Take a point on the surface as origin. Let $(x_{1}, x_{2}, x_{3}, x_{4})$ be rectangular coordinates in the tangent plane (four-dimensional) at the origin; and let the fifth rectangular axis along the normal be~$z$. Then by Euclidean geometry \[ -ds^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2} + dz^{2}, \Tag{(65.1)} \] imaginary values of~$ds$ corresponding as usual to real distances in space. The four-dimensional surface will be specified by a single equation between the five coordinates, which we may take to be \[ z = f(x_{1}, x_{2}, x_{3}, x_{4}). \] If the origin is a regular point this can be expanded in powers of the~$x$'s. The deviation from the tangent plane is of the second order compared with distances parallel to the plane; consequently $z$~does not contain linear terms in the~$x$'s. The expansion accordingly starts with a homogeneous quadratic function, and the equation is of the form \[ 2z = a_{\mu\nu} x_{\mu} x_{\nu}, \Tag{(65.2)} \] correct to the second order. For a fixed value of~$z$ the quadric~\Eq{(65.2)} is called the \emph{indicatrix}. \index{Indicatrix}% The radius of curvature of any normal section of the surface is found by the well-known method. If $t$~is the radius of the indicatrix in the direction of the section (direction cosines $l_{1}$, $l_{2}$, $l_{3}$, $l_{4}$), the radius of curvature is \[ \rho = \frac{t^{2}}{2z} = \frac{1}{a_{\mu\nu} l_{\mu} l_{\nu}}. \] In particular, if the axes are rotated so as to coincide with the principal axes of the indicatrix, \Eq{(65.2)}~becomes \[ 2z = k_{1}\, dx_{1}^{2} + k_{2}\, dx_{2}^{2} + k_{3}\, dx_{3}^{2} + k_{4}\, dx_{4}^{2}, \Tag{(65.3)} \] and the principal radii of curvature of the surface are the reciprocals of $k_{1}$, $k_{2}$, $k_{3}$,~$k_{4}$. Differentiating~\Eq{(65.2)} \[ dz = a_{\mu\nu} x_{\mu}\, dx_{\nu},\quad dz^{2} = a_{\mu\nu} x_{\mu}\, dx_{\nu} · a_{\sigma\tau} x_{\sigma}\, dx_{\tau}. \] Hence, substituting in~\Eq{(65.1)} \[ -ds^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2} + (a_{\mu\nu} a_{\sigma\tau} x_{\mu} x_{\sigma})\, dx_{\nu}\, dx_{\tau} \] for points in the four-dimensional continuum. Accordingly \[ -g_{\nu\tau} = g_{\nu}^{\tau} + a_{\mu\nu} a_{\sigma\tau} x_{\mu} x_{\sigma}. \Tag{(65.4)} \] Hence at the origin the~$g_{\mu\nu}$ are Euclidean; their first derivatives vanish; and their second derivatives are given by \[ \frac{\dd^{2} g_{\nu\tau}}{\dd x_{\mu}\, \dd x_{\sigma}} = -(a_{\mu\nu} a_{\sigma\tau} + a_{\sigma\nu} a_{\mu\tau}), \] by~\Eq{(35.5)}.