The isotropic system could have been found directly by seeking particular solutions of Einstein's equations having the form~\Eq{(38.12)}, or \[ ds^{2} = -e^{\lambda}\, dr^{2} - e^{\mu}(r^{2}\, d\theta^{2} + r^{2}\sin^{2}\theta\, d\phi^{2}) + e^{\nu}\, dt^{2}, \] where $\lambda$, $\mu$, $\nu$ are functions of~$r$. By the method of \SecRef{38}, we find \[ \left. \begin{aligned} G_{11} &= \mu'' + \tfrac{1}{2}\nu'' + \frac{2}{r}\mu' - \frac{1}{r}\lambda' + \tfrac{1}{2}\mu'^{2} - \tfrac{1}{2}\lambda'\mu' - \tfrac{1}{4}\lambda'\nu' + \tfrac{1}{4}\nu'^{2}\Add{,} \\ G_{22} &= e^{\mu-\lambda} \bigl[1 + 2r\mu' + \tfrac{1}{2}r(\nu' - \lambda') + \tfrac{1}{2}r^{2}\mu'' \\ &\qquad\qquad+ \tfrac{1}{2}r^{2}\mu'(\mu' + \tfrac{1}{2}\nu' - \tfrac{1}{2}\lambda')\bigr] - 1\Add{,} \\ G_{33} &= G_{22} \sin^{2}\theta\Add{,} \\ G_{44} &= -e^{\nu-\lambda}\left[\tfrac{1}{2}\nu'' + \frac{1}{r}\nu' + \tfrac{1}{2}\nu'\mu' - \tfrac{1}{4}\lambda'\nu' + \tfrac{1}{4}\nu'^{2}\right]\Add{.} \end{aligned} \right\} \Tag{(43.5)} \] The others are zero. Owing to an identical relation between $G_{11}$, $G_{22}$ and~$G_{44}$, the vanishing of this tensor gives only two equations to determine the three unknowns $\lambda$, $\mu$,~$\nu$. There exists therefore an infinite series of particular solutions, differing according to the third equation between $\lambda$, $\mu$, $\nu$ which is at our disposal. The two solutions hitherto considered are obtained by taking $\mu = 0$, and $\lambda = \mu$, respectively. The same series of solutions is obtained in a simpler way by substituting arbitrary functions of~$r$ instead of~$r$ in~\Eq{(38.8)}. \PageSep{95} The possibility of substituting any function of~$r$ for~$r$ without destroying the spherical symmetry is obvious from the fact that a coordinate is merely an identification-number; but analytically this possibility is bound up with the existence of an identical relation between $G_{11}$, $G_{22}$ and~$G_{44}$, which makes the equations too few to determine a unique solution. This introduces us to a theorem of great consequence in our later work. \index{Identities satisfied by $G_{\mu\nu}$}% If Einstein's ten equations $G_{\mu\nu} = 0$ were all independent, the ten~$g_{\mu\nu}$ would be uniquely determined by them (the boundary conditions being specified). The expression for~$ds^{2}$ would be unique and no transformation of coordinates would be possible. Since we know that we can transform coordinates as we please, there must exist identical relations between the ten~$G_{\mu\nu}$; and these will be found in \SecRef{52}. \Section{44.}{Problem of two bodies---Motion of the moon} \index{Moon, motion of}% \index{Problem!of two bodies}% \index{Two bodies, problem of}%