The equations of the gravitation-field free from matter must thus be in every case satisfied when all B^{ρ}_{μστ} vanish. But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformed _away_ by any choice of axes, _i.e._, it cannot be transformed to a case of constant _g__{μν}’s. Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical tensors B_{μν} deduced from the tensors B^{ρ}_{μστ} should vanish. We thus get 10 equations for 10 quantities _g__{μν} which are fulfilled in the special case when B^{ρ}_{μστ}’s all vanish. Remembering (44) we see that in absence of matter the field-equations come out as follows; (when referred to the special co-ordinate-system chosen.) (47) $$ \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x_{\alpha}} + \Gamma^{\alpha}_{\mu\beta} \Gamma^{\beta}_{\mu\alpha} = 0 $$ $$ \sqrt{-g} = 1 $$ $$ \Gamma^{\alpha}_{\mu\nu} = - \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} $$ It can also be shown that the choice of these equations is connected with a minimum of arbitrariness. For besides B_{μν}, there is no tensor of the second rank, which can be built out of _g__{μν}’s and their derivatives no higher than the second, and which is also linear in them. It will be shown that the equations arising in a purely mathematical way out of the conditions of the general relativity, together with equations (46), give us the Newtonian law of attraction as a first approximation, and lead in the second approximation to the explanation of the perihelion-motion of mercury discovered by Leverrier (the residual effect which could not be accounted for by the consideration of all sorts of disturbing factors). My view is that these are convincing proofs of the physical correctness of my theory. §15. Hamiltonian Function for the Gravitation-field. Laws of Impulse and Energy. In order to show that the field equations correspond to the laws of impulse and energy, it is most convenient to write it in the following Hamiltonian form:— (47a) δ∫ H_d_τ = 0 H = _g_^{μν} γ^{α}_{μβ} γ^{β}_{να} √(-_g_) = 1 Here the variations vanish at the limits of the finite four-dimensional integration-space considered. It is first necessary to show that the form (47a) is equivalent to equations (47). For this purpose, let us consider H as a function of _g_^{μν} and _g_^{μν}_{σ} (= ∂_g_^{μν}/∂_x__{σ}) We have at first δH = Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2_g_^{μν} Γ^{α}_{μβ} δΓ^{β}_{να} = - Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2Γ^{α}_{μβ} δ(_g_^{μν}Γ^{β}_{να}). But