Page:Relativity (1931).djvu/125

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

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: