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in its right side or edge ; then with these lengths, and the paper trammel, mark off a few points at either end, and through them draw the elliptic arcs to the right and left edges of the front face of the frustum, and they will be the lines of penetration of the solids seen in elevation. For the plans of these lines on the upper surface of the sphere, let fall projectors from points 1, 2, in No. 1, to cut the corresponding slant edges of the frustum (in No. 2) in points 1 1', 2 2' ; and for points b &', 5 5', in the same diagram, set off from 01, on the two lines drawn through it at right angles to each other, the distance that b or &', in No. 1, is from the axial line or point 5 ; then curved lines drawn through the points thus found, as shown in No. 2, will be the lines of penetration on the upper surface of the sphere by the frustum. The corresponding lines on its under surface, if required, are found in the same way. If the sides of the frustum are equally inclined to the VP, as in the case of the square prism in Fig. 185, and an elevation of the solids in this position be required, then, as in previous problems, transfer the plan No. 2 to the required position shown in No. 3, and from it and the elevation No. 1 find by direct projection the view given in No. 4. The dotted lines and projectors show how the major axes of the elliptical portions of the intersections are obtained, the line c c, in which the minor ones lie, being a projector drawn through din No. 1. 78. As a sphere penetrated centrally by a cylinder or cone would in each case give circles as the lines of intersection, becoming in projection either straight lines, circles, or ellipses, according to their positions with respect to the planes of projection, it is considered