In Fig. 524, let M N P O be the side elevation of the larger frustum and F1 G1 S R the side elevation of the smaller, the two joining upon a line between the points R and S, which line must be obtained before the patterns can be developed. Produce the sides S G1 and R F1 until they meet in the point E. At any convenient place on the line of the axis of the smaller frustum draw the profile H F K G, corresponding to the end F1 G1. Divide this profile into any convenient number of equal parts, as shown by the small figures 1, 2, 3, etc., and from these divisions, parallel to the axis of the cone, drop points on to F1 G1 From the apex E, through these points in F1 G1, carry lines cutting the side F P of the larger frustum, and producing them until they meet the center line, or the base O P, all as shown by B A. C A1 and D A1.

The next step is to construct sections of the larger frustum as it would appear if cut on each of these lines, from which to obtain points of intersection with the lines of the smaller frustum for determining the miter line from R to S in the elevation. Draw the plan of the base of the larger frustum, as shown by T U V W. and divide one-half of it in the usual manner. From these points carry lines vertically to the base O P of the frustum. Produce the sides O M and P N until they meet in the point L. From the points in the base line obtained from the plan carry lines toward the apex L, cutting the section lines. A B,

Fig. 524.   Patterns of the Frustums of Two Unequal Cones Intersecting Obliquely.

Fig. 524. - Patterns of the Frustums of Two Unequal Cones Intersecting Obliquely.

A1 C and A2 D, as shown. Parallel to A B and of the same length, at any convenient point outside of the elevation, draw A3 B1, and from the points in A B project lines at right angles cutting A3 B1, as shown, in the points 4, 3 and 2. In like manner make A4 C1 equal and parallel to A C, and from the points in A C project lines at right angles to it, cutting it as shown, giving the points i, 3 and 2, Also make A5 D1 equal to the section line A2 D of the elevation, and cut it by lines from the points in A2 D, obtaining the points 3 and 2, as shown. In order to complete these several sections, the width of the frustum through each of the points indicated is to be set off on corresponding lines drawn through A3 B1, A4 C1 and A5 D1. To obtain the width through these points first draw an end elevation of the larger frustum, as shown by M1 N1 P1 O1. Produce the sides, obtaining the apex L'. Draw a plan and divide it into the same number of spaces as that shown in TUV W, and commence numbering at a corresponding point, all as indicated by V U1 T1 W. From the points in the plan carry lines vertically to the base O1 P1, and thence toward the apex L1. Place the blade of the T-square at right angles to the axis of the cone, and, bringing it successively against the points in the section line A B in the side elevation, draw lines cutting the axis of the end elevation, and cutting the lines corresponding in number to the several points in A B, all as shown by a a, b b and c c. Make the length of the lines drawn through A3 B1 equal to the corresponding lines thus obtained, as shown by a1 a1, b1 b1, cl c1 and d1 d1 and through these extremities trace a line, as shown by d1 B1 d1, which will be the section through the cone when cut on the line A B. In like manner complete the sections l1 C1 l1 and f1 D1 f1.

As remarked in the previous problem, it is only necessary that these sections should be developed far enough from the points B1, C1 and D' to receive the lines representing the sections of the smaller frustum. Produce A3 B1, making B1 E1 equal to B E of the elevation, and B1 X3 equal to B X1 of the elevation. In like manner make C1 E2 equal to C E, and C X1 equal to C X. Make D1 E3 equal to D E, and D1 X' equal to D X2. Through X3, at right angles to B1 E1, draw a line in length equal to the line 2 2 drawn across the profile F K G H, with which this section corresponds, as shown by 2' 2'. Through X4 draw a line equal to H K, as shown by H1 K1, and through X5 draw 4' 4', in length equal to the line 4 4 drawn through the profile F K G H. From E1, through the extremities of 2' 2', draw lines cutting the section. In like manner draw lines from E2 through the points H1 K1, and from E3 through the points 4' 4'. From the points at which these lines meet the sections, as a1 a2 in the first, o o in the second and m m in the third, carry lines back at right angles to and cutting the corresponding section lines in the elevation. A line traced through the points thus obtained, as shown by R S, is the miter line in elevation formed by the junction of the two frustums.

Having thus obtained the miter line in elevation, proceed to develop the patterns as follows: From the points in R S, at right angles to A1 E, which is the axis of the smaller cone, draw lines cutting the side E S, as shown by the small figures 1, 2, 3, 4 and 5. These points are to be used in laying off the pattern of the smaller frustum. From E as center, with radius E G1, describe the arc F2 G2, upon which step off the stretchout of the profile F K H G, numbering the points in the usual manner. Through the points, from the center E, draw radial lines indefinitely. From the same center, E, with radius E 1 (of the . points in E S), cut the radial line numbered 1, and in like manner, with radii E2, E3, etc., cut the corresponding numbers of the radial lines. A line, R1 S1, traced through the several points of intersection thus formed will be the larger end of the pattern for the small frustum, thus completing the shape of that piece, all as shown by R1 S1 G2 F2

Fig. 555.   The Envelope of the Larger Frustum Shown in Fig. 524.

Fig. 555. - The Envelope of the Larger Frustum Shown in Fig. 524.

Pattern Proplems.

To avoid confusion of lines, the manner of obtaining the envelope of the large frustum is shown in Fig. 525, which is a duplicate of the side elevation and plan shown in Fig. 524, the miter line R1 S1 and the points in it being the same. Similar letters refer to corresponding parts in the several figures. From L2 as center, with radius L2 O2, describe an arc, as shown by V Z, and from the same center, with radius L2 M2 describe a second are, as shown by y x. Draw

Y y. and upon Y Z lav off the stretchout of the plan

V3 W2 T2 , all as shown. Draw Z z. Then ZayY will be the envelope of the large frustum. Through the points in the miter line R1 S1 draw lines from the apex of the cone to the base, and from the base continue them at right angles to it until they meet the circumference of the plan. Mark corresponding points in the stretchout Y Z, and insert any points which do not correspond with points already fixed therein. From each of the points thus designated draw a line across the envelope already described to the apex, as shown by 3 L1, x L2 and 1 L2. Also, from the points in the miter line R1 S1 draw lines at right angles to the axis of the frustum cutting the side L2 O2, as shown. From L2 as center describe arcs corresponding to each of these points and cutting the radial lines drawn across the envelope of the cone. A line traced through the points of intersection between arcs and lines of the same number, as shown by h R2 h1 S2, will be the shape of the opening to fit the base of the smaller frustum.