# Page:Grundgleichungen (Minkowski).djvu/42

and have the value given on the right-hand side of (79). Therefore the general relations

 (80) $S_{h1}S_{1k} + S_{h2}S_{2k} + S_{h3}S_{3k} + S_{h4}S_{4k} = 0$

h, k being unequal indices in the series 1, 2, 3, 4, and

 (81) $S_{h1}S_{1h} + S_{h2}S_{2h} + S_{h3}S_{3h} + S_{h4}S_{4h} = L^{2} - Det^{\frac{1}{2}}f Det^{\frac{1}{2}}F$

for h = 1,2,3,4.

Now if instead of F and f in the combinations (72) and (73), we introduce the electrical rest-force $\Phi$, the magnetic rest-force $\Psi$ and the rest-ray $\Omega$ [(55), (56) and (57)], we can pass over to the expressions, —

 (82) $L=-\frac{1}{2}\epsilon\Phi\bar{\Phi}+\frac{1}{2}\mu\Psi\bar{\Psi}$,
 (83) $S_{hk}=-\frac{1}{2}\epsilon\Phi\bar{\Phi}e_{hk}-\frac{1}{2}\mu\Psi\bar{\Psi}e_{hk}$ $+\epsilon(\Phi_{h}\Phi_{k}-\Phi\bar{\Phi}w_{h}w_{k})+\mu(\Psi_{h}\Psi_{k}-\Psi\bar{\Psi}w_{h}w_{k})$ $-\Omega_{h}w_{k}-\epsilon\mu w_{h}\Omega_{k}\qquad\qquad (h, k = 1,2,3,4)$

Here we have

 $\Phi\bar{\Phi}=\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}+\Phi_{4}^{2},\ \Psi\bar{\Psi}=\Psi_{1}^{2}+\Psi_{2}^{2}+\Psi_{3}^{2}+\Psi_{4}^{2}$, $e_{hh}=1,\ e_{hk}=0(h\gtrless k)$.

The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4✕4 element on the right side of (83) as well as $S_{hk}$, represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case $w_{1} = 0,\ w_{2} = 0,\ w_{3} = 0,\ w_{4} = i$. But for this case $\mathfrak{w} = 0$, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and $\mathfrak{e}=\epsilon\mathfrak{E},\ \mathfrak{M}=\mu\mathfrak{m}$ on the other hand.

The expression on the right-hand side of (81), which equals

$=\left(\frac{1}{2}(\mathfrak{mM}-\mathfrak{eE})\right)^{2}+(\mathfrak{em})(\mathfrak{EM})$

is $\geqq0$, because $(\mathfrak{em})=\epsilon\Phi\bar{\Psi},\ (\mathfrak{EM})=\mu\Phi\bar{\Psi}$, now referring back to 79), we can denote the positive square root of this expression as $Det^{\frac{1}{2}}S$.

Since $\bar{f}=-f,\ \bar{F}=-F$, we obtain for $\bar{S}$, the transposed matrix of S, the following relations from (78),

 (84) $Ff=\bar{S}-L,\ f^{*}F^{*}=-\bar{S}-L$.

Then is