Page:LorentzGravitation1915.djvu/13

 for this in his hist paper by introducing a quantity which he calls $$H$$ and which is a function of the quantities $$g_{ab}$$ and their derivatives, without further containing anything that is connected with the material or the electromagnetic system. All we have to do now is to add to the left hand side of equation (12) a term depending on that quantity $$H$$. We shall write for it the variation of

$\frac{1}{\varkappa}\int QdS,$

where $$\varkappa$$ is a universal constant, while $$Q$$ is what calls $$H\sqrt{-g}$$, with the same or the opposite sign. We shall now require that

not only for the variations considered above but also for variations of the gravitation field defined by $$\delta g_{ab}$$, if these too vanish at the limits of the field of integration.

To obtain now

$\delta\mathrm{L}+\frac{1}{\varkappa}\delta Q+\Sigma(a)K_{a}\delta x_{a}$

we have to add to the right hand side of (17) or (40), first the change of $$\mathrm{L}$$ caused by the variation of the quantities $$g$$, viz.

$\Sigma(\overline{ab})\frac{\partial\mathrm{L}}{\partial g_{ab}}\delta g_{ab},$|undefined

and secondly the change of $$Q$$ multiplied by $$\tfrac{1}{\varkappa}$$. This latter change is

$\Sigma(\overline{ab})\frac{\partial Q}{\partial g_{ab}}\delta g_{ab}+\Sigma(\overline{ab}e)\frac{\partial Q}{\partial g_{ab,e}}\delta g_{ab,e},$|undefined

where $$g_{ab,e}$$ bas been written for the derivative $$\frac{\partial g_{ab}}{\partial x_{e}}$$.

As

$\delta g_{ab,e}=\frac{\partial\delta g_{ab}}{\partial x_{e}}$|undefined

we may replace the last term by

$\Sigma(\overline{ab}e)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,e}}\delta g_{ab}\right)-\Sigma(\overline{ab}e)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,e}}\right)\delta g_{ab}.$|undefined

§ 14. As we have to proceed now in the same way in the case