Draw a small sphere surrounding the point~$P$ which is being considered. Let $g_{\mu\nu} = \delta_{\mu\nu} + h_{\mu\nu} + h_{\mu\nu}'$, where $\delta_{\mu\nu}$~represents the Galilean values, and $h_{\mu\nu}$~and $h_{\mu\nu}'$ represent the fields of force contributed independently by the matter internal to and external to the sphere. By \SecRef{36} we can choose coordinates such that at~$P$ $h_{\mu\nu}'$~and its first derivatives vanish; and by the symmetry of the sphere the first derivatives of~$h_{\mu\nu}$ vanish, whilst $h_{\mu\nu}$~itself tends to zero for an infinitely small sphere. Hence the cross-terms which are of the form \[ h_{\sigma\tau}'\, \frac{\dd h_{\mu\nu}}{\dd x_{\lambda}\, \dd x_{\rho}},\ \frac{\dd h_{\sigma\tau}'}{\dd x_{\lambda}}\, \frac{\dd h_{\mu\nu}}{\dd x_{\rho}},\ \text{and } h_{\sigma\tau}\, \frac{\dd h_{\mu\nu}'}{\dd x_{\lambda}\, \dd x_{\rho}} \] will all vanish at~$P$. Accordingly with these limitations there are no cross-terms, and the sum of the two solutions $h_{\mu\nu}$ and $h_{\mu\nu}'$ is also a solution of the accurate equations. Hence the values~\Eq{(46.5)} remain true. It will be seen that the limitation is that the coordinates must be ``natural coordinates'' at the point~$P$. We have already paid heed to this in taking~$\rho$ to be the proper-density. We have assumed that the matter at~$P$ is not accelerated with respect to these natural axes at~$P$. (The original particles had to be \emph{continually} at rest, otherwise the solution~\Eq{(46.15)} does not apply.) If it were accelerated there would have to be a stress causing the acceleration. We shall find later that a stress contributes additional terms to the~$G_{\mu\nu}$. The formulae~\Eq{(46.5)} apply only strictly when there is no stress and the continuous medium is specified by one variable only, viz.\ the density. \PageSep{104}