Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/120

$$\xi ={\frac {\partial x}{\partial s}},\qquad \eta ={\frac {\partial y}{\partial s}},\qquad \zeta ={\frac {\partial z}{\partial s}}.$$

The extremities of all shortest lines of equal lengths ${\textstyle s}$ correspond to a curved line whose length we may call ${\textstyle t.}$ We can evidently consider ${\textstyle t}$ as a function of ${\textstyle s}$ and ${\textstyle \theta ,}$ and if the direction of the element of ${\textstyle t}$ corresponds upon the sphere to the point ${\textstyle \lambda '}$ whose coordinates are ${\textstyle \xi ',}$ ${\textstyle \eta ',}$ ${\textstyle \zeta ',}$ we shall have

$$\xi '.{\frac {\partial t}{\partial \theta }}={\frac {\partial x}{\partial \theta }},\qquad \eta '.{\frac {\partial t}{\partial \theta }}={\frac {\partial y}{\partial \theta }},\qquad \zeta '.{\frac {\partial t}{\partial \theta }}={\frac {\partial z}{\partial \theta }}.$$

Consequently

$$(\xi \xi '+\eta \eta '+\zeta \zeta ')\,{\frac {\partial t}{\partial \theta }}={\frac {\partial x}{\partial s}}.{\frac {\partial x}{\partial \theta }}+{\frac {\partial y}{\partial s}}.{\frac {\partial y}{\partial \theta }}+{\frac {\partial z}{\partial s}}.{\frac {\partial z}{\partial \theta }}.$$

This magnitude we shall denote by ${\textstyle u,}$ which itself, therefore, will be a function of ${\textstyle \theta }$ and ${\textstyle s.}$

We find, then, if we differentiate with respect to ${\textstyle s,}$

$${\begin{aligned}{\frac {\partial u}{\partial s}}&={\frac {\partial ^{2}x}{\partial s^{2}}}.{\frac {\partial x}{\partial \theta }}+{\frac {\partial ^{2}y}{\partial s^{2}}}.{\frac {\partial y}{\partial \theta }}+{\frac {\partial ^{2}z}{\partial s^{2}}}.{\frac {\partial z}{\partial \theta }}+{\tfrac {1}{2}}\,{\frac {\partial \left\{({\tfrac {\partial x}{\partial s}})^{2}+({\tfrac {\partial y}{\partial s}})^{2}+({\tfrac {\partial z}{\partial s}})^{2}\right\}}{\partial \theta }}\\&={\frac {\partial ^{2}x}{\partial s^{2}}}.{\frac {\partial x}{\partial \theta }}+{\frac {\partial ^{2}y}{\partial s^{2}}}.{\frac {\partial y}{\partial \theta }}+{\frac {\partial ^{2}z}{\partial s^{2}}}.{\frac {\partial z}{\partial \theta }},\end{aligned}}$$

because

$$\left({\frac {\partial x}{\partial s}}\right)^{2}+\left({\frac {\partial y}{\partial s}}\right)^{2}+\left({\frac {\partial z}{\partial s}}\right)^{2}=1,$$

and therefore its differential is equal to zero.

But since all points [belonging] to one constant value of ${\textstyle \theta }$ lie on a shortest line, if we denote by ${\textstyle L}$ the zenith of the point to which ${\textstyle s,}$ ${\textstyle \theta }$ correspond and by ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z}$ the coordinates of ${\textstyle L,}$ [from the last formulæ of Art. 13],

$${\frac {\partial ^{2}x}{\partial s^{2}}}={\frac {X}{p}},\qquad {\frac {\partial ^{2}y}{\partial s^{2}}}={\frac {Y}{p}},\qquad {\frac {\partial ^{2}z}{\partial s^{2}}}={\frac {Z}{p}},$$

if ${\textstyle p}$ is the radius of curvature. We have, therefore,

$$p.{\frac {\partial u}{\partial s}}=X\,{\frac {\partial x}{\partial \theta }}+Y\,{\frac {\partial y}{\partial \theta }}+Z\,{\frac {\partial z}{\partial \theta }}={\frac {\partial t}{\partial \theta }}(X\xi '+Y\eta '+Z\zeta ').$$

But

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

because, evidently, ${\textstyle \lambda '}$ lies on the great circle whose pole is ${\textstyle L.}$ Therefore we have

$${\frac {\partial u}{\partial s}}=0,$$