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