Translation:Principles of the Dynamics of the Electron (1902)

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Principles of the Dynamics of the Electron  (1902) 
by Max Abraham, translated from German by Wikisource
In German: Prinzipien der Dynamik des Elektrons, Physikalische Zeitschrift 4 (1b), S. 57–63

Principles of the Dynamics of the Electron

M. Abraham (Göttingen)

Already in January this year, I have published a treatise on the dynamics of the electron in the "Nachrichten der Göttinger Gesellschaft der Wissenschaften". The agreement of the theories developed there, with the experimental results of Kaufmann, makes the assumptions on which the theory is based, look appropriate; it furthermore shows, that the inertia of electrons is purely electromagnetic in nature. While I at first still used a "material" mass independent of electric charge, it becomes necessary now, to establish the dynamics of the electron from the outset in an electromagnetic way. From that, remarkable analogies of the principles of the electron on one hand, and the principles of ordinary dynamics of material bodies on the other hand, are given, analogies which may become important for the coming electromagnetic foundation of the whole of mechanics.

We ascribe to the electron, the atom of negative electricity, a charge e (measured electrostatically). The free electron moving in the cathode- and Becquerel rays shall be (as we suppose) a sphere of radius a, where electricity is uniformly distributed on its volume with density \varrho. Electricity shall be connected to the volume elements of the electron, as matter to the volume elements of a rigid body, i.e. the kinematic fundamental equation shall apply to the electron.

1) \mathfrak{v}=\mathfrak{q}+[\vartheta\mathfrak{r}].

The kinematic fundamental equation determines the velocity \mathfrak{v} of any point of the electron, whose distance from the center is indicated by a vector \mathfrak{r}, when the velocity \mathfrak{q} of the center and the angular velocity \vartheta around the center are given; we write them in vectorial form, and we use Grassmann's symbol of the outer product.

The electromagnetic field agitated by the electron, is determined by the Maxwell-Lorentz field equations:

2) \begin{array}{ll} \frac{1}{c}\cdot\frac{\partial\mathfrak{E}}{\partial t}=curl\ \mathfrak{H}-\frac{4\pi\varrho}{c}\cdot\mathfrak{v}, & div\ \mathfrak{E}=4\pi\varrho,\\ \\ -\frac{1}{c}\cdot\frac{\partial\mathfrak{H}}{\partial t}=curl\ \mathfrak{E}, & div\ \mathfrak{H}=0.\end{array}

there, \mathfrak{E},\ \mathfrak{H} denote the electric and magnetic field strength, c the speed of light.

It is to be emphasized, that Lorentz's theory calculates with absolute velocities.

The electron shall now be located in a given external electromagnetic field, of field strengths \mathfrak{E},\ \mathfrak{H}_{h}. For the determination of the motions carried out by it, also a third system of fundamental equations is required, the system of "kinetic" or "dynamic" fundamental equations. In the course of stating these equations, we will be guided by the following reasoning. H. A. Lorentz and E. Wiechert have shown, that one can derive the forces acting upon resting and streaming electricity in the electric or in the magnetic field, from the electron theory by making the following assumption for the force acting on the individual electron:

\mathfrak{R}=e\cdot\mathfrak{F}_{h},\ \mathfrak{F}_{h}=\mathfrak{E}_{h}+\frac{1}{c}[\mathfrak{q}\ \mathfrak{H}_{h}].

There, the electron is interpreted as a point charge. We have to distinguish the volume elements of the electron; therefore we define the "outer force" by

\mathfrak{R}=\int\int\int dv\ \varrho\cdot\mathfrak{F}_{h},\ \mathfrak{F}_{h}=\mathfrak{E}_{h}+\frac{1}{c}[\mathfrak{v}\ \mathfrak{H}_{h}]

and furthermore introduce the "outer angular force"

\Theta=\int\int\int dv\ \varrho[\mathfrak{r}\ \mathfrak{F}_{h}].

According to Maxwell and Hertz, however, the "principle of the unity of electric and magnetic force" applies; according to this principle, the separation of an "outer field" and an "inner" field excited by the electron itself, is artificial; in reality only one field exists, of field strengths:

\mathfrak{E}_{h}+\mathfrak{E},\ \mathfrak{H}_{h}+\mathfrak{H}.

