Page:Zur Elektrodynamik bewegter Systeme II.djvu/12

Since we measure electromagnetic forces with respect to bodies which are at rest with respect to Earth or moving slowly at least, the value which $f$ assumes for $u=p=const.$ is most important. We obtain it in the most vivid form, by again introducing local time $t'$ by means of (6). Then, it follows from (29) by means of (7), or more simple by means of:

${\frac {\partial }{\partial x}}={\frac {\partial }{\partial x'}}-p_{x}{\frac {d}{dt'}}$ etc.

and

${\frac {\bar {d}}{dt}}={\frac {d}{dt}}={\frac {d}{dt'}}$

directly from (28):

$f_{x}=-{\frac {d\Sigma _{x}}{dt'}}-[\epsilon {\mathsf {E}}\cdot {\mathsf {P'(E)}}]_{x}+\Gamma '(\epsilon {\mathsf {E}})\cdot {\mathsf {E}}_{x}-{\frac {1}{2}}{\mathsf {E}}^{2}{\frac {\partial \epsilon }{\partial x'}}$

$-[\mu {\mathsf {M\cdot P'(M)}}]_{x}+\Gamma '(\mu {\mathsf {M}})\cdot {\mathsf {M}}_{x}-{\frac {1}{2}}{\mathsf {M}}^{2}{\frac {\partial \mu }{\partial x'}}$

$-p_{x}{\frac {d}{dt'}}(q+q_{xx})-p_{y}{\frac {d}{dt'}}(q_{xy})-p_{z}{\frac {d}{dt'}}(q_{xz})$.

Here, it is by I'b to IIIb:

$-{\mathsf {P'(E)}}=\mu {\frac {d{\mathsf {M}}}{dt'}}$

${\mathsf {P'(M)}}=\epsilon {\frac {d{\mathsf {E}}}{dt'}}+\Lambda$;

thus it follows:

${\begin{matrix}f_{x}=f_{0x}+{\frac {d}{dt'}}\left\{(\epsilon u-1)\Sigma _{x}+p_{x}\left({\frac {1}{2}}\left(\epsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2}\right)-\left(\epsilon {\mathsf {E}}_{x}^{2}+\mu {\mathsf {M}}_{x}^{2}\right)\right)\right.\\\\\left.-p_{y}(\epsilon {\mathsf {E}}_{x}{\mathsf {E}}_{y}+\mu {\mathsf {M}}_{x}{\mathsf {M}}_{y})-p_{z}(\epsilon {\mathsf {E}}_{x}{\mathsf {E}}_{z}+\mu {\mathsf {M}}_{x}{\mathsf {M}}_{z})\right\}\end{matrix}}$

(30)

where

$f_{0}=\Gamma '(\epsilon {\mathsf {E}})\cdot {\mathsf {E}}-{\frac {1}{2}}{\mathsf {E}}^{2}\cdot \nabla '\epsilon +\Gamma '(\mu {\mathsf {M}})\cdot {\mathsf {M}}-{\frac {1}{2}}{\mathsf {M}}^{2}\cdot \nabla '\mu +[\Lambda \cdot \mu {\mathsf {M}}]$

(31)

The value in (31), considered as a function of relative coordinates and local time, doesn't explicitly depend on $p$; but also not implicitly, since according to § 3, also E, M and $\Lambda$ are functions (independent from P) of the same four variables. With respect to stationary states it becomes $f=f_{0}$. Furthermore, it is irrelevant for the representation of these states, as to whether we use local time or general time. Thus the approach is given: the forces of the stationary field in relatively resting bodies are in all rigor independent from Earth's motion. It additionally gives the amount of these forces in the well-known form, which forms the expression of all certain experiences.