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 $P,$  $Q,$  $R$ respectively. If $ds$  is an element of the curve; $\lambda$  the point upon the auxiliary sphere, which represents the direction of this element; $L$  the point giving the direction of the normal as above; and $\xi,$  $\eta,$  $\zeta;$ $X,$  $Y,$  $Z$  the coordinates of the points $\lambda,$  $L$  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 $d\xi,$  $d\eta,$  $d\zeta.$ And since $P,$  $Q,$  $R$  are proportional to the quantities $X,$  $Y,$  $Z,$  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 $\lambda,$ which represents the direction of the linear element, and the point $L,$  which represents the direction of the normal to the surface. The two are evidently $90^{\circ}$  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, $\mathfrak{L},$ which represents the direction of the normal to the plane, and $\lambda',$  which represents the direction of the normal to the element of the curve in the plane. In this case, therefore, $\mathfrak{L}$  was a fixed point and $\lambda,$  $\lambda'$ were always in a great circle whose pole was $\mathfrak{L}.$  In generalizing these considerations, we shall retain the notation $\mathfrak{L},$  $\lambda',$  but we must define the meaning of these symbols from a more general point of view. When the curve $s$ is described, the points $L,$  $\lambda$  also describe curved lines upon the auxiliary sphere, which, generally speaking, are no longer great circles. Parallel to the element of the second line,