Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/35

Also it is evident that any finite part whatever of the curved surface will retain the same integral curvature after development upon another surface.

Surfaces developable upon a plane constitute the particular case to which geometers have heretofore restricted their attention. Our theory shows at once that the measure of curvature at every point of such surfaces is equal to zero. Consequently, if the nature of these surfaces is defined according to the third method, we shall have at every point

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

a criterion which, though indeed known a short time ago, has not, at least to our knowledge, commonly been demonstrated with as much rigor as is desirable.

13.

What we have explained in the preceding article is connected with a particular method of studying surfaces, a very worthy method which may be thoroughly developed by geometers. When a surface is regarded, not as the boundary of a solid, but as a flexible, though not extensible solid, one dimension of which is supposed to vanish, then the properties of the surface depend in part upon the form to which we can suppose it reduced, and in part are absolute and remain invariable, whatever may be the form into which the surface is bent. To these latter properties, the study of which opens to geometry a new and fertile field, belong the measure of curvature and the integral curvature, in the sense which we have given to these expressions. To these belong also the theory of shortest lines, and a great part of what we reserve to be treated later. From this point of view, a plane surface and a surface developable on a plane, e.g., cylindrical surfaces, conical surfaces, etc., are to be regarded as essentially identical; and the generic method of defining in a general manner the nature of the surfaces thus considered is always based upon the formula

$${\sqrt {E\,dp^{2}+2F\,dp.dq+G\,dq^{2}}},$$

which connects the linear element with the two indeterminates ${\textstyle p,}$ ${\textstyle q.}$ But before following this study further, we must introduce the principles of the theory of shortest lines on a given curved surface.

14.

The nature of a curved line in space is generally given in such a way that the coordinates ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ corresponding to the different points of it are given in the form of functions of a single variable, which we shall call ${\textstyle w.}$ The length of such a line from