Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/28

 conditions will be fulfilled at once if the tangent plane at this point be taken for the $xy$ -plane. If, further, the origin is placed at the point $A$  itself, the expression for the coordinate $z$ evidently takes the form

$$z = \frac{1}{2}T^{\circ}x^{2} + U^{\circ}xy + \tfrac{1}{2}V^{\circ}y^{2} + \Omega$$

where $\Omega$  will be of higher degree than the second. Turning now the axes of $x$  and $y$ through an angle $M$  such that

$$\tan 2M = \frac{2U^{\circ}}{T^{\circ} - V^{\circ}}$$

it is easily seen that there must result an equation of the form

$$z = \tfrac{1}{2}Tx^{2} + \tfrac{1}{2}Vy^{2} + \Omega$$

In this way the third condition is also satisfied. When this has been done, it is evident that

I. If the curved surface be cut by a plane passing through the normal itself and through the $x$ -axis, a plane curve will be obtained, the radius of curvature of which at the point $A$ will be equal to $\frac{1}{T},$  the positive or negative sign indicating that the curve is concave or convex toward that region toward which the coordinates $z$  are positive.

II. In like manner $\frac{1}{V}$  will be the radius of curvature at the point $A$ of the plane curve which is the intersection of the surface and the plane through the $y$ -axis and the $z$ -axis.

III. Setting $z = r \cos\phi,$ $y = r \sin \phi,$  the equation becomes

$$z = \tfrac{1}{2}(T\cos^{2}\phi + V\sin^{2}\phi) r^{2} + \Omega$$

from which we see that if the section is made by a plane through the normal at $A$ and making an angle $\phi$  with the $x$ -axis, we shall have a plane curve whose radius of curvature at the point $A$  will be

$$\frac{1}{T\cos^{2}\phi + V\sin^{2}\phi}$$

IV. Therefore, whenever we have $T = V,$ the radii of curvature in all the normal planes will be equal. But if $T$  and $V$ are not equal, it is evident that, since for any value whatever of the angle $\phi,$  $T\cos^{2}\phi + V\sin^{2}\phi$  falls between $T$  and $V,$  the radii of curvature in the principal sections considered in I. and II. refer to the extreme curvatures; that is to say, the one to the maximum curvature, the other to the minimum,