Page:Grundgleichungen (Minkowski).djvu/42

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

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

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, —

Here we have

The right side of (82) as well as L is an invariant in a 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),

Then is