Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/34

If we substitute these different expressions in the formula for the measure of curvature derived at the end of the preceding article, we obtain the following formula, which involves only the quantities ${\textstyle E,}$ ${\textstyle F,}$ ${\textstyle G}$ and their differential quotients of the first and second orders:

$${\begin{aligned}\displaystyle &\qquad 4(EG-F^{2})k=E\left({\frac {\partial E}{\partial q}}.{\frac {\partial G}{\partial q}}-2{\frac {\partial F}{\partial p}}.{\frac {\partial G}{\partial q}}+{\biggl (}{\frac {\partial G}{\partial p}}{\biggr )}^{2}\right)\\&+F\left({\frac {\partial E}{\partial p}}.{\frac {\partial G}{\partial q}}-{\frac {\partial E}{\partial q}}.{\frac {\partial G}{\partial p}}-2{\frac {\partial E}{\partial q}}.{\frac {\partial F}{\partial q}}+4{\frac {\partial F}{\partial p}}.{\frac {\partial F}{\partial q}}-2{\frac {\partial F}{\partial p}}.{\frac {\partial G}{\partial p}}\right)\\&+G\left({\frac {\partial E}{\partial p}}.{\frac {\partial G}{\partial p}}-2{\frac {\partial E}{\partial p}}.{\frac {\partial F}{\partial q}}+{\biggl (}{\frac {\partial E}{\partial q}}{\biggr )}^{2}\right)-2(EG-F^{2})\left({\frac {\partial ^{2}E}{\partial q^{2}}}-2{\frac {\partial ^{2}F}{\partial p.\partial q}}+{\frac {\partial ^{2}G}{\partial p^{2}}}\right)\end{aligned}}$$

12.

Since we always have

$$dx^{2}+dy^{2}+dz^{2}=E\,dp^{2}+2F\,dp.dq+G\,dq^{2},$$

it is clear that

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

is the general expression for the linear element on the curved surface. The analysis developed in the preceding article thus shows us that for finding the measure of curvature there is no need of finite formulæ, which express the coordinates ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ as functions of the indeterminates ${\textstyle p,}$ ${\textstyle q;}$ but that the general expression for the magnitude of any linear element is sufficient. Let us proceed to some applications of this very important theorem.

Suppose that our surface can be developed upon another surface, curved or plane, so that to each point of the former surface, determined by the coordinates ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z,}$ will correspond a definite point of the latter surface, whose coordinates are ${\textstyle x',}$ ${\textstyle y',}$ ${\textstyle z'.}$ Evidently ${\textstyle x',}$ ${\textstyle y',}$ ${\textstyle z'}$ can also be regarded as functions of the indeterminates ${\textstyle p,}$ ${\textstyle q,}$ and therefore for the element ${\textstyle {\sqrt {dx'^{2}+dy'^{2}+dz'^{2}}}}$ we shall have an expression of the form

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

where ${\textstyle E',}$ ${\textstyle F',}$ ${\textstyle G'}$ also denote functions of ${\textstyle p,}$ ${\textstyle q.}$ But from the very notion of the development of one surface upon another it is clear that the elements corresponding to one another on the two surfaces are necessarily equal. Therefore we shall have identically

$$E=E',\quad F=F',\quad G=G'.$$

Thus the formula of the preceding article leads of itself to the remarkable

Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.