connecting these points produce a rectangle, showing the true shape of the flat surface in the throat of the last piece of the elbow. If this surface were allowed to taper, the oval pipe to which the elbow is joined would not fit properly. Since the flat surface is to disappear at the throat of the first piece, straight lines may be drawn from the extremities of line N to the point H, thus establishing widths at M and K.
The widths of the flat surfaces on the back of the elbow, as shown by Fig. 308, are developed in exactly the same manner, and the description given above may be used again by substituting the word back and the letters that correspond.
Since the major axis (long diameter) of the oval is greater than the diameter of the circular profile a gradual increase must also be made in the major axes of the sections formed by the several miter lines. A stretchout of the center line of Fig. 306 is made as shown by Fig. 309. On the horizontal line T the long diameter of the oval profile is placed (one-half on each side of point T), while on line P one-half of the diameter of the round profile is placed. Straight lines are then drawn connecting these points, thereby establishing the major axes of the sections formed by miter lines DM and CK.
The patterns for the first and last pieces can be drawn by the method described in Chapter III since they are pieces of regular five-piece elbows.
The intermediate pieces of the elbow must be developed separately, either by triangulation or sectional development. The third piece of the elbow has been selected for treatment in this description, but the second and fourth pieces are developed in exactly the same manner.
Figure 310 shows the third piece removed from Fig, 306, in order to avoid confusion of lines. At the center points R and S perpendicular center lines are erected. Upon the center line at S one-half of the major axis of section DM, Fig. 309, is set off; and upon the center line at R, one-half of the major axis of section CK, Fig. 309, is set off. These points are lettered 4 and R as shown. Perpendiculars are now erected at points D, M, K, and C. Upon these lines the following lengths are placed: line D–1 is made equal to one-half of line D in Fig. 308; line C–U one-half of line C in Fig. 308; line M–7 one-half of line M in Fig. 307; and line K–Z one-half of line K in Fig. 307. Arcs are now drawn connecting
