It is often convenient to use the mixed tensor~$T_{\mu}^{\nu}$ in place of~$T^{\mu\nu}$. For Galilean coordinates we obtain from~\Eq{(53.5)}\footnote {E.g.\ $T_{2}^{1} = \rho_{\sigma 2} T^{\sigma 1} = 0 - T^{21} + 0 + 0$.} \[ \begin{array}[t]{r} T_{\mu}^{\nu} \\ \MuNuarrow \end{array} = \begin{array}[t]{@{}c@{,\quad}c@{,\quad}c@{,\quad}c@{}} -p_{xx} - \rho u^{2} & -p_{yx} - \rho vu & -p_{zx} - \rho wu & \rho u\Add{,} \\ -p_{xy} - \rho uv & -p_{yy} - \rho v^{2} & -p_{yz} - \rho wv & \rho v\Add{,} \\ -p_{xz} - \rho uw & -p_{yz} - \rho vw & -p_{zz} - \rho w^{2} & \rho w\Add{,} \\ -\rho u & -\rho v & -\rho w & \rho\Add{.} \\ \end{array} \Tag{(53.91)} \] \PageSep{119} The equation equivalent to~\Eq{(53.82)} is then \[ \frac{\dd T_{\mu}^{\nu}}{\dd x_{\nu}} = (-\rho X,-\rho Y,-\rho Z, \rho S). \Tag{(53.92)} \] That is to say $\dd T_{\mu}^{\nu}/\dd x_{\nu}$ is the rate of creation of negative momentum and of positive mass or energy in unit volume. \Section{54.}{New derivation of Einstein's law of gravitation} \index{Continuous matter, gravitation in}% \index{Einstein's law of gravitation!in continuous matter}% We have found that for Galilean coordinates \[ \Chg{\dd T^{\mu\nu}/\dd x_{\nu}}{\frac{\dd T^{\mu\nu}}{\dd x_{\nu}}} = 0. \Tag{(54.1)} \] This is evidently a particular case of the tensor equation \[ (T^{\mu\nu})_{\nu} = 0. \Tag{(54.21)} \] Or we may use the equivalent equation \[ (T_{\mu}^{\nu})_{\nu} = 0, \Tag{(54.22)} \] which results from lowering the suffix~$\mu$. In other words the divergence of \index{Divergence!energy@of energy-tensor}% the energy-tensor vanishes. Taking the view that energy, stress, and momentum belong to the world (space-time) and not to some extraneous substance in the world, we must identify the energy-tensor with some fundamental tensor, i.e.\ a tensor belonging to the fundamental series derived from~$g_{\mu\nu}$. The fact that the divergence of~$T_{\mu}^{\nu}$ vanishes points to an identification with $(G_{\mu}^{\nu} - \frac{1}{2}g_{\mu}^{\nu} G)$ whose divergence vanishes \emph{identically} (\SecRef{52}). Accordingly we set \[ G_{\mu}^{\nu} - \tfrac{1}{2}g_{\mu}^{\nu} G - 8\pi T_{\mu}^{\nu}, \Tag{(54.3)} \] the factor~$8\pi$ being introduced for later convenience in coordinating the units. To pass from \Eq{(54.1)} to \Eq{(54.21)} involves an appeal to the hypothetical Principle of Equivalence; but by taking \Eq{(54.3)} as our fundamental equation \index{Identification, Principle of}% \index{Principle!of identification}% of gravitation \Eq{(54.21)}~becomes an identity requiring no hypothetical assumption.