Page:Grundgleichungen (Minkowski).djvu/20

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and the equations (III) and (IV), are

(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}

§ 8. The Fundamental Equations for Moving Bodies.

We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively.

The first Axion shall be, —

When a detached region of matter is at rest at any moment, therefore the vector \mathfrak{w} is zero, for a system x, y, z, t — the neighbourhood may be supposed to be in motion in any possible manner, then for the spacetime point x, y, z, t the same relations (A) (B) (V) which hold in the case when all matter is at rest, snail also hold between \varrho, the vectors \mathfrak{s,e,m,E,M} and their differentials with respect to x, y, z, t.

The second axiom shall be : —

Every velocity of matter is < 1, smaller than the velocity of propagation of light.

The third axiom shall be : —

The fundamental equations are of such a kind that when x, y, z, it are subjected to a Lorentz transformation and thereby \mathfrak{m},\ -i\mathfrak{e} and \mathfrak{M},\ -i\mathfrak{E} are transformed into space-time vectors of the second kind, \mathfrak{s},\ i\varrho as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.

Shortly I can signify the third axiom as ; —

\mathfrak{m},\ -i\mathfrak{e} and \mathfrak{M},\ -i\mathfrak{E} are space-time vectors of the second kind, \mathfrak{s},\ i\varrho is a space-time vector of the first kind.

This axiom I call the Principle of Relativity.