This problem differs from the preceding one only in the shape of the pipe having the elongated profile, which profile in the preceding problem consists of two semicircles joined by a straight part, whereas in this case its curve is continuous throughout; its pattern therefore will consist throughout of a series of triangles having short bases instead of having a large flat triangular surface uniting its curved portion as in the previous ease.

In Fig. 696, D C B A represents the elevation of the offset, C F E. B that of a portion of the round pipe with which it is required to conned at its upper end and H D A G that of the elliptical pipe joining it below. M P N is the half profile of the round pipe and K J L that of the elliptical pipe. The plan or top view is not shown,' and is not necessary to the work of obtaining the pattern. Since the profiles given necessarily represent sections on lines at right angles to the respective pipes, as at F E and H G, it will first be necessary to derive from them sections on the joint or miter lines C B and DA, from which to obtain correct stretchouts of the two ends of the pattern of the offset piece.

Fig. 696. - Elevation and Sections of Offset, Showing Method of Triangulation.

As the pattern required consists of symmetrical halves, one-half only will be given, and one-half of the profiles only need be used. Therefore divide the half profile M P N into any convenient number of equal spaces, as shown by the small figures, and from the points thus obtained draw lines at right angles to F E, cutting M N and C B. To avoid confusion of lines a duplicate of C B is shown at the left by C1 B1 From the points on C1 B1 draw lines at right angles to it indefinitely, and upon each of these lines, measuring from C1 B1, set off the lengths of lines of corresponding number in the profile M P N measured from M N. Thus make the distance of point 2' from C1 B1 equal to the distance of point 2 from line M N, the length of line 3' equal to that of line 3, measuring from the same base lines as before, etc. A line traced through the points of intersection, as shown by C1 O B1, will be the correct section on the line C B of the elevation. The method of obtaining the section on the line D A, shown at D1 I Av, is exactly the same as that just described in connection with the round pipe, all as clearly shown in the lower part of the engraving.

The next operation will consist of dividing the surface of the transition or offset piece into measurable triangles, making use of the spaces used in the profiles; therefore connect points in C B with those of similar number in D A by solid lines, as 1 with 1', 2 with 2', etc., and connect points in C B with those of the next higher number in B A by dotted lines, as 1 with 2', 2 with 3', 3 with 4', etc. The surface of the transition piece is thus divided into a scries of triangles the lengths of whose bases or short sides are found in the two sections C1 O B1 and D1 I A1.

As the hights of corresponding points in the two sections, measuring from their center or base lines, differ very materially, it will be necessary to construct two diagrams of sections from which the lengths of the various solid and dotted lines can be obtained. In Fig. 697 is shown a diagram of sections through A B C D taken on the solid lines drawn across the elevation, in which the base line P Q represents the surface of a plane dividing the offset into symmetrical halves. At P erect a perpendicular, P R, upon which set off the hight of the points in the profile K J L or the section D1 I A1, measuring upon the straight lines joining them with the base line K L, as shown by the small figures. From P, upon P Q, set off the lengths of the various solid lines drawn across the elevation. also shown by small figures, and at each of the points thus obtained erect a perpendicular, which make equal in hight to the distance of point of corresponding number in profile M P N from M N, measuring on the perpendicular line. Thus, make line 2 of Fig. 697 equal in hight to the distance from point 2 of profile to the line M N, line 3 equal to the length of line 3 of profile M P N. Now connect points of corresponding number at the two ends of the diagram by straight lines, as shown, then will these oblique lines be the correct distances between points of corresponding numbers connected by the solid lines drawn in the elevation.

Fig. 697. - Diagram of Sections on Solid Lines of Elevation.

The diagram in Fig. 698 is constructed in an exactly similar manner. The distances S 1, S 2, S 3, etc., on the base line are in this case made equal to the lengths of the dotted lines of the elevation, and the perpendiculars erected at points 2, 3, etc., are the same as those used in the previous diagram. The perpendicular S U is also an exact duplicate of P R in Fig. 697. In drawing the oblique dotted lines, point 1 at the right end of the diagram is connected with that of the next higher number (2') on the line S U, 2 at the right with 3' on the line S U, etc., all as shown.

Fig. 698. - Diagram of Sections on Dotted Lines of Elevation.

The oblique dotted lines will then be the correct distances between points of corresponding numbers connected by the dotted lines in the elevation.

To develop the pattern, first draw any straight line, as D C in Fig. 699, which make equal in length to D C of Fig. 606. From D as center, with a radius equal to 1 2 of the section D1 I A1, strike a small are, which intersect with another small arc Struck from C as center, with a radius equal to 2' 1 of Fig. 698, thus establishing the location of point 2' of pattern. From 2' of pattern as center, with a radius equal to 2' 2 of Fig. 697, strike a small arc, which intersect with another small are struck from C of pattern as center, and a radius equal to 1 2 of section C1 O B1, thus establishing the position of point 2 of pattern. So continue, using alternately the dotted and the solid oblique lines in Figs. 698 and 697 to measure the distances across the pattern, the spaces from the section D1 I A1 to form the stretchout of the lower end (D A) of pattern, and the spaces from the section C1 O B1 to form the stretchout of the upper end (C B) of the pattern. Lines traced through the points of intersection, as from C to B and from D to A, will complete one-half the required pattern.