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dotted in No. 1 the same method of procedure is repeated as in finding those which are seen. As a vertical plane would be tangent to the extreme back surface of both cylinders, the point x in the elevation No. 1 would be where the lines of penetration of the two solids would cross. A plane passing through the vertical cylinder tangent to the front surface of the inclined one gives the two points a b in plan and elevation, as those where that surface enters and leaves the vertical cylinder. Should the plane of the axis of the inclined cylinder, repre- sented by the line a i in No. 2, be otherwise than parallel with the YP say, as in No. 3 the lines of intersection of the two solids would be found in the same way as that explained in Fig. 179, the difference in their appearance as seen in No. 4 being the result of the changed position of the solids with respect to the plane of their projection, the vertical cylinder having been turned on its axis through a certain angle, carrying the inclined cylinder with it. With the assistance of the few projectors shown, and bearing in mind that each point in the lines of intersection found in No. 1 has, by the turning of the vertical cylinder, passed through the same angle horizontally, the student should be able to find without trouble the actual lines due to the altered position of the solids which are shown in No. 4 (Fig. 180). As the intersections of cylinders in any position only require the correct use of section planes as exhibited in this chapter to determine them, no further examples are given in this connection, but we pass on to the solution of one or two problems in which the intersections of the cone and cylinder are involved. Problem 78 (Fig. 181). A cylinder in a vertical position is pene- trated at the middle of its height by a cone, having its axis horizontal