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

(80) |

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

(81) |

*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, —

(82) | , |

(83) |

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),

(84) | . |

Then is