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If for \delta_{1}Q and \delta_{2}Q the expressions (48) and (44) are taken, the equation

dQ-\delta_{2}Q=\delta_{1}Q (50)

is an identity for every choice of the variations.

It will likewise be so in the special case considered and we shall also come to an identity if in (50) the terms with the derivatives of \xi are omitted while those with \xi itself are preserved.

When this is done \delta Q reduces to

-\frac{\partial Q}{\partial x_{h}}\xi

and, taking into consideration (44) and (48), we find after division by \xi

-\frac{\partial Q}{\partial x_{h}}+\sum(abe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,e}}g_{ab,h}\right)+\sum(abef)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,fh}\right)-\\
-\sum(abef)\frac{\partial}{\partial x_{e}}\left\{ \frac{\partial}{\partial x_{f}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,h}\right\} =-\sum(ab)G_{ab}\mathfrak{g}^{ab,h}
\end{array} (51)

In the second term of (44) we have interchanged here the indices e and f.

If for shortness' sake we put, for e\ne h

\mathfrak{s}_{h}^{e}=\sum(ab)\frac{\partial Q}{\partial g_{ab,e}}g_{ab,h}+\sum(abf)\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,fh}-\sum(abf)\frac{\partial}{\partial x_{f}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,h} (52)

and for e=h

\mathfrak{s}_{h}^{h}=-Q+\sum(ab)\frac{\partial Q}{\partial g_{ab,h}}g_{ab,h}+\sum(abf)\frac{\partial Q}{\partial g_{ab,fh}}g_{ab,fh}-\sum(abf)\frac{\partial}{\partial x_{f}}\left(\frac{\partial Q}{\partial g_{ab,hf}}\right)g_{ab,h} (53)

we may write

\sum(e)\frac{\partial\mathfrak{s}_{h}^{e}}{\partial x_{e}}=-\sum(ab)G_{ab}\mathfrak{g}^{ab,h} (54)

The set of quantities \mathfrak{s}_{h}^{e} will be called the complex \mathfrak{s} and the set of the four quantities which stand on the left hand side of (54) in the cases h=1,2,3,4, the divergency of the complex.[1] It will be denoted by div\mathfrak{s} and each of the four quantities separately by div_{h}\mathfrak{s}.

The equation therefore becomes

div_{h}\mathfrak{s}=-\sum(ab)G_{ab}\mathfrak{g}^{ab,h} (55)
  1. Einstein uses the word "divergency" in a somewhat different sense. It seemed desirable however to have a name for the left hand side of (54) and it was difficult to find a better one.