Page:Prinzipien der Dynamik des Elektrons.djvu/4

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:

$\mu _{s}=\mu _{r}=\mu _{0}={\frac {4}{5}}{\frac {e^{2}}{ac^{2}}},$

the experiment gives:

${\frac {|e|}{c\mu _{0}}}=1,865\cdot 10^{7}.$

Thus we obtain

$a={\frac {4}{5}}\cdot {\frac {|e|}{c}}\cdot 1,865\cdot 10^{7}.$

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