This principle leads us, so as to put the vector

\mathfrak{F}_{h}=\mathfrak{E}_{h}+\frac{1}{c}[\mathfrak{v}\ \mathfrak{H}_{h}]

next to the vector

\mathfrak{F}=\mathfrak{E}+\frac{1}{c}[\mathfrak{v}\ \mathfrak{H}],

and to speak about an "inner force"

\int\int\int dv\ \varrho\mathfrak{F}

\int\int\int dv\ \varrho[\mathfrak{r\ F}].

The "dynamic fundamental equations" now say:

The resultants of the inner and outer force and angular force are vanishing:

3) \begin{array}{rl} \int\int\int dv\ \varrho\mathfrak{F}+\mathfrak{R} & =0,\\ \int\int\int dv\ \varrho[\mathfrak{r\ F}]+\Theta & =0. \end{array}

The kinematic fundamental equation (7), the field equations (2) and the dynamic fundamental equations (3), these are the foundations upon which the dynamics of the electron is built on. It may be emphasized, that the word "force" is at this place only an abbreviating denotation for certain vectors defined by the field strengths and by the velocity of translation and rotation of the electron; from ordinary mechanics, we only incorporate the purely geometric-kinematic concept into the foundations of the dynamics of the electron. But we choose the denotation of the derived quantities so that the analogy to ordinary mechanics clearly emerges.

Before passing to the treatment of special types of motion, two general theorems that follow from the field equations, may sent before. The first theorem formulates the energy principle, it reads:

4) \int\int dv\ \varrho(\mathfrak{v\ F})+\int\int do\ \mathfrak{S}_{v}=-\frac{dW}{dt}.

There, o denotes the boundary of the field, which can be determined by other bodies or by an (only imagined) surface; (\mathfrak{v\ F}) is the inner (scalar) product of vectors \mathfrak{v} and \mathfrak{F}, and the first member of the left-hand side is the "power of the inner forces". The second member contains the normal component of "Poyntings radiation vector".

\mathfrak{S}=\frac{c}{4\pi}\cdot[\mathfrak{E\ H}]

and gives the radiation sent through the boundary by the electron. Power of the inner forces and radiation appear at the cost of the scalar quantity W, which we will call "electromagnetic energy", and which occupies the field with the density


Eq. (4) thus corresponds to the theorem of the living force. In a corresponding way – as it was first noticed by H. Poincaré – the theorems of momentum, or the "momentum theorems" can be obtained from Lorentz's theory. We have:

5) \begin{array}{cc} \int\int\int dv\ \varrho\mathfrak{F}+\int\int do\ \mathfrak{P} & =-\int\int\int dv\frac{1}{c^{2}}\frac{\partial\mathfrak{S}}{\partial t},\\ \\ \int\int\int dv\ \varrho[\mathfrak{r\ F}]+\int\int do\ [\mathfrak{r\ P}] & =-\int\int dv\left[\mathfrak{r},\ \frac{1}{c^{2}}\frac{\partial\mathfrak{S}}{\partial t}\right].\end{array}

Here, \mathfrak{P} denotes the force, exerted by the so called "Maxwell tensions" of the field excited by the electron upon the unit surface of the boundary surface o, [\mathfrak{r\ P}] is the static moment of this force, related to the center of the electron. According to equations (5), the forces exerted by the field upon the electron on one side, and on the boundary on the other side, are in general not canceling each other throughout; thus they contradict the third axiom of Newton. Eq. (4) would also contradict the energy principle in quite the same way, if we hadn't maintained it by the assumption of a new "electromagnetic energy". Also the third axiom can be saved by us, when we introduce a new "electromagnetic momentum", distributed over the field with the density \tfrac{1}{c^{2}}\cdot\mathfrak{S}. Eq. (5) is then to be interpreted as follows.

We imagine a framework rigidly connected with the electron. In all points of the field, at which the density of the electromagnetic momentum increases with time, a corresponding force acts at the framework. If all these individual forces are composed according to the rules of the statics of rigid bodies, one obtains the force and angular force, exerted by the field (limited by surface o) upon the framework. The consideration becomes simpler, when it is allowed to move surface o into infinity, and to omit the surface integrals in Eq. (5) and (4); this is then the case – the proof would lead too far at this place –, when the influence of other bodies upon the electron, as far as it is not considered in vector \mathfrak{F}_{h}, becomes unnoticeable. If we furthermore assume this condition as satisfied, then we can fully replace the "inner forces" by means of Eq. (5) by the dynamic effects of the electromagnetic momentum.

