Page:Grundgleichungen (Minkowski).djvu/15

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(23) \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 f_{23},\dots f_{34}. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors

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

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

(24) \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 f'_{23},\dots f'_{34} depend solely on f_{23},\dots f_{34} and the coefficients \alpha_{11},\ \alpha_{12},\dots \alpha_{44}.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes 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 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 \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 \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 \varrho\mathfrak{w},\ i\varrho is to be transformed as a space time vector of the 1st kind, \mathfrak{m},\ -i\mathfrak{e} is to be treated as a vector of the 2nd kind, or more significantly, —

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