Page:Prinzipien der Dynamik des Elektrons.djvu/3

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

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

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,