In the more general case we continue to call the invariant~$G$ the Gaussian curvature although the interpretation in terms of normal curvatures no longer \index{Normal, $6$-dimensional}% holds. It is convenient also to introduce a quantity called the \emph{radius of spherical curvature}, viz.\ the radius of a hypersphere which has the same \index{Gaussian curvature}% Gaussian curvature as the surface considered\footnotemark.\footnotetext {A hypersphere of four dimensions is by definition a four-dimensional surface drawn in five dimensions so that \Eq{(65.6)} applies to it. Accordingly if its radius is~$R$, we have $G = 12/R^{2}$. For three dimensions $G = 6/R^{2}$; for two dimensions $G = 2/R^{2}$.} Considering the geometry of the general case, in $10$~dimensions the normal is a six-dimensional continuum in which we can take rectangular axes $z_{1}$, $z_{2}$,~\dots\Add{,} $z_{6}$. The surface is then defined by six equations which near the origin take the form \[ 2z_{r} = a_{r\mu\nu} x_{\mu} x_{\nu}\quad (r = 1, 2\Add{,} \dots\Add{,} 6). \] The radius of curvature of a normal section in the direction~$l_{\mu}$ is then \[ \rho = \frac{t^{2}}{2\Chg{\surd(z_{1}^{2} + z_{2}^{2} + \cdots + z_{6}^{2})} {\sqrt{z_{1}^{2} + z_{2}^{2} + \cdots + z_{6}^{2}}}} = \frac{1}{\Chg{\surd\{(a_{1\mu\nu} l_{\mu} l_{\nu})^{2} + \cdots + (a_{6\mu\nu} l_{\mu} l_{\nu})^{2}\}} {\sqrt{(a_{1\mu\nu} l_{\mu} l_{\nu})^{2} + \cdots + (a_{6\mu\nu} l_{\mu} l_{\nu})^{2}}}}. \] It is, however, of little profit to develop the properties of normal curvature, which depend on the surface chosen to represent the metric of space-time and are not intrinsic in the metric itself. We therefore follow a different plan, introducing the radius of spherical curvature which has invariant properties. \PageSep{152} Reverting for the moment to five dimensions, consider the \emph{three-dimensional} space formed by the section of our surface by $x_{1} = 0$. Let $G_{(1)}$~be its Gaussian curvature. Then $G_{(1)}$~is formed from~$G$ by dropping all terms containing the \index{Curvature!quadric of}% suffix~$1$---a dimension which no longer enters into consideration. Accordingly $G - G_{(1)}$ consists of those terms of~$G$ which contain the suffix~$1$; and by~\Eq{(65.53)} and~\Eq{(65.54)} we have \[ \tfrac{1}{2}(G - G_{(1)}) = -G_{11}. \Tag{(65.71)} \] Introducing the value $g_{11} = -1$ at the origin \[ G_{11} - \tfrac{1}{2}g_{11} G = \tfrac{1}{2}G_{(1)}. \Tag{(65.72)} \]