The terms arising out of the two last terms within the round bracket are of different signs, and change into one another by the interchange of the indices μ and β. They cancel each other in the expression for δH, when they are multiplied by Γ_{μβ}^α, which is symmetrical with respect to μ and β, so that only the first member of the bracket remains for our consideration. Remembering (31), we thus have:—
δH = -Γ_{μβ}^α Γ_{να}^β δg^{μν} + Γ_{μβ}^α δg_{α}^{μβ}
Therefore
{ [part]H/[part]g^{μν} = -Γ_{μβ}^α Γ_{να}^β
(48) { [part]H/[part]g_{σ}^{μν} = Γ_{μν}^σ
If we now carry out the variations in (47a), we obtain the system of equations
(47b) [part]/[part]x_{α} ( [part]H/[part]g_{α}^{μν} ) - [part]H/[part]g^{μν} = 0,
which, owing to the relations (48), coincide with (47), as was required to be proved.
If (47b) is multiplied by g_{σ}^{μν},
since
[part]g_{σ}^{μν}/[part]x_{α} = [part]g_{α}^{μν}/[part]x_{σ}
