In Fig. 512, E G H F represents an elevation of the cylinder, and M N L K an elevation of the frustum of a cone intersecting it. F1 Z Q represents the profile or plan of the cylinder, to which it will be necessary to add a correctly drawn plan of the frustum before the miter line in elevation can be obtained. At any convenient point on the axial line T O of the cone construct the profile V Y X W, which represents a section through the cone on the line M N. Divide the section V Y X W into any convenient number of equal spaces in the usual manner, as shown by the small figures 1. 2, 3, 4, etc. From each of the points thus established drop lines parallel with the axis of the cone cutting the line M N, From the intersections in M N thus obtained drop points parallel with the side G H of the cylinder and continue them indefinitely, cutting the line F O1, which is drawn through the center of the plan of the cylinder at right angles to the elevation, all as shown in the engraving. Make Y1 W1 equal to Y W of the first section c instructed. In like manner measure distances from the center line V X of the first section to the points 2, 3, 4, etc.. and set off corresponding spaces in the plan view, measuring from M1 N1, upon lines of corresponding numbers dropped from the intersections in M N, already described. Then a line traced through these points will represent a view of the upper end of the frustum as it would appear when looked at from a point directly above it. Produce the sides of the frustum K M M and L N until they meet in the point O. From O drop a line parallel to the side G H of the cylinder, cutting the line F1 O1 in the point O1, thus establishing the position of the apex of the cone in the plan. From the point O1 thus established draw lines through the several points in the section M1 Y1 N1 W1. which produce until they intersect the plan of the cylinder in points between Z and Q, as shown in the engraving. From O, the apex of the cone in the elevation, draw lines through the several points in M N already determined, which produce until they cross G H, the side of the cylinder, and continue them inward indefinitely. Intersect these lines by lines drawn vertically from the points of corresponding number between Z and Q of the plan just determined. Then a line traced through these intersections, as indicated by K T L, will represent the miter between the frustum and cylinder, as seen in elevation.

Fig. 512.   The Frustum of a Cone Intersecting a Cylinder of Greater Diameter than Itself at Other than Right Angles.

Fig. 512. - The Frustum of a Cone Intersecting a Cylinder of Greater Diameter than Itself at Other than Right Angles.

To lav off the pattern proceed as follows: From O as center, with O N as radius, describe the are P R. on which set off a stretchout of the section V V W X in the usual manner. From O, through the several points in P R thus obtained, draw radial lines indefinitely. From the several points in the miter line KT L draw lines at right angles to the axis O T of the cone, producing them until they cut the side N L. From O as center, with radii corresponding to the distance from 0 to the several points in N L just obtained, describe arcs, which produce until they intersect radial |in3S of corresponding number drawn through the stretchout P R. Then a line traced through these points of intersection, as indicated by S L2 U, will be the lower line of the pattern sought, and P S L2 U R will be the complete pattern. The pattern for the cylinder and the opening in the same to lit the intersection of the cone is really a problem in parallel forms, with which problems (Section 1) it should properly be classed. F1 Z Q is the profile of the cylinder, and L T K is the miter line. The stretchout R D is drawn at right angles to E F, the direction of the mold or cylinder. The points between Z2 and Q2 of the stretchout are duplicates of those between Z and Q of the plan. Place the T-square at right angles to the cylinder, and, bringing it successively against the points in the miter line K T L, cut lines of corresponding numbers. A line traced through the points of intersection thus formed, as shown by Z1 K1 Q1L1, will be the shape of the required opening in the cylinder.