Let S P R T in Fig. 491 be the elevation of the cylinder, and a G K H b the elevation of the frustum. Draw the axis of the cylinder, A B, which prolong, as shown by C D, on which construct a profile of the cylinder, as shown by C E D F. Produce the sides of the frustum, as shown in the elevation, until they meet in the point L, which is the apex of the cone. Draw the axis L K, which produce in the direction of O, and at any convenient point upon the same construct a plan of the frustum at its top, a b.

In connection with the profile of the cylinder draw a corresponding elevation of the cone, as shown by K1 a1 b1 K2. Produce the sides K1 a1 and K2 b1 until they intersect, thus obtaining the point L1, the apex corresponding to L of the elevation. Draw the axis L1 E, as shown, which produce in the direction of N1, and upon it draw a second plan of the frustum at a b, as shown by M1 O1 N1. Divide the plans M O N and M1 O1 N1 into the same number of equal parts, commencing at corresponding points in each, as shown. With the T-square set parallel to the axis of the cones, and brought successively against the points in the plans, drop lines to the lines a b and a1 b1, as shown.

From L' draw lines through the points in a1 b1, cutting the profile of the cylinder, as shown in K1 E K2 and in like manner from the apex L draw lines indefinitely through the points in a b. Place the T-square parallel to the sides of the cylinder, and, bringing it against the points in the profile K1 E K2 just described, cut corresponding lines in the elevation, as shown at H K G. A line traced through these points of intersection, as shown by H K G, will form the miter line between the two pieces as it appears in elevation.

This miter line is not necessary in obtaining the pattern, but the method of obtaining it is here introduced merely to show how it may be done, should it be desired under similar circumstances in any other ease. The development of the pattern in this case could be most easily accomplished by using L1 as a center from which to strike ares from the various points on the line a1 K1. The same result is accomplished, however, by continuing the lines drawn from K1 E K2 until they meet the side a G of the cone prolonged, as shown from G to Z. Thus a Z becomes in all respects the same as a1 K1.

Fig. 491. - The Pattern of a Frustum of a Cone Intersected at Its Lower End by a Cylinder, Their Axes Intersecting at Right Angles.

From L as center, and with radius L a, describe the arc b2 a2, upon which lay off a stretchout of the plan M O N of the frustum. Through each of the points in this stretchout draw lines indefinitely, radiating from L, as shown, Number the points in the stretchout a2 b2 corresponding to the numbers in the profile, commencing with the point occurring where it is desired to have the seam. Set the compasses to

L Z as radius, and, with L as center, describe an are cutting the corresponding lines drawn through the stretchout, as shown by 1, 5 and 1. In like manner reduce the radius to the second point in G Z, and describe an arc cutting 2, 4, 4 and 2. Also bring the pencil to the third point and cut the lines corresponding to it in the same way. Then a line traced through the points thus obtained, as shown by H1 K3 G1, will be the pattern of the frustum.

PROBLEM 141. The Pattern for a Conical Boss.

The principles and conditions in this problem are exactly the same as those in the one immediately preceding (that is, the frustum of a cone mitering against a cylinder, their axis being at right angles), but its pro-portions are so different that it is here introduced as showing that the same application of principles often produces results so widely differing in appearance as to be scarcely recognizable.

Fig. 492. - The Pattern for a Conical Boss.

Let A B C D of Fig. 492 represent the elevation of the boss that is required to fit against the cylindrical can, a portion of the plan of which is shown by the arc A B. The plan at the smaller end of the boss is represented by K F G H. Continue the lines A D and B C until they intersect at K, which is the apex of the cone of which the boss is a frustum. An inspection of the elevation will show that it is only necessary to describe one-fourth of the pattern, the remaining parts being duplicates. Divide one-quarter of the plan into any convenient number of parts, in the present instance four, as shown by the points in H E.. Drop lines from these points to the base D C, as shown. Draw lines from K through the points in the base until they intersect the arc at A B, which represents the body of the can. These points can be numbered to correspond with the points in the plan from which they are derived. At right angles to the line F K draw lines from the points on A B until they strike the line A K, where their true distances from K can be measured. With K as a center, and K D as radius, strike the arc L M N, equal in length to the circumference of plan.

If the whole pattern of boss is to be described from measurements derived from elevation it will be necessary to reverse the order of the numbers for each quarter, as shown. From K draw lines extending outwardly through these points, as indicated by the small figures. With K as center, draw an are from the point 1' until it. intersects radial lines 1 drawn from K, as shown at O, Z and R. In the same manner draw an are from 2' to lines 2, etc, as shown. A line traced through these points will produce the desired patterns, as shown by LOSZVRN.

PROBLEM 142. Pattern for the Lip of a Sheet Metal Pitcher.