Page:Grundgleichungen (Minkowski).djvu/37

 For this matrix I shall use the shortened from lor.

Then if S is, as in (62), a space-time matrix of the II. kind, by lor S' will be understood the 1✕4 series matrix

$$\left|K_{1},\ K_{2},\ K_{3},\ K_{4}\right|$$

where

When by a transformation $$\mathsf{A}$$, a new reference system $$x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$$ is introduced, we can use the operator

$$lor'=\left|\frac{\partial}{\partial x'_{1}},\ \frac{\partial}{\partial x'_{2}},\ \frac{\partial}{\partial x'_{3}},\ \frac{\partial}{x'_{4}}\right|$$

Then S is transformed to $$S'=\bar{\mathsf{A}}S\mathsf{A}=\left|S'_{hk}\right|$$, so by lor' Sis meant the 1✕4 series matrix, whose element are

$$K'_{k}=\frac{\partial S'_{1k}}{\partial x'_{1}}+\frac{\partial S'_{2k}}{\partial x'_{2}}+\frac{\partial S'_{3k}}{\partial x'_{3}}+\frac{\partial S'_{4k}}{\partial x'_{4}}\qquad (k=1,2,3,4)$$

Now for the differentiation of any function of (x y z t) we have the rule

so that, we have symbolically

$$lor'=lor\ (\mathsf{A}$$

Therefore it follows that

i.e., lor S behaves like a space-time vector of the first kind.

If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements