Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/123

 and also

$$p' = X\, \frac{\partial \xi'}{\partial s}  + Y\, \frac{\partial \eta'}{\partial s}   + Z\, \frac{\partial \zeta'}{\partial s}.$$

Further [we obtain], from the result obtained by differentiating (8),

$$-p' = \xi'\, \frac{\partial X}{\partial s}   + \eta'\, \frac{\partial Y}{\partial s}    + \zeta'\, \frac{\partial Z}{\partial s}.$$

But we can derive two other expressions for this. We have

$$\frac{\partial m\xi'}{\partial s} = \frac{\partial \xi}{\partial \theta},\qquad \left[ \frac{\partial m\eta'}{\partial s} = \frac{\partial \eta}{\partial \theta},\qquad \frac{\partial m\zeta'}{\partial s} = \frac{\partial \zeta}{\partial \theta}, \right]$$

therefore [because of (8)]

$$mp' = X\, \frac{\partial \xi}{\partial \theta} + Y\, \frac{\partial \eta}{\partial \theta} + Z\, \frac{\partial \zeta}{\partial \theta}.$$

[and therefore, from (7),]

$$-mp' = \xi\, \frac{\partial X}{\partial \theta} + \eta\, \frac{\partial Y}{\partial \theta} + \zeta\, \frac{\partial Z}{\partial \theta}.$$

After these preliminaries [using (2) and (4)] we shall now first put $m$ in the form

$$m = \xi'\, \frac{\partial x}{\partial \theta} + \eta'\, \frac{\partial y}{\partial \theta} + \zeta'\, \frac{\partial z}{\partial \theta},$$

and differentiating with respect to $s,$ we have

$$\begin{aligned} \frac{\partial m}{\partial s} &= \frac{\partial x}{\partial \theta}. \frac{\partial \xi'}{\partial s} + \frac{\partial y}{\partial \theta}. \frac{\partial \eta'}{\partial s} + \frac{\partial z}{\partial \theta}. \frac{\partial \zeta'}{\partial s} + \xi'\, \frac{\partial^{2} x}{\partial s\, \partial \theta} + \eta'\, \frac{\partial^{2} y}{\partial s\, \partial \theta} + \zeta'\, \frac{\partial^{2} z}{\partial s\, \partial \theta} \\ &= mp'(\xi'X + \eta'Y + \zeta'Z) + \xi'\, \frac{\partial \xi}{\partial \theta} + \eta'\, \frac{\partial \eta}{\partial \theta} + \zeta'\, \frac{\partial \zeta}{\partial \theta} \\ &= \xi'\, \frac{\partial \xi}{\partial \theta} + \eta'\, \frac{\partial \eta}{\partial \theta} + \zeta'\, \frac{\partial \zeta}{\partial \theta}. \end{aligned}$$