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 , the magnetic rest-force and the rest-ray [(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 Lorentz transformation, and the 4✕4 element on the right side of (83) as well as , 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 . But for this case , we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and on the other hand.
The expression on the right-hand side of (81), which equals
is , because , now referring back to 79), we can denote the positive square root of this expression as .
Since , we obtain for , the transposed matrix of S, the following relations from (78),