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

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

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

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