We call the integral extended over the infinite space:

\mathfrak{G}=\frac{1}{c^{2}}\int\int\int dv\ \mathfrak{S}

the "momentum" of the electron, furthermore

\mathfrak{M}=\frac{1}{c^{2}}\int\int\int dv\ [\mathfrak{r\ S}]

the "angular momentum" related to the center of the electron, in quite the same way as we are accustomed to call the integrals over the whole infinite field

W_{e}=\int\int\int\frac{dv}{8\pi}\mathfrak{E}^{2},\ W_{m}=\int\int\int\frac{dv}{8\pi}\mathfrak{H}^{2}

as the electric and magnetic energy of the electron.

If we now introduce the relations (5) into the dynamic fundamental equations (3), then we immediately obtain the "equations of motion" of the electron.

6) \begin{align}{c} \frac{d\mathfrak{G}}{dt}=\mathfrak{R}, \\ \frac{d\mathfrak{M}}{dt}+[\mathfrak{qG}]=\Theta. \end{align}

An outer force causes a temporal change of momentum. The outer angular force is not required for changing the angular momentum; no, the outer angular force must also then act, when the electron is equipped with a constant angular momentum and a inclined momentum with respect to the direction of translation. Indeed, in this case, with respect to a point fixed in space, the static moment of momentum increases or decreases, and exactly for that the effect of an outer angular force is necessary. Additionally, the equations of motion (6) fully correspond to those, which were derived for the motion of rigid bodies in an ideal liquid. There, however, momentum and angular momentum are linear functions of the velocity or angular velocity, respectively. Here, on the other hand, momentum and angular momentum, as defined by integrals over the whole field, are in general depending on the prehistory of the electron, i.e. on its velocity from the start up to now. By that, a far greater complication of the electrodynamic problem is caused, by which a general solution of the problem appear to be hopeless. One has to confine himself, to choose certain classes from the manifold of motions and fields, that are accessible to mathematical treatment; fortunately, exactly the mathematically simplest motions of electron don't seem to be noticeably different from the ones actually taking place with respect to cathode- and Becquerel rays.

When we are interpreting the cathode- and Becquerel rays as swarms of moving electrons, we have to view the first axiom of Newton as true. Namely, as long as no external force is acting, the motions takes place rectilinearly and with constant velocity. Also the second axiom of Newton was found to be experimentally confirmed in the sense, that with increase of the deflecting or accelerating external force, the amount of the transverse or longitudinal acceleration increases in the same ratio as the force; thus one was allowed to ascribe to the electron an inertial mass, a mass that is growing with increased velocity according to Kaufmann. To deduce such a behavior from electromagnetic theory, we have to look for motions of electrons, at which the translatory velocity \mathfrak{q} remains constant without the influence of an external force or angular force. With such motions satisfying the first axiom, we have to start; we have to alter them by external forces, to come to the second axiom and to the concept of electromagnetic mass.

Any translatory motion of our spherical electron satisfies the first axiom of Newton. Because it is given from the field equations, that the electron (when its velocity \mathfrak{q} is constant) simply drags its field, furthermore that the angular momentum \mathfrak{M} is zero, and momentum is directed parallel to the direction of motion. The equations of motion (6) are thus satisfied without the assumption of external forces. Now we imagine the motion as altered by an external force \mathfrak{R}, for instance by a homogeneous electric or magnetic field. Angular forces and rotation are excluded by us; we satisfy the equations of motion, when we accordingly alter the momentum vector \mathfrak{G} of force \mathfrak{R}, and let the motion be parallel to the respective direction of the momentum vector. If we presuppose, that the amount G would only depend on amount q of the velocity, then it follows, that under longitudinal acceleration, the mass has to be accounted for:

7) \mu_{s}=\frac{dG}{dq},

however, in transverse direction the mass:

7) \mu_{r}=\frac{G}{q}.

These formulas for longitudinal and transverse electromagnetic mass, have been already derived in my earlier report. At slow motion, i.e. \beta^{2}=\tfrac{q^{2}}{c^{2}} z can be neglected, \mathfrak{E} is independent from the velocity, \mathfrak{H} is proportional to it. And the density of the electromagnetic momentum, as well as momentum G, is proportional to velocity q. Therefore, the longitudinal mass becomes equal to the transverse mass at this place, a result experimentally established with slow cathode rays, but only become understandable by formulas (7) in the sense of electromagnetic theory.

