Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/106

or

$${\begin{aligned}{\frac {1}{r}}&={\frac {\xi ^{2}T+2\xi \eta U+\eta ^{2}V}{-\xi 't-\eta 'SIC{\mu }{u}+\zeta '}}\\&={\frac {Z(\xi ^{2}T+2\xi \eta U+\eta ^{2}V)}{X\xi '-Y\eta '+Z\zeta '}}\\&={\frac {Z(\xi ^{2}T+2\xi \eta U+\eta ^{2}V)}{\cos L\lambda '}}.\end{aligned}}$$

10.

Before we further transform the expression just found, we will make a few remarks about it.

A normal to a curve in its plane corresponds to two directions upon the sphere, according as we draw it on the one or the other side of the curve. The one direction, toward which the curve is concave, is denoted by ${\textstyle \lambda ',}$ the other by the opposite point on the sphere. Both these points, like ${\textstyle L}$ and ${\textstyle {\mathfrak {L}},}$ are ${\textstyle 90^{\circ }}$ from ${\textstyle \lambda ,}$ and therefore lie in a great circle. And since ${\textstyle {\mathfrak {L}}}$ is also ${\textstyle 90^{\circ }}$ from ${\textstyle \lambda ,}$ ${\textstyle {\mathfrak {L}}L=90^{\circ }-L\lambda ',}$ or ${\textstyle =L\lambda '-90^{\circ }.}$ Therefore

$$\cos L\lambda '=\pm \sin {\mathfrak {L}}L,$$

where ${\textstyle \sin {\mathfrak {L}}L}$ is necessarily positive. Since ${\textstyle r}$ is regarded as positive in our analysis, the sign of ${\textstyle \cos L\lambda '}$ will be the same as that of

$$Z(\xi ^{2}T+2\xi \eta U+\eta ^{2}V).$$

And therefore a positive value of this last expression means that ${\textstyle L\lambda '}$ is less than ${\textstyle 90^{\circ },}$ or that the curve is concave toward the side on which lies the projection of the normal to the surface upon the plane. A negative value, on the contrary, shows that the curve is convex toward this side. Therefore, in general, we may set also

$${\frac {1}{r}}={\frac {Z(\xi ^{2}T+2\xi \eta U+\eta ^{2}V)}{\sin {\mathfrak {L}}L}},$$

if we regard the radius of curvature as positive in the first case, and negative in the second. ${\textstyle {\mathfrak {L}}L}$ is here the angle which our cutting plane makes with the plane tangent to the curved surface, and we see that in the different cutting planes passed through the same point and the same tangent the radii of curvature are proportional to the sine of the inclination. Because of this simple relation, we shall limit ourselves hereafter to the case where this angle is a right angle, and where the cutting