Page:Relativity (1931).djvu/125

$v$-curves and to attach numbers to them, in such a manner, that we simply have:

$ds^{2}=du^{2}+dv^{2}$.

Under these conditions, the $u$-curves and $v$-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. Here the Gaussian co-ordinates are simply Cartesian ones. It is clear that Gauss co-ordinates are nothing more than an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring points “in space.”

So far, these considerations hold for a continuum of two dimensions. But the Gaussian method can be applied also to a continuum of three, four or more dimensions. If, for instance, a continuum of four dimensions be supposed available, we may represent it in the following way. With every point of the continuum we associate arbitrarily four numbers, $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ which are known as “co-ordinates.” Adjacent points correspond to adjacent values of the coordinates. If a distance $ds$ is associated with the adjacent points $P$ and $P'$, this distance being measurable and well-defined from a physical point of view, then the following formula holds:

$ds^{2}=g_{11}\ dx_{1}^{2}+2g_{12}\ dx_{1}\ dx_{2}\ \ldots +g_{44}\ dx_{4}^{2}$,