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section parallel to an axial one is a similar figure. Therefore, if No. 1 (Fig. 178) be the given plan of the two solids, to find the lines of inter- section of their surfaces, first get an elevation of them as if entire. As only the front halves of the two cylinders will be seen in elevation, it is with them that we have to deal. If, then, a series of vertical section planes parallel to each other and the axial plane ap of the solids, be assumed to pass simultaneously through their front halves, the points of intersection of the sections of the cylinders by these planes will be points in their lines of penetration. To find these lines, draw in No. 1 (Fig. 178) parallel to the line ap, as many lines say three at convenient distances apart, as it is in- tended to use section planes. Find by projection from the points 11', 2 2', 3 3' where these lines cut the plan of the vertical cylinder the sectional elevations produced by them, as shown in dotted linejs in No. 2. For the corresponding sectional elevations of the horizontal cylinder by the same planes, on its axial line x y, No. 2 produced to the left with point x as centre and x a as radius, describe the semi-circle a x'a. This will be one-half of the end of the horizontal cylinder turned on its vertical diameter a a as a hinge. In this semi-circle set off from x in the line xx the distances that the lines 1 1', 2 2', 3 3' in No. 1 are from the line ap, and through the points thus found draw lines parallel to a a to cut the semi-circle in points 11', 22', 33'. Faint lines drawn through these points from end to end of the horizontal cylinder, parallel to x y, or the IL, will give the corresponding sections of it made by the same planes as used in the vertical cylinder. Then the points where the edges of the corresponding sections of both cylinders cut each other, as shown in No. 2, are points in their lines of intersection, which, when