It must be admitted, that this introduction of the energy-tensor of matter cannot be justified by means of the Relativity-Postulate alone; for we have in the foregoing analysis deduced it from the condition that the energy of the gravitation-field should exert gravitating action in the same way as every other kind of energy. The strongest ground for the choice of the above equation however lies in this, that they lead, as their consequences, to equations expressing the conservation of the components of total energy (the impulses and the energy) which exactly correspond to the equations (49) and (49a). This shall be shown afterwards. §17. The laws of conservation in the general case. The equations (52) can be easily so transformed that the second member on the right-hand side vanishes. We reduce (52) with reference to the indices μ and σ and subtract the equation so obtained after multiplication with ½ δ_{μ}^σ from (52). We obtain, (52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = -κ(_t__{μ}^σ + T_{μ}^σ) we operate on it by ∂/∂_x__{σ}. Now, ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α) = -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β} + ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})]. The first and the third member of the round bracket lead to expressions which cancel one another, as can be easily seen by interchanging the summation-indices α, and σ, on the one hand, and β and λ, on the other. The second term can be transformed according to (31). So that we get, (54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α) = ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ} The second member of the expression on the left-hand side of (52a) leads first to - 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.