When space-time is not flat we can introduce coordinates which will be approximately Galilean in a small region round a selected point, the $g_{\mu\nu}$ being not constant but stationary there; this amounts to identifying the curved space-time with the osculating flat space-time for a small distance round the point. Expressing the procedure analytically, we choose coordinates such that the $40$~derivatives $\dd g_{\mu\nu}/dx_{\sigma}$ vanish \emph{at the selected point}. It is fairly obvious from general considerations that this will always be possible; but the following is a formal proof. Having transferred the origin to the selected point, make the following transformation of coordinates \[ x_{\epsilon} = g_{\epsilon}^{\mu} x_{\mu}' - \tfrac{1}{2} \{\alpha\beta, \epsilon\}_{0}\, g_{\alpha}^{\mu}\, g_{\beta}^{\nu}\, x_{\mu}' x_{\nu}', \Tag{(36.4)} \] \PageSep{78} where the value of the $3$-index symbol at the origin is to be taken. Then at the origin \begin{align*} \frac{\dd x_{\epsilon}}{\dd x_{\mu}'} &= g_{\epsilon}^{\mu}, \Tag{(36.45)} \\ \frac{\dd^{2} x_{\epsilon}}{\dd x_{\mu}'\, \dd x_{\nu}'} &= -\{\alpha\beta, \epsilon\}\, g_{\alpha}^{\mu}\, g_{\beta}^{\nu} \\ &= -\{\alpha\beta, \epsilon\}\, \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'} \quad\text{by \Eq{(36.45)}.} \end{align*} Hence by~\Eq{(31.3)} \[ \{\mu\nu, \rho\}'\, \frac{\dd x_{\epsilon}}{\dd x_{\rho}'} = 0. \] But \[ \{\mu\nu, \rho\}'\, \frac{\dd x_{\epsilon}}{\dd x_{\rho}'} = \{\mu\nu, \rho\}'\, g_{\epsilon}^{\rho} = \{\mu\nu, \epsilon\}'. \] Hence in the new coordinates the $3$-index symbols vanish at the origin; and it follows by~\Eq{(27.4)} and~\Eq{(27.5)} that the first derivatives of the~$g_{\mu\nu}'$ vanish. This is the preliminary transformation presupposed in \SecRef{4}. We pass on to a somewhat more difficult transformation which is important as contributing an insight into the significance of~$B_{\mu\nu\sigma}^{\epsilon}$.