n') and (l", mn", n") have been taken in the directions of increasing 9 and 0 respectively.
Lastly, by (138) and (140), § 89, we get

picos

ryi,

pi

=

1Z

Wi,

z==

i,

i3i)

F(n)-= J

J(

(oiPi)j 0

Wi(e)

=

II

J1

say.
Thus we find by means of (144) and (145)

d=h,

[

11= 2

cr

(

,,

L

pi

C

i

/sin

{coi

(t

-

r/c) + Si}

(t

-

c

+ 2 J1' ( tPz)c cos
Si'

f coi

r/c) + Si}]

. .. (174).

115. These equations simplify considerably in our problem, where we are
dealing with circular and linear vibrations alone.
(1)

Circular

vibrations-right-handed.
a,=bi,

In this case
Hence, by (136),
By (146)
(2)

Here

C/'=

Circular

pi

=

aicos

9,

c =0,
ai sin 9,
i =

i=i3i+r.
Si = a

0,

0C'=0, 5S'l=l-ai.

vibrations-left-handed.
a = b, c,=O, a =/-j17r.

4).

pi= asin0,=a+i
Cti =aicos90,

(3)

4).

-

Si'=0,

CI'=0,

Axial vibrations.
ai=b,=0, pi=cicos9,

Sill=+ai.

Si=yi.

146

ON THE DISTANT FIELD DUE TO A MOVING CHARGE

[CII.

The circular vibrations produce elliptically polarized waves, reducing to
circularly polarized waves on the axis, and to linearly polarized ones on the
equator. The direction of rotation about the axis is the same as that of the
generating vibration.
The linear vibrations give waves completely polarized at right angles to
the meridian, the vibration in the wave being as nearly parallel to the
generating vibration as transversality will permit.
Thus the general character of the waves emitted is the same as that of the
generating vibration.
The arrangement of the lines on a scale of frequency is easily seen from
the table in § 113.
The right-handed circular lines form a symmetrically placed triplet; so
also do the left-handed ones, and the linearly polarized ones. The last are
symmetrically placed on either side of the original line, q; the former are
symmetrically placed about the lines q ±+ , which are themselves symmetrically placed about q; thus the whole group of nine lines is symmetrically placed
with respect to the centre line, q.
116. As regards intensity there is in general no symmetry, except that
the two lines, q ±+, are always equally intense. There is symmetry of
intensity with respect to the centre provided D = E; but the triplets of
circular lines can never possess symmetry of intensity with regard to their
own centres, q +±. The condition for symmetry of intensity requires, either
that A = B, or that a = /; in either case the generating vibration must take
place in a plane through the axis of symmetry OC of the system of charges.
This condition is both necessary and sufficient.
An examination of the known cases of Zeeman effect shows that the
symmetry of intensity is always present*. Hence if the Zeeman effect be due
to precessional motion of a symmetrical atom or ion under the influence of the
magnetic field, then all vibrations causing lines possessing a Zeemam effect of
this type must take place in a meridian plane of the atom or ion.
117. As regards the spacing of the lines, it is to be noticed that the
distance between the components of each of the three triplets, (1), (2) and (3),
is the same, namely n. This arrangement is the usual one in all cases of
Zeeman effect showing triplets.
A considerable variety of arrangements of the nine lines may be got by
choosing suitable values for the ratio u/n; the cases where
2, or ,
are all known.

p/n=3,

* Since this was written cases of asymmetry have been discovered.

ON THE DISTANT

VIII]

Runge's rule*,
of the group are
distance, obviously
important result is

147

FIELD DUE TO A MOVING CHARGE

that the distances of the several lines from the centre
small multiples of a small sub-multiple of a standard
requires that /n should be a ratio of small integers. This
due to Ritz.

118. We shall now consider the relative intensities of the several lines on
the assumption that there is symmetry, that is, that D = E. The table in
§ 113 gives in this case, for the amplitude of lines,
,
W3,

5:

4 (1 + cos9 ) D
.C

-sin

6:

(1-cos )D

2, 4 :
w7, W:

...................... (175),

sin 8. D

.C
the lines being supposed to be observed along the equator, that is, in a direction at right angles to the axis Oz, or at right angles to the direction of the
magnetic force causing the precession.
cos

S. :

In this case we see by (173) and (174), § 114, that the factor, by which the
amplitude of the generating vibration must be multiplied to give that of the
wave emitted, is practically the same for each line. Thus (175) give the
amplitudes of the lines emitted, as well as of the generating vibrations.
For this direction we have seen that all the lines are plane polarized;
W,, o9,

(o0

vibrate parallel to the axis, i.e. parallel to the lines of force (p); the

others perpendicular to them (s).
In

general, the lines (p) are the strongest, and w9 > W7, showing that

cost 0. C > sin 0. D, by (175).
In general also ol <

(3

<

, showing that 0 >

The two conditions together require that

r, and

0 > 3r.

The case where w9 > W7, and at the same time o > (03 >
requires

02 > tan2 0. D, and 02 <

cot2

9

tan 2

2<

2,

.D

also occurs ; it

. D , together with 9 < ir.
2

They

require that 9< r.
In certain cases some of the lines are absent.
D = 0 gives

co3, w9,

e alone, a normal triplet. This can only be due to axial,

without orbital, vibrations.
C=0 gives the sextet W7,

ws,

o,

0W5,
)2,

W4; it is equally spaced when

= 2n. It is due to orbital vibrations, i.e. vibrations perpendicular to the
axis of symmetry.
9 = 0 gives the triplet

0,

0i

,

(eo,

with abnormally wide spacing.

It is due

to spinning about the line of force, without precession.
* Phys. Zeitsch. 8, p. 232, 1907.

10-2

148

ON THE DISTANT FIELD DUE TO A MOVING

0 = 7 gives the triplet w2 ,

(9,

w4,

CHARGE

[CH. VIII

with abnormally narrow spacing.

It is

due to spinning about the negative direction of the line of force, without
precession.

0 = 17r gives the octet, with the central line w9 missing. It is due to
precession with the axis of symmetry perpendicular to the line of force.
We may also get exceptional cases by a suitable
ratio : n.

choice of the

For instance, n = 0 gives a triplet;
= n gives a quintet, wo and (o4 coinciding with o9, and 3o,, so respectively with 7, cs. If at the same time
C= 0, and 0 = 0, only o,, (o5 are left, and we get a doublet.

Thus the hypothesis that the Zeeman effect is due to precessional motion,
explains a good many of the observed cases; but there are some cases of
resolution into 10 and 15 components which it does not explain very readily*.
This hypothesis appears to be due to Ritz (loc. cit. § 112).
* It is unlikely that these lines could be accounted for by means of sum and difference vibrations of higher order. It is true that, for example, a line of frequency q + + 2n could be got by
making s1 =s 7 =1, s89=-1, giving w 1+ (w7-ow
9, which has the right value; but the intensity of
such a line would be excessively small.

CHAPTER

IX

PSEUDO-PERIODIC AND APERIODIC MOTIONS

119.

THE treatment of polyperiodic motions given in the last chapter was

restricted to strictly periodic motions, that is, to motions involving no aperiodic
components. Such a limitation is however undesirable for the purposes of
Physics, because many of the most important physical problems deal with
motions which are not strictly periodic, such as damped free vibrations,
"pseudo-periodic motions," or with motions which involve no periodic components at all, such as motions in orbits with infinite branches, "aperiodic
motions." Many of these motions are discontinuous, that is to say, they do
not follow the same law for all time; for instance, damped free vibrations
imply a time at which they were started by the application of some
disturbing force. The beginning marks a time at which the functions representing the coordinates are discontinuous in form.
In deducing equations (140) and (141), § 89, we made the assumption that
the Fourier Integrals of the type
S=

f

()-

(
(t-r/c-T)f(.) dT d/

could be replaced by

or what is the same thing, by f (n) (t - r/c).
Here

f(T) is a function of the coordinates and velocity of the moving

charge at time T.

The transformation is meaningless unless f(n) (t) be finite,

except for singular values of t; but this condition is always satisfied for every
value of n in a physical problem.
The transformation is obviously justifiable when f(t) is continuous in
form, so that all its differential coefficients are continuous, and not merely
finite, for all values of t. This occurs for all periodic motions, and indeed for
all motions which follow the same law for ever, so that the coordinates of the
moving charge are always given by the same expressions. But many cases

150

PSEUDO-PERIODIC AND APERIODIC MOTIONS

[CH.

occur where this is not the case. For instance, when the charge starts moving
from rest under the action of a given finite force, the function f (t) is constant,
and all its differential coefficients are zero for times before the commencement
of the motion. But immediately afterwards those of its differential coefficients
which involve the acceleration of the moving charge, assume certain definite
values different from zero, but finite.
In cases such as these, where the
motion is discontinuous, in the sense that the coordinates of the moving
charge are given as functions of t by expressions of different form before and
after the instant of time at which the discontinuity occurs, we cannot apply
the transformation without further examination.
The coordinates and velocity components are continuous in value, but
one at least of the higher derivatives is discontinuous in value, and we must
enquire what effect this discontinuity produces.
120. We shall use the method developed by Stokes* in Section II of his
memoir on the critical values of the sums of periodic series.
The integral

f

f(t) =

Em

(t-r/c-T) f(r)

dr d

is convergent, because f(7) is finite, except for singular values of r, but some
of the integrals to be derived from it are not. For this reason we use the
equivalent integral

f(t) =-J

0

0

f
cos ' (t - r/c - r) () drd,

00

which is the limit to which the integral

.-

_1f

h'cos

(t

-

r/c - r)

f (r)drd

tends as h is indefinitely diminished. The last integral is absolutely convergent provided that h be positive. This we assume to be the case.
Again the integral under examination, I, may be regarded in the same
way as the limit when h = 0 of the absolutely convergent integral
-f

_F -

t

cos F' (t - r/c - T)f (r) drd .

The inner integral may be written in the form

Integrating by parts and replacing ~- by

f-

0cos F' (t -

-t,

we get

r/c - r) (n) dr
f (r)

* Collected papers, Vol. i. p. 271.

~PSEUDO-PERIODICMOTIONS15
AND APERIODIC

ix]

IX]

(Y)T

S Lim.

+ I
i=14-1

at

trc-T)+
1

O'

f c(T)cs

n--i

a

- Z
Li.

151

-

r/c -T

00~(3~

Here t, denotes one of the instants at which a discontinuity occurs, and
the summation S is for all these times of discontinuity. The square bracket
denotes that the difference of the quantity inside is to be taken for times
ti ± e, where E is to be made equal to zero in the limit. It should be noted
that the infinite limits together are to be treated like the times of discontinuity. The result obtained is the equivalent of the equations (55) and (63)
of Stokes (loc. cit.).
Substituting the value just obtained for the inner integral in the
expression for I, and noticing that all the single integrals are absolutely con-

-hI,

vergent on account of the factor
I

=

J0 00
0f
i=n-1

+ I

cos
/a

(t - r/c -

e-h

tl t,

a

i=n-1

Lim.

T)f ()

()Td

n-i-11l00

SLim.at-)

i=O

-

,t

we get

[cos (tr/c-T)f(i)(r)]d

o

(at

n-i-11

?-

fEC00

[cos

(t

-r/c-T)

f(i)('r)] d1

... (176).

The double integral in the first line remains convergent when h =0, and
then reduces to the Fourier Integral forf(n (t - r/c), that is to say, to

(atnIfo

fcos

(t - r/c -)f()dd.

If the transformation previously used is allowable, the remaining, or complementary, terms in the expression (176) for I must vanish. We have to
determine under what conditions this takes place.
121. It is obvious that it cannot take place at all unless the complementary terms approach a definite limit as h is diminished to zero, and this
limit must be zero quite independently of the discontinuities of the functions
If we can prove that the limit is definite, though not zero, then the
complementary terms give the correction which is necessary on account of the
discontinuity.

a

Let us consider one of the single integrals in (176) due to the discontinuity
t1. We may put T = t1 + e, and make equal to zero in the limit. Thus we
have to consider the limiting value of a term of the type

(i) J
,

"

cos

(t - r/c -

ti

- e) f(')

(t1 +I e)S.

~PSEUDO-PERIODIC
AND APERIODIC

152

152

C.

MOTIONS

[OH.

It may be written in the form

f

() (t 1 +

E)

+f (i' (t, +

fE-h

)

J0 Eh

E)()at

cos
'

t

{cos

(t - r/c - t,) dpt
(t -r/c

-

t

cos

-

(t

-r/c

-

t1)} d .

The integral in the last line is equal to
Ehsin
E "(t-r/c-t-

-e-

that is, to

)sinl

feh

at

cos

t

r/c-

-

(

/~lc

-tl

E

2

1esin 2/
1

-1

2)

Ed.

E

It is clear that the factor sin 21 aE/2tE improves the convergence of the
last integral for large values of , however small E and h may be. Since E
occurs as a factor outside the integral, the term is ultimately negligible.
Hence the term in question reduces ultimately to

+ 0)

fi> t1

(a)

t,)dp
,.t(t
ilZLcos ,
-r/

E'f:

-

and the two complementary terms due to the discontinuity reduce to
a

7r
1

n\-i-1

ao

c-h
os

(t

-

r/c-- t1 ) dq

()0

where the square bracket now denotes f () (t, + 0) -f

(i)

(t,

-

0).

This complementary term vanishes when either factor vanishes.
The second factor vanishes when f (i)(t, + 0)

=f

(i)

(t,

-

0), that is, when

there is no discontinuity at t1. This of course must be so, and ,justifies the
presence of the factor [f (W(t)].
It also vanishes when f

W(T()

is zero ; this frequently occurs for T =+±GO

In all other cases it is necessary to consider the first factor
atf

-l

1~

cos Ft(t

E h

-

/

-

t,) d t.

When h is greater than zero we have

f

ch" cos Ft (t - r/c

-

t) d t = h/{h 2 + (t

f

E-h

r/c

-

t,) d t

- r/c

- t)}

and therefore

(a)lt

1

cos Ft (t
n~n-i-1 (n
(+)

-

-1

)

sin {(n -2i) tan-, (h/(t - r/c
1{h_+.(t rInv/c _21(-2
f-

-

t,))}

.

PSEUDO-PERIODIC

IX]

153

AND APERIODIC MOTIONS

As h approaches zero, t-r/c-t 1 remaining constant, the last expression oscillates between finite limits until h becomes less than (t - r/c - t1) tan w/2 (n - i),
and thereafter diminishes monotonously to zero. Hence the limit for h = 0
is definite, and ultimately zero, whenever t - r/c - t1 differs from zero by a
finite amount. This is all the more the case when t1 = oo, on account of
the infinite denominator.
Thus the complementary terms in (176) due to the two infinite limits of
the T-integral vanish, because fi' (7) is finite.
The terms due to a discontinuity at t1 vanish practically whenever t differs
from t1 + r/c by an appreciable amount.
But when t becomes very nearly equal to t1 + r/c the complementary
terms execute large oscillations because the denominator
2
{h + (t - r/c- t)2

(n-i)

becomes very small in the limit. The series in (176) become divergent and
useless for purposes of calculation. In this case we must either calculate the
complementary term by a different method, or if that prove to be impossible,
give up this method of calculating the integral I altogether.
In conclusion we may sum up our discussion as follows:
When t differs from any of the values t1 + r/c, ... corresponding to the
discontinuities of the motion by a finite amount, we may write
122.

J

(t
(t - r/c - r)f () drdi = f (n)

cos

This is equivalent to taking the operator

-

r/c) ... (177).

8 outside the signs of
t

integration.
But when t is nearly equal to t1 + r/c, where t1 is any one of the times of
discontinuity of the motion, (177) ceases to give a sufficient approximation;
we must add the complementary term
-i-1

i=n-1

I
i=0

Lim. Lim.
h=o0

1

E-h cos

-

)(f
()

(t - r/c - r) d

.

to

=ti

Unless this complementary term can be proved to converge to the limit
zero as h converges to zero, the expression (176) is no longer true.
123. As examples we shall deduce two important expansions for the
potentials due to a point charge.
In the first place, let us expand the exponential in (131), § 86, in powers
of p by means of Taylor's Theorem; we get, on changing the limits of ,
0a
= e
'rr .o 8! Jo
8

-'rcos
.

(t - r/c -

T).p

dd .

154

PSEUDO-PERIODIC

AND

[CH.

APERIODIC MOTIONS

Each term of this expansion is an integral of the type I just considered.
Hence we get from (177)
eS=" 1 a 8' S
cat) {p (t - r/c)}8 ................. (178).

os=

= r

This expansion is valid, provided that it converges, and that t is not
nearly equal to t1 + r/c, where t1 is one of the times of discontinuity. The
first condition may be assumed to be satisfied for physical reasons in all
problems to which the expansion would be applied.
When however t is nearly equal to t1 + r/c, we must evaluate the complementary term, which becomes
e

So

i=s-1l

C7rr=1 i=o

-i-1

i

SLim. Lim.
!

h=O

t t=

{p (()}

-hO).

CaT,(/-

In precisely the same way we get from (132)
a=

S=oo

i=S -

CT s=1

i=O

e

1

)' (

1 a

(

y.p
8 ! cat

(t - r/c) v (t- r/c).....(179)

which holds under the same conditions as (178), and in the case of failure
must be corrected by the addition of an analogous complementary term.
124. Again, applying (12), § 9, to a point charge, by omitting the
integration with respect to de, and introducing a factor e instead, we get in
the same way as before, for a unit charge (e = 1),
=

S=

(-1)s

a

. ...

s=o

-

R-

cat/

s!

..................... (180),

where R is the distance of the fieldpoint from the position of the point
charge at the actual time t.
When t differs appreciably from t1, one of the times of discontinuity of the
motion, the expansion (180) is valid whenever it is convergent, that is, whenever R is small enough compared with the radii of curvature and torsion and
similar leringths characteristic of the path of the moving charge.
But when t is nearly equal to t1 we must evaluate a complementary term
which is found to be
c7rr =1

i=0o

( s!' )"Lim. Lim.
h=-0
o=t,

Rs"-

c

/cat

o(

-h Oos

(t -

T)

d

where R is taken for time T.
In the same way and under the same conditions we get from (13), § 9,
ca=
o

s!

cat

,..................(181),

where R and v both correspond to the time t.

,

IX]

PSEUDO-PERIODIC AND APERIODIC MOTIONS

155

125. The fact that all series for the potentials of the types (177)-(181)
fail when t is nearly equal to tl is susceptible of a simple physical
interpretation.
We know that the moving charge is continually emitting waves of
disturbance, by means of which the field is propagated to all parts of the
surrounding medium with the velocity of light. In particular, at a time t1 ,
at which its motion changes its form discontinuously, a wave is emitted
which separates space into two regions in which the potentials are determined
by different mathematical expressions. Such a wave may be called a wave of
discontinuity, for the potentials change their form discontinuously there,
although their numerical values do not necessarily suffer any change. At
points which are not close to a wave of discontinuity the expressions (177)(181) are valid, and have a physical meaning. But at points on a wave of
discontinuity they lose all meaning, unless the complementary terms belonging
to the wave can be evaluated. In general this cannot be done, and it is
better to resort to the integral expressions from which the series have been
derived. This failure is quite analogous to the failure of the point laws
under similar circumstances, of which we have met several examples in Ch. V.
Since all distributions of charge met with in nature appear to be extended
distributions and not point charges, the failure of our series can only affect an
infinitely small element of charge at one and the same time, namely the
charge of that part of the system which is enclosed between two infinitely
close spheres, whose radii are infinitely nearly equal to r, and whose centres
are at the fieldpoint. Any difficulty arising from the discontinuity will disappear when the integration for the charge is extended to the whole system,
for the discontinuity is then, as it were, spread over a layer of transition of
small but finite thickness.
It should be noted that the series (180) and (181) are the developments
by Lagrange's Theorem of the point potentials (26) and (27), § 13. They
imply, in the first place, that the characteristic equation t = T + R/c has but
one root less than t, i.e. that the velocity of the charge is less than that of
light; and secondly, that all the differential coefficients of the coordinates
with respect to the time are finite.
126. The series (180) and (181) have been discussed by Schott* for
the particular case of rectilinear motion. From his discussion the following
results follow:
In the equations (178)---(181) the series in the first lines represent the
potentials due to the given motion regarded as continuous. For instance, if
we are required to find the potentials due to a charge which starts from rest
* Ann. der Phys. 25, p. 63, 1908. See also A. W. Conway, Proc. Roy. Irish Acad. Vol. xxviii.
A. p. 5, 1910.

156

PSEUDO-PERIODIC AND APERIODIC MOTIONS

[OH.

at time zero according to a prescribed law, e.g. with uniform acceleration,
they give the potentials due, not only to the given motion for all positive
time, but in addition that due to a motion taking place for all negative
time according to the same law, e.g. with uniform retardation in the reverse
direction.
The complementary terms in the second lines of the equations give the
correction to be applied in order to neutralize the effect of the motion for
negative time, which is continuous with the given motion, but in the case of
a discontinuous motion actually does not take place. These terms are due
to waves which would be emitted at negative times if the actual motion were
really continuous.
In the language of Ch. III, § 14, the first line represents the effect of the
whole continuous characteristic curve; the second line represents the correction to be applied in order to replace the negative branch of the continuous
characteristic curve, which is not required, by the negative branch of the
discontinuous curve, which is actually present.
Thus we see that for discontinuous motions the only general method is to
use the integral expressions (12) and (13), § 9, or their equivalents, the point
laws, (26)-(27), § 13. At great distances from the charge the former may
be replaced by (131) and (132), § 86.
For all continuous motions with a velocity less than that of light we may
use the series (178) and (179) at a great distance from the charge, and (180)
and (181) close to it, provided that in either case they be convergent. This
point requires special investigation.
It is possible that these series may also be used in cases of discontinuous
motion, if the complementary terms can be determined. In this case however the value of these terms must be found before we make h equal to zero,
in order to make sure of the convergence of the integral.
We shall now pass on to the consideration of some problems in illustration
of the methods of this chapter.
127. Problem 1. To find the distant field due to a monoperiodic motion of variable amplitude. We assume that the motion
involves a single period, but that its amplitude varies, so that it is not
strictly periodic. For instance, we may have a charge executing a damped
vibration in a straight line, or moving in a logarithmic spiral, or some similar
curve. The potentials cannot be worked out by means of equations (140) and
(141), § 89, because such a motion must have had a beginning, before which
the charge was either held at rest, or was moving according to some different

AND APERIODIC
~PSEUDO-PERIODICMOTIONS15

ix]

IX]

1.77

law. We must therefore use (131) and (132), § 86, which we may write in
the form
rmA

t=

~
r
cr

K=cos
0fI 00a
00ef

(t - r

cos
=

o

ir

-

c r + p/c) di-d

(t - r/c -

,

T+ p/c) vdrd

-0

o

t.

If we suppose the given pseudo-periodic motion to start at time zero, the
part of the integral from 7 = 0 to T = cc represents the effect of the required
motion, the remaining part the effect of the preceding state of rest, or of the
preceding motion, which is discontinuous with the required one in form.
The values of the displacement and velocity of the charge are necessarily
continuous, but those of the acceleration and higher differential coefficients
will generally be discontinuous (one at least must be so). The effect of the
discontinuity must be found by Stokes' method. If this be done, we may
take the time integral from 0 to , hnd the result will give the potentials for
any time t greater than r/c.

co

P

Suppose that

where f () is for the present arbitrary.
the times mentioned

Cos /

r=r

We get at once for the potentials at

t- r/c-r+f
(T)sin'(r+

sin
0cos itt-r/o-r+f()
well known equations*

=je1

a

Now we have the

cos (x sin

.... (182),

=f () sin (WT + a)............

4) =

)} dd

.,

(WT+ca)lv dd

t.

2J,, (x) cos 2s(f),

!'

s=o

sin (x sin40)=Y 2J+ 1 (x) sin (2s + 1) r).
Hence
2e 5 =o f

f)-

c
iJ28{

'

£"J+1{ f()
J j hfio t(r)
0

2e
-

s

0

s=00

a=- 8rr Jo J
cr~'
=0
2e 8 00
2e

fO

{

o

_

(t -r/c - )cos 2s (err+a)d'rd

tf (T) cos

°°

cr~r s=IJo

sin

f(r)} cos

(t

-

+1)

r/c -r) sin (2s

(t -r/c-

) cos 2s

(w

(WT+ac) d~d,

+ a) vdrd

°

J

2+1
*..a.ad.

{i4(r)} sin Ft (t - r/c - T-) sin (2s
ateweselFucton,.

18.

+ 1) ((or + ac) v dTd

t.

158

158

~PSEUDO-PERIODIC
AND APERIODIC

MOTIONS

[H

[OH.

These integrals can be transformed further. For we have

128.

0J

(
Jo ()cos

4d

=

cos

x} for 0<x< 1

{2s -x2)
1(ix sin
1

forx >1,

=0

"0J(

and

8

sin

(x)

d

=-

sin {(2s + 1) sin-'

fo

\(I

xfor

0<x<1,

X)

-

forcx>1.
=0
It is obvious that the integrals for negative values of x are got by simply
changing its sign. They are continuous at x =0.
Put

4= f(T),

x=(t-r/c- i-)/f(T).

The critical values of x, ± 1, obviously correspond to T =
where
=.......~...............(183).
'r~ ±f(ri)
Tl

'T

T

and Tr=T

+f(T)=t-r/crl

-f(T2)

t

=

r/)

-

By (182) we can always arrange matters so that f () is essentially
positive, by allowing p to be negative as well as positive, according to the
sign of sin (er + C).

Then we have by (183)
'T< t -

r/o <

'T.

Two cases arise; in the first case Tr is negative, and the lower limit of the
T integral becomes zero. In the second 'r is positive and itself gives the
lower limit of the integral.
We now find on substitution, for case
I.

< 0.

T

2e,9=00(T2cos

=

2s (WTr ±

a) cos

sCy(T2

r7' s o oE!(r]{
sin (2s + 1) (cor
-2e
o JO

-r'rrs

2s sin-' [(t - r/c - T)/f (T)] d'r
- (t - r/c - T)2}
±+ca) sin (2s ± 1) sin-' [(t - r/c
V{[f')- (t - r/c - T)2}

-

T)/f (T)] dT
(184),
...

.. . . . . . . .

7

- 2e8=00f cos 2s (err + ax) cos 2s sin-' [(t

r/c - T)/f (T)] yd

-

(t - r/c- T)2}
_2e 8= °f T sin (2s + 1) (err + ax) sin (2s + 1) sin-1 [(t
cr~r
o[f

cr~r

~o

(T)]2 -

V{[f ( )]2

80so

/

-(t

-

r/c -

T)/f (T)] vdT

)21

..................
(185),
......
with

II.

T27-f
T,>

(2)

=

t

-?/C

0.

The same expressions with the lower limit equal to 'ri, where
T1
*Nielsn,

+±f = t
(T)

-

r/c.

Cyi ndenrfun.
ktio n,1p.

5.

ix]
PSEUDO-PERIODIC

IX]

AND APERIODIC MOTIONS15

159

The conditions as to the limits are best seen from the diagram.

AB is

the line Tr=t - r/c, CD is the curve (1)
T1

+f

T2

-

r)

f(2)

=

t - r/c, and JIFH the curve (2)

=

t - ./C.

(1)
D

B(2 )

JI and IC are each equal tof(O).
When t- r/c <f(0), we have case (I),
and the limits are from E to F.

-

When t - rlc >f(0), we have case (II),
and the limits are from G to H.

J7-

The second case represents, as it were, A
the permanent regime and is obviously the
Fig. 38.
most important; the first case represents a
transition period during which the permanent regime is being established.
It is clear that this transition period lasts only a very short time, for f(O) is
merely the initial value of p/c (by (182)), and thus is of the order of the time
required by light to travel a distance of the same order as the initial amplitude of the vibration. For this reason we shall henceforth only consider the
permanent re'gime (II), for which the limits of the integral are T, and T2.
129. We can now transform the integrals (184) and (185) into a form
more suitable for calculation. Write
t

~V[f (T)]2-(t

g
The limits for

T

f)

dr

Weget

= cos x.....................(186).
_

-

d

d

r/c - T1)2}1_dx Jf(T) sin x~

x are :
x=O,

t-r/c -i-=f(r,),

for Tr=T1,
forTr= T2

- r/c -

)

t --r/c-

=

2

f(

x= i

2 ),

Thus the limits become independent of t in the permanent regime.
Moreover we get
cos 2s

sin-' [(t

sin (2s + 1) sin-' [(t

r/c

-

-

T)/f (T)] =

(- 1)8 cos

r/c

-

T)/f (T)]

(- 1)11 cos (2s

=

2x

+ 1)

Lastly we require to express the functions
cos 2s (WT'+a) dr - d fTcos2s(car+a)dr
f (T)
dyjdxJo
f(r)

sin (2s+

1)Qor+a)d'r

f (T)

d

-

d f T sin(2s+ 1)Qo'r+a)d'r

d

f (T)

X.

160

PSEUDO-PERIODIC AND APERIODIC MOTIONS

X.

as functions of
We have by (186)

[CH.

This is best effected by means of Lagrange's Theorem.

F(T) =F(t -r/c) - csxft-rc
+

'(

) csx tn-1
1!a!

(

/)+.
- r/c)}n F' (t - r/c)] ..

[ft

Thus we get
fTcos 2 s (wr + a)

t-rc cos 2 s (CWT
+

d7

f(r)

0

0f

C

+.. +- )n

a)dT

tx

[{ft-

~ cs~xn

I

a}

( )

with a similar series for the second function.
orc
co_

t t-/C)+
-

-cscs2s

r/c)}n-1 cos 2s {w (t -r/c) +a}]

Hence
[ {f (t -r/c)}n cos 2s {wo(t -r/c) + c]

with a corresponding series for the second function.
Substituting in (184) we get first a series of terms of the type
cs
{ t
2e (- 1)n+s a/cn
cohxos2d.
atn [{J(t - r/c)}n cos 2s{c(t-r)+a]
2e n
The integral vanishes when
When

nmis

even,

=

2j it is

it

is odd.

equal

to 2j'.j

2 + s!

when s

j-s

<j,

otherwise

zero.
Secondly, we get a series of terms of the type

2e (-

1)n8+s+1 anl
2
mn! at[f(t-r/c)}nsin( s+l)to(t-r/c)ca1]

r7r

Jo

SH x cos (2s + 1) x dX.

The integral vanishes when n is even.
When nis odd, =2j +1, it isequal to

22j+1

2
"+l+!

1

s!,

provided that

s <j, otherwise it is zero.
We put n = 2s + 2k, or 2s + 2/k + 1 as the case may be, and sum from
k =0 to k =oo. We get a first series
2e
r (_

k=00
)s

1

1

2s+2k

2s+k2 + k ! k !

at2

" f(t
[
U

- r/c)}2s+2k, cos 2s

too al]
(t - r/c) +

and a second

2e

k= ll

1

/a28±2k+1l

[_f!.-_rc)}2±Sk1-.sn(s+

){

t-1/)+~]

~PSEUDO-PERIODICMOTIONS11
AND APERIODIC

ix]

IX]

161

which may be included in a single series by introducing a - 17r for a.
we get

8=00k=°0

2e

r

8=0

1

Hence

a\+2kk

k=o2+1 . s+k!! kat,
x [f(t-r/c)}8+2k. cos s {w (t - r/c) + a

.(187).

-

In the same way we get

2es=oo k=oo

1

/a\S+2k

2+.s +k! k!

o=
0 kO

x [f(t - r/c)}8+2x . coS

at)
tw

(t - r/c) + a

-

fir} . v (t

r/c)].. .(188).

-

These equations could have been deduced from the series (178) and (179),
by neglecting the complementary terms. But the present process is
more rigorous, provided that the velocity be less than that of light, and has
the further advantage of showing that the effect of the discontinuity,
represented by the complementary terms, disappears when the permanent
regime is reached.

§ 123,

130. So long as the series (187) and (188) converge, we may differentiate
them with respect to t, and derive the electric and magnetic forces by means
of the equations
d=a (rl-a),
cat(
a=

)

cata]
We get at once

4.

2e s=ook=

c8=
0

a
ays+2k+1

f{f(t~r/c)}+2k o
s2k.s+I

w(t

-\r

)

(

zr

V .r1

v)

k=O0a

..................
(189),
......
2e

c
= cr

8=00 k=c
=o
°/

s+2k+1

8

c)+2
{f t! cos stw(t r/c) + ai-7} .[v .rl]}
k'
s2.s+

.............................
(190).

k

The series for
gives the harmonic of order s, and frequency sco. It is
of order s in the amplitude, and therefore generally small. We shall now
consider an example.

131.

Example.

Damped rectilinear vibration.
=aE-KTsin (wr+a),

forr)>0.

Using polar coordinates (r, 9, c0) we get
p=

n

AR 9
os

KT sin (..,r+4a)

We write

~PSEUDO-PERIODICMOTIONS[CI
AND APERIODIC

162

162

[en.

whence by comparison with (182), § 127,
a cos 0
-TE

~)

Substituting in (187) and replacing the cosine by an exponential, of which
the real part only is to be taken in the usual way, we get
cos

a

a

co 9

2e
2e

The

s=oo

9)s+2k

)cs+2k2

{

c E-K (t--r/c)

modulus

of

(SO + ((ss+
2k)

I~

2cE

k=°°

2

the

general

2

term

E,1c~o
-K(t-r/c)

of

v

r

this

{s2W2

+

seresi

(s
S

k2

s+k!k!

Ultimately it behaves like
2e

{rca cos9

s+2k!

E-K (t-r/c)V

C(s

r'-r (s + 2) 2+2k s+k!k!

1+

82(02

+ 2k) 2

fl+2k
Ic25I

It is therefore convergent for all values of 9 provided that
log (EKa/c)

t - r/c >.......

................

(192).

If 1 be the logarithmic decrement and X the wave-length of the vibration,
Elca

we have--

24la

Unless the damping be enormous this is very small
Condition (192) may be written
c (t

-

r/c)> log (Eica/c)

Ea

EKa/c

= 1,

The greatest value of the right-hand member occurs for K~a/c
or
21a/X = 1, and its magnitude is
Hence we must have c (t - r/c)> a,

1/c.

which is precisely the condition we obtained on p. 159, for the existence of
the permanent regime.
As a matter of fact, in all practical cases the convergence of the series is
very rapid indeed, because /cealo =2Ela/X is an exceedingly small quantity.
For this reason the only terms which contribute appreciably to the k-sum
are those for which k is small. For these terms we may neglect 2krc in the

~PSEUDO-PERIODICMOTIONS16
AND APERIODIC

Ix]

I%]

quantity tsw - (s+ 2k)
K.
gives

163

The k-sum in (191) can then be effected and

(w + tic)

E~L8m Isa cos 8
J.

FttJ8

(tr/C)
1

-EK

C

and we get
2e

cos 9 (C +

8_a°s

M)

0c
8=

r

the real part alone being taken.
132.

The Bessel Function is easily developed.
K/o=tanE,

Write

wa/c=

The Bessel Function may be written
J 8 {s/3 sec e cos 9. e-K(t-r/c) . ete}
sica 1
cos89/c

(sa

COS

(t-rlc)

8/C)2

2

J

t-rc

s/3 sec

Js

-c(t-rc)

(S+2o9/)2

JS (s/3 se eacos 9.E- K(t

e cos 9.

s

r/c)} Ese

e

6 -t

2

C

-K(t-r/c)} e(8+ )e

On substitution in (193) we get, taking the real part,
2e8g=
b= r

(

J8

{s/3 secea cos 9 .

e - (t -r/C)} .

cos 8 tco (t -

r/c) + a + e}

srca cos 8/c ,-K(t-r/c)
.cos [s {o(t

+

-

r/c) + a+e} + e -

(sKa cos 9/C)2E- 2K(t- r/c).

2J!

.~

r]
Kt rfc)
(t-

[c,(t
a +} + 2-?]
.......................... (194).

c os [s

+..

3sceco
-r/c)+

The mode of formation of successive terms is now obvious.
If the damping be so small that the terms involving Kcas a factor may be
neglected, and the principal term only retained, we see that we can take
account of damping by giving to the amplitude the value it had at the time
t - r/c, namely aE- (t -r/c), that is, we may use the instantaneous value of the
amplitude.

K be

In fact if
zero, equation (194), or (193), leads to precisely the values
we found previously for the forces in the absence of damping in problem 2,
p. 120.

164

PSEUDO-PERIODIC

AND APERIODIC MOTIONS

[CH. IX

If on the other hand the motion be aperiodic, so that o = 0 and a = 0, we
get from (193)
2e _
C sKa cos9 -(t-ric)}
= real part of

=2e8 Z-(- 1
1)8

7

e' J,s

I,

2sca cos

s=o

061
e-(t-ri,) .............. (195),

C

where 1, (x)= -8SJ 8 (ix) in the usual way.

We shall refrain from calculating the vector potential and electric and
magnetic forces, for our chief object in this problem is not so much to determine the field in all its details, as to investigate the character of the
deviation produced by a small variation from strict periodicity, as for instance,
a small amount of damping. This question is sufficiently illustrated by the
expression just calculated for the scalar potential 0. The chief result is that
in the permanent regime simple harmonic components exist, but with slowly
varying amplitudes.

CHAPTER X
ON THE FIELD NEAR THE ORBIT OF A MOVING CHARGE OR GROUP

133. [FoR the purpose of studying the motion of a charge, and more
particularly of a group of charges, we shall require to know the field at points
on or near the orbit, not necessarily close to any one of the charges. In
Problem 2, Appendix A, an investigation is given of the field close to a
charge moving in any prescribed manner, but the results obtained there are
insufficient for our purpose. Our present object is to calculate the field at a
point, which is on or near the orbit of the electron producing it, but at a
distance from the latter of at least several of its diameters, so that it may be
treated as a point charge.]
The most rigorous and complete method would be to use the equations
(18) and (19), § 10. This however would be a complicated process, and is
not necessary for our purpose. Nor can we introduce our simplification that
the electron is a point charge, directly into these equations, because the
integrals with respect to T would then diverge.
For this reason we shall use a different method.
134. Potentials. We have just stated that in order to find the equations of motion of a group of electrons, not merely in its permanent state,
but also when it is slightly disturbed, we require expressions for the field due
to the permanent motion and the small disturbances from it at points close
to the orbit as well as actually on it. It is convenient to treat the whole
problem at once; for this purpose we must develope the potentials as far as
small quantities of the second order. We start from equations (12) and (13),
§ 9; for a unit point charge they may be written
0=

27r

a=2Jr
In future we shall suppose

8

00

-

1

R
R

,

drd,, .................. (12),
vdrd , ............... (13).

1,
R, and v to refer to the undisturbed

motion and (x, y, z) to be a point on the undisturbed path of the charge.
We shall denote by prefixed to any symbol the variation by which we pass

166

ON THE FIELD NEAR THE ORBIT OF A MOVING CHARGE OR GROUP

[CH.

from the permanent to the disturbed motion, and from the point (x, y, z)
actually on the undisturbed path to a point close to but not on it. For
example, SR represents the vector whose components are (6x - 6, 8y 8z - 8)), and SR, the unit vector in the direction of SR. For shortness write
p = (R, . sR), so that p is the projection of sR on the vector R drawn from
2 _p
( , , ) to (x, y, z); and write q=
so that q is the length of
the perpendicular let fall on R from the end of the vector SR. Then we
have

V{J(8B)

2
(R +8R) 2 =(R+p) +q2 ,
S+ R =1 + p + q2/2R,

1-

1

p+

R + R R

-

E {t-(R+8R)cT} -1_-(p+-9i1

(t+p
R/C-)

=

{

2

2

R3

+22(1a)2}

catl

2 2
(-R- 2+-q' ) cat

- q

2R3

R

2p 2 -q

2R1 cat

.

Hence

2

R

(t

- R/c-T)

2R-(cat)I2ELq2aIa
a

22ct2

All these expansions are correct as far as terms of the second order.
for 2p2 - q2 its value 3p2 - (8B)2 , and substituting in (12), we get

q5 +

0Jl1 -

=21J"0

()
i:JL{-

-

32 - 8(
2

p

R

ca\2c_

2
'U

It

+

2(8)2

pB+A

(

2R

(t-1/c-T)

(t-R/c-T)

E

(

Putting

dTd~

d

drd .

We treat the integrals just in the same way as the integrals (12) and (13),
S9,
which the point laws were deduced. Changing the variable from Tr
to t', where t' = T + B/c, we find that

from

A+ 3kF1

p

+p

L KR KR2 +

2

-(8B)2

2KB3

J

a

[p

cat LKB

-3p2--(8R)2~
2KB 2

+ (c)2 2KB]
As before K

is the

Doppler

factor

at,

and the functions of

J
.'."(196).

Tr
within

the

square brackets are to be transformed from functions of the time of emission,
Tr, to functions of the time of reception, t, by means of the characteristic
equation t =T + B/c. It is assumed that this equation has but one root T less
than t, which implies that the velocity of the charge is less than that of light.

X]

ON THE FIELD NEAR THE ORBIT OF A MOVING CHARGE OR GROUP

167

It is assumed that the fieldpoint (x, y, z) is at a sufficient distance from
the charge to permit of our treating the latter as a point charge; otherwise
we must divide the charge into sufficiently small elementary charges and
integrate for the whole.
In precisely the same way we get from (13)

c(a+a)=
( +
)

pv

av

KR

KR

a

-

pv

cat KR

+p

+3p2

2

-

(8R)2

2KR3

3p2.(SR) 2
2KR2

av

v+ KR
+ pv
KR

P 8V

KR2

... 197).

2p

cat

2KRJ

135. Electric and magnetic forces. The introduction of the displacement (Sx, Sy, 8z) of the fieldpoint enables us to deduce the forces very
simply, so far as they involve differentiations with respect to the coordinates
(x, y, z). For now we may treat these coordinates as invariable, and instead
differentiate with respect to (&x, Sy, 8z). The terms of first order in the
potentials then give the parts of the forces which depend on the permanent
motion, and these for points on the path; the second order terms give the
parts of the forces due to the disturbance from the permanent motion, and to
displacements of the fieldpoint away from the permanent path.
In differentiating with respect to the time we get the corresponding parts
of the forces due to the permanent motion from zero order terms, and the
parts due to disturbance and displacement from the first order terms, while
we may neglect the second order terms altogether. It is however convenient
to delay the differentiations with respect to the time as long as possible.
In the equations giving the forces, that is (VII) and (VIII), § 3,
h + h = curl (a + sa),

d + d = - grad. ( + 80) -

(a +a

the operations grad. and curl now refer to (8x, 8y, Sz) which occur in p and
8R, while x, y, z and R must be treated as constants. We easily find that
grad. p = R 1 , grad. (SR)2 = 2sR, curl (p. v) = [R 1 . v],
curl {(SR) 2 . v} = 2 [BR. v], curl (p. sv) = [R1 . Byv],
while such quantities as grad. (KR), curl

BY,

and so on, are to be considered

as vanishing. The terms in the forces which would arise from them are in
fact already included in those derived from the first order terms.
In this way we get, from (196) and (197),
d + Sd

=[KR2

cat

KR

a [BsR - 3pR
[sBR - 3pR1
+cat L KR2
KR3
[ K

+

-KRaI

pv

v

+cKR cKR
) J......... (198),

168

ON THE FIELD NEAR THE ORBIT OF A MOVING CHARGE OR GROUP

h+

=

[CH.

R1] + a [v.R]
+[v.
KR2 J

cat L cKR

+ [[v . (sR - 3pR1)] + [sv. R 1]
cKR2
cKR3
( 2 p[v.R,]
[Sv. R11]
a[[v. (R -- 3pR )]
1
kcatLcKR
cKR2
BcKB
+ catL
The first line in each equation gives the force due to the permanent
motion at a fieldpoint on the permanent path, and the remaining ones give
the forces due to disturbance and for a fieldpoint close to but not on the
permanent path.
A notable simplification of the work arises from the fact that certain
combinations of symbols, such as [v. R 1], SR -. 3pR 1, and [v. (SR - 3pR 1)],
occur with varying factors, and affected by a varying number of differentiations with respect to the time t. Thus in the expression for d +-Sd the
combination sR - 3pR1 occurs undifferentiated with the factor 1/KB 3, and
differentiated once with the factor 1/KB 2 , and so on.
As an illustration of the method we shall consider the problem of a point
charge moving uniformly in a circle.
. ."(199).

136.

Problem. A point charge moves in a circle with uniform
velocity. Required to find the electromagnetic field at a point on
the circle. With the notation previously employed we write
=pcos(W'r+6),
=psin((O +8 ),
Further, the coordinates of the fieldpoint are
x=pcos4,
With these values we get
where

x

,S=cp/c.

y=psinc, z=0.

B = {(x - )2 + (y -17212= 2 p sin -- (WT +8
is a new variable defined by the equation

In the same way write

=0,

-4

=

2

p sin x,

x =' (WTr+8~-4)4

Then the characteristic equation t = Tr + B/c gives

~k= x+/3sinx......................(200),
and the Doppler factor is given by

K=at -*#1 +13cs

a7 ax

The transformation from the time of emission Tr to the time of reception t
by means of the characteristic equation, indicated by the use of the outer
square brackets in (196)-(199), now reduces to a change of variable from x

x]

ON THE FIELD NEAR THE ORBIT OF A MOVING CHARGE OR GROUP

169

For the purposes of the present problem we only require the zero order
terms in (196)-(199), which involve the vectors
R1, v and [v. R 1]. We require their compoV !
nents along the inward * radius vector P0 atEE
the fieldpoint, perpendicular to it along theE
T
tangent PT, drawn in the direction of motion,
and perpendicular to the plane of the orbit.
In Fig. 39 the orbit is the circle PE'E.
P is the fieldpoint (x, y), so that x OP =
E' is the position of the point charge at time T,
so that POE' = 2x; and E is its position at
5
time t, so that PO E=2-4. The vector R1 is
Fig. 39.
along E'P, v is along TE', and [v. R 1] perpendicular to the paper and
upwards, along Oz. We get for the components of these vectors in the

directions p, (P0),

0,

(PT) and z

Rip=-sing, RIO=-cosx, R 1 =0,

2X,

v=c$3sin2x, v4=c cos
[v

.

