only amounts the 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.)
Discussion.
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