However, at larger velocities where G is not proportional to q anymore, both masses are depending in different ways on velocity.

The theory gives, at slow motion:


the experiment gives:


Thus we obtain

a=\frac{4}{5}\cdot\frac{|e|}{c}\cdot 1,865\cdot10^{7}.

If for e, the charge of a monovalent ion is taken, we obtain a=10^{-13} for the radius for the electron, a result that is only to be viewed as an indication of the order of magnitude with respect to the uncertainty in the determination of e.

Concerning the area of applicability of formulas (7), still some words have to be said. That the electron can move force-free and stationary in any direction, is caused by the symmetry that we attributed to it. For instance, if the electron would be an ellipsoid uniformly charged with electricity, then force-free stationary motion would only be thinkable parallel to one of the 3 major axes, since only here, the momentum vector is directed parallel; also, of these three directions only the one parallel to the major axis is stable, in the sense that when the direction of translation is changed, an inner angular force occurs, tending to adjust the major axis into the new direction of translation. Here, at least with respect to weakly curved paths, formulas (7) are applicable. As to motions with inclined momentum with respect to the direction of motion, it is principally inadmissible to speak about an electromagnetic mass; since already here, the axiom becomes invalid.

As regards the presupposition, that the amount of momentum should only depend on the respective velocity, it is only satisfied for such motions, which I have called "quasi-stationary motions". These are such motions, under which the velocity doesn't experience very sudden changes. The electromagnetic mass indeed corresponds to self-induction in the theory of electric oscillations; one calculates self-conduction from the magnetic field of the current, as if the current were stationary; this is allowed, as long as the current fluctuations take place sufficiently gradual, as long as the current is "quasi-stationary". At very rapid fluctuations of current intensity, for instance with respect to oscillations of Hertz, the thus calculated self-induction is not sufficient anymore. The relations are quite similar at this place; at very rapid accelerations of the electron it becomes inadmissible to calculate with electromagnetic mass, especially then, when the speed of light is reached or even exceeded. Though we still may apply the theory of quasi-stationary motion when the velocity remains behind the speed of light only by a few kilometers per second, and when the the acceleration assumes the values attainable in the strongest fields.

We recapitulate: Energy theorem and momentum theorems could be generally deduced from the fundamental equations of the dynamics of the electron. To derive the second axiom of Newton, we confined ourselves to quasi-stationary translational motion. There, the concept of mass experienced an extension; the electromagnetic mass is not a scalar, but a tensor with symmetry of an ellipsoid of revolution; both masses, longitudinal and transverse, are functions of velocity. Now it's known, that in analytical mechanics, starting from Newton's axioms, other formulations of the dynamical principle are derived, the Lagrangian equations and the Hamiltonian principle, that make the dynamics of the system dependent on a single function, the difference of kinetic and potential energy (the Lagrangian function). But this derivation is based on the presupposition, that the potential energy is independent from the velocity, and the kinetic energy is a homogeneous function of second degree of the velocity components. It is near at hand, to let the kinetic energy correspond to the magnetic one, and the potential energy to the electric one, and to set the Lagrangian function to L=W_m -W_e. At slow motion, W_e is indeed independent from q, and W_m is proportional to the square of q. At larger velocities, however, this is not true anymore; here, the derivation of the Lagrangian equations given in analytical mechanics, becomes invalid. If we want to test the Lagrangian equations and the Hamiltonian principle, we have to resort to the fundamental equations of the dynamics of the electron.

This shall happen now; there, the assumptions concerning motion shall be stated somewhat more generally; we confine ourselves, not on translatory motions, but consider rotations as well. We introduce a coordinate system connected with the already previously mentioned framework, which is rigidly fixed with the electron. We write

\frac{\partial'\mathfrak{E}}{dt},\ \frac{\partial'\mathfrak{H}}{dt}

the temporal changes of the field strengths, assessed from this framework, and related to the axis-intersection fixed in the framework. The two first field equations then assume the simple form:

8) \begin{array}{cc} \frac{1}{c}\frac{\partial'\mathfrak{E}}{\partial t}=curl\ \mathfrak{H}', & \mathfrak{H}'=\mathfrak{H}-\frac{1}{c}[\mathfrak{vE}],\\ \\ -\frac{1}{c}\frac{\partial'\mathfrak{H}}{\partial t}=curl\ \mathfrak{F}, & \mathfrak{F}=\mathfrak{E}+\frac{1}{c}[\mathfrak{vH}].\end{array}

