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XXV

GAUSSIAN CO-ORDINATES

ACCORDING to Gauss, this combined analytical and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see Fig. 4) drawn on the surface of the table. These we designate as $u$-curves, and we indicate each of them by means of a number. The curves $u=1$, $u=2$ and $u=3$ are drawn in the diagram.
Fig. 4.Between the curves $u=1$ and $u=2$ we must imagine an infinitely large number to be drawn, all of which correspond to real numbers lying between 1 and 2. We have then a system of $u$-curves, and this ‘‘infinitely dense” system covers the whole surface of the table. These $u$-curves must not intersect each other, and through each point of the surface one and only one curve must pass. Thus a perfectly definite value of $u$ belongs to every point on the surface of the marble slab. In like manner we

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