First Case. - In Fig. 542 is shown the elevation and plan of the article drawn in proper relation to each other, as shown by the lines of projection. In this case the top of the article is located centrally with reference to the bottom, as shown in the plan. From O, the center of the top, erect the perpendicular O o, cutting the line of the top in elevation, and from P erect the perpendicular P p, cutting the line of the base. Since LMN and F H G of the plan arc semicircles lying in parallel planes, that part of the pattern of the article shown by L FH GNM must be one-half the envelope of the frustum of a scalene cone.

Fig. 542. - Plan, Elevation and Pattern of Flaring Article with Pound Top and Oblong Base, the Top being Centrally Located.

To ascertain the apex of the cone, prolong the side line C D of the elevation indefinitely in the direction of X. Through the points p and 0 draw p o, which produce until it meets C D prolonged in the point X. Then X is the apex required. From X drop a perpendicular, cutting the center line of the plan at X1, thus locating the apex in plan. Divide F H G into any convenient number of equal spaces, as shown by the small figures. Should lines be drawn from each of the points thus obtained to X1, they would represent the bases of a set of right angled triangles, of which Y X is the common altitude, and whose hypothenuses will give correct distances from the apex of the cone to the various points assumed in the base.

The simplest method of obtaining these hypothenuses is as follows: From X1 as center draw arcs from each of the points in F H G, cutting the center line K H of the plan. From each point in P II erect a perpendicular to p C, as shown. From the points thus obtained in p C carry lines toward the apex X, cutting o D, as shown. From X as center strike arcs from each of the points in p C indefinitely. Assume any point, as G1, upon the arc struck from point 1 as the first point in the pattern of the base, from which draw a line to X. Set the dividers to the space used in stepping off the plan, and, commencing at G1, step to the second arc, and from that point to the third arc, and so on, as shown in the engraving. A line traced through these points will be the boundary of a lower side of the semicircular end. From each of these points just obtained draw a line toward the center X. Place one foot of the dividers at X, and, bringing the pencil point successively to the points in 0 D, cut radial lines of corresponding number just drawn. A line traced through these points of intersection, as shown by N1 L1, will form the upper edge of the pattern of the end piece. From the point L1, which corresponds to L of the plan, as center, with L1 F1 as radius, describe the arc Fl E1, and from F1 as center, with radius equal to F R of the plan, intersect it at R1, as shown. Draw L1 R1. Then L1 R1 F1 is the pattern of one of the sides. To L1 R1 add a duplicate of the end piece already obtained, all as shown by L1 R1 S1 N2, and to S2 N2 add a duplicate of the side just obtained,' as shown by S1 N2 G2, thus completing the pattern.

Second Case. - This case differs from the first only in the fact that the top of the article, being located near one end. is drawn concentric with the semicircle of the near end. As the result of this condition, that portion of its pattern shown by S E R L K N in the plan, Fig. 543, becomes one-half the envelope of the frustum of a right cone, the method of developing which is given in Problem 123 of the previous section of this chapter.

In that portion of the article shown by R F H G S N the conditions are exactly the same as in the first case. In Fig. 543 corresponding parts have been lettered the same as in Fig. 542, so that the demonstration given above is equally applicable to either figure. In the final make up of the various parts of which the near One End.

Complete pattern is composed, the part E1 S1 N2 L1 is of course obtained as in Problem 123 instead of being a duplicate of L1 F1 G1 N1 all as clearly shown in the engraving. The seam in this case has been located upon the line N S of the plan instead of upon N G as before.

Fig. 543. - Plan, Elevation and Pattern of Flaring Article with Round Top and Oblong Base, the Top being Located