Let A B C D of Fig. 493 represent the side elevation of a pitcher top having the same Bare all around, and E F G H the plan at the base. By producing the lines A D and B C until they intersect in the point K, the apex of the cone of which the pitcher top is a tion will be obtained. Divide one-half of the plan into any convenient number of equal spaces, as shown by the points in G H E. From these points drop lines to the base D C, as indicated. Then draw radial lines from K, cutting the points in D C, and producing them until they intersect the curved line representing the top of the pitcher, otherwise an irregular cut through the cone, as shown by A B. For convenience in subsequent operations, number these points to correspond with the numbering of the points in the plan from which they arc derived.

Place the T-square at right angles to the axial line of the cone H K, and, bringing it against the several points in A B, cut the line D A, as shown in the diagram. By this means there will be obtained in the line K A the length of radii which will describe hits corresponding to points in the top line of the lip A B. With K as center and K D as radius, describe the arc L M X, which in length make equal to the circumference of the plan by stepping off on it spaces equal to the spaces originally established in the plan, all as indicated by the small figures. From K through each of the points in L M X thus established draw radial lines, extending outwardly indefinitely, as shown. Then with K as center and K°, K,1 K2, etc., as radii, strike arcs, which produce until they intersect radial lines of corresponding number just drawn, all as shown in the diagram. Then a line traced through the points thus obtained will be the required pattern, all as shown by L O P R N M.

The method above described is a strictly mathematical rule for obtaining such shapes when a design embodying the necessary curve at the top is at hand. As by the nature of the problem, this part of the pattern does not require to be fitted or joined to any other piece, it would be much easier to obtain, by the foregoing method, the principal points in the outside curve of the pattern and finish by drawing the remainder to suit the taste of the designer. In other words, after the arc L M X has been drawn and stepped off into spaces, draw radial lines from K through the points representing the highest and the lowest parts required in the top curve, as 0.5 and 12, upon which lines the required lengths can he set off. Then these points can be connected by any curve suitable for the purpose.

The principle involved in the foregoing is exactly the same as that of a hip or sitz bath given in the following problem, the difference in the finished article being a matter of size and shape and not of principle. Thus the sides could be made less flaring by placing the point K much further away from the base line, the hight A D could be increased and the curve A B could he altered to one more desirable; but the various steps necessary to perform the task would remain exactly the same.

Fig, 493. - Pattern for the Lip of a Sheet Metal Pitcher.

PROBLEM 143. Pattern for a Hip Bath of Regular Flare.

Let C B D E. in Fig. 494, be the elevation of the bod; of a hip bath having an equal amount of flare on all sides, the plan of which is a circle. In describing the pattern for the body it will be considered as a section of a. right cone, the plane C E being at right angles to the axis and the base being represented by the curved line B D, as shown. The sides E D and C B can be extended until they meet at A. Then A will be the apex of a cone of which C B D E is a frus turn having an irregular base B D.

Fig. 494. - Pattern for a Hip Bath of Regular Flare.

At any convenient distance above D draw J K parallel to C B to be used as a regular base upon which to measure the circumference of the cone. Parallel to J K draw E H, and from a center obtained on F H by prolonging the axis A N draw a half-plan of the frustum as shown by F G H. Divide this half-plan into any convenient number of equal parts, and from the points thus obtained carry lines parallel to the axis until they cut the line J K, and from there extend them in the direct ion of the apex A, thus cutting the curved line B D. Place the T-square parallel with J K, and bringing it against the several points in the curved line B D, cut the side E D, as shown. From A as center, with

A K as radius, describe the arc K1 K2, on which lay off a stretchout of either one-half or the whole of the plan, as may be desired. In this case a half is shown. From the extremities of this stretchout, as K1 and K2 draw lines to the center, as K1 A and K2 A, and from the several points in the stretchout draw similar lines, as shown by 1, 2, etc. With one point of the dividers set at A bring the pencil point to the point D in the side A K, and with that radius describe an are. which produce until it cuts the corresponding line 12 in the stretchout, as shown at D1. In like manner, bringing the pencil point up to the several points between D and E in the elevation, describe arcs cutting lines of corresponding numbers in the stretchout. Then a line traced through these intersections will form the upper line of the pattern. From A as center, with A E as radius, describe the arc C E1, cutting A K1 and A K2, as shown by C1 and E1, forming the lower line of pattern. Then C B1 D1 E1 will be half the pattern for the side of the hip bath.

As a feature of design, the form produced in the pattern by a curved line B D drawn arbitrarily may not be entirely satisfactory. If, for instance, that part. of the pattern lying between lines 9 and 12 should not appear as desired, it can be modified upon the patten at will, as this edge of the pattern is not required to lit any other form. Such a modification is shown by the dotted lines a1 K2 of the pattern and a K of the elevation. 'The foot of the tub is a simple frustum of a right cone, the pattern for which is obtained in the manner described in Problem 123. Different forms of bathtubs in which the flare is irregular will be found in Section 3 of this chapter.