The shape shown in Fig. 580 differs from that treated in Problem 176 principally in the fact that the round end is larger than the rectangular end instead of smaller as in Fig. 566; the conditions involved are, however, exactly the same as in the other problem and consequently the method of obtaining the pattern must be similar. F G H J, in Fig. 580, represents the plan of the base, K L M N that of the top, and A B C E the elevation of a side of the article.

Through O, the center of the circle of the base, draw the diameters G J and F H parallel to the sides of the top. From the four points thus obtained in the circumference of the base draw lines to the angles of the top, as shown by G M and H M, H N and J N, etc. It will be seen from this that the envelope of the article consists of four flat triangles, of which L G M is a plan and B D C the elevation, and four rounded corners, either one of which, as J N H, is a portion of an oblique cone of which J H is the base and N the apex.

Fig, 580. - Plan and Elevation of a Flaring Article, Round at the Base and Square at the Top.

To obtain the pattern first divide the quarter plan of base J H into any convenient number of parts, as indicated by the small figures, and connect these points with N, as shown. To obtain the distance from points in J H of base to N of top it will be necessary to construct the diagram of triangles shown in Fig. 581. Draw any line, as R P, in length equal to the hight of the article, as shown by S C in Fig. 580. At right angles to R P draw P Q, and on P Q lay off the lengths of lines in J H N. Thus make P 1 equal to N 1 of the plan, P 2 equal to N 2, etc. Connect the points in P Q with R. The hypothenuses thus obtained give the true distances from the points in the base to N in the top.

. From any convenient point, as N in Fig. 582, as center, with radius R 1 of Fig. 581, describe an are, as shown by 1 7. In like manner, with radii R 2. R 3 and R 4 of Fig. 581, describe arcs, as shown. Draw a straight line from N to any convenient point upon the arc 1 7, as shown by N H. Set the dividers to the space used in stepping off the plan of the base and, starting with H, lay off the stretchout, stepping from arc to arc, as shown. A line traced through these points will form the pattern for as much of the article as shown by J N H of the plan. With N of pattern as center, and N K of plan or B C of elevation as radius, describe a small are, K. which intersect with an arc struck from J of pattern as center and J N as radius. Connect J K and K N, which completes the pattern for J K N of the plan. J K F of pattern is the same as J N H, and can be obtained in the same manner, or by any convenient means of duplication. With N as center, and N W of plan as radius, describe a small arc, which intersect with one struck from H as center, and E C of elevation as radius. Connect H W and W N, thus producing the part of pattern corresponding to N W H of the plan.

Fig. 581. - Diagram of Triangles Used in Obtaining Pattern of Article Shown in Fig. 580.

Fig. 582. - One-Half of Pattern of Article Shown in Fig. 580.

F K V of pattern is obtained in a similar manner. Then V K N W H J F is the pattern for one-half of article.