Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/114

and since we always choose the point ${\textstyle \lambda '}$ so that

$$\lambda 'L<90^{\circ },$$

then for the shortest line

$$\lambda 'L=0,$$

or ${\textstyle \lambda '}$ and ${\textstyle L}$ must coincide. Therefore

$${\begin{aligned}\rho \,d\xi &=X\,ds,\\\rho \,d\eta &=Y\,ds,\\\rho \,d\zeta &=Z\,ds,\end{aligned}}$$

and we have here, instead of ${\textstyle 4}$ curved lines upon the auxiliary sphere, only ${\textstyle 3}$ to consider. Every element of the second line is therefore to be regarded as lying in the great circle ${\textstyle L\lambda .}$ And the positive or negative value of ${\textstyle \rho }$ refers to the concavity or the convexity of the curve in the direction of the normal.

14.

We shall now investigate the spherical angle upon the auxiliary sphere, which the great circle going from ${\textstyle L}$ toward ${\textstyle \lambda }$ makes with that one going from ${\textstyle L}$ toward one of the fixed points ${\textstyle (1),}$ ${\textstyle (2),}$ ${\textstyle (3);}$ e.g., toward ${\textstyle (3).}$ In order to have something definite here, we shall consider the sense from ${\textstyle L(3)}$ to ${\textstyle L\lambda }$ the same as that in which ${\textstyle (1),}$ ${\textstyle (2),}$ and ${\textstyle (3)}$ lie. If we call this angle ${\textstyle \phi ,}$ then it follows from the theorem of Art. 7 that

$$\sin L(3).\sin L\lambda .\sin \phi =Y\xi -X\eta ,$$

or, since ${\textstyle L\lambda =90^{\circ }}$ and

$$\sin L(3)={\sqrt {X^{2}+Y^{2}}}={\sqrt {1-Z^{2}}},$$

we have

$$\sin \phi ={\frac {Y\xi -X\eta }{\sqrt {X^{2}+Y^{2}}}}.$$

Furthermore,

$$\sin L(3).\sin L\lambda .\cos \phi =\zeta ,$$

or

$$\cos \phi ={\frac {\zeta }{\sqrt {X^{2}+Y^{2}}}}$$

and

$$\tan \phi ={\frac {Y\xi -X\eta }{\zeta }}={\frac {\zeta '}{\zeta }}.$$