Page:Zur Elektrodynamik bewegter Systeme II.djvu/11

The forces which exert this work, consist of the translational force

$f_{1}=-{\frac {\overline {d\Sigma }}{dt}}$

(26)

and of a system of deformation forces, which are entirely in agreement with Maxwell's tensions. They can be decomposed into a universal tension

$q=-{\frac {1}{2}}(\epsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})$

(27a)

besides the tensions

$q_{ik}=+(\epsilon {\mathsf {E}}_{i}{\mathsf {E}}_{k}+\mu {\mathsf {M}}_{i}{\mathsf {M}}_{k})$.

(27b)

The motions of the material particles are thus determined by the equivalent system of translational forces $f$, whose components are:

$f_{x}=f_{1x}+{\frac {\partial q}{\partial x}}+{\frac {\partial q_{xx}}{\partial x}}+{\frac {\partial q_{xy}}{\partial y}}+{\frac {\partial q_{xz}}{\partial z}}$; etc.[1]

(28)

If one substitutes the values of (26) and (27), then one obtains

${\begin{matrix}f=-{\frac {d\Sigma }{dt}}-[\epsilon {\mathsf {E}}\cdot {\mathsf {P(E)}}]+\Gamma (\epsilon {\mathsf {E}})\cdot {\mathsf {E}}-{\frac {1}{2}}{\mathsf {E}}^{2}\cdot \nabla \epsilon \\\\-[u{\mathsf {M}}\cdot {\mathsf {P(M)}}+\Gamma (\mu {\mathsf {M}})\cdot {\mathsf {M}}-{\frac {1}{2}}{\mathsf {M}}^{2}\cdot \nabla \mu \end{matrix}}$

(27)

This is the most general expression for the forces.

We notice at first, that it applies for vacuum:

$u=0$, $\lambda$=0, $\epsilon =\mu =1$ and thus ${\mathfrak {E}}={\mathsf {E}},\ {\mathfrak {M}}={\mathsf {M}},\ \Lambda =0$;

furthermore $\Gamma {\mathsf {E}}=\Gamma {\mathsf {M}}=0$. Thus the last four terms in (29) individually vanish, however, the three first ones give zero according to I' and II'. Force $f$ is thus identical with zero at all space locations, where a material substrate of forces is unknown to us. This theorem is a logical postulate, as long as one doesn't ad hoc substitute (into vacuum) a medium with properties of matter. On the other side, this follows from our equations only be means of presupposing $u=0$. Thus one can conceptually define the reference system, for which the fundamental equations are valid, so that it is at rest with respect to empty space. By that, however, not the least is gained for the representation of experience.

1. As regards the recent developments, see Lorentz, Math. Enc. V, p. 251 ff. However, it is to be noticed, that Lorentz has neglected terms with $u^{2}$ throughout.