Page:Zur Elektrodynamik bewegter Systeme I.djvu/4

3. The motions which are carried out by a material particle of the progressing system under the action of forces $F$ in space $r$, are different from the motions, which the same particle in the case of rest is carrying out under the forces $F_{0}$ in space $r_{0}$, only by the fact that the process is slowed down in a constant ratio. Corresponding distances are traversed in times $t$ and $t_{0}$, which are connected by the equation

$t=kt_{0}\,$

(4)

If (with respect to a certain particle) $r_{0}\ t_{0}\ F_{0}$ forms a connected system of distances, times, forces in the case of rest, then $rtF$ – being in agreement with equations (1) to (4) – also belong together as a system of values that represent a possible state in the case of translation.

We apply the following theorems: in the theory of electrons it is given by definition, that

the "electric force" $E+[wH]$ is the spatial average of the force upon a particle, charged by the quantity of electricity 1;

the "electric moment of unit volume" $P={\tfrac {\Sigma (er)}{\tau }}$, where $r$ is the relative displacement of $e$, $\tau$ denotes a volume, and the sum is to be extended over all $e$ in this volume;

the "convection current" $J={\tfrac {\Sigma (eu)}{\tau }}$, where $u$ is the relative velocity of $e$.

Now, let $E_{0}\ P_{0}\ J_{0}$ be the connected values in the case of rest. According to (3),

$E+[wH)=\left(1,\ {\frac {1}{k}},\ {\frac {1}{k}}\right)E_{0}$

are corresponding to them for $e=e_{0}$ in the case of translation; furthermore, since $\tau ={\tfrac {\tau _{0}}{k}}$ according to (2), and $u=\left({\frac {1}{k^{2}}},\ {\frac {1}{k}},\ {\frac {1}{k}}\right)u_{0}$ according to (2) and (4), we have:

$P=(1,\ k,\ k)P_{0}$

$J=\left({\frac {1}{k}},1,1\right)J_{0}$.

Thus, if it is given for the state of rest:

$P_{0}=\eta E_{0}\,$ $J_{0}=\sigma E_{0}\,$

then consequently it follows for the translation: