# Help on Hamiltonian tensor equations in Einstein's original General Relativity papers

Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie published in 1916's Annalen Der Physik, I came across Equations 47b) regarding the gravity contribution to the stress-energy-momentum-tensor through the Hamiltonian:

\begin{align} \frac{\partial}{\partial x_{\alpha}} \left( \frac{\partial H}{\partial g^{\mu\nu}_{,\alpha}} \right ) - \frac{\partial H}{\partial g^{\mu\nu}} = 0 \tag{47b} \end{align}

According to the discussion in the following paragraphs, this should be equivalent to:

$$\frac{\partial}{\partial x_{\alpha}} \left( g^{\mu\nu}_{,\sigma}\frac{\partial H}{\partial g^{\mu\nu}_{,\alpha}} \right ) - \frac{\partial H}{\partial x_{\sigma}} = 0$$

but you can see that in the original equation there is a term:

$$- \frac{\partial H}{\partial g^{\mu\nu}}$$

not appearing in the modified equation, which doesn't seem to be vanishing.
Can you explain why?