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 ${\textstyle xy}$-plane. If, further, the origin is placed at the point ${\textstyle A}$ itself, the expression for the coordinate ${\textstyle z}$ evidently takes the form

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

where ${\textstyle \Omega }$ will be of higher degree than the second. Turning now the axes of ${\textstyle x}$ and ${\textstyle y}$ through an angle ${\textstyle M}$ 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 ${\textstyle x}$-axis, a plane curve will be obtained, the radius of curvature of which at the point ${\textstyle A}$ will be equal to ${\textstyle {\frac {1}{T}},}$ the positive or negative sign indicating that the curve is concave or convex toward that region toward which the coordinates ${\textstyle z}$ are positive.

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

III. Setting ${\textstyle z=r\cos \phi ,}$ ${\textstyle y=r\sin \phi ,}$ 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 ${\textstyle A}$ and making an angle ${\textstyle \phi }$ with the ${\textstyle x}$-axis, we shall have a plane curve whose radius of curvature at the point ${\textstyle A}$ will be

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

IV. Therefore, whenever we have ${\textstyle T=V,}$ the radii of curvature in all the normal planes will be equal. But if ${\textstyle T}$ and ${\textstyle V}$ are not equal, it is evident that, since for any value whatever of the angle ${\textstyle \phi ,}$ ${\textstyle T\cos ^{2}\phi +V\sin ^{2}\phi }$ falls between ${\textstyle T}$ and ${\textstyle V,}$ 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,