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CCORDING 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. 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