Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/107

 plane, therefore, is passed through the normal of the curved surface. Hence we have for the radius of curvature the simple formula

$$\frac{1}{r} = Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).$$

11.

Since an infinite number of planes may be passed through this normal, it follows that there may be infinitely many different values of the radius of curvature. In this case $T,$  $U,$  $V,$  $Z$ are regarded as constant, $\xi,$  $\eta,$  $\zeta$  as variable. In order to make the latter depend upon a single variable, we take two fixed points $M,$  $M'$ $90^{\circ}$  apart on the great circle whose pole is $L.$  Let their coordinates referred to the centre of the sphere be $\alpha,$  $\beta,$  $\gamma;$  $\alpha',$  $\beta',$  $\gamma'.$  We have then

$$\cos\lambda(1) = \cos\lambda M. \cos M(1) + \cos\lambda M'. \cos M'(1) + \cos\lambda L. \cos L(1).$$

If we set

$$\lambda M = \phi,$$

then we have

$$\cos\lambda M' = \sin\phi,$$

and the formula becomes

$$\begin{aligned} \xi  &= \alpha\cos\phi + \alpha'\sin\phi, \end{aligned}$$

and likewise

$$\begin{aligned} \eta &= \beta \cos\phi + \beta' \sin\phi, \\ \zeta &= \gamma\cos\phi + \gamma'\sin\phi. \end{aligned}$$

Therefore, if we set

$$\begin{aligned} A &= (\alpha^{2}T + 2\alpha\beta U + \beta^{2}V)Z, \\ B &= (\alpha\alpha'T + (\alpha'\beta + \alpha\beta')U + \beta\beta'V)Z, \\ C &= (\alpha'^{2}T + 2\alpha'\beta' U + \beta'^{2}V)Z, \end{aligned}$$

we shall have

$$\begin{aligned} \frac{1}{r} &= A\cos^{2}\phi + 2B\cos\phi \sin\phi + C\sin^{2}\phi \\ &= \frac{A + C}{2} + \frac{A - C}{2}\cos 2\phi + B\sin 2\phi. \end{aligned}$$

If we put

$$\begin{aligned} \frac{A - C}{2} &= E\cos 2\theta, \\ B &= E\sin 2\theta, \end{aligned}$$