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. fig. 04 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 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.
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