Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/112

$$d.{\frac {\frac {dx}{dw}}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}},\qquad d.{\frac {\frac {dy}{dw}}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}},$$

$$d.{\frac {\frac {dz}{dw}}{\sqrt {\left({\frac {dx}{dw}}\right)^{2}+\left({\frac {dy}{dw}}\right)^{2}+\left({\frac {dz}{dw}}\right)^{2}}}}$$

must be proportional to the quantities ${\textstyle P,}$ ${\textstyle Q,}$ ${\textstyle R}$ respectively. If ${\textstyle ds}$ is an element of the curve; ${\textstyle \lambda }$ the point upon the auxiliary sphere, which represents the direction of this element; ${\textstyle L}$ the point giving the direction of the normal as above; and ${\textstyle \xi ,}$ ${\textstyle \eta ,}$ ${\textstyle \zeta ;}$ ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z}$ the coordinates of the points ${\textstyle \lambda ,}$ ${\textstyle L}$ referred to the centre of the auxiliary sphere, then we have

$${\begin{array}{c}dx=\xi \,ds,\qquad dy=\eta \,ds,\qquad dz=\zeta \,ds,\\\xi ^{2}+\eta ^{2}+\zeta ^{2}=1.\end{array}}$$

Therefore we see that the above differentials will be equal to ${\textstyle d\xi ,}$ ${\textstyle d\eta ,}$ ${\textstyle d\zeta .}$ And since ${\textstyle P,}$ ${\textstyle Q,}$ ${\textstyle R}$ are proportional to the quantities ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z,}$ the character of the shortest line is such that

$${\frac {d\xi }{X}}={\frac {d\eta }{Y}}={\frac {d\zeta }{Z}}.$$

13.

To every point of a curved line upon a curved surface there correspond two points on the sphere, according to our point of view; namely, the point ${\textstyle \lambda ,}$ which represents the direction of the linear element, and the point ${\textstyle L,}$ which represents the direction of the normal to the surface. The two are evidently ${\textstyle 90^{\circ }}$ apart. In our former investigation (Art. 9), where [we] supposed the curved line to lie in a plane, we had two other points upon the sphere; namely, ${\textstyle {\mathfrak {L}},}$ which represents the direction of the normal to the plane, and ${\textstyle \lambda ',}$ which represents the direction of the normal to the element of the curve in the plane. In this case, therefore, ${\textstyle {\mathfrak {L}}}$ was a fixed point and ${\textstyle \lambda ,}$ ${\textstyle \lambda '}$ were always in a great circle whose pole was ${\textstyle {\mathfrak {L}}.}$ In generalizing these considerations, we shall retain the notation ${\textstyle {\mathfrak {L}},}$ ${\textstyle \lambda ',}$ but we must define the meaning of these symbols from a more general point of view. When the curve ${\textstyle s}$ is described, the points ${\textstyle L,}$ ${\textstyle \lambda }$ also describe curved lines upon the auxiliary sphere, which, generally speaking, are no longer great circles. Parallel to the element of the second line,