This section is from the book "The New Metal Worker Pattern Book", by George Watson Kittredge. Also available from Amazon: The new metal worker pattern book.

To obtain the pattern for the middle portion in one piece further calculations, however, will be required. This, of course, could be obviated by making a slip joint in the middle portion of the pipe, by means of which the two elbows could be made separate, and then simply turned upon each other till the required angle is obtained. But as it might be desirable to make the pattern of the middle portion in one piece some means must be employed of ascertaining just how far one elbow would have to be turned upon the other were they made separately. As the seam in pipe containing elbows is usually made at either the shortest or the longest point of the miter, it may be easily seen, by an inspection of Fig. 336, that a line from the shortest point, or throat, b of the upper miter of the piece in question, would not meet the longest point, or point a, Fig. 337, in the miter of the other end, and some means must be devised for obtaining the real position of these points, of which the following is perhaps the simplest: From either of the points D or B, Fig. 339, draw a line through the point b, continuing it to the further side of the triangle, as indicated by the line B X. Lay off the distance D X upon the line D C of the side view, Fig. 336, thereby locating the position of the point x in that view. A line connecting this point with point B must intersect the miter line 22, 42, in this view at the same point which it does in Fig. 339, thereby locating its position just as much as in Fig. 339. This point having been obtained, its equivalent upon the lower miter may be found by means of a line drawn parallel to the center line of the middle portion, intersecting it at, the point g. from which point it can be carried vertically to the plan, as shown by Z, where its distance from other points can be measured with accuracy. The position of the point a in Fig. 337 will readily be seen to be at point h in the plan of the front view, Fig. 336. By transferring the point Z from the plan of the side view to the plan of the front view, which can be done by measuring its distance from either of the points 2 or 3, the relative position of the points h and Z upon the same circle will be apparent. Fig. 340 shows a diagram, in which a correct side view of the two elbows is shown, giving the proper distance between the points B and C. Considering the lower one to be in its proper and fixed position, the profile is constructed and divided into points for the purpose of obtaining a stretchout and the miter pattern according to the usual method, the stretchout being shown upon the line E F in the profile and point 8 will readily be seen to correspond with point h in the plan of the front view. The position of the point Z in the same plan can be obtained by measuring its distance from point h and transferring it to Fig. 340, as indicated by M. As the point b of the upper elbow is in relation to the highest point, or a, of the lower elbow as the point M is to the point 8 in the profile, it becomes necessary to place the point 8 in the stretchout of the upper elbow as far from the point 8 on the stretchout of the lower one as the distance from 8 to M in the profile, which is shown by m in the stretchout. The stretchout of the upper elbow is thus moved, as it were, in its relation to the stretchout of the lower elbow, that portion of it which extends beyond the point 1 at the left end being added to the other so as to make the seam continuous. The points are then dropped from the profile to the two miter lines, and thence into measuring lines of corresponding number in the stretchout. Lines traced through the points of intersection, as shown by Y P R X, will be the required pattern. The miters for the upper and lower sections would, of course, be inverted duplicates of the adjacent ends of the middle piece.

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