R]P

=

,

[V.

R1]o

=

,

;=0,
$snX

[V.R]z=

Substitute in (196)-(199), replace sin' 2 x by 2 sin x cos x and cos 2 x by
1 - 2 sin2 x, and notice that the expression cos x/sine x (1 + $ cos x), which

(it_).

occurs in do, may be written in the form - a

After a little

reduction we get
1__
2psin x(1 1+,8cos

a=1 -

x)'

1

aP=P

(1

hz4p2

sin

137.
functions.

a L4p2 sin x(1

'

$sinx

2p sin x(1 +3cos

d

1 3 o)

dP=-4p2sin x(1

a

x)

3

+,8cos

P (1+,8cosx)'

+

X2sin

x)+2p2 (1

x

+,8cos x)j

+,8cos x) + a4p2 (1+-}-cos x)l

l........

x 13cos x)+4a*[4p2(1 +3cos

"

"(
201).

It is to be noticed that these expressions only involve three distinct
Two of

these, 11

Series with
z
*~

~
~ ~

acsx d

4'1$o variable ; the
as in Fourier

13si
x

are at once

+,os si
third,

1+

a
d
Th1naddreto
~ a ~ ~~~
ae ~teai+zrgh-addrlaietch

i

expansible

osx
otowqs,

being
pan

170

ON THE FIELD NEAR THE ORBIT OF A MOVING CHARGE OR GROUP

[CH.

infinite at the limits 0 and it is not so expansible. But its excess above the
function /sin Jr vanishes at both limits, and may be expanded. Write
1
sin
+

sin

=1+

1

+acos X

1

jzCi

1
P X
C

(l+3

jr
x
2=00As
ij
2C "sinj#.
n
"=
l+1acosx

(

cos

'

)

i-

2B'cosj's,

j=1

2A

sin

+

We get, by the usual method, using (200),
A

r

1

A=
3

o

sin (1 + cos~dr
x)-sin
1-1sn

d

i

In the first integral change the variable from 1Jr to x, remembering (200),
and in the second change the notation by writing x in place of fr. We get
A
injo1

j'sinj (x +13 sin x)
sinx

f dx

-j

f

sinx

x

f

cosj(x + xsinx) d =(-1)j

Bj = (- 1)j Jj (j$),

Similarly

sinjxd

Cj = - (-1)i J,'

J"(jx) dx.

(j3).

138. Substitute in the expressions for the potentials and forces, perform
the necessary differentiations with respect to 1fr (which are obviously
permissible), and supply the charge e as a factor. We get
e.70

e

_

(-1)sin lf

n)

2

2psin#
=

2psin

Jj (Jx) dx

Pj=1

e13
+2e/3
p

j=1

()sinj2)r

,x

Pi=1

e (1I3'2)Cos#jr4p2 sine

d

I'

e2 '

(_
._(=1

iycosj#'jj2
Lj)

J'

-

-1
sin if
4p~2
=
=4

sin#
p2e13

(j/

2j

)

Jj
O (jx) dx1
j

J-

l~j
1sin j*

(-

J(jx)

p2 j=18)+.ijx,

_- 2 2
p

Y.(-1j
\1

)

sn '-

F
/J
jL

(/)

'

Jo

d

J" ('x)dx]

The remaining force components vanish identically.

(203).

i,0

from

These series are available so longs as the distance of the fieldpoint
the charge, 2p sin J4r, is at least equal to several multiples of the diameter of
the charge. This restriction is not serious in problems dealing with the
circular

motions of electrons, even when the diameter of the circle is much

X]

MOVING CHARGE OR GROUP

ON THE FIELD NEAR THE ORBIT OF A

171

139. Group of n charges. When we have n charges moving uniformly
round the circle at equal angular distances apart, we may take -Jr, which is
equal to 2 (wt + 8 - 0), to be one-half of the angular distance from the fieldpoint to any one of the charges, which we may number zero. For the ith
charge we must replace 8 by 8 + 2'ni/n, and *r by iJr + 'ni/n, and in order to
get the field due to the whole group at a point on the circle we must sum
(203) fromi =to

1.

i= nIE

We have

7ri/n)

kj(1+

-

neL2
2sn%,
for

i=0

=

j=
2sn,

0, for other even values of]j,
2

E ( + ) (i-it/2n)
=

In this way we get
(1

=-e

-

-,/2)

2

i=n-~1 cos

6k
i+

ti(
s

2 s+l)/

j= 2s+--1.

for

2

in-/n)

4sine+(ijri/n)

2ne 1 [sni32J;8fl(N2sn$,)

s2n2 (1-

-

.

/32)

(2snx) dx] cos 2sn*

Pes=o[
xjf0 J

sin=n) i
(*4,+
ne24 [sn/3J (2sn/3) +

1

{(2s+1) }dx
+1x
x+

sin (2s +1)(t

'

sn(2s+1)i

S2j=

eP811

+

s2
Y[

+

2s + 1) /3J

s

s1{(2s

J2

(snx) dx] sin 2sn*j

\

) fa=
+ 1) /3} + (28+ 10J

1t(2s + 1

cos(2s+1)

h
r
}d
-7

s i n2n
hs=iZni(m+an/n

co(2s+1)ir1
x2

sin-21

. . 20 )
(0)

172

ON THE FIELD NEAR THE ORBIT OF A MOVING CHARGE OR GROUP

[CH. X

In these equations the variable Jr may be taken to lie between the limits
0 and nr/n without any loss of generality, but a very small range at either
limit must be excluded, because of its nearness to one of the electrons. If it
were not for the necessary exclusion of these portions of the range, we might
be tempted to find the limit of the forces for infinitely large values of n.
We cannot however use our results for values of n beyond a certain limit,
which depends on the ratio of the diameter of the electron to that of
the orbit.
For small velocities the harmonic terms vanish, and the remaining ones
reduce to the values of the forces for static charges, such as those employed in
the theories of Nagaoka and J. J. Thomson.
In general all the odd harmonics are present, as well as even harmonics of
order 2sn. For anything like large values of n the latter have exceedingly small
amplitudes, but the amplitudes of the former for small values of s are not
small except when
is small.

$3

These equations will be of use later in finding the mechanical force acting
on one charge of a group. We shall not consider them further here.

CHAPTER XI
THE MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

140. HITHERTO, in calculating the electromagnetic field due to electric
charges moving in an assigned manner, we have only made use of the
equations (I)-(V) of Ch. I, § 2, and of the equations which follow directly
from them. We have already pointed out that the determination of the
mechanical forces acting on a charge involves an additional hypothesis, and
have adopted that due to Lorentz. This assumes that the mechanical
connection between the charge and the aether is such that the resultant
mechanical force on the charge is completely determined by the electric and
magnetic forces on the one hand, and the velocity of the charge on the other,
according to the equation

f= d + [vh]/c .............................. (VI).
The assumption expressed by this equation implies that, whatever the
mechanical connection between the charge and aether may be, that part of
the reaction due to it, which is not included in (VI), reduces to a system of
stresses which can consume and do work, but contributes nothing to the motive
force because it is self-equilibrating.
That such stresses must generally exist is obvious when we consider the
case of the deformable electron, for example that of Lorentz. It is well
known that, when this electron is accelerated, its gain of kinetic energy is
not all accounted for by the work done by the motive force, but is in part
derived from work done by the aether in compressing the electron, according
to the hypothesis of Poincar *, a difficulty first pointed out by Abraham.
141. The total force on a charge is given by (XIX), Ch. I, § 6, which
may be written in the form
F =

-

t

fade - f{V (0

- (va)/c) + xa/c - [coa]/c} de ... (XIX').

Here wois the angular velocity of rotation at the element de, and xa is
the vector whose components are
(ear + cay + baz, ca., + fay + aaz, bax + aay + gaz),
* Comptes Rendus, cxi. 1905, p. 1504.

174

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

[CH.

where (e, f, g, a, b, c) are the six components of velocity of pure strain at the
element.
For a perfectly rigid electron, such as the sphere of Abraham, the strain
is zero. Its angular velocity is found to contribute to the resultant force
only terms of higher order in the dimensions of the electron than the
remainder, so long as the velocity of the charge is less than that of light.
For a deformable electron the corresponding terms are important. For
instance, in the case of the Lorentz electron, which is well known to be an
oblate spheroid, with equatorial axis a, and minor axis a /(1 - v/C2), lying

in the direction of motion, there is a homogeneous compression parallel to
the direction of motion at the rate - Vv/(c2 - v2 ), together with a rotation

about the binormal at the rate v/p. Hence the pure strain is given by
e = - I33/(1 - /32), with the remaining five components all zero, and the rotation
is given by oe = v/p, with the remaining two components zero. Thus the
vector xa/c - [wa]/c has the components

-c-1c

a + ay, - -as,0,
p

-)

P

where , = v/c as usual.
It follows from (XIX') that on the hypothesis which we have made, the
mechanical force is completely determined when the motion of the charge
and the potentials 4 and a of the field are given. In practice it is often
more convenient to work with the forces d and h, and use (VI) directly.
142. Whichever method of working be adopted, it is convenient to
separate the mechanical force acting on one of a system of discrete charges
into three parts:
(1) The first part is the resultant of all the internal forces, due to
interactions between the elementary charge under consideration and the
remaining elements of the electron; it is got by a double integration over
the whole electron, and will be denoted by the symbol F,.
(2) The second part is the resultant of the actions of the external field
on all the elements of the charge, so far as that external field is due to
charges outside the particular charge considered but belonging to the same
system. We shall denote it by the symbol F 1.
(3) The third part is the resultant of the actions of the external field
on all the elements of the charge, so far as it is due to charges outside the
system. We shall denote it by the symbol F2.
The distinction between the second and third parts of the force is
somewhat arbitrary, but it is convenient when the system of charges forms
a group. When we do not wish to draw any such distinction, we shall use
the symbol Fe to denote the whole external force, whatever its origin, so
that Fe = F1 + F2.

XI]

175

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

The resultant internal mechanical force F. leads to a determination of
the electromagnetic momentum and mass of the charge, and of the mechanical
reaction due to its radiation. For this reason we shall consider it first. The
external force will be studied later.
143. The internal mechanical force.
The force exerted by a
moving charge on itself has been the subject of a good deal of investigation,
first by Abraham*, then by Sommerfeldt, and lastly by Lindemann+, in each
case for the rigid spherical electron, always on the basis of the Lorentz
theory with (VI) as a foundation§.
In the case of quasi-stationary motion-where the terms involving the
accelerations of higher orders of the electron, such as the torsion of the path,
are small compared with those involving the acceleration proper and the
curvature-we need only retain the terms of the two lowest orders. We have
(1) a principal term of the form
dG
d (my)
dt ........................... (205),
dt
where G is the electromagnetic momentum and m the "transverse mass."
For the rigid spherical electron of Abraham of radius a
3e 2 (1 /3 2
+al20)
-

52m

32

,

l

log 1 -

3

. . . . . . . . . ......

(206).

For the deformable spheroidal electron of constant volume of Bucherer,
with longitudinal semiaxis a (1 -

32)j,

and transverse semiaxes a (1 -/32)

-,

2

4e

m

=5ca /(1 -

/32)...........................(207).

For the deformable spheroidal electron of Lorentz, with longitudinal
semiaxis a (1

- /)2,

and transverse semiaxes a,
2

4e
m = 5c2a(1-/32 ........................... (208),

where = v/c as usual.
All these values are for uniform volume distributions; for the corresponding
surface distributions the values are five-sixths of these. In addition we have
* Ann. der Phys. 10, p. 105, 1903.

Theorie der Strahlung, p. 136, 1905.

t

Gott. Nach. pp. 99 and 363, 1904; p. 201, 1905.
1 K. Bay. A kad. ii. Kl. xxiii. Bd. II. Abt. pp. 235 and 339, 1907.

§ [It is calculated by a different method in Appendix D below, without any restrictions as to the
form and structure of the electron, but with the velocity restricted to be less than that of light.
The result agrees with that of Abraham and Sommerfeld for the rigid spherical electron, and with
that got by Einstein on the basis of the Relative Theory for the Lorentz electron.
Recently Walker (Phil. Trans. A, 210, p. 145, 1910) and Nicholson (Phil. Mag. [6], 20, p. 610,

1910) obtain a different result for the cases of a conducting and of a dielectric sphere.

They

however start from fundamentally different assumptions, in so far as the distribution of the charge

on their sphere is not rigid, but depends on the motion.
p. 640, 1911, and 21, p. 169, 1911.]

See also Livens, Phil. Mag. [6], 20,

176

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

[CH.

(2) a small term due to radiation, given by
C
K=2 2
3

__2

___

(ViV)v

+

(1-32

3c

++

c2(1-'32)2

3(i)

3/

c2(1-/2

(209).

32

2

from

Abraham, in his Theorie der Strahlung, deduces this expression
the
conditions that the force be equal to the rate of gain of electromagnetic
momentum due to radiation, and its activity to the rate of gain of energy
due to radiation, both rates of gain being of course negative. [It is deduced
by a direct method in Appendix D.]
144. Resolving the two forces (1) and (2) along the tangent (s), the
principal normal towards the centre of curvature (p), and the binormal (n),
we get the following expressions for the components in these directions of
the internal mechanical force Fi :
/m,

d
F3

-

3

C21-2+c

2e 2

V2
m

-

3.../32)3
1

38,

3cp(1

p

c(

,C3)

2e2

d

+
2)2

C2

2

21-2

)

2(

2

(-

)2)

2e2 3
F

n3PT(1

__

$2)

where p and T are the radii of curvature and torsion*.
145. Energy relations. The activity of the force Fi is equal to (vFi),
that is, to vFi. We get from (2101)
(vF) =

v+d
-

vdm/

(m$) + 2

(d (c2m,62)
d/3

-2 m)

(1 _-

/32)2

{

02 (1

2e2$

d

3

dt

-/32)+

P2

CmIN/

2ce2

-3

3
3
3$212$4

C

dC2(l-/2)2+C2(1--2)3

/3

p2

)

3c (1-,2)2~

(21)

$2
.................

(

1

)

The behaviour of the terms in the first and second lines of this equation
is quite different. Since m is a function of /3 alone, the first term, like the
second, is a complete differential coefficient with respect to the time of
*

[Let the direction cosines of the moving axes (s, p, n) with respect to the fixed axes (x, y, z)

be given by the annexed scheme.
Then we have i = l'v/p,

'=

l"v/r - lv/p,

Ii

1

with similar equations for y and z.
From these equations we get

x

Y

z

p

l'

m'

n'

n

l"

in"

n"

with similar equations for m

and n.
Moreover, we get
3 2
2
x= lv, x = + l'v /p, x = (v3 r /p ) ± ' (3vtfp

s____
-

3
v2Pi/p2 ) + l"v /pry

XI]

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

177

a function of the motion; but the third term cannot be expressed in this
form.
When we form the time integral of the activity, and so obtain the work
done by the internal force during any interval,

(vFi)dt, the terms in the

second line of (211) give a quantity which can be expressed as the difference
between the initial and final values of a certain function of the motion,
namely,
3d
3c / ) ]1........(212).
d -c2m
2
d

(c2,")dP

s83

2

-f

But the terms in the third line give rise to a time integral which
cannot be reduced until the motion is completely known, namely

-C2

+

)
3 (1

-

(

2

dt ............... (213).
2

&)

The quantity (212) is reversible; by this is meant that it would change
sign if the motion were completely reversed, and would vanish if the initial
and final motion were identical. It represents work done by the external
forces and consumed by the internal force, which is recoverable on reversing
the motion, and may be regarded as stored in the charge in virtue of its
motion. Therefore the function of the motion inside the bracket in (212)
may be regarded as the kinetic energy of the charge.
The function under the sign of integration in (213) is essentially positive,
and accordingly the integral never vanishes. It represents work done by the
external forces and consumed by the internal force, which is not recoverable.
It is not stored in the charge, but radiated away from it. In fact, the
function under the sign of integration in (213) is equivalent to Lidnard's
expression for the rate of loss of energy from a moving charge owing to
radiation [in accordance with the result already obtained in § 5, p. 8].
Returning to the kinetic energy we see from (212) that it involves two
terms: (1) a principal term, derived from the mass m and depending only
c

on the velocity; (2) a small term, - 2e2/3

2(1-

/32)2,

depending on the

acceleration as well as on the velocity.
146. If we adopt the usual convention that the kinetic energy depends
on the velocity only, and not on the acceleration, we must exclude the second
small term from the kinetic energy, and regard it as reversible radiant
energy. The distinction is not of much importance; but as a matter of
convenience we shall define the kinetic energy, T, by the equation
T= c2 m/
s.

-

f

c2md

=

cfm'3d3...............(214).
12

178

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

[CH.

From (206)-(208), § 143, we get:
For the Abraham electron
2

/1
-(2

6e

T = 5

log

5a \23 0g

1+0 l

1

f-

1

....................

(215).

For the Bucherer electron
e2

T=5a
For the Lorentz electron
4e2
a

3 +)3

1

1
(1 -

...................... (216).

- 3

2

2

)-

1 .........

.......... (217).

The last agrees with the expression given by Einstein on the basis of the
Relative Theory;
[The small acceleration term omitted from the kinetic energy is precisely
the supplementary term, which must be added to Lienard's expression for the
radiation, when we adopt a mechanical theory of the aether, as we found in
Ch. I, § 5, p. 9.]
147. In calculating the energy we have completely ignored the rate of
working of the internal stresses. The work done by the external force is
necessarily the same as before, for it is the work consumed by the internal
force Fi, and by the fundamental assumption of § 140, Fi does not depend on
the internal stresses of the electron, which are self-equilibrating.
Nevertheless, if the electron be deformed, work is done and its potential
energy is altered without any expenditure of work by the external forces.
This difficulty occurs with the Lorentz electron, as was first noticed by
Abraham. It can only be resolved by assuming with Poincard that the
electron is held together by a pressure exerted by the aether, which supplies
the work done in compressing the electron. This is indicated by the form of
equation (XX), § 6.
148.- Equations of Motion. The equations of motion are easily
written down on the basis of our fundamental assumption, that the resultant
mechanical force is completely expressed by the terms Fi + Fe derived
from (VI), § 140.
By Newton's Laws of Motion the resultant force must be equal to the
rate of increase of the ordinary (non-electromagnetic) momentum of the
charge, Mv. We shall suppose M zero, in accordance with the most recent
measurements of the specific charge of the electron, which makes the introduction of any non-electromagnetic mass unnecessary.
Hence we have Fe = - Fi, and get by means of (205) and (209), § 143,
+

+

+

=F

(218)

XI]

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

§ 144, we get
d (cm3) +
2e2$3
2e2/3
dt
+3p2(1 - /32)2 3c2 (1 -

179

Or by using (210),

2m
P

I

+3c

2262p

FS

2

2(1i1!r)

c21
P

2e2 3/32
C (1 _/3
2

2

-c

2e $8
(_

2=

p...(219),

2

2e2 3

where the suffix e has been omitted from F as no longer necessary.
We shall now consider some examples in illustration.

149.

Example 1. Uniform circular motion. Here we have f = 0,
3=0, = 0, T = co. For the sake of generality we shall admit the possibility
0
of a slow secular expansion of the electron, expressed by a change in the
parameter a at the rate a, and leading to a slow change in m at the rate m,
even when (3 does not alter.
With cylindrical coordinates (z, w, p), the directions (s, p, n) in (219)
coincide with the directions (0, -a, z) respectively. Hence we get
ch/3 +

2e2$3

2e

F4

2

c2r$

2

F

...............

(220).

150. Example 2. Disturbed uniform circular motion. We may
represent the rectangular coordinates of the disturbed electron by expressions
of the form
x =p(1 =wt

) cos (4)+X), y =p(1- p.) sin (4 +X), z =PV} .(2)
+.(21)

where p is the radius of the circular orbit and wv uniform angular velocity
the
of the undisturbed electron, and (X, , v) are functions of t, so small that their
squares and products may be neglected. Obviously p (X, Iz, v) are the components along the tangent, inward radius and axis of the displacement of
the electron from its undisturbed position at the same time t, when its
azimuth would have been 0.
We shall denote by the prefix 8 the change produced by the disturbance
in any quantity, whether scalar or vector *, and shall apply the operation 8 to
both sides of (218).
* [For instance, if F be the external force in the undisturbed motion at time t, with
components (F1 ,, Fp , Fs) along the tangent, inward radius and axis for the undisturbed position

180

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

[CH.

We get in the first instance

(

X 2
p (C
\co

_

,

,

i

Sv=CO

2!
,

3(3

v

o

2

w

3

+~~2 -i

...(222)*.

'-'

!

(

2

3

-3+2L

The first triplet of equations gives

80=)

-

S

c

=

dm

c-

(223).

.........

Denoting by m the secular rate of change in m, with /3 constant, due to
a possible change in the parameter a which determines the size of the
electron, we find
d
m

(

y)

18 o

m)-;

dd

m

md +

+)

i

_

dm

(

Substituting from (222)-(224) in
directions (X, Fti v), we get the following
2e23
'/2 d (c'm/) k+/d
3 2
p (1 )32)2
+
d$
3

+

2
2e 3
32

p 2 (1

)2

2

jt

/3d(c2m32)

d

v

... (224).

(218), § 148, and resolving in the
equations for the disturbed motion :
(cr/)+2e2/33(3+ $2)(
d
:3p2(1_)32)3Jw

d-d(cm

+)
d/21± 2e2$3 (1+33)

w

F

3

/3

2

m/32

AC+c

o

d$

w2+ p

P2(1 -1"2)2

+

3P2(1/-'2)w3

-

AInc

v +-

/3 d (cam/3) X.

2e2/33

- 210

d$3

P

2

c$

- -

2

2e3

v

-}-c

m/3

2

+182))

$. + 2e2$3 (3 2 2
}cmp2+3p 2
(l-/3 )

V**(

$ +

2e#/35

o

/ 2 d (c 2 m/3)
p

S-

2)23

d/3M

-

v

(225).
...

.. . . . . . . .

axis for the disturbed position of the electron, then the disturbing force is 8F, and its components
6
are ( FA , 8F,
in the directions (X, , iv), where 6F,=F,'
- 8F = F ' - FP%, 6Ff,= F,,' -Fz.]
*[These equations are easily proved as follows :
The direction cosines of the moving axes of (X, u, v) are given by the annexed scheme.
Let v' denote, for the moment, the velocity in
the disturbed motion, and so on. Then we have
x
y
z

,,,)

FO,,

We

vx +v'P(y - i) a-(4 +A), vv,=~
where (x, y, z) are to be got from (221).
We have y

(0

sn /+A)

-1) e
+a); differentiating
and hearing in mind that P= wp/c, we get
-

c x-=

p(

/

cos (q5 + X)

n h

xreso

frV'+

sin (p + X)

0

y-x C - +c E(+k.v0

Susiutn

cos (b+ A)0
-

I'~adeqaigreladwmgnryprswwe

0

1

XI]

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

181

These equations rest on the assumption that all the quantities
X, £/W, X/WI2 /, 3, ... are small. When the disturbance is periodic, with the
period T and wave-length A, this assumption requires that X (2irp/A) 3,
be all small compared with /3. When these conditions are satisfied, the
disturbed path nowhere has large curvature or torsion, and the change in the
acceleration of the electron always remains small.
...

The equations cannot be applied to orbits with short and steep corrugations, such as the looped and cusped paths described by the vertex of a top,
when it is disturbed from steady precessional motion.
It is important to notice that the equations are linear, but of the third
order, not of the second, as in ordinary dynamics. Hence generally the
frequency equations of a ring of electrons are at least of the third degree.
The equations (225) were first given by Schott* with a slightly different
notation.
As a further illustration of the use of the vector equation of motion we
shall now work out the motion of a /-particle in a uniform electrostatic
field.
151. Problem. A /-particle is projected in any manner in a
uniform electrostatic field. Required to find the motion.
We shall make the following assumptions:
(1) The mass is entirely electromagnetic, and is connected with the
velocity, v = c$, by the mass formula of Lorentz
m,=

m/V(1 -

32)...........................(226),

where m is now written for the mass corresponding to zero velocity.
This formula has been verified experimentally to a very high degree
of accuracy by Bucherer and his pupil Wolz t. The latest result gives
e/cm = 1"77.10' electromagnetic units.
(2) The motion is quasi-stationary, so that the small radiation terms can be
neglected. Since the value of 2e2/3cm is about 0"0035, this condition admits
of a value for / which is very close to unity, unless the acceleration and
curvature of the orbit become abnormally great. The degree of the approximation will be examined a posteriori.
* Phil. Mag. [6], Vol. xvr. p. 180, 1908.

=-d

(m, is used by mistake
[In the equations (12) of this paper the longitudinal mass,
instead of the transverse mass m in the coefficients of tj/w and i/f. in the last two equations.]
t Ann. der Phys. [4], 28, p. 513, and 30, p. 273, 1909.

182

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

152.

[OH.

Let the electric force be X electrostatic units in the direction of

At time t let the particle be moving with the velocity c3 in the plane

xOy,

and in a direction making an angle 0

with Oy.
Neglecting the small radiation terms for
the present, we may write the equations of
motion in the form
dt

e

dt

dt

-'

(227).
.... .....

The last two equations
my'= mvv cos 9 = eXa,
where a is a constant.
continues to take place in
since x~ = v sin 9,
a

give
(228),
Fig. 40.
Thus the motion
the xy plane. The fi
First equation (226) now gives,
mvi

= ...

d tan 9
dt=1, whence t=atan

(229),

.............

where the constant of integration is chosen so that 9 = 0 when t = 0.
Writing
b = cm/eX ....................... (230),
we find from (226), (228) and (230)

a

(231).

V~a2+..c...9..................

Hence we get

2
2
x=fvsin Odt= c V(a + b2 +t )
l

dt

2
2+ f

.(232),

where the constants of integration have been chosen so as to make x equal to
c V (a2 + b2) and y equal to zero when t = 0. These equations give
2
+ b2) sinh (y/ca)...(233).
x= c V(a 2 + b2) cosh (y/ca), t =
Let us choose the axis of y so that y and t have the same sign, and therefore y is positive. Then we see from (228) and (233) that a, /(a2 + b2) and x
all have the sign of e, that is, of b.

V(a

Initially we have x = c <(a 2 + b2), y =
initial value of fi we find that a = b$3 //(1
0
Hence we get, by (230) and (233),
cosh
eX\(10
m

-eX

t=

-

cm

<(1

-

2

eX<(1

=
-

2
0,=ca/,<(a+ b2). If /o be the
2
2
1302) , and </(a + b ) = b/V(1 - 132).

<Xi(1

2)

yeX~

c~/3
0

1802) sinh yeX<V(1-)302)
c m13
2

0

.. . .. .. . .. ..

(234).

.

XI]

183

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

153.

The first equation represents the curve known as the "menoclinoid f,"
(l132) and c2m13/eX V(1 - /3o2),
so that their ratio is 1 : ,30. The curve is got from a common catenary of
parameter c2in/eX V(1 - $3) by compressing it at right angles to its axis Ox
If the velocity at the vertex were equal to the velocity of
in the ratio 1
remained finite, the
light, and at the same time X so large that X (1
path would be the common catenary; if the velocity at the vertex were zero,
it would be the part of the x-axis beyond the point x = c2m/eX traversed
to and fro. The last case gives x = /(k 2 + c2t2), where k = c2m/eX, which is
precisely the first equation (81) of Problem 4, Ch. V.
of which the two parameters are c2n/eX

:,80

-,802)

[Fig. 41 represents on a scale of 1:80 the path of a negative
electron, in a field of intensity 50 E.s.U., or 15000 volt/cm., and
for the value $ = 0.508. Thus the minimum velocity, which
occurs at the vertex A, is nearly one half of that of light.

Xx
Y
A

The value of e/cm is taken to be -1.77.107 E.M.U. This
makes the two parameters - 39.4 cm. and - 20 cm. respectively.
Thus the distance of the vertex from the origin, QA, is 39.4 cm.,
and the curve lies wholly on the negative side of the x-axis.]

154.

The error made in neglecting radiation. In

order to estimate the error to which the results just obtained
[and also the results obtained in Appendix G below] are liable
on account of the neglect of the small radiation terms, we return
to equations (219), § 148. The terms neglected in the first
equation may be written in the form

gV(1
3) {;g~ )

3

Fig. 41.

g

/2)}....(235).

Those neglected in the second equation are

3

-&V(1/3.)...c..

......(236).

Those in the third equation are identically zero, because the path is a plane
curve.
From (229) and (231),

V a+2+t2
_

we find

1d_#2=
l(
)\(a

a2 +b2
+t2 '

c
p

§ 152,

cdO _
a
v dt $ (a2 +t 2 ) '
*

Loris,

b
2+ b2 +t 2 )'11

c
p Y/(1

3 =aV
_
032)}-

08

_

V(a 2 +t 2)

2)

2

2

b

2

(a +b +t )
b3

Ebene Kurven, p. 580, 1902.

184

MECHANICAL FORCES ACTING ON ELECTRIC CHARGES IN MOTION

[CH. XI

Hence the terms (235) become
2e2a2

2e4X2/32
3c4 m 2 (1

3c2=b4

and the terms (236) become
2 2

3c b

first

_,2)18)

2e2at

V(a

2 + t2 )
/(a 2

2
+ b + t2)

The
quantity is greatest when /9 has its least value 00, and then is
equal to 2e 4X2/9°/3c4rn2 (1 -I02), that is, its greatest value is numerically
equal to 2e3X, 0/3c 4n2 (1 - f 02) times the external mechanical force on the
electron, eX.
The second quantity has its greatest value when
t2 =

a 'V(a2 + b2 ) = b2$ 0/(1
4 2

4

-

)302)

and then it is equal to - 2e X f 0/3c m2 (1 + X30), that is, its greatest value is
equal numerically to 2e3Xf 0 /3c4m2 (1 + i80) times the external mechanical force.
It is clear that both values are of the same order of magnitude, but the
first is the greater on account of the factor 1 - $ in the denominator. Hence
we need only consider it in estimating the error.
For a 8-particle we have e/cm =1.77. 10' E.M.Uv. numerically, and
e/c = 155 . 10-20 E.M.U. Hence 2e3/3c4rn2 = 1.08.10 16 If further X be equal
to F volt/cm., we find that the reaction due to radiation is at most equal to
36. 10-19

F)3 0
1(1_-,/32) times

the external mechanical force.

Even if the field

were as strong as one million volts per centimetre, and the minimum velocity
were as great as 999/lOO0ths of that of light, the error would be less than two
ten thousand millionths.
Hence the effect of the reaction due to radiation is quite inappreciable in
this, and probably in all practical cases.

CHAPTER XII
THE MOTION OF GROUPS OF ELECTRIC CHARGES

155. IN the last chapter we left over for future study the external
mechanical force, Fe, which acts on one of a system of charges, whether it
be due to other charges of the same system, or to charges outside it. We
shall take up the consideration of this part of the force in the present
chapter, and shall then show how to deduce the equations of motion of a
group of charges.
In calculating this force from the external field by means of (VI), § 140,
we shall assume that the linear dimensions of each charge are small compared with its distance from its nearest neighbours. Then we get

Fe =

e {d + [vh]/c}

......................

(237)

where d and h are the electric and magnetic forces at some conveniently
chosen point of the charge, and v is the velocity of that point relative to
the observer *.

Resolving along the tangent, principal normal drawn towards the centre
* [We may obtain an idea of the degree of approximation attained in (237) as follows: let

de be any element of the charge, and fe the mechanical force per unit charge for the element, so
far as it depends on charges other than the charge considered, whether belonging to the same
group or not. Then we have

re=ffede.

Take any origin in or near the charge and moving with it, and let v be its velocity relative
to the observer. Also let r be the radius vector drawn from it to the element de, and let u be
the velocity of de relative to it. Then r and u are small quantities of the order of the linear
dimensions of the charge, and fe is a function of r and u, as well as of the position and velocity
of the origin. Hence
fe=f + (rV) f +t(uD) f -}-....,
where f is the value at the origin, where r=u = 0, and V and D denote vector differentiation with
respect to the coordinates and velocity components respectively.
Let us choose the origin so that Jrde = 0; since de is independent of the time we have jude = 0
also. The point thus found may be called the "electric centre" of the charge, from analogy with
the mass centre. It does not generally coincide with any particular element of charge, but is never
far from any element. When we treat the charge as a point charge, we may suppose it concentrated
at the electric centre. Then (237) holds when d, h and v refer to the electric centre, and we
neglect the squares of the linear dimensions of the charge.]

186 MOTION OF GROUPS OF ELECTRIC CHARGES
THE

186

[CH.
[R

of curvature, and the binormal drawn so as to make a right-handed system
of moving axes (s, p, n), we get
FS = ed8
F~o =e(d -. h)

..................

(238),

Fn = e (d + ih8)j
where the suffix e has been omitted as no longer necessary to distinguish the
external force F.
We now proceed to the consideration of an illustrative problem.
156. Problem. A group of n electrons moves uniformly in a
circle. Required to find the mechanical force on one of them due
to the rest, and the equations of motion. We shall use the notation
of the problem, § 136, and shall take the 0th electron of the group as the one
whose motion is required.
We must write down the electric and magnetic forces due to the ith
electron at the electric centre of the 0th, as given by (203), § 138, and

i=n-1

sum from i =1 to

[not from i =0 to i = n-1

The azimuth of the 0th electron is
value of 0f.

wt

as

in § 139].

+ S; this is to be taken as the

The azimuth of the ith electron is wt + S + 27r1/n, because it forms one
of the same group. Hence we get
=

r

I (cot +

8-

+2

7 ri/n) =

7ri/n.

Substituting this value in (203) we get for the electric a nd magnetic
forces due to charges of the same group
o

h
ae

4 2

p

i=1 sin2e'n-/n

iron-1

-

Henc

3

2

i=1

j=10

e f n1

4P
2 i=1

d18
-

e

1)i
fO

(

R

j,2/

ji

-

re

(1-/2)

j2

J3

(jxr)

d.Ixjcos

1

sinnrj/n

22e/3(-1) j3J
2_1 ,j=1

105'

,(fi-;>
x)dx1sin'n'

(/91i

I

As regards
the sums with
respect to i we
have
i=n-1

cos

i=n-1

i/n

i~isine 'in/n

-'

1'n

i=1

sin 'in/n-

-4K.
.(239),

,

XII] MOTION OF GROUPS OF ELECTRIC CHARGES18
THE

XIIl

with the notation of Maxwell*.

187

Moreover

i=n-1

cos 32

1

i=1

=n-1,forj=2sn,

n

=

-

1, for other even values of j,

= 0, for odd values of j,

C sinr2 =0, for even values of j,

i=1

n

t( 2 Sl+ 1)7,for=2s+1.

=c
0

2n

The change of order of the summations is allowable because the series are
convergent; hence we get
dl
=

2e80

2

s

'

L()(

2

P

(1-

(2sx)dx

J

2)

fS

s-lif

Ps=d]

p

Z

2%

eK

-e
+

(

Sp

s+=

)

e

2

[JU

+

)

S

cot
(2s

O

(s

+

+1),7r'

1cot

f

+fJ 8 +1 (2s +1) x dx]
J81{2

8=0

p

(2s

(4

2nL

8=0

+'2

P

o

s=O

e
=

(2s+ )
28 +1 1(2+1)

( l)
I/J
(2s+ 1)'s

sDcot

-(s+
c2
hiz

-2

(8=o1

)8

2n

157. By comparison with (129), § 84, we see that the first series in d1,,
when multiplied by 2ce2/3/p2, gives the loss of energy due to radiation from a
single electron describing the given circle with the given velocity, and the
second series, when multiplied by 2ce2/3n2/p2, gives the total radiation from the
group of n electrons.
The radiation from a single electron has been calculated already and is
s

given by (130),

§

85;

1~

hence the sum of the first series is /33/3 (1-

/32)2.

The second series has not been summed, nor have the remaining series in
(240). For the sake of brevity we shall write
U= 2

7

[sn/32J;sn(2Sn/3)

(s+s-))ot
2n

- s2fl2 (1[/3(1

/32)

fJ~

(2snx) dx].........(241),

/3)J~+ 1 {(2s+1)$8}

+(1+,82)foJ+ 1 1(2s+1)x}dx]...(242).
*

Collected

Papv'esVol.

Tp.14.I

188

THE MOTION OF GROUPS OF ELECTRIC CHARGES

NCH.

Remembering that the directions (s, p, n) here coincide with (., -Wr,
we get from (238) and (240)
2
e 2nU
2e3
2

F

3p2 (1
S-

eK

-

p

2)2

(1 + 2

z),

)

e2

P2

.........

.......

(243).

p

Fz =0
The last equation follows from the fact that dz and h, are both zero.
158. The physical interpretation of the first equation is interesting.
Comparing it with (2101), § 144, we see that its first term on the right is precisely
the negative of F1 in the present case, where /$= 0, that is, it is just
$=
8
sufficient to balance the radiation pressure on the electron caused by its own
radiation. In fact, it gives the radiation from the electron when it is
multiplied by the velocity c$. Moreover, the negative of the second term
on the right of (2431), when multiplied by the velocity c3, is equal to 1/nth
part of R, the radiation from the ring of n electrons, as is seen at once by
referring to (129), § 84.
Hence the work done on any one electron by the tangential pull exerted
by the remaining electrons of the ring is just equal to the loss of energy due
to radiation from the electron less the 1/nth part of the loss from the whole
ring. This result may be expressed in a different way by saying that of all
the energy lost by radiation from any one electron, the major part is reabsorbed by the remaining electrons of the ring, and only a small portion is
lost to the system by radiation into space. The mechanism of this process
acts through the tangential pull exerted on one electron by the rest.
159. Equations of motion of the ring. We apply (220), § 149,
where the external force F is the sum of F 1 due to the rest of the ring and
,

given by (243), and F2, due to charges outside the ring.

We get at once

crh = - e2nU + F
c2 rm

P

2

__

eK (1 S 82)
2
+

PP2

+

e2

F .

.............. (244),

where the suffix 2 has been omitted from F as no longer necessary to distinguish

the impressed force external to the ring as a whole.
These equations were first given by Schott*.
* Phil. Mag. [6], Vol. xii. p. 21, 1906.

189

THE MOTION OF GROUPS OF ELECTRIC CHARGES

XII]

A little consideration shows that in the first equation F4 must vanish.
For its magnitude remains the same for all positions of the moving electron,
although its direction changes. Thus it is periodic relative to fixed space,
and its period is 2 r/o.
In Ch. VII we saw that a force of this type only arises from one of the
harmonic waves emitted by an electron, or group of electrons, describing a
circle coaxal with the given one, with uniform angular velocity
where k
is an integer.

Wo/k,

Any such electron, or group of electrons, emits waves capable of interfering
with those from the given group, and therefore should be reckoned as forming
part of that group.
For this reason we shall omit the term F4 from (2441).
160. Let us now consider this equation more closely, and for the sake of
generality let us admit the existence of a small acceleration, or retardation, 3,
such that /3/ is of the same order as the ratio of the small radiation terms to
the principal terms in (219), § 148. Then the radiation terms in (2191) involving
3 and $ will be of the second order, and may be neglected in comparison with
the two first terms on the left. The same thing applies to the radiation terms
in (2192), which only involve the small quantities p and 3. Hence to this
approximation (2441) becomes
d (m)
e2 nU
S+
................
(245),
while (2442) and (2443) are unaltered.
We saw in § 157 that U, when multiplied by ce2/3n2/p,

is equal to the

radiation from the ring, and therefore is an essentially positive quantity*.
Hence the right-hand member of (245) is always negative. An approximate
value of it is easily got by taking only the first term in U, for s = 1, and only
the lowest power of $in it: we get U=

2n + 1!

Remembering that d (m$)/d, is what is usually called the longitudinal
mass, m', of the electron, we get the following approximate equation from

(245)

$
m'5
m

_/

+m

2e2 n2 (n
cmp2

2n + 1!

* [A direct proof of this statement may be given.
differential equation of the function J

d

(U§=)=

2 s7

s(1

1)(n)2"

.............. (246).

From (241) we get by means of the

g
2

) 'n (2sn/) + s2h12m(2snp)].
J2

J2 m(2snp) and J'2 ,, (2snp) are both positive so long as jS is less than unity, and U vanishes with
and
jS. Hence as 8 increases from zero to unity, U/(1 - p52), with it U, increases from zero and
remains positive.]

190

[OH.

THE MOTION OF GROUPS OF ELECTRIC CHARGES

On the left-hand side of this equation the factor of //f, namely m'/m, is unity
for very small velocities and increases to infinity as / increases to unity.
The term m/m can be replaced by - i/a, where a is the radius of the
electron supposed spherical, or its equatorial radius, if ellipsoidal, or generally,
a length determining the size of the electron. We see that this is so from
(206)-(208), § 143, and it can be proved generally from a consideration of
dimensions.
On the right-hand side the function 12 (n + 1) (n/) 2 3/2n + 1! diminishes as
/3 diminishes, all the more rapidly the greater n is. The first few values are,
for
n = 2: (2/3)4/10;

n= 3: (3/3)6/140;

n =4: (4/3)8/4536.

For large values of n we get by Stirling's formula ( e/3)2n.n3/2e/

4

/7r.

For small values of n this formula gives a value about twice too large, but
still of the right order.
The numerical factor, 2e2 /cmp 2, is greatest for small velocities, and
diminishes to zero as 8 increases to unity, on account of the increase in m.
E.M.U., for e the value 4"8.10 - 10
Choosing for e/cm its greatest value,

1-77.107

4.
E.S.U., and for p the value 10 - 8 cm., we get 2e2/cmp = 0-86. 10

161. Let us begin by assuming that the electron does not change in
linear dimensions, that is, that a is constant, and therefore mh= 0.
Then (246) shows that 3 is necessarily negative. Thus a steady motion
without a supply of energy from without is impossible.
But that is not all; under these conditions the size of the ring and its
speed of rotation are indeterminate. In fact, the tangential equation of motion,
(2441) or (246), serves only to determine the retardation in speed of the ring;
the radial equation, (2442), supplies but one relation between the speed and
the radius. One of the two is quite arbitrary; given the number of electrons
in the ring, its properties are indeterminate. In particular, it has no definite
periods of oscillation, and an assemblage of such rings would produce a banded,
and not a line spectrum, in the absence of any mechanism sufficient to confine
its structure within narrow limits.
In order that the ring should be fairly permanent, the acceleration /3 must
be small. The form of (246) requires that /3 be small, all the smaller the
smaller the value of n. For instance, in order that /3//3 may be less than
1/100, we must have /3 less than 9.10 - 1 for n= 2, 1.7.10 - 3 for n = 3, 7.3. 10 - 3
for n = 4, and so on.
162. As an alternative let us admit a change in size of the electron, and
consequently also a change in mass. We can then satisfy (246) by supposing
/3 to be zero, and /3 to be constant.

XII]

THE MOTION OF GROUPS

OF ELECTRIC CHARGES

191

Since the right-hand member is always negative, we must suppose ih
negative. Hence the mass of the electron diminishes, and its size increases,
quite independently of temporary changes due to temporary changes of speed.
The change of mass need only take place very slowly provided that the
velocity of the ring be small enough. In order that it should not contradict
our experience as to the constancy of the properties of electrons and atoms,
it is sufficient that rh/m be less than, let us say 10- 1 , or one thirty millionth
part per annum, but of course it might be very much smaller. What is
contemplated is a slow secular diminution in the mass of the electron,
superposed on all the changes it undergoes when its speed changes.
Replacing rh by its value - m/a in (245) we get
nU cmp2d
e2a ..... .................... (247).
Given the value of d/a as a constant, or as a slowly varying function of the
time, this equation constitutes a second relation between /3 and p, which
together with (2442) determines both completely as functions of n, the number
of electrons in the ring, and of the parameters fixing the external field of
force in which the ring moves. In addition/3 and p will involve the constant,
or slowly varying, parameter a/a.
The form of U as a function of / and n is such that even for small values
of n quite considerable variations in the value of n U/$ have little influence
in changing the value of /. This is particularly true when n is large; for
instance, when n = 5, doubling the value of n U/fl only increases that of/3
by 7 per cent.
Hence we may alter the values of 6/a, of m, or of p considerably in (247)
without changing /9 very much. Two consequences follow from this fact:
in the first place, the value of / depends mainly on that of n, so that we
can calculate it even when we have only an approximate value of p at our
disposal; the radial equation, (2442), then enables us to calculate p exactly,
and if necessary we can correct /3.
Secondly, any slight changes in the values of &/a, or of m, produce
practically no effect on the value of /3; thus it is immaterial whether d/a
and m be constant, or be undergoing slow secular changes. Any such changes
in the values of d/a, or of m, will produce very much slower changes in the
values of /3 and p, that is to say in the properties of the ring of electrons,
so that its motion is steady to a very high degree of approximation.
[163. It is interesting to examine the mechanism by means of which
the expansion of the electron regulates the structure of the ring of electrons.
In (246) the first term, m'/3/m/3, represents the reaction which the electron
opposes to any change of its speed, in consequence of the electromagnetic

192

THE MOTION

OF GROUPS OF ELECTRIC CHARGES

[CH.

XII

actions between its parts, the second, ri/m, the reaction which it opposes in
virtue of a diminution in size and consequent increase in mass, and the third
the part of the radiation pressure on the electron, which is not balanced by
the pull of the rest of the ring. Owing to the mutual repulsion of its parts,
the electron tends to expand, and will do so, unless it is prevented by some
external system, for instance by the aether on Poincard's hypothesis. If
the mutual repulsions overbalance the pressure of the aether, the electron
expands, and its mass diminishes, and then we get a negative second term
in (246), which is equivalent to a forward pull on the electron. If this pull
exceeds the drag due to radiation, the speed of the ring increases and the
drag increases with it, until it is just sufficient to balance the pull due to
expansion. If the pull is less than the drag, the reverse process takes place.
If through any cause the speed of the ring be increased beyond the
value necessary for steady motion, the drag due to radiation increases
beyond the amount which balances the pull due to expansion, and produces
a retardation of the ring, and vice versa. Thus the drag due to radiation
supplies the mechanism for regulating the speed of the ring; its great
effectiveness for the purpose is due to the fact that it increases very greatly
for small increases of velocity, particularly when the number of electrons in
the ring is large.
When the motion of the ring is steady, the drain of energy due to
radiation is supplied through the pull due to expansion. This occurs at
the expense of the electromagnetic energy of the electron, which has also
to supply the work done in pushing back the aether opposing the expansion.
Another alternative altogether is to assume that the electron is deformed
so as to become unsymmetrical. We shall see in Appendix D, § 230, that in
this case an additional term arises on the left-hand side of (244), namely
- m,,v2/p, where m, is of the nature of a transverse mass.

