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 *w*.

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

### § 12. The Differential Operator Lor.

A 4✕4 series matrix

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

- 1) the alternating matrix
*f*, which corresponds to the space-time vector of the II. kind, —

- 2) the product
*fF*of two such matrices, for by a transformation , it is replaced by ,

- 3) further when and are two space-time vectors of the 1st kind, the 4✕4 matrix with the ,

- 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 employ the 1✕4 *series matrix, formed of differential symbols*, —

or

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