d, e, f, g, h, i, j, and draw lines from these points to the apex at A. They will cut the diagonal b-b at d'e'f'g'h'i'j', then carry horizontal lines from these points to meet the slant edge B'A in d", e", f", g', h", i", j". Then the lengths Bd", Be", Bf", Bg", Bh", Bi”, Bj", will be the actual lengths of the lines dd', ee', ff' gg', hh', ii", jj" in Fig. 142. To describe the pattern take the length AB as a radius and strike an arc of a circle as ABCD. From the point C set off to the right and left the points J, I, H, G, F, E, D, B, using the lengths of

the points of division in Fig. 141. Draw lines from the points of division to A. On these lines set off the projected lengths from Fig. 141, thus: Take the length CC' and set off from C to C' in Fig. 142. Take the length from B to j' and set it off from j to j' in Fig. 142. Take the length B to i" and set it off from i to i', and so on. A curve drawn through the points c', j', i', h′, g', f', e', and d' to right and left will give the shape of the development.

semicircle ABC representing the diameter of the larger pipe, and extend the line DA to M and CA to L, take the radius OF of the smaller semicircle and from A mark off the same distance to M, then take the half length of the base of the smaller pipe as K to H or H to J and mark off a like distance from A to L. A quarter of an ellipse is required as shown from L to M, the radius of which may be obtained in the following manner: Draw a line from M to N, also one from L to N at right angles, and draw the diagonal line LM; draw a line from the point N to cut the diagonal LM at right angles, producing the points c and b. With c as a center and radius cL draw a curve from L to a. With b as a center and radius ba, draw the remainder of the curve from a to M. Divide the curve from L to M into three equal parts, and then draw perpendicular lines from these points to intersect the semicircle ABC, as df, eg, Mh. Draw lines parallel to AD from f to i, g to k, and h to 1. The points where these lines are intersected by the lines drawn from the semicircle on the smaller cylinder will be the points through which to draw the curve r, s, t, w, v. Draw the line GG, Fig. 146, equal to the circumference of the smaller pipe, or to twice the number of divisions in the small semicircle. Divide onehalf of GG into six equal parts, as Fm, mn, no, op, pq, and qG. Draw perpendicular lines from F, m, n, o, p, q, and G. Take the length of the lines in

Fig. 145, as FH, mr, ns, ot, pu, qv, and GJ, and transfer their lengths to the perpendicular lines marked by corresponding letters. Draw a curve through the points thus obtained as H, r, s, t, w, v, and J. This will give the half-pattern for the

A

side of A in Fig. 147 on the line DB, as f, g, and h, and draw lines from these points parallel to HJ. Draw a perpendicular line from the point K to R in Fig. 145, and transfer the lengths KH and KJ, from A to H and A to J in Fig. 147, also the distances xr and xv from f to r and f to v in Fig. 147, and the distances y to s and y to u, from g to s