If m, be positive,

and the velocity suitably chosen, this term may balance the radiation term
on the right-hand side.]

APPENDIX A
ON THE DOPPLER EFFECT

164. A radiating system of charges moves with small uniform
velocity and without change of constitution. To find the frequencies
and intensities of its radiations relative to a stationary distant
observer (Doppler effect).
This problem, with the limitations mentioned, is obviously the problem
of finding the Doppler effect due to the motion. That the limitations are
necessary will be seen during the course of the investigation.
For a distant observer the potentials are given by the equations (131)
and (132), § 86, and the electric and magnetic forces by (133) and (134), § 87.
The latter are
d =2e __
2

ra

-

e(t-r/c+p/c-y
--

(r - v/c) drd

...... (133),

Here p is the projection of the displacement of the charge at time T on
the radius vector r drawn to the observer from some fixed point, whose
distance from the charge always remains small compared with r. Thus,
while the distance travelled by light in a certain time is comparable with r,
the distance travelled by the charge in the same time is of order p, small
compared with r. Hence the average velocity of the charge must be assumed
to be small compared with that of light when we apply (133) and (134) to
the case of a progressive motion of the charge.
In reality this progressive motion cannot have continued for all negative
time, but we saw in Ch. IX that the influence of the initial conditions may
be disregarded as soon as the permanent regime has been reached. As
regards values of T later than the time of emission t - r/c, we have already
seen that the corresponding motions have no influence on the field at the
time t, because the waves due to them have not yet arrived. Hence we may
apply (133) and (134) without fear of error to the case of a motion which is
6.

13

194 THE DOPPLER EFFECT[AP
~ON

194

GYP.

partly progressive, although the condition that p be small compared with r
will certainly be violated sooner or later, because the values of T, for which
it is violated, have no effect on the field which we wish to find.
165. We shall suppose the motion of the charge to be compounded of
two motions: (1) a uniform progressive motion with velocity u at an angle 9
with the radius vector r; (2) a motion relative to this with velocity v.
Choose the origin so that the progressive displacement of the charge vanishes
at the time T = 0; then its value at the time T is c/3-, where 8 = u/c. In
order to satisfy the condition as to the smallness of the projection p we must
have small, for the part of this projection due to the progressive motion
is c/ cos 9. T. Let us now change our notation and henceforth denote by p
the part of the projection of the whole displacement which is due to the
relative motion alone, so that the whole projection is now denoted by
p + c$ cos 9.T. Then in (133) and (134) we must replace v by v + u and p
hy p + c$ cos 9. .
Thus we get

$

J_

d= .2;cr~j
27rcar

i( t- r/C+p/c-(1-

at

2'n-c r atf
2

cOSO)T}(r

11
f00 LM r/c-p/C-(1-COSO)7} [r 1

,

y+u)drd
(v + u)]

These expressions can be reduced to the forms (133) and
of the substitution

drd u.

(134)

by means

t=(1-I3cosO)t'
1 -3 cos..
- 3cosO0) r', p
-cos0)p'

T=.(1
We get v

=

= (1

v', because we must still have w' = °J(rlv').

Now we find

2ircr' (e_/3cos
2l'

at''K

Of)3't'"I

(1 e d cos 9)3 atf

VU

00

xe

)r 1~v

r_

u]d~

1

(249).
.................. ......
Denote by 0', d', 1h'what 4), d, h become respectively, when we replace
t and r by t' and r'.
Notice that the integral which multiplies u/c is simply 4)'.
166.

Then we get at once
d=
-

d'

os

h'

h=

cs~)

_

3C
_

2(1+

u f'

/ cos 8)3 ............

[rl u] c'

~1/3o~)

.. .... ...

(5)

(5)

.. . (251).

THE DOPPLER EFF'ECTr19
~ON

A]

A]

195

These equations contain the whole theory of the Doppler effect; they
enable us to express the field due to a moving system when we have found
that due to a modified resting system, obtained from the given one by the
transformation (248).
The direction of the. vector r' is the same as that of r, although its
magnitude is changed in the ratio 1 - /3 cos 9: 1. Hence d' and h' are to
be calculated by (133) and (134), § 164, for the modified system. It follows
that we have
(rid')

=

0,

(r h') = 0,

= [rid'].

h'

Thus we get from (250) and (251), since (riu) = c/3 cos 9,
(rd)-

/3cos6

c(1 -/p cos

h=

].

)[

3

(2)

Hence the magnetic force due to the moving system at a great distance
is still transverse and perpendicular to the electric force, but the latter is
not transverse except at the equator.
167.

The Poynting vector is given by
c [dh]

=

cd 2 r-

c(rid)d

Hence
c {d2 - (r d)2}

(sri)

Lr

=cd-d'
=c (rid)

_

c2d'2 _ 2 (d'u)

o'+

$2sine
9)6

4wc(1-3cos

3cosO4h '

,h

by (252).
9

2

1$cos9'0o

(253).
........................
These equations show that the flow of energy is not entirely radial, but
that there is a component flow in the direction of the magnetic force hi, that
is a circulation of energy along the parallels of latitude. This obviously
contributes nothing to the radiation and therefore nieed concern us no
further. The rest of the flow is entirely radial, but is of a different form
from that due to a system at rest.
On the line of motion we have 9 = 0, or 9 = ir, according as the motion
is towards, or away from the observer. Here u is along the radius vector.
and therefore (d'u) = 0 ; hence, on the line of motion,
(sr1 ) = hcd'2
... (254),
)6 ..................
(1
=
}according as the motion is towards

(away from) the observer,

196

ON THE DOPPLER EFFECT

[APP.

On the equator we have 0 = r, and d' is parallel to u, so that (d'. u)/c
is equal to ±+ d', according as the directions of d' and u are the same, or
opposite.
Hence, on the equator,

(sr)

.
.............. ... .. . .

d'

. .

(255).

47r

We must now consider in what way the force d' differs from its value
when the system is at rest. There are two kinds of changes to be investigated:
(1) the change of frequency of any constituent vibration; and (2) its change
of intensity.
168. Change of frequency. We must remember that d' is given by
(133), § 164, except that in accordance with (248) and (249), § 165, t, r and
p are replaced by t', r' and p'.
When the system is polyperiodic, with frequencies wl, w, ... , the radiation

from it when at rest consists of an infinite number of simple harmonic sum
and difference vibrations, given by equations such as (144)-(146), § 91, the
argument being of the form
s1, s2 , ... are integers.

2 (t -

r/c) + A, where

2 = sw1

+ s2

2

+ ... , and

If we suppose the constitution of the system to be unchanged when set in
w2 , ... to be unaltered, and
therefore also
But as we have seen, t and r are to be replaced by t' and r',
so that the argument of the vibration for a distant stationary observer becomes
of the form 12 (t' - r'/c)+ A, and this by (248) may be written
uniform rectilinear motion, we must suppose w,

12.

12
let us say

fl'

(t -

r/c) + A.

Thus the frequency is changed from

12

to

12',

where
1'=1/(1 -

cos 0) ....................... (256).

This equation expresses the well known Doppler effect, as observed in
canal rays. The fact that it agrees with experiment shows that the constitution of the canal ray ion is unaltered by its motion, at any rate to the first
order. According to the Theory of Relativity we may expect it to be altered
to the second order, but such a change, though theoretically determinable by
observation of the canal rays in a direction transverse to their motion, would
be difficult to establish by experiment *.
169. Intensity of the radiation. We shall for simplicity consider only
the most important case, where the radiation is observed in the line of motion,
in front of, or behind, the moving system, that is where 0 = 0 or r. In this
* Einstein, Ann. der Phys. 23, p. 197, 1907.

Schott, Phys. Zeitech. Vol. 8, p. 292, 1907.

A]

197

ON THE DOPPLER EFFECT

case, as we have seen in § 167, (254), the radiation (sr)= cd'2/4w (1 +1)6.
In
order to have a perfectly definite problem before us, let us suppose that the
moving system is a uniformly rotating ring of n equidistant electrons.
The field due to such a ring is given by (125) and (126), § 82. To avoid
confusion we shall however replace the polar distance 0 of the radius vector rl
by X, and the velocity of the rotational motion by cy. The general term of
order s is given by
d ,_2ery' cot x . n 2
d=

2ey'con

s J(sn7'sinX) sin sn {(t'-r'/c)+8-

n
- 2e)rp S J n (sn
r'p'

=_

sin

--

COS sn

+

},

--

where we have replaced t, r, p and 7 by their values t', r', p' and 7' in the
progressive motion of the ring. It must be remembered in the first place
that p sin x in the arguments of the Bessel Functions represents p, and therefore, when the ring has a progressive motion, must be changed in accordance
with (248), § 165, and secondly that 7y= wp/c, so that 7 must also be changed
cos 0)7', because o is not altered by the progressive
to y', where y= (1 = 7/p and y'/r'= 7/r. Hence the changes in amplitude
motion. Thus
of d,' and de' depend essentially on the changes of J, (sn 7' sin x)/r', and of
J',, (sn y' sin X). If we are content with a first approximation, we may assume
Hence to this approximation the
sin X) to be proportional to 7y').
J4 (s
amplitudes both of dx ' and d,' are got from those of dx and d,, corresponding

$
y'/p

ry'

to a ring at rest, by dividing by (1 - 8 cos

)8n-1.

It now follows from (254), § 167, that for our ring, the radiation in the
direction of motion is increased in the ratio
1 : (1 -

3)2n+4 for the lowest harmonic (s = 1),

1 :(1 -/)4n+4

And in the opposite direction it is diminished in the ratio

and so on.
1 (1 +

for the second (s = 2),

13)2+4

for the lowest harmonic, 1 :(1 + 3)4n+4 for the second, and

so on.

In other words, the radiation in the direction of the motion is to that in
the opposite direction in the ratio

(1')

1+18

for the lowest harmonic,

for the second, and so on.

Thus the effect of the motion is to destroy the centro-symmetry of the
radiation of the stationary ring, by causing it to radiate more strongly in the
direction of motion than in the opposite direction. The asymmetry produced

198

ON THE DOPPLER EFFECT

[APP.

is more pronounced the higher the order and frequency of the harmonic, and
the greater the number (n) of electrons in the ring. A more complete investigation shows that the total radiation is increased, but only to the second
order *.

170. As a second example, consider the oscillations of our ring, the field
of which is given by (166) and (167), § 102. The general term is of a type
such as
e coteX hnA' mly'J (ly' sin)
cot
A

sin

[(q + mo)

(t' - r'/c) + m

-

+

7r/2) + a,

where A is the amplitude of the oscillation selected as a type.
represents a displacement it is subject to the same transformation,
A = (1 -f

Since it

cos 8) A',

to which t, r and y are subject. Hence we have A'/r' = A/r, and y'/p'= /p,
so that the change of amplitude due to the progressive motion depends on
the change in Jm (ly' sin x). To the same approximation as before this changes
in the ratio 1: (1 - 8 cos 6)' . Now we have m = k + sn, where k is the class
of the oscillation selected, that is the number of nodes in the wave which
disturbs the ring; and the only vibrations which produce appreciable spectrum
lines are those for which s = 0, and k is a small number. For these m is small,
and the change of amplitude due to progressive motion is small in consequence.
171. These two examples are sufficient to show that the vibrations due
to systems of electrons fall into two classes as regards the distribution of
intensity in the radiation produced when they are in slow progressive
motion:
(1) Vibrations due to cyclic motions taking place with a velocity comparable with that of light; for example, the rotation of a circular ring of
equidistant electrons, or the motion of an elliptic group, and other motions of
the same kind. These vibrations may exhibit considerable asymmetry of
intensity of radiation, even for a slow progressive motion, particularly when
there are many electrons in the group.
(2) Small oscillations of groups of electrons about their state of permanent
motion. The only waves emitted, which are strong enough to be observed,
are of an order too low to show any appreciable asymmetry of radiation
intensity due to progressive motion. The harmonics of higher order, which
possess the asymmetry, are too weak to be observed.
* Phil. Mag. [6], Vol. xxII. p. 657, 1907.

A]

ON THE DOPPLER EFFECT

199

Recently, Stark* has examined the canal rays of hydrogen and of mercury.
He finds that there is little, if any, asymmetry in the case of the hydrogen
line Hg, while there is considerable asymmetry, in the direction required by
theory, for the mercury lines X 4359, X 4047, 3650, but none for the mercury
line X4078. These experiments are by no means decisive, and it would be
premature to draw any very definite conclusions from them, but so far as they
go they seem to indicate that the hydrogen line Hp and the mercury line
X 4078 may be due, either to small oscillations of some kind or other, or to
motions of large amplitude, but of small groups of electrons, whilst the remaining three mercury lines may be due to motions of large amplitude and
of groups of many electrons.

A

* Phys. Zeitsch. Vol. 11, p. 179, 1910.

APPENDIX B
ON THE DISTURBED MOTION OF A RING OF ELECTRONS

172. THE calculation of the motion of a ring of electrons, when it is
slightly disturbed from its state of steady motion, affords a good illustration
of the methods of Chapters X-XII. The investigation naturally falls under
three heads: (1) the calculation of the field due to a single point charge
disturbed from its uniform motion in a circle; (2) the deduction of the field
due to a ring of electrons, and of the mechanical force on one of them due to
the rest of the ring, when a wave of disturbance travels round the ring;
(3) the deduction of the equations for the free and forced small oscillations of
the ring. We shall consider these three problems in order.
173. Problem 1.
A point charge is slightly disturbed from
uniform circular motion. Required to find the field close to the
orbit. We shall use the same notation as in Ch. XI, § 150. In Fig. 42
P is the point (x, y, z) on the circle, and Q is the projection on its plane of
the point (x + x, y + Sy, z + az), at which the field is required.

Fig. 42.

APP. B]

ONT THE DISTURBED

OF A RING

MOTION

20

OF ELECTRONS

201

E is the position ( ,~) of the point charge in the uniform motion at the
time of emission Tr, and F is the projection of its position at the same time

T in

+
the disturbed motion, that is the projection of the point ( + 8 , qn
from E to P, and
± 6'). The vector R in the undisturbed motion is drawn
in the disturbed motion from (x + &x, y + 6y, z + 6z) to
the vector R +
+ &,, + 6).
±
Thus FQ is the projection of R +
(4+ 6 ~,

SR

7

SR.

We shall write as before

x+ &x

-')sin(Cor+6+aX'),

+±8n=p(1

(Wt± - b),

'=

y+ 6y=p(1 -)sin(f±X),

p(1 -)cosQ/+X),

Y+&?PV'

#X+IsinX

-),

x=(wr+

z+Sz=pv~

............... .......
(257 ),

)a',

v') are small quantities whose squares and products
where (X, t, v) and (X',
are to be neglected. Hence we get

8x=-p(,cos0+Xsin4), 6y=p(-- zsinc/+Xcos4),Sz~p
1
8= - p {a' cos (2 x + b + X'sin (2x + cP)},

87

p{

...(258).

+'os2+),S=pv/

si(

is pX, PN
42 NVQ is p,a, DF is pX', ED is p

In Fig.

', and pv and pm! are

fieldpoint and disturbed point charge above their respective

the heights of the
projections, Q and F.

As before we have

lk=

2p sinK = 1+, COSX,

a

and

i

8

a"hec

substituting these values in (198) and (199), § 1'35, and taking only the
small terms, we get

d

SR- 3pR 1

Bpsin'x~l

+

a

8cox)

aS

3 (SR

18afp2+ cos x)
8p sine x7J1 +,3

_a

2

sin x(1l+ /3 cos X))

sh sinx

[v . (sR - 3pR,)]
8cp 3
(1 + /3cos

a

+

x)

8cp sin x (1

+,8

cos x

1 2pR 1

_a2

/38v
a*4cp

3pR 1 )

3 2pv

+p
+

-

8p3sin~X(l+I3cosx)f

a 28p sin x(1
[v. (SR -3pR
J/3
3

+±flcos

x)).(25.

1 )]

dlJt(8cp sin x (1+ /3 cos x)
R,]
±a8/[Sv.
[sv.R 1 ]
E)*fr 4 cpsinx(1l+,8lcos x)f
+4p2sinsx(1-+-/3 cos x)

a*

8p3sin

x (1 +13 cos

x

.. (5)

O
ON

202

THE DISTURBED

MOTION OF A RING OF ELECTRONS

[AP

[APP.

174. Our object will be to express these forces in series, proceeding
according to sines and cosines of multiples of ifr, by means of the characteristic
equation * = x + 0 sin
For this purpose it is best to reduce all the terms
to one or other of three fundamental types

X.

f(x)
sinX(l +I$cos

f(x)

)' 1 ±flcosx'

andf(y)sinx
1 ±$cosy'

and their first or second differential coefficients with respect to
Each of
these three types can be expanded in the required form when f(x) is an
exponential function; this is the case in our problem, where f(x) represents
one of the disturbances x', ... or one of their differential coefficients. The first
step in the reduction is the expression of the numerators, such as
(SR - 3pR 1 ), in powers of cos x and sin

X.

We shall resolve each vector into rectangular components, respectively in
the directions PU, P0 and normal to the plane of the circle; these we shall
denote by suffixes 0, p and z as before.
As before we have
R10= 4=

RIP= - sinX

cos x,

c/cos2X,

vP

,Rz=0,

=c/3sin 2 x,

z=

0

Further we get
8

-

8vP= -

3vz

=

8 sin 0 + 1 cos
8

p ('

-

cos 4 - 87 sin 4 = p ('

-

cw')sin 2 X

2 },

O')c+X')sin

=

( 1 '+±s')cos

pt'.

In order to find the components of SR we notice that FQ in Fig. 42 is
the projection of R ± SR on the plane of the circle, while EP is R ; hence
the projection of SR on the plane of the circle is the geometric sum of the
vectors PQ and FE, that is. of the vectors FD, DE, PN and NQ. And its
normal component is 8z - 8
Thus we find

SR

'cs x+

X

6Rz=p°(v

sn2)

R=p( -XA' 2x - 'cos 2x),
sin

-v').

T
Lastly, p is the sum of the projections of the vectors ED, DE, PA and
NQ on the vector EP. Thus

p =-p

{(X

-

X') cos

x+ (/I+

,L')

sinx}.

With these values weget the following expressions for the components in

B]

20

ON THE DISTURBED MOTION OF A RING OF ELECTRONS

203

the directions 0, p and z of the several vectors which occur in the numerators
of (259), § 173 ; all are arranged according to ascending powers of sin x :
(6b1?

-

(8RN

-

(6Rz

-

3pR4)/p = - 2(X - X') -(3 +± ') cos xsin x +(3X - X')s5if2 x,
3pR 1 p)/p = - ' - (3X - X') cos x sin x - (3,ut + 'i) sin2 x,
3pR 1z)/p

= v -v'

cos xsin2 x
+ 2r3( +,a) sinsx,
2$( tt±')cosXsin2X+ 2 3(X-X')sin3X,
pvp/Cp =- 2$3(X- X') sinxpv,/cp =-$/3(X

pvz/Cp

-

X')cosx~-3

=0,

2 (A' + oX') cos xsin x- 2 (X' (X' -cGil') cosx~sinxy- 2(A' +

SVOjp = X'- c'Vp/p =

t3 Vz/ p

± ,*') sin x+2/3(X -X')

(

=

ftc'+±&,'+±2

sin2

GiL)

x

X') sin2 x,

v'

PR1I/P = X- X'+ (

pR1 p/p = (X

cos xsin x- (X -X') sin2 x,

,a')

xsin

-X')cos

x+±(

sine x,

±/)

pR1 z/p =0,

[v . (sR- 3pR,)] / cp = 2/3 (v - v') cos x sin x,
[v. (SR - 3pR,)1p/cp = - /3 (v~ - v') + 2/3 (v - v') sin2 x,
[v . (SR -. 3pR,)]z/cp = /3( - ') +/3(X -3X') co x sin x+C3 ( L+ 3 L)sin2X,
[By.R,=,5/p/vsin x, [By. R]p/p = -v"cosx

[B

. R,]z/p =

p [v . R]z/cp=

(Ac'
+wX')

cosx~+ (X'

-

/kL+

-$/(X-X')cosxsinX-

Il') sin

x,

'L)sin2X.

175. An examination of (259), § 173, shows that when these values are
substituted in the numerators of those equations, all the terms .reduce to one
or other of the three standard types

f (x)

f(x)

sinx(l +/3cosx)'

f (x) sin x

1 +±/3cosy' 1 ±/3cosx'

where f (x) is a linear combination of the small displacements X, , v, ,',,u', v',
and the first differential coefficients of the last three, or they reduce to a term
of the type /f(x) cos x/(1 + /3 cos x), or lastly they reduce to one or other of
the following four combinations :
sins

f

x (1 + /3 cos x)

+a

f/

j'

+ *

sin2

o

(1+/

)........(a),

f cos x
± a_
f /3cosy
)
sin 2 x(l+/3cosx)
fr l sin x(1 +/3 cosx)....(b
a

f

snx

a2

Dq1+,C o x+a it
X
a f$3cos x

a*1

o'

+ a2
a2

{.()
1+,3co

2
f Q sin2()C

±/csy..........x)
f 2 sin

..........

xcos)

1

O

v

.........

(d).

2 THE DISTURBED MOTION OF A RING OF ELECTRONS
ON

204

[AP
APP.

The middle type at once reduces to the second standard type, for it may
be written in the form f-f/(1 + /3 cos x) ; the last four are reduced by means
of the lemmas A, B, C and D, which may be verified by differentiating out
S1
the brackets, remembering that
They are
g+r l±Iacosxax
Lemma A.
.f

+a

.faP

1

sinsX(1+'3cosX) a#{sin2 x(
2

a
a#

f+f''

sinx(l + /cosX)
+_2__

f'

_

ff

_-

sinx (1+I3cosx)

1

+,8

2 {
(1

sin

+/$ cosxd

2)

2(1+lcos x)

f/3sin x

+

2+,
(1

cos x)

cosX

Lemma B.
fcosx
sin2 (1 +

3cosx) +rasin

f

(

a

sin x(1 +f cos x)

fa COs
x(l+

cos

)

f

i

{sinx

+

o

Lemma C.
a

SfI

a+

sin X ) + a2 fa 2 sin2 y)
I cosx)
ar k1 +'8lcosx{
a {+ f lfsinx +
2f'

+

cos

Icsa 1+r
2 X

1

a

c

f(1-/3 2 )
+

s

Lemma D.

*
2f'

1+/ 3 cosX

x

cos x

+ aSf 'asin x_
2f__a2
4r ,1±/3 cosX 1 +/cosx)

_____si__

a#2(1 +!3cosXfl

It is important to notice that all the differentiations with respect tor4
occurring in these lemmas, as well as in the equations (259), § 173, represent
the partial differentiations with respect to t, which occur in the equations
(198) and (199), § 135. During these differentiations the fieldpoint is kept
and v, are
unchanged, so that x, y, z, p, 0, 8x, Sy, &z, and therefore also X,
to be treated as invariable, whilst X', 1a' and v', which are functions of -r and
= 2 (CO
such
therefore of x, are to be varied. Further, since
or
functions as f ', f", where f stands for X', or ~', v', are equal to 2f/av, 4f/w, 2,
the dots as usual denoting differentiations with respect to the time Tr.

IL

X

176.
r~

-sb)

Substituting the expressions for the numerators in (259), § 173,
necr to
the" sta1
,4.
ndard tc
:ypes of terms in;1the m"
ann"e

redc i whenre

B]

ON THE DISTURBED MOTION OF A RING OF ELECTRONS

20

205

indicated, and collecting corresponding terms together, we find the following
expressions for the components of the electric and magnetic forces :
2X' -

do
&4=2

2i

Way'+

)2X

+,8 COS x)

x( 1

2
)
8Wp2 sin x(1

a 4X' +(1l+'3
+ a*

3

'"/(

+,8

a#*2(8p 2 sin x(1+,flcos x)
4 f' 4wX' +W~2(3it' + 5
l 6 W22s in
fcox)
-

-'2)

a't

4

±

cos x)
8p 2

13 ' - 3wX
12 sinxc~
Wp2 (1+,8cos x) + 2 p2 (L + ,3cosX)

(1+,8 cos x)

4

p2 (1

,cosx)j

)

1+8CSPx(l

2

a (2(1 -/3 )ft-o'(3-23
8
+ a
op2 sin x(l +
-'32) (

-((1

-+ a2

ah2116p2 sin x (1

2

$

)

coX

cos x)

+l

_

- Imo)

8 2

+ 8 cos x)

-

4/3X+313ro(,u6

wvp (l

1(x -X')

p (1

2

+'83cos

+I8cos x)
+

x)

2

')

+

2 sin x(

2

sinyX

2

p (l +,8cos x))

3
+± / )

16p2 (1 +I3cos x)f

2

8d= W (vii v') -4v'
za 1 6W2 p2sinX(l +flcosx)
2

+a

1

±*

0p

i

(1-18

)i'

(1+13 cos

a+f2 (l6p2 sin x(1+,83cos

x)

-

4
8h-13(w2v...32v'- 1')

Pl6w2p2sinx(l 13 cos x)
2

l6p 2 (1
a

4 Wp2 sin x(1+,acos x)

a'9r

l6p2 (1±13 cos x)J

x)

1IQ./
=

1(v -v)

+

+

a*
a j

a+

16p 2 sinx(1 +1C3cos x) +l

+'8cos x)
13Q(v -v')

4p2 sin x(1 +13cosx)
312(v-v)

$127

4WP2 (1+I cosx)$

)I

tl6p2(l -- 3cosx)i
6 2

p (l +13 cos x))

2
(X-X')
-,(4 :'+-30) /h-F 2,/) -aa#1 8 p2 sin3x(1 +fl cos x) 1w2p2 sin x(1l+,Q3cos x)
l 13

1
a#*216p2in x(1 +)3 csx)

_

1 (3/t+p')}
32
l6p2 (l +flcos x)

8p2 (1 +13cosx) +16p2 (1 +'3cos X)
.............................
(260).

The parts of these expressions which involve the quantities X, , v can be
de~veloped in Fourie~r Series by the methods used in § 137, for A, , v are to
be treated as independent of x and 1Jr ; but the development of the remaining
' and v', necessarily requires a knowledge of these
terms, which involve
quantities as functions of the time T, or what comes to the same thing, as
functions of X. The case which is of importance for a study of the stability
of rings of revolving electrons is that in which X', t' and v' are expressible as
su11ms of' damped imprle harmonic viratinsc,.acsetrepYresenting awave, of

X"~,

206

ON THE DISTURBED

MOTION OF

A RING

OF ELE(CTRONS

[APP.

disturbance, forced or free, travelling round a ring of electrons. What is
wanted is the mechanical force on any one electron of the ring due to the
rest. Since we are considering only terms of the first order in the disturbance,
this force is the sum of the forces due to each wave separately, and there is
no loss of generality in supposing the disturbance due to a single wave.
Hence we are led to consider the following problem.
177. Problem 2. A group of n electrons in uniform circular
motion is disturbed by a wave travelling round the circle. To find
the disturbing force on any one electron due to the rest. As in § 98
and § 136 we take the electron under consideration to be the 0th, its azimuth
in steady motion to be given by 4 = wt + 6, and its disturbance from steady
motion to be given by (X, 1, v)= (s4, , V) exp. pt, real parts as usual being
taken. The quantity p is complex of the form q + rc, so that q is the frequency
relative to the revolving ring, and is the damping.

K

We take another electron to be the ith, and assume its azimuth to be
wrT + 6 + 2ri/n, and its disturbance to be given by
(X',

',

v/) = (s4,

) EL(p-kli/)...............(261),

where k is an integer, which we may without loss of generality suppose to
lie between ±+ n. The expression gives (X', ', v') = (,
, v) when i= 0, as
it should do; it shows that (A', ', v') at time t + k2ri/nq assume the values
which (X, , v) had, or will have, at the time t, according as k is positive or
negative, except for the presence of a damping factor. Hence (261) represents
a damped wave of disturbance, of which the phase travels round the rotating
ring from the 0th to the ith electron in the time k27ri/nq. Thus the angular
velocity of the wave relative to the rotating ring is q/k; relative to a stationary
observer however it is (q + klw)/k, the velocity in each case being reckoned
as positive in the direction of rotation of the ring. Further, (X', ~', v') vanish
for k azimuths 0, 0 + 2w7r/k, 0 + 47r/k, ... 0 + (k- 1) 27r/k, where 0 is some angle
between 0 and 2r/k, and have maxima for the halfway azimuths; thus the
wave has k nodes and k loops travelling round the ring and affecting the
electrons in turn. An observer rotating with the ring notices q/k times k,
that is q nodes passing per second, a stationary observer (q + kco)/k times k,
that is q + kco nodes; so that the frequency relative to the ring is q, but
that relative to a stationary observer is q + kw. This distinction between
relative and absolute frequencies is fundamental in the theory of the forced
and free vibrations of a rotating ring and of the waves radiated from it.
178.

The angles r and x have been defined by the equations
=
X=2(T+6(),

9(o

-k

B]

B
ON THE DISTURBED

MOTION OF A RING

20

OF ELECTRONS

207

and the characteristic equation consequently reduces to

x

-=X±Isin .................
x

(262).

Writing for the sake of brevity
= pW......

63

we get from (261)
(X',

~, = (,
v'1)

c

.

s)

(264).

This may also be written in the form

p,

(',

v')=

(X, t)

E -L(OasX+k27ri/n...............(265).

The second form shows that we may pass from (X', ,')
to (X, , v) by
putting a-= 0 and k =
but when we wish to differentiate a function of
(X', ', v') partially with respect to t, the coordinates of the fieldpoint remaining constant, we must employ form (264), in which f is to be treated as
constant while f and x are variable. Now in equations (260), § 176, the
differentiations with respect to (r represent partial differentiations with
respect to t, where X', ', v' and x must all be varied, while c and
,v
where expressed explicitly, must be treated as constant, because 4 and
v,
wherever they occur explicitly, represent the fixed coordinates of the fieldpoint,
which happens to coincide with the position of the 0th electron at time t.
After these differentiations have been performed we may replace (X', pc" v') by
their expressions (265).

0;

x,

>,O,

179. Now (264) shows that (X',
v"), are of the form E"
the usual theory of symbolic operators we may write
(gX',
p',

v')=

E

~ a

+

to"

tE

,

-Lcad(x',

and using the second form (265), we get, for any function

{F()

X', / v)

(,ji

)+

a

-c

f (x); hence by

(f~in

F (X),
+k2wtnf)

Fx)1
................ ......
(266),

which is a very useful formula of transformation.
It may be remarked that x is a supernumerary coordinate introduced
merely for convenience. We have in fact written 0 + for the azimuth of
the 0th electron in its disturbed position, where the single letter 0 would
have sufficed. By doing this we secure the advantage that the angle Jfr,
which is one half the difference in azimuth of the 0th and ith electrons at
time t in the undisturbed motion, may be put equal to wri/n without any
further correction in the disturbed motion, for the change in J4r completely
is
represented by 1 (X' -X)

X

2ON

208

THE DISTURBED

MOTION OF A

RING OF ELECTRONS

[AP
[APP.

180. The mechanical force. The substitution of the values of
(X', ', v') in (260), § 176, and the subsequent development involve some
complicated analysis. Since the mechanical force on the 0th electron is all
that is of interest, half the labour will be saved by using the expressions for
its three components in place of the six equations (260).
The mechanical force is given generally by
f= d +[v. h]/c

.

...................

(VI),

whence for a small disturbance,
8f= Bd + [v

.

8h]/c +

[By . h]/c.

In our problem the directions (0, p, z) form a right-handed system to
which this formula is directly applicable. The velocity components are
3,
v9,/cIf VP/cO,

(X

Sv,/c( =

Ct)/w,

-

C

=

(

Vz/C,
+
+c3/w.

The components of h are (0, 0, he), the last being given by (201), § 136, or
(203), § 138; those of 8d and sh are given by (260), § 176. Hence we get

sf4 =ado±+3( +
f = 6d,6f=

6d

3hz
+

-

o)hz/w,

$ (X -

w)h/w

6h.

181. A further simplification is possible; instead of resolving the force in
the directions (0, p, z), which correspond to the undisturbed position of the
0th electron, we may resolve along the tangents to the circle and the perpendicular radius through the electron in its disturbed position. The components
f and fP of the mechanical force in the steady motion now contribute first
order terms in the new tangential and radial directions, namely Xfp to the
to the new radial force. Indicating the components
new tangential, and in the new system by suffixes X, ,c and v, v being parallel to z, we get

Xf,

f =

6f

=

&1, + X (fp + flhz) + /,3fth /w,
8 - I3ahz - af, + f8hzt- 3Xhz/w,

182. We notice that f~, + I$hz = dP, and f ,, = d ,; these, as well as kz, are
given by (201), § 136, while &d,,&d &z Bh and c3Ii are given by (260),

§ 176.
p2 X' =4-woaFurther, we have X cps, = z twaX, ... and X
With these results we use (266), § 179, and write for shortness
-

0-'

=

sin

x

2

X'.

±k2-ri/n..................------(267).

ON THE DISTURBED MOTION OF

B]

A RING OF

20

ELECTRONS

209

Then we get the following expressions :

f2L(

+

a)

42 {

l4(1+/3 cos )8 sin(1 +I cosX)
32
si sin
)x(
in(1+-/$2
osxE j
axa

-

x(1 +

+a
[~32h4 (1+,Q +c+s)18

-( ± )
a+~)

+2

-}-

++10sX)}2

{(4

{8sinf

/2L

a

-a

r~'~U

$32
sin x
)+18 (1

*

1+132
a (
+fr f 8 sin x(1l±/ 3

+

(1-)8)

8(1+3cos x)la2

a

-+

+/3c1X)}- cosi(+/cs)

)2t18(1a/3$

+

+

cos

8(1 +/cos

(t

-L0toa

cos x)}
+)&3 -

+

x (1 +,8~

cos

x

x)i
l+sinX81+13+oso)

2
N1 3 3 18 E to
a
a+ 1V86(1+,8 OS

~21sin

x)f
E

(8 sin

d
*

(= +t ){/2hz

a
- \a#

{8(2+/3eo-X)}

cosax)}

8 i

P

8 cos x)}

(1 +

a

a#18(+

x)+

(-

32L

$Colsn(1x)s)

16(+/8csnx

cos )

os

1+)J

]
s )
E
2(332)(1
a,/(132)
a*sin(+/ 3 cosx)j
a*22(snl 6 (l +/8cos x))+

+

+ + a*

a*p
+

2

+a*f 2

+

(

1+$)E

/3(1++132)

1

16 sin

a

os}+

o

1

Cs)+

$(1 + 3 cos X
(l±/3 2)a )J

jr

x(1

+,8cos)
2E

L'

a')

4 sin x 1

,2

+22 1
$
16 (1 +/

(3+3/2)(1l-

to

+,cosx)

i

Cosx)J

E-to)1

1sn

~ si x ~l
a1

-119

2 (1±+/12) E

-

a

to

l6 sin x(1+,83cos x)

a

a*

) E
$

+1

-t

16 (1+,8cos x)

3
top( 16 ( + c
2

a

Xe

sin

2$2(1-,2)

a

I+$ox

i-/o
c

X

1 +cjX
2Et0.laf6

......................
(268).

2 THE DISTURBED MOTION OF A RING OF ELECTRONS
ON

210

When a-=0 and k =0 we have

e= o

A.

LAPP.

=0 and

also, and when

e=0

these expressions become very greatly simplified, showing that most of the
terms which do not involve the exponential E- can be derived from those
which involve it by means of this substitution, a fact which will be of use
later.
183. Expansion in series. The first stage in the process of reduction
is to expand the functions of x on the right-hand sides of (268) in Fourier
For
series proceeding according to exponentials of integral multiples of tr.
this purpose we notice that the terms involving hz and f, are already known
or we may use (243), § 157.
from equations (203), § 138, because f, =
Further we find by means of the results used in the problem of §

136:

2

1
2 sin X(1 +±13cosX)+

1 21w
2

sin

32 (1
(

Lf2(1

sin
j1(

's )

- )2) sinX
(1+,8cosx)

a"#2}4

1

f0(ix)dxl
.............(269).
......

)j2J(j1

m

0'""

This equation gives the expansion of the first line in the coefficient of
in the equation for 8f.
All the remaining terms are of the kind just mentioned, which can be
determined when the terms involving O are known, so that the latter alone
remain to be considered. Since we have O = a-$lsin ± k2wi/n,by (267), § 182,
they depend on the following three functions of x:
E

sin X

3

L T- sin Xsn

-tosiX

1+/3cosx

l-i-1cosx

sinX(l+I cosX)

c)

$3

Of these (b) can be derived from (a) by differentiating with respect to the
unaltered wherever they occur separately,
combination a-]l, leaving a- and
the result by t ; in the same way we can derive (c) from
and multiplying
(b), so that we need only expand (a).

Now (a) is infinite at both limits, when x = 0 and
X7an
47;btsne
X =xi(1a+d1k

-LOPsin

X

cos

x)butsin1

1fr

= 0, as well as when

vanishes at both limits and

can be expanded in a series of exponentials, differentiable both with respect
to o-fl and to 4~r. Write
1

E-LtapSifX

sin x(1 +

$cosx)

We get in the usual way, since
A,

=d/{i

EL

2

osx-sn

sin*
fr = x

i_00i

; _ _o

AeL&

+13 sin x,

E~

-

-

G(-

1)

- 17

(a-+j)118.

B]

THE

ON

DISTURBED

21

MOTION OF A RING OF ELECTRONS

211

Jj {(o +j) 0} as usual denotes the Bessel Function of the first kind, and
the index - 1 means that it is to be integrated once between the limits 0 and
(a- +j)
The functions (b) and (c) can be derived by differentiating the
Bessel Function with respect to its argument and multiplying by once or
twice as the case may be. Multiplying byec-k2il we get the following
expansions :

,8

t,

-to

(a)

(b)

8cos

=

I+3cosX
x

sin

E

-

E

j=
X)

-

Lkknr/n

j

_-

sin X(1 +

j=-0

= a

sin
(- 1)

=-

1/
Et

1)

(-

G

q

k2,i/n)

J'

{(a

2

EL(itik

+j)

7ri/n)J-{(a-+j))31

$}

j=0

o

(270).

.. . . . . . . . ...

When we put a=0 and O = 0 we get the expansions already used in
the problem of § 136.
The second stage is to substitute for

Summation of the series.

184.

r its value in steady motion, 'ni/n, which is allowable since we have introduced the auxiliary quantity X to represent the displacement from steady
It is of course necessary first to
motion, and to sum for i from 1 to n
develope all the differential coefficients with respect to Jr, which obviously
offers no difficulty in the case of the exponential series, for we need only

- 1.

by j in the usual way.

replace the operator

But the terms derived from

the term (a), (270), § 183, which involves 1/sin 4, require separate treatment.
An inspection of (268), § 182, shows that there are four functions of O to
be considered, which we shall denote by 61, e , 63 and 04 in order. These
2
are :
In the coefficient of X in 8f,

(a*

4

~i(,+a

2

X)}

6-iXE40/2EL
1 +cosX)}....................................(271).

+*I28i(

In the coefficient of
0 2

sin x(1 +3cos
-8

(1+,8cosx)

{

=-(a* + a-){ 8 (1

(

+

+,8

in 8f4. and that of
cosX)}
.2-0

4

X in 4,,

% + a)

_+

a

8(1

lsX)o

E-

0(22}

ON THE DISTURBED
2

212

~
in

In the coefficient of
03

+ ta2)j'l 2 (l-

-

~

a

a2 j
In the
_4

04

a

X3c)sin)xE-®

sinx2(1 +/$2cosx0

(/32 (1

2

+

+,8cos

of v in
-

6

1

sin

(1

+,8cos

"""")"(7)

8f,

0
2
X3 ) SiyE-L

16 (l+3 cos

to-ali"

coxa-)la*
If++

a2{

Csx

1snx1+

2
(l±+/1 ) E-c(3+$3)e_'®..........(7)

coefficient

[APP.

