Page:LorentzGravitation1915.djvu/14

 of a material and in that of an electromagnetic system we need consider only the latter. The conclusions drawn in § 11 evidently remain valid, so that we may start from the equation which we obtain by adding the new terms to (43). We therefore have

When we integrate over $$S$$, the first two terms on the right hand side vanish. In the terms following them the coefficient of each $$\delta g_{ab}$$ must be 0, so that we find

These are the differential equations we sought for. At the same time (48) becomes

§ 15. Finally we can derive from this the equations for the momenta and the energy of the gravitation field. For this purpose we impart a virtual displacement $$\delta x_{c}$$ to this field only (comp. §§ 6 and 12). Thus we put $$\delta x_{a}=0,\ q_{a}=0$$ and

$\delta g_{ab}=-g_{ab,c}\delta x_{c}$

Evidently

$\delta Q=-\frac{\partial Q}{\partial x_{c}}\delta x_{c}$|undefined

and (comp. § 12)

$\delta\mathrm{L}=-\left(\frac{\partial\mathrm{L}}{\partial x_{c}}\right)_{\psi}\delta x_{c}$|undefined

After having substituted these values in equation (50) we can deduce from it the value of $$\left(\dfrac{\partial\mathrm{L}}{\partial x_{c}}\right)_{\psi}$$.

If we put

and for $$e\ne c$$

the result takes the following form