Page:Relativity (1931).djvu/124

imagine a system of $v$-curves drawn on the surface. These satisfy the same conditions as the $u$-curves, they are provided with numbers in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of $u$ and a value of $v$ belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates). For example, the point $P$ in the diagram has the Gaussian co-ordinates $u=3$, $v=1$. Two neighbouring points $P$ and $P'$ on the surface then correspond to the co-ordinates

${\begin{aligned}P:\qquad &u,v\\P':\qquad &u+du,\ v+dv\end{aligned}}$,

where $du$ and $dv$ signify very small numbers. In a similar manner we may indicate the distance (line-interval) between $P$ and $P'$, as measured with a little rod, by means of the very small number $ds$. Then according to Gauss we have

$ds^{2}=g_{11}\ du^{2}+2g_{12}\ du\ dv+g_{22}\ dv^{2}$,

where $g_{11}$, $g_{12}$, $g_{22}$, are magnitudes which depend in a perfectly definite way on $u$ and $v$. The magnitudes $g_{11}$, $g_{12}$, and $g_{22}$ determine the behaviour of the rods relative to the $u$-curves and $v$-curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the $u$-curves and