Sf

(1+13cosx)
~~~~16

1sinX(1

a#

AP

MOTION OF A RING OF ELECTRONS

siX+Icx}

2
R)cL(e

12 (1 +
+x 16 sinx 1l+
a
G132EL
a

cos

x)5

4sinx(1+13cos

(1 +

l 6 sinx(1l+8cosx)

a 21
sin(1+$os

x
(7)

.......

E-c0(+38)Ec332)

'"."""(7)

We shall denote by 01 ... the functions got by putting o- = 0 and 0 = 0
in 01 ... , which are clearly derived from the parts of the electric and magnetic
forces involving (X, , v). Then the coefficient of
Al in 'Sf~involves

01~- 0,

1,

in Sf~ involvesE) +6e
2 2
Xin 3fN. involves - 02 ± 020,
Sin 8ff involves 03 - 030 ,
vi in c6fv involves 04- 040.
We shall consider the Bessel Function parts of these functions first, and
Jfr.
then the parts derived from 1

/sin

185. Bessel Function series. Using (271)-(274), we easily find for
the Bessel Function parts of the functions

0:

j=00

(- 1)J EL(?41k2nIrin) .a {j, (a-+

0=-G
02

Y

=

(- 1)j Et

(qk2ffi/n) .

b {j, (o-

j)1}

+j)/3}

...............................

(-1)j

03 =£

E c(j

-k27ri/nl)

. . ......

C {j,

(a- +

(275),

j) /$}

j=-00
j=00o

04=-G£

(-

1)j

2

cqJ -k ?Ti/n).-

d{j, (a- ±

j=-oo

j))/31

where a, b, C,d are four generalized Bessel Functions defined by the equations
a {m, 1/3} = 412I32Jm/2 (1/3) -. (m2 .12132) J '" 1(1$3)

1,81

b {m,
= (m
c {m, l$3} =Ti6I(

1/32) lIJm (1/3)

-

13$2)

+1s {(1

+

± s(rn

1212Jm' (113) +i

)32) (m2 +

1232)

+1132)Jm'

(1$3)

(3 +1$2) 1/3Jm (1$13.(276).

-4m1)32--

3)32} m (/)

21

B
ON THE DISTURBED MOTION OF A RING OF ELECTRONS

B]

In these equations l = a-+ rn, where a-= 2p/w.

When

I=

a-= 0 we

213

therefore

have
m, and the three functions a, c and d become odd functions of m,
while b becomes an even function. We shall denote them by a zero suffix,
e.g.

a , b0,
0

co, do.

We must now put 1 =wri/n in (275) and sum fromi=1 toi=n-1; we
have

Cc
(

i=X-1

i=1

2

k) ni/l=t

cot\(2+l)wr
-

2k + 2s +

o1

2n

1

1, for j = 2k + 2s, s not a multiple of n,

-

= n -1, for j = 2k + 2sn.
Hence we get, for the Bessel Function parts,
S1=-4A+

8= 00

S02=-t

J+B-i1

8-00.

i=n-1

I

b{2s,(a-±2s)$}

s= -00

i=1

....

..

4(277),

sS~ao

03 =--V-C

C

i=1

c{2s, (a-+2s)$}

c
s=

i=n-1

-

00

_s=o

ZU
i=1S

s

)3}

s= -0o

i=n-1

where

a2

c

i=1

4 =-

00
cot

s

(2s l

- D +

d{2s,(a-+2s) 3}

I- a {2k + 2s + 1,(a-2k+2s+)}

S=00..(7

A=n

)

a{2k+2sn, (o-+2k+2sn)/3}

with similar expressions for

,2,

B, C and D.

a-

Denoting the values of a, ... for
= 0 by &/o, ... we see that go = 0
= C0= D = 0, since a0 , co and do are odd functions and b is an even
0
function of ni.

and A0

186. The third sum in each of equations (277) requires no special
symbol, since it can be evaluated as follows. By Taylor's Theorem we have

J2s1
8

{(a- + 2s) $9} = J! 1 (2s$3) + 6$/3J 2s (2s$3) + 2 a2/32J'28 (2s3) + ..
2

2s),8~1

with similar expressions for J28 {(a- +
and J' 28 {(a- + 2s) /3} . In this
manner the functions a23 {2s,
+ 2s) $3} may be expanded in a series of terms
of the types 8 qJ28 (r) (2s$3), where q = 0, 1 or 2, and r = - 1, 0, 1,.The
last

(a-

800

terms of 01,

...

thus reduce to sums of the type

s J 28
8
s

(r)(2s$).

00

Now all1 sumsof this typei vanish identically, unless q - r is even and

214

ON THE DISTURBED MOTION OF A RING OF ELECTRONS
8o00

at the same time q

r, and further

>

[APP.

J,, (28)/(28)2 = 1,2 *, as we saw

s=1

before. Differentiating this equation repeatedly with respect to 3, and using
the differential equation of the Bessel Function where necessary to eliminate
differential coefficients of the higher orders, we get
8 00

J28 (2s/$)_'32/2

I

S=1

(1

132),

-

s=00

whence

cJ

2
8

(2s$)= 1/(1 -

132)

(2s$)

$2)/(1

S=-o4

By the same process we get
S=30

8

2J

,

J2

and by integration with respect to

AL
S= -

J-S1

1

(2s$3)

=

(1 +

12

from 0 to
-

-

32)4,

1

we get

-

132)3.

233/3 (1

00

From these all the necessary sums may be derived by differentiation with
respect to 13, and we get without difficulty
00

S=

I
S=

-

2

sJ

-}

1 {(

J~s

1

,8}

=1

t±2s)13}=

S =0s J 2
S

2s)

'+
2 s)
{(0o+-2)$}

1-2)3

1

(1
1-

=
=

+2(1.2)2'
(

± 13 2)

'+ Q3/33 ( 1
'

12(1

24)

+3

32 )

_32)3

From these the remaining sums required are got by differentiating with
respect to a-.

Substituting in (277) and using (276) we get
i=n-1

(133

333132)

i=n-i

f

_

3

a

-)32)32

i=iLLf(

1

*

22O

3)

1(1

5

ion-i11

Q

121

'2)

N3±3

3(1

a-1

83

_

01

1322

3

R)

(1

Nielsen, Cylinderfunktionen, p. 303.

)

i

ON THE DISTURBED

B]

MOTION OF

A RING

21

OF ELECTRONS

215

187. Sums derived from 1 /sin 4*. The terms in the functions 6E,..,
1
1/sin *J, by (270), § 183, are seen to be derived from
which are derived
terms of type (a), which involve sin x in the denominator. From (27 1)-(274),
LIT~i/n, are
S184, we see that these terms, apart from the common factor
given by the equations

from

e

a
0-2$32_S
wi2

618

sin

,

a

81fra*
si
6
-$
SQ2

4

a2

l

4 sin

}+r)a

8sin

-$

32 (1±+1$2)

i8

+~a

sin<

1

(1 -12)2)
ailr(6 sin M

24-a2

4fr

sin

_3+132

2

w-3(1-_32)1

a

16 sin

-32)

-fr)'

a
ar {8 sin

1+$)+
l 6 sin#

(1

l18

a2

a

Mfr

r2

l 6 sin 1r '

1±3)32

((1 - 32)2)
16 sin Mr

l6 sin#*

Since
a

-

___

and

os*

a2

1+

1__

ost fr

we get the following expressions, which occur in (268), § 182, by supplying
the factor E- ''i/1 subtracting, or adding the functions 610 ... as may be
necessary, putting * = 'ni/n, and summing from i = 1 to i = n - 1:
.a

i=1

(6
1

10

)--(1

-2)

N+±2$2M +2(K-2H)-

2

i=1
i-n-i

++$12 (1 + $2)

(K 2H)o 2

n-1i

i=1I
+11)32 (1

+

$2)

(K1

-

2H)0'2
(280),
....

... .... .

where with Maxwell's notation* we have
i-n-i

N

i=1

sine k'ni/n

=

i=-i

.

(1 + cost 'ni/n) cosec 3 wri/n

sin 21c'ni/n . cot 'ni/n . cosec 7i/n

Mi!= I
i-i
i

as before

K

=

n-i

cosec 'ni/n

I
i=1

and in addition H

i~n-i1'
=

f
i=1

sin2 k'ni/n. cosec 'ni/n
(281).
............................

*

Coleceda

pei7s--,
..

p. 314.

2ON

216

THE DISTURBED MOTION OF A RING OF ELECTRONS

[AP

[APP.

The following series are useful for purposes of calculation and may
conveniently be noted here :
27K=

n

{o.(/)+
log, n + ngE (/ir)n.2! '57 72}
{lg.

N= 2k2K -

B127r2

-

2

(2k2s1)cot(

81

2

s=o

M=kK-

(2

2
B 7r

3
n 3 .2.4!
-1)

..

)

n~)

.8=0(2k-2s-1)cot(sl)n

s=k-1

H 2

-

jcot

s=0

(2s +1)
2
2

7r-

......
..............(282).
188.

Summing (269), § 183, from Jr

The remaining terms.

=

7n-/n to

r= (n -1) 7r/n, we get

2sin x(1

i=n1[

+,8cosx)

a 2 {4 (1

+.$cos2)s

i]

...................................

(283),

where, as in § 157,

8=00

V2= Y (s +1) cot

(2s±+1)
2

7r-

s-o

,3 - $2)J 28+1 {(2s +1) 9a}
1 (1
1

+ (1 +1/2)f0 J2+1 {(2s + 1) x}4 dx]

. .. (242).

Further we get from (2031), § 138, on summation, for unit charge

where, as in § 157,
U= 0r0 snI32J' 23n( 2sn t)

U=28=

-

s2n2

(1 -

12
)J

J

Sn

(2snx)

dx]

(241).

Lastly, we get on summing (2033), § 138,

p2h3 = f(K

+

~6),

where

2U=I(s + j) cot (2n

2w+1{(2s + 1)131 -f

J 28 +i {(2s

+ 1) x}adj

.............. ....
. .(284).
Substituting from (279) and (280), and the equations just found, in (268),

§182. and supplying the factor e2 we get for the disturbing force due to the
res of +theringr

B]

ON THE DISTURBED MOTION OF' A RING OF' ELECTRONS

Xe2[(1

2

-$

P

)N-2M

2(

-

+182)

3

+- B

I

- n U-

+

=

-1

TT

(1

_

12(1

$2)2

(1-3

2

) Mo

$2

l+3182H-

2)N+I

2

18
/3

R2)

)3(3

S}

-J

-,

+3)H

(

) N+

3

6F

K-H+

2

2 2

C

e2
e21j2/

2)2

3 (1_$2)3

M++

2

I 1

+1-

12(11H+'3-{J
_
K

{(1+ 32 )M+082 (K-H+62&)a

+ B

[t{(1

Xe 2

+a-a0

3P

{( /
A- BB(12 -)3( 2)3
F XP,

K2H)0s2

21

217

$2(1

-

2)

Mo

PL

+

+38
(1+

3

(1

-

/32)

(K
P2)

-

2H) a2
12

(1

-

,

§

+g

......................
285).
It is to be remembered that by § 178,
2p

2(q+ c)..............(26.3),

where q is the frequency and icthe damping relative to the rotating ring.
But for slight differences of notation these equations agree with those
given by Schott *, except that in the coefficient of in the second of equations
(14) the signs of the last three terms are wrongly given.
189. Problem 3. A ring of n equidistant electrons is slightly
disturbed from its state of steady motion in a circle. Required to
find the equations of motion. We found the equations of disturbed
motion for a single electron in Ch. XI, (225), § 150. The quantities
(3 FA, 8F, 8F,,) are the components in the directions (X, ,a, v) of the disturbing
force, which includes a part due to the remaining electrons of the ring, given
by (285), § 188, and a part due to electrons outside the ring and constituting
the external force proper. Before proceeding to the study of the equations
of motion we must consider the external force briefly.
*

Phil. Mag. [6], Vol. x. p. 181, (14).

218

[APP.

ON THE DISTURBED MOTION OF A RING OF ELECTRONS

190. The external field. We must take into account two parts of
the external field: (1) the steady part, already used in Ch. XII, § 159, in
considering the steady motion of the ring, and (2) the external disturbing
force proper, due to electromagnetic waves generated outside the ring. Both
parts produce small mechanical forces disturbing the steady motion of the
ring.
We have already seen in § 159 that the steady part of the tangential
mechanical force, FO, may be assumed to vanish in every case that is at all
likely to occur. Moreover, the steady part of the radial force, F,, is constant,
the same in every position of any one electron, and the same for all the
electrons. Lastly, the axial component, Fz, vanishes.
These conditions require that the external field, so long as it is undisturbed,
be steady and symmetrical about the axis of the circle in which the electrons
move.
Hence the electric force is derivable from a scalar potential 4, and the
magnetic force from a vector potential ao, whose direction is along the
and a, are independent of t and 4 and
parallel of latitude; and both
functions of z and w alone.

*

Thus the steady part of the forces of the field are given by
d = 0,
h,=O,

, d,=

d, =h

_(

aa ,

F=O0, F. = e

)a¢
'

},

-+-

Fz= e

{

ww/c

+

w

for steady motion in a circle of radius = p with
Now we have $ =
angular velocity co. Moreover the differentiations with respect to z andW
are performed on the supposition that 4 is kept constant. Let us regard co,
or what amounts to the same thing, //w, as constant during these differentiations, and let us write

D = e (+

-

gra/c),

a

P,

DzQ
=

............ (286).

Then we get
F4,=0, F, = - P, Fz= Q....................(287).
With this notation the equations of steady motion, (244), § 159, may be
written in the form

c7+e

2

nU= 0

p2
c2mn2 e2{(1+i 2) K - }} p ................. (288).
p
p2
0=Q
+
2
=

B]

ON THE DISTURBED

MOTION OF A RING OF ELECTRONS

w)

219

In these equations (z,
are supposed to have been put equal to 0 and p
respectively. The last two may be regarded as the equations of the steady
orbit.
191. In calculating the disturbing forces we must bear in mind that the
displacements of the electrons are supposed to be so small that their squares
and products may be neglected. Indeed, it is impossible to obtain manageable
equations on any other assumption, and thus we are compelled to make it
by the necessities of the analysis. It implies that the disturbing forces
themselves are small. Accordingly we shall treat both the disturbing forces
and the displacements and changes of velocity of the electrons of the ring as
small quantities of the first order, and neglect their squares and products.
This procedure leads to the following simplifications:
In the first place, in calculating the parts of the disturbing forces due
to the displacement and change of velocity of the electron under consideration
itself, we may treat the external field as if it were in the condition corresponding to the steady motion. This process gives the first parts of the
disturbing forces.
Secondly, in calculating the effect of the disturbances in the external
field itself, we may treat the electron as if it were executing its steady
motion. In this way we get the second parts of the disturbing forces.
192. (1) As before let the component displacements of the electron
considered, the 0th, from its position in steady motion be p (X, u, v),
perpendicular to and along the inward radius vector, and parallel to the axis.
The electric and magnetic forces at the disturbed position are slightly changed,
and they are in the (new) meridian plane, which is the plane of p and v.
The tangential displacement, pX, has no effect on the magnitudes, but only
changes the direction of the forces, because the steady external field is
symmetrical about the axis.
The changes in the components of the
mechanical force on this account are given by

I,=
where

e

adz

ah,

-i

+

w

pve

-/3

h

Iadz

is to be put equal to p, and z to zero.

In addition we have terms depending on the change in velocity, of which
the components are c/3

-

.,

,

in accordance with (222), § 150.

By

2ON

220

THE DISTURBED MOTION OF A RING OF ELECTRONS

[AP

LAPP.

means of (VI), § 140, we find for the corresponding changes in the components
of the mechanical force
8F17
r

e/(3Ahz+

h),

- e$Q

6F

h

- )hzUF=eI(X-

................. ......
(290).
The terms (289) and (290) together give the first parts of the disturbing
forces.
Consider the terms in both sets of equations which involve the displacements (X, , v) themselves.
In the component

8F, we

find terms peQe 3

z

remembering that

+ e/hz;

W= p, $ = cop/c, and that c is to be treated as constant, we may write these

a (zhz/c),
w

terms in the form poe

that is,

component U',, we find terms p e/3
written

in

the

ah,

pea

+ e

Similarly,ithe

.

h

which

together may be

at (lag,) Similar considerations
wie iteo eaa
form

terms involving v.
write

-

Pe

azar

Let us use the function t

a

W

,

defined by (286),

a =T.

ayz

a2

apply

.........

to the

§ 190, and
(291)

az

aw

co

with the same understanding as before, that
is to be treated as constant
in the differentiations, and that z and w are to be replaced by their steady
motion values, 0 and p.
Then the first parts of the disturbing forces, depending on the displacement and change of velocity of the electron, are given by
hz+

F~e$3

7'h)

6F, =-.......v....e.....h....................(292).

8F,,==pS-pvT -e
193.

(2)

Let us denote by

(

8

,

3

8d

h~
d)

and

(8hO, 8h, 8h,) the

w,

deviations of the electric and magnetic forces of the external field from the
values corresponding to steady motion for the point (z,
f). In calculating
the corresponding mechanical forces on the 0th electron, which is at the point
(0, p, cot + 8) in the steady motion at time t, we are to neglect its displacement and change of velocity, since these would only give rise to terms of the
scond nrder. Hnce in the geoneral expressins for the deitozns of the
o

B]

ON THE DISTURBED MOTION

OF A RING OF

ELECTRONS

221

forces we must put z = 0, w = p, 4 = wt + 8, and take the velocity of the
electron to be c$ in the direction 0. Hence we get by (VI), § 140,
Fa = e6d,

8F, =

- e (8d, + P38h), 6F, = e (8dz -

38h,)...(293).

These give the second parts of the disturbing forces, that is, the external
disturbing forces proper. They are given functions of t.
We must now
194. The equations of the disturbed motion.
substitute for the forces (8 FA, SF,, 8F6) on the right-hand sides of (225),
§ 150, the sums of three sets of terms:
(1) the disturbing forces on the 0th electron, due to the disturbances of
the remaining electrons of the ring and given by (285), § 188;
(2) the changes in the forces of the steady field, due to the displacement
and change of velocity of the 0th electron from its steady motion and given
by (292), § 192;
(3) the external disturbing forces, due to changes in the external field
and given by (293), § 193.
The forces under the first head have been calculated for a damped simple
harmonic oscillation of the type (A, B, C) exp. t (pt - k2 i/n), where (A, B, C)
are arbitrary constants, generally complex, i is the number fixing the electron
in the ring, p = q +

tLc,

q is the frequency and

K

the damping, both relative

to the rotating ring, and k is an integer, the "class," lying between the
limits ± n and determining the mode of the vibration.
The forces under the second head are obviously expressible by means of
sums of terms of the same type.
Hence we must express the forces under the third head, the external
disturbing forces, in the same way.
They are due to electromagnetic waves of the most general type, and can
always be expressed by means of sums of terms of the form f(z, w, 4) exp. t Nt,
where N is generally complex with positive imaginary part. Its real and
imaginary parts give respectively the frequency and damping of the external

disturbing forces relative to an observer outside the rotating ring and fixed
with respect to its centre.

w,

The amplitude, f(z,
0), is a periodic function of 0, with the period
r, and can be expanded in a Fourier series of terms of the type
F(z, r) exp. (- ik4), where k is an integer.
2

For the ith electron of the ring we have
z= 0,
= p, and
=at+ 8 +2r2i/n.

222

ON THE DISTURBED MOTION OF A RING OF ELECTRONS

[APP.

Hence we can express the external disturbing forces on it as sums of terms
of the type F exp. L(pt - k27ri/n), where p = N- kw. We can, if we wish,
without any loss of generality,
suppose k to lie between the limits ±
by slightly changing the meanings of the symbols N and p. Thus the
external disturbing forces can always be expressed in the required form.

in

The frequency of any term relative to the rotating ring, which we denoted
by q, must be carefully distinguished from that relative to an outside observer,
which is q + kw.
195.

Since a- = 2p/w = 2 (q + tc)/w, by (263), § 178, we must replace

2
such quantities as 5 /w, X/w, X/w9, by to-aX, - Jo-X, -

3
to-a , respectively on the

left-hand sides of (225), § 150. When we compare the resulting expressions
with the right-hand members of (285), § 188, which determine the parts of
the disturbing forces due to the rest of the ring, we see that all those terms,
which do not involve the mass mnin the former expressions, just cancel equal
terms in the latter.
For example, we find on the left-hand side of the third equation (225),
the terms
e2 3

e25
3p2(1 -

2)2

+12p2

)
2(1

v.

These terms also occur on the right-hand side of the third equation (285),
and therefore cancel out. The same thing occurs in the first two equations
of motion.
The terms in question have a simple physical meaning: they represent
the reactions on the electron due to its own radiation, so far as it is due to
the disturbance from steady motion. When the electron forms one of a
group, the radiation per electron is much reduced, and accordingly the
corresponding reaction is partially balanced by the forces exerted by the
remaining electrons of the ring. The small unbalanced part left over
corresponds to the reduced radiation from the ring and is represented by
the terms involving the Bessel Function Series A,... in (285). The complete
cancelling out of the large terms in the reactions due to radiation serves as
a verification of the correctness of both (285) and (225).
The terms in (225) which involve m can be expressed more cond(m3)
; for instancewe
veniently by means of the longitudinal mass m' =d d
have d(a)

=_m'rn/m.

We might eliminate m and

'h

by means of the

equations of steady motion (244), § 159, but the ratio m'/m would still
remain, so that there is little advantage in doing so.

22

ON THE DISTURBED MOTION OF A RING OF ELECTRONS

B]

223

On substituting the values of the three sets of disturbing forces
from (292), (293) and (285) in (225), we find that the equations of disturbance
may be written in the following form :

196.

p,+f1 v =e
f 2 X+f 1
3
u ±f22v =-e
f 2 lX +f
f

31

X +f32ti+f 2 3 v

=

)38h,))

(294).

(bd,±8hz) ...........

e (8d 2-

The nine coefficients are given by

=(12 -

l)/31 (B+N -

e

(

-2Ho

-

m

m

2('

cmbm+{o3z

2A

+

f21

U)M$
-

n)32

P

~I

pm

1,31=

-2

te3h~-

1+

(

-2

Bp

1

P
f22

are2

i

e~jI(1
)2N+1(3

c2m/3

2

2

12 o-+ 121+0)K

2

22 THE DISTURBED
ON

224

MOTION

OF A

fm,

OF

RING

[AP

ELECTRONS

[APP.

2
l/3}_= I~(in
1232JIm A(1/)
1282) jmj (1/3)
b {m(13}= (i- 1R2) 13Jm(l/3) +
±+1/32) f
(l3)
22 J /(1/3) +
c {in, l/3} = AT (1 - /31) 1
(3+ 32 l3Jn(1/3)
at

I(rn

2
+ fi {(18/2) (in + 12/32) - 4mrn/
/}
(1 - /32) 12/2J' (1/3) +-F

2

&

di {21,8

+ {(1 +

(flt2 +

/3)

1)7r

G~(2s+

Scot

n

A =n~a

12/32)

34--f"2}f

(1 +

4m1/321

-

a {2/c±
2s

+1,

3/32)

-

33=1

(f+2

1

(1/3)"!
l/3Jm 3
1
Jmz (/3)
"

s+1

3

{21c±+2sn, (+ 2k +2sn)3}
(278),
....

'

where

. . . . . . . . . . .

=
2p/w =

and

Jm'

with similar expressions for R

2 (q +Fc)/o ..................

dy,

W,,
(y)

Jm

=

The values of these

B, C and D.

,

(263),

functions for o- = 0 are denoted by a suffix 0.

Again, we have
U= 2

%

[8n/32J'S8 t (2sn/3)

1)cot

130 (s+
=

s2n 2 (1-

-

(2s+ 1)w

F2}

2n

8=0

f

J28., (2snx) dxii

J2+,{(2

+ 1)

~L"~ .. 3)t2±)3
±(1
1w

s=0

/32)

""(241),

18

+X32)f J2s+ {(2s±1)x} dxl ...(242),

FPT

{(2s

+1)/3I1 -Uf

1)x}
8+l f(2s + dx]

J

................... ......
(284).

Lastly, we have
i=n-l

4cosec 7wi/n

K
=

H = 2=I

sine kwri/n

Y

1M1=

Schott*.
*

wri/n
.9..(281).

i=1

$ sin 2kwri/n~. cot
sine

kwri/n . (1

The equations (294) and (295)

given by

cosec

"

4

N

.

Tn/n. cosec wi/n

+ COS

together

2

wi/n
are

cosec 3

i2/n

equivalent

They are however somewhat more

Phil. May. [6], Vol. xv. pp. 180-182, 1908, eqs.

to

the ,equations

general,

12-15.

in so far

.

B]

ON THE DISTURBED

225

MOTION OF A RING OF ELECTRONS

as in the present case the steady field in which the ring moves is not
merely electrostatic, as was assumed in the paper just cited, but includes a
magnetostatic field as well. Consequently the values of the potential ( and
its differential coefficients are somewhat altered by the addition of terms
involving the vector potential a4, of the magnetostatic field, as shown in (286),
§ 190. Moreover, the equations of the disturbed motion (291) and (295)
involve the components h, and hz of the steady magnetic field explicitly.
The additional terms introduced in this way are gyrostatic terms as usual,
that is to say, no work is done on their account*.
197. The following points should be noticed:
(1) The explicitly imaginary parts of the coefficients f12 and f21 are equal,
but of opposite sign; so also those of f3 and31, while those of f2 and f82 are
zero. The corresponding terms in (294) represent gyrostatic motional forces,
which do not consume or do work. " Explicitly" imaginary merely refers to
the occurrence of c as a factor; as a matter of fact when
is complex these
terms have real parts.
(2) The explicitly real parts of f f21, f, , and the explicitly imaginary
i,
parts off11, f, and J, all involve functions of the type A, together with terms
which have mh for a factor. The first equation of steady motion (244), § 159,
enables us to replace ihh by the series U, which is of the same type as A. An
examination of the expressions found for the radiation from the ring, namely
(129), § 84, and (168), § 105, shows that the radiation due to steady motion is
of the order of U, and that due to disturbance is of order A,... Thus we may
conclude that the corresponding terms in (294) represent damping forces due
to radiation. When the radiation, and consequently the damping, is small,
these terms are small.
(3) The effect of the external field is represented by two sets of terms:
(a) the gyrostatic motional terms involving the magnetic forces h, and h ;
(b) the conservative terms derived from the potential
namely pR in f2,
pT in f33, and the terms J =f23, - pS. When the external field is purely
=
electrostatic, the first set of terms disappears. When the electrostatic field
is due to a continuous homogeneous distribution of electric charge, as in the
case of J. J. Thomson's well known model of the atom, S vanishes identically.
The same thing occurs when the field is due to a central point charge, as in
Nagaoka's model. In these cases the third equation (294) is independent of
the first two, but this is not generally so.

a-

(F,

(4)

The functions

s4, ...

A, ... are infinite series of Bessel Functions

involving ra- their arguments, that is, they are functions of the frequency of
in
infinite degree.
* Slight errors in the equations (12)-(15) just cited have been corrected; namely in the last
two of (12) the term pnhi/m has been replaced by its proper value oh, and in the second of (14)
the sign of - B + B0 has been changed.
s.

15

226

ON THE DISTURBED

MOTION

OF

A RING OF ELECTRONS

[APP. B

Hence, when the external disturbing forces on the right of (294) are
absent, so that the ring is executing its free vibrations, the frequency equation
got by eliminating (X, a, v) is of infinite degree for each class of vibration.
Thus the ring has n sets, each of an infinite number of free periods, and from
this point of view must be regarded as possessing an infinite number of degrees
of freedom. We need feel no surprise at this result, when we reflect that
each electron of the ring is linked with the electromagnetic aether, and so
cannot be compared with the particle of ordinary dynamics.
In this connection it is interesting to notice that Herglotz and Sommerfeld* some years ago found that a single spherical rigid electron can execute
an infinite number of free rotational vibrations, and therefore may be regarded
as possessing an infinite number of degrees of freedom. But these vibrations
differ from those of our ring of electrons in so far as their wave-lengths are
only of the order of the diameter of the electron, while ours are of the order of
the diameter of the ring or larger. Moreover for a surface charge, Sommerfeld's
vibrations are undamped and therefore do not radiate, while for a volume
charge they are damped and radiate.
The calculation of the free periods of a ring of electrons revolving in a
given steady field presupposes a knowledge of the properties of the Kapteyn
series of Bessel Functions, which appear in equations (276) and (278), § 196.
Series of this type have hardly been studied yet; consequently any detailed
discussion of the equations of disturbance would be far beyond the scope of
this essay. For this reason we shall content ourselves with the deduction of
the equations of steady motion and disturbance; they will form the basis of
any theory of atomic structure which is founded on the hypothesis that the
atom contains circular rings of electrons in revolution.
- Gtt. Nach. 1904, p. 434.

APPENDIX

C

ON THE FIELD CLOSE TO A POINT CHARGE

IN MOTION

198. IN order to calculate the mechanical force exerted by a moving
electron upon itself, with a view to deducing expressions for its electromagnetic
energy, momentum and mass, we must develope a method of finding the field
close to a moving point charge. The mechanical force exerted by one element
of the electron upon another can then be found, and hence the mechanical
force exerted by the whole electron on one of its elements, and also on itself,
can be obtained by integration over the whole charge.
This method has already been used by Schwarzschild*, Sommerfeld (loc.
cit. § 143), Lindemann (loc. cit. § 143), Walker (loc. cit. § 143) and other
writers, but with certain restrictions as to the form and structure of the
electron, imposed so as to render the necessary integrations possible.
For the purpose of our investigation in the following Appendix D it is
undesirable to introduce definite assumptions respecting the form and structure
of the electron at an earlier stage than absolutely necessary; but on the other
hand we shall find it necessary to restrict ourselves to speeds less than
that of light, and to times when the motion is permanent in character. For
instance, if the motion of the electron changes discontinuously, our results
will not apply to the interval embracing the discontinuity, during which the
electron is still disturbed by the waves due to it.
What occurs during the establishment of the permanent regime has been
the subject of investigations by Lindemann, Schott and Walker. Walker in
particular has shown that the effect of the discontinuity mainly consists in
small displacements and changes of velocity of the electron, due to strongly
damped waves generated by the discontinuity. Owing to the smallness of the
electron these effects are practically negligible.
199. For the sake of simplicity we shall take the whole charge of the
electron as unity. When it is equal to e electrostatic units we need only
supply e as a factor in the potentials and forces of the field due to the electron,
' Gott. Nach. 1903, pp. 126, 132 and 245.

15-2

228

ON THE FIELD CLOSE TO A

POINT

CHARGE IN MOTION

[APP.

and e2 as a factor in the electromagnetic energy, momentum and mass, and
generally in the resultant mechanical force of the electron on itself.
We shall base our investigation on the series (180) and (181), § 124, but
shall neglect the complementary terms. The use of the series limits us to
velocities less than that of light, and to points near the charge, and the
omission of the complementary terms implies that the permanent r4gime has
been established.
The result will be expressed in the form of series, which can be integrated
over the electron and converge for points not too far away from it. The
conditions of convergence imply certain restrictions on the values of the
accelerations of all orders of the electron.
200. It will be convenient to develope the expression for a, (181), § 124,
so as to put the successive differential coefficients of the velocity v in evidence
explicitly. Write
+ kU (k)= s=: (- 1)s asR
... .......... ........ (296).

a

s=O S1 C

Then we get in place of (180) and (181)

(o)

4= U
k=(_

a=

k=0
k=0

..............

-

1)
k!
k!

(k)V

'k+

...

................ (297),

(k)

.

................... .(298).
(298).
.

k+1

Here v(k) denotes the kth differential coefficient of v with respect to t, or
as we may call it, the acceleration of the kth order of the charge at the
time t.
We shall whenever necessary denote the coordinates of the moving charge
by ( ,
), and those of the fieldpoint by (x, y, z), both referred to rectangular
axes fixed with respect to the observer; also we shall denote the radii drawn
from the fixed origin to the moving charge and to the fieldpoint by p and
r respectively.
Thus we have v= p, and generally v(k) = p(k+i).
These
quantities may be supposed to be given quite arbitrarily.

q,

201. Expansions for U(k). The expansion (296) is inconvenient because it involves partial differentiations with respect to t. We desire to
replace it by an expansion involving partial differentiations with respect to
(x, y, z), so as to make the velocity and accelerations of various orders appear
explicitly in the coefficients. We have
R = (x) + (y -)2 + (z where (f, t, () may for the present be regarded as given functions of t, but
independent of (x, y, z), the latter being themselves independent of the

operator

at

C]

29

ON THE FIELD CLOSE TO A POINT CHARGE IN MOTION

229

Let 0 prefixed to any function of t denote the increase of the function
when t is changed into t + r. Then we may write

(R +

Lim

asR8+k-1

=Lim.-

atx

=0

sk
atb

But by the symbolic form of Taylor's Theorem we have
(R

+

A R)+

=(

A

-

)2 + (y

17A9)

-

-

2

+ (z

-

- 0)2}(s+k_1

-a(pV) Rs+k-1,

where V denotes vector differentiation with respect to (x, y, z), and (pV)
denotes the scalar product of p and V in the usual way. Since p is independent of (x, y, z), V does not operate on it, and the order of the factors can
be interchanged. The same result holds for all products of the types (v(k)
[V(k) V], scalar as well as vector.
Hence we get

V),

asRs+k-1

i

ats

=

as

(

Lim. a S E(PV) Rs+k-............(299).

In this equation T only occurs in the exponent 0 (pV), so that the
symbolic operator can be found by developing the exponential in powers of
T, differentiating and proceeding to the limit. Thus we need only find the
coefficient of TB/s
the exponential.

!in

Now we have
11=00

)

{

( -

E-(pV)
n=O

(pV)}.............(300).

Further we have

a

(}pv) (t+ )ax+(t+

a-aTa-aj)ay+

where v, v, v, ... all refer to the time

a

azyax

to

Substituting in (300), and collecting the several terms, we find that the
coefficient of T8 in E-° (pv) is equal to

+s(s-1)(s-2)(s-3)

(vC)S-4(iV)2

Substituting this value for s ! asE--(p') in (299), and using the result in

(296), we get,
U~k)
_

1

-

1

V

(s -1)

S -

s(2!)

(s -2)

23s=0S1
V)S2
(s -3)

i

(V\-)8-4

+
(V)2.

s(

3!

-2)vV)r

}..R.

3

((

vV

.

(30 1) .

2
ON

230

THE FIELD CLOSE TO A POINT CHARGE IN MOTION

[AP

[APP.

This result may be written in the form
U (k)-1V(k)(

where

V (k)

s

S=0

s

(vV)
3

T k+2+

2C2

6c

V (k +(k+4)

.....

...
!

(802),

±.

8c4

cs

The expansion expressed by these equations is particularly convenient
because the series V(k) can be summed and expressed in a finite form and the
series U (k) reduced to single series. It is easy to see that the symbolic
operators in (302) are the terms involving C 2, c 3 , c 4 , ... and so on, in the
scalar part of the expansion of exp. (-

+

2c2 +6c

4

-24c

V, where V is

+

treated as a vector.
202. Expression for V(k).
V (k)is a function of the coordinates (x, y, z)
and ( , q, ') and of the velocity v, but is independent of the accelerations
of all orders. Hence (302) gives as it were an osculating expansion for U(k),
and the motion which osculates the given motion at the time t is a uniform
rectilinear motion, in which the charge is in its actual position and is moving
with its actual velocity. The value of U(k) for this osculating motion is
precisely the function V(k). In particular, the value of
for the osculating

U

motion is V (0)

But U is the scalar potential of the osculating motion, by (297), § 200,
and we know that this potential is given by (47), § 23, since the velocity is less
than that of light in our present problem, otherwise the series (180) and (181),
§ 124, would not hold.
Hence, when the axis of x is chosen to lie in the direction of the instantaneous motion, and the axes of y and z are any two mutually perpendicular
lines normal to the path of the charge, for instance, the principal normal and
binormal, we have
1 ............................................. (304),
IS

V(O)
-

where

S

=

{(x

-

)2

+ (1 -

2

((y

V2C

- X7)

+ (z

-

21_

This value may also be deduced from (303) ; for with the present choice
of

axes, the symbolic operator

(vV)reduces

0

V )-(

)
s

Now we have identically
a28R28-1 12. "2 .. (2s
=

-

=acI

1)2

to va
ax

-v/)

aR

s

_

J(.

7-'/)

R2S+1

and therefore

2

--

+

-

(z-

')2}S
-ax28+1

028+1

RU~-0
=

.0

C]

23

ON THE FIELD CLOSE TO A POINT CHARGE IN MOTION

231

Hence substituting we get
=x 1.3....(2s-1)v2t(y
i~v2s
2.4.
... (2s)

V(0)=

.

sO

The value of

V(k)

1

2}s

)2±(ze~

cS.

may be deduced as follows.

In the first place, we proceed to find V(1).

We have

a28 R2s
=

alb'+I es+1
=

2s

12. 32. ...

\(2s±1 )2

j

xl

-

s

s+ -

(x

!3

_1!

R

(

axe+

2t

:33

5?

!

R

Substituting these values in the series (303), we get
(v/c)(x-E )±S
(1 - Vc2/2).

.....

=,l)

.

(

)

We next proceed to establish an equation of differences for V(k);we have
identically
asRs+k-1

ax.,

as-2 Rs+k-3

asRsk-s
=

R2

+ 2s (x

-

-

)

Substituting in the series (303),
V(k)

=

2 (v/c) (x

1

-

)"-

§ 201, for V(k) we get
1B
-1)+

)T7

V2/C2

a8-2Rs+k-1

+(8

1

-

2

Vz -

v/C

Solving this equation in the usual way we find
V(k)

=

Ak + Bbk,

where A and B are arbitrary constants, and a and b are the roots of the
quadratic
2(v/c) (x

1

Thus
Making

+

a=(/)(

1

ic= 0

- 4)

-V 2/c 2

R

and

V2/C2

-

1

a

2
0

-V2/C2

b= (v/c

- ) -S

-/c)(x

V2/C2

and 1 in succession, we find that A

=

1/S while B =

0;

hence
V1tv2/_2)kS...................(306).
Without reference to any special system of axes we may write this
equation vectorially in the form
V
where

(k) S

(VR)/c + S}k
(1 _ v2/C2)k 5...........................(307),

= {(vR)2/c2 +

(1

-

V2/C2) R2},

and R is the vector drawn from the point (P_n_,L")

to

the fleldoint (.,yz)

232

2
ON

THE FIELD CLOSE TO A POINT CHARGE IN MOTION

[AP

[APP.

If 0 be the angle made by R with the direction of motion, and 83= v/c,
we have
(vR)/c= R$3cosO, S=Rx/(1 $2sin29).

Hence

cos

! ,C

8

(1

+

Vk= /3 $
cos-9±-V(1(1 - ' )k \V( 1

(k)
V k,

S in9)
/3P2

This expression shows that

si29)kRh-.........(308).
2
/,

=

/32

sin

0)

V(k) is of the order k

in the distance R.

-1

203. First form of the expansions for the potentials. Substituting
the expression (302), § 201, for U(k) in (297) and (298), § 200, we get
6c3
"'

C

a
-V

--

VV

2c(

{V(1)

V(2) +
2
2C

(VV}

Vo)

-

V

V(.3,

+

~.

)

6c3

(V)V3

3

V( ) +

8cv
+(-

8c4

3

+2c YjV(2)2c-3...

V()O

We notice that
is of the order - 1 in B, by (307),
of order 1, V(3) of order 2, V(4 of order 3, and so on.

(4

.

l

.

...

(0)

(310).

V Mof order 0, V
is

Further, the operator V reduces the order of its operand by one unit, so
that the function (4v) V(2) is of order 0, and the functions (v V(()
V(s)
and (i7)2 V(4) are all of the order 1 in R, the quantities v, v, ... not counting
in establishing the order, since they are quite arbitrary. Hence the expansions (309) and (310) are series proceeding according to terms of ascending
order in B. The principal terms, depending on
are of order
and
involve the coordinates and velocity only; the terms next in importance,
depending on V(' and
are of order 0, and involve the acceleration
coordinates and velocity ; the next, depending on V(2), V V(3
in addition to the
and V(4) are of order 1, and involve the acceleration of the second order in
addition, and so on.

'7)

V(1),

- 1,

V V(",

204. Second form of the expansions. The expansions ,just found
are convenient when the expressions for the functions V(k) are comparatively
simple, particularly when we are dealing with a point charge. But in other
cases, particularly when we are dealing with an extended charge and have to
integrate throughout a given volume, the presence of the higher functions V(k)
gives rise to troublesome integrations. For instance, this difficulty arises
when we wish to find the mechanical force exerted by an electron on itself
due to the interaction of its several elements. It is possible to replace the
higher functions V (k) by functions of lower order in the following way.
In this second method of development we regard U(k) as a function of the
indpendePnt variabhles(!.
z,
, Y), 9f-(v1Yv
;)_.v;)1... all the

)(xm

7v_)_(41-,

23

C
ON THE FIELD CLOSE TO A POINT CHARGE IN MOTION

C]

233

components of the velocity and of the accelerations of various orders being
treated as independent since the motion is arbitrary.
Let D denote the partial differential operator whose components are

(at,
'a a ),so that D bears the same relation tov that V does to the
vector r whose components are (x, y, z). From what has been said we must
suppose that D operates only on functions of v, and not at all on (x, y, z),
, , f'), (ix), , i!z), or any component of a higher order acceleration. In
particular it does not operate on R, while D (vV) V.

=

§ 201;

Operate with D on the series (303),
DV(k)==

s=Qo

V

we obtain

S~
s-1}C'

(VY~)-'

s=1

Replacing s by s + 1 in the series and using (303), we get
V

cDV(k)

VT(k+1)=

.................

(311).

This relation is fundamental; by means of it we can express the function
U(k) in terms of functions of lower order. We find from (302), § 201,
U(k = V(k)

-

(YD)
2

V(k+++

+(D)2V+
((

c(312).

.

205. Substituting from this equation in (297) and (298),
the following expansions for the potentials:
T(U- D)V

(1)

+

a = v{ V (0) j) U(1) +
_ v

62

(I

(vD) +.....
V(''
...

V2
JT

.

§ 200,

±( 8o> V(2) +.......31
'
D
+
..

f
(')-...

72

we get

(313).

+ ..

........

(314)

As before we have
Vk-{(vR)/c + S}k
(1v2/c2)k

where

S

=

...............................

(307),

2
{(vR) 2/c2 + (1 - V /c2) R2}&.

- 1,

When we regard 1? as of the first order we must treat V(°) as of order
as of order 0, V(2) as of order 1, and so on. Moreover, the operator D leaves
the order of its operand unaltered.

V(1)

Thus the approximations in (313) and. (314) are carried to the order 1.
The principal terms are of the order - 1 and involve the velocity, but not the
accelea rat+ins.

234

ON THE FIELD CLOSE TO A POINT CHARGE IN MOTION

[APP. C

The terms next in importance are of order 0 and involve the acceleration as
well as the velocity, but not the accelerations of the second and higher orders.
The smallest terms written down are of order 1, and involve the accelerations
of the two lowest orders as well as the velocity, but not those of the third and
higher orders, and so on.
The principal terms give the potentials due to the osculating uniform
rectilinear motion, in which we have a point charge occupying the same
position, and moving with the same speed and in the same direction as the
actual charge at the time considered, but without acceleration. The terms
of the two lowest orders together give the potentials due to an osculating
motion in the circle of curvature with the actual speed and acceleration, and
SO on.

The series (309) and (310), § 203, or (313)and (314), are equivalent to the
series obtained by Schwarzschild *. They converge provided that R be less
than a definite multiple of the distance cT, where T is an upper limit to the
interval of time for which the developments of the coordinates of the point
charge in powers of t are absolutely convergent. This condition excludes
times for which an acceleration, of whatsoever order, becomes infinite; that
is to say, it excludes times at which the motion of the charge becomes
discontinuous, as was to be expected. Moreover, it assumes that all the
functions V(k) remain finite; this requires that the denominators, and therefore also the quantity 1- V2/C2, be different from zero.

In fact convergence

requires that the speed of the charge be always less than that of light.
loc. cit. pp. 249 and 250, IV a and IV b.

APPENDIX D
THE MECHANICAL

FORCE EXERTED BY AN ELECTRON ON ITSELF

206. OUR object in this appendix is to calculate the internal mechanical
force, Fi, on a moving electron, the expressions for which have been given
in § 143, Ch. XI, for several types of electron. This will be done with as
few restrictions as possible on the form and structure of the electron, as a
preparation for an investigation--to be carried out in Appendix E-of the
conditions under which a mechanical explanation of the electron is feasible.
We shall start from the expansions (313) and (314), § 205, for the
potentials due to a moving point charge, calculate the mechanical force due
to one element of the electron on another by means of the general formula of
Lorentz (VI), § 140, and sum for the whole electron, thus obtaining the
resultant mechanical force exerted by it on one of its elements. This will be
required for the investigation of Appendix E.
For the purpose of verifying the expressions given in § 143, we shall
integrate the expression just mentioned over the whole electron and in this
way obtain an expression for the internal mechanical force Fi. It need
hardly be pointed out that the force Fi only vanishes for an electron which
moves uniformly without any relative motion of its parts, because Newton's
Third Law does not hold for two electric charges in relative motion.
207. The convergence of the series (313) and (314) requires that the
greatest distance between any two elements of the electron be small compared
with such lengths as the radius of curvature of the path of either element,
and that their speeds be less than that of light. When there is relative
motion, due to rotation or deformation of the electron, these two conditions
require that the relative velocity of the two elements be small compared
with that of light; consequently the velocities of rotation and of strain must
all be finite.
For the present we shall make no further assumptions
respecting the form or structure of the electron.
208. Notation. In order to express the relative motion explicitly we
shall use a moving origin, namely the electric centre already introduced in

2:36

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

the footnote to § 155. We shall denote its radius vector and velocity relative
to axes fixed with respect to the observer by r and v respectively.
Let del denote an elementary charge of the electron, and r1, v1, u, its
radius vector and velocity relative to fixed axes, and relative to the centre
respectively. Then we have from the definition of the electric centre
er = fr 1de1

=

ev v1 de
1
. . . . . . . . . . . . . . . . . . . . (315).
0 = Ju 1de1
If we expand u1 in powers of the coordinates of del relative to the electric
centre and substitute in the last equation (315), the linear terms disappear
on integration on account of the first equation (315). Hence the constant
term of the expansion must just neutralize the terms of the second and
higher degrees. This means that the electric centre moves about in the
electron, because its velocity differs by second order quantities from that of
the element of charge with which it coincides for the moment.
209. Denote by R the vector drawn from del to de2, so that R = - r1.
Also denote by V 1 (k), or V2(k), as the case may be, what the function V(k
defined by (307), § 202, becomes when v is replaced by v 1, or v 2 , and (x, y, z)
and ( ,
) by (x2, y2, z2) and (x1, yl, z1), or (x1, y 1, z) and (x2, y2, z2)
respectively. Then we get from (307)

q,

V (k) ={(

(1

R)/c+SS1 }

1

(v 2R)/c

(k)-

S

/C)kS,

(1

-

v

2

S}

+

+(1

1 R)2/c2

={(v

-

V

1

/2)

1...(316).

=

S("

22)k

(V2 R) 2 /+

2

Thus V 1 (k) and V2(k) differ in two ways : (1) because the sign of R is
changed ; (2) because v1 and v2 are interchanged, that is to say, on account
of the relative motion.
210.

Let

01,2

0P21

and a 12 , or

and a 21 , as the case may be, denote the

potentials at the position of de2 due to unit charge at del, or at del due to
unit charge at de2. Also let D1, or D2, denote vector differentiation with
respect to the components of v1, or v2, respectively. Then (P12 and a12 are got
from (313) and (314), § 205, in the form
2cDI

= -V(°1l)V

a
12

C

=v!1

)±

2c )
V,1 (01 - (9l Dl)

1c

1

1 1

V,{V
sufi11

)

4

(+

6C
+ (yD 1 )
(l)

(+1

(

1

)
1D

V

(2)+

)..31

c
T 1(2 +(PD)2V2)+..

D1) V

}{(2)
-...

V

(

-

.

.(318)

D]

237

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

211. Let f12, or f21 , denote the mechanical force on a unit charge at del
due to a unit charge at del, or on a unit charge at del due to a unit charge at
de 2 as the case may be. They are found most easily from
,
§ 6,

(XV),

fV= -b+V\7 (va)_da

+

cdt'

c

where Va operates only on a, not on v.

V

Let 1 or V2, denote vector differentiations with respect to the com,
ponents of r 1, or r2, as the case may be.
Putting

0=

12,

a = a1 , v = v2 and V =V, we get by means of (317) and
2

(318)
Ciy)

12_

{vv

()-2

Vvv)
D) 'J7111+

(\

1 1

V
(\iD1)p 2)
6c,
-

c"

d_
(i IC

V11)

9 V V 2,) ..
1 +(iID)V,4+
P

c4

_ (1 y) (1

V Vi)

8C2-

V2

V1(2)

+..

.+

II+

2

±}(-

