Fig. 509. - Diagram of Angles for a Five Piece Elbow.

In this problem, as in the two immediately preceding, the various pieces necessary to form the elbow may be cut from one cone, whose dimensions must be determined from the dimensions of required elbow. The first essential will be to determine the angle of the cutting lines, which may be done the same as if the elbow were of the same diameter throughout.

Such an elbow of five pieces would consist of three whole pieces and two halves; therefore, if it is to be a right angle elbow, divide any right angle, as A' B C' in Fig. 509, into four equal parts, as shown by the points 1, 2, 3. Bisect the part A' B 3 by the line A B and transfer the portion A' B A to the opposite side of the figure, as shown by C B C.

This gives the right angle ABC divided into the same number of pieces and half-pieces as would be employed in constructing an ordinary five-piece elbow,

Fig. 510. - A Fivt-Piece Elbow in a Tapering Pipe.

The division lines in this diagram are of the correct angle for the miter lines in the elbow pattern, and therefore can be used upon the diagram of the cone, out of which are to be obtained the pieces to compose the required elbow.

It is assumed that the amount of rise and projection are not specified, therefore after having got the line of the angle or miter it becomes a matter of judgment upon the part of the pattern cutter what length shall be given to each of the pieces composing the elbow.

Fig. 511. - Elevation of Five-Piece Tapering Elbow.

In Fig. 510, let A 13 represent the diameter of the large end of the elbow. From the middle point in the line A B, as C, erect a perpendicular line, as indicated by C N, producing it indefinitely. On the line C N, proceeding upon judgment, as already mentioned, set off C X to represent the length of the first section of the elbow measured upon its center line. With X thus determined, draw through it the line D E, giving it the same angle with A B as exists between B C of Fig. 509 and the horizontal B C. This, in all probability, can most readily be done by extending B A indefinitely beyond A and letting E D intersect with B A extended, producing at their intersection an angle equivalent to C B C of Fig. 509. From the point X set off the distance X V, also established by judgment, thus determining the position across the cone of the miter line of the next section. Through Y draw G F at the sanv angle us D E, already drawn, but inclined in the oppo site direction. In like manner locate the two other miter lines shown in the diagram, finally obtaining the point Z. From Z set off the width toward N of the last section of the pattern, and through the point N thus obtained draw the line M O at right angles to G N, making it in length equal to the diameter of the small end of the elbow and placing its central point at N. Through the points A M and B O of the figure thus constructed draw lines, which produce indefinitely until they intersect the axis in the point P. Then P will be the apex of the required cone.

Construct a plan of the base of the cone or large end of the elbow below and in line with the diagram, as shown in the drawing, which divide into any convenient number of spaces, as indicated by the small figures, and from the points thus obtained carry lines vertically, cutting the base of the cone A B. From A B continue them toward the apex of the cone, cutting the several miter lines drawn. With the apex P of the cone for center, and with P B as radius, describe the arc T U, upon which set off a stretchout of one-half the plan, all as indicated by the small figures. From the points thus established in T U carry lines to the center P. With the T-square placed at right angles to the axis NC of the cone, and brought against the points of intersection in the several miter lines made by the lines drawn from points in the base of the cone to the apex, cut the side O B of the cone, as shown. Then from P as center, with radii corresponding to the distance from P to the several points on O B, as mentioned, strike arcs cutting the lines of corresponding numbers in the pattern diagram, as shown. Then lines traced through the points thus obtained, as indicated by D1 E1, F1 G1, etc., will cut the pattern O WU T of the frustum in such a manner that the sections will constitute the half patterns of the pieces necessary to form the required elbow. In Fig. 511 is shown an elevation of the elbow resulting from the preceding operation.