The form EFKLJ shown in Fig. 557, the lower line of which is an irregular section through an elliptical cone, is introduced here, not as representing any particular article or class of forms, but because it embodies a principle somewhat different from other sections of cones previously given, which may be useful to the pattern cutter.

B A C is the side elevation of a cone having an elliptical base, one-half of which is shown by B1 H C1. Divide one quarter of the plan, as H C1, into any convenient number of equal parts, as shown by the small figures. From each of the points of division draw lines to the center D2, and also erect lines cutting the base of the cone B C, from which carry them toward the apex, cutting the lines E F and J L K. The first operation will be that of obtaining the envelope of the complete cone in the same manner as described in previous problems.

Construct a diagram of triangles, as shown at the right, in which A1 D1 is equal in hight to A D, and at right angles to G1 W and D1 V, extensions respectively of E F and B C. Upon D1 V, measuring from D1, set off the distances from D2 to the several points in H C1, as shown. From each of the points thus obtained draw lines toward A1, cutting G1 W. Also from each of these points, with A' as center, describe arcs indefinitely. Take between the points of the dividers a space equal to that used in dividing the plan H C, and placing one foot upon the arc drawn from point 1 in the line D1 V, step to arc 2, thence to arc 3 and so continue till one quarter of the stretchout is completed at 7, and, if desirable, continue the operation, taking the arcs in reverse order, thus completing the outline of one-half the envelope of the cone, as shown in the engraving. From each of the points in this outline or stretchout draw measuring lines toward the center A1. Place one point of the compasses at point A1, and, bringing the pencil point successively to the several points of intersection on the line G1 W, cut measuring lines of corresponding number, as shown from G1 to G2. Place the T-square parallel to the base B C, and. bringing it successively to the several points of intersection previously obtained in the curved line L K, cut lines of corresponding number drawn from the points in D1 V to A1, as shown from X to Y. Finally, with one foot of the compasses at A1, bring the pencil point to each of the points of intersection last obtained and cut corresponding measuring lines in the pattern. Then lines traced through the points of intersection, as shown from L1 to L2 and from G1 to G2, will complete the pattern of one-half the frustum E F K L J.

Fig. 557.   Patterns for the Frustum of an Elliptical Cone Having an Irregular Base.

Fig. 557. - Patterns for the Frustum of an Elliptical Cone Having an Irregular Base.

Should it be desirable to cut a pattern to fill the end J L K of the frustum, as for a bottom in the same, it will first be necessary to obtain a correct plan of the line JKL. To accomplish this, set off on the lines D2 7, D2 6, etc., of the plan the lengths of the several lines of corresponding number drawn from the line G1 D1 to the intersections between X and Y, thus obtaining the desired line L3 K2. Extend the center line B1 C1 of the plan, as shown at the right, upon which lay off a stretchout of the line L K, taking each of the spaces separately as they occur, all as shown by

Z K3, through which draw measuring lines at right angles. Place the T-square parallel to B1 C1, and, bringing it to the several points in the line L3 K2, cut corresponding measuring lines. Then a line traced through the points of intersection, as shown by L4 K3, will be the pattern of one-quarter of the desired piece, which may be duplicated as necessary for a half or for the entire pattern in one piece.