# Page:LorentzGravitation1916.djvu/32

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

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

By means of the equations

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

this may be decomposed into two parts

 ${\displaystyle dQ=\delta _{1}Q+\delta _{2}Q}$ (42)

namely

 ${\displaystyle \delta _{1}Q=\sum (ab)\left\{{\frac {\partial Q}{\partial g_{ab}}}-\sum (e){\frac {\partial }{\partial x_{e}}}{\frac {\partial Q}{\partial g_{ab,e}}}+\sum (ef){\frac {\partial ^{2}}{\partial x_{e}\partial x_{f}}}{\frac {\partial Q}{\partial g_{ab,ef}}}\right\}\delta g_{ab}}$ (43)
 ${\displaystyle {\begin{array}{c}\delta _{2}Q=\sum (abe){\frac {\partial Q}{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,e}}}\delta g_{ab}\right)+\sum (abef){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\delta g_{ab,e}\right)-\\\\-\sum (abef){\frac {\partial }{\partial x_{e}}}\left\{{\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\right)\delta g_{ab}\right\}\end{array}}}$ (44)

The last equation shows that

 ${\displaystyle \int \delta _{2}QdS=0}$ (45)

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

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

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

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

${\displaystyle (dQ)=\delta Q}$

Moreover we can show that the equalities

${\displaystyle \left(\delta _{1}Q\right)=\delta _{1}Q,\ \left(\delta _{2}Q\right)=\delta _{2}Q}$

exist separately.[1]

1. Suppose that at the boundary of the domain of integration ${\displaystyle \delta g_{ab}=0}$ and ${\displaystyle \delta g_{ab,e}=0}$. Then we have also ${\displaystyle \delta {\mathfrak {g}}^{ab}=0}$ and ${\displaystyle \delta {\mathfrak {g}}^{ab,e}=0}$, so that

${\displaystyle \int \left(\delta _{2}Q\right)dS=0,\ \int \delta _{2}QdS=0}$

and from