The principles involved in the solution of this problem are exactly the same as those of the problem immediately preceding, to which the reader is referred for a more full explanation of the operation. The details or conditions differ only in the angle at which the cylinder joins the side of the cone.

Fig. 521. - A Cylinder Joining the Frustum of a done at an Oblique Angle.

In Fig. 521, let C B L be the elevation of a right cone of which C c l L is a frustum, and let M T U N represent the cylinder which is to join the frustum, making the angle U N L greater than a right angle The first operation will he to determine the shape of miter line M N of side elevation. Draw V \V X. the section of the cylinder, which divide into any convenient number of equal parts, as indicated by the small figures, and from these points carry lines parallel with N X cutting the side B L of the cone in the points A. G and E, producing them until they cut the axis B Y in the points H. F and D. Draw a plan under the side elevation of cone, as shown by OSQ P, winch divide into any convenient number of equal parts. From points 1 to 4 in S Q P carry lines vertically to the base C L, and thence toward the apex B, cutting the lines D E, F G and H A.

Draw a second elevation of teh one. as shown by

C1 B2 L1. which represents the cone as turned quarter way round. Draw a corresponding plan under the end elevation, as shown by S1 Q1 P1 O1. Divide this plan into the same number of equal parts, commencing to number them at the same point as in the other plan that is, at the point Q. From the points 1 to 4 inclusive in S1 Q1 P1 of plan carry vertical lines to the base C1 L1. and thence to the apex B2.

The next step is to construct sections of the cone as it would appeal if cut upon the planes represented by the lines H A. F G and D E. For this purpose 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 the base C L with D E, cut corresponding lines in the second elevation, and through the points of intersection thus established trace a line, as shown by Nl E3 N2

Fig. 522. - Envelope of Cone Shown in Fig. 521.

Continue the axis Y1 B2 as may be convenient, upon which set off spaces equal to those between the points in D E, and through these points draw lines at right angles to D1 E1. Place the T-square parallel to the axis Y1 B2, and, bringing it against the several points in N1 F3 N2 cut the lines drawn through D1 E1, as shown, and through these points of intersection trace a. line, as shown by N4 E4 N3. Then N4 E1 N3 is a section of the cone as it would appear if cut on the line D E. Sections corresponding to F G and H A can be obtained in a similar manner.

Having obtained sections of the cone corresponding to the several lines D E, F G and H A. it will next be necessary to project a plan at right angles to the axis of the cylinder, in which each of these sections shall find its place. Therefore, from all the points of the cylinder and of its intersections with the sides and axis of the cone project lines at right angles to N X indefinitely, through which at any convenient point draw a line, as D2 X1, parallel to N X. Upon this line, as a center of the plan about to be constructed, place the oblique sections just obtained so that each may be in line with the line in the elevation which it represents, and their center lines shall all coincide with D2 X1. all as shown. Make T2 U2 equal to T U and complete the plan of the cylinder, opposite the end of which draw a profile, as indicated by V1 W1 N1, commencing the divisions at the point V. From the several points in the profile V1 W1 X1 drop lines paral-lel with the center line D2 X1 against the several profiles d E2 d1,f G2 f2 and h A2 h 1 and hence drop the points back on the elevation, cutting corresponding lilies in it. Thus, from the intersection of the line drawn from point W1 (3) with G2 f1 of section cut the line F G, which in the elevation corresponds to the point 3 in the profile V W X. From the intersection of a line drawn from point 2 in V1 W1 with A2 h1 of section cut the line H A, and so on, as indicated by the dotted lines. A line traced through these points of intersection, as shown by the curved line M X, will he the miter line in elevation, from which the patterns can be obtained as follows:

For the pattern of cylinder shown in elevation by M T D N. on T U extended lay off a stretchout of profile V W X, through the points in which draw the usual measuring lines. Place the T-square parallel with T U, and, bringing it against the points in the miter line M N. cut measuring lines of corresponding number. Trace a line through the points thus obtained, as shown from m to m'1 Then m t t' m' is the pattern of the cylinder to fit against the cone, as shown in elevation by M T U N.

To obtain the pattern of the frustum carry lines from each of the points in the miter line M N horizon-tallv across the elevation, cutting the side of the frustum c C, as shown by a1 b1 and d1; also through the same points draw lines from the apex B, cutting the base line C L, and thence drop them on the plan, as shown by 1, 2, a and b. From any convenient point, as IV in Fig 522, as a center, with radii equal to B c and B C, describe arcs, as shown by O Q O1 and c1l1 Make O Q O1 equal in length to the plan of cone O S Q P and upon it set off each way from the point Q spaces equal to those upon the plan between Q and S. From these points draw lines indefinitely toward the center B2. With B2 as center describe arcs whose radii correspond to B M, B a1, B b1 B d1 and B N, cutting lines of corresponding number or letter. Then a line traced through the intersections thus obtained will be the shape of opening to cut in the envelope of frustum where it joins the cylinder, and lines drawn from 0 and O1 toward B2 till they cut the arc c1 l1 in the points c1 and ll will complete the pattern of the frustum.