It is not possible to make the second derivatives of the~$g_{\mu\nu}$ vanish at the selected point (as well as the first derivatives) unless the Riemann-Christoffel tensor vanishes there; but a great number of other special conditions can be imposed on the $100$~second derivatives by choosing the coordinates suitably. Make an additional transformation of the form \[ x_{\epsilon} = g_{\mu}^{\epsilon} x_{\mu}' + \tfrac{1}{6} a_{\mu\nu\sigma}^{\epsilon}\, x_{\mu}' x_{\nu}' x_{\sigma}', \Tag{(36.5)} \] where $a_{\mu\nu\sigma}^{\epsilon}$~represents arbitrary coefficients symmetrical in $\mu$,~$\nu$,~$\rho$. This new transformation will not affect the first derivatives of the~$g_{\mu\nu}$ at the origin, which have already been made to vanish by the previous transformation, but it alters the second derivatives. By differentiating~\Eq{(31.3)}, viz.\ \[ \{\mu\nu, \rho\}'\, \frac{\dd x_{\epsilon}}{\dd x_{\rho}'} - \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \{\alpha\beta, \epsilon\} = \frac{\dd^{2} x_{\epsilon}}{\dd x_{\mu}'\, \dd x_{\nu}'}, \] we obtain at the origin \[ \frac{\dd}{\dd x_{\sigma}'}\{\mu\nu, \rho\}'\, \frac{\dd x_{\epsilon}}{\dd x_{\rho}'} - \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \frac{\dd x_{\gamma}}{\dd x_{\sigma}'}\, \frac{\dd}{\dd x_{\gamma}} \{\alpha\beta, \epsilon\} = \frac{\dd^{3} x_{\epsilon}}{\dd x_{\mu}'\, \dd x_{\nu}'\, \dd x_{\sigma}'}, \] since the $3$-index symbols themselves vanish. Hence by~\Eq{(36.5)}\footnote {For the disappearance of the factor~$\frac{1}{6}$, see~\Eq{(35.6)}.} \[ \frac{\dd}{\dd x_{\sigma}'} \{\mu\nu, \rho\}' · g_{\rho}^{\epsilon} - g_{\mu}^{\alpha} g_{\nu}^{\beta} g_{\sigma}^{\gamma}\, \frac{\dd}{\dd x_{\gamma}} \{\alpha\beta, \epsilon\} = a_{\mu\nu\sigma}^{\epsilon}, \] which reduces to \[ \frac{\dd}{\dd x_{\sigma}'} \{\mu\nu, \epsilon\}' - \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\} = a_{\mu\nu\sigma}^{\epsilon}. \Tag{(36.55)} \] The transformation~\Eq{(36.5)} accordingly increases $\dd \{\mu\nu, \epsilon\}/\dd x_{\sigma}$ by~$a_{\mu\nu\sigma}^{\epsilon}$. Owing to the symmetry of~$a_{\mu\nu\sigma}^{\epsilon}$, all three quantities \[ \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\},\quad \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \epsilon\},\quad \frac{\dd}{\dd x_{\mu}} \{\nu\sigma, \epsilon\} \] \PageSep{79} are necessarily increased by the same amount. Now the unaltered difference \[ \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \epsilon\} - \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\} = B_{\mu\nu\sigma}^{\epsilon}, \Tag{(36.6)} \] since the remaining terms of~\Eq{(34.4)} vanish in the coordinates here used. We cannot alter any of the components of the Riemann-Christoffel tensor; but, \index{Riemann-Christoffel tensor!importance of}% subject to this limitation, the alterations of the derivatives of the $3$-index symbols are arbitrary.