Page:Grundgleichungen (Minkowski).djvu/6

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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