Having gone over the construction of ovals, we can now turn our attention to the development of oval equal-tapering articles, or those in which the overhang for the sides is the same as for the ends. Such an article is shown in Fig. 136. It is best to think of the surface of this article as being built up of parts of the surfaces of two different-sized cones, but whose taper is the same. Thus, referring to Fig. 137, the large dotted circles represent the bases of the pones, part of whose

Fig. 136.

Surfaces go to form the sides of the oval vessel. The small dotted circles show the bases of the cones from whose surfaces the end portion of the oval article is formed. The plan of the axes of these cones, it will be seen, coincides with the points P and Q.

The fitting together of the cone parts is exhibited in

Fig. 138, which is a sketch of a model showing a part of one large cone with the two parts of the smaller cones fitting on to form the end portion of the oval object. The front side, which would be the part of the other large cone, it may be imagined, is removed to better explain the fitting together of the side and ends. The three upright lines show the axes of the cones. It should be noticed that the small and large cone surfaces join together in a common line (shown by the dotted lines at the back): hence the two

Fig. 137.

Fig. 138, curved surfaces fit together without showing lump or hollow. Part of the small oval which forms either top or bottom of the article, as the case may be, is also shown on the model in this figure.

When it is thoroughly understood how the surface of the oval equal-tapering article is built up, the development of the pattern is not at all a difficult matter.

It will, perhaps, be easier to follow if we fix some definite dimensions, and work out the problem completely from these. Thus suppose an oval article is 32 in. by 20 in. at the top, 22 in. by 10 in. at the bottom, and 7 in. perpendicular depth. It should be noted that the dimensions must be such as to give the same overhang all round, and these can be checked by using the following rule: - "The length of bottom deducted from the length of top must be the same as the width of bottom deducted from the width of top." In this case it will be seen that the overhang is -

32-22 / 2 or 20-10/ 2 = 5 in. and for the ends -

Calculating the radius for the sides of the large oval by the before-mentioned rules, this will be -

8 x 32 - 5 x 20 / 6 = 26. in.

4 x 20 - 32 /6 = 8 in

As each quarter of the oval is exactly the same, there is no need only to set out just one quarter, and this can be done in the usual way (Fig. 139). The same centres can be used for marking out the quarter of small oval for the bottom, the radii for sides and ends being in each case 5 in. less than those used for the top. For purposes of getting out the body pattern, there is really no need to set out the quarter-oval for the bottom, its only use being to obtain the size and shape of the bottom plate.

Fig. 139.

Having marked out the oval, the depth 7 in. should be set up from the point G, and also along from the point H, as shown in Fig. 139. B is joined to K, and produced until it meets a perpendicular through Q in S. Also D should be joined to M and produced until it meets a line which is square to D P in L. By referring again to Fig. 138, it will be seen that the points L and S (Fig. 139) represent the apexes of the large and small cones respectively. Half the large cone being given as a side elevation on D P, and half the small cone being shown as a front elevation on Q B. The line S B then will give us the slant height of the small cone, and thus the radius for the development of its surface; the line L D serving the same purpose for the large cone.

The pattern can now be struck out. With centre L and radius L D draw an arc, and along it mark off a portion, B B, equal in length to twice the arc D F. Join the points B B to L, then with centre L and radius L M draw the bottom curve, K K. The part of the pattern thus set out will give the side portion of the article, or we may imagine it to be the development of the part of large cone. The ends can now be added by opening the compasses to the slant height of small cone B S, as shown in the elevation, and marking it along the lines B L in the pattern, thus obtaining the centres, S S. The curves for the end part of pattern are now set out from these centres, using radii S B and S K. The outside curves, which are shown marked B F, are now cut off equal in length to the curve B F on the plan. This is best accomplished by bending a piece of wire along the curves, as before mentioned. Particular notice must be taken that the points F are joined to the centres S. There is no need to trouble about the length of curve for the bottom of pattern, as this will be cut off to the correct proportion by the radial lines as drawn. This may be tested by measuring the length of the curve, and seeing if it is equal to twice the length of the bottom quarter-oval.