In Fig. 544, A B C D shows a side elevation of the tub, L M N O P the plan at the top, and E F G H K the plan at the bottom, an inspection of which will show that the head, as shown by H O or C D, has more flare than the sides, whose flare is shown by A J or A B, the flare of the sides and foot being the same. Inasmuch as the article is tapering in plan, the conical part of the pattern will include a little more than a semicircle, as shown. The points showing the junction between the straight sides and the conical part are to be determined by lines drawn from the centers by which the top and bottom were struck, perpendicular to the sides of the article. Therefore lay off in the plan T N and T P, drawn from the center T of the curved part of the plan of the top of the article, perpendicular to the sides M N and L P respectively.

And in like manner from S, the center by which the curved part of the bottom of the article is struck, draw S U and S K.

Since the top and bottom of the tub are parallel, as shown by the side elevation, and their circles are not concentric in the plan, it follows that the part P O

Fig. 544.   Plan, Eleration and Pattern of Flaring Tub with Tapering Sides and

Fig. 544. - Plan, Eleration and Pattern of Flaring Tub with Tapering Sides and

Semicircular Head.

N G H K is part of the envelope of a scalene cone. To find the apex of this cone, first drop lines from the points T and S vertically, cutting respectively the top and bottom lines of the elevation, as shown at T1 ami S1 A line connecting T1 and S1 will give the inclination of the axis of the cone in that view, which continue indefinitely in the direction of R1 until it intersects the side C D continued, as shown at R1. Then

R1 is the apex of the cone. From R1 draw R1 R verti cully, cutting the center line of the plan at R. Then R shows the position of the apex of the cone in the plan. As the pattern of the curved portion consists of two symmetrical halves when divided by the center line of the plan, divide the curve N O into any convenient number of equal spaces, as shown by the small figures. Lines drawn from each of these points to R would represent the bases of a series of right angled triangles whose common altitude is V R1, and whose hypothenuses when drawn will represent the correct distances from the apex to the various points assumed in the base of the cone. The simplest method, however, of measuring these bases is to place one foot of the compasses at the point R, and, bringing the pencil point successively to the points in N O, draw arcs cutting the center line, as shown between T and O. Now place the blade of the T-square parallel to R R1 and drop lines from each of these points, cutting the line A D as shown. From the points obtained upon A D draw lines toward the apex R1, cutting the bottom line of the tub B C. These lines drawn from the points in A D to R1 will be the desired hypothenuses and may be used in connection with the spaces of the plan in developing the envelope of the scalene cone.

Therefore from R1 as center, and radii corresponding to the distance from R1 to the several points in T1 D, describe a set of arcs indefinitely, as shown. Assume any point upon the arc 0, as N1, as a starting point, from which draw a line to R1. With the dividers set to the space used in dividing the plan N O, place one foot at the point N' and swing the other foot around, cutting the arc 1. Repeat this operation, cutting the arc 2, and so continue to step from arc to arc until all the arcs have been reached, which will complete the outline of one-half the pattern. The operation of stepping from arc to arc can be continued, stepping back from arc 5 till arc 0 is reached at P1, thus completing the top line of the pattern of the entire curved portion of the tub. From each of the points thus obtained draw lines toward the apex R1, as shown. Place one foot of the compasses at R1, and, bringing the pencil point successively to the points in the line S1 C previously obtained, cut radial lines of corresponding number in the pattern, as shown from G1 to K1. Lines traced through the several points in the two outlines, as shown by G1 H1 K1 and N1 O1 P1, will complete the pattern of the conical part of the tub. The patterns of the sides and foot may be obtained as described in Problem 74 and as indicated in the upper part of the engraving.