Fig. 252. - Method of Deriving the Pattern from the Drawing.

A right cone having an elliptical base might seem to belong in the same class with regular tapering forms, but as the distance from its apex to the various points in the perimeter of its base is constantly varying, it is therefore placed in the class with irregular forms in the following section of this chapter, where it will be duly discussed. But as tapering articles of elliptical shape are of frequent occurrence, and as the circle is much easier made use of than the ellipse, such articles are usually designed with approximate ellipses composed of ares of circles. This method is in many cases especially desirable, as articles so designed have an equal amount of Hare or taper on all sides, which would not he the case if they were cut from elliptical cones. It will thus he seen that an article designed in that manner is the envelope of a solid composed of as many portions of frustums of right cones as there were ares of circles used in drawing its plan.

In Fig. 253 is shown the usual method of drawing the plan and elevation of an elliptical flaring article, the outer curve of the plan A C B D being the shape at M N of the elevation, while the inner curve I G V J is the plan at the top K L. As many centers may he employed in drawing the curves of the plan of such an article as desired, all of which is explained in the chapter on Geometrical Problems (Chap. IV.), Problems 73, 76 and 78. To simplify matters only two sets of centers have been employed in the present drawing, all as indicated by the dotted lines drawn from the various centers and separating the different arcs of circles. Reference to the plan now shows that that portion of the article included between the points E

Fig. 253. - Usual Method of Drawing an Elliptical Flaring Article.

C W V G H is a position of the envelope of a cone the radius of whose base is D C and whose apex is located at a point somewhere above D; and likewise that that portion included between the points X A E H I J is part of a cone the radius of whose base is A F and whose apex is somewhere upon a line erected at F. Thus four sectors cut from cones of two different sizes go to make up the entire solid of which the article shown in Fig. 253 is a frustum,

To determine the dimensions, then, of such cones it is necessary to construct a diagram such as that shown in Fig. 254, which is in reality a section upon the line E D of the plan, in which P O and P R are respectively equal to E D and E F of the plan. At points 0 and R, Fig. 254, erect perpendicular lines O J and R Z indefinitely. Upon O J set off OS equal to the Straight bight O K of the frustum, Fig. 253, and draw S U parallel to O P, which make equal in length to D H. A line drawn through the points P and U will then represent the slant or taper of the frustum, as shown at M K of the elevation, and if continued till it intersects with the perpendiculars from O and P will determine the respective bights of the two cones, as shown by Z and J. Then P J O is the triangle which, if revolved about its vertical side J O, will generate the cone from which so much of the figure as is struck from the centers C and D in Fig. 253 is cut; and P Z R is the triangle which if revolved about its vertical side Z It will generate the cone from which the end pieces of the article are taken. To present this before the reader in a more forcible manner, several pictorial illustrations are here introduced in which the foregoing operations are more clearly shown. In Fig. 255 is shown a view of the plan of the base A C B D of Fig. 253 in perspective, in which the reference letters are the same as at corresponding parts of that plan, and upon which is represented, in its correct position, a sector of the larger cone from which the side portions of the frustum are taken. Thus the triangular surfaces. F D E and F D W, being sections of the cone through its axis, correspond to the triangle J O P of the diagram, Fig. 254. In Fig. 256 two additional sectors from the smaller cone previously referred to are represented as standing upon the adjacent portions of the plan from which their dimensions were derived. Thus C F and H D, the center lines of their bases, correspond respectively to A F and Y B of the plan. Fig. 253, and the triangles L F G and M H K, being radial sections of the cones, correspond with the triangle Z R P of the diagram. In Fig. 257 is presented the opposite view of the combination seen in Fig. 256, showing at C B and D A the joining of their outer surfaces or envelopes.

Fig. 254. - Diagram Constructed to Determine the Dimensions of the Cones, Portions of which are Combined to Make up the Article Shown in Fig. 253.

Fig. 255. - Perspective View of the Plan in Fig. 253, with a Sector of the Larger Cone in Position.

As previously remarked, two sets of centers only were employed in constructing the plan, Fig. 253, for the sake of simplicity. Had a third set of centers been made use of the arrangement of sectors of cones shown in Figs. 256 and 257 would have been supplemented by a pair of sectors, cut from a cone of intermediate size, which would have been placed on either side of the large sector between it and the smaller ones, all being joined together upon the same general principle as before explained. Reference to Fig. 220 in the chapter on Geometrical Problems shows at 3 L P, P S W and W U V what the relative position of their bases would be. If it be desired to complete the solid, which has been begun in Fig. 256, it will first be necessary to cut away that portion of the middle sector which stands over the space F H B, Fig. 256. Such a cut might be begun upon the line F H, and passing vertically through the points L and M would finish through the curved surface of the further or curved side of the sector. The cut thus made between the points L and M is shown at D C in the other view. Fig. 257, and is by virtue of the conditions a hyper bola. (See Def. 113, Chap. I.) The piece necessary to complete the solid would then be a duplicate of the shape remaining after making the above described cut, the outer surface of which is shown by A B C D of Fig. 257. The complete solid would then have the appearance shown in Fig. 258.