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

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),

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:

$P=(1,\ k,\ k)P_0$

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

then consequently it follows for the translation: