$$ \delta(g^{\mu\nu} \Gamma^{\beta}_{\nu\alpha}) = - \frac{1}{2} \delta \begin{bmatrix}g^{\mu\nu} & g^{\beta\lambda}\end{bmatrix} (\frac{\partial g_{\nu\lambda}}{\partial x_{\alpha}} + \frac{\partial g_{\alpha\lambda}}{\partial x_{\nu}} - \frac{\partial g_{\alpha\nu}}{\partial x_{\lambda}}) $$ 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} Therefore (48) ∂H/∂_g_^{μν} = -Γ_{μβ}^{α} Γ_{να}^{β} ∂H/∂_g__{σ}^{μν} = Γ_{μν}^{σ} If we now carry out the variations in (47a), we obtain the system of equations (47b) ∂/∂_x__{α} ( ∂H/∂_g__{α}^{μν} ) - ∂H/∂_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 ∂_g__{σ}^{μν}/∂_x__{α} = ∂_g__{α}^{μν}/∂_x__{σ} and consequently _g__{σ}^{μν} ∂/∂_x__{α} (∂H/∂_g__{α}^{μν}) = ∂/∂_x__{α} (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν}) - ∂H/∂_g__{α}^{μν} ∂_g__{α}^{μν}/∂_x__{σ} we obtain the equation ∂/∂_x__{α} (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν}) - ∂H/∂_x__{σ} = 0 or { ∂_t__{σ}^α/∂_x__{α} = 0 (49) { -2κ_t__{σ}^{α} = _g__{σ}^{μν} ∂H/∂_g__{α}^{μν} - δ_{σ}^{α} H. Owing to the relations (48), the equations (47) and (34), (50) κ_t__{σ}^{α} = ½ δ_{σ}^{α} _g_^{μν} Γ_{μβ}^{α} Γ_{να}^{β} - _g_^{μν} Γ_{μβ}^{α} Γ_{νσ}^{β}. It is to be noticed that _t__{σ}^{α} is not a tensor, so that the equation (49) holds only for systems for which √-_g_ = 1. This equation expresses the laws of conservation of impulse and energy in a gravitation-field. In fact, the integration of this equation over a three-dimensional volume V leads to the four equations (49a) _d_/_dx₄_ {∫_t__{σ}^4 _d_V} = ∫(_t__{σ}^1 α₁ + _t__{σ}² α₂ + _t__{σ}³ α₃)_d_S where α₁, α₂, α₂ are the direction-cosines of the inward-drawn normal to the surface-element _d_S in the Euclidean Sense. We recognise in this the usual expression for the laws of conservation. We denote the magnitudes _t_^α_{σ} as the energy-components of the gravitation-field. I will now put the equation (47) in a third form which will be very serviceable for a quick realisation of our object. By multiplying the field-equations (47) with _g_^{νσ}, these are obtained in the mixed forms. If we remember that _g_^{νσ} ∂Γ^α_{μν}/∂_x__{α} = ∂/∂_x__{α} (_g_^{νσ} Γ^α_{μν}) - ∂_g_^{νσ}/∂_x__{α} Γ^α_{μν}, which owing to (34) is equal to