- ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β} + ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})]. The expression arising out of the last member within the round bracket vanishes according to (29) on account of the choice of axes. The two others can be taken together and give us on account of (31), the expression -½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ} So that remembering (54) we have (55) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = 0. identically. From (55) and (52a) it follows that (56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0 From the field equations of gravitation, it also follows that the conservation-laws of impulse and energy are satisfied. We see it most simply following the same reasoning which lead to equations (49a); only instead of the energy-components of the gravitational-field, we are to introduce the total energy-components of matter and gravitational field. §18. The Impulse-energy law for matter as a consequence of the field-equations. If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to §15, remembering that _g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes, the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0 or remembering (56) (57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0 A comparison with (41b) shows that these equations for the above choice of co-ordinates (√(-_g_) = 1) asserts nothing but the vanishing of the divergence of the tensor of the energy-components of matter. Physically the appearance of the second term on the left-hand side shows that for matter alone the law of conservation of impulse and energy cannot hold; or can only hold when _g_^{μν}’s are constants; _i.e._, when the field of gravitation vanishes. The second member is an expression for impulse and energy which the gravitation-field exerts per time and per volume upon matter. This comes out clearer when instead of (57) we write it in the form of (47). (57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β. The right-hand side expresses the interaction of the energy of the gravitational-field on matter. The field-equations of gravitation contain thus at the same time 4 conditions which are to be satisfied by all material phenomena. We get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another. D. THE “MATERIAL” PHENOMENA.