(1v2){V

v2 2 2
1
V(

31-

c

f 21 is got by interchanging the suffixes

I and 2.

Since the operator V2 lowers the order of its operand by one unit, the
principal term in f1. is that involving V2 V1(°), and it is of order - 2. The
terms next in importance are those which involve V1(°) and V2 V1(1),
and they
are of order - 1. All the remaining terms are of order 0, and those not
written down are of order 1, or higher.
212.

The mechanical force exerted on the element del by the rest of the

electron may be written in the form Fi2 de2 , where
Fi =f f 1 2 del .... . . . . . . . . . . . . . . . . . . . (320).
This is the force required for the investigation of Appendix E.
the suffix 2 may be omitted as unnecessary.

As a rule

The resultant internal force exerted by the electron on itself is given by
F = f Fiede2 = f Fitde 1

f(f

12

+ f21) del

de2 ..

. . . .. . .

The last, symmetrical, form is the most convenient for use.
integrations are extended over the whole electron.

(321).

In it both

This is the force for which expressions have been given in Ch. XI, § 143.
It does not generally vanish, because the relation required by Newton's Third

238

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

The method just outlined for finding the mechanical force, based as it is
on the series (319), involves troublesome expansions, but the work is very
much simplified owing to the fact that a large number of terms cancel out
from (321) by symmetry, and those left in happen to be easily amenable to
calculation.
We might have deduced the force by an application of the Principle of
Least Action, based on the electrokinetic potential of Schwarzschild, but the
calculation of the Action involves more troublesome integrations than the
direct method, which for that reason has been preferred here.
213. Before integrating the expression (319) we shall find it convenient
to expand it in terms of the relative velocity u, but as a preliminary to this
process we shall transform (319) slightly. Since the function V 1
involves v, and the differences of the coordinates (x2 - x1, y2 - y,, z2 - z1) we
have
(1)only

=

Also from (311),

V,'
(=(1D,)

+

§

204, we get V,V

(y1 Dl)()
2c

_

-d

dt

({v2

vJ

-

(' = cD

(V l
2c)

V) V11'.
Hence

V,).

y

v'>1

1_0

)

2

A precisely similar relation holds for V,2V().
Substituting these results in (319), and using (311),
possible, we find after some simple transformations

p

f=-

-

(v'y2)}

{V 2V 1
n+

2

v }D

v

QV

) '+

§ 204,

wherever

')2DfV,'1
+(y1 D1)
+8c

(v,i, )

-

(i

c'

L-{

+1

? DV(°

2 ) .v(~

(+1

(iD1) DT

2C3

10)+

(vl v,)) D1V1 0)(

vV1(°)

1

______ly2

1

+J((tv2 -

+2C3
v1}

1

DV

~

D1 11

*

'

D1) V1(°)} +!01 Vl(1)
+ v1 d V1(1)]

..

(322).

This form for f12 has the advantage that, by means of (311), all the higher
functions V,(k) have been replaced by the two lowest, V1(°) and V11) so far as
the terms of the three lowest orders are concerned ; the functions V1(2), .. .

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

D]

239

214. Expansion in powers of the relative velocity. We must now
expand (322) in powers of the relative velocity. In doing this we must bear
in mind that, in accordance with § 207, the relative velocities u1 and u2 are to
be reckoned as small quantities of the first order. To be quite exact, the
ratios u1/c and u2/c are to be treated as of the same order as the ratio of the
distance R to the radius of curvature of the path, or any similar length. We
shall begin with the expansion of the function V1(k). Replacing v by vi and
V by V2 in the series (303), § 201, we get

TJ
1(k)=

S=00

(VIV2).g

s=0

s!tOCs

S=

(

(Iv

s+k-i=

8=0

(+

~1ivV 8+k1.
. Cs

Now V does not operate on v or u 1 ; hence, expanding the symbolic
2
operators, arranging according to powers of u1, and using (303) and (311), we
find
(323),
V
........
V(k)= 1 + (u 1D) + (u 1D) 2 +...V
where D may operate on u 1, provided that we interpret (u1D) 2 to mean
(u1 (u1D) D), and so on, always taking care to keep the velocity u1 in front of
all the operators D whenever it depends on v.
We see from (316), § 209, that the function V(k) is defined by (307),
If we separate the rational and irrational parts of this equation we
can write
215.

~ 202.

[Fk
+
17Vk 14k
=

V

(vR)k C-k +

=(k)

1 (VR)k

(k)2 82 (vR)k 2 c-k2
(1..2324C)k.S
1 '2c)

i1

-

(k) 3S''(vR)k

±+

k-1

(1

c

-k-;I

±

v2/C2)k

Vk is an irrational homogeneous function of the differences of the coand Wk- 1 is an integral
ordinates (x2 - x 1 , 1y2 - yl, z 2 - z 1) of the degree k -

1,

rational homogeneous function of the same degree.
The first two pairs of functions are
IT

=C

(1

-

1
.................
2/2)

8'

'1-

........

(325)

/

We see from (316) that the corresponding expansions for the functions
V2(k)are got from (:323)-(325) by interchanging the suffixes 1 and 2, that is,
by replacing u1 in (323) by u2, and changing the signs of all the odd functions

240

[APP.

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

216. Operate on both sides of (323) with V2, bearing in mind that V2
does not operate on u1, nor on v, and is commutative with D. Then we get
1+ 1D)+i(uiD)2+ ... }V 2V(k.......(326).
(u
V 2V(k =
replace V2 V1(k) by D1 V2(k)
Remembering (311), § 204, we see that we
scalarly by v2 - v1 , which is the same
and V 2 V(k) by DV(k). Multiplying
thing as u, - u 1, we get as far as terms of the second degree,

may

({v 2 - v1} D 1) V2 (k)={({u2

-

u 1} D) + ({u2

-

u1 } (u1D) D)}V(k)

...

(327).

Adding one half of this to (323) we get

1+

(u(uDD)
......
...............(328).
Operating on (327) with V2 , remembering that V2 operates on v2 and 12,
V (k) +

(V
v

v1 D1))V (k)=
1

-

f(

+12)+

but not on ul, or v or v1, and using (323), we easily see that we may replace
V1(k)in (328) by VV2 (k), provided that we replace V(k) by V2 V(k).
217. Transformation of the force. We shall now apply the results
just obtained in % 214-216 to the expansion of the expression (322), § 213, in
powers of the relative velocities, beginning with the large terms of orders - 2
and - 1, which are conveniently taken together. These terms are the first
and second in the first line and the first and second in the last. We get by
means of (323), (326) and (328)
+(u1+21D))+
{1~j2

:1
CL

-

vlv2))

1±+(ul DDVO

u 2 uDD)}VV

u2}D)}

- vn{1 +il+

Vo

........... (329).
.....
Consider the first line of (329).
-(vl

y2)

_~-v,

C2

Since v 1 = v + 111, v, = v +
(jui +

C2C2C

u2}V)

_(111112)

Hence retaining only squares and products ot u, and
first line of (329)
1 -

1-

(112

(uD)
2

,~jU~
D) V- 9V. +

u.., we have

112,

we get for the

11221)(u

U+21 V) ({u,

+

11211D)

'V2V, +

(11V1V2

which reduces to
-

1+N({ul + 1121 D)) ± j(112 (ulD) D)} (1_ ) (11
(132- ~1u~1))V9V . . . . . . . .

V2
.

. . .

.

(30)

D]

241

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

because D (1

- V2/c2)

=-

2v/c2, and consequently

D) (1 V2/C2),
+
1121 v)/c22 -~({u1
and
(1121u2)/02 =- j (u 2 (u 2 D) D) (1 - V2/02).
In the same way we get for the second line of (329)
({u1 +

-

u12}

2c2

+?

VP-

-

(331).

L iIJD)LD'Voj...

2

l

2

218. The remaining terms in (322), § 213, are all small of the order 0,
and we may use first approximations.
In the first term of the second line we may replace Dl V1(°) by its first
approximation D V0 , and in the factor neglect the products (ulu~ and (iilu 2).
2)
Thus we get({'2-i}v-11-11
11

({2

2

v

fu02

}

.... . . . . . .
...

yV
0

"D

(332).

In the remaining terms, which involve V (1), we may replace v1 i, vvl, Dl by
2
2
the symbols v, v, v and D, and V1 (1)by its first approximation V1 + IW1, where

V1 and W have the values (325), § 215. In this way we find
1
(vv) (vD) (vv)]DdF v
(D) (vD)2
- C2
6C + c 5+
203
+203J
d t C3

q2

+

2 (vv) v

_2v_+

+3c

(2 -

2

jv )13C

(c

2

-

+2(vv)

2
V)2

v

C0(02 -

v

+2
2

v dV1
203dtj
v

(v)2

v

2cCl02-

2

3)

..

)2 333),

where the terms in the last line are derived from Wf, and are got by differenti2
ating the functions in the first two lines, but with V2 replaced by Wf, or
1
(1

-

V/02)-1.

219.

Collecting together all the terms found in (330)-(333), we get

}

+ u2 D)±+AI(u 2 (u, D) D)} (i_

f12 = -{1+(u

+(12 ullyv)
+ 2c2
(112-11

0

+

2

2

-.

1v)

2

3c (02 - uv )
2Ji)(v)

dF vvvV2 dt
2+

D1 -

j

+(u2-u}
)DV

2,2c21V°+1c
+
-

VV
2

D) V2V°

dt 1
+I ({u+dtu2}
3 (2

C)

C(D

V°)
v)(v)+(v

0

2 .

.

.

.

.

(3)
16

242 .

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

This is the desired expansion in powers of the relative velocities up to
and including zero order quantities. The corresponding expression for f21 is
got by interchanging the two elements of charge de1 and de2. For this
purpose we must interchange ul and u 2, and change the sign of the vector R,
which amounts to leaving V0 unchanged, but changing the signs of V2 V0 and
V1. The terms in (334) fall under two heads: (1) terms which are unchanged, namely, those in the third and eighth lines; and (2) terms which
merely change sign, including all the remaining lines. The latter terms
constitute the reversible, Newtonian, part of the force, to be denoted by f,,
in future, the former the irreversible part, fZ.
Again, as regards the order of magnitude of the terms, we see from (325),
that V0 is of the order - 1, and V1 of the order 0 in the linear
dimensions of the electron. Moreover, the operator V2 lowers the order of its
operand by one unit, so that V, V0 is of the order - 2. Lastly, u1 and u 2 are
2
of the first order.
Hence it follows that the principal term of f, is of the order - 2, that of

§215,

fi only of the order - 1.

We shall only require the complete expression for f12 for the investigation
of Appendix E, and that only as far as the two lowest orders; hence we can
neglect zero order terms in f,, which amounts to keeping only the first two
terms in the first line of (334).
On the other hand we shall require the value of f, in order to find the
resultant mechanical force of the electron on itself, and with it to form the
equations of motion. These involve external forces, which are of the order
zero; hence we must retain zero order terms in fi.
Collecting together the

220. Working expressions for the forces.
necessary terms from (334) we get
f=

-

+
f

1 +

+ u2} D)}1i-

2 (vv) v

2v

3c(c

n=f+f,
1
s

2((u1

2

v

2

V

2 (vy) v

2 (vi)2

+
+
+
2
- v2) 3c (C2 - v2)2 C(C -v 2) 2 C
f2 =- fn + fI
1

/{(1 -

V2/c2)

2

+ (vR)/c 2 } f R V/(1

23

.. (

335

)

1
-

2 sin 0)

The suffix 0 has been omitted from V0 as no longer necessary. The last
equation follows from (325), § 215, and (308), § 202;
= v/c as usual, and 0 is
the angle between R and v.
For the purpose of calculating f or f2, for use in Appendix E, we need
only retain the first term in fi, neglecting all the remaining zero order terms.

a

D]

243

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

221. The relative velocities. Before proceeding further with our
investigation we shall find it useful to consider what expressions are possible
for the relative velocities.
Obviously the forces cannot be calculated from the working expressions
(335) until the expressions for the relative velocities u1 and u 2 as functions of
the relative coordinates of the two elements of charge and of the time have
been assigned. On account of the smallness of the electron we can expand
the relative velocity u of any element in a series of powers of its relative
coordinates (x, y, z), as we have already pointed out in § 208. It has been
shown there that the constant term in this expansion is small of the second
order, because we have taken the electric centre as origin. In our working
expressions (335) the relative velocities u, and u 2 only occur in the small
terms; hence it is sufficient to calculate them to a first approximation and
retain only linear terms. Thus we may write
fy, uz)

(u,

= (,,

1x

+

,12y +

, Z,
13

or in vector notation

,
21

x +

+ 0,
2y

031 x+
x

+32 33

U = ar

..... (336).
The quantities a are independent of (x, y, z), but generally depend on
t; usually they are functions of the velocity v of the electron and of its
derivatives.
Hence to our approximation we may regard the electron as homogeneously
strained. The strain operator a- involves nine coefficients, because the strain
is generally rotational. The configuration of the electron is completely
determined when we are given its configuration in some standard state, and
the strain as a function of the time. It is convenient to take the state of
rest as the standard.
222. The bounding surface. The bounding surface of the electron at
any time is completely determined under these conditions. In fact if its
equation be F (x, y, z, t) =0, F must satisfy the equation
aF

a-+ ux
Thus, if

0 (x, y, z, t) = constant,

aF
aF
aF
= 0 ................. (337).
- + u.v + uz
- (x, y, z, t) = constant, and

y (x,

y, z, t) = con-

stant be three independent integrals of the system

0' x +.
±

dx
1y
2

dy

-

_

a2 .y + a-z
the equation of the bounding surface is
F (,,
x)=0
11

+ 0-13 Z

a-21 x +

dz
a- x ±a- 32 y +0
+
31

= dt
33Z

r

...

(338),

where F is an arbitrary function, and is to be determined so as to satisfy the
initial conditions.
16-2

244

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

Vice versa, if the bounding surface of the electron be given at all times,
the strain is no longer arbitrary, but must satisfy (337) at every point of the
surface.
So far we have supposed the origin to be moving, but the axes to remain
parallel to their original directions. If however the axes be rotating in any
assigned way, the equations (336) and (337) still hold, provided only that
u and a now denote the velocity and strain relative to the moving axes.
223. For example, to make this quite clear, consider the Lorentz
electron. Take Ox in the direction of motion of the electric centre, which
is taken as origin as before. Then, if (x0,, y, z0)be the relative coordinates
of any element when the electron is at rest, and (x, y, z) its coordinates when
the velocity is v, we have
2

v )/c,

x = Xo
0(c2
Hence

Y= O

uy =

= 0,

2

U, = - vx/(c2 - v ),

z

01n =

0

.

- v/(c2 -v),

and all the other relative strain coefficients are zero.
The equation (337) becomes
aF
vi
aF
0L:St -------- x
c" v
=0.
ot C - 2? 2 xax
The equations (338) are
dx
vi
c

s

-

dy
0

dz
0

xV

Three independent integrals are
Cx
-

constant,

y = constant,

z = constant.

Hence the equation of the bounding surface is
F (VC 2

), yz

=0,

F(x, y, z) = 0 being its equation in the state of rest.
For instance, if in the state of rest the bounding surface is the sphere
2
-a
= 0, then when the electron is moving with velocity v, the
bounding surface is the spheroid
2x2+
y2 + z2 a2 = 0, with axis along the
x2 + y 2 + z 2

line of nmotion.'
Vice versa, if the bounding surface be assigned as the spheroid given

D]

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

245

above, the relative strain a.
must satisfy the condition (337) at every point of
the surface; that is, we must have
2vic'2x2
2

(C

-

v

2

2c2x
(X

O13z) 2

-12Y

(

2

21

2

+- 022Y

-

23

Z)

2

y

+ (a31x + 32Y + a'3 z) 2z = 0,

for all values of x, y, z which satisfy the equation
2
c2x

2
2
+2 +z - a =0.

y
c0 - v2 + 2

From these equations we get

'731+"0.13

0"23+0.32=0,

0.1+

2

vi

-v
c -

C

2

-

V

2

C-

- O, 021

+0.12 C

-

0,

2

O0' 22-

0,

0-33 =0.
21

Thus any possible strain is compounded of the Lorentz contraction, of
a shear 2c /(

2
2
1

+ .312) arbitrary amount in any plane through the axis, of
of

a determinate rotation 1 -

2
21/(0.2 + a-312) in the plane of the shear, and of
2

an arbitrary rotation round the axis.
224. The resultant internal force on the electron. The resultant
internal force on the electron, that is, the resultant of all the mutual actions
and reactions between the several elements of the electron, is found from
(335), § 220, in the form
Fi= I ff(f 12 +

f 2 1) del de2 = fffde1 de 2 . . . . . . . .. . . . (339).

The factor 1/2 is introduced for the sake of symmetry, so as to allow us
to extend each of the two integrations over the whole electron; fiis the
irreversible component of the electromagnetic force and is given by the
second of equations (335), § 220.
By the principle of the conservation of electric charge we have
d
dt d

d
1=

e

=

de 2 = 0;

hence the double integration of the first line of fican be effected behind the
d
operator dt
Again, the terms in the second line of f%are completely independent of
the coordinates, so that the double integration of this line is simply effected
by supplying the factor e2, where e is the total charge of the electron as
before, every element being supposed to have the same sign.

246

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

Hence we get
Fi= [

ff didel
+

1+

2e2i
3c (c2

-

+

v2)

({ui+u

2

2e2 (vi)v
2

3c (c - v)

-

D)D(

2

+

2e2 (vv)v+
(c

2

2e2 (v)2 v

+ 2 (c

- v

2

(340).
(340).

225. For the purpose of comparing this equation with the equations
given in Ch. XI, § 143, we shall write it in the form
dG
Fi =
+K
dt

where

G= K+

ffdeide2 {,1+

2
2e (v2) v

2e2~
2

({ui+u 1D)} D(-)
2

2

3c (c

3c (c - v )

2

+

2
2e (vi) i

- V2)2

V

... (3 4 1).

2e2 (v )2v
+ c( 2
23
)

Comparing these equations with (205) and (209), § 143, we see that
G represents the electromagnetic momentum of the electron, while K is the
radiation pressure, the expression for which was first published by Abraham*.
This agreement between his result, obtained by an indirect method, and the
present one, found directly, may serve as a verification of our analysis.
The radiation pressure may be dismissed very briefly. It is a small
quantity of the order zero, and depends only on the magnitude of the charge
and its mean motion, not at all on its configuration nor on the relative
motion of its parts. It is not difficult to prove that the expression for
K which is in question, can be obtained by means of the Lorentz-Einstein
transformation from the well known expression for the radiation pressure on
an electric charge e, which is vibrating with high frequency but small
amplitude, so that its velocity is always very small while its acceleration is
finite. The radiation pressure on such a charge is equal to
2e2 d3r'
3c3 dt'3
where r' denotes its radius vector and t' the time. If we regard this system
as moving relative to a fixed system with velocity v, considered as constant
for the time being, and transform by the method of Lorentz and Einstein,
we obtain the expression (3413) for K, for a charge e moving with the velocity
v, now regarded as variable.
226. The electromagnetic momentum. The expression (3412) for
the electromagnetic momentum G shows that there is a principal term of
order - 1, namely
-

.ffdedeD) V
1 -

.................... (342),

* Theorie der Strahlung, p. 123, eq. 85, 1905.

D]

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

247

together with a small term of order 0, namely
-f

+
Jdede ({ui u2 lD)D
2

The latter is easily disposed of.

V=

-)V.........(343).

For we have by (3355),

220,

2
2
{(1 - v2/C2) R + (vR)2}-1 ,

while D denotes vector differentiation with respect to the velocity v. Thus
V and all its derivatives D V, D 2V, ... are even functions of R, and therefore
do not change sign when the sign of R is changed.
Moreover ul and u 2 are linear vector functions of the radii vectores rl and
r 2 respectively, and change sign when the signs of these radii vectores are
changed. This follows from (336), § 221.
Hence if the configuration of the electron at any time be centrosymmetrical, so that to any element de with radius vector r there corresponds
an equal element with radius vector - r, the electron will always remain
centro-symmetrical, at any rate to our approximation where we neglect
squares of the radius vector. It follows further that the small relative
motion term (343) vanishes on account of the centro-symmetry. In other
words, the electromagnetic momentum of every centro-symmetrical electron
depends only on its mean motion and not at all on the relative motion of
its parts.
All the types of electron hitherto suggested, such as the rigid spherical
electron of Abraham and the deformable spheroidal electrons of Bucherer
and of Lorentz, are centro-symmetrical. There does not appear to be any
fact known which requires us to consider types of electron not possessing
this kind of symmetry; for this reason, and to avoid needless complications,
we shall in what follows neglect the small relative motion term (343).
227. The electromagnetic momentum of the centro-symmetrical
electron. Bearing in mind the value of V and the meaning of the operator
D, explained at the commencement of § 226, we find
2
D(1 -v2)= V {2 (vR) +(c2 v2 2
- 2 )(vR)R
3
2
4
2c (1 - v2/c2) R + (vR)2/c2 /2
c2
Substituting this value in the expression (342) we find
2 v2 22R + (c -2
2
(vR) R
2
G = U {2 (vR) ± (c - - v2/c)2) R 2v+ (vR)2/cZ,) /2
2c
de 2
de(1de ... (344).
3

This equation shows that the electromagnetic momentum is the resultant
of two components: (1) a component in the direction of the velocity v; (2) a
component which is itself the resultant of a large number of radial forces,
and does not necessarily lie in the direction of the velocity.

248

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

Forming the scalar product of G by the unit vector v, we find for the
component in the direction of the motion
U

(v2G)

24

f

(C

(c)R2 + (vR) /c j}/
2

212)(yV1R

'(C2"+

'

_2

2(

e, de2 .. . . . . . (345).

2

1 l-

2:

{(+

Again, forming the vector product of G by v1 we find for the component
perpendicular to the direction of motion

G]
[v =(C

2

1

2
-

4
2C

)

jf 1
I{(

(
-v

'c2)

d

R)[R]

fR2

+ (vR)2/c

.

(46)

}~/

228. In order to understand clearly the physical meaning of the
equations (345) and (346), let us take Ox parallel to the direction of motion
for the time being, and apply the transformation
-V 2/c 2),

=(1

y=iz=

...........

(347).

It is obvious that these equations merely express the fact that the actual
electron can be derived from the transforned electron, that is the electron in
the ( , m, ') system, by applying the Lorentz-Fitzgerald contraction.
Of
course we are not bound to identify the transformed electron with the actual
electron when at rest; such an identification involves the acceptation of the
Lorentz-Fitzgerald contraction hypothesis, to accept which may or may not
be desirable on other grounds.
Applying the transformation (347) to (345) and (346) we get

V(+
-

-

)±(12

v(- (-dd
±
(t2 -7~

Gv

2

2c .~(~2-~

2

+

-

)2

++q(f-lq)-

+

1j
)21

,)de,de

~

(q 2-

X-R

2

ql)

+ ( '2

-

1)2}32

(4)

The last two of these equations show that the transverse component of
the electromagnetic momentum vanishes when one or other of the following

two conditions is satisfied, namely
(1)

when the electron is symmetrical fore and aft, or

(2) when it possesses two longitudinal planes of symmetry.
In the first case, to any element de at ( , 2), ) corresponds an equal
element at (

,~

)

D]

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

249

Both conditions are satisfied by the electrons of Abraham, Bucherer and
Lorentz; accordingly their electromagnetic momenta have no transverse
components.
Both conditions are violated by a spheroidal electron whose axis is
inclined to the direction of motion; accordingly its electromagnetic momentum
has a transverse component, a result which will be verified in example 4,
infra.
229. The electromagnetic mass. There is no difficulty in defining
the electromagnetic mass in the usual way, as the ratio of the electromagnetic
momentum to the velocity, so long as the electron is centro-symmetrical. In
the general case this definition is inconvenient, because the ratio is one
of two differently directed vectors, and therefore neither a vector nor a scalar,
but a quaternion. A more useful way is to define the electromagnetic mass
as the ratio which the electromagnetic momentum bears to the speed, that is
the tensor of the velocity, by the equation
G = vm ................................ (349).
This definition makes the electromagnetic mass a vector quantity, whose
components are given by
+_)

()21

X
=

1

2c2
1
=
2c2

) (,2

('-

=2

f

12+

1)

-

1

2

1

e, de2

2

(
2

2- +)±(1+

2~2
(

de de2

3/2

dede2

21

2- (2 ) (2
2 2

(,)
-

-......

+ (.
(7(2-

4)
)1)2

1 2

3/2
Y

(350).

In the same way we can define the longitudinal mass m' by the equation
am
aG
m'=-- v= m + v v......................(351).
This definition makes the longitudinal mass also a vector.
When the electron is unsymmetrical, in the sense that it violates both
conditions of § 228, both masses have transverse components. But for the
electrons of Abraham, Bucherer and Lorentz, both masses lie in the direction
of motion, and their absolute values reduce to the transverse and longitudinal
masses as usually defined.
230. We can now write down expressions for the components of the
external force acting on the electron, so far as they depend on the electromagnetic momentum. This force is of course equal to dG where the
-'-,

250

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

differentiation is total.

Let us take as our axes of ( , q,

[APP.

) the tangent,

principal normal and binormal to the path of the electric centre of the
electron, just as in Ch. XI, § 144; then we have
dG

aG
+ [ G ] ........................... (352),

dt

where

denotes differentiation relative to the rotating axes, and w is their
G tG
angular velocity about themselves. Thus we have at, while the
t

c

components of are (v/r, 0, v/p). Bearing in mind (349) and (351), we find
for the components of the part of the external force with which we are
dG
concerned, that is for -, the values
dt

m'

'

v2
- m,-,

2

2

m,'vi + m

P

- m-.
P

2

m

+ m,-......(353).
'3.

Thus we have
(1) A component m', proportional to the longitudinal acceleration v,
and in the direction of the vector m'.
(2) A component (- m,, mt, 0) v2/p, proportional to the centripetal
acceleration v2/p, in the osculating plane and perpendicular to the vector m.
(3) A component (0, - m , m,) v2/7, proportional to the torsion 1/T, in the
normal plane and perpendicular to the vector m.
When the electron is symmetrical, either fore and aft, or with respect
to any two mutually perpendicular longitudinal coordinate planes, the first
and second components reduce to the usual longitudinal and transverse ones,
while the third vanishes.

When the electron is asymmetric this is not so. For instance, in order to
keep it moving in a straight line with acceleration ), we require transverse
forces, m,' and ms'v, in addition to the longitudinal force, rnmt.
Again, in order to keep it moving in a circle of radius p with the uniform
speed v, we require a longitudinal force, - mv2/p, in addition to the centripetal force, mev2/p.

This case has been referred to already at the end of Ch. XII.
speed and radius be so adjusted that
p2

If the

p2

this part of the external force is just supplied by the drag on the electron
due to radiation from the ring.
Hence a steady motion is possible provided that m, be positive, and the
speed of the ring be properly adjusted to its size.

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF
]

D]

251

231. The energy relations of the electron. For the sake of comparison with (211), § 14 5, we shall begin by finding the activity, (vFi), of the
resultant internal force Fi. We find from (341), § 225,
(vF

(v

-vd

By means of a few simple algebraic transformations we deduce the
following form of this equation
_

vd =-()

(vF)

tj=3(~-)

-

2ce 2 (vi)

+

GG

wher
........................... (
2c 2
R

where

3

(C2

........ (354).

v2(°

2e

=

2 2
-,U

)

C2 _v2f

1 is Lienard's* value for the rate of loss of energy due to radiation from
the electron. When the electron is symmetrical, either fore and aft, or with
respect to any two longitudinal planes at right angles, so that G is in the
direction of v, the first equation reduces to (211).
When there is relative motion of the parts of the electron, the quantity
(vFi) does not represent the whole activity of the internal electromagnetic
forces. We shall call the latter the internal activity and denote it by the
symbol Ai.
232. In order to calculate it, we notice that the activity of the electromagnetic forces exerted between the elements del and de, is equal to

,
2

{(v 2 f12 ) + (v 1 f 21)} delde

on the basis of the usual definition of work. Putting v1 = v ± u2 , and
V2= v + u2, using (333), § 220, neglecting squares and products of u1 and u 2,
and integrating, we find by (339), § 224,
Ai= 2ff{2 (vfi) + ({u + u2} fi) + ({u2 - u1} fn)} delde 2
1
= (vFi,) +
where

Ar

Ar

=ijfdel

de2{l

-

u2 } {1 + 2 ({u1 + u2} D)}J1(+ifu1+

C2)

u2

v 2 v)
{D(1

V})
.. (355).
.............
2

)

The factor 1/2 is supplied because in integrating over the whole electron
with respect to both del and des we count the pair of elements del and
twice over.

de2

*

L'Lclairage electrique, 16, . 5,.198.
t

252

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

Thus the internal activity Ai exceeds the activity of the resultant internal
force F2 by the amount A,., which depends on the relative motion and
vanishes with it. We shall call A, the relative activity.
233.

The expression (3553) for A, can be simplified.

Since V is a function of the differences of the coordinates of the elements
1= 12 -1,
we find

del and del, as well as of the velocity v, and v 2 1

--

V

_

dt

so that

vD

=

2

1

(

c2

({112-12}2V

(i
2)

V

-

({v2

c2

V ={D

1-

2

vV2)

}

V

C

(i-

V.

Similarly
(u+ +

1121

v})

D(1i

=

({u + u,}(D)D) 1 ({u2-

+u2

-(tul

V

u2} V2)D) (i
D

v

Hence the integrand of (3553) is equal to

{
('r 1+ (l+1D}D(1+

-1 u21
j([u

)

1+

2}

V)-d
D) V2 -

{u,

1vt(
+ 121 Gu,

-1121

V2)D2(
-cV.

When the second line is expanded, care being taken to keep the relative
11 and u2 in front of the operators, it is at once seen to vanish
identically because D and V2 are commutative with each other. By comparing the first term of the first line with the expression (3412), § 225, we see
that it is equal to the scalar product of v by the integrand of G ; the second
term however is new.
velocities

Multiplying by 2 del de2i integrating and bearing in mind that dei and de2
are invariable, we find

................
.......... 356).
where

't= 2(iv-v2

ddj

When G is in the direction of motion, it is easily proved by means of
(356) and (344), § 227, that

G=......................................(357),

D]

253

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

234. In order to see precisely what the function
the transformation (347), § 228.

represents, let us use

Using the expression for V given in (3355), § 220, we find
del del2
1) () + (2-)2}1/
S= (1 2 '/c

... (358).

Thus cP is equal to /(1 - v2/c2) times the electrostatic energy of the
transformed electron, of course for the same total charge. This energy
generally depends on v, because the limits do so; it is independent of v in
the particular case of the Lorentz electron.

235. In the preceding articles we have found three different activity
equations, namely
(vF) =

(vG) - 3 (c - v2)

-

Ai= (vFi) + A
A,

=

-

(.

G)(35f),

+

(4G) - R ... (354), § 231,

.......................
dC
dt

(355), . 232,

........................... (356),§ 233.
233.

From these we deduce a fourth, namely
AZ =--

(vG) +

4

-

(...........(359).
R

These equations are true for any electron, whether it be centro-symmetrical

or not. The first of them takes the place of the energy equation (211), § 145,
which is only true when the electromagnetic momentum is in the direction
of motion, and is merely a particular case of (354).
The external mechanical force impressed on the electron, which balances
the resultant internal force F whenever the mass is wholly electromagnetic,
does work at the rate - (vF), equal to A, - Ai, by (355).

Of the work thus

available, the internal electromagnetic forces consume a portion at the rate
- Ai, part of which is irretrievably lost at the rate R owing to radiation,
while the rest is stored in the electron and can be recovered if the motion be
completely reversed, in accordance with (359). We may regard the part
which is independent of the acceleration v as the electromagnetic energy of
the electron; it is given by
........................... (360).
W = (vG ) +
The remaining small acceleration

term, - 2ce2 (vi)/3 (c2 _ v2)2, is the

supplementary reversible radiation term, just as in § 146.
236. The remainder of the available external work, namely the part At,
cannot be accounted for by means of the electromagnetic energy or the
radiation. The work thus unaccounted for is equal to the work done by the

254

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

internal electromagnetic forces during the relative motion of the parts of the
electron, and must be consumed by forces of other than electromagnetic
origin.
Substituting the value of G from (349), § 229, in (356), we find
v3
A,.= - mvvi- m, p

d
dt

dI?
d
dt fmvdv +

(}

- m,

... (361).

Thus a part of the work in question is stored as non-electromagnetic energy
to the amount
E= - fmgvdv ........................... (362),
the rest is lost at the rate - rmv 3/p.
For the rigid spherical electron of Abraham the non-electromagnetic
energy E is zero, because there is no relative motion; for the spheroidal
electron of Bucherer E is zero because there is no change of volume; but for
the spheroidal electron of Lorentz, E has a finite value different from zero.
It is well known that this fact has been urged as an objection to the Lorentz
electron by Abraham *, and that an explanation has been given by Poincar t.
We shall see in Appendix E that Abraham's difficulty is already involved in
the more fundamental difficulty of accounting for extended electrons at all in
view of the mutual repulsions between their parts.
237.

When we substitute the value (349) for G in (354), § 235, we find

Mv

d
- (vF) =

2
(

dt

-

-

2ce2 (vy)

mvdv mdv
-

_
- +

3

(2

.

2

This equation shows that part of the external work done is lost to the
electron at the rate R - m,v/p, owing to the combined effect of the drag due to
radiation and the pull due to asymmetry. Another small part, depending on
the acceleration and derived from the radiation pressure, is stored to the
amount - 2ce2 (vv)/3 (c2 - v2)2. The remainder depending on the speed is
stored as kinetic energy to the amount
T= mntv 2 - f mtvdv .......................... (363).
From (360), (362) and (363) we see that
T= W + E ............................. (364),
so that the kinetic energy T is partly electromagnetic, to the amount W, and
partly non-electromagnetic, to the amount E. When the electron is symmetrical, so that m reduces to the ordinary transverse mass m, T reduces to
the kinetic energy defined by (214), § 146.
We shall now consider some examples in illustration of the results just
obtained.
Phys. Zeitsch. 5, p. 576, 1904.
t Comptes Rendus, cxI. p. 1504, 1905.
*

D]

255

D
THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

238. Example 1. The Abraham electron. Xe assume that the
electron is a rigid sphere, of radius a and charge e, and of uniform volume
density. For uniform surface distribution we must multiply the mass by the
factor 5/6.
The transformed electron is bounded by the prolate spheroid
(1

2+

-12)

2

iq

+

=a

2

,

where 1= v/c as usual. It is of uniform volume density and its total charge
is e. Its electrostatic energy gives, by (356), § 233,
3e2 (1-,32) log 1+
log1-18.................(365).
= l1a

d1 only depends on the speed v and has no reference, explicit or implicit,
to the direction of motion of the electron. Hence, by (357), § 233,
a

3e

av =v1 5ca
1

G=-v

2
3

y2/3lg 1±+3 -1).......(66).
/1+,82

1 1 3

G is in the direction of motion, so that m, = m, rnt= i', and the other
components of the vectors m and m' are zero. Hence
2
m 5 (2 'log

d (m,8)

d,3

=

6e2
5c2aP (1

...... (367),

_I -1)....
1 _ 1-/32

log1±+3

21

,32)k

The electromagnetic energy is given by (360), § 235, whence

W = my2

+ P=

1

log

l) .- .........

(369).

In the present case there is no relative motion, so that the relative activity
A,. vanishes. Accordingly the non-electromagnetic energy E reduces to a
constant, and we get by (363), § 237,
T

3=
fm',8d$ 6e

log

1_.

(370),

,. ......

-1i).

which only differs from W by a constant.

§ 143,

The values of mn and T agree with those given in Ch. XI, (206),
and

(215), § 146.
239.

Example 2.

The Bucherer electron.

Heaviside ellipsoid with semiaxes a

(1

-,12)1/3,

a (1

-

The electron is a
132)-1/6,

a (1

-

2
13

)'

/

16, of

uniform volume density and total charge e. If the charge were an equilibrium surface distribution the mass would have to be multiplied by the
factor 5/6.

2 56

[APP.

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

The transformed electron is a sphere of radius a (1volume density and with total charge e.

f 2) 16, of
/

The usual expression for the electrostatic energy gives by (356),

(D=
:e 2 (1

-

uniform

§ 233,

12)2/3

5a

(371).

.......................

1 involves only the speed v explicitly, but it also involves the direction
of motion implicitly, on account of the want of spherical symmetry of the
original electron. Hence we must use (:348), § 228. We find that the second
term in G4 is one-third of the first, while Q, and G vanish identically. Hence
we find
= 5G
5c'2a

in'

4e 2 v
(-l

1(1

2
/32)

-

)..................(372).

The vectors m and m' are in the direction of motion, so that
= m', m, = m = my,'
' = O. We obtain

=n

c24(I
_-d

(n)
m

.
2

_54e

2

-

a (1

2)

/c

-

The electromagnetic energy is given by (360),
W =nv2 +

(373),

..

(1

v=m

..

....

§ 235,

(374).

whence

e2(,).................(375).
5a

The kinetic energy is given by (363),

(1-132

§ 237,

whence
3............(376).

T~fn/vvL/(31 -......

T differs from W by a constant, so that the non-electromagnetic energy
E is constant, agreeing with the fact that the relative activity vanishes.
This is easily verified by means of (356), § 233. (371) and (372) give
4e2, 3/3
Ar =
that

0.

(i )so

d('P

Thus the relative activity vanishes although there is relative motion of
the parts of the electron on account of its rotation and deformation. Since
the strain takes place without change of volume, it would seem that the
relative activity depends on changes of volume.

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

D]

257

The electron is a
The Lorentz electron.
240. Example 3.
Heaviside ellipsoid with semiaxes a (V(1 - 32), 1, 1), of uniform volume
density and charge e. For an equilibrium surface distribution the mass
would have to be multiplied by the factor 5/6.
The transformed electron is a sphere of radius a, of uniform volume
density and with total charge e. The usual expression for its electrostatic
energy gives by (356), § 233,

~3e -2
e

7z'

(1

/32):

(

2 ...

... .........

.(377).
.5a...

As in the last example we must use (348), § 228, to find G; we get
4e 2v
2
5c a .(1 ................. (378).
As before m and m' reduce to their tangential components, so that n _=
=rn', rnq = m = m,I'= rnZj==0. We find
4e2
(1 - 32)

=5c2a

d (rn1)

d$3

(379),

..........
4e

5c'a (1-

The electromagnetic energy is given by (360),
W
2+

T=fmid2=

§ 235,

(380).

whence

e2(3±/32 )

W~mv + '5a
The kinetic energy is given by (363),

.....

/2)3/2.

(1-/32)..............(31)

237, whence

{....)}.....(382).
)

W and T differ by a quantity which depends on /3. Accordingly we
find that the relative activity Ar is different from zero. In fact we have
from (377) and (378)
_ 4e2$$/
4 d'c

1-32

(i!G)=5a
Hene,.

3 dt

~dt, and a non-electromagnetic energy E exists. By (362),

S236, we find
E=-fmvdv

-

D=

e{(1

_32)-41 .........

(383).

The relation T = WV + E is easily verified. Of course arbitrary constants
may be added to W and E, but since T represents kinetic energy, the
constant in T has been chosen so as to make
vanish when the speed
vanishes.

it

S.

17

258

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

[APP.

241. Example 4. Deformed Lorentz electron. As an example
of an asymmetric electron we shall take the case of an electron, which is
got from the Lorentz electron by a slight deformation symmetrical about an
axis inclined to the direction of motion.
Take axes of (x, y, '), such that Ox makes the angle f with O,
measured towards 077, so that
= x cosr - y sin 9, = x sin + y cos
......... (384).
Let us suppose that the transformed electron is given by
r = a + P (), i>0 ........... .......... (385),
where 1 = cos 0, and (0, f0) the colatitude and longitude referred to Ox as
are
polar axis and any convenient initial meridian. We shall assume that E is
so small that its square may be neglected. The electron deformed in this
way is no longer centro-symmetrical in the case of i an odd integer, and
therefore the small relative motion term in the electromagnetic momentum,
(343), § 226, does not vanish. As however it is small of zero order in any case,
it becomes absolutely negligible when e is small. The volume of the transformed electron is the same as that of a sphere of radius a, and the volume
density of the uniform electrification is 3e/47ra3 as before. Any double integral
of a function of the coordinates can be reduoed by means of the equation

f.

f.

deide=

dede+

8

(

.P(pi)dSdV

... (386).

The integrals on the right are taken over the sphere r = a; the second
integral gives the effect of the thin shell between the sphere and the surface
(385). For an element of this shell we must put del = 3eePi ( i) dS1 /4ra,
where dS is a surface element of the sphere, and de2 = 3ed V2/47ra, where
d V2 is a volume element. In this way we find one half of the second
integral; the other half is got by interchanging del and de2, and obviously
has the same value as the first half.
242. We shall first calculate the scalar integral 1 from (356), § 233; for
this purpose we put f= (1 - 2)'1/
13 2/2R, where R is the distance between the
elements del and de2 of the transformed electron. The value of the first
integral has been found already in § 240, and is given by (377); hence its
value is 3e2 /(1-

2)/5a.

The second integral becomes

9e2e /(1 - /32)

ff P(1 ) dSdV2
167r 2a6
R
taken for a sphere of radius a. This integral vanishes, for it is the mutual
electrostatic energy of the sphere and of a shell whose thickness is EPi ( ).
In fact the potential due to the sphere at every point of the shell is the
same, and the total volume of the shell is zero. Hence we find
A3
=a

.......................... (387).

259

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF
]

D]

243. We must now calculate the scalar integral G, or, what amounts to
the same thing, the vector m by means of (350), § 229. For this purpose
we require to find the integrals
:1)2 2,
dede

2 -

ff(

3

ff(e2 -

1) (i2

'

h1)

delde2

R3

f(2

-

-

) delde2*

R

,

The last vanishes identically because the axis of symmetry has been
taken to lie in the plane rj. In the absence of any deformation the first
integral is one third of the integral
d e taken over the spherical transformed electron, that is, it is equal to 2e2/5a, and the second integral vanishes
by symmetry. Hence we find by (386), § 241,
(

~2 1
-

dd

f(
f(2

-

-

del de2

2

(a

) 2 P. (

-

6

+ 87r a

-

7

1)delde2- -

) (12-

2
e

2e2 +9e

R3

) dS 1 dV 2

R

82
9e2E
w

(ffPi

dSd V 2

"8

R

87r~a6

Both the integrals in (388) can be expressed in terms of a single integral
by means of the transformation (384), § 241. For this integral we shall choose
the following
y1)2

I(y2

(2

+

l)2R

-

Pi.(.)..........
......

............

(389).

By symmetry we have
[f(X2-

xl)2

I1X1 -

ii(X
2-

P1

@') dS 1 d V2
P (l)2
(V

Pi ()d1) dSd d V-

x1)

(Y2

-

- )P1 (,)dSidV _
2 0

Hence we find by (384)

ff
ff2

1)

2

Pid( 1) dS 1dV 2

) (t7l2 -,l)

11(3

-

(y-___________

R3aI

cs2#

1

1)

. 1 ) dS1 dV 2 -}Iin.cs

P (

244. Calculation of the integral I.
no longer necessary. We have identically

The first

2

=

x-x

1

term, 1/B, gives the integral

-

ff

.
.(390).

We shall omit the suffix 2 as
1+
ax R
(
Pi (u1)

1

1

dS1 d V/R, which vanishes

260

[APP.

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

Since the element dS1 is on the surface of the sphere, and the element
d V is inside, we have r < a ; hence
n

1

rnP (cos y)

Y

R

-

1 +412(1

=

'Y

+1

U

l)cs()-c)

Put dS1 = a2dp 1d4, dV = r2 drddck in (389),

§ 243,

and notice that

a
and ax ' 'ar + i- rap a
an
Thus the integrand of (389) only involves 4 and 01 in the factors Pn (cos ry);
hence the integrations with respect to 0 and f1 can be performed at once
=
x-x 1 =rr - -a 1

by using the addition theorem for spherical harmonics.

n- a4
,

=0

an

2
1

Pi
J- lar(1r) dr
2

_

-1

(,a

We have

a ) (fa

-

+ 1a~

r

(2n + 1) FLP

(,)

=

+

-2

a

(

ar

and

d d i (rp

p

-

We get

)rP()
~?
(

=

nrn-1P_ 1

)

(n + 1) Pn+i
,()

)
(

+ nPn-1 ()

Substituting and integrating with respect to r, we find
j n=00PI(± (n - Pn 2 -(FL)}P
)
n=O (n±+3)
(2n-1) -1 -1
n=
47r2a4n
1 1
+P
j
(n + )>
n-O (n + 2) (2n + 1)

1)

(

1)

)d d

P(

o

(

The only values which yield a result different from zero are given by

n

=

i = 2 for the first integral, and n = 1, i = 2 for the second.

find I =

-

647r2a4/2 25 for i

Substituting these values in (388) and (390),

f

Re
)('12 - q)

f( 2 -

~25a

For other values of i the terms in
245.

2

sin *cos#*,

for

39)

i=2

e are absent.

The electromagnetic masses.

§ 229,

§ 243, we find

5a+ 2 5 a2( 3 cosjl ), for i=2
delde2 _1 2e2

R3

and (391) in (350),

Hence we

2, zero for all other values of i.

=

Substituting from

(387), § 242,

we get, for i = 2,

4e2

~~5c2a \/(1

-I'2)

+10ac

21)}

(

2

,

D]

THE MECHANICAL FORCE EXERTED BY AN ELECTRON ON ITSELF

261

From (351), § 229, we find
m =1

'

2'

rm,, m = 0 .................. (393).

For all values of i other than 2 the terms involving are absent. Thus
the only axially symmetrical deformation of the transformed Lorentz electron,
which has any effect on the masses, is that of order 2, for which
r = a + j e (3 cos -1).
This changes the sphere of radius a into a spheroid of revolution, with
semiaxes (a + e, a - je, a - Ie), and with its axis of symmetry at an angle f
to the direction of the motion. The proportional change in mE, and also in
me', is independent of the speed, and equal to one fifth of the proportional
change of the radius vector drawn in the direction of motion. It is a
maximum when the axis of the deformation coincides with the direction
of motion, a minimum when it is at right angles, and zero when it makes an
angle cos-' 1/V3 with it.

The values of m, and m,' are independent of the speed, vanish when
the axis of deformation is along or perpendicular to the direction of motion,
and are greatest when it makes an angle 450 with it.

246. The energy. We see from (363), § 237, (392) and (393) that the
kinetic energy is proportional to that of the Lorentz electron, and given by
1ea

4e2

{({-1

1

(3cos21-11).........(394).

1+

For the electromagnetic energy we find from (360), § 235,
W=5a(

3+

+3 5a( COS2

-1).........(395).

For the non-electromagnetic energy we get from (362), § 236,
E =

