In Fig. 514, let B G E D F A C be the elevation of the required article. Draw the plan in line with the elevation, making like points correspond in the two views, as shown by M O S T U P N. Let D M E be a half section of the cylinder in the elevation and D1 M1 E1 a corresponding section in the plan.

Divide these sections into any convenient number of equal parts, commencing at the same point in each, as shown by the small figures, and draw the center line of the cylinder in plan D1 R. From each of the points in the section shown in elevation carry lines parallel to D

Fig. 514. - A Cone Intersected by a Cylinder of Less Diameter than Itself at Right Angles to Its Axis.

F cutting the side of the cone, and extend them some distance into the figure for further use. From the several points of intersection with the side of the cone, as shown by a, b, c, d and e, drop lines parallel to the axis of the cone cutting the line D1 R of the plan, giving the points a1,bl,c1,dl and e1 and through each of these points, from R as center, describe an arc, as indicated in the engraving. From the points in the profile D1 M1 E1 of the plan draw lines parallel to the sides of the cylinder, producing them until they meet the arcs drawn through corresponding points, giving the points indicated by 11, 21, 31, 41 and 51. From these points carry lines vertically to the elevation, producing them until they meet the lines drawn from points of corresponding numbers in the profile of the cylinder in the elevation, giving the points 12, 22 32, 42 and 52 A line traced through these points, as shown from G to F, will be the miter line in elevation formed by the junction of the cylinder and the cone.

Fig. 515. - Half Pattern of the Cone Shown in Fig. 514.

To obtain the envelope of the cone with the opening to fit the intersecting cylinder proceed as follows: From any convenient point, as A1, Fig. 515, draw A1 B1, in length equal to A B of the elevation. Set off points e2, d2, c2, b2 and a2 in it, corresponding to e, d, c, b and a of A B, Fig. 514. From A1 as center, with radius A1 B1, describe the arc B1 V, Upon which lay off the stretchout of the plan of the cone, as indicated by the small figures outside of the pattern. (But one-half of the envelope of the cone is shown in the engraving.) From the same center-A1 describe arcs from the points e2, of, c2, b2 and a2 From the center R of the plan draw lines to the circumference through the points 21, 31, 41, etc., giving the points in the circumference marked 22, 33, 43, etc. Set off by measurement corresponding points in the arc B1 V, as shown by 34, 24, 44, 54, etc. From these points draw lines to the center A1, intersecting the arcs of corresponding number drawn from a2, b2, e2, etc. A line traced through these points of intersection, as shown by F1 O1 G1 P1, will be the shape of the opening to be cut in the side of the cone to fit the mitered end of the cylinder.

The pattern for the cylindrical part is shown above the elevation, and is obtained in accordance with the principles demonstrated in the first section of this chapter, which need not be here repeated.