In the plan, Fig. 496, let A B C D F represent a frustum of the cone B C G, B H O being the half profile of cone at its base and A D J the plan of the cylinder. In line with the cylinder in plan draw the elevation, as shown by R S T U. With the T-square placed parallel to the sides of the cylinder, cany a line from the point G in plan to any convenient point, as G' of elevation. At right angles to G G' draw G' Q indefinitely, and extend B C through Q G', cutting same at O. With O as center, and E C of plan as radius, describe the semicircle L Q M, representing one-half of the profile of the cone at the larger end. Divide L Q M into any convenient number of equal parts, as indicated by the small figures. From the points thus obtained carry lines at right angles to L M, cutting that line as indicated. From the points thus obtained in L M carry lines to the apex G', as indicated by the dotted lines in the engraving. Divide B H C into the same number of equal parts as was L Q M, numbering them to correspond with the elevation, as shown, and from the points thus obtained carry lines at right angles to B C, cutting that line as indicated. From the points in B C carry lines to the apex G, cutting the plan of cylinder as shown. With the blade of the T-square placed parallel with G G', and brought successively against the points thus established in the plan of the cylinder, cut lines of corresponding number drawn from the points in L M to the apex G', as indicated from K to N, and extend these lines to the line M G'. A line traced through these points of intersection, as shown by K P N, represents the intersection of cone with cylinder in elevation, as shown by A F D in plan.

For the pattern proceed as follows: From G' as enter, with G' M as radius, describe an arc, as indicated by l 0 m, and, starting from l, step off the stretchout of the half profile LQM, as indicated by the small figures. If the entire pattern is required in one piece extend the arc lo m, and from in set off a duplicate of l o m, numbering the points in inverse order. From the points thus obtained draw radial lines to G', as indicated. Then with G' as center, and with radii corresponding to the distance from G' to the points established in M G', describe arcs, producing them until they cut lines of corresponding number drawn from G'. A line traced through these points of intersection, as shown by k p n, will with l o m give the pattern of part of article shown in elevation by L M N P K.

Fig. 496.   Pattern of a Cone Intersected by a Cylinder at its Upper End.

Fig. 496. - Pattern of a Cone Intersected by a Cylinder at its Upper End.

It will be easily seen that the pattern might have been obtained directly from the plan without the trouble of drawing the elevation, as in Problems 141, 142 and 143. Should it be desirable, however, to cut an opening in the side of the cylinder to fit the frustum of the cone, the hights of all points in the perimeter of such opening must be obtained from the line NPK of the elevation, while width of the opening upon lines corresponding to these points must be measured from F toward D or A upon the circumference of the cylinder.