By means of the results~\Eq{(43.5)} the~$G_{\mu\nu}$ can be calculated for either Einstein's or de~Sitter's forms of the world. De~Sitter's equation~\Eq{(67.33)} is of the standard form with $\chi$~substituted for~$r$, and \[ e^{\lambda} = R^{2},\quad e^{\mu} = R^{2} \sin^{2}\chi/\chi^{2},\quad e^{\nu} = R^{2} \cos^{2}\chi, \] thus \begin{gather*} \lambda' = 0,\quad \mu' = 2\cot\chi - 2\chi,\quad \nu' = -2\tan\chi,\displaybreak[0] \\ \mu'' = -2\cosec^{2}\chi + 2/\chi^{2},\quad \nu'' = -2\sec^{2}\chi. \end{gather*} Hence by~\Eq{(43.5)} we find after an easy reduction \[ G_{11} = -3,\ G_{22} = -3\sin^{2}\chi,\ G_{33} = -3\sin^{2}\chi \sin^{2}\theta,\ G_{44} = 3\cos^{2}\chi. \] These are equivalent to \[ G_{\mu\nu} = \frac{3}{R^{2}}\, g_{\mu\nu}. \Tag{(69.11)} \] De~Sitter's world thus corresponds to the revised form of the law of gravitation \[ G_{\mu\nu} = \lambda g_{\mu\nu}, \] and its radius is given by \[ \lambda = \frac{3}{R^{2}}. \Tag{(69.12)} \] Einstein's form~\Eq{(67.12)} gives similarly \[ e^{\lambda} = R^{2},\quad e^{\mu} = R^{2}\sin^{2}\chi/\chi^{2},\quad e^{\nu} = 1, \] from which by~\Eq{(43.5)} \begin{gather*} G_{11} = -2,\ G_{22} = -2\sin^{2}\chi,\ G_{33} = -2\sin^{2}\chi \sin^{2}\theta,\ G_{44} = 0, \Tag{(69.21)} \\ G = 6/R^{2}. \Tag{(69.22)} \end{gather*} It is not possible to reconcile these values with the law $G_{\mu\nu} = \lambda g_{\mu\nu}$, owing to the vanishing of~$G_{44}$. Einstein's form cannot be the natural form of empty space; but it may nevertheless be the actual form of the world if the matter in the world is suitably distributed. To determine the necessary distribution we must calculate the energy-tensor~\Eq{(54.71)} \[ -8\pi T_{\mu\nu} = G_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}G + \lambda g_{\mu\nu}. \] %\PageSep{160}