In Fig. 672, let A G C B H D represent the elevation of elbow, A F D the half profile of elliptical end and C E B the half profile of round end. The first step will be to establish a section on the ratter line G H. Since the width at A D. one-half of which is shown by K F, is greater than J E, one-half the width at C B, it is proper that the width at L should be a medium between the two. Therefore from K, on K F, set off the distance J E, as indicated by K m. Bisect F m in n, and take K n as the width at L. The section at G H will then be an ellipse, of which G H is the major axis and K n one-half of the minor axis.

Fig. 672. - Elevation of Elbow, with Half Profiles of the Two Ends.

In Fig. 673, A G H D is a duplicate of the part hearing the same letters in Fig. (572. Against A D is placed a half profile, A F D, of the elliptical end. while against G H is placed one-half of the elliptical section, constructed as above described and as shown by G L H. Divide G L H into any convenient number of equal parts, and from the points thus obtained drop perpendiculars cutting G H, as shown. Also divide A F D into the same number of parts, and from the points thus obtained drop perpendiculars cutting A D. Connect the points in A D with those in G H, as indicated by the solid and dotted lines.

The next step is to construct sections on each of the solid and dotted lines drawn across the elevation by means of which to obtain the true distances between the points in A D and those in G H as though measured upon the finished article. In Fig. 674 is shown a diagram containing sections upon the solid lines, which is constructed in the following manner: Draw any two lines, as M N and M P, at right angles to each other. Upon M N set off the hights of the several points in the profile A F D; thus make M 13, M 12, M 11, etc., respectively equal to k 13, j 12, h 11, etc., of Fig. 673. Upon M P set off from M the lengths of the several solid lines of the elevation; thus make M a, M b, etc., respectively equal to f a, g b, etc., of Fig. 673. and at the points a, h. c, etc., thus obtained, erect perpendiculars, each equal in hight to the bight of the corresponding point in the profile G L H of Fig. 673 from the line G H. Thus make a 2, b 3, etc., of the diagram respectively equal to a 2, b 3, etc., of Fig. 673, and from the points 2, 3, 4, etc., thus obtained, draw solid lines to points 9. 10, 11, etc., in the line M N, all as shown. Then the distances 9 2, 10 3, 11 4, etc., will be the true lengths represented by corresponding solid lines drawn across the elevation.

Fig. 678. - Elevation of Lower Piece of Elbow, Showing Method of Triangulation.

The true distances represented by the doited lines drawn across the elevation are obtained in the same manner by means of the diagram shown in Fig. 675. RS is drawn at right angles to R T and upon it are set off the hights of the points in A F D the same as in M N of Fig. 674. Upon R T set off from R the lengths of the several dotted lines drawn across the elevation, as shown by corresponding letters, and from the points thus obtained erect perpendiculars also as in Fig. 674. Finally connect by dotted lines such points as correspond with those connected by dotted lines in the elevation. Thus from 9 in R S draw a line to point 1 in the base line, corresponding to the line f 1 of the elevation, Fig. 673. Lines from 10 to 2 and from 11 to 3 of the diagram will correspond respectively to g a and h b of Fig. 673.

Fig. 674. - Diagram of Sections Upon Solid Lines of Fig. 673.

To develop the pattern from the dimensions now obtained proceed as follows: At any convenient place draw the straight line H D in Fig. 676, in length equal to H D of Fig. 673. From H of pattern as center, with radius 1 9 of Fig. 675, describe an arc, which intersect by a second arc struck from D as center, with radius D 9 of profile, thus establishing the point 9 of pattern. Then with radius H 2 of G L H, Fig. 673, from H of pattern as center, describe an arc, which cut with another arc struck from point 9 of pattern as center, and 9 2 of Fig. 674 as radius, thus establishing point 2 of pattern. With point 2 of pattern as center, and 2 10 of Fig. 675 as radius, describe an arc, which intersect with another arc struck from point 9 of pattern as center, and 9 10 of profile as radius, thus establishing point 10 of pattern. With point 10 of pattern as center, and 10 3 of Fig. 674 as radius, describe a small arc, which intersect with one struck from point 2 of pattern as center, and 2 3 of G L H as radius, thus establishing point 3 of pattern. Continue this process, locating in turn the remaining points in pattern, as shown. Lines drawn through the points thus obtained, as indicated by G H and D A. will be one-half of the required pattern. The other half of the pattern can be obtained in a similar manner, or by tracing and transferring. The pattern for the other part of elbow, as shown in Fig. 672 by G C B H, can be obtained by the same method. The shape G L H of Fig. 673 is to be drawn to the left of the miter line G H, and the operation continued, using the same process, as shown in Figs. 674, 675 and 676.

Fig. 676. - Pattern for Lower Piece of Elbow.