Page:LorentzGravitation1915.djvu/10

 § 10. We shall define a varied motion of the electric charges by the quantities $$\delta x_{a}$$ and we shall also vary the quantities $$\psi_{ab}$$, so far as can be done without violating the connections (25) and (26). The possible variations $$\delta\psi_{ab}$$ may be expressed in $$\delta x_{a}$$ and four other infinitesimal quantities $$q_{a}$$ which we shall presently introduce. Our condition will be that equation (12) shall be true if, leaving the gravitation field unchanged, we take for $$\delta x_{a}$$ and $$q_{a}$$ any continuous functions of the coordinates which vanish at the limits of the domain of integration. We shall understand by $$\delta w_{a}$$, $$\delta\psi_{ab}$$, $$\delta\mathrm{L}$$ the variations at a fixed point of this space. The variations $$\delta w_{a}$$ are again determined by (13) and (14), and we have, in virtue of (26) and (25),

$\delta\psi_{aa}=0,\ \delta\psi_{ba}=-\delta\psi_{ab},\ \Sigma(b)\frac{\partial\delta\psi_{ab}}{\partial x_{b}}=\delta w_{a}=\Sigma(b)\frac{\partial\chi_{ab}}{\partial x_{b}}.$|undefined

If therefore we put

we must have

$\vartheta_{aa}=0,\ \vartheta_{ba}=-\vartheta_{ab},\ \Sigma(b)\frac{\partial\vartheta_{ab}}{\partial x_{b}}=0.$|undefined

It can be shown that quantities $$\vartheta_{ab}$$ satisfying these conditions may be expressed in terms of four quantities $$q_{a}$$ by means of the formulae

Here $$a'$$ and $$b'$$ are the numbers that remain when of 1, 2, 3, 4 we omit $$a$$ and $$b$$, the choice of the value of $$a'$$ and that of $$b'$$ being such that the order $$a, b, a', b'$$ can be derived from the order 1, 2, 3, 4 by an even number of permutations each of two numbers.

§ 11. By (31), (36) and (37) we have

Here, in the transformation of the first term on the right hand side it is found convenient to introduce a new notation for the quantities $$\bar{\psi}_{ab}$$. We shall put

$\bar{\psi}_{ab}=\psi_{a'b'}^{*}.$