We shall try to satisfy this by breaking it up into two equations \[ G_{\mu\nu} = \tfrac{1}{2}g^{\sigma\rho}\, \frac{\dd^{2} g_{\mu\nu}}{\dd x_{\sigma}\, \dd x_{\rho}} \Tag{(57.31)} \] and \[ 0 = g^{\sigma\rho} \left(\frac{\dd^{2} g_{\sigma\rho}}{\dd x_{\mu}\, \dd x_{\nu}} - \frac{\dd^{2} g_{\mu\sigma}}{\dd x_{\nu}\, \dd x_{\rho}} - \frac{\dd^{2} g_{\nu\rho}}{\dd x_{\mu}\, \dd x_{\sigma}}\right). \Tag{(57.32)} \] %\PageSep{129} The second equation becomes, correct to the first order, \begin{align*} 0 &= \delta^{\sigma\rho} \left( \frac{\dd^{2} h_{\sigma\rho}}{\dd x_{\mu}\, \dd x_{\nu}} - \frac{\dd^{2} h_{\mu\sigma}}{\dd x_{\nu}\, \dd x_{\rho}} - \frac{\dd^{2} h_{\nu\rho}}{\dd x_{\mu}\, \dd x_{\sigma}}\right) \\ &= \frac{\dd^{2} h}{\dd x_{\mu}\, \dd x_{\nu}} - \frac{\dd^{2} h_{\mu}^{\sigma}}{\dd x_{\nu}\, \dd x_{\rho}} - \frac{\dd^{2} h_{\nu}^{\rho}}{\dd x_{\mu}\, \dd x_{\sigma}}, \end{align*} where \[ h_{\mu}^{\rho} = \delta^{\sigma\rho} h_{\mu\sigma};\quad h = h_{\rho}^{\rho} = \delta^{\sigma\rho} h_{\sigma\rho}. \] This is satisfied if \[ \frac{\dd h_{\mu}^{\alpha}}{\dd x_{\alpha}} = \frac{1}{2}\, \frac{\dd h}{\dd x_{\mu}} \] or \[ \frac{\dd}{\dd x_{\alpha}} (h_{\mu}^{\alpha} - \tfrac{1}{2} \delta_{\mu}^{\alpha} h) = 0. \Tag{(57.4)} \] The other equation~\Eq{(57.31)} may be written \[ \Wave h_{\mu\nu} = 2G_{\mu\nu} \] or \[ \Wave h_{\mu}^{\alpha} = 2G_{\mu}^{\alpha}, \] showing that $G_{\mu}^{\alpha}$~is a small quantity of the first order. Hence \begin{align*} \Wave (h_{\mu}^{\alpha} - \tfrac{1}{2} \delta_{\mu}^{\alpha} h) &= 2(G_{\mu}^{\alpha} - \tfrac{1}{2} g_{\mu}^{\alpha} G) \\ &= -16\pi T_{\mu}^{\alpha}. \Tag{(57.5)} \end{align*} This ``equation of wave-motion'' can be integrated. Since we are dealing with small quantities of the first order, the effect of the deviations from Galilean geometry will only affect the results to the second order; accordingly the well-known solution\footnote {Rayleigh, \Title{Theory of Sound}, vol.~\Vol{II}, p.~104, equation~(3).} may be used, viz.\ \[ h_{\mu}^{\alpha} - \tfrac{1}{2} \delta_{\mu}^{\alpha} h = \frac{1}{4\pi} \int \frac{(-16\pi T_{\mu}^{\alpha})'\, dV'}{r'}, \Tag{(57.6)} \] the integral being taken over each element of space-volume~$dV'$ at a coordinate distance~$r'$ from the point considered and at a time $t - r'$, i.e.\ at a time such that waves propagated from~$dV'$ with unit velocity can reach the point at the time considered.