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shown, but they will not be symmetrical curves as in the problem just solved, but such as are seen in full lines in No. 4 (Fig. 179), the dotted ones being the return lines of intersection of the penetrating or horizontal cylinder, with the surface of the vertical one, which is out of sight. As the correct finding of the intersecting lines shown in full, and dotted in No. 4 of the two cylinders is but a matter of careful projection by the student, no further assistance than that given in the diagram will be required by him. 73. The next case is that of the intersections of unequal cylinders, at an angle other than a right angle, and having their axes in different planes at a given distance apart. The problem is Problem 77 (Fig. 180). Two cylinders of unequal diameter, with their axes in parallel planes, intersect each other at an angle of 30 ; it is required to find the projections and lines of penetration of the solids, when the axis of one of them is vertical, and the other paral- lel to the VP. 170 FIRST PRINCIPLES OF MECHANICAL AND ENGINEERING DRAWING 171 As one of the cylinders is inclined, an elevation of both as if they were entire must first be drawn, as shown in No. 1. From this obtain by direct projection the plan of the solids as given in No. 2. Then to find the visible lines of intersection, assume parallel planes represented by the lines 1, 2, 3, in No. 2, to pass vertically through both solids. An elevation of their sections made by these planes will show where they intersect, and will give the points through which the required lines will pass. For the return lines of intersection, or those out of sight shown