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the prism at an angle of 45 with the IL or VP, as in No. 1. Then find by projection the elevation of them as if entire. As the axes of the sphere and prism are coincident, their lines of intersection in plan are in the planes of the sides of the prism, and therefore coincide with the four lines in No. 1 representing its plan. Then, in the elevation No. 2, as the sides of the prism are equally inclined to the vertical plane of projection or the VP the lines of penetration of the sphere by the prism will all be portions of ellipses, or parts of vertical plane sections of the sphere taken through it at the sides or faces of the prism seen at an angle of 45. To draw in these lines, find from the plan No. 1 the major and minor axes of the ellipses into which the circular sections of the sphere are projected in elevation, and by means of the paper trammel previously explained find the points through which the lines of intersection are drawn, as shown in No. 2 in the figure. As the lines of intersection of a prism having any number of sides with a sphere, are but a series of circular or elliptic arcs, or both, obtained in the same way as those in the case of the square prism, further examples of their penetrations are unnecessary ; and as the bounding edges of the faces of a prism are parallel, the next problem is a variant from this, or one in which the sides and edges of the penetrating solid incline equally to its axis. The problem is Problem 82 (Fig. 186). A sphere is penetrated by the frustum of a square pyramid, having its base edges parallel and perpen- dicular to the VP ; required the lines of penetration of the solids, in plan and elevation, when their axes coincide and are in a vertical tion.