Page:LorentzGravitation1916.djvu/32

 derivatives and we shall determine the variation it undergoes by arbitrarily chosen variations $$\delta g_{ab}$$, these latter being continuous functions of the coordinates. We have evidently

$\delta Q=\sum(ab)\frac{\partial Q}{\partial g_{ab}}\delta g_{ab}+\sum(abe)\frac{\partial Q}{\partial g_{ab,e}}\delta g_{ab,e}+\sum(abef)\frac{\delta Q}{\partial g_{ab,ef}}\delta g_{ab,ef}$|undefined

By means of the equations

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

this may be decomposed into two parts

namely

The last equation shows that

if the variations $$\delta g_{ab}$$ and their first derivatives vanish at the boundary of the domain of integration.

§ 35. Equations of the same form may also be found if $$Q$$ is expressed in one of the two other ways mentioned in § 33. If e.g. we work with the quantities $$\mathfrak{g}^{ab}$$ we shall find

$(\delta Q)=\left(\delta_{1}Q\right)+\left(\delta_{2}Q\right)$

where $$\left(\delta_{1}Q\right)$$ and $$\left(\delta_{2}Q\right)$$ are directly found from (43) and (44) by replacing $$g_{ab}$$, $$g_{ab,e}$$, $$g_{ab,ef}$$, $$\delta g_{ab}$$ and $$\delta g_{ab,e}$$ etc. by $$\mathfrak{g}^{ab}$$, $$\mathfrak{g}^{ab,e}$$ etc. If the variations chosen in the two cases correspond to each other we shall have of course

$(dQ)=\delta Q$

Moreover we can show that the equalities

$\left(\delta_{1}Q\right)=\delta_{1}Q,\ \left(\delta_{2}Q\right)=\delta_{2}Q$

exist separately.