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 $s,$ 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 $x,$  $y,$  $z$ 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

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