In Fig. 548, let H ALNO be the elevation of the bath, of which D1 G E1 B1 is a plan on the line D

E, Let the half section A2 M C1 B2 represent the flare which the hath is required to have through its sides on a line indicated by A B in elevation. By inspection of the elevation it will be seen that three patterns are required, which, for the sake of convenience, have been numbered in the various representations 1, 2 and 3.

Since the plan of piece No. 1 on the line D B, which is at right angles to its axis A F, is a semicircle, as shown by G D1 B1, and since its flare at the side, as shown by C1 B2 A2, is the same as at B D H, its pattern will be a portion of the envelope of a right cone. Patterns of this class have been treated in the previous section of this chapter - to which the reader is referred - where, in Problem 143, an exactly similar subject has been treated. The operation of obtaining this pattern is fully shown in Fig. 549, and, therefore, need not be here described.

Fig. 648. - Plan, Elevation and Section of a Hip Bath.

Piece No. 2, as shown in Fig. 548, is so drawn as to form one-half of the frustum of an elliptical cone. As its section at A B (shown at the right) must necessarily be the same as that of piece No. 1, against which it fits, the point F is assumed as the apex of the elliptical cone, and consequently the flare at the foot, E L, is determined by a continuation of the line drawn from the apex through E. Should it be decided to have more flare at the foot than that shown by E L, the point L may be located at pleasure, and the plan of the top, K L1 A1, be drawn arbitrarily; after which its pattern may be developed by means of the alternating triangles alluded to in the introduction of this section (page 306), examples of which will be found further on in this section.

Fig. 549. - Pattern of Piece No. 1 of Hip Bath.

The plan G E1 B1, from which the dimensions of the pattern are to be determined, maybe a true ellipse, or may be composed of arcs of circles, as shown, according to convenience. Divide one-half the plan G E1 into any convenient number of equal spaces, as shown by the small figures, and from each point thus obtained draw lines to the center C. To avoid confusion of lines a separate diagram of triangles is constructed in Fig. 550, in which M C is the hight of the tub or frustum and M F the hight of the cone. Draw M L, and C E, each at right angles to M F. Upon C E set off from C the lengths of the several lines C 1, C 2, etc., of the plan. Through each of the points in C E draw lines from F, cutting the line M L. From F as center draw ares indefinitely from each of the points in C E and also from the points in M L.

Fig.550 - Pattern of Piece No. 2 of Hip Bath.

Fig. 551. - Diagram of Radii for Pattern of Foot.

Through any point upon arc 1 of the lower set, as G, draw a line from F and extend it till it cuts arc 1 of the upper set at K; then K G will be one side of the pattern. With the dividers set to the space used in dividing the plan G E1, place one foot at the point G of the pattern and step to arc 2, and so continue stepping from one arc to the next till all are reached, as at E, and repeat the operation in the reverse order, finally reaching B and completing the lower line of the pattern. From each of the points in G E B draw lines radially from F, cutting arcs of corresponding number drawn from M L. Lines traced through these points of intersection will complete the upper line of the pattern. Then G E B A L K will be the required pattern of piece No. 2.

Fig. 552. - Pattern of Foot of Hip Bath.

As the plan D1 G E1 B1 has been drawn entirely from centers (C, P, S and P1), the pattern of piece No. 3 is exactly similar to that described in Problem 134 of the previous section of this chapter, to which the reader is referred. In Fig. 551 is shown a diagram for obtaining the radii taken from dimensions given in Fig. 548, while Fig. 552 shows the pattern described by means of the radii given in Fig. 551.