In Fig. 623, let A B D F H G E C be the side view of a three-piece elbow, the middle piece (C D F E) of which is made tapering. The piece C D F E may also be described as an offset between two round pipes of different diameters. A half profile of the upper and smaller of the two pipes is shown by a m b, while g n h shows that of the larger pipe. The straight portions A B D C and E F H G are in all respects similar to many pieces whose patterns have already been described in the first section of this chapter in Problems 38 to 45 inclusive. It will therefore be unnecessary to repeat the description in this connection.

Since an oblique section through a cylinder is an ellipse, an inspection of the drawing will show that the sections C D and E F, the upper and lower bases of the middle piece, must be elliptical. The first operation, therefore, will be to develop the ellipses, which may be done in the following manner: Divide the profile a m b into any convenient number of equal spaces, as shown by the small figures. From the points thus obtained drop lines vertically to a b and continue them till they cut the line C D. From the intersections on C D carry lines at right angles to the same indefinitely, as shown. Through these lines draw any line, as c d, parallel to C D. Upon each of the lines drawn from C D, and measuring from c d, set off the lengths of lines of corresponding number in the profile a m b, measuring from a b. A line traced through the points thus obtained, as shown by c k d, will be the required elliptical section. The section upon the line E F may be obtained in the same manner, all as shown by epf.

Fig. 623. - Elevation of a Three-Piece Elbow, the Middle Piece of which Tapers.

In Fig. 624, C D F E is a duplicate of the middle piece of Fig. 623, below which is drawn its half plan made up of the elliptical sections just obtained, all as shown by corresponding letters. The piece thus becomes an irregular flaring article or transition piece, the envelope or pattern of which may be obtained in exactly the same manner as described in Problems 184 or 186, to which the reader is referred. The elevation in Fig. 623 is so drawn that C D and E F are parallel, and C E is at right angles to both. Should the elevation, however, be so drawn that C D is not parallel with E F the conditions will then become the same as in Problem 193 succeeding, which see; and should C E be drawn otherwise than vertically, the plan would then resemble that shown in Problem 194.

Fig. 624. - The Middle Piece of Elbow in Fig. 623, with Plans of its Bases Arranged for Triangulation.