Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/119

We shall here consider 4 figures:

1. an arbitrary figure upon the first surface,
3. the figure on the auxiliary sphere, which corresponds to the zeniths of the previous figure,
5. the figure upon the second surface, which No. 1 forms by the development,
7. the figure upon the auxiliary sphere, which corresponds to the zeniths of No. 3.

Therefore, according to what we have proved, 2 and 4 have equal areas, as also 1 and 3. Since we assume these figures infinitely small, the quotient obtained by dividing 2 by 1 is the measure of curvature of the first curved surface at this point, and likewise the quotient obtained by dividing 4 by 3, that of the second surface. From this follows the important theorem:

"In the transformation of surfaces by development the measure of curvature at every point remains unchanged."

This is true, therefore, of the product of the greatest and smallest radii of curvature.

In the case of the plane, the measure of curvature is evidently everywhere zero. Whence follows therefore the important theorem:

"For all surfaces developable upon a plane the measure of curvature everywhere vanishes,"

or

$$\left({\frac {\partial ^{2}z}{\partial x\,\partial y}}\right)^{2}-\left({\frac {\partial ^{2}z}{\partial x^{2}}}\right)\left({\frac {\partial ^{2}z}{\partial x^{2}}}\right)=0,$$

which criterion is elsewhere derived from other principles, though, as it seems to us, not with the desired rigor. It is clear that in all such surfaces the zeniths of all points can not fill out any space, and therefore they must all lie in a line.

17.

From a given point on a curved surface we shall let an infinite number of shortest lines go out, which shall be distinguished from one another by the angle which their first elements make with the first element of a definite shortest line. This angle we shall call ${\textstyle \theta .}$ Further, let ${\textstyle s}$ be the length [measured from the given point] of a part of such a shortest line, and let its extremity have the coordinates ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z.}$ Since ${\textstyle \theta }$ and ${\textstyle s,}$ therefore, belong to a perfectly definite point on the curved surface, we can regard ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ as functions of ${\textstyle \theta }$ and ${\textstyle s.}$ The direction of the element of ${\textstyle s}$ corresponds to the point ${\textstyle \lambda }$ on the sphere, whose coordinates are ${\textstyle \xi ,}$ ${\textstyle \eta ,}$ ${\textstyle \zeta .}$ Thus we shall have