Denoting the angles of this triangle simply by and the perpendicular drawn on the sphere from the point to the side by we shall have
and
the upper sign being taken for the second case, the lower for the third. From this it follows that
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 represents six times the volume of the pyramid formed by the points and the centre of the sphere. Whence, finally, it is clear that the expression expresses generally the volume of any pyramid contained between the origin of coordinates and the three points whose coordinates are
3.
A curved surface is said to possess continuous curvature at one of its points if the directions of all the straight lines drawn from to points of the surface at an infinitely small distance from are deflected infinitely little from one and the same plane passing through This plane is said to touch the surface at the point 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 which is also called the normal to the curved surface at the point We shall represent the direction of this normal by the point on the auxiliary sphere, and we shall set
and denote the coordinates of the point by Also let be the coordinates of another point on the curved surface; its distance from