The form of the field equations makes it near at hand, to emphasize a class of preferred motions, namely such ones, whose field is stationary with respect to that framework. To the "preferred" motions, pure translation belongs, as well as the pure rotation of the spherical electron, eventually also translation connect with rotation around the direction of translatory motion; as regards rotation around an inclined axis with respect to the direction of motion, however, the velocity vector won't possess a fixed location in the framework, and the field won't be stationary with respect to the framework. For preferred motions, \tfrac{\partial'\mathfrak{H}}{\partial t} vanishes; vector \mathfrak{F} (decisive for the inner forces according to (8)) is thus the gradient of a scalar \varphi; this will be called "convection potential". The "Lagrangian function" is connected to it by the relation:

9) L=W_{m}=W_{e}=-\int\int\int dv\frac{\varrho\varphi}{2}.

(In my earlier report, when considering this formula as a generalization of a formula of ordinary potential theory, I have

U=\int\int\int dv\frac{\varrho\varphi}{2}

as the "force function" of the electron.)

For "preferred" motions, the momentum can now be derived from the Lagrangian function. So, for pure translation, the equations apply:

9a) G=\frac{dL}{dq}
9b) \frac{d\mathfrak{G}}{dt}=\mathfrak{K},

which are denoted in analytical mechanics as first and second line of the Lagrangian equations. Furthermore, the known relation from analytical mechanics apply with respect to energy:

9c) W=-L+q\cdot\frac{dL}{dq}.

These formulas are already implicitly contained in my previous report; for the Lagrangian function of the spherical electron, one has to set

10) L=-\frac{3}{5}\frac{e^{2}}{a}\cdot\left(\frac{1-\beta^{2}}{2\beta}\right)\ln\left(\frac{1+\beta}{1-\beta}\right),\ \beta=\frac{q}{c}.

From (10a), (9a) and (7), the formula follows especially for the transverse electromagnetic mass:

10a) \begin{array}{c} \mu_{r}=\mu_{0}\cdot\frac{3}{4}\cdot\psi(\beta)\\ \\\psi(\beta)=\frac{1}{\beta^{2}}\left\{ \left(\frac{1+\beta^{2}}{2\beta}\right)\ln\left(\frac{1+\beta}{1-\beta}\right)-1\right\} \end{array},

which experienced such an illuminating confirmation by Kaufmann's experiments.

(It is remarkable, that when one assumes homogeneous surface charge instead of volume charge, the Lagrangian function as well as \mu_0 obtain the factor \tfrac{5}{6}, though formula (10a) remains valid.)

Also pure rotation belongs to the "preferred motions". Here it follows

11) L=C+\frac{p\vartheta^{2}}{2}, where C, p are constants.

Thus angular momentum becomes

11a) M=p\cdot\vartheta,

analogous to a material rigid body; for p, the "electromagnetic moment of inertia", one obtains

11b) p=\frac{1}{7}\mu_{0}a^{2},

while with respect to a sphere homogeneously filled with a material mass M, the moment of inertia is P=\tfrac{2}{5}\cdot M\cdot a^{2}

\left\{ \mathrm{With\ surface\ charge\ one\ obtains}\ p=\tfrac{1}{3}\mu_{0}a^{2},\ P=\tfrac{2}{3}Ma^{2}\right\}.

At pure rotation, nothing remarkable occurs. At slow translatory motion, one only commits an error of order \beta^2, when one sets \mathfrak{G}=\mu_{0}\mathfrak{q}, \mathfrak{M}=p\vartheta. Angular forces occur at the passage of cathode rays through inhomogeneous fields. Though the angular force is very small, since the electron's radius a is so small; thus according to my estimation, the energy in the angular motion emerging in the fields of the greatest attainable inhomogeneity, only amounts the 10^{-23th} part of the translatory energy. At Becquerel rays, the consideration of rotation is far more difficult; here, translatory and rotatory motion is, so to speak, coupled. I have confined myself to treat rotation around the translatory direction, a motion belonging to the preferred ones. The general rotation problem of large velocities is not solved for the time being; I consider its treatment as of little value, since the difficulties are considerable, and since nothing forces us thus far, to assume an essential role of rotation with respect to Becquerel rays.

