Let B A K in Fig. 519 be the elevation of a right cone, perpendicular to the side of which a cylinder, L S T M, is to be joined. The first operation will be to describe the miter line as it would appear in elevation. Draw the section U V of the cylinder, which divide into any convenient number of equal parts, as indicated by the small figures, and from these points drop lines parallel to L S, cutting the side A K of the cone in the points H, F and D, producing them until they cut the axis A X in the points G, E and C. In order to ascertain at what point each of these lines will cut the envelope of the cone it will be necessary to construct sections of the cone as it would appear if cut on the lines G H, E F and C D. Draw a second elevation of the cone, as shown by B1 A1 K1, representing the cone turned quarter way round; the first may be regarded as a side elevation and this as an end elevation. Draw a plan under the side elevation of the cone, as shown by NRPO, which divide into any convenient number of equal parti, and in like manner draw a corresponding plan or half plan under the end elevation, as shown by R1 P1 O1. Divide this second plan into the same spaces, numbering them to correspond with the other plan. From the points 1 to 5 in plan NRPO carry lines vertically to the base B K and thence toward the apex A, cutting the lines C D, E F and G H. In like manner, from the same points (1 to 5 inclusive) in the plan R1 Pl O1 carry vertical lines to the base B1 K1 and thence toward the apex A'. Place the T-square at right angles to the axes of the two cones, and, bringing it against the points of intersection of the lines from X to K with C D, cut corresponding lines in the second elevation, and through the points of intersection thus established trace a line, as shown by C3 C4 Produce the axis X1 A1 to any convenient distance, upon which set off C1 D1, in length equal to C D, in which set off the points corresponding to the points in C I), and through these points draw lines at right angles to C1 D1. Place the T-square parallel to the axis X1 A1, and, bringing it against the several points in C2 C4, cut the lines of corresponding number drawn through C1 D1, as shown, and through the intersections thus established trace a line, as shown. Then C1 D1 is a section of the cone as it would appear if cut on the line C D.

In like manner carry lines from the points upon

E F across to the end elevation, intersecting them with lines of corresponding number, as shown from E3 to E4, and thence carry them parallel to the axis, cutting lines drawn through E1 F1, which with its points has been made equal to E F. The resulting profile E1 F1 is a section of the cone as it would appear if cut on the line E F. Also use the points in G H in like manner, establishing the profile G1 H1, which represents a section of the cone as it would appear if cut on the line G H. (Some of the lines indicating the operation in connection with sections E1 F1 and G1 H1 are omitted in the engraving to avoid confusion.)

Fig. 519. - A Cylinder Joining a Cone of Greater Diameter than Itself at Right Angles to the Side of the Cone.

Having thus obtained sections of the cone corresponding to the several lines C D, E F, G H, it will next be necessary to project a plan of the cone, with its intersecting cylinder, at right angles to L S. or as viewed in the direction of A K, which plan shall include all of these sections. To do this extend the line A K to a convenient distance above the elevation, and project lines from all other important points parallel to the same, as shown. At right angles to A K draw any line, as C2 V1, as a center line of the new plan. As the points D1, F1 and H1 of the oblique sections of the cone are all in the line A K, transfer these sections to the new plan, so placing them that their center lines shall coincide with the center line of the new plan, and the points D1, F1 and H1 shall be at the intersection of A K with the center line of the plan, all as shown. Opposite the end of the cylinder draw a section, as indicated by U1 V1, which divide into the same number of equal parts as used in the divisions of U V, commencing the division at corresponding points in each. As both halves of the cylinder and of the cone, when divided by a vertical plane passing through the axis of each, are the same, only one-half of the section of the cylinder has been numbered. From the points in U1 V drop lines parallel to C2 V1, each line cutting its corresponding section, as shown from x toy, and then carry them parallel to A K back to the elevation, cutting lines of corresponding number in that view. That is, from the intersection of the line drawn from point 4 in U1 V1 with the profile C2 L3 cut the line C D, which in the elevation corresponds to the point 4 in the profile U V, and from the intersection of a line drawn from 3 with E2 L3 cut the line E F, and so on, all as indicated by the dotted lines. Then aline traced through these points of intersection, as shown by L M, will be the miter line in elevation, from which the patterns may readily be obtained.

Fig. 520. - Envelope of Cone. Shown in Fig. 519.

The intersections in the plan above give all that is necessary to obtain the pattern of the cylinder, which can be done as follows: Lay off a stretchout of the profile IP V opposite the end S2 T2, through the points in which draw the usual measuring lines. Place the T-square at right angles to the same, and, bringing it against the points in the miter line LM (or the points of intersection in x y in the plan from which L M was obtained), cut the corresponding measuring lines. Then a line traced through these points, as shown from L1 to M1, will be the shape of the pattern of the cylinder to fit against the cone.

For the pattern of the cone proceed as follows: From each of the points in the miter line L M carry lines horizontally across, cutting the side A B of the cone, by means of which their distance from the apex A may be accurately measured; also through these points draw lines from the apex A cutting the base B K, continuing them vertically into the plan N R P O, as shown. It may be noted that the line from point 4 on L M falls at point 2 in the plan of the cone; likewise that the line from 3 on L M falls at 3 in the plan of the cone, while the line from 2 falls upon the plan of the cone at a point marked a. From any convenient point, as A2 of Fig. 520, with a radius equal to A B, describe the arc B2 K2 B3, which in length make equal to the circumference of the plan of the cone, setting off in the same all the points of the plan, as indicated by the figures and letters, and from these points draw lines toward the center A2, all as shown. From A2 as center, with radii corresponding to the distances A L, A 2, A 3, A 4 and A M of the elevation, strike ares intersecting corresponding lines just drawn. Then a line traced through the intersections thus obtained will be the shape of the opening to be cut in the envelope of the cone to fit the end of the cylinder.