Page:Zur Elektrodynamik bewegter Systeme I.djvu/5

${\begin{cases}P=(1,k^{2},k^{2})\eta \{E+[wH]\}\\J=\left({\frac {1}{k}},k,k\right)\sigma \{E+[wH]\}\end{cases}}$

(L2)

or written in a different manner:

$\left.{\begin{matrix}P=(\eta )\{E+[wH]\}&&(\eta )=\left(1,\ k^{2},\ k^{2}\right)\eta \\&where\\P=(\sigma )\{E+[wH]\}&&(\sigma )=\left(1,\ k,\ k\right)\sigma \end{matrix}}\right\}$

(L'2)

The values from (L'₂) are to be included into (L'₁). The differential equations for $E$ and $H$ with given coefficients, will follow. These equations have, as independent variables, the coordinates $xyz$ and the time $t$.

Instead of them, we will introduce new variables $x_{0}\dots t_{0}$ by the equations

$x_{0}=kx;\ y_{0}=y;\ z_{0}=z;\ t_{0}={\frac {t}{k}}.$

(L3)

Furthermore, we denote by $\rho _{0}$ the magnitude ${\tfrac {\rho }{k}}={\tfrac {de}{dx_{0}\cdot dy_{0}\cdot dz_{0}}}$, as well as by $P_{0}$ and $\Gamma _{0}$ the operators, which formally correspond to $P$ and $\Gamma$ in the system of the new variables. Eventually, we define the two new vectors E and M by:

${\begin{cases}{\mathsf {E}}=(1,k,k)\{E+[wH]\}\\{\mathsf {M}}=(1,k,k)\{H-[wE]\}\end{cases}}$

(L4)

from which it conversely follows due to (1)

${\begin{cases}E=(1,k,k)\{{\mathsf {E}}-[wM]\}\\H=(1,k,k)\{{\mathsf {M}}-[wE]\}\end{cases}}$

(L'4)

Then it arises:

${\begin{cases}&{\mathsf {P}}_{0}(M)=\sigma {\mathsf {E}}+{\frac {d{\mathfrak {E}}}{dt_{0}}}\\&{\mathsf {P}}_{0}({\mathsf {E}})=-{\frac {d{\mathfrak {M}}}{dt_{0}}}\\&\Gamma _{0}({\mathfrak {E}})=\rho _{0;}\ \Gamma _{0}({\mathfrak {M}})=0\\\\\mathrm {where} \\&{\mathfrak {E}}=(1+\eta ){\mathsf {E}}-[w{\mathsf {M}}]\\&{\mathfrak {M}}={\mathsf {M}}+[w{\mathsf {E}}]\end{cases}}.$.

(C')

From (C') the following can be derived: E and M are functions of $x_{0}\dots t_{0}w$, which contain $t_{0}$ and the translation velocity $w$ only in connection with:

$t_{1}=t_{0}-(w\cdot r_{0})$