Page:Das Relativitätsprinzip und seine Anwendung.djvu/5

Thus one can treat the phenomena of motion in two different ways, either with the Minkowski- or with the Newtonian force. In the latter case, the equations of motions read:

${\mathfrak {F}}=m_{1}{\mathfrak {j}}_{1}+m_{2}{\mathfrak {j}}_{2}{,}$

here, ${\mathfrak {j}}_{1}$ means the ordinary acceleration into the direction of motion, and ${\mathfrak {j}}_{2}$ the ordinary normal acceleration, and the factors

${\begin{array}{rl}m_{1}&={\frac {m}{\sqrt {\left(1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}\right)^{3}}}},\\&\\m_{2}&={\frac {m}{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}}\end{array}}$

are called the "longitudinal" and "transverse mass".

In the same way as the Minkowski force, also the Newtonian forces occurring in nature must satisfy certain conditions, when the relativity principle should be satisfied. This is e.g. the case, when (independently from motion) a normal pressure of constant magnitude $p$ is acting per unit area; in the transformed system, a normal pressure of the same magnitude is acting upon the corresponding surface element in motion.

Since we already recognized the invariance of the field equations, then the question, as to whether the motions in a system of electrons are in agreement with the relativity principle, only tantamounts to the experimental test of the formulas for longitudinal and transverse mass $m_{1},\ m_{2}$; although the experiments of Bucherer and Hupka seem to confirm these formulas, we have not arrived at a definite decision.

Regarding the mass of the electron, it is to be considered that they are of electromagnetic nature; thus it will depend on the distribution of the charges in the interior of the electron. Therefore, the formulas for the mass can only then be correct, when the charge distribution and thus also the shape of the electron are variable with velocity in a certain way. One has to assume, that in consequence of the translation of the electron, which is a sphere when at rest, the electron becomes an oblate ellipsoid in the direction of motion; the amount of oblateness is

${\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}.$

If we assume, that shape and magnitude of the electron are regulated by inner forces, then they must (to be compatible with the relativity principle) have such properties by which this oblateness