In Fig. 540 is shown an elevation and plan of the article, in which E F G I is the plan of the bottom and E J K L that of the top, the two being tangent at the point E. In Fig. 541 the elevation and a portion of the plan are drawn to a larger scale and conveniently located for describing the pattern.
Since the top and the base of the article are both circular and are parallel, the shape of which the pattern is required becomes a frustum of a scalene cone, and lines drawn upon its surface from any set of points assumed in the circumference of its base to its apex-will divide the circumference of the top into similar and proportionate spaces. Therefore, the first step is to extend the lines of the sides B A and C D until they meet at M, the apex. Next divide the plan of the base, one-half of which, E H G, only is shown, into any convenient number of equal spaces, as shown by the small figures. As it is necessary to ascertain the distance from each of these points to the apex of the cone the simplest method of accomplishing this is as follows: From E, the position of the apex in plan, as a center, with E 6, E 5, E 4, etc., as radii, describe ares cutting E G. Carry lines vertically from each of the points in E G, cutting the base line A D; thence carry them toward the apex M, cutting the line of the top B C, all as shown.
With M as center describe arcs from each of the points in the base line A D, and extend them indefinitely in the direction of O. In the same manner draw arcs from the points of intersection in B C, as shown. From the apex M draw any line to intersect the arc from A or 7 of the base line, as M N, which will form one side of the pattern, corresponding to B A of the elevation. Set the dividers to the space E 6, used in dividing the plan of the base, ana starting from N step from one arc to the next, thus laying out the stretchout of the base E H G, and at the same time locating each point at its proper distance from the apex M. A line traced through these points, as N Q O, will be the bottom of one-half of the pattern. From the points in the line NQO draw lines to M, cutting the arcs of corresponding number previously drawn from BC; then a line traced through these points of intersection, as shown by RP, will be the bop of the pattern, and PRNQO will thus be one-half the required pattern.
Fig. 540. - Plan and Elevation of Frustum of a Scalene Cone.
Fig. 541. - Method of Obtaining Pattern of a Scalene Cone.