e5

(1 -

)

4 -

1

-

5a

-

(3 c os

- 1) ... (396).

We see from (394) and (395) that for a positive value of c, that is, for

a small elongation along the axis, both the kinetic and the electromagnetic
energies are greatest when the axis of deformation is along the direction of
motion, and least when it is at right angles.
For a negative value of e, that is, for a small compression along the axis,
the reverse is the case.
The non-electromagnetic energy behaves just in the reverse way.
The electromagnetic forces tend to produce contraction in the direction
of motion beyond the amount, appropriate to the speed, given by the Lorentz
ratio 1: V(l - /2), while the non-electromagnetic forces resist it.

APPENDIX E
THE MECHANICAL
247.

EXPLANATION

OF THE ELECTRON

OUR investigations of App. C and D respecting the mechanical

forces on the electron have been avowedly based on the assumption that
the electric charge occupies a finite region of space, whether it be a small
volume, a small closed surface, or even a small closed curve. This assumption-that the electron has parts-compels us to face the problem of
accounting for its continued existence in spite of the mutual electromagnetic
repulsions between those parts. We must seek some system of forces of
non-electromagnetic origin*, which shall equilibrate the resultant of all the
electromagnetic forces on each element, whether they be due to the remaining
elements of the electron, or to an impressed external field. Moreover, since
the electron is capable of motion in any arbitrary curve with any arbitrary
acceleration under the action of the appropriate external field, the system of
non-electromagnetic forces must be adaptable to every possible motion of the
electron as a whole and consistent with the results of observations on the
electromagnetic momentum and electromagnetic mass.
Two alternatives respecting the nature of the non-electromagnetic forces
suggest themselves: they may be actions at a distance, or actions between
neighbouring elements, equivalent to stresses in an elastic medium. The
first alternative may be dismissed in a few words; if actions at a distance
are to be admitted at all, we may as well regard the electron itself as a centre
of force, and the electromagnetic force due to it as an action at a distance.
On this view the electron is an indivisible whole and has no parts; no
explanation of its existence can be given, nor should it be expected. This
hypothesis is no doubt feasible, but it has difficulties of its own; it does not
appear to afford much scope for explaining the differences between positive
and negative charges, or for constructing systems of charges sufficiently
stable and permanent to serve as models of the atom.
* During the passage of this book through the press, H. T. Wolff, Ann. d.Phys. 36, p. 166,1911,
has suggested a way of attaining the same end by modifying the law of electric force. His
additional force, as regards mathematical form, is included in the more general expression found
below.

APP. E]

THE MECHANICAL

EXPLANATION

OF THE ELECTRON

263

248. We shall adopt the second alternative as more in keeping with the
general trend of this essay, and accordingly shall assume that the nonelectromagnetic forces are due to stresses in an elastic medium. At the
present stage we need not enter into the question whether this elastic
medium is the electric charge itself, or is different from it, but mechanically
connected with it so that the stresses are transmitted to it practically
unchanged. All that we do in fact assume is the formal equivalence between
the mathematical expressions for the non-electromagnetic forces of the
electron and those of the forces due to stresses in an elastic medium.
In this form the problem is too wide to require any special investigation;
for it is possible to find a system of stresses in an elastic medium which
shall equilibrate any system of forces whatever, whether applied throughout
a certain volume, or to a certain surface. In particular a type of stress can
always be found which shall equilibrate the electromagnetic forces of the
electron, whatever its motion may be, but its investigation would not be
of much interest.
We shall content ourselves with determining the conditions under which
the stress system reduces to a distribution of hydrostatic pressure, without
concerning ourselves with the physical causes producing it. An investigation
of this kind will be useful as a basis for some future physical theory of the
electron, and of its connection with the electromagnetic aether, if the aetherhypothesis should be adopted. In the latter case the rotational displacements
of the aether would be needed for the explanation of the electromagnetic
forces, and only the irrotational displacements, involving hydrostatic pressure,
would be available in accounting for the non-electromagnetic forces. Even
if the aether-hypothesis were rejected, a distribution of hydrostatic pressure
would naturally be chosen as being the simplest stress system available.
249. The equation for the pressure. In order to express our
hypothesis of the hydrostatic pressure mathematically, let us denote an
element of charge of the electron by de, its coordinates referred to the
electric centre by (x, y, z), and its electric volume density by e, so that
de = e dxdydz. We shall treat a surface distribution as a limiting case of
a volume distribution.
For the sake of generality we shall assume that the electric charge has
associated with it a certain mass of non-electromagnetic origin, which may
be intrinsic to it, or may arise from its mechanical connection with the
elastic medium producing the hydrostatic pressure. It is only to be expected that one effect of this connection may be equivalent to a loading
of the charge, to an increase of its effective inertia as it were, and may as
well be provided for. Moreover measurements of the specific charge of the
electron do not entirely exclude the presence of non-electromagnetic mass,

264

THE MECHANICAL

EXPLANATION

OF THE ELECTRON

[APP.

provided that it be but a small fraction of the electromagnetic mass. We
see from § 229 that the latter mass is of the order - 1 in the linear
dimensions of the electron, so that the former is of the same order, but
much smaller. We shall denote its value for the whole electron by M, and
the non-electromagnetic mass density by t.
250.

The electromagnetic forces acting on the element de are twofold:

(1)
We have the resultant of all the electromagnetic forces due to the
remaining elements of the electron. In this Appendix we shall denote it by
per unit charge; although in Appendix D and in Ch. XI this symbol was
used for the resultant internal force on the whole electron, no inconvenience
will result. In (335), § 220, we are given the value of f12, the electromagnetic
force exerted by unit charge of the element del on unit charge of the element
de2 ; if we omit the suffix 2, multiply by del, and integrate over the whole
electron, we shall obtain the value of Fi. The principal term of f, is of the
order - 2 in the linear dimensions of the electron, and it still occurs in Fi,
although it disappears on averaging Fi for the whole electron. We shall go
one step further and retain terms of the order - 1, neglecting all terms of the
order 0, such as the terms involving the relative velocities u1 and u2 , and the
d
radiation terms in the expression of fi, (3352). Since the operators d and D

F

do not affect the charge de, on account of its invariability, we may integrate
behind these operators. Thus we find
Fi=-

de, 1-+({u+uD)

'1v

v+

d

Dfde

1-

V...(397.

251. (2) We have the electromagnetic forces due to the external field,
whose resultant will be denoted by Fe per unit charge. The integral

f (Fi +

Fe) de

taken over the whole electron represents the resultant electromagnetic force
acting on it due to all causes, and produces non-electromagnetic momentum
at the rate Mi, because the non-electromagnetic forces are self-equilibrating
by the fundamental assumption of Ch. XI, § 140. Hence we have by
Newton's Second Law
Mi.
fFede= - fFde+
The integral f Fide is the resultant internal force on the electron, which
in § 224 was denoted by Fi simply, and is of the order - 1. Moreover My is
at most of this order, so that Fe is of the same order.
Hence the variations of Fe from point to point of the electron are only
of the order 0, and can be neglected, so that the integral fFede may be
replaced by eFe. By means of (340), § 224, we find, neglecting the small
zero order quantities,
d
dede 1 V + M.(398).
Fe = -t2
ede 1
V+Mi.......(398).

THE MECHANICAL EXPLANATION

E]

265

OF THE ELECTRON

252. The non-electromagnetic force due to the hydrostatic pressure, p, is
equal to - Vp per unit volume. This; together with the resultant of the
internal and external electromagnetic forces, amounting to E (Fi + Fe) per
unit volume, produces non-electromagnetic momentum at the rate v per
unit volume. Hence we find by Newton's Second Law
i

=

- Vp + e (F + Fe).

Substituting the values of Fi and Fe from (397), § 250, and (398), § 251,
respectively, we find

1+
Vp=-efde, l

+d

dfde

({u+uD) (1 -

(1 -

V-

VV

ffdede,

(1

- v

V +

-

v

...... (399),
where V= 1/S = 1//{(1 - v2/C2) 12 + (vR)2/c 2 }, by (3354), § 220.
This is the desired relation for the hydrostatic pressure; it will be seen
that we have eliminated the external electromagnetic force, Fe, by means of
(398), § 251, so that the pressure gradient is expressed solely in terms of the
configuration and motion of the electron at the instant considered. The
motion of the electric centre, which is determined by the values of the
velocity v and acceleration Q, can be anything we please, so that v and are
quite arbitrary. On the other hand the configuration of the electron, the
distribution of its electric charge and non-electromagnetic mass, and the
relative motions of its parts are at our disposal, subject to the restrictions
that the resulting expressions for the electromagnetic momentum and mass
shall agree with the results of experiments on the specific charge of the
electron, and that the non-electromagnetic mass shall admit of a mechanical
interpretation.
Our problem may now be stated as follows:
It is required to find all configurations of the electron which shall make
an equation of the form (399) possible for all values of v and v, subject only
to the restrictions just indicated.
253. Uniform rectilinear translation. In order to gain a clear
notion of the nature of the problem, we shall first consider the particular
case where the electron is moving as a whole with uniform speed in a fixed
direction. The acceleration, v, and the relative velocities, u and u,, all
vanish, so that all the small terms in (399), § 252, disappear, and only the
principal term is left. Hence we have

VP=- EVfde

1-

V ..................... (400).

266
THE

266

MECHANICAL EXPLANATION

OF THE ELECTRON

[AP

[APP.

The operator V has been taken outside the sign of integration because it
does not affect either of the quantities v or del.
Let us apply the transformation of Lorentz, expressed by
these equations may be written in the form
xIK,

y=fl,

z=

(347), § 228;

=1/(1-v2/c2)............(401),

with similar equations for (xi, yl, z1), provided that the line of motion of the
electric centre be taken as the axis of x.
We shall assume with Lorentz that the elements de at (x, y, z) and del at
(xi, yl, z1) in the actual electron correspond to equal elements de at
and del at (
) in the transformed electron.

,, q,

Kc:

1, so that the
The corresponding elements of volume are in the ratio
corresponding volume densities are in the inverse ratio 1 :K; hence the
volume densities at (E, , ') and (,, i, 1) in the transformed electron are
equal to KE and iEce respectively.

By means of (399),

§ 252, and (401) we find

V= 1/,cP,

P = 1/{(

F1)2 +

-

n)2 + (

(n/-

-

X1)2}..(402).

P is the distance between the elements de and del of the transformed electron.
We now find
de (i-

V=

where

............

-

(403).

-1f=del

'P is the electrostatic potential of the transformed electron at the point
( , '), which corresponds to (x, y, z) in the actual electron. We shall
q,
denote its mean value for the whole electron by
so that

T1,

1lfde dei)
~.

1 ff dedei (i-

and

e ..............
.P

(404).

v) V = ik'

A reference to (356), § 233, shows that

K'1'

is equal to 24)/e, where

4)

is

the function on which the electromagnetic momentum of the electron and
the relative activity of its internal forces depend.

254.

Returning to (400), § 253, and applying (403), we find
Vp =- ceV'P.......................(405).

This equation requires that
[Vice. VNP] =0 ...................... (406).
Hence we find

p = A - fKEd!P.....................

where A can onlydpnd onn the ime t_

(407),

El

THE MECHANICAL

EXPLANATION

OF THE ELECTRON

267

We see from (406) and (407) that a hydrostatic pressure, p, can only
exist when the volume density, ice, of the transformed electron involves the
coordinates only in so far as they enter into the expression of the electrostatic potential, 'i, and then p has the same property. Both ice and p will
generally also depend on the time, usually through the medium of the
speed v*.
Hence the equipotential surfaces of the electron are also surfaces of
constant volume density and constant hydrostatic pressure.
In particular, if the bounding surface of the electron should happen to be
an equipotential surface, it will also be a surface of constant pressure. In
this case the electron, whether at rest or in uniform motion in a straight line,
can exist provided that a uniform external pressure be applied to its surface.
In general this pressure depends on the speed of the electron; if the
speed change so slowly that the effect of the acceleration is small, our
present solution may be considered as a first approximation. The external
pressure needed to ensure the continued existence of the electron will be
variable.
If however the configuration of the transformed electron be invariable, its
volume density, e, and electrostatic potential, ', do not depend on the time,
and in (407) the constant of integration, A, may also be taken to be independent of the time, so that the hydrostatic pressure, p, is invariable.
Hence in this particular case the external pressure on the electron is not
only uniform over its surface, but does not alter as the speed changes.
255. We see from (350), § 229, that the invariability of the transformed
electron ensures that the tangential component, ml, of the electromagnetic
mass vector there defined shall follow the Lorentz mass formula. In order
that rnm shall reduce to the ordinary electromagnetic mass, in consequence
of the vanishing of the transverse components, m,, and m , it is necessary in
addition that the electron either be symmetrical fore and aft, that is with
respect to the yz plane perpendicular to the direction of motion, or be
symmetrical with respect to two longitudinal planes intersecting in the line
of motion of the electric centre.
* It is worth noticing that, when

KE

is the same everywhere inside the electron, p is a linear

function of 4 and therefore satisfies the equation

(1
-2)

+ a2+

=0

referred to axes moving with the electron, or the equivalent equation
Sp--

referred to axes fixed relative to the observer.
for p is much more complicated.

1 82p
c2

t2

When

Ke

varies inside the electron the equation

268

THE MECHANICAL

EXPLANATION

OF THE ELECTRON

[APP.

The conditions as to invariability and symmetry are all satisfied in the
case of the Lorentz electron, because the surfaces of constant density, as well
as the bounding surfaces, whether external or internal, of the corresponding
transformed electron are concentric spheres of invariable radius.
Hence the most general electron of the Lorentz type not only leads to
the Lorentz mass formula, but also can exist provided that a uniform and
invariable pressure be applied to its external surface, if it be solid, or to each
of its surfaces, internal as well as external, if it be hollow.
256. We cannot however assert that the Lorentz electron is the only
type satisfying these conditions. From the mathematical point of view the
problem of finding an electron, capable of existing under the action of an
external pressure uniform over its surface, is the same as the problem of
finding an equilibrium form of gravitating uniformly rotating liquid in the
limiting case when its angular velocity vanishes. Gravitational attractions,
it is true, are replaced by electrostatic repulsions, but since the angular
velocity vanishes, this only results in a change of sign of the pressure
gradient.
Thus when we have found a series of equilibrium forms of gravitating
liquid, including a limiting form for zero angular velocity, this limiting form
corresponds to a possible type of electron. For instance the sphere, which is
one limiting form of the series of MacLaurin's ellipsoids, corresponds to the
Lorentz electron. The other limiting form of the same series, the infinitely
long and infinitely thin circular cylinder, obviously does not correspond to
a possible type of electron.
Whether, amongst the series of equilibrium forms of gravitating rotating
liquid, it is possible to find one, which shall lead to a possible type of electron
other than that of Lorentz, or not, it is not easy to say. But it is possible to
assert that any type which may be found to satisfy all the necessary conditions as to stability and finiteness, will also lead to the Lorentz mass
formula, provided that the surface pressure be invariable. For this ensures
the invariability of the transformed electron; and according to Poincar*
every possible equilibrium form of gravitating rotating liquid possesses the
necessary symmetry. Hence we deduce the following proposition:
Every type of electron, which can exist under the action of a uniform and
invariable pressure applied to its bounding surface, whether it be at rest, or in
uniform rectilinear motion, will also lead to the Lorentz mass formula.
The converse proposition is also true.
* Vide Tisserand, Micanique Celeste, Vol. 2, p. 168.

E]

THE MECHANICAL EXPLANATION

OF THE ELECTRON

269

257. In order to gain some idea of the magnitudes of the pressures
involved, we shall consider briefly the case of the ordinary solid Lorentz
electron.
The corresponding transformed electron by § 240 is a sphere of invariable
radius a and uniform volume density ice = 3e/4wa3. Its electromagnetic mass
for zero speed is given by m = 4e2/5c2a.

From (403), § 253, and (407), § 254, we find
e
s
2a (3a2
2),
p=B

p

where

s2 =

e2 + n2 +2

=2

3e 2
87ra 6

...

. .. .. .. .. . .

(408),

y2 +

and B is a function of t alone, or constant.
The difference of pressure between the centre and the external surface
(s = a) is equal to

3e2
87ra4
With the values e/c = 16.10-

1875c4 cm
20487re2 Ke/
20 E. M. U.

4

and e/cm=1-77.10' E.M.U. we

find that the difference is about 1031 dynes per sq. cm., or 1025 atmospheres.
For Lorentz electrons of variable volume density, and for the Lorentz
electron with surface charge, the difference of pressure has a different value,
but is of the same order of magnitude.
258. Motion of the most general type. We shall now consider the
case of an electron whose electric centre is moving in any prescribed curve in
any prescribed manner, and shall determine the effect of the acceleration and
motion. Partly in order to obtain definite results, partly for the sake of
simplicity, we shall however limit our investigation to an electron of the
type considered in § 256, for which the Lorentz mass formula holds. We
are all the more justified in doing so, in so far as the Lorentz mass formula
agrees with the more recent measurements of the ratio e/m for cathode ray
and particles, and is besides the only formula which allows us to treat the
motion of electrons mathematically with a reasonable hope of obtaining
simple results, as we shall see below in App. F and G.

,8

We have already seen that the acceleration and relative motion terms in
(399), § 252, are small of the order - 1. Therefore they will produce small
changes in the hydrostatic pressure, which will react on the electron producing
small changes in its configuration, generally small changes in its volume
density as well as slight deformations of its bounding surface.
We shall denote the small increase of hydrostatic pressure by Sp, that of
the volume density by 6E, and that of the electrostatic potential by 6'.
The
last is the electrostatic potential of the volume distribution 6e, and of a

270

THE MECHANICAL

EXPLANATION

[APP.

OF THE ELECTRON

surface distribution representing the deformation of the bounding surface;
the volume and surface distributions must be such as to leave the total
charge of the electron unaltered.
We shall try to determine the conditions which must be satisfied in order
that a hydrostatic pressure 8p may be possible.
259. We notice at once that the small changes in the configuration of
the electron will only alter the small acceleration and relative motion terms
by small amounts of order 0, which are negligible. Thus in calculating
these small terms we may treat the electron as exactly of the type considered
in § 256; that is, we may assume the corresponding transformed electron
to be invariable, so that the coordinates (, q, ') and the potential 'P are
independent of the time and do not involve the velocity v. In order to use
these coordinates we must however refer the motion to rotating axes as
in Ch. XI, %§141 and 144, so that the axes of (x, y, z) are the tangent to
the path, the principal normal drawn towards the centre of curvature, and
the binormal. Then the angular velocity, w, of the axes about themselves
has the components (v/7, 0, v/p), where p and are the radii of curvature and
torsion respectively. In (399), § 252, on the other hand the velocities u and
d
and the total differentiation d are relative to axes fixed in direction.

7

u,

In this Appendix we shall denote a unit vector drawn in the direction of
the velocity v by v1 ; hence we have

v=

vv + [ov] ...................... (409),
a relation by means of which we can eliminate w.
By the usual formula of rigid dynamics we have

u = i + [w r] .............................

(410),

where r denotes the radius vector whose components are (x, y, z), or (ic, , )
by (401), § 253. Since ( , n, ) do not change owing to the assumed invariability of the transformed electron, while K/ -(1
- v2/c2), we find
.

v(vr)

r = -C-2

v (vr)

u= ----

v,

with similar equations for ir and u 1 .
(uv)=

(Cu -

v 1 + [wr]..............(411),

By means of (409) we find from (4112)
2v2)

c -K

(v 1r)- (r)

..................

(412).

260. The relative motion term in the first line of (399), § 252, becomes
by (402), § 253,
- ie

del ({u+ u1} D) KV p = 2

de ({u-+uv)V

,

MECHANICAL EXPLANATION
~THE

E]

E]

OF THE ELECTRON21

271

because P is independent of v and therefore not affected by the operator D.
Using (412), § 259, we may write this in the form
E(c22v2)f

vt

2C4,3

= E

+r

P

r

2C2k

+V(

[(Cl -2v 2 )v2{()

de]

+r)

del1f
})

VP'+ V (viAj}

-1{(+r)

.......... (413),
......
where we have used (403), § 253, and put
P
fr...........................(414).
del

rl

A=

A is of the nature of a vector potential due to current r1del in the transformed electron.
Since we have identically
V

(vir)

V

=v,,

(i)

=vVT

0

we can write (413) in the more convenient form
)(c

EV [
L

2f

r
(

'Iv'-+)
-(

P--

+)(v
(C2+

2

)
20

+r(P\-YP)

--

~2C

K
/C3

V~
2..](415).
20/c

Again, the first term in the second line of (399), § 252, may be treated in
the same way; by means of (403) and (404), § 253, we find that it becomes
d Dc(q
t

-EdP-P
dt 22/C

)

Lv2 4 C3
('2C

v1 +

2c

..

because PV and PV are to be treated as independent of the time t.

The small changes Sp, SE and SP
calculating the large terms in (399),

will have to be taken into account in

V 8p, and that of the first principal term

member is simply
1s - KSEVPI - isEV8P1.

(416) we find
V~p = - FCVP+
-E

)'1-

V[C4 k3
1

§ 252. The increase of the left-hand
on the right side

Hence taking account of the small terms (415) and

EVS'P

{v ) P

) +(v 1A)}

-2C2kC

{(*r) (P-'P) +(*A)

++ M

261. Effect of a linear distribution of volume density. Before
discussing the equation (417) just obtained for the pressure Sp, we shall
prove a lemma relating to linear distributions of volume density, which
enales us to transform theequationto a more convenient form.

272
THE

272

Lemma.

MECHANICAL

EXPLANATION OF THE ELECTRON

[AP

[APP.

If a volume distribution be given by

Se= e (br)
........................ (418),
where b is any vector independent of the coordinates, but possibly depending
on the time, then
8EV

+
+

eP

= E [(br) V1'P + V (bA)]
=

ev

[(br)

(1Y

b)

-'1)+

-e(

-Y)

.

(1)

8E

The volume density of the transformed electron, corresponding to
for
the actual electron, is K(SE by § 253; hence we find by means of (403), § 253,
and (414),

§ 260,
Sp

fic(Se dpd

dri

f

i

de,

-

j

(420.

el(bA).

Operating with V, multiplying by E, and adding the term
SeV T - e (br) V1Y,
we obtain (419).
262. The equation for the increase of pressure. In order to
simplify (417), § 260, we may use the lemma of § 261 to eliminate either all
the quantities in the second line of (417), or the first term in the third line.
We prefer the first method as it leads to simpler results. Let us write
8e= (1E +2E
(C 2

SNP= 81Y'+8

,
2

2

P
....... (421),

)'

We sawin 2(v58r)

2at

where*81 'P is the electrostatic potential due to the volume distribution 8i .
e saw in § 2.58 that the total additional charge due to acceleration and
relative motion vanishes ; that due to S1E also vanishes, because
is a
homogeneous linear function of the coordinates relative to the electric centre.
Hence the total charge due to 8S together with that due to the surface
6,
2
distribution, if any, also vanishes.

8SE/e

Substituting from (421) in (417),

§ 260, and

using the lemma of § 261, we

find
v 8p

= -

K(

2 EVP -

E8'

VVE6(-PY)
4 3

22C /c

v

{

E(P-P)M

2c2,c

-

+ee-

v.(2)

2'

2cK +2c2 c2
1

....................

so that d is independent of the coordinates. Adding
Vice {82 + (P1'-'I) (dr)}
to both sides of (422), and noticing that

V

dr)=d,

Ve

E

VT,

(423),

THE MECHANICAL EXPLANATION OF THE ELECTRON

E]

on account of (406),

V [Sp+Ke{

2

§ 254,

P+T'(P

273

we obtain finally

)(dr)}]-

[-ae

#

+K

62+c

We shall determine the conditions under which an equation of this form
is possible, and shall then find an expression for 8p and determine the further
conditions which are necessary to ensure that it vanishes at the surface of
the electron.
263. Let us eliminate 6p by taking the curl of (424), § 262, and multiply
the resulting equation scalarly by VP. The first and second terms on the
right both disappear, and we obtain
(vV.V)=0......................(425).
This equation must be true for all values of v, which is quite arbitrary.
We see from (403), § 253, that PT does not depend on v at all, nor can do
so, for a non-electromagnetic mass depending on the acceleration cannot be
explained mechanically. Hence (425) requires that

[V. V]=0
Thus
264.

t,

..................

(426).

like e, must be a function of P and t alone.

Let us subtract a term

from each side of (424),

§ 262,

bearing in mind that

'U

V

Ve= a

-

because e and a involve the coordinates only in so far as they enter into the
expression for P. Then we find

[8P +KE{8T+(P-P)dr)}-Me-

y

...................
(427).
....
This equation is only possible when each of the functions enclosed in
square brackets reduces to a function of
and t alone. Hence we have

SP=-'
T
/CE8 2

-ks 2 e+K

where

f

3 2P=-

-

KE

P

(P

-

P) (dr) +

ap

is an arbitrary fuinction of P'

)+

and

(4

a

(+r) +f

Ea

...

(

e...(2)
p

(428),

t alonne, and therefore doesnot

274

EXPLANATION

THE MECHANICAL

OF THE ELECTRON

[APP.

involve any terms such as (dr) or (+r). This fact shows that f cannot
represent an effect due to acceleration and relative motion, but results from
other causes, which are not under consideration. Moreover it is always
possible to make it vanish at the surface of the electron owing to the fact
that P is constant there. For these reasons we may leave it out of account
in what follows.
The volume density ee and potential 62P are not independent; since
I8,2 is the volume density of the transformed electron which, together with
some possible surface distribution, produces the electrostatic potential 6 2 ',
these two quantities are connected by Poisson's equation
+

r+

2+

4

e=0 ............... (430).

7r

From (429) and (430) we find, omitting f

+

+
=

82

2 + 47tc

-

(dr)±'r

47 i

e

(r)

which holds everywhere, both inside the electron, where e and
finite values, and outside it, where they vanish.

c

...

(431),

have given

When the value of 6 2D obtained from (431) is substituted in (428) we
obtain 6p. It generally differs from zero, even at the surface, so that the
uniformity and invariability, which were found in % 253-257 to be
characteristic of the surface pressure of our present type of electron when
the motion is a uniform rectilinear translation, no longer exist when the
motion is general.
Hence we conclude that for the type of electron in question, which gives
the Lorentz mass formula, a hydrostatic pressure exists even when there is
acceleration and relative motion, but it is no longer uniform and invariable
at the surface in general, that is to say, without further restriction of the
type of the electron.
265. Additional conditions for a uniform and invariable surface
pressure. Let us examine the potential equation (431), § 264, more closely.
The right-hand member involves two scalar products of the radius vector
r by constant vectors, namely (dr) and (fr); they occur in the first degree,
have this
and their coefficients are functions of P alone, because e and
property by (406), § 254, and (426), § 263.
We see from (423), § 262, that
d

v2 )

-

2 C4K4v1

+

1

2c2,c2

"THE MECHANICAL EXPLANATION

El

275

OF THE ELECTRON

Thus d has a component in the direction of the velocity except in the
particular case when the speed v is constant. Since both the velocity and
acceleration of the electron are perfectly arbitrary, d and v are two perfectly
arbitrary and independent vectors. Hence in (428) and (431), § 264, the
terms involving (dr) and (fr) must be treated separately. Moreover, (431)
cannot be satisfied for every value of d and v, unless its left-hand member
reduces to the sum of two terms involving (dr) and (+r) as factors respectively.
Therefore 2 P must be of the form

8

82. = A (dr)+ B (r) + ..................... (432),
where A, B and C are functions of the time and coordinates, but do not
involve either d or i. Then A (dr) must satisfy (431) when (+r) is put
equal to zero, B (ir) when (dr) is zero, and C when both are zero.
Beginning with the term A (dr), we find
2
A
a2A
a2A
aE AA
+dA
A

(a

+a2

+

+

A

-

(dr) +

2

a+

= - 47rr

E(

+

t

-

) (dr)

... (433).

This equation must be satisfied for every value of d, so that the second
term in the first line must reduce to the form F (dr), where F is some
function of the coordinates and time, but does not involve d. This occurs
when, and only when
BA aA aA aA

as =..................(434),

Sa

where s =

2

V(+ + n2 + '2) as before.
da + d,

Then we have

+d

a(dr) ............... (435),

and (433) gives after reduction

Os

+ss

-as+ 4rk A

- 4 rk

(

N9=

99.(436).

T;

266. The equation (436), § 265, only involves the quantities s, t and
hence
must be a function of s and t alone, and e and u have the same
property.
Thus we arrive at a necessary condition to be satisfied by every type of
electron for which the surface pressure is uniform and invariable when there
is acceleration and relative motion, namely, the volume density, non-electromagnetic mass density and electrostatic potential of the corresponding
transformed electron must be functions of the radius s only (they cannot
involve the time t because the transformed electron is invariable). The
surfaces of constant density of the transformed electron are concentric spheres,
and therefore those of the actual electron are concentric Heaviside ellipsoids
to a first approximation.

P

18-2

276

THE MECHANICAL

EXPLANATION

OF THE ELECTRON'

[APP.

All this may be expressed simply by saying that the transformed electron
possesses spherical symmetry, or that the actual electron is a generalized
Lorentz electron (of that shape, but not necessarily of uniform density).
This agrees with Poincar6's well known solution of the energy difficulty of
Abraham.
267. Treating the term in (432), § 265, which involves (+r), in the same
way as the first, we find
a2B- 4 B +
e
(M
B = 47,r
e
s+ 47rc
as2

s

as

aY

aE
_

e

)

......... (437).
(437)

This equation has the same first member as (436), § 265, but a different
second member; hence the particular integrals occurring in A and B are
different, but the complementary function is the same. Of course the
values of the two arbitrary constants involved in it will be different in the
two cases. It follows from the theory of differential equations that the
solution of either equation entails that of the other as a consequence.
The third term, C, of (432), § 265, is a function of s and t alone, and when
substituted in (428) and (429), § 264, merely gives rise to functions of the
form off, which has been already ignored for reasons mentioned in § 264.
268. Substituting the value of 821 given by (432), § 265, with C
omitted, in (430), § 264, and using (435) and (436), § 265, and also (437),
§ 267, we find
1

a2A

1,e

4 rK(

= A

s\2

+E(--

4

8a2B

A

as

(dr)+
( r+
)+

+

4

aBB

+ -4B((- r)
s

s2
+

(dr) +

B

MEa(

r...(
4

3

8)

which might also have been obtained from (429), § 264.
As we have already pointed out, 82E does not represent the whole effect
on the charge, because the potential 82 is generally due in part to 8,c, but
also in part to distributions on the surfaces of the electron, both internal and
external. In order to find these, we must calculate the part of 21 due
to 82e.

T

The quantities (dr) and ('r) are solid spherical harmonics for the transformed electron, and of the first degree. The theory of these functions gives
for the potential due to the part of 826 involving (dr), by (438),

=(dr) {A -Aa-*aAa' + 3s3Ab

E]

'THE MECHANICAL EXPLANATION

277

OF THE ELECTRON

where b and a are the internal and external radii of the transformed electron,
JA
Aa and Ab the corresponding values of A, and Aa' and Ab' those of -.
In
obtaining this expression it must be borne in mind that the density for the
transformed electron is !cse, because 62 e belongs to the actual electron. A
similar expression is obtained for (fr); adding the two expressions together
and subtracting the result from the expression (432), § 265, we find that the
part of the potential ~21 due to surface distributions is equal to
A+ +

.

a a

-3s3

(dr) + B +

aB

3s 3
B

(ir)

. .(439).

The form of this expression shows that the terms independent of s are
due to a distribution, on the outer surface of the transformed electron, of
surface density
3
4a

aA,') (dr) + (Ba +

{(Aa +

The terms involving ssurface density

aBe') (r)}.........(440).

3are due to a distribution, on the
4
-

inner surface, of

{Ab' (dr) + Bb' (r)}.....................(441).

These surface distributions are equivalent to normal displacements of the
two surfaces, obtained by dividing each surface density by the volume density
of the undeformed electron at the corresponding surface.
In addition to the volume density 62 e we have the volume density 81E
given by (421), § 262. These two volume densities together are equivalent
to a displacement of charge inside the electron.
269.

Substituting the value (432), § 265, of 82P in (428), § 264, we find
SP= - KE(A

+

-

T) (dr) +

-

eEB+

e -E

(r)

... (442).

Since the differential equations for A and B are of the second order, so
that A and B each involve two arbitrary constants, we can choose these four
constants so as to make each of the coefficients in (442) vanish for any two
values of s, for instance, for s = a and s = b. In that case 6p vanishes at each
surface of the electron.
The necessary and sufficient conditions for this to occur are
Aa= I'-Pa,
BaM
e

a
Ea

Ab=

B

b

M

b

e

.................. (443),

b

where the suffixes a and b denote values taken at the two surfaces respectively, just as before.

278

THE MECHANICAL EXPLANATION

OF THE ELECTRON

[APP.

Hence it is always possible to equilibrate the electromagnetic forces
inside the electron by means of uniform and invariable pressures applied to
its internal and external surfaces, whatever its path, velocity and acceleration
may be, provided only that it be of the generalized Lorentz type defined in
§ 266.
270. The surface pressures and electromagnetic mass. We have
found that for the generalized Lorentz electron the surface pressures are
unaltered by the motion. From (407), § 254, we see that the pressure
involves an arbitrary constant, so that one of the two surface pressures is
indeterminate; their difference however is determinate and invariable.
Consequently it is characteristic of the particular electron selected, like
the charge, e, and electromagnetic mass for zero speed, m. We shall denote
the excess of the external above the internal pressure by P.
In studying the dependence of m and P on the structure of the electron
we shall find it convenient to use the function E defined by the equation
47ricEs2ds ......................... (444).

E =

Thus E is the total charge inside the sphere of radius s in the transformed electron, or the corresponding Heaviside spheroid in the actual
electron. It satisfies the boundary conditions
Eb = 0,

= e ........................... (445).

Consequently it is given at both limits.
271. In order to calculate the electromagnetic mass we must know the
function 1Qdefined by (358), § 234, which, as we saw in § 253, is also given by

(D=
j;ceP =

;KfTdE.

For the generalized Lorentz electron we see from (350), § 229, and (358),
that, on account of the spherical symmetry of the transformed electron,
we have

§234,

m=4

41

3c2

2

2 fedE.

3c

2

J0

P is determined by the conditions
E

e

82'

-

a8

a

Hence we find by partial integration
m

2e2

3c2 a

2

3c2

a E 2ds

........................

(446).

E]

THE MECHANICAL

EXPLANATION

279

OF THE ELECTRON

Similarly we find from (407), § 254,
P=

2

1

e2

a4 +

a E ds447

.,.s(44).

Putting b = a in (446) and (447) we obtain the usual expressions for the
Lorentz electron with surface charge. Putting b = 0 and E = esS/a 3 we obtain
those for the solid Lorentz electron of uniform density.
It is not difficult to show that P is stationary for b = a, subject to the
conditions that e and m be given. It seems probable from the form of (447)
that this value of P is an absolute minimum; at any rate it is less than that
belonging to the uniform solid electron. Whether this be so, or not, we see
from (447) that P is essentially positive; that is, the external pressure is
always greater than the internal.
Hence we may suppose that the internal pressure is zero; but the external
pressure can never be less than a certain positive minimum value, which is
probably that belonging to the surface charge. From (446) and (447) we see
that this value is equal to
81c8m4
1287re6

"

With the values e/c = 1"6.10-20 E.M.U. and e/cm = 177

E.M.U., we

.107

find that the pressure in question is 7"2.103o dynes per sq. cm.
272. Example. The solid Lorentz electron. As an example of the
determination of the effects of acceleration under the influence of a uniform
and invariable surface pressure, we shall take the solid Lorentz electron of
uniform density, without any non-electromagnetic mass. With the notation
of § 257 we find
e

( a2 - s2)........................(448).

The equation (436), § 265, becomes
a2A

+- 4

A
=- 47r
+s2sas

It gives

A

=

es2
10a

3e
-.
a3

3

= O-

..
........... . . . . . .

. .. .

. .. .

(449).

A term of the form D/s3 cannot be present because it becomes infinite at
the centre.
The second function B is not required because M and

are zero.

Determining C from (443), § 269, for s = a, we find
0=-,

e
2'/

A=

2

e

.n

-(a-~~

..................... (450).

280

EXPLANATION

THE MECHANICAL

OF THE ELECTRON

[APP.

Substituting from (448) and (450) in (442), § 269, we find for the change
of pressure
3e2 2
Sp =

(dr)

5ra (a -2)

................

(451).

This vanishes at the surface and at the centre, and is greatest for
s = a//3.

In order to find the change of density we must use (421), . 262, and
(438), § 268.

Since b

vanishes, we find
e = e (dr).

Adding this to ,e and using (423),
1
2c22

we obtain

eVx

ev

e=

§ 262,

(v

1 r)

EV

2c2 C2

2c2

..............

(452).

The surface densities are given by (440) and (441), § 268.
Since by (450) A, +

aAa' and A 0' both vanish, there is no surface

distribution and no deformation of the surface.
Thus we see that the only effects produced by the acceleration in the
uniform solid Lorentz electron are the change of pressure given by (451) and
the redistribution of charge given by (452). The latter is equivalent to a
displacement of part of the charge, on the whole in the direction of motion,
or the opposite direction, according as the speed is increasing, or diminishing.
273. Conclusion. Premising that the terms electric density, nonelectromagnetic mass and potential in what follows refer to the transformed
electron, we may summarize the chief stages and conclusions of our argument
thus:
(1) In the case of uniform rectilinear motion the electromagnetic forces
can be balanced by a distribution of hydrostatic pressure when, and only
when, the electric density is a function of the potential and time (or speed)
alone (§ 254).
(2) In the last case the pressure is uniform and invariable at the surface
of the electron when, and only when, the potential is constant at the surface,
and the transformed electron is invariable (invariable here means independent
of the speed). These conditions are equivalent to the condition that the
electromagnetic mass of the electron shall obey the Lorentz mass formula
(§§ 255 and 256).
(3) When the electron is moving as a whole in any path according to
any law, a balancing distribution of hydrostatic pressure exists when, and
only when, the non-electromagnetic mass, if any, associated with each element
of charge depends only on the potential at the element and the time (§§ 263
and 264).

E]

THE MECHANICAL EXPLANATION

OF THE ELECTRON

281

(4) In the last case the pressure at the surface is uniform and invariable, and has the same value as for uniform rectilinear motion when, and
only when, the transformed electron is spherically symmetrical (§ 265 and
266).
(5) The only effects of the acceleration are to produce (a) a small
redistribution of the charge inside the electron, and (b) small deformations
of its surfaces (§ 268 and 269).
(6) In the type of electron to which we have been led under (4), the
external always exceeds the internal pressure, so that the internal pressure
may be zero, but the external pressure must have a finite positive value

(§ 271).
274. In this Appendix we have neglected the small terms due to squares
and products of the relative velocities and to radiation. The corresponding
terms in the expression for the electromagnetic force (334), § 219, are very
complicated, and the investigation of their effect would be troublesome, even
though the problem is very much simplified by the result that the transformed electron must be spherically symmetrical. But this result having
been once established, probably a simpler method of treating the whole
problem ab initio can be found, and it is hardly worth while to carry our
approximation to a higher stage.
275. So long as we neglect the effects of acceleration and relative
motion, our results are in complete agreement with the Postulate of
Relativity. The formula for the mass is the Lorentz mass formula, when
the pressure at the surface is uniform and invariable, a result which is to be
expected on the Relative Theory, for which the pressure is an invariant, that
is independent of the speed.
But when the effects of acceleration and relative motion are taken into
account, this agreement is absent. In § 272 we found that even in the case
of the ordinary solid Lorentz electron, moving under the influence of a
uniform and invariable surface pressure, such as the Postulate of Relativity
requires, there is a slight redistribution of the charge, which could hardly
fail to be detected by an observer moving with the electron and using the
coordinates and time appropriate to its speed at the moment. Thus the
electron would not appear to be invariable to this observer; he could detect
its acceleration in spite of the Postulate of Relativity.
Thus it appears that the Postulate of Relativity, when applied with
absolute strictness, excludes a mechanical explanation of the electron. But
we must bear in mind that the Postulate was only advanced in the first
instance in connection with uniform relative motions. Its extension to
motions involving acceleration goes far beyond what is required by experimental facts.

APPENDIX F
THE MECHANICS OF THE LORENTZ ELECTRON

276. WE have just seen, in Appendix E, that the Lorentz mass formula
is superior to any other in two respects, both of which are very important from
the theoretical point of view. In the first place, it is in agreement with the
Principle of Relativity, which we know affords the simplest explanation of
the absence of any effect of the earth's motion on optical and other electromagnetic phenomena.
Secondly, it is consistent with any possible mechanical explanation of the
electron which accords with the symmetry observed experimentally.
Moreover, it is in better agreement with recent measurements of the
mass of 8-particles than any other formula; but even if the experimental
evidence were not so much in its favour as it actually is, the theoretical
advantages would be decisive.
In the present Appendix we shall deduce the equations of motion of the
electron on the basis of the Lorentz mass formula, leaving out of account
the terms of higher orders; in other words, we shall confine ourselves to
quasi-stationary motions, to use Abraham's expression. This limitation is
necessary in order to secure manageable equations yielding definite results.
It is of little practical importance except in cases of discontinuous motion.
These are cases where the acceleration, or some differential coefficient of the
velocity of higher order, becomes infinite, so that the expressions for the
coordinates and velocity-components become discontinuous in form, but not
in value, at one or more instants of time. An example is that of an electron
starting its motion from rest.
The effect of such discontinuities of motion has been examined by
Schott* on the basis of the electron theory, and by Walkert on the several
assumptions that an electron behaves like a conducting and an insulating
sphere respectively. The general result of these investigations is that the
* Ann. d. Phys. [4], 25, p. 63, 1908.