Coming to the end, I summarize the results. The problem of the dynamics of the electron is the simplest problem of electromagnetic mechanics. The electron moving translatory only, corresponds to a material point, the rotating one corresponds to the rigid body of ordinary mechanics. Also, we have made the assumptions on shape and charge distribution of the electron as simple as possible – exactly the simplest assumptions are in agreement with experiment –, nevertheless the dynamics of the electron is far more complicated than the corresponding problems of ordinary mechanics. Only for a special class of motions, for "preferred motions", it was possible to deduce the Lagrangian equations in the form known from analytical mechanics; to such motions, also that formulation of Lagrangian mechanics applies which is called "Hamiltonian principle". If the applicability of analytical mechanics is restricted in some way, it again experiences an essential extension in other ways. Because ordinary mechanics of material bodies is related to very small velocities, while the dynamics of the electron is valid close up to the speed of light. Also here, the Lagrangian system of mechanics is proved to be true; but the ones applying to electromagnetic mechanics, are more complicated types of the Lagrangian function, types, that pass (at slow motion) to those considered as valid in ordinary mechanics.

This extension of the realm of analytical mechanics is confined to the ordinary, three-dimensional, euclidean space. It is the only one that takes into account (and by that it is preferred over the other proposed extensions) those physical properties of space, that find their mathematical expression in the Maxwell-Hertz differential equation. It may be emphasized that our theory still assumes continuous space-occupation of the aether, i.e. exact validity of those differential equations, for distances that are small against the radius of the electron, i.e. against a trillionth of a millimeter, and for field strengths that trillion-times exceed the ones accessible to our measurement. The agreement of the theoretical results with the experimental results of Kaufmann demonstrates the justification of this assumption. Thus: atomistic structure of electricity, but continuous space-occupation of the aether! That shall be our solution!

(Self-lecture of the reader.)


Planck (Berlin): Anyone, who was engaged with those things, will be satisfied that both gentlemen have succeeded in solving this difficult question in a principally simple way. When all of this is confirmed this way, then we may hope, that those investigations will be connected with an essential advancement of electrodynamics. Of the many questions, that are excited by this lecture, only two I want to pose to the reader. The first question is related to the meaning of these things for electrodynamics as a whole. Those statements are only of importance for Lorentz's theory, as they are based on Lorentz's equations throughout. Now it is known, that also other fundamental equations exist, that claim to be in agreement with facts, the equations of Cohn for instance. The execution of the calculations would probably be very hard, but still interesting, because one can probably find out by that, whether the theory of Cohn is still admissible at all.

The second question is as follows: It would be interesting for me, to learn, within which limits a quasi-stationary state can still be used as a basis. I would like to know, by which way one can learn more details about that.

Abraham: With respect to the first question I have to remark: certainly it would be important to investigate, whether Cohn's theory explains the facts, but this would be the task of the originator of the theory. I see the advantage of Lorentz's theory mainly in the fact, that one can derive the electromagnetic momentum from it. This simplifies the mathematical treatment and makes the analogy to ordinary mechanics possible. With respect to the second question I have made the following consideration. At rapid current fluctuations, Hertz's oscillations for instance, emitted energy comes into consideration; then it is not allowed anymore to calculate with a quasi-stationary current. The matter is similar with respect to rapidly accelerated motion of electrons. The radiation of energy and momentum can be calculated by the aid of the Wiechert-Des Coudres point theorem in the Lorentz-Festschrift. I have executed the calculation, and from that I found the basis for the determination of the area of applicability of quasi-stationary motion.

Meyer (Königsberg): I have a concern that you can maybe solve without further ado. These double integrals indeed occur very often. There, you have an ideal surface, that is moved into infinity by you, and then you say that the integral becomes equal to zero.

Now I would like to ask, whether the proof, that the integral vanishes when passing the limit, was quite easy. It was alluded to the fact, that at this place, physicists sometimes calculate not precisely and make things easy for them. Neumann has considered it necessary, to make a development with respect to spherical functions, in the cases where he wanted to prove it. Now, is the proof so simple in your case, that the integral vanishes when passing the limit?

Abraham: The assumption, that the boundary surface of the field lies in infinity, becomes irrelevant, when, for instance, the motion takes place very close to a conducting surface, as for example when cathode rays are reflected. If such cases are excluded, then it is allowed to treat the problem, as if the electron would be located alone in space. The proof, that the considered surface integrals then vanish, is not very hard, as long as one only offers a certain physical understanding for those things.