$$ \delta(g^{\mu\nu} \Gamma^{\beta}_{\nu\alpha}) = - \frac{1}{2}
\delta \begin{bmatrix}g^{\mu\nu} & g^{\beta\lambda}\end{bmatrix}
(\frac{\partial g_{\nu\lambda}}{\partial x_{\alpha}} +
\frac{\partial g_{\alpha\lambda}}{\partial x_{\nu}} - \frac{\partial
g_{\alpha\nu}}{\partial x_{\lambda}}) $$
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} δ_g_^{μν} + Γ_{μβ}^{α} δ_g__{α}^{μβ}