Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/113

we draw a radius of the auxiliary sphere to the point ${\textstyle \lambda ',}$ but instead of this point we take the point opposite when ${\textstyle \lambda '}$ is more than ${\textstyle 90^{\circ }}$ from ${\textstyle L.}$ In the first case, we regard the element at ${\textstyle \lambda }$ as positive, and in the other as negative. Finally, let ${\textstyle {\mathfrak {L}}}$ be the point on the auxiliary sphere, which is ${\textstyle 90^{\circ }}$ from both ${\textstyle \lambda }$ and ${\textstyle \lambda ',}$ and which is so taken that ${\textstyle \lambda ,}$ ${\textstyle \lambda ',}$ ${\textstyle {\mathfrak {L}}}$ lie in the same order as ${\textstyle (1),}$ ${\textstyle (2),}$ ${\textstyle (3).}$

The coordinates of the four points of the auxiliary sphere, referred to its centre, are for

$${\begin{alignedat}{4}&L\qquad &&X\quad &&Y\quad &&Z\\&\lambda &&\xi &&\eta &&\zeta \\&\lambda '&&\xi '&&\eta '&&\zeta '\\&{\mathfrak {L}}&&\alpha &&\beta &&\gamma .\end{alignedat}}$$

Hence each of these ${\textstyle 4}$ points describes a line upon the auxiliary sphere, whose elements we shall express by ${\textstyle dL,}$ ${\textstyle d\lambda ,}$ ${\textstyle d\lambda ',}$ ${\textstyle d{\mathfrak {L}}.}$ We have, therefore,

$${\begin{aligned}d\xi &=\xi '\,d\lambda ,\\d\eta &=\eta '\,d\lambda ,\\d\zeta &=\zeta '\,d\lambda .\end{aligned}}$$

In an analogous way we now call

$${\frac {d\lambda }{ds}}$$

the measure of curvature of the curved line upon the curved surface, and its reciprocal

$${\frac {ds}{d\lambda }}$$

the radius of curvature. If we denote the latter by ${\textstyle \rho ,}$ then

$${\begin{aligned}\rho \,d\xi &=\xi '\,ds,\\\rho \,d\eta &=\eta '\,ds,\\\rho \,d\zeta &=\zeta '\,ds.\end{aligned}}$$

If, therefore, our line be a shortest line, ${\textstyle \xi ',}$ ${\textstyle \eta ',}$ ${\textstyle \zeta '}$ must be proportional to the quantities ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z.}$ But, since at the same time

$$\xi '^{2}+\eta '^{2}+\zeta '^{2}=X^{2}+Y^{2}+Z^{2}=1,$$

we have

$$\xi '=\pm X,\quad \eta '=\pm Y,\quad \zeta '=\pm Z,$$

and since, further,

$${\begin{aligned}\xi 'X+\eta 'Y+\zeta 'Z&=\cos \lambda 'L\\&=\pm (X^{2}+Y^{2}+Z^{2})\\&=\pm 1,\end{aligned}}$$