We see that the pressure vanishes at $r = a$, and would become negative if we attempted to continue the solution beyond $r = a$. Hence the sphere $r = a$ gives the boundary of the fluid. If it is desired to continue the solution outside the sphere, another form of~$ds^{2}$ must be taken corresponding to the equations for empty space. Unless $a > \Chg{\surd(8/9\alpha)}{\sqrt{8/9\alpha}}$ the pressure will everywhere be finite. This condition sets an upper limit to the possible size of a fluid sphere of given density. The limit exists because the presence of dense matter increases the curvature of space, and makes the total volume of space smaller. Clearly the volume of the material sphere cannot be larger than the volume of space. \PageSep{170} For spheres which are not unduly large (e.g.\ not much larger than the stars) this solution corresponds approximately to the problem of the equilibrium of an incompressible fluid. The necessary conditions are satisfied, viz.\ (1) The density is uniform. (2) The pressure is zero at the surface. (3) The stress-system is an isotropic hydrostatic pressure, and therefore satisfies the conditions of a perfect fluid. (4) The pressure is nowhere infinite, negative, or imaginary. Further equation~\Eq{(72.4)} determines the pressure at any distance from the centre. But the problem is only solved approximately, and the material here discussed is not strictly incompressible nor is it a perfect fluid. The values of $T_{1}^{1}$, $T_{2}^{2}$, $T_{3}^{3}$, $T_{4}^{4}$ refer to the particular coordinates used; these are arbitrary and do not correspond to natural measure. So long as the sphere is small, the difference does not amount to much; but for large spheres the solution ceases to correspond to a problem of any physical importance since it does not refer to natural measure. It is unfortunate that the solution breaks down for large spheres, because the existence of a limit to the size of the sphere is one of the most interesting objects of the research. Clearly we need a solution in which the density referred to natural measure is constant throughout; i.e.\ $T$~constant, instead of $T_{4}^{4}$~constant. The condition for a perfect fluid also needs modification. (But it would be of considerable interest to find the solution for a solid capable of supporting non-isotropic stress, if the problem of the fluid proves too difficult.) So far as I know, no progress has been made with the exact solution of this problem. It would throw interesting light on the manner in which the radius of space contracts as the size of the sphere continually increases.