It can be shown that one exact solution is the same as in \SecRef{38}, viz.\ \[ e^{-\lambda} = e^{\nu} = \gamma = 1 - 2m/r. \Tag{(62.6)} \] For taking the partial derivatives of~\Eq{(62.3)}, and applying \Eq{(62.6)} after the differentiation, \begin{align*} \frac{\dd\mf{K}''}{\dd\lambda} &= -\tfrac{3}{2} \mf{K}'' + 4 \frac{(1 - e^{-\lambda})}{r^{4}} · 2e^{\frac{1}{2}(\lambda + \nu)} \sin\theta = \left(-72 \frac{m^{2}}{r^{4}} + 16 \frac{m}{r^{3}}\right) \sin\theta,\displaybreak[0] \\ % \frac{\dd\mf{K}''}{\dd\lambda'} &= 2e^{\frac{1}{2}(\lambda + \nu)} \sin\theta \bigl\{2e^{-2\lambda} \lambda' + r^{2} e^{-2\lambda}(\tfrac{1}{4}\lambda'\nu' - \tfrac{1}{4}\nu'^{2} - \tfrac{1}{2}\nu'')\nu'\bigr\} \\ &= \left(24 \frac{m^{2}}{r^{3}} - 8 \frac{m}{r^{2}}\right) \sin\theta,\displaybreak[0] \\ %\PageSep{143} \index{Einstein's law of gravitation!alternatives to}% \frac{\dd\mf{K}''}{\dd\lambda''} &= 0,\displaybreak[0] \\ % \frac{\dd\mf{K}''}{\dd\nu} &= \tfrac{1}{2} \mf{K}'' = 24 \frac{m^{2}}{r^{4}} \sin\theta,\displaybreak[0] \\ % \frac{\dd\mf{K}''}{\dd\nu'} &= 2e^{\frac{1}{2}(\lambda + \nu)} \sin\theta \bigl\{2e^{-2\lambda} \nu' + 4r^{2} e^{-2\lambda}(\tfrac{1}{4}\lambda'\nu' - \tfrac{1}{4}\nu'^{2} - \tfrac{1}{2}\nu'') (\tfrac{1}{4}\lambda' - \tfrac{1}{2}\nu')\bigr\} \\ &= \left(-40 \frac{m^{2}}{r^{3}} + 8\frac{m}{r^{2}}\right) \sin\theta,\displaybreak[0] \\ % \frac{\dd\mf{K}''}{\dd\nu''} &= -2e^{\frac{1}{2}(\lambda + \nu)} \sin\theta · 2r^{2} e^{-2\lambda}(\tfrac{1}{4}\lambda'\nu' - \tfrac{1}{4}\nu'^{2} - \tfrac{1}{2}\nu'') \\ &= \left(16 \frac{m^{2}}{r^{2}} - 8\frac{m}{r}\right) \sin\theta. \end{align*} On substituting these values, \Eq{(62.5)}~is verified exactly. The alternative law $\Ham K'/\Ham g_{\mu\nu} = 0$ is also satisfied by the same solution. For \[ \delta(G_{\mu\nu} G^{\mu\nu} \sqrt{-g}) = G_{\mu\nu}\, \delta(G^{\mu\nu} \sqrt{-g}) + G^{\mu\nu} \sqrt{-g}\, \delta G_{\mu\nu}, \] hence the variation of $K' \sqrt{-g}$ vanishes wherever $G_{\mu\nu} = 0$. Any field of gravitation agreeing with Einstein's law will satisfy the alternative law proposed, but not usually \Foreign{vice versa}. There are doubtless other symmetrical solutions for the alternative laws of gravitation which are not permitted by Einstein's law, since the differential equations are now of the fourth order and involve two extra boundary conditions either at the particle or at infinity. It may be asked, Why should these be excluded in nature? We can only answer that it may be for the same reason that negative mass, doublets, electrons of other than standard mass, or other theoretically possible singularities in the world, do not occur; the ultimate particle satisfies conditions which are at present unknown to us.