f Phil. Trans. A, 210, p. 145, 1910.

APP. F]

THE MECHANICS OF THE LORENTZ ELECTRON

283

approximation used in this Appendix fails during an interval of time, which
is comparable with the time required by an electromagnetic wave to pass
across the electron and includes the instant at which the discontinuity
occurs. Further, the subsequent motion is affected to a certain extent, but
the relative error committed when we neglect this effect is only of the order
of the ratios which the radius of the electron bears to the radius of curvature
of its path, and to the distance within which its velocity would have been
acquired if its acceleration had been uniform.
277. On the basis of this approximation the equations of motion can be
expressed in Lagrange's form whatever mass formula is adopted. When the
external electromagnetic field is steady, an energy integral can be deduced
in the usual way. When it is symmetrical about an axis, we can deduce an
integral which expresses the Principle of Conservation of Areas for the axis
of symmetry, whether the field be steady or not. When the field is steady
as well as symmetrical about an axis, both integrals exist. To this extent
all the various mass formulae are on a par.
When the Lorentz mass formula is used, further progress is possible.
When the energy integral exists, it can be used to reduce the equations of
motion to a form in which they are little more complicated than those of
ordinary particle dynamics. For instance, they may be written in Hamilton's
form, so that the whole of Jacobi's theory and its subsequent developments
become applicable.
When there is symmetry about an axis, the angular coordinate belonging
to that axis may be ignored, and the problem reduced to that of motion in
two dimensions.
When the mass formulae of Abraham, or of Thomson, or of Walker, or
any of the more complicated ones are used, these reductions are practically
impossible, because the necessary eliminations cannot be effected.
The simplifications which can
mass formula are so great as to
problems of practical importance.
ably superior to all other formulae
to justify its use as a basis for the

be effected in this way with the Lorentz
make numerical results possible in many
The Lorentz mass formula is immeasurin this respect; this fact alone is sufficient
mechanics of the electron.

278. The equations of motion of the electron in the Newtonian
form. We shall use the symbols m and m~ to denote the mass of the
electron for zero velocity and for velocity v respectively, just as before.
Neglecting the radiation terms in (218), § 148, we get for the vector equation
of motion
d (m v)

dt=

=

e

e (d + [vh]/c).

MECHANICS OF THE LORENTZ ELECTRON[AP
~THE

284

284

[APP.

The Lorentz mass formula gives
cm

2 2
V(c - v ).

{

Hence we find

-

e d + [vhc

=

.454

This is the first Newtonian form of the vector equation of motion.
A second form can be obtained by expressing the external electric and
magnetic forces in terms of the scalar potential 0 and vector potential a
by means of (VII) and (VIII), § 3. We get

d+
Now

[vh]/c

cat + [V

[Va]]/c.

-

=

[v [Va]]

V

=

V (va) - (vV) a,

-

where V does not operate on v, that is to say, the coordinates and velocity
components are treated as independent variables, just as in ordinary
mechanics. Moreover, since fixed axes are presupposed in (VII) and (VIII),
aada
d
we have hve
+-(vV7)
hr
d denotes total differentiation. Substinte
we s +(vV a= d

a=dawhered
dt

dt

at

tuting in (454), we find
d

cmv

dtt,\/(v)

aa

ea

v2+

c2

C

=

{e4p

-

-

ecVa).........(455).
I

This is the second Newtonian form of the equation of motion.
279.

Lagrangian equations.

Let (x, y, z) be the coordinates of the

electron referred to fixed rectangular axes.

(t, ,

Then the components of v are

z ).

The potentials 0 and a are functions of x, y, z and t, but not of d,,i
they are not independent, but satisfy the equation
ao+

cat

;

div. a = 0.

This equation of condition is however of no particular moment for what
follows. Write
L
Then

we

2

find

with two similar

(456).

±m a

equations;

that is, the three

the components of the vector quantity
left-h..ad side oVf (455),

47

2

e
- v )+
'cmV ..................(va) .................

278.

\/(.2

quantities

-v 2) + e

Reembeing tat

-

C

,a

ax 'ay

and

-L

az

are

which occurs on the

Vdoesnot~-operate-on

A,
F]

THE MECHANICS OF THE LORENTZ ELECTRON

285

y and z, we see that the equations of motion, written in vector form in
(455), § 278, may also be written in the form

aL
dt-... ax .............................

d aL

(458),

with two similar equations.
We may regard the three equations of type (458) as Lagrange's equations
for the Lorentz electron, and L as the Lagrangian function. On this view
the components of momentum of the electron are given by (457), and the
two similar equations. The vector cmv/V(c2- v2) is the momentum of the
electron of the ordinary type, depending on its motion; the term ea/c
represents additional momentum due to the presence of the external
field.
280. That a Lagrangian function should exist for the system consisting
of the electron and the external field need not surprise us when we
remember that Schwarzschild* has already proved it to exist for any system
of moving charges. From its existence we may conclude, just as Maxwell
did for electric currents, that a mechanical explanation of the system of
electron and external field is possible, even though no such explanation has
yet been found. This agrees with the results of our last Appendix; but it is
to be borne in mind that those results go much further, in so far as they
indicate that the Lorentz electron is consistent with that particular type of
mechanical explanation, which ascribes the forces acting on the electron to
stresses in an elastic medium.
A Lagrangian function exists for other types of electron also, for example
for the Abraham electron; and similar conclusions as to the possibility of
mechanical explanations follow from the fact of its existence. But for these
types of electron any possible mechanical explanation is necessarily more
complicated than it is for the Lorentz electron.
The usual consequences follow from the existence of a Lagrangian
function. The equations of motion can be deduced by varying the integral
L dt, and therefore the Principle of Least Action holds.

We can introduce

Lo

generalized coordinates (ql, q2, q3) in place of rectangular coordinates, and in
this way transform our equations as we please. When a coordinate does not
occur explicitly in the transformed Lagrangian function, that is to say, when
it is a speed coordinate, the corresponding momentum is constant, and the
coordinate may be ignored by using a modified Lagrangian function.
* Enc. Math. v. 14, p. 160.

286
THE

286

MECHANICS OF THE LORENTZ ELECTRON[AP

[APP.

281. Field symmetrical about an axis. We shall consider this case
a little more fully as an illustration of the ignoration of coordinates.
Take the axis of symmetry as x-axis in a system of cylindrical coordinates
Let v' be the component velocity in the meridian plane, and let
ax denote the component of the vector potential a perpendicular to the
meridian plane. Then we have
(x, W,

X).

(va)=(v'a) + xza

v2=V2 + 2722,

(459).

..

The potentials 0 and a are independent of x; hence the Lagrangian
function L does not involve x explicitly, and x is a speed coordinate and may
be ignored.
The modified Lagrangian function L' is defined by the equation
L'=L-Now from (456), § 279, and (459) we get
aL

Cmp232

where

L--

ewac

/(c 2 - v2 )

a

m=k...............(460),

c

K is the cyclic momentum and is constant.
2
cm (c -

L'=--

V(C2

-

Hence

e (v'a)

'2))

v2)+

C

Equations (459) give in succession
C2m72&x2 -

cm2(C2

Then we find

m'

eax)2

(kea)2

- /2)

2

Write

(k

=

V/m2+

L'= cn'V C2-

2-v2 k
(

-

ea)}...................(461).

V'2)+ e (v'a)..........(462).
C

Comparing (462) with (456), § 279, we see that L' is of the same form
as L. It differs in so far as the constant mass factor m in L is replaced by
the variable mass factor m' in L'.
The similarity in form of the original and modified Lagrangian functions
is due entirely to the fact that the Lorentz mass is proportional to
and the corresponding first term in the Lagrangian function

(c2 - V2) -1

therefore

proportional to (c2 - v2) . Consequently the elimination of the velocity x is
si Nmle, and le
adsA o hereltmetioned.YAV
?'1V Y1T7A11!" Ithas
-4k
of oth eNas
A?
NA

F]
THE MECHANICS OF THE LORENTZ ELECTRON28

F]

287

formulae the elimination is very complicated, and usually impracticable;
for example, for the Bucherer mass formula, which makes the mass proportional to (C2 - v2) - 3, we have to solve a cubic equation, and in consequence

the result of the elimination becomes so complicated as to be useless in
practice. Again, for the mass formulae of Abraham, of Thomson and of
Walker the equation to be solved is transcendental, and in consequence the
elimination cannot be effected in finite terms at all. The superiority of the
Lorentz formula in this respect is decisive for all problems in which actual
results are aimed at. We shall find below that this is also true for the more
general Hamiltonian transformation.
282. Hamiltonian equations. Let us use generalized coordinates
(q1 , q2, q3), and let pi be the generalized momentum corresponding to qi as
usual. Then

Pi=a.. . . . . . . . . . . . . . . . .(463).
aL
agi

The Lagrangian equations may be written in the form

dpi aL
dt aqi

4

H = p1q1 +p2 2 +p3 3q

As usual write

L............(464),

where (ql, g2, q3) are to be eliminated by means of (463).
We find in the usual manner

dqi - aH

dpi

..

dt

so

.

aQ ' dt - -aq

..

.46)
..

..

.....

(6 )

In order to understand what form the function H takes in our problem,
let us consider the matter more closely. The coordinates (x, y, z) are given
as functions of q1, q2, q3 and t by the equations of transformation.
Differentiating these equations totally with respect to the time t we find

ax

,.ax . ax
. ax
q 1 + q2 g+q 3 aq3'

x- =t+
with two similar equations.
v2 = x2 +
wher

whr

2"z2

Hence we get
= b0

bo=

0

2bqll...

S(axN

Sat
bS ax
b11
a,

ll-I-+h

1 2 1 q 2 +-.

b 1 =Sax ax

2

at aq1 '

'

=Sax ax
8 b1 = aq
Saq
2

2

2

.\

~...(466).

288

THE MECHANICS OF THE LORENTZ ELECTRON

[APP.

The sign S denotes a sum of three terms similar to the one written down.
Again,
(va)

xtx + ' ay, + ictz

=

=

Saxax+

x+ .........

a

(467).

The values of v2 and (va) must be substituted in the expression (456),
we obtain
Differentiating it partially with respect to
S279, for

b +b13 q3) +
22

+

(b 1 + b11

-crn

_Cm (v2
=

+p

3 3

Hence

1 - b2q
(C2 -V2)

-b1

-b°

2

-b343)

+e

Sxa

find

b,3q)

(C2b° b q1b 2 2

=cm

(va) _e

C+C-xSat

Substituting in (464) and using (456), § 279, we
H

46)

aq,....

C

Similar expressions hold for p2 and p3.

+P24 q

Sax

e

v2)

,1(C2 -

plq 1 +~ 2 +

41

4,

L.

a

e

ax

and

q3 between (468) and (469). This is
It remains to eliminate ql, q2
most easily carried out in two stages. Firstly, eliminating the three
quantities 41//(C2 v2), ... between (468) and (469) we find
-

2

b°-C

1/(C2 -v 2)

+H-e4~+- 2eSa

b1

1/(C2

-

Cm

Cm

p

_

v2)

1

ax

_+_e

x aq1 '

2
Cm

cm

b)
2

b

a

x at

1

b11 ,bb

b
2
1

1

b2__eSaxax

1(C2

-

2
V)

c
~
q2
p 3 + e Sax

3
.(C2

-

ax

2me
Cm______

2

b12,

b22

b23

b13, b23,

a

2
Cm

CM

v)

b

,

b33

0.

aq 2 '

For the sake of brevity we shall write

e Sax

=ed

at

c2m

cm

Pi

e Saxx.....(7)
aq1. .

' --

cm

cm

Thus the last equation may be written
1\(C2

-

v 2)

cm

b1

b1
2

)
C2_v - P1,

A

1/(c2 -'v2)

_

2

)

22

1

b11, b12,

b13

b22,

b23

b1 ,
2

0....(7)

. (7)

MECHANICS OF
~THE

F]

F]

THE

LORENTZ ELECTRON28

289

of

p', p2, and p3
v2) in terms
Secondly, we must find the value of (0(
and substitute it in the last equation. For this purpose we use (470) to
reduce the first equation (466) to a linear equation;with' the previous
notation we obtain
btu

02

C2

V

Eliminating
(468), we find
b0

'1/V(c

02

-

2

-

(-2,)

v 2), ... between this equation

+P

,

1

)

V(02

bb1

V

(C.

and the three equations

_

_bb_

V2

b1

q

-v1
V+oV

1

2

2)

)+P 2 1V(c2
2,

2

b12,

b221,2

- P3 ,

b13,

b ,b
23

Z
2
ZZ

b_

33

Let B denote the determinant ± b11 bj ,3 that is the discriminant of the
quadratic part of v2, and let B1, Bx2, ... be its first minors. Then the last
1
equation gives
B+ B,1 P12 + BP 2 + 2B 12PP 2
C2J
2
2
2
-- V. - B(c2 - b0)+ B 1 b1 +B22b 2 +"2B, bb 2 +

+(2

With the same notation we find from (471)
B

C

B (c

P+Bl

+ B 1 b1
-

12

b) + B1 1b12 . 2B 2 bb 2 + .
V(62

-

(b1"P 2 + b 2 P[) +)"""
1

V2)

VtB((21 2 b©) -+B11b1 + 2B12bx bz + .I.}B
B

This equation shows that H is of the
f1 (p1 y

p2

BP
11

2

+ 2B 1 2P 1 P2 + ..

form

)P3) +v'f2 (

respectively

+

P

2

....(473).

pa)......(7)

linear and quadratic, not homogeneous,
dfare
fucino'h
oenaP,~
ar and
P2 with
which are functions
root must be taken, because
q, q
of q, an t.Thepositive sign of the
Prepresenting as it does the variable factor in
^2 v2 ) is esncnf.nItillypitie,

wherf

-

coefficients

290

290

MECHANICS OF THE LORENTZ ELECTRON[AP
~TE

[APP.

ordinary

283. The equation of Hamilton'and Jacobi. Just as in
dynamics, so here it is possible to make the solution of the problem of
motion depend on the determination of the complete integral of a partial
differential equation of the first order in the four variables q 1 , q 2, q3 and t.
Fr this purpose we need only make the usualsutitonn(4),
2:

a_
waw

aw
aq 1

at

__a

P

aq

When the equation is rationalized, we get

an

equation of the second

degree, but it must be borne in -mind that this also includes the case-where
the negative sign is taken in (474), § 282, and
therefore give rise to
solutions which do not belong to our problem.

will

284.

Equations of transformation free from the time.

particular

case

-

-

-=0; hence by (466),

-282,

In this

b=b =b2 =b3 =O.
1

Using these values in (470) and (473), 282, we find
l=e+y/{(c411c2+

(OpI
3
B12

2

c

--

eSac)

eSax q)C/P

)..:}...(47 5).

esax

The interpretation of this equation is easy
(469),
with b0, b1, ... all zero the latter

§282;

+

aea

a\/;
a

1

when we compare it with

equation

becomes

C3(C
i
"")

Bearing in mind that e is that part of the energy of the electron which
is due to the external electric field, and that by § 240 and 278, with our
present notation, c3rn/V(c2 -- v2) is the variable part of the
electron, due to its own field and to its motion, we see that H
part of the total

energy of the electron, due both

energy of the
is. the variable

to itself and to the external

field.
The

expression

(475) shows that, when

H

momenta, it will involve t explicitly unless the

is- expressed

potentials

(P

in

terms

words

a

is not

zero in

general even when

te

the

and a are both

independent of t, that is to say, unless the external field is steady.

aH.

of

In other

equa~ftio of transcformatio;n

MECHANICS OF THE LORENTZ ELECTRON29
~THE

F]

F]

291

285. Steady external field-the energy integral. When the external field is steady we can always choose our equations of transformation
so that

Then the Hamiltonian equations (465),

aH = 0.

H = constant.

By (476),

§ 284,

§ give as usual
282,

this integral gives
................................................

(477),

where It is the arbitrary constant value of the energy.
In this case the Hamilton-Jacobi equation becomes
)w_
a
e

B,,

B

(C,

2

x

B1IV

eSa

-_cW-

aa)+q B(cqe

x

a

+

q

2

= (h -

eo)2

---

in1.. .(478).

In particular for rectangular coordinates referred to fixed axes of (x, y, z)
we have

q1= x,

q2

=

,

B = B,, =B 23 B 3

q3 = z,

1, B123,= B,,

=

0.

Hence the equation becomes
/13W

Cx-

aW
eax)+yc~a

\

2

7

aWIV

eay)+yc

- ea3 )

(Ii - eo)' -

=

c 4 n 3'

... (479).

A complete integral of the Hamilton-Jacobi equation leads to a solution
of our problem just as in ordinary dynamics, but in many cases it is easier to
work with the equations of motion themselves. Thus the interest in the
Hamilton-Jacobi equation is largely theoretical.
When an energy integral of the Hamiltonian equations of motion exists,
it can be used a~s in ordinary dynamics to chanige the independent variable
and so reduce the order of the
from t to one of the coordinates, say
Hamiltonian system by one unit, that is from three to two, so that the system
after reduction consists of only four equations in place of six.

q1,

This reduction however requires us to solve the energy equation (475),
p3, q1) q2, q3, h) then the
§284, for the momentum p,; if we get p, =reduced system is

K(p,,

1373-

aq3_3 K

aK

1q, 1 p
l

1-

'-a3'

aI}2 __
cl

-

3K

aJ),,

aq2

aq1

_-_6-K

13.

On account of the somewhat complicated form of the function K, and
the particular form of the energy equation
proeed
follows.

as

(477),

it is more convenient to

292

292

MECHANICS OF THE LORENTZ ELECTRON
~TE

LAP

[arr.

286. Reduction of the equations of motion when an energy
integral exists. It is simplest to return to the second Newtonian form
of the vector equation of motion (455), 278, and change the independent
variable from t to T, where
c

dt_

dr

2

-/(C

_h-eo

2

- v )-

C.

by the energy integral.
When the problem has been solved, the coordinates of the electron are
known functions of t ; hence (480) enables us to find the relation between t
and 7 by a quadrature, and therefore we may also regard the coordinates as
functions of T, to be found by the solution of the transformed equations of
motion. When this has been accomplished we get t from the equation
TVU
c
)
fT (her
(81)
..

...

where the constant of integration has been chosen so that t and T-vanish
together.
Let us denote total differential coefficients with respect to T by dashes,
e.g. x',.... Then we have
VC=2(C2 -

72)

I'

___xv...2........................(482),

with two similar equations.
Also let w denote the resultant of
v
We get c/V/(c2

- V2)

/(c2

x',

y', z'; then

-

..................... (483).

)w

C

+ w2)/C, and therefore

f

c2+w

dr............

....

(484).

Multiplying (455), 278, by C/V,(C2 - v2), using (480) and remembering
that V does not operate on x, y, z, but only on the coordinates as contained
in

0 and a, we easily

find

d

7nw

dT

+ea}_
+c

{(h -eo)2
=

m

+

........

ce..(wa)).(

485),

where V now does not operate on x, y', z', the components of w.
This is the vector equation of motion for a charge e of constant mass m,
moving in a field of which the scalar potential is equal to - (h - e4)2/2C2rne
and the vector potential retains the value a. In other words, the motion of
the Lorentz electron in the given steady electromagnetic field is the same as
that of an electron of equal charge, but of constant mass equal to that of the
Lorentz electron for zero velocity, when the electric field is modified in the
way indicated. The change in the electric field only depends on the field
itself and not on the motion of the electron.

F]

F]

MECHANICS OF THE LORENTZ ELECTRON29
~THE

293

A similar transformation is theoretically possible for any of the other
mass formulae, but the necessary eliminations are impracticable.
Since our problem has been reduced to that of finding the motion of a
particle of constant mass in a certain field of force, all the results of ordinary
mechanics apply without modification. In particular, the Lagrangian fuuction is given by
e (wa)
(h -.
(4)
L

L=~,w

=

2 mw2

.

2c2nm

+

c

+

.

+..........(8)

When there is symmetry about Ox, the cyclic coordinate x is a speed
coordinate as in § 281, and the cyclic momentum is constant. We have
,

aL=

2x +

+X~k....

W2

(487),

.............

ew

x
X
where K is an arbitrary constant.
The modified Lagrangian function is given by
L=L

X'

-1

12LW

(hL 2c=L-n
e)J-I

2+

7255

'
cm 2 (h -e

ax
2
where w' is the velocity (referred to

2X

7)

2

c)-(cK-e

a)

2

(488),
...

of the electron in the meridian plane.

The Hamiltonian function in the general case is given by
H

(It - e
c...................(489).
.-..
W Ill,
(heo)

H ~=,nw
-_-1
2

Since H does not involve 7 explicitly, the Hamiltonian equations admit
the energy integral in the form II = constant, but in virtue of the definition
of Tr the constant is not arbitrary. By comparing (489) with (477), § 285,
and using (48:3), we find that the energy integral may be written in the form
i

When there is symmetry

(h 22c
Comparing (491) with (488), we
plane is that of a particle of constant
the force function is given by
-

1 c2

............... (490).

m

about Ox we get

H=1mw'2

U

_

mw-

2 . . ... .... .
(491).
- ewaX)
2c2 mvs 2
rn
see that the motion in the meridian
mass unity in a field of force of which

e4) 2 + (cK

2

2
(Ih - e4) - (cK

e

-

waX)

2
..

For instance, with cylindrical coordinates
a'

(x,

,,~_auV

...

....

w, x), we have

. ...

(492).

294

294

MECHANICS OF THE LORENTZ ELECTRON[APF
~THE
[APP.

F

Again, with polar coordinates (, , x), where the axis of symmetry is the
initial line from which 0 is measured, we have
7,,-9"=aU

?'ra

2r-'6Y+rV

.494).
(

-

Changing to 6 as variable in the usual way, and writingtfor l/r, and
for
(in the present ease there need be no confusion with the first
meaning of i), we get

It

7.20'

pa

=

dh-

)

r26'=h

aU

ec21,Ve'

I

d'dh

±id0+
2,

-A-

dua

These equations admit the energy integral; by
(492) we see that it can be written in the form

1.2
When

u

{(d

+) 202+ =>U
2

and h have been found

-

c2 .

........

(495).

Itt

comparing

(490) and

. . . . . . . . . . . . . . . . . .

as functions

.

.

(.496).

of 6, we get T from the

equation
de--(497),

and then t is found from (480), § 286.

it

It is worth noting that when there is no magnetic field, and the electric
force is a function of the radius only, U is a fnnction of
only, and the
problem reduces to that of central forces. For in this case the spherical
symmetry of the electric field allows us to choose the axis of x so that the
initial motion takes place in the meridian plane, and consequently the cyclic
momentum is zero. This reduces the expression (492) for U to its first term.
This case includes the important problems of an electron moving in the field
due to a fixed point charge, and in that due to a spherically symmetrical
distribution of charge, for instance the positive sphere of uniform density in
J. J. Thomson's model atom.

APPENDIX G
PROBLEMS ILLUSTRATIVE OF THE MOTION OF THE
LORENTZ ELECTRON

287. WE shall now consider some problems of motion of the Lorentz
electron, selected partly on account of their intrinsic physical interest,
partly because of their suitability as illustrations of the methods developed
in Appendix F. At the same time they will serve to show that the
mechanics of the Lorentz electron is hardly more complicated than that of
the bodies of invariable mass postulated in ordinary mechanics, and at any
rate is sufficiently simple to allow of our obtaining definite numerical results.
As we have already pointed out, this fact distinguishes the Lorentz mass
formula from all others which have been proposed hitherto, and makes it the
only one of any use for practical purposes.
We have already considered the case of a Lorentz electron which is
projected in a uniform electrostatic field (Ch. XI, % 151-154). Its motion
in a uniform magnetostatic field takes place with uniform velocity, and
therefore differs in no wise from that of a charged body with constant mass.
The path is a helix of constant pitch, with its axis parallel to the lines of
force, in accordance with the results of ordinary mechanics. We need not
consider this problem here, and shall confine our discussion to more general
cases of motion.
Our first three problems concern the motion of a Lorentz electron in a
steady and uniform electromagnetic field. In the first problem the electric
and magnetic forces are parallel; it involves the theory of Kaufmann's
experiments on the specific charge of i-particles as a special case. The
general case suggests a new form of the experiment, which offers great
advantages, provided only that fairly intense electrostatic and magnetostatic
fields can be secured throughout a space of moderate extent.
In the second problem the electric
to each other; the particular case in
electrostatic units, is numerically less
magnetic units, involves the theory of
charge of ,8-particles.

and magnetic forces are perpendicular
which the electric force, measured in
than the magnetic force, measured in
Bucherer's experiments on the specific

296

PROBLEMS

ILLUSTRATIVE OF THE

[APP.

The third problem is the most general, in so far as the electric and
magnetic forces are inclined at any angle.
The fourth problem treats of the motion of a Lorentz electron in a steady
electromagnetic field possessing an axis of symmetry. It is important
because it involves the theory of the Zeeman effect for a single electron for
any velocity less than that of light. The particular case, where the electric
force is central and proportional to the radius, is worked out in detail.
288. Problem 1. An electron moves in a steady and uniform
electromagnetic field, with electric and magnetic forces parallel to
each other. Required to find the motion. The simplest way of
attacking this problem is to notice that the field is symmetrical about every
line of force. Hence any convenient line of force can be taken as the axis of
symmetry in a system of cylindrical coordinates of (x, r,
x). Since the field
is steady, an energy integral exists, and since it is symmetrical about Ox, the
total angular momentum about Ox is constant; hence we can use the
simplification represented in equations (492) and (493), § 286.
For the electric potential we may write
.= -d.x .............................. (498),
where d is the electric force, supposed to act in the positive direction of the
x-axis and to be measured in electrostatic units.
By a proper choice of the yz plane we can reduce the energy constant h
to zero, and as before write the energy integral in the form
c

V(c2 -v2

-

ed.x

2

........................ (499).

In the present problem the vector potential a reduces to the component
at perpendicular to the meridian, and it is easily seen that in this, and
similar cases where the lines of magnetic force lie in the meridian planes,
ax is equal to */,r, where f is the stream function. Hence we have

ax=

h

.............................. (500),

where h is the intensity of the magnetic field. It is reckoned positive, or
negative, according as the magnetic force is in the same direction as the
electric force, or the opposite, and is measured magnetically.
We now choose our axis of x so as to pass through the point of projection
of the electron. Thus w vanishes initially, and therefore the cyclic constant
Sis identically zero. Hence equations (492) and (493), § 286, give
.- e2d2x2 - e2h2w 2
U ,, U
,
0
2c2m . 8c2m 2
='
ax e' ra

4

Therefore

e2d 2
X"=2
2

e2h2

=

w .................. (501).

G]

MOTION OF THE LORENTZ

297

ELECTRON

Also (487), § 286, gives, by (500),
eh
S=- 2cm
Lastly, (481), § 286, gives

I'

.xdT

ed

t =f ed " d
o

................... (502).

..

......................... (503).

c2m

289. Equations (501)-(503) determine the motion when the initial
conditions are given. In order to simplify the results as much as possible
we shall suppose the origins of space and time chosen so that the direction
of projection is perpendicular to Ox, and is along the initial line from which
X is measured. Hence when r = 0 we have x = 0, w = 0, x'= 0. Therefore
we get from (501) and (502), § 288,
x=

ed.7
,
cm

ehr
......... (504)
2cm

.ehr
= 2b sin=

a coshe

2cm'

The arbitrary constants a and b remain to be determined. Let v0 be the
initial speed, which is essentially positive. From (499), § 288, we find
c3m
a = ed (c2 - vo
Also ro = v 0; hence we obtain from (480),
wo'=

:o (dt

therefore by (504)

0

or

b=

§ 286,

o ' =Co//(c
0

(c-

eh

c(c2

.................. (505).

....

2

-vo2

.................

(506).

V02)

-

Lastly, (503), § 288, gives
Sed.r
ct = a sinh
.....................
cm

(507).

Eliminating r between (504) and (507) we get
=acosh(2/X,
ct = - a sinh

(h

#=-2bsinXl

............... (508).

x)

The first and third give

(a2 + c-t2) ........................... (509),
where the upper, or lower, sign must be taken according as a, that is
according as e, is positive, or negative.
x =+

The energy equation (499), § 288, gives

c2 - v

a'
2

(c2

v02)

....................... (510).

The equations (508)-(510) determine the path and the mode of its
description completely.

298

PROBLEMS ILLUSTRATIVE

OF THE

[APP.

290. We notice first that a is positive, or negative, according as e is
positive, or negative; thus a positive electron moves entirely on the positive
side of the x-axis, a negative one entirely on the negative side. The relation
between x and t, namely (509), is precisely the same as in the problem at the
end of Ch. XI.
Secondly, b is positive, or negative, and X has the opposite sign to
t, or the same sign, according as eh is positive, or negative.

and

The second equation (508) represents a circle, of radius + b, touching
B
the initial line Oy at the origin. Fig. 43 represents it
viewed from the positive side of the axis of x, for the
case where eh is negative, e.g. for a negative electron
c
when the electric and magnetic forces are in the same
direction. In this case x, starting from zero, increases
as t increases, and the circle is described in the positive
x
y
o
direction, right-handedly with respect to the magnetic
Fig. 43.
force. If the electron were positive, but the magnetic
force in the opposite direction to the electric force, eh would still be negative,
and the direction of description of the circle the same, but it would be lefthandedly with respect to the magnetic force.
It is important to notice that the sign of the product ed alone determines
on which side of the yz plane the electron moves, while the sign of the
product eh alone determines the direction of description of the path around
the lines of force.
Let s denote the arc OP of the circle in Fig. 43; then s = 2bx .
2dx/h = sd/bh = cs/voa by (505) and (506), § 289.

Hence

Thus the first equation (508), § 289, may be written in the form
x=a

cosh

c}

vo

=

2
ed V/(c2 - vo )

cosh

e)
c

..........

c2mvo

-

(511).

)s

Comparing this with (234), § 152, we see that the path of the electron
lies on the right circular cylinder which stands on the circle r = - 2b sinx
as base, and further that if it be supposed unwrapped from the cylinder it
developes precisely into the menoclinoid of the last problem of Ch. XI. In
other words, given an electric field, the electron describes a menoclinoid
determined by the value of the velocity at the vertex. If the electric field
and the velocity at the vertex be kept the same, while a parallel magnetic
field is superposed, the menoclinoid is bent round a right circular cylinder,
which has the axis of the menoclinoid for a generator, and whose radius is
numerically equal to vod/ch times the first parameter a of the menoclinoid.

G]

MOTION OF THE LORENTZ

299

ELECTRON

Each half of the menoclinoid gives rise to a spiral curve. The two
spirals are continuous at the vertex, and are images of each other in the
diametral plane through the vertex. They cut each other in a series of
nodes, given by X = nw7/2, where n is an integer. The nodes for which n is
even lie on Ox; those for which it is odd lie on the opposite generator of the
right circular cylinder. The positions of these nodes are given by
x,= a cosh (nrd/h) ........................ (512).
The distances x, ultimately form a geometrical progression whose ratio
is 'rd/h. Hence the pitch of the spirals becomes infinitely great ultimately.
In fact we see from (508), § 289, that the rate of revolution of the
electron around the cylinder is equal to ch/2dx, that is ultimately
zero, while the velocity parallel to Ox is ultimately equal to the
.velocity of light.
As a numerical example let us consider the numerical
example of the problem of Ch. XI, with a magnetic field of

hd

100 gauss superposed.

As before we have e/cm= - 177.107,

d = 50 E.S.U. = 15,000 volt/cm.,

and vo/c = '508, giving a = - 39"4 cm. Equation (506), § 289,
gives b = 10 cm., hb/d = 20 cm. Fig. 44 gives the projection of
the path on the xy plane, on a scale of 1: 80, the same as that
of Fig. 41, § 153.
The point A is the vertex, B, C are the first two nodes.
The thick line refers to those portions of the path, which are
in front of the right circular cylinder, of which the traces are
shown by fine lines, the broken line refers to the parts of the
path on the back of the cylinder. The arrows show the direction
of motion.
The figure is drawn for the case where h is positive, so that
a and b are negative, while x is positive.
291. The circumstances of our present problem are precisely
those of Kaufmann's well known experiments on the specific
charge of f-rays; in fact the part of the path described by the
/f-particles in passing through his condenser is a small arc PAP'
of the curve of Fig. 44, § 290. Consequently his electric deflection especially was very small, and measurable with difficulty.
c
If however we could realise a considerable portion of the curve,
such as the arc AP'B between the vertex A and the first node
Fig. 44.
B, we should have a considerable deflection. Let the coordinates
of B, referred to the vertex A as origin, be ( , q, ); they are given by
putting x equal to ±+n/2 in (508), § 289, and noticing that
=x-a,

=y=O,

- -I-.

300

PROBLEMS ILLUSTRATIVE

hence we get

= 2a sinh2 ('rd/2h), 4 = 0,

[APP.

OF THE
'=

2b...............(513).

Moreover, if 9 denote the inclination of the path at B to the
get, by (511), § 290,
ds

dx

"plane, we

vo

.(514).

c sinh (d/h).............

14).

Eliminating vo between (513) and (505) and (506), § 289, we find
2

ee

2

e2h

_d2

4

2

2

4

4c4m sinh (w d/2h) 4c m 2
Thus the locus of the first node B is a hyperbola in the L plane, with
its real axis along the line of force at the vertex A. Its real semiaxis is
equal to 2c2m sinh" ('rd/2h)/ed, its transverse semiaxis to 2c2m/eh, and its
eccentricity to

1 + 2 cosech 4

.d

It is important to notice that the transverse semiaxis can only be
reduced to a reasonably small value by using a powerful magnetic field. The
magnetomotive force, or fall of magnetic potential, between A and B is
constant and equal to 2c2m/e.
For a negative electron 2c2m/e is numerically equal to 3430 gauss cm.;
a magnetic force equal to 100 gauss would give a transverse semiaxis 34"3 cm.
On the other hand the real semiaxis can be reduced as much as we please
by making d small enough. The electromotive force, or fall of electric
potential, between A and B is equal to 2c2m sinh 2 (nd/2h)/e. For a negative
electron this is 3430 sinh2 (rd/2h) E.S.U., or 1"03.106 sinh 2 (wd/2h) volts.
With d = jh we should require 777,000 volts. In this case (514) gives
tan 0 = "43vo/c, so that a particle for which v0 = jc, would make an angle of
about 11° with the &plane. By increasing h to the utmost and making d/h
smaller, say about one fifth, we could diminish the transverse axis, and the
difference of electric potential, while the real semiaxis could be reduced to
a reasonable value, and the angle 0 increased so that the rays should not
strike the E plane too obliquely. In this way it should be feasible to realize
a considerable arc of the hyperbola (515) on a photographic plate placed in
the
plane, without getting a trace too wide to measure accurately,
were used.
provided a sufficiently fine cylindrical pencil of

$-rays

By measuring the trace on the plate and testing how far it agreed with
the hyperbola (515) of eccentricity

1

+d

cosech 4 -),

we could test the

accuracy of the Lorentz mass formula; this test would only require us to
know the ratio d/h.
Further, by calculating the real semiaxis of the hyperbola and measuring
d and h, or by calculating the conjugate semiaxis and measuring h, we could
determine the specific charge e/cm.

G]
MOTION OF THE LORENTZ ELECTRON30

G]

301

If the Lorentz mass formula be assumed to be true, so that the curve is
assumed to be a hyperbola, the last method only requires an accurate
determination of h, not of d.
292. Problem 2. An electron moves in a steady and uniform
electromagnetic field with electric and magnetic forces at right
angles to one another. Required to find the motion. We shall take
Ox in the direction of the electric force d, Qy in that of the magnetic force h,
and Oz in that of the Poynting vector [dh]. By a proper choice of the
constant involved in the electric potential we may take
.................

c=-d.x..

The energy integral becomes
c
ed(xk)57
2
V(c- - v )
c2m
where k is an arbitrary constant to be determined later.
gives for the time
tfo

fed

(x

+

k

dT~

c~~kmd

(516).

Then (481),§286,

................... (518).

Again, we may take for the vector potential a
ax=hz, ay=az=0
...............
(519),
= 0 and satisfy all other necessary
h,
for these values give h ,y= li =
conditions.

T

Using the energy integral for the reduction of the equations of motion
to as independent variable, we find for the Lagrangian function, by (486),
§ 286,
ehzx' e 2d2 (xv2 + lk)2
2o m
L
+
L- ~m(x'2+y'2+z'2)+c C,

=

Then the equations of motion become
eh
eh
e2d2 (xk
c m2
cm
=0

.......

(...520).

cmr
The last equation gives
z

c(x-x°) ................... (521),
li

4°

where the zero suffix denotes initial values. We can express z 0 ' in terms of
the initial velocity by means of (482), § 286, which gives

with similar equations for

x~'

= Vo -v0 2 ).....................(522),
and ym,

~PROBLEMS
ILLUSTRATIVE

302

302

OF THlE[AP

APP.

Substituting from (521) in the first of (520) we get
//+e2 (h2 - d2)
L'

e2d21+e'l2xo

x=

+

c~m

c2m2

-

el

i

-z

......

.

.
(2

)

cm

Cm

The constants ic and x0 are not independent, but are connected by the
energy equation (517), which gives
ed(k+x)

__2

V(c 2 - V02 )

cn

Subject them to the further condition
e2d'k + e2h2x0 eli-ehz
'eh
cm2
c2m2
cm
m V(C2-v02)'
which makes the right-hand member of (523) vanish. Thus we get
2
c2m (cd - z h)
mh (iod - cl)
k
ed (d2 - 2) \/(C' 2 - v2)'
e'(d 2 - h2)V(c' - v 02)
.(524)
Then (521) and (523) give
2
//e
(h2 -d 2 )
.....(525
-ec

_=0..........

c-m22

eh
cmx
By means of (482),

cd (z 0d - cl)
(d2 v'(c 2 - v 2).............(526).
L2)

§ 286, we get from the second of (520)
Y = Yo/T =

T .. .... .... ... .

co2

.. .

(527),

where we have taken o = 0, which implies no limitation of generality.
Lastly, by means of (520) we get from (518)
ctcm

=ed x

,

+li

c2mx 0
V(c 2 - v

dzed

2

)

li
-d

..........

(a28).

Before proceeding any further we must distinguish two cases according
as d h~l.
293.

Case I.

d < li.

This case is of importance for the theory of the experiments of Bucherer
and Wolz on the measurement of the specific charge of /3-rays.
Let us write

6= e

a

e

V(li

2

cm

V(li 2
-

d2)7T
c-m...........(529).
d2)

V'(c 2 -

2

v0 )

4

is an angle, and a is a time ; a is positive, or negative, according as the
charge e of the electron is positive, or negative.
Moreover, let us choose the initial values z0 and to to be zero, which
i mpies no limitatFiono n erliy

OF THE LORENTZ ELECTRON30
~MOTION

G]

G]

We get from (524) and (527),

§ 292,

a(ih-cd)
0

2
V (h

-d

2

=0,z
y0

)

()

=

303

=0,t

0 =0

(d)=a;,(d)= a.

,

o=
0

Hence we get from (525)-(528), §292, and (529)
S = a {o h - ed c
2
)co
-_d
x= a

y=a' 0 c
ah
V=(h2 -d 2 )

ct =

+
-s

sin

h-cd.d(ch0
h2 d2)sin f+x

0

(1

-cosc)+h

ad
z 0h
h2 - d2) t/(h2 - - cd sin 4 + x 0 (1 - cos
d)

zod)
2

d

2

h(ch-zd)
2

) +d

h

-

d)

.....
........... (530).
These equations show that the origins of space and time can always be
In this
- cd = 0.
special case the choice of d0 = 0 would limit the generality of the problem;
hence it is best to consider it separately.
chosen so that do = 0, except in the special case when i0h

294. Special case. z =cd/h. This is the case of the "compensated
rays " in Bucherer's experiment. We get
x = anY sin 4
0
t =a

h2-d2)(1- cos)+cP}
When the electron is projected at right angles to the electric force, so that
x= 0, its path is the straight line
y = z4h/cd........................(532),
which is described with uniform velocity v0. Since io = cd/h, while v0 must
be less than c, yo must be less than c V/(h 2 - d2 in numerical. value. Hence
0
all the compensated rays lie within the sector which is limited by the lines

)/h

Y

=

±z

Vl(h 2

-

d2)/d.

The motion is such that z is positive, that is in the positive direction of the
Poynting vector.
When 'io is not zero, the path is a tortuous curve, but for large values
of 0 it approximates to the straight line (532), and z is positive. Hence the
motion is in the positive direction of the Poynting vector on the whole.

304
PROBLEMS

304

ILLUSTRATIVE OF THE[AP

[APP.

The projection of the path on the plane of xy, the plane of the electric
and magnetic forces, is a sine curve, whose base is along Qy, parallel to
magnetic force, amplitude equal to ax0, and wave-length to 2'nay.
The projection on the plane of yz, perpendicular to the electric force, is a
sheared cosine curve. Its base is the line z =
V(h2 - d2), parallel to
the magnetic force, its amplitude is equal to a6oh/V(h 2 - d2), and wave-length
to 2ra&. It is sheared parallel to Oz through the angle tan' (cd/hy?).
0
The projection on the plane of xz, perpendicular to the magnetic force, is
a curve of the nature of a trochoid. It is derived from a trochoid, whose
base is parallel to Oz, that is to the Poynting vector, by reducing all the
2
ordinates in the ratio h:V(h - d2). The radius of the rolling circle of the
trochoid is acd/h, and the distance of the tracing point from its centre
aoh/V(h2 - d2). The trochoid is curtate, or prolate, according as 'i is
0
2
greater,. or less, than cd V(h2 - d2)/h , and the projection has loops, or no
loops, accordingly. The path itself however has no loops, because there are
none in its other two projections.
As a numerical example, let us suppose that h = 2d, so that
Further let d0 = c, yo= Jc. Then

the
Qy,

Jacsin4o,

y-

1

ac,

z=Jac

The resultant initial velocity is

v,

=

0+

c.

.

1cos4)}

The third

rojection has

loops.

[dh lz

[dh

d

h

Fig. 45.

45.

The scale of the diagrams is
The three projections are shown in Fig.
determined by the value of the length z ac; by (529), § 293, we have

_cam

Jac= 2e Vh

2

-

2

d

c2m
) 'c

2

-

2

v0

) =eciV'3

For a negative electron e/cwm = -1l'77 .10' E.S.U. If d be 98 E.S.Uv.,
i.e. 29,400 volt/cm., we find that Jac is equal to - 10 cm.
As a second example, suppose that da = Ic, all the other quantities
retaining their former values. We get

X=4c

in o,

y =4-acco, z = 2ac{P+4V3p1 -cos

4)}.

G]
MOTION OF THE LORENTZ ELECTRON30

G]

305

The resulting initial velocity is v0 = c. The third projection has no
loops. The three projections are shown in Fig. 46.
In all the figures the path is indicated by a thick line, and the direction
of motion by arrows. They are all drawn for a positive electron; for a
negative one we need only suppose the directions of the vectors d and h to
be reversed.

[dbz

[dhz

Ii

d

Fig. 46.

d/h. By shifting the origins of space and
295. General case. ,
time we can without any loss of generality ensure that 'h =0. Thus we find

from (530), § 293,
Zoi,-

x~a
x(h 2

-

d
cos4~
d2 )

d (ch - od){f
Z(t

z=

2-d2

ct=ah(ch-zd)
0?
into 0-

h (ioh-od).sin.(533).
ch

d

-

odsi

{1+2chsin q}

7r/2 changing the origins we can write
and shifting
d

h-h2By

these

equations in the same form as (531), § 294, s0 that the three projections
are the same as in the special case, but the constants are different. The
constant x0 is now replaced by (4h - cd)/V/(h 2 - d2), and the constant c, which
multiplies 4 in the bracket in ct, and do~/h in the bracket in z, is replaced by
h (ch - zod)/(h 2 - d2).

46:

of the first type
Hence the path is of the types shown in Figs. 45 and
loop, or of the second without loop, according as h (i0 h - cd) is greater,
with
or less, than d (oh - zod) numerically. Now we have

Thus the first type occurs when zo is negative, and when it is positive and
2
grreater than cdh /(d7 +h 2 )
I,20
S.

~PROBLEMS OF
ILLUTSTRATIVE

306

306

296.

Case II.

THE[AP

[APP.

d > h.
work and may be

This case has not yet occurred in experimental
dismissed briefly. Write
e /(d2 - h2 )
7

C............(534).

12

h2)V(C
2

/(d2 -

e

2

2

-v

)

In the present case we have the same expressions for x0
except that /(h 2 - d2) is replaced by V/(d 2 - h2). Hence

a(cd2 -. °h)
S= h2 ) ' Yo 0(d
dx

I

o

~

(d)=ay,
dy

~o

=0,t

dz

as before,

...

0 = 0,
d

(d)

=a;

0,

(dd)=a.

Since z° I < c and d > h, x0 cannot vanish. Hence we can always shift
the origins of space and time so as to make x0 vanish. Then we get
xc=a
Y

=

cd -

co

(d2
0 b

h2)

cosh ~

)4

z=ah (cd -i 0 h)
S

d2

2

-

( 0 d -ch)
h+d
i, h(cd - 0 h)(

+

sinh

,
ct=ad(cd -h)
d2

n

-h

h2

..

... 53)

h(zd-h)
d

These equations can be found from (533),
into -- ta, and V.h 2 - d2) into t Vd
h2).

(d-z

§ 295, by

changing 4) into

to,

a

When 0 is very large, whether positive, or negative, we get z = cth/d, so
that the motion is in the positive direction of the Poynting vector on the
whole, as in case I.
The projection of the path on the plane of xy, of the electric and
magnetic forces, is a menoclinoid, whose axis is Ox, parallel to the electric
force, and whose parameters are a (cd - z 0 h)/V/(d 2 -- h2) and ay0, the former
being the greater of the two. This curve has already occurred in connection
with problem 1, § 290.
The projection on the yz plane, perpendicular to the electric force, is got
from a trepsiclinoid °, of axis Oz and parameters ah (cd - ioh)/(d2 - h2) and
by shearing it parallel to Oz through an angle

ago',

tn'd(ez
tan (d
1

*

d-ch)
02
2
-

" )"

Loria, lc. cit. p. 580.

G]
MOTION OF THlE LORENTZ ELECTRON30

G]

307

The projection on the xz plane, perpendicular to the magnetic force, is
similar to the hyperbolic cycloid of Laisant *. It can be generated by a
point which moves along one or other branch of the hyperbola
h2

)

Z

-

z2-a2(d

h2

-

x2-d2

d2-2

with uniform areal velocity about the centre, while the hyperbola moves
parallel to Oz with uniform velocity. When d (zod - ch) is negative and
numerically greater than h (cd - I h), i changes sign when

d (ch -izd)

P =cosh-l~ -ih)
and the projection has a loop.

This occurs when z is negative.

Z [dhj

d xh

[dh] z

h

d

y

A

'

Fig. 47.

N N'
h

_

_

_d

N

N

Y

AX

Fig. 48.

As numerical examples we shall take the cases where d = 2h, Jo = jc,
and i0 =jc and - c respectively. The projections of the path are shown in
Figs. 47 and

48;

they are given by

5ac
x =4
and

x

=2

5ac
y =jIaccf,

accosh

4
*

z= 12 (sinh0 +

y =j1ac4,

8cosh 4

z =y5(sinh.P-k)

TLoria,

c. cpit. p. 476.

),

~PROBLEMS
ILLUSTRATIVE

308

308

OF THE[AP

[APP.

As before, the path is shown by a thick line, and the direction of motion
by arrows. The charge is negative.
In these figures A is the vertex, and QA is equal to a, which is negative,
because the charge is negative.
In Fig. 48 N is the node in the xz projection. It does not occur in the
