# Page:Grundgleichungen (Minkowski).djvu/40

### § 13. The Product of the Field-vectors fF.

Finally let us enquire about the laws which lead to the determination of the vector w as a function of x, y, z, t. In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

${\displaystyle f=\left|{\begin{array}{cccc}0,&f_{12},&f_{13},&f_{14}\\f_{21},&0,&f_{23},&f_{24}\\f_{31},&f_{32},&0,&f_{34}\\f_{41},&f_{42},&f_{43},&0\end{array}}\right|,\ F=\left|{\begin{array}{cccc}0,&F_{12},&F_{13},&F_{14}\\F_{21},&0,&F_{23},&F_{24}\\F_{31},&F_{32},&0,&F_{34}\\F_{41},&F_{42},&F_{43},&0\end{array}}\right|}$

are of much importance. Let us write.

 (70) ${\displaystyle f\ F=\left|{\begin{array}{llll}S_{11}-L,&S_{12},&S_{13},&S_{14}\\S_{21},&S_{22}-L,&S_{23},&S_{23}\\S_{31},&S_{32},&S_{33}-L,&S_{34}\\S_{41},&S_{42},&S_{43},&S_{44}-L\end{array}}\right|}$

Then (71)

 (71) ${\displaystyle S_{11}+S_{22}+S_{33}+S_{44}=0}$

Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by

 (72) ${\displaystyle L={\frac {1}{2}}(f_{23}F_{23}+f_{31}F_{31}+f_{12}F_{12}+f_{14}F_{14}+f_{24}F_{24}+f_{34}F_{34})}$

Then we shall have

 (73) ${\displaystyle {\begin{array}{c}S_{11}={\frac {1}{2}}(f_{23}F_{23}+f_{34}F_{34}+f_{42}F_{42}-f_{12}F_{12}-f_{13}F_{13}-f_{14}F_{14})\\S_{12}=f_{13}F_{32}+f_{14}F_{42},\ u.s.f.\end{array}}}$

In order to express in a real form, we write

 (74) ${\displaystyle S=\left|{\begin{array}{cccc}S_{11},&S_{12},&S_{13},&S_{14}\\S_{21},&S_{22},&S_{23},&S_{23}\\S_{31},&S_{32},&S_{33},&S_{34}\\S_{41},&S_{42},&S_{43},&S_{44}\end{array}}\right|=\left|{\begin{array}{cccc}X_{x},&Y_{x},&Z_{x},&-iT_{x}\\X_{y},&Y_{y},&Z_{y},&-iT_{y}\\X_{z},&Y_{z},&Z_{z},&-iT_{z}\\-iX_{t},&-iY_{t},&-iZ_{t},&T_{t}\end{array}}\right|}$

Now