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occurs in practice or that where a cone is penetrated by a cylinder, for a further problem in this connection, it would be solved on the same principles, though in a different way. Problem 79 (Fig. 183). A cone with its axis in a vertical position is penetrated by a cylinder ; required the lines of intersection of the solids in plan and elevation, when the axis of the cylinder is hori- zontal, and parallel to the V P, and in the same plane as that of the cone. Draw in first the plan and elevation of the solids as if entire. Then as the axes of both are in one and the same plane, an axial section 176 FIRST PRINCIPLES OF of them will at once give 1, 2, 3, 4 in No. 2 as the extreme points in the penetration. To find others through which the lines sought will pass ; at one end of the cylinder say the left give a side elevation of the front halves of the two solids as shown. Assume the right-angled triangle a a &, and the semi-circle on c d, to be half-sections of both solids swung round on the axis of the cone until parallel with the "VP. From these it will be seen that a part of the front and back surfaces of the penetrating cylinder lie wholly within the cone. To find how much, and thus determine the lines of intersection sought, proceed as follows : Through the vertex of the right-angled triangle a a 6, draw straight lines to its base ; the first one tangential to the semi-circle on c d, and the others cutting it at suitable points, as shown. These lines are the edge views of the section planes passing through both solids, which will give the points required, as the sections produced by them will be triangles for the cone and rectangles for the cylinder. The rectangles are at once