# Page:Grundgleichungen (Minkowski).djvu/15

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 (23) ${\displaystyle {\begin{array}{c}f_{23}(x_{2}u_{3}-x_{3}u_{2})+f_{31}(x_{3}u_{1}-x_{1}u_{3})+f_{12}(x_{1}u_{2}-x_{2}u_{1})\\\\+f_{14}(x_{1}u_{4}-x_{4}u_{1})+f_{24}(x_{2}u_{4}-x_{4}u_{2})+f_{34}(x_{3}u_{4}-x_{4}u_{3})\end{array}}}$

with six coefficients ${\displaystyle f_{23},\dots f_{34}}$. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors

${\displaystyle x_{1},\ x_{2},\ x_{3};u_{1},\ u_{2},\ u_{3};f_{23},\ f_{31},\ f_{12};f_{14},\ f_{24},\ f_{34}}$

and the constants ${\displaystyle x_{4}}$ and ${\displaystyle u_{4}}$ at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject ${\displaystyle x_{1},\ x_{2},\ x_{3},\ x_{4}}$ and ${\displaystyle u_{1},\ u_{2},\ u_{3},\ u_{4}}$ simultaneously to the Lorentz transformation (21), the combination (23) is changed to.

 (24) ${\displaystyle {\begin{array}{c}f'_{23}(x'_{2}u'_{3}-x'_{3}u'_{2})+f'_{31}(x'_{3}u'_{1}-x'_{1}u'_{3})+f'_{12}(x'_{1}u'_{2}-x'_{2}u'_{1})\\\\+f'_{14}(x'_{1}u'_{4}-x'_{4}u'_{1})+f'_{24}(x'_{2}u'_{4}-x'_{4}u'_{2})+f'_{34}(x'_{3}u'_{4}-x'_{4}u'_{3})\end{array}}}$

where the coefficients ${\displaystyle f'_{23},\dots f'_{34}}$ depend solely on ${\displaystyle f_{23},\dots f_{34}}$ and the coefficients ${\displaystyle \alpha _{11},\ \alpha _{12},\dots \alpha _{44}}$.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes ${\displaystyle f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}}$ with the condition that when subjected to a Lorentz transformation, it is changed to a new system ${\displaystyle f'_{23},\ f'_{31},\ f'_{12},\ f'_{14},\ f'_{24},\ f'_{34}}$ which satisfies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I) — (IV), — which are the fundamental equations for Æther.

If x, y, z, it (space co-ordinates, and time it) is subjected to a Lorentz transformation, and at the same time ${\displaystyle \varrho {\mathfrak {w}}_{x},\ \varrho {\mathfrak {w}}_{y},\ \varrho {\mathfrak {w}}_{z},\ i\varrho }$ (convection-current, and charge density × i) is transformed as a space time vector of the 1st kind, further ${\displaystyle {\mathfrak {m}}_{x},\ {\mathfrak {m}}_{y},\ {\mathfrak {m}}_{z},\ -i{\mathfrak {e}}_{x}\ -i{\mathfrak {e}}_{y},\ -i{\mathfrak {e}}_{z}}$ (magnetic force, and electric induction × i) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced info the system.

These facts can be more concisely expressed in these words: the system of equations (I, and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where ${\displaystyle \varrho {\mathfrak {w}},\ i\varrho }$ is to be transformed as a space time vector of the 1st kind, ${\displaystyle {\mathfrak {m}},\ -i{\mathfrak {e}}}$ is to be treated as a vector of the 2nd kind, or more significantly, —

${\displaystyle \varrho {\mathfrak {w}},\ i\varrho }$ is a space time vector of the 1st kind, ${\displaystyle {\mathfrak {m}},\ -i{\mathfrak {e}}}$ is a space-time vector of the 2nd kind.