path, and is shown undoubled at NN' in the xy projection.

297. Problem 3. An electron moves in a steady and uniform
electromagnetic field. Required to find the motion. This is the
general problem, of which those considered so far are particular cases. As
before let d and h be the electric and magnetic forces, measured in
electrostatic and magnetic units respectively, and let 9 be the angle between
them.
We shall take the plane of d and h as plane of xy, and the positive
direction of the Poynting vector [dl] as Oz at some arbitrary origin 0. Let
Ox make an angle a with d, measured in the positive direction, that is
towards h.
Then we may write
..............

c =-d(xcosa- ysina)

(536).

The equation of energy becomes
c
ed
(xcosa - ysina+).........(537),
whr(C2is
constant.
an

arbitrary

Again, we may write for the vector potential a
ax =1hsin (9 -a) z,

a y= -h cos (9 -a) z,

a z=O0...(538).

Hence the Lagrangian function becomes
2
L~im(x+y'2+z' )+e

{ x'sin(9-a)-y'cos(O-a)}z
2 2
+e d

(xcos a

-y

2

2c n

+

sin a ±k)

The equations of motion are
eh sin (

,,

- a)

,

e2d2 cos a(x cos a

CM

Y

-ehcos(9-a)

/

=-e2d2sina(xcosa-y sin a+k
2259)

-.

z"= eh {x' sin

ysin a kI)

2 2
c m

(0 -a) -y' cos (0-

a)}

2

G]
MOTION OF THE LORENTZ ELECTRON30

G]

309

The last equation gives

where

z' = h{xsin (8 - a))- y cos (9-cL)}+-hA
cm
cm
wher.4
eh
ehA
mo{x sin (0 a)-y 0
( -a)}

cos

so that A is arbitrary. Substituting in the first two equations we get two
linear differential equations of the second order, both of which contain terms
in x and y, as well as constant terms.
The coefficient of y in the first equation and that of x in the first are the
same and equal to
e2 {d2 sin 2a - h2 sin 2 (9 - a)}
2c2m2
They can be made to vanish by choosing a so that

29

sin 2a = h2 sin 28/V(d 4 + 2d2 h2 cos
+ h4)
2
4
2 2
cos 2a = (d2 + h cos 2)/(d + 2d h cos 29 +h4"f.

(541).

With this value of a the coefficient of x in the first equation reduces to
- e2f2/c2m2, and that of y in the second to e2g2/c2rn2, where

2f 2 = V/(d4+2d 2h2cos 2+h

4

)+d 2 -

2g2 = (d 4 + 2d2h2 cos 29+ h4

2

d2 +h 2

-

.

...

(542).

f

and g are both real and may be taken to be positive, or negative, as may
be most convenient. We shall take f to be positive, and choose the sign of g
so that
fg=dhcos....
................ (543),
which obviously agrees with (542).
2
/2 + g =

V(d

4

We see at once that

f2

g 2 = d 2 - h2....(544).

V

2d2A2 cos 20 + /Z4),
±+

"

Then we easily find from (541)
co

cos (- a)=

N~4

sny

r

d

2

,sin

8 -

_f2I....(545).
/d 2

=

\

d\/f+g2

h

f2 +g2)

Again, the constant terms in the equations for x and y vanish when

Ah 2sin(0-a)-1cd2cosa=0,

Ah2cos(9-a)-Icd2sina=0.

The determinant of these two equations in Ah 2 and kd2 reduces to cos 9,
2 a case already treated in problem 2. Hence
and only vanishes when 9
we may suppose cos 9 different from zero, and must put A and k equal to
zero. Remembering that z0 = CZo/V/(C2 - v02) we get from (540)

=7r/ ,

w sin
0

(9

-a)

-yZocos

(90-a)

=

e

c

2

-

V0

2

~PROBLEMS OF
ILLUSTRATIVE

310

310

Putting k = 0, x
we find

cr, y
0

=

=

x° cos a

Solving for

x°

and

x°

c

The

a

-?/osifl

cam

~2

e

2

-v

(542)-(545)

2+ 2
c'ni {c V(d g )
2

=

[APP.

and v = v° in the energy equation (537)

y°

and using

y°

THE[AP

we get

V(d

-f/ 2 )1
z°
2
efV(f + g2) \/(C -v° 2)
2
_f 2 ) - Z° 'V(d 2 ± g2)}1(
1V(d
2

wm{c

I

(546).

..

0

first two equations (539) become
e'f- x= 0,

x-

. . .
ce 22.

y+

. .. . .. .. . .

.. (5 47 ).

Using (545) we find from (540)
of d2f~x-eVf2+g2

z

Lastly, remembering that
g

t
d

=

=

.4)

2y...........

c/Vc(

2

v 2) ) and

-

using (545),

we

get

from (537)
ct

d2 +g 2

ef

=

cm
Remembering that x°'

from (547)

cx /V/(c2

-

5t+
Xw= x°cosh ejfT
cm
/

0

eg
m

-'2g
-

2

-f

2.

.

and yo' = cy 0/V(c

v°2)

.

2

.

-

.

(4)

V2), we find
0

c mx0
ef V'(C2 - Vol) sn cm
eg'r
c~m4.n..........(550).

___T

eg PAc

C5cm

/d

f~~2

-v°

2

)

Substituting these values in (548) and (549), integrating and shifting the
origins of space and time, we obtain
2_efT
fr
/f 2 J
cosh e
x0sinh e-T+efc mx0
2
2
C
g2 ° cm ofV(c - v°)
2
egTr
.
c2m'°
V/ Id + g2 YO egT
sin
cos
2
cmj
egV\(c2 - Vo)
c
'f 2+g
..
eT
cmx
d2+g 2
2 °sihcm
Ct~ f'+g
(551)
5 ho
-

J

-f2

The2

.efT

Yo.inegi

q+ /(2

_
T

45)

vc2m)cosg

Teequations (550) and (551) completely determine the path and its
decrripntionn
terms of
arameter 'r_

modeP of

in

thep

G]
MOTION OF THE LORENTZ ELECTRON31

G]

298.
x2

-

311

From (546), § 297, we find

V(d

4 2 2
4 2
m ti
e~f2C(C20 v 2) - Cm [J{c
-0

>

2

m2 [{Ic

2
+ g2)2 - z0 V(d2 -f 2)}22- x~ (.f2 + g2)]
e 2f (/2 +g 2 ) (C2
vp) 0

V(d 2
c2

0,
because

2) -

-_f

>

'z

+

x2

V(d' +

ey' 2 (f

2

+

g2)}2 + (C2 - x2-

g2) (C2

-v

0

i 2 (f 2 + g 2)]
)

2)

Zi2.

Thus we may write
2
X2--=a
e2f2(C2-v2)..............(5)
c mX

4
C m2j 0

S+

YO e 2g 2 (
where

0

2

both real.

a and b are

.

-

u2)
°

Moreover, write
ef-+ tanh-l
~2 6
cm
f \/(C - v02) x0

g

ertan-'-c
cm
tan-'

so that

0)y

2

~2'

an
+
c my 0
f
eg \V(C 2 v 2)+-9o

c mxp

fl~c

-

Then

we

find,

in

place

x= a cosh
y

b cos

of

d

-

)

(550) and (551), § 297,

f)

ainf4)-

+ g2

d2 +

g7'f2+gZ2

2

a sinhf{-/

.(..).

2

fbi(

(554).

\

)

E

g

fg gf2

These give

0

g
(0b-E)

d 2 -f 2
\/2+g2

c=

*(53)

--

e /c

x + y + Z - ct =a 2 +b2 ... . . . . . . . . . . . . . . (555).
2

2

2

2

Equations (554) give the simplest form of the solution.
299.

By means of (546),

§ 297, and (552), § 298, we find
f 2a 2

-

g2 b2

= C4n2/e2.......................

(556),

showing that a and b are not independent.
The first equation (554), § 298, shows that x always has the same sign,
that of a and x 0; from (546), § 297, we see that ce has the sign of e. Hence
0
x and a are positive, or negative, according as the charge e is positive, or
negative.
Again, the form of the second equation (554), § 298, shows that b may be
to be poi tive
itou
nlss of grenerali ty.

taken

312
PROBLEMS

312

ILLUSTRATIVE OF THE[AP

[APP.

For large values of 0 we get z = ct V(d2 -)/V(d 2+ g 2); hence z increases,
and on the whole the motion is in the positive direction of the Poynting
vector, just as it is in the particular cases already examined.
Equations (554),
of two components:
(1)

§ 298,

x--acosh{4,
g

show that the actual displacement is the resultant

z-Th\f2±g22
/d

2

f

asinhfq0,whence

2"+J2"

2

_f+

-

d2

(2) y=bcos( -

2

2
2 =a2....................(5 57).
)
__

x2' 2

z),

b sin (-

y2 +

f2 +g 2 .2=
2. .
2+g

dz

), whence
.

.... ...................

(558).

+gg

Equation (557) represents a hyperbola, whose real semiaxis is a, along
Ox,

conjugate semiaxis a

along Oz, and

2f2
+

2

'

g

4

aog

Equation (558) represents an ellipse, whose major semiaxis is b
along Oz, minor semiaxis b, along Qy, and eccentricity
g
gy
Y
at once
ld2

Using (545),

297, we

get(

-

+ g2

eccentricity\f

d2

+

+
A!.+g

,g2

2

We find

+g2~

-i

zcos(9a

2

+g =

-

d

2

h

f 2 +g 2
an
from (554), § 298, we see that b has the sign of fa/g. Hence, since f and h
are essentially positive, we see that (y~z - '?z) cos
- a~) has the opposite sign
to a, or e. Thus the direction of revolution of the electron round the positive
direction of the magnetic force is positive, right-handed, when the charge is
negative, and negative, left-handed, when it is positive.
g (

(y

)

(-

(9

For a numerical example, take a negative electron, for which
e/cmn= -1.77.10'

E.M. U.,

moving in a field where d = 214.5 E.s. u., h
We find

f=

g

=

151.7,a

=

Assume initially x~ = 0, yo
an=--15 mhb= 10m.

30°,
-

=

214.5 E.M.U. and 9= 60°.

ef /c2m = eg/c2m = - 0.0894.

"544 c, io = "577 c, r. = "793 c. We get

MOTION OF

THE LORENTZ ELECTRON31

By means of (546), 297, we find that
find from (554),

15sin
12--

=

in a

e :r/

2 .Hence

=

we

y=10sin4 ,
+10 3
- 12h COS4t,

The path is drawn.in Fig 49

roughly

0, so that

§298,

w = -- 5cosh
z

o

313

plane. through:

1513

10

ct=-12Binh 4+ -cos

4.

as seen from an infinite distance, looking

the- Poynting vector

a:t:.300, to

the magnetic force

and 1200to the Poynting vector.

6

Fig. 49.

Path of a negative electron.

Path shown by the thick line, full in front, broken behind.
of motion.

Generating ellipse in

various positions

the

Arrowheads show the direction

( = r etc.) indicated by fine lines, full in
;direction of revolution. Guiding hyperbolasown

front, dotted behind. Arrowheads show
by dotted line. Boundaries of the tubular surface generated by the ellipse shown by fine full lines.
small circles -and radii.
Successive positions of the electron shown

by

300.

Problem

4. An elctron moves under the. combined ln-

a.function

.of the -distance alone,
fluence of a central force, which is
nd uniform magnetic field of small Intensity.
and o astay
sa
Required to find the moton. This problem has ,been

:chosen

exapleofthecae

field

is symmetrical

wer anenrg

integral eit,

about an axis.

I

and the electromagnetic

spariulry itrsigi

ofra

314

3I4

~PRO13LEMS
ILLUSTRATIVE

tOFTHEtA.

ARP.

it supplies an extension of the Lorentz theory of thenoralZeman.effectto
cases where the motion is not circular, and the velociyneed not be smal
coparewit thatflt.
The
effect is-knownto be a linear
funtinof the intensity of the external
reasonw
may neglect terms involving the square of the :intensity. This produces a
very g ts Y. -.. g: fthe work;need,:the inclusion of these terms
fti
would renderteequations well igritractabl.
Withthe notation of'Appendix F e .shall denote by U the force function
when the externalm: agneticfield is bsent.Byhyothesis U isafunction
0
of r alone, and thereforeits possible to make the constant ofanular
momentum, K,
by choosing the axis of : to lie in the plane of
projection of the electronhrou
theradius ector
euation (9)
286, for the force function, in the absence of
eternal
geticeld reduces to
(r. .e )2 e 2a

Zeeman
magnetic field;,forthis

vanish

Hence e

the

The energy constant, -, is arbitrary,.' dependingon

Uo

on the =ciricumaces

of projection.
order that
may be a.functon.of aone for ivalues
of I,
an a, must each be functions of r a.
possibility of a continuous distribution
of electricchargeinthesstem'to
which the electric potential is due, 4)can.be anyfunction of r alone.:But
is not true for the vector potential a ; for sincthe field isstady,.thee
are no electric currentsin the system, and therefore
muststisfyh
equation

this

2a

:a:ac

Ia

This equation is incompatible with the requirementtht

of

ralone other than zero ;

external

be a function

hence ax must vanish in the absence off the.

magnetic field.

when the

.about.

external field is applied, the electromagneti ieold as a-whole
is symmetrical
the line of magnetic force through the centre of force..
this line as ,axis -of x of our cyindric coordiates :(
) I h

We take

intenityof the field be H, the vector potential: s givenby
aa

~,c=HG.......(6)

]

315

OTION OF THE LORENTZ ELECTRON

The equations of motion (494), § 286, give
e
9U+
2
d
r"~9'2
0
(r6')
r r m2r3sin2' dr

3U

We also have the equation of energy, by (496),
r'2+2r2'2=k2
U

2
r p_____ sin2
=2

k2cos

-

§ 286,

0 ± eL 2...............(563),

as before; further, the second equation of motion gives
r 40' 2 J

K
c
m2 sin2

(564),
where h is an arbitrary constant. This equation shows that k must be less
than m21t2, otherwise the motion could not be real; hence write
K= mh sin.... .................
(565).
.................

2

We get

r'

-,

h(sin2 6 - sin2 6)
sin 6
..................

(566).

rte = h

We ge

This equation suggests the substitution
cos

= cos 6 cos
r2' = h.....

whence

(567)

.................

............... (568).

Eliminating r2 between (561) and (566) we find
eH

+

9'

eli
m

sin

8

9s/(Sin2
0 -sin2

sin

8)

6

sin2 6V(co

-

-_cot

0)

Integrating this equation, we get

(
eHlr N
cosyx+
2 - Y)= tan S cotu0.............(569),

where

ry is an arbitrary constant.

Eliminating 6' between (563) and (564) we find
h2

r2-2U

rO-U

2

f

whence

Tr= ±+2j

eli-2

±+cm 2

~k

-

dr
"""""""""(570).

Again, combining this equation with (568), we get

~= ±

r2V(Uo+~~~c2...........(51.

Lastly, in order to determine t we must remember that d
by (480),

§ 286.

Now with the notation of

§ 286 we

have

=

c/V/(c 2 -v)

316

[APP.

PROBLEMS ILLUSTRATIVE OF THE

But from (483), § 286, we find that w = cv//(c2 - v 2); hence
2

C

=
- = (2 U )..........................(572),

Vc

which is another form of the energy integral.
1/(2

ct = +

Uo)

Hence we find from (570)
.

dr

H
2 Uo - + e

-

............... (573).
c2

Equations (567), (569), (570) and (571) determine the orbit, and (573)
determines the mode of its description. The two first equations admit of a
geometrical interpretation.
301. Let a sphere of unit radius be described with the origin as centre.
In Fig. 50 let rectangular axes of (x, y, z) cut it in
X, Y, Z, and let the radius vector to the electron cut
x
it in P.
x
Then

XP= 0,

YXP= X.

8,

With X as centre describe a small circle of radius
and draw the great circle PA to touch it at A. Then
XA =

8, and A

Hence

A

Fig. 50.

is a right angle.

cos 0 = cos S cos AP,

P

cos AXP = tan cot 0.

Comparing these equations with (567) and (569), § 300, we see that

6eHTAP=
AXP= x+ -

AP=b,
and therefore

_
YXA =, 7

-

,

2cm

Thus (571), § 300, may be regarded as the equation of the orbit, 0 as the
longitude measured from OA as initial line, and the plane OAP as the
plane of the orbit. When there is no external magnetic field YXA = y, a
constant, and the plane of the orbit is fixed in space. But when there is an
external magnetic field YXA increases with time at the rate
while AX retains the constant value

. Now

=

c-v)

-edT

2cm dt'
and the

instantaneous mass of the electron is given by the Lorentz mass formula as
equal to m,, = cm/v(c 2 - v2 ).

Hence the angular velocity of the plane of the

orbit is equal to - eH/2cm,. In other words the electron describes its orbit
relative to the plane OAP, while this plane precesses about Ox with angular
velocity - eH/2cm,,. This result is analogous to that obtained in the
elementary Lorentz theory of the Zeeman effect, where the angular velocity

G]

317

MOTION OF THE LORENTZ ELECTRON

of precession is equal to - eH/2cm, and therefore uniform. Here it is
variable, because the zero mass m is replaced by the actual mass m, which
varies as the velocity of the electron varies. This result is important,
because it indicates that the normal separation in the Zeeman triplet is to
be expected only in those cases, where the electrons generating the spectrum
lines are moving with velocities which are small compared with that of light.
Otherwise the separation will be less.
302. The two equations of angular momentum, (561) and (568), § 300,
remain true even when the electromagnetic field is not steady, that is,
during the variable period while the external magnetic field is being
established, provided only that the force due to the permanent field
remains central. On account of the great importance of this result we
shall give a proof from first principles.

w,

X) we can by (454), § 278, write the
With cylindrical coordinates (x,
equations of motion in the following forms, bearing in mind that centrifugal
forces must be added on account of the rotation of the w-axis with angular
velocity X:

dt {c(

d

- V2) = - Po cos 0,
2

dt /(cd

cU

)

ca

2
_

/(c2- v) -

v

- Po sin

+cm

'

eH.

efH

cr

cax

eH .

0+e-

2
)
2cm -cm
(C
dt 1(c2 -v2)
Here the additional terms in the left-hand members of the second and
third equations represent the centrifugal forces, P 0 is due to the central
attraction of the original field, the terms in H represent the motive force
due to the electrodynamic action of the external field, and the term in H
represents the inductive action due to its establishment.

Multiplying the third equation by w and integrating we find

+ el
((c2- v2)

Remembering that dt = c

+e = -- , a constant.

2cm

n

and that

c2

we get (561), §300,

d

proved without any use being made of the equation of energy. Multiplying
dt
the first two equations of motion by d, and using the integral just
obtained, we find

d2x

cPo cos 0

d
d2

cPo sin 0

2

d,-r2

\(C1(c

v2)

-

2

v)

e2 H2

2

ic
m

2

2

-

4c

m

2

'

318

where the last term is to be neglected.
we get
r" - r'

=

[APP.

ILLUSTRATIVE OF THE

PROBLEMS

Transforming to polar coordinates

.

+

rod

cos 0=
2 2

m r sin' 0
n 0
m 2r
r/(c2 - v2)
The second of these gives (564), § 300, as before. Hence we see that the
two quantities k and h are constant throughout the motion, whether the
external field be variable or steady, and therefore the inclination 8 of
the plane of the orbit to the lines of magnetic force, as well as the angle ry
in (569), § 300, are constant. Thus, while the field varies, the inclination of
the plane of the orbit to the lines of magnetic force remains the same, but
it precesses round Ox with the angular velocity - eH/2cm,, appropriate to
the intensity of the external magnetic field and the velocity of the electron
at the moment considered.

303. The energy equation cannot be obtained in the form (563), § 300,
for although Po is a function of r only, when the internal field is steady, yet
2
the factor c//(c - v2) depends not only on r, but on the time T, because
work is done on the electron by the external electric force of induction,
which exists as long as the external magnetic field varies. Therefore the
arbitrary constant, which is involved in the energy equation (563), § 300,
and included in the force function U0, generally has a different value when
the external magnetic field has become steady, from that which it had before
the field was applied.
We shall now use the equations (567)-(573), § 300, to determine the
orbit of the electron relative to the precessing plane OAP. For this
purpose we must make definite assumptions respecting the form of the
force function Uo. We shall only consider the case of an electron moving
inside a fixed sphere of uniform density and opposite sign to that of the
charge of the electron.
304. Example: The electron moves inside a fixed sphere of
uniform density, of charge - e' and radius b.
The charge is taken to be - e', so that ee' may have the positive sign.
There is no magnetic field other than that of intensity H.
potential is given by

The electric

e'r2
.=

..
2........

................... (574).

The energy integral (477), § 285, may be written in the form
ee'(a2 -r )
c
..................
- v2) .c 2cmb
where a is the arbitrary constant, and is real because ee' is positive.

(575),

MOTION OF THE LORENTZ ELECTRON31

319

The force function U is therefore given by
0
=

e

r2)2

_

(a2

e2e 2

(576).

...-..................
8~c2

m2

b6

From (570), § 300, we find

I

TJ

dr

2
le (a2
e'2

-

r 2)2

h2

2rn 2b6

__7)

2

2

m

_r

4C

The choice of the positive sign for the square root clearly involves no
loss of generality; it is merely a question of a convenient choice of the zero
from which is measured.

T

The function under the square root is positive for r2 = o, negative for
r2= a2 (unless eHKc/c3m2 be too large), and negative for r2 =0. Hence it
vanishes for some value of r greater than a. The energy equation (575)
however limits r to values less than a. Therefore r2 must have an upper
limit less than a2, and a lower limit greater, or in the limit, equal to zero,
and both of these values must make the function under the square root
vanish. Hence this function has three real positive roots; we shall denote
them by a2e1, a2e2, a2e3 respectively, where
e>1>e >e 3 >0
2

Writing for brevity

... .

r2 = a2

(579)

d
d

2cmb3

where e2 >

(578).

. .. . .. .. . .. . .. .. .

..

ee'aT

we findeeTd

................

)

e-

JeV

(e

(

-

e3).........(580),

-

e 3.

Identifying (577) with (580) we find
e2e'2 (a 2

-

r2 ) 2

2

4C2m b6
whence

h2

e2e'2 (a2 e1 - r2 ) (a2 e2

_eH_

1-c3m2}

C

r2

2

6

e. 2 e+e~e+e~ 6
2 =1-~e'~

(

eHK

2

=e~2b

r2 ) (r 2 - a2e3)

m b r2

=40

e1 +e 2 +e 3 =2

e1 e2e3

-

I

......... (581).

We can reduce the elliptic integral to the normal form by wriing

S=e 3 + (e

Then we get

2 -e 3

) sn2Q(u,

k),

/e 2 -e 3 . .. . . . . . . . (582).
el- e3

k=

from (575), (579) and (580)

r 2 =...a2e..a2{e3+ (e2 - e3)Sfl2U}

_ 2cmb3a
Teea 1V(e 1 - e3 )
2

c

V/(o 2

-v

2

)

ee'a
2

fL3_e
1

3

2

-

-(e

e3 ) sn2u}

)

.....

(583).

320
PROBLEMS

320

ILLUSTRATIVE OF THE

AP

[APP.

In order to find an expression for the time t we use (480),

t =fTj 1

2)

fo V(

a
ae

)

1

./(

3)

§

eu.

0

286;

hence

Using Jacobi's notation we get
a 1/(e1c

t=

f

The longitude
we find

e3) { (K-el - 1 )
el -e a
e

§

is determined by (568),
_

d

(8)

Zu

300;

with the help of (583)

fa_0r
~/

e 3u

1-e

e-

e d

nu...(586),

3

where a is an arbitrary constant, and the upper, or lower, sign must be taken
according as It is positive, or negative. Write
sn (In,

k)

=c

that(is,

e1-

')

st(n

.(587).

e

Reducing in the usual way and using Jacobi's notation we find
Ve

=e±{
+

(e,

-

e3 ) ± l (

eleZ -

o

(u

I, logk)

3)

-

2

k

K}

tn, k)

....................

(588).

The equations (585)-(588) completely determine the orbit of the electron
and its mode of description relative to the precessing plane OAP.
305. Since the Zeta and Theta functions are periodic functions of utof
period 2K, each of the quantities t and 0 consists of a progressive term,
increasing proportionately to u, together with a periodic term. For the
sake of brevity we shall write
ee'2cmb3K
e'V(e - e3)

T =a {(e - e3 ) E- (e-1)K}

(8)

....

c V(e -e3)t
1

{V'e=e2e3

+ Z(nk) K +........58)
e3)
(,k).
2'
Then we find from (583), (585) and (588), § 304,

= /e(e

3 )KZ()

uKt-_aV/(e1--e

T-e t
7
1
+

T

1-

aV (e1 - e 3) pZu..(9)
cT
Z(u)
_____1-e3

a-+aV(eie3iZ

.59)

O

(
0u

(u)±Ilog

-

9.4

i

lk

Gin,

G]

321

MOTION OF THE LORENTZ ELECTRON

Z(u) is an odd function of u, O (u) is even, and therefore log o (u + t k)s i
(u , k)
odd; all are periodic functions of period 2K. Therefore the right-hand
members of (590) are odd periodic functions of u of period 2K. The first
equation shows that they may also be regarded as odd periodic functions of t
of period 2T,and accordingly can be expanded in series of sines of integral
multiples of the argument 7rt/T.
Again, we know that k 2sn2u - (K - E)/K can be expanded in a series of
cosines of integral multiples of ru/K. Hence we see from (583), §304, that
r2 - a2 e1K - (e1 - e3) E}/K can be expressed in the same form; therefore it
is an even periodic function of t of period 2T,and can be expanded in a series
of cosines of integral multiples of the argument 7t/T. The mean value of r2
is clearly equal to a2 {e1K - (e1 - e3) E}/K.
Thus the motion relative to the plane OAP is compounded of a uniform
rotation in the plane with the angular velocity ± t/T, and an oscillation of
the period 2T. The orbit has an infinite number of apses, the apsidal
distances are a /e and a /e3, and the apsidal angle is
The precessional motion of the plane OAP is determined by (567) and
(569), § 300. We saw from Fig. 50, § 301, that the angle YXA between the
fixed meridian plane XY and the meridian plane XA through the initial
line OA,which we shall call r, is given by * = - eHr/2cm. From the
second equation (590) we find
eHae /(e - e3)(
eHO
............ (591).
Z (u)
2c2m T
S+ 2cmT t - 7

(.

The right-hand member of this equation is an odd periodic function of t
of period 2T, and can be expanded in a series of sines of integral multiples of
the argument 7rt/ T .

Thus the motion of the plane OAP consists of a uniform precession about
Ox with angular velocity - eHO/2cmT, compounded with an oscillation of
period 2T.
306. In order to get the most convenient representation of the motion,
we choose a system of moving axes of (x', y', z').
X
Let the Eulerian angular coordinates, which define
X
the position of the moving axes relative to the axes
of (x, y, z), be (0', c', q'), wherea,

f

'T'A
eHO
'

P

AA

t .(592).

45/T,

z
Fig. 51.

Then the axes of (x', y', z') rotate about Ox'
and precess about Ox with uniform
with uniform angular velocity +
21

322

PROBLEMS ILLUSTRATIVE

[APP.

OF THE

angular velocity - eHO/2crnT. The plane 0 Y'Z' cuts the plane of the orbit
OAP in the radius OA', such that XA'= XA = 8. The arc AA' and the
angle PA'Y' are periodic functions of t of period 2T, and vanish whenever t
is an integral multiple of T. The coordinates (x', y', z') of the electron at P
relative to the moving axes are periodic functions of t of period 2T, and
therefore can be expanded in series of sines and cosines of integral multiples
of the argument n't/T. Thus the motion is a particular case of that studied
in problem 6, Ch. VIII, p. 143, such that
eHe n=+_-T,'q- s
,=2r-,cmT'

T'

where s is an integer, which may be supposed positive without loss of
generality.
307. Accordingly all the results obtained respecting the distant radiation from the electron apply in the present case. In particular, we see from
§ 113 that each value of s gives rise to a nonet of vibrations:
(1) A triplet of right-handed circular vibrations about Ox, that is, about
the positive direction of the magnetic force, with frequencies
(sr±
and

(

- eHO/2cm)/T

(sr - eHO/2cm)/T.

(2)
(sw +
r
(3)
(swi +

A triplet of left-handed circular vibrations about Ox, with frequencies
+ eHe/2cm)/T and (sWr eHO/2cm)/T.
+
A triplet of linear vibrations parallel to
and sr/T.

()/T

Ox, with frequencies

When the external field is absent (H = 0) the three triplets coalesce, so
as to form three elliptic vibrations with frequencies (si ± Co)/To and sWr/To,
where To and 0 correspond to the value H= 0.

(D

Thus the effect of the magnetic field is two-fold: (1) it splits up each
component of the triplet into three, one linear and two circular vibrations,
the separation being eHO/2cmT; (2) the central vibration of each triplet is
displaced whenever T and C differ from To and I 0.
When the charge e is negative, the right-handed circular vibrations have
the greatest frequencies, when e is positive, the least.
In general, there is symmetry of position in each triplet produced by the
field, but there is not symmetry of intensity.
These results are in qualitative agreement with the results of experiments
on the Zeeman effect, if the charge e be supposed to be negative in the
normal effect. In order that the agreement may be quantitative, corresponding to separation strictly proportional to H, and no shift of the central
vibration, it is necessary that T, C and be independent of H.

0

G]

MOTION OF THE LORENTZ

323

ELECTRON

and O as functions of the
Now equations (589), § 305, determine T,
energy constant a, and of the three quantities e1, e and e3, which are given
2
by equations (581), § 304, as functions of a, h, K and H.
We have seen in § 302 that the constants of angular momentum, h and K,
are the same whether the external magnetic field is on or off, but that the
energy constant a is generally different, on account of the work done on the
electron owing to induction during the variable state.
Hence it becomes necessary to examine under what conditions, if any, the
change in the value of a, and the presence of H in the second equation
(581), § 304, are without appreciable effect on the quantities T, Q and O.
308. We shall begin by examining the effect of the presence of H in the
second equation (581), § 304, where it occurs in the factor 1 - eHK/c 3in 2 on
hence from
the right-hand side. Now by (565), § 300, we have k= mh sin
the third equation (581), § 304, we find

8;

eH

eH sin 6 ee'a3 /(ele 2e3)

c3m 2

2c 2mb 3

c27n

Let ro be the least apsidal distance, a /e3, and let vo and v1 be the
velocities at the apses; from the first equation (581), § 304, and the last
equation (583), § 304,
eHK
c3m2

eH sin 8

V2

c
-

+

v0 2

c
-

e2
v12)

Here ro must be less than the atomic radius, let us say less than 10-8 cm.;
for H = 30,000 gauss, and e/cm = - 177.107 E.M.U., eH sin 6/c2m is numerically
less than 17-7; and the last factor is less than unity. Hence eHK/c3m2 is
comparable with 10- 6, unless the velocity of the electron at the nearer apse,
where it is greatest, is nearly equal to the velocity of light.
Hence the direct effect of the external field on the periods will be less
than one millionth, unless the velocity of the electron at some stage of its
motion becomes nearly equal to that of light.
As this effect is already near the limit of accuracy of wave-length
measurements, we shall neglect the term eH/c3m2 in the second equation
(581), § 304. Then it gives
ee'a2
= (1 - ee3 - ee 1-e1e2)
2c2mb 3

- 1

............. (593).

309. Again, in considering the effect of changes in the value of a, we
must bear in mind that even when there is no external magnetic field, the
energy constant a, as well as the constant of angular momentum h, can have

324
PROBLEMS

324

ILLUSTRATIVE OF THE[AP

[APP.

various values. If the motion under consideration is to account for fine
spectrum lines, in spite of the fact that waves fall on the optical receiving
instrument, which have been emitted by large numbers of electrons, projected
in similar fields but under widely different circumstances, one of two
conditions must hold: either (1) the periods of the waves emitted by an
electron do not depend on the values of the constants a and h to any
appreciable extent; or (2) some cause is at work which confines the values
of these constants within very narrow limits for all electrons, which are
contributing to the particular line under observation.
In the present problem the first condition holds, provided that the
velocity of the electron always remains small compared with that of light.
in fact, we know that under these circumstances the electron can describe
ellipses of various shapes and sizes, but always with the same period.
The third equation (583), § 304, gives for the two apses
c
,%(C2 - v02)2-

ee'a2 (1 - e3)
2c2 b3 '

c
/(c2

__

ee'a2 (1 -e)
2c 2 nb

v12)-

-

3

Hence v0 and v1 differ appreciably, and one of them at least is comparable
with c, unless both e2 and e3 are small.
Neglecting squares and products of e2 and e3, we find from (581) and
(586), 304, (589), § 305, and (593), § 308,
ee'a2
2c'mb 3 _= 1 + e2 + e3
2c1mb3
1-e e2 e)...........(595).

ee.T,/

V-

T=iw

q)=2

1mb

3

,

+

(e2 +e3)}

w{1+V(e 2 e3)}

Since the apsidal distances are given by r0

from (595) approximately

e2= 8 2T2
and from (594)

v

=

c

V/2e2,

=

a 1Ve 3,i and r1 = a\/e,, we find

e3= 8 2T2.....................(596),
v1 =

c ,V2e 3

... . . . . . . . . . . . . . .

(597).

In order that the fundamental period 2T should be comparable with
that of the D lines of sodium, the third equation (595) requires that
b = 46.10-8 cm. for e' = e, and that it be larger than this for e' > e. In
order that r0 and r1 may be less than b, it is necessary that e2 and e3 be less
10-3 (597).
nd v 0/c and v. /c less than
than 3.01

(596),

2-5.

G]

325

MOTION OF THE LORENTZ ELECTRON

With these values of e2 and e3, we see from (595) that O, T and c differ
by less than one part in five hundred thousand from their mean values,
a variation just about small enough to allow of the observed fineness of
spectrum lines. To the same approximation the separation in the Zeeman
effect then reduces to the normal value eH/2cm.
When the velocity of the electron exceeds the limiting value just found,
namely one four hundredth of the velocity of light, it is necessary to invoke
the second hypothesis in order to account for the observed fineness of
spectrum lines.
Moreover, the second hypothesis is required in every case where the
potential 0 is not a quadratic function of the radius vector, for instance,
when the field is due to a fixed point charge. A hypothesis tantamount to
this was suggested by J. J. Thomson in his book on the Corpuscular Theory
of Matter, pp. 157 et seq.

21-3

INDEX
The numbers refer to the pages
Abraham ix, xiii, 173, 175, 176, 178, 246
Abraham electron xiv, 255; its electromagnetic mass 175; its kinetic energy
178
Accelerated motion of the electron, cf. the
particular motion
Acceleration of the electron, determines the
radiation 8, 177; its effect on the pressure
and configuration xiv, 269; its relation
to the Postulate of Relativity 281; - of
the second order determines the radiation
pressure 246
Activity of the forces on the electron 176;
internal - 251; relative - 252
Aether, disturbing effect of the structure
of an electric charge is confined to its
immediate vicinity 1; energy of the - 5;
- hypothesis vii; use of the - in preserving the electron 263
Angular momentum of the Lorentz electron
286, 293, 297, 315, 317
Aperiodic motions 149
Assumptions, of the electron theory 1, 14,
173; - of the present essay vii; - of
the mechanical explanation of the electron
262; - of the relative motion of the
electron 235
Asymmetry, of the electron, its effect on
the electromagnetic momentum 246, and
on the steady motion of a ring 192;
- of the deformed Lorentz electron xvi,
258; - of intensity in the Zeeman effect
146; - of radiation in the Doppler effect
198
Asymptotic formula of Duhamel 108
3-particle, its energy vi; its motion in a
uniform electrostatic field 181
Bessel functions, addition theorem 109;
asymptotic formula of Duhamel 108;
Gegenbaur's theorem 20; integral for109, 136, 158; Kapteyn
mulae for series of - for the radiation from a ring
110, 138, and for the mechanical forces
187, 212, 216, 224; products of - in the
potentials and forces at a distance due to
polyperiodic motions 114, 115, 117; series
of - for the potentials and forces at a

distance due to periodic and pseudoperiodic motions 107, 108, 120, 123, 132,
133, 134, 138, 140, 163, 164; the same
for a circular ring near the circle 170,
171; summation formulae for - 110, 121,
125, 214
Bounding surface of the electron determined
from the strain 243
Bucherer, his experiments on e/m xv, 181;
their theory 302
Bucherer electron xiv, 255; its electromagnetic mass 175; its kinetic energy 178
Centre of the electron 155, 236
Centro-symmetrical electron 247; its electromagnetic mass 175; its electromagnetic
momentum 247
Characteristic curve 24; general discussion
of - 88; examples of - 38, 46, 54, 59,
65, 85
Characteristic equation 22; its geometrical
interpretation 29; examples of - 38, 4],
64, 84
Charge, specific - of the electron, Bucherer
181, 302; Kaufmann 299; new method
299; Wolz 181
Circular motion of an electron, disturbed 179; equations of - 180; distant field
due to - 126, special case of axial oscillations 132, of orbital oscillations 129;
near the circle 200
field ofCircular motion of an electron, uniform 179; equations of- 179; field due to anywhere 83, 105; distant field of - 106,
119; field of - on the circle 168
Circular ring of electrons, disturbed motion
of - 200; disturbing force on one of the
electrons 206; part of the same due to
the external field 218; part due to the
rest of the ring 209, 217; equations of
disturbed motion of - 217, 223; field
due to - anywhere 106; distant field
of - 133; radiation from - 135
Circular ring of electrons, uniform motion
of - 186; equations of uniform motion
of 188; mechanical force on one
electron due to the rest 186; distant
field of - 108; radiation from - 109

328

INDEX

Compensated rays in Bucherer's experiment
303
Complementary functions representing the
effect of charges outside a given region 4
Continuity in general of potentials and
forces vii
Convergence of series 110, 153, 170, 172,
228, 234, 235
Conway 155
Cusp-locus 35; its geometry 36; examples
of - 44, 57, 59, 86
Deformation of a moving electron under
superficial pressure 269, 277, 281
Deformed Lorentz electron xiv, 258; its
energy 261; its masses 260
Degrees of freedom, cf. freedom
De la Rive 22
Density, electric, cf. electric; - of nonelectromagnetic mass 263; the same a
function of the potential and timealone 273
Descoudres xi, 61
Diffraction of X-rays 92
Discontinuities of a motion, their effect xi,
152; - of the potentials for speeds greater
than that of light 26
Distant field of a motion, allowable approxi112;
calculation of mations 98;
examples of - 99, 106, 111, 128, 156;
its forces 100, 108, 116, 120, 123, 140,
145, 161 ; its nature 101, 108, 116, 134, 141
Doppler effect, asymmetry of radiation in
the - 198; change of frequency in the
- 196; forces in the - 194; radiation
in the - 196
Doppler factor 22; examples of - 37, 41,
64, 83
Duhamel 108
Einstein 28, 90, 196
Electric centre of the electron 155, 236
Electric charge, assumed to be extended
viii, ix, 14, 235, 262; its structure without
effect on its radiation 1, 111
Electric density, effect of a linear distribution of - on the potential of the electron
271; Fourier integral expressions for 15; surface - produced by acceleration
of the electron 277; - of volume changed
by motion 267, 272, 276, 280, 281
Electric force, cf. forces
Electromagnetic energy, of the electron 253,
255, 256, 257, 261; - of the field 5
Electromagnetic field, equations of the - 2;
comparison of methods of calculation of
the - 92; - near a moving electric
charge 227; - near its path 165; - of
various motions, cf. forces and potentials,
also distant field; sequence of events in
the - 31, 47, 55, 61
Electromagnetic mass 249; - of various
types of electron 175, 255, 256, 257, 260;
- ill defined for speeds near thatof light 90

Electromagnetic momentum 12, 175, 246;
- of centro-symmetrical electron 247; longitudinal in certain cases of symmetry
248
Electron, its electromagnetic energy 253;
its electromagnetic mass 175, 249, 278;
its electromagnetic momentum 175, 246;
energy equation of the - 176; its energy
relations 251; its kinetic energy 177, 254;
mechanical explanation of the -- x, 262;
non-electromagnetic energy of the - 254;
its non-electromagnetic mass 263, 267; its
internal pressure 263, 266, 273, 277; its
superficial pressure 12, 267, 278; conditions of its invariability and uniformity
267, 268, 274, 277; various types of 175, 178, 255, 257, 258
Electron theory, its fundamental assumptions 1; its equations 2
Electrons, ring of -, cf. circular ring
Electrostatic field, motion of a -particle in
a uniform - 181
Elliptic motion 122; its radiation 125
e/m, Bucherer 181, 302; Kaufmann 299;
new method 299; Wolz 181
Energy, equation of - for the electron 176;
for the field 5; its verification for a
point charge 6; expression for the - of
the field 5; Macdonald's supplementary
term in the - on the mechanical theory 5
Envelope, cf. wave-envelope
Epitrochoidal motion, of an electron 140;
- of a group 142
Equation, of energy, cf. energy;
- of
pressure for the moving electron 263,
272; - of Hamilton and Jacobi for the
Lorentz electron 290
Equations, of the electron 178; - of disturbed and uniform circular motion 179;
- of a circular ring, disturbed 223, and
uniform 188; - of the Lorentz electron,
Hamiltonian 287, Lagrangian 284, Newtonian 284
Equilibrium forms of gravitating liquid,
their application to the mechanical theory
of the electron xxii, 268
Equipotential surfaces inside the electron
in motion 267
Error made in neglecting the radiation
pressure on a moving electron 183
Expansion of the electron steadies the
motion of a circular ring 191
Expansions of the function U according to
accelerations of increasing order 228; of the function V according to powers of
of the forces and
the velocity 23; potentials, cf. forces and potentials
Explanation of the electron, mechanical x, 262
Expressions for the function V in finite
terms 231
Extension of the electron vii, ix, x, 14,
262

INDEX

Field, electromagnetic of a moving
electric charge, cf. electromagnetic; action
of an external - on an electron 174, 285,
and on a circular ring 218; a steady
leads to an energy integral 291; asymmetrical about an axis leads to an
integral of angular momentum 286
Force, mechanical - on the electron, external 174, 185, 235, and internal 175,
235, 245, 250; - on one element of an
electron due to another 237, 241; equation
of Larmor and Lorentz for the - viii, 2,
173; working expressions for the - in
an electron 242; resultant 9; its
resolution 12; - on one electron of a
circular ring due to the rest 187, 206, 217
Force-function of the Lorentz electron 293,
296, 314
Forces, electric and magnetic
due to
moving electric charges, anywhere 18, 23,
66, 73, 97, 112, at a distance 77, 100, 107,
116, 120, 123, 128, 133, 144, 161, 194, near
the path 167, 170, 171, 187, 201, 205
Freedom, degrees of - of an electron and
of a circular ring 226
Frequency, change ofin the Doppler
effect 196; - equations of a ring of
electrons 181, 226
Function, force -,

cf. force; -

U 228; -

V 230, 242; Hamiltonian -, cf. Hamiltonian; Lagrangian -, cf. Lagrangian
Gegenbaur 20
Geometrical construction of the waveenvelope 34, cf. also wave-envelope; interpretation of the characteristic equation 29, and of the point-laws 31; representation of the field due to an
electron moving in a uniform electrostatic
field 66, and of the point-law potentials 33
Graf and Gubler 108
Gray and Mathews 109, 157
Group of charges 102; its radiation reduced
by interference 103
Haga 92
Hamilton and Jacobi, equation of - for the
Lorentz electron 290
Hamiltonian equations of motion of the
Lorentz electron 287; - function 287,
290, 293
Harmonic motion, cf. rectilinear motion
Harmonic waves, of a single electric charge
98, 101, 116, 120, 123, 141; - of a group
103; - of a circular ring disturbed 134,
and in uniform motion 108; of a
precessing system 146
Heaviside ellipsoid xiv, 275
Herglotz 90
Hertz vii, 1, 2
Indeterminateness of structure of a ring of
electrons and its removal xii, 190

329

Integral of angular momentum of the
Lorentz electron 286, 293, 297, 315, 317;
- of energy 291, 296, 301, 308
Integrals for forces and potentials 17, 18,
20, 21, 111, 112
Intensity, of radiation in the Doppler effect
196; - of components in the Zeeman
effect 146
Jacobi 290
Kapteyn series of Bessel functions 110, 138,
187, 212, 224; convergence of - 110;
need for their study vii
Kaufmann xv, 299
Kirchhoff 4
Lagrangian equations for the Lorentz electron 284; - function for the same 285,
301; - series for the potentials xi, 154
Laisant, hyperbolic cycloid of307
Langevin 143
Larmor, Sir J. vii, viii, ix, x, 2, 3, 5, 122
Liapounoff xxii
Lienard 8, 22, 24, 110, 175, 251
Lindemann 16, 21, 28, 89, 175, 227
Liquid, equilibrium forms of gravitating
xxii, 268
Livens 175
Lorentz, H. A. vii, viii, ix, x, 2, 4, 5, 28,
173
Lorentz electron 257; its electromagnetic
mass 175; energy difficulty of the - 173,
178; kinetic energy of the -178;
the
mechanical explanation of the electron
leads to the - 268, 276, 278; its pressure
and structure 279; mechanics of the xiv, 282; problems on the motion of the
- 295; its motion in a central electrostatic and uniform magnetic field 313,
special case of the electric force proportional to the radius 318, and its application to spectrum lines 323, and the
Zeeman effect 322; motion of the - in a
uniform and steady electromagnetic field,
with forces at any angle 308, parallel 296,
at right angles 301
Loria 183, 298, 306, 307
Macdonald 5
MacLaren xxii
Magnetic force, cf. forces
Mass, electromagnetic -, cf. electromagnetic; non-electromagnetic - 263, 273
Maxwell vii, viii, 2, 3, 9, 126, 187, 215
Mechanical explanation of the electron, cf.
explanation; - force, cf. force
Menoclinoid curve 183
Momentum, cf. electromagnetic
Monoperiodic motion 95; its distant field
101; - of variable amplitude 156
Motions, various - and their fields, cf. the
particular motion

330

INDEX

Nagaoka 91, 172
Neumann's addition theorem 109
Newton's Second Law assumed viii, x,
265; - Third Law fails for electrons
235, 237
Newtonian equations of motion for
Lorentz electron 283
Nielsen 110, 158, 214
Notation, for forces and potentials 2; relative velocities of the parts of
electron 235

178,
xiii,
the
for
the

Pohl 92
Poincar6 xiv, 173, 178, 192, 268
Point charge hypothesis ix, 262
Point-laws for the forces 23; - for the
potentials 22, 24; their geometrical interpretation 31, and graphic representation
33; examples of - 37; cf. also forces
and potentials
Polarization of waves emitted by electric
charges 101, 108, 118, 120, 124, 134, 146
Polyperiodic motion 112; its distant field
116
Postulate of Relativity vii, ix, xiv, 281
Potentials due to a moving electric charge,
inside 266, 275, 279; - outside, anywhere
3, 4, 18, 96, 105, 111, at a distance 99,
103, 107, 114, 115, 154, 158, 162, 163, 164,
near the path 154, 166, 170, 171, 228, 232,
233 ; point-law - 39, 40, 52, 55, 60, 65, 85;
- due to a spherical charge inside it 20, 21
Poynting vector, examples of the - 102,
103, 109, 121, 125, 195; its identification
with the radiation flux 5; its verification 6
Precessional motion of a vibrating system
143; application to the Zeeman effect 146
Pressure, component of the resultant
mechanical force 12; - inside the electron
263, 266, 277, 278; equation of - 265,
266, 273
Pseudoperiodic motion 149
Pulse generated by a (3-particle 69; application to X-rays 83; its ultimate nature
77, effect 82, energy 79, strength 80; thin
- due to a motion with speed greater
than that of light 92
Quasistationary motion 175
Radiation, from a single electron, formula
of Lidnard for the - 8, 110, 122, 177,
251; - pressure ix, 176, 246, error due
to its neglect 183; - from a ring 109,
136; supplementary term in the - 9,
177, 253; unitary theory of - vii, viii
Rays, X and y - 91; diffraction of - 92;
pulse theory of - 83
Rectilinear motion of an electric charge,
distant field, due to aperiodic -- 164,
simple harmonic - 120, damped vibra-

CAMBRIDGE :

PRINTED BY

JOHN CLAY,

tory - 161; general field, due to uniform
- 37, uniform- generated by a uniform
acceleration 53, uniform - generated by
a uniform electrostatic field of finite
extent 69, uniformly accelerated - 40
"Relatiftheorie" vii, 28, 90, 281; Abraham's
objection to the - xiii
Relative motion of the parts of the electron
10, 243
Relativity, Postulate of - vii,
xiv, 281
Ring of electrons, disturbed motion of a -,
equations of - 223, radiation due to 136; uniform motion of -, equations of
- 188, distant field due to- 103, radiation due to -

109

Ritz vi, xi, xv, 143, 148
Runge xv, 147
Schott 17, 21, 62, 89, 100, 104, 108, 134,
155, 181, 188, 196, 198, 217, 224, 282
Schwarzschild 227, 234, 285
Searle 126
Series, Kapteyn - vii,
110, 138, 187, 212;
Lagrangian - xi, 154
Sommerfeld xiii, 20, 28, 37, 61, 83, 90,
16,
92, 175, 227
Specific charge of the electron, theory of
the experiments of Bucherer 303, of
Kaufmann 299, new method 299
Stark 199
Stokes 112, 150
Stress inside the electron 262; Maxwell's
-

in the field 9

Thomson, Sir J.J. vi, 103, 104, 172, 325
91,
Uniform motions, cf. circular, rectilinear
Uniformly accelerated motion, cf. rectilinear
Unitary theory of radiation vii, viii
Walker 175, 227, 282
Walter 92
Wassmuth xxii
Wave-envelope 29; examples of - 40, 44,
57, 59, 81; geometrical construction for
the -

34; potential near the -

37, 92

Waves emitted by a vibrating system, cf.
harmonic; number of simultaneous from one source 34
Wiechert 22
Wien 139
Wind 92
Wolff 262
Wolz 181, 302
X and y rays, diffraction of theory of -

92; pulse

83

Zeeman effect, extension of the Lorentz
theory 314; precessional theory of Ritz
xi, xv, 146, 322

M.A.

AT THE UNIVERSITY

PRESS.

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