Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/121

or ${\textstyle u}$ independent of ${\textstyle s,}$ and therefore a function of ${\textstyle \theta }$ alone. But for ${\textstyle s=0,}$ it is evident that ${\textstyle t=0,}$ ${\textstyle {\frac {\partial t}{\partial \theta }}=0,}$ and therefore ${\textstyle u=0.}$ Whence we conclude that, in general, ${\textstyle u=0,}$ or

$$\cos \lambda \lambda '=0.$$

From this follows the beautiful theorem:

"If all lines drawn from a point on the curved surface are shortest lines of equal lengths, they meet the line which joins their extremities everywhere at right angles."

We can show in a similar manner that, if upon the curved surface any curved line whatever is given, and if we suppose drawn from every point of this line toward the same side of it and at right angles to it only shortest lines of equal lengths, the extremities of which are joined by a line, this line will be cut at right angles by those lines in all its points. We need only let ${\textstyle \theta }$ in the above development represent the length of the given curved line from an arbitrary point, and then the above calculations retain their validity, except that ${\textstyle u=0}$ for ${\textstyle s=0}$ is now contained in the hypothesis.

18.

The relations arising from these constructions deserve to be developed still more fully. We have, in the first place, if, for brevity, we write ${\textstyle m}$ for ${\textstyle {\frac {\partial t}{\partial \theta }},}$

(1)

${\textstyle \displaystyle {\begin{alignedat}{3}{\frac {\partial x}{\partial s}}&=\xi ,\qquad &{\frac {\partial y}{\partial s}}&=\eta ,\qquad &{\frac {\partial z}{\partial s}}&=\zeta ,\quad \end{alignedat}}}$

(2)

${\textstyle \displaystyle {\begin{alignedat}{3}{\frac {\partial x}{\partial \theta }}&=m\xi ',\quad &{\frac {\partial y}{\partial \theta }}&=m\eta ',\quad &{\frac {\partial z}{\partial \theta }}&=m\zeta ',\end{alignedat}}}$

(3)

${\textstyle {\begin{alignedat}{4}&\xi ^{2}&&+\eta ^{2}&&+\zeta ^{2}&&=1,\end{alignedat}}}$

(4)

${\textstyle {\begin{alignedat}{4}&\xi '^{2}&&+\eta '^{2}&&+\zeta '^{2}&&=1,\end{alignedat}}}$

(5)

${\textstyle {\begin{alignedat}{4}&\xi \xi '&&+\eta \eta '&&+\zeta \zeta '&&=0.\end{alignedat}}}$

Furthermore,

(6)

${\textstyle \displaystyle {\begin{alignedat}{4}&X^{2}&&+Y^{2}&&+Z^{2}&&=1,\end{alignedat}}}$

(7)

${\textstyle \displaystyle {\begin{alignedat}{4}&X\xi &&+Y\eta &&+Z\zeta &&=0,\end{alignedat}}}$

(8)

${\textstyle \displaystyle {\begin{alignedat}{4}&X\xi '&&+Y\eta '&&+Z\zeta '&&=0,\end{alignedat}}}$

and

[9]

${\textstyle {\begin{aligned}&\left\{{\begin{alignedat}{2}X&=\zeta \eta '&&-\eta \zeta ',\\Y&=\xi \zeta '&&-\zeta \xi ',\\Z&=\eta \xi '&&-\xi \eta ';\end{alignedat}}\right.\end{aligned}}}$