Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/20

Denoting the angles of this triangle simply by ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L''}$ and the perpendicular drawn on the sphere from the point ${\textstyle L''}$ to the side ${\textstyle LL'}$ by ${\textstyle p,}$ we shall have

$$\sin p=\sin L.\sin LL''=\sin L'.\sin L'L'',$$

and

$$\lambda L''=90^{\circ }\mp p,$$

the upper sign being taken for the second case, the lower for the third. From this it follows that

$${\begin{aligned}\pm \Delta &=\sin L.\sin LL'.\sin LL''=\sin L'.\sin LL'.\sin L'L''\\&=\sin L''.\sin LL''.\sin L'L''\end{aligned}}$$

Moreover, it is evident that the first case can be regarded as contained in the second or third, and it is easily seen that the expression ${\textstyle \pm \Delta }$ represents six times the volume of the pyramid formed by the points ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L''}$ and the centre of the sphere. Whence, finally, it is clear that the expression ${\textstyle \pm {\frac {1}{6}}\Delta }$ expresses generally the volume of any pyramid contained between the origin of coordinates and the three points whose coordinates are ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z;}$ ${\textstyle x',}$ ${\textstyle y',}$ ${\textstyle z';}$ ${\textstyle x'',}$ ${\textstyle y'',}$ ${\textstyle z''.}$

3.

A curved surface is said to possess continuous curvature at one of its points ${\textstyle A,}$ if the directions of all the straight lines drawn from ${\textstyle A}$ to points of the surface at an infinitely small distance from ${\textstyle A}$ are deflected infinitely little from one and the same plane passing through ${\textstyle A.}$ This plane is said to touch the surface at the point ${\textstyle A.}$ If this condition is not satisfied for any point, the continuity of the curvature is here interrupted, as happens, for example, at the vertex of a cone. The following investigations will be restricted to such surfaces, or to such parts of surfaces, as have the continuity of their curvature nowhere interrupted. We shall only observe now that the methods used to determine the position of the tangent plane lose their meaning at singular points, in which the continuity of the curvature is interrupted, and must lead to indeterminate solutions.

4.

The orientation of the tangent plane is most conveniently studied by means of the direction of the straight line normal to the plane at the point ${\textstyle A,}$ which is also called the normal to the curved surface at the point ${\textstyle A.}$ We shall represent the direction of this normal by the point ${\textstyle L}$ on the auxiliary sphere, and we shall set

$$\cos(1)L=X,\quad \cos(2)L=Y,\quad \cos(3)L=Z;$$

and denote the coordinates of the point ${\textstyle A}$ by ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z.}$ Also let ${\textstyle x+dx,}$ ${\textstyle y+dy,}$ ${\textstyle z+dz}$ be the coordinates of another point ${\textstyle A'}$ on the curved surface; ${\textstyle ds}$ its distance from ${\textstyle A,}$