The most symmetrical way of imposing further conditions is to make a transformation such that \[ \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\} + \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \epsilon\} + \frac{\dd}{\dd x_{\mu}} \{\nu\sigma, \epsilon\} = 0. \Tag{(36.7)} \] There are $80$~different equations of this type, each of which fixes one of the $80$~arbitrary coefficients~$a_{\mu\nu\sigma}^{\epsilon}$. In addition there are $20$~independent equations of type~\Eq{(36.6)} corresponding to the $20$~independent components of the Riemann-Christoffel tensor. Thus we have just sufficient equations to determine uniquely the $100$~second derivatives of the~$g_{\mu\nu}$. Coordinates such that $\dd g_{\mu\nu}/\dd x_{\sigma}$~is zero and $\dd^{2} g_{\mu\nu}/\dd x_{\sigma}\, \dd x_{\tau}$ satisfies~\Eq{(36.7)} may be called \emph{canonical coordinates}. By solving the $100$~equations we obtain all the $\dd^{2} g_{\mu\nu}/\dd x_{\sigma}\, \dd x_{\tau}$ for canonical coordinates expressed as linear functions of the~$B_{\mu\nu\sigma}^{\epsilon}$. The two successive transformations which lead to canonical coordinates \index{Canonical coordinates}% \index{Coordinate-systems!canonical}% \index{Fundamental velocity!tensors}% are combined in the formula \begin{multline*} x_{\epsilon} = g_{\mu}^{\epsilon} x_{\mu}' - \tfrac{1}{2}\{\mu\nu, \epsilon\}_{0}\, x_{\mu}' x_{\nu}' \\ - \frac{1}{18}\left[\frac{\dd}{\dd x_{\mu}}\{\nu\sigma, \epsilon\} + \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \epsilon\} + \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \epsilon\}\right]_{0} x_{\mu}' x_{\nu}' x_{\sigma}'. \Tag{(36.8)} \end{multline*} At the origin $\dd x_{\epsilon}/\dd x_{\mu}' = g_{\mu}^{\epsilon}$, so that the transformation does not alter any tensor at the origin. For example, the law of transformation of~$C_{\mu\nu\sigma}$ gives \begin{align*} C_{\mu\nu\sigma}' = C_{\alpha\beta\gamma}\, \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \frac{\dd x_{\gamma}}{\dd x_{\sigma}'} &= C_{\alpha\beta\gamma}\, g_{\mu}^{\alpha} g_{\nu}^{\beta} g_{\sigma}^{\gamma} \\ &= C_{\mu\nu\sigma}. \end{align*} The transformation in fact alters the curvature and hypercurvature of the axes passing through the origin, but does not alter the angles of intersection. Consider any tensor which contains only the~$g_{\mu\nu}$ and their first and second derivatives. In canonical coordinates the first derivatives vanish and the second derivatives are linear functions of the~$B_{\mu\nu\sigma}^{\epsilon}$; hence the whole tensor is a function of the~$g_{\mu\nu}$ and the~$B_{\mu\nu\sigma}^{\epsilon}$. But neither the tensor itself nor the~$g_{\mu\nu}$ and $B_{\mu\nu\sigma}^{\epsilon}$ have been altered in the reduction to canonical coordinates, hence the same functional relation holds true in the original unrestricted coordinates. We have thus the important result---