The elbows of the smaller pipes in the problem here presented are such as would, if each were completed independently of the other, form six-piece elbows. The junction between the two elbows occurs between the fifth pieces, which pieces unite to form the transition from the smaller diameters of the elbows to the diameter of the larger pipe, or sixth piece. A pictorial representation of the finished work is shown in Fig. 667, in which, however, the upper section, or larger pipe, is omitted to more fully show the shape and junction of the transition pieces. A front view or elevation of the various parts is shown in Fig. 668. The side view given in Fig. 669 shows more fully the amount of lateral flare of the transition piece necessary to form a union between the varying diameters of the larger and smaller pipes.

The drawing of that portion of the elbows in the smaller pipes from the horizontal parts up to the line a h t in Fig. 668 is exactly the same as that employed in drawing a six-piece elbow. The piece A G h a. occupying the place of what would otherwise be the fifth piece of the elbow, becomes in this case an irregular shape, the lower end or opening, a h, of which is nearly circular while its upper end, A G, is a perfect semicircle. This piece unites with its mate G D t h on the line G h, thus forming the complete circle at A D, a plan of which is shown immediately below the elevation. The relative proportion between the diameters of the larger and smaller pipes is such that the junction between the elbows is carried somewhat below the fifth pieces, mitering the fourth pieces for a short distance, as shown from h to L. The method of cutting the lower parts of the elbow, however, is the same as that employed in all elbow patterns where the pipe is of a uniform diameter throughout, numerous examples of which are given in Section 1 of this chapter, to which the reader is referred.

As the section or profile of all the parts forming the elbow is a perfect circle when taken at right angles to the sides of the pipe, as at Q F or M N, it will be seen that a section on the line a h will be somewhat elliptical; it will therefore be necessary to obtain a correct drawing of this section from which to obtain the stretchout of the lower end of the piece A G h a.with which it joins, and also a drawing of it as it will appear in plan. Therefore between two parallel lines drawn from M and N at right angles to M N construct a profile or section, as shown below at the left, which divide 'into any convenient number of equal spaces, as shown by the small letters a. b,. c, etc. From each of these points carry lines back to M N at right angles to the same, and continue them in either direction till they cut the miter line a n of the elevation, as shown by the small letters, and the center line a n of the section. From the points in a n of the elevation draw lines at right angles to the same indefinitely, as shown above the elevation, across which at any convenient point draw a line, as B1 C1, at right angles to them. From B1 C1 set off on the lines last drawn distances equal to the distances from the circumference to the diameter on corresponding lines in the section below, all as shown by a2, b2. c2. etc. A line traced through these points will be the correct section on the miter line a n. It will be noticed that the section has not been carried further than the point h2. the balance of the curve not being required by reason of its intersection with the corresponding piece in the other elbow.

Fig. 667. - Perspective View of the Junction of a Large Pipe with the Elbows of Two Smaller Pipes.

Below the elevation and in line with the same, as shown by the center line G T, is drawn the plan of the larger pipe ABC D. It will be necessary to add to this the plan of the curve on the line a h of the elevation, in order that the horizontal distances between the points assumed in the two curves may be accurately measured. Therefore from the points on the miter line a h drop lines vertically through the plan, cutting the transverse center line X Y. Prom X V set off distances on these several lines equal to the distances of corresponding points from the line a n of the original section, as shown by a', b', etc., from X to S. A line traced through these points will give the correct position of the intersection of the smaller pipe as seen from above. This entire line is shown in the plan, although the part from S to Z will not be required, for the reason given above. An inspection of the plan will show that the side of the plan from V T to the right would be an exact duplicate of the left side if it were completed, and that therefore the plan consists of four symmetrical quarters, one of which, X R T, is completely shown in the plan. Hence the pattern for this quarter will suffice by duplication for the entire transition piece.

Divide the quarter of the plan of the larger pipe P T, adjacent to the curve X S, into the same number of equal spaces as arc found in the inner curve from X to S, as shown by the small figures 1, 2, 3, etc. Connect corresponding points in the two lines as shown by the solid lines h' 8' .g' 7, .f' 6, etc. Next subdivide the four-sided figures thus obtained by their shortest diagonal, as shown by the dotted lines g' 8' ,f 7, etc. These solid and dotted lines across the, plan represent the bases of a scries of right angled triangles whose altitudes can easily be obtained from the elevation, and whose hypothenuses when obtained will give correct distances across the finished piece between points connected on the plan. These lines have also been drawn across the elevation from corresponding points in the same for illustrative purposes, but such an operation is not necessary to obtain the pattern. Neither is the side view shown in Fig. 669 necessary to the work, but is here introduced merely to assist the student in forming a more perfect conception of the operations described. From the points a, b, c, etc. on the miter line a h of the elevation carry lines horizontally across, cutting the vertical line G L, as shown by the points from s to h. The distances of these points from G will then represent the vertical distances of corresponding points in X S of the plan from the plane of upper base of the transition piece shown by A D of the elevation and V P T of the plan.

To obtain the hypothenuses of the various triangles above alluded to, or in other words, the true lengths of the lines dividing the surface, as shown in the two elevations and plan, it will be necessary to construct a series of diagrams, as shown in Fig. 670. Therefore draw any two lines, as h 8 and h h', at right angles to each other; make h 8 equal to h' 8 of the plan, Fig. 668, and h h' equal to h 8 of the elevation, and draw h' 8. Next draw any two lines, as g 8 and gg', at right angles to each other, making g 8 equal to the dotted line g 8 of the plan and g 7 equal to the solid line g' 7 of the plan. Make g g' equal to the distance of point g from the line A D as measured by its corresponding point on the line L G. Draw g 8 and g 7. So continue till all the triangles have been constructed. Then the solid hypothenuses will represent the true distances across the pattern indicated by the solid lines of the plan and elevations, and the dotted hypothenuses the true distances on corresponding dotted lines in those views. In describing the pattern, work can be begun at either end of the pattern most convenient. Draw any straight line as 1 a of Fig. 671, which make equal to the line 1 a' of the diagram of triangles, Fig. 870. From 1 as a center, with a radius equal to 1 2 of the plan, describe a small arc, which intersect with another small are drawn from a as center, with a radius equal to a' 2 of the diagram of triangles, thus locating the point 2 of the pattern. From point 2 as a center, with a radius equal to b' 2 of the diagram of triangles, describe a small arc, which intersect with another small arc struck from a of the pattern as center, with a radius equal to a' b2 of the section on line a h of elevation shown above, thus establishing the position of the point b of the pattern. Proceed in this manner, using the spaces in P T in the plan of the larger pipe to form the upper edge of the pattern and the spaces from the section B1 C1 to form the lower edge of the pattern, measuring the distances between the same by the alternate use of the solid and dotted hypothenuses of corresponding number and letter taken from the diagram of triangles in Fig. 670. A line traced through the two series of points and a straight line from 8 to A will complete the pattern for onequarter of the transition piece required. The remaining three-quarters can be obtained by any means of duplication most convenient

Fig. 668. - Front Elevation, Plan and Sections, Shotting Method of Triangulation.

Fig. 669. - Side Elevation.

Fig. 670. - Diagram of Triangles.

Fig. 671. - Pattern for One-Quarter of Connecting Piece.