Page:The principle of relativity (1920).djvu/114

 Now by comparing with Φ = ?]-ωF, the relation (E) can be brought into the form

{E} s + (ω[=s])ω = - σωF,

This formula contains four equations, of which the fourth follows from the first three, since this is a space-time vector which is perpendicular to ω.

Lastly, we shall transform the differential equations (A) and (B) into a typical form.

§12.

A 4 × 4 series matrix 62) S = | S_{11} S_{12} S_{13} S_{14} | = | S_{kh} |
 * S_{21} S_{22} S_{23} S_{24} |
 * S_{31} S_{32} S_{33} S_{34} |
 * S_{41} S_{42} S_{43} S_{44} |

with the condition that in case of a Lorentz transformation it is to be replaced by ĀSA, may be called a space-time matrix of the II kind. We have examples of this in:—

1) the alternating matrix [function], which corresponds to the space-time vector of the II kind,—

2) the product [function]F of two such matrices, for by a transformation A, it is replaced by (A^{-1}[function]A·A^{-1}FA) = A^{-1}[function]FA,

3) further when (ω_{1}, ω_{2}, ω_{3}, ω_{4}) and (Ω_{1}, Ω_{2}, Ω_{3}, Ω_{4}) are two space-time vectors of the 1st kind, the 4 × 4 matrix with the element S_{hk} = ω_{h}Ω_{k},

lastly in a multiple L of the unit matrix of 4 × 4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero.

We shall have to do constantly with functions of the space-time point (x, y, z, it), and we may with advantage