Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/124

If we differentiate again with respect to ${\textstyle s,}$ and notice that

$${\frac {\partial ^{2}\xi }{\partial s\,\partial \theta }}={\frac {\partial (pX)}{\partial \theta }},\quad {\text{etc.}},$$

and that

$$X\xi '+Y\eta '+Z\zeta '=0,$$

we have

$${\begin{aligned}{\frac {\partial ^{2}m}{\partial s^{2}}}&=p\left(\xi '\,{\frac {\partial X}{\partial \theta }}+\eta '\,{\frac {\partial Y}{\partial \theta }}+\zeta '\,{\frac {\partial Z}{\partial \theta }}\right)+p'\left(X{\frac {\partial \xi }{\partial \theta }}+Y{\frac {\partial \eta }{\partial \theta }}+Z{\frac {\partial \zeta }{\partial \theta }}\right)\\&=p\left(\xi '\,{\frac {\partial X}{\partial \theta }}+\eta '\,{\frac {\partial Y}{\partial \theta }}+\zeta '\,{\frac {\partial Z}{\partial \theta }}\right)+mp'^{2}\\&=-\left(\xi \,{\frac {\partial X}{\partial s}}+\eta \,{\frac {\partial Y}{\partial s}}+\zeta \,{\frac {\partial Z}{\partial s}}\right)\left(\xi '\,{\frac {\partial X}{\partial \theta }}+\eta '\,{\frac {\partial Y}{\partial \theta }}+\zeta '\,{\frac {\partial Z}{\partial \theta }}\right)\\&{\phantom {={}}}+\left(\xi '\,{\frac {\partial X}{\partial s}}+\eta '\,{\frac {\partial Y}{\partial s}}+\zeta '\,{\frac {\partial Z}{\partial s}}\right)\left(\xi \,{\frac {\partial X}{\partial \theta }}+\eta \,{\frac {\partial Y}{\partial \theta }}+\zeta \,{\frac {\partial Z}{\partial \theta }}\right)\\&=\left({\frac {\partial Y}{\partial \theta }}\,{\frac {\partial Z}{\partial s}}-{\frac {\partial Y}{\partial s}}\,{\frac {\partial Z}{\partial \theta }}\right)X+\left({\frac {\partial Z}{\partial \theta }}\,{\frac {\partial X}{\partial s}}-{\frac {\partial Z}{\partial s}}\,{\frac {\partial X}{\partial \theta }}\right)Y+\left({\frac {\partial X}{\partial \theta }}\,{\frac {\partial Y}{\partial s}}-{\frac {\partial X}{\partial s}}\,{\frac {\partial Y}{\partial \theta }}\right)Z.\end{aligned}}$$

[But if the surface element

$$m\,ds\,d\theta$$

belonging to the point ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ be represented upon the auxiliary sphere of unit radius by means of parallel normals, then there corresponds to it an area whose magnitude is

$$\left\{X\left({\frac {\partial Y}{\partial s}}\,{\frac {\partial Z}{\partial \theta }}-{\frac {\partial Y}{\partial \theta }}\,{\frac {\partial Z}{\partial s}}\right)+Y\left({\frac {\partial Z}{\partial s}}\,{\frac {\partial X}{\partial \theta }}-{\frac {\partial Z}{\partial \theta }}\,{\frac {\partial X}{\partial s}}\right)+Z\left({\frac {\partial X}{\partial s}}\,{\frac {\partial Y}{\partial \theta }}-{\frac {\partial X}{\partial \theta }}\,{\frac {\partial Y}{\partial s}}\right)\right\}ds\,d\theta .$$

Consequently, the measure of curvature at the point under consideration is equal to

$$\left.-{\frac {1}{m}}\,{\frac {\partial ^{2}m}{\partial s^{2}}}.\right]$$