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 ${\textstyle m}$ 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 ${\textstyle s,}$ we have^[1]

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

1. It is better to differentiate ${\textstyle m^{2}.}$ [In fact from (2) and (4)
   $$m^{2}=\left({\frac {\partial x}{\partial \theta }}\right)^{2}+\left({\frac {\partial y}{\partial \theta }}\right)^{2}+\left({\frac {\partial z}{\partial \theta }}\right)^{2},$$
   therefore
   $${\begin{aligned}m\,{\frac {\partial m}{\partial s}}&={\frac {\partial x}{\partial \theta }}.{\frac {\partial ^{2}x}{\partial \theta \,\partial s}}+{\frac {\partial y}{\partial \theta }}.{\frac {\partial ^{2}y}{\partial \theta \,\partial s}}+{\frac {\partial z}{\partial \theta }}.{\frac {\partial ^{2}z}{\partial \theta \,\partial s}}\\&=\left.m\xi '\,{\frac {\partial \xi }{\partial \theta }}+m\eta '\,{\frac {\partial \eta }{\partial \theta }}+m\zeta '\,{\frac {\partial \zeta }{\partial \theta }}.\right]\end{aligned}}$$