Consider the expression \[ \rho_{0}\, \frac{dx_{\mu}}{ds}\,\frac{dx_{\nu}}{ds}, \] where $dx_{\mu}/ds$~refers to the motion of the matter, and $\rho_{0}$~is the proper-density (an invariant). The matter is at rest in the coordinates hitherto used, and consequently \[ \frac{dx_{1}}{ds},\ \frac{dx_{2}}{ds},\ \frac{dx_{3}}{ds} = 0,\quad \frac{dx_{4}}{ds} = 1, \] \PageSep{103} so that all components of the expression vanish, except the component $\mu$,~$\nu = 4$ which is equal to~$\rho_{0}$. Accordingly in these coordinates \[ T^{\mu\nu} = \rho_{0}\, \frac{dx_{\mu}}{ds}\,\frac{dx_{\nu}}{ds}, \Tag{(46.8)} \] since the density~$\rho$ in~\Eq{(46.7)} is clearly the proper-density. Now \Eq{(46.8)}~is a tensor equation\footnotemark,\footnotetext {When an equation is stated to be a tensor equation, the reader is expected to verify that the covariant dimensions of both sides are the same.} and since it has been verified for one set of coordinates it is true for all coordinate-systems. Equations \Eq{(46.6)} and \Eq{(46.8)} together give the extension of Einstein's law of gravitation for a region containing continuous matter of proper-density~$\rho_{0}$ and velocity~$dx_{\mu}/ds$. The question remains whether the neglect of~$m^{2}$ causes any inaccuracy in these equations. In passing to continuous matter we diminish~$m$ for each particle indefinitely, but increase the number of particles in a given volume. To avoid increasing the number of particles we may diminish the volume, so that the formulae~\Eq{(46.5)} will be true for the limiting case of a point inside a very small portion of continuous matter. Will the addition of surrounding matter in large quantities make any difference? This can contribute nothing directly to the tensor~$G_{\mu\nu}$, since so far as this surrounding matter is concerned the point is in empty space; but Einstein's equations are non-linear and we must consider the possible cross-terms.