# Page:Grundgleichungen (Minkowski).djvu/6

i.e. the components of the convection current $\varrho\mathfrak{w}$, and the electric density multiplied by $\sqrt{-1}$.

Further I shall write

$f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$

for

$\mathfrak{m}_{x},\ \mathfrak{m}_{y},\ \mathfrak{m}_{z},\ -i\mathfrak{e}_{x},\ -i\mathfrak{e}_{y},\ -i\mathfrak{e}_{z}$,

i.e., the components of $\mathfrak{m}$ and $-i\mathfrak{e}$ along the three axes; now if we take any two indices h, k out of the series

$f_{kh} = -f_{hk}$,

therefore

 $f_{32} = -f_{23},\ f_{13} = -f_{31},\ f_{21} = -f_{12}$, $f_{41} = -f_{14},\ f_{42} = -f_{24},\ f_{43} = -f_{34}$,

Then the three equations comprised in (I), and the equation (II) multiplied by i becomes

 (A) $\begin{array}{ccccccccc} & & \frac{\partial f_{12}}{\partial x_{2}} & + & \frac{\partial f_{13}}{\partial x_{3}} & + & \frac{\partial f_{14}}{\partial x_{4}} & = & \varrho_{1},\\ \\\frac{\partial f_{21}}{\partial x_{1}} & & & + & \frac{\partial f_{23}}{\partial x_{3}} & + & \frac{\partial f_{24}}{\partial x_{4}} & = & \varrho_{2},\\ \\\frac{\partial f_{31}}{\partial x_{1}} & + & \frac{\partial f_{32}}{\partial x_{2}} & & & + & \frac{\partial f_{34}}{\partial x_{4}} & = & \varrho_{3},\\ \\\frac{\partial f_{41}}{\partial x_{1}} & + & \frac{\partial f_{42}}{\partial x_{2}} & + & \frac{\partial f_{43}}{\partial x_{3}} & & & = & \varrho_{4}.\end{array}$

On the other hand, the three equations comprised in (III) multiplied by -i, and equation (IV) multiplied by -1, become

 (B) $\begin{array}{ccccccccc} & & \frac{\partial f_{34}}{\partial x_{2}} & + & \frac{\partial f_{42}}{\partial x_{3}} & + & \frac{\partial f_{23}}{\partial x_{4}} & = & 0,\\ \\\frac{\partial f_{43}}{\partial x_{1}} & & & + & \frac{\partial f_{14}}{\partial x_{3}} & + & \frac{\partial f_{31}}{\partial x_{4}} & = & 0,\\ \\\frac{\partial f_{24}}{\partial x_{1}} & + & \frac{\partial f_{41}}{\partial x_{2}} & & & + & \frac{\partial f_{12}}{\partial x_{4}} & = & 0,\\ \\\frac{\partial f_{32}}{\partial x_{1}} & + & \frac{\partial f_{13}}{\partial x_{2}} & + & \frac{\partial f_{21}}{\partial x_{3}} & & & = & 0.\end{array}$

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations