In Fig. 338 is shown a front and side view of a somewhat complicated arrangement, of elbows such as sometimes occurs when pipes have to be carried around beams or through limited openings. An inspection of the drawing will show that once the correct angle of the different elbows is ascertained the development of the miters will be quite simple, and is the same as those occurring in several of the problems preceding this. The lower section of the pipe rises vertically to the first elbow, B, from which it must be carried upward a distance equal to C M, to the left a distance equal to B M, as shown in the front view, and back a distance equal to o C, as shown by the side view.

Fig. 336. - Elevations of Double Elbow.

Fig. 337. - Correct Side View of Lower Elbow.

Fig. 338. - Diagram Used in Obtaining Correct Side View of Upper Elbow.

Fig. 339 - Correct Side View of Upper Elbow.

Fig. 340. - Method of Obtaining the Pattern of Middle Portion in One Piece.

The Pattern for the Intermediate Piece of a Double Elbow Joining Two Other Pieces Nut Lying in the Same Plane.

From the elbow C it then rises vertically, as seen in front, but really toward the observer as shown by the side view. The problem then really consists in finding the correct angles of the elbows, and becomes a question of draftsmanship rather than of pattern cutting. Some suggestions then with regard to the methods employed in drawing the two views shown in Fig. 336 will be of assistance to the pattern cutter. According to the principles of projection each individual point must appear at the same hight in both elevations, and at the same distance right or left and forward or back, with reference to the center lines of the plan. As front and side views are here required, begin by first placing the given plan in two positions, turning those sides of it to the bottom which correspond to the sides required in the elevations, and proceed by erecting the center lines of the different pieces in their proper positions and building the pipe around them, so to speak. The plan being a circle, the different sides can only be indicated by numbering the points, as will be seen by referring to the plans, point 2 appearing in front in the front elevation, and point 3 appearing in front in the side elevation. The plans having been so arranged and corresponding parts in both given the same Dumber, proceed now to erect the center line of the lower section, making the hight of the first bend, B, the same in both views, as indicated by the dotted horizontal line. From this point the center line is continued in both views, giving it its proper inclination to the left in the front view, and to the right in the side view, all according to the specified requirements, thus establishing the point C, making it agree in hight in both views. From this point the pipe appears inclined only in the side view, which means that it leans toward the observer in the front view. Next draw the outlines of the pipe at equal distances from the center line and on either side of it throughout the entire course of the pipe in both views, deriving them from the points of plans 1 and 3 in the front view and 2 and 4 in the side view. Their intersection in the front view will give definitely the positions in the miter of points 1', 12, and 3', 32, and in the side view of points 2', 22 and 4', 42 As point 3' has been established in the front view, if a line be carried horizontally across till it intersects the line from point 3 of the side view, it will give the hight of point 3' in the miter, as shown in the front view. In the same manner a horizontal line from 1' in front, intersecting the perpendicular from point 1 in plan of side, will give the true hight of point 1' in the side view. A careful inspection of the dotted lines of Fig. 336 will make the subsequent operations necessary to the com-pletion of the elevations clear to the reader without further explanation. Since neither of the views gives a true side view of the intermediate piece, one must be constructed from the facts now known, so as to get the true angle of the elbow B. By dropping a vertical line from the point C of the front view into the plan it will appear that the horizontal distance between the points C and B would be measured by the line E P of the plan; but by further reference to the side elevation the position of the point C is found to be to the right of its center line by a distance equal to B C' of the plan; therefore, if this distance be set off on the vertical line from the point E in the plan below the front view, which is indicated by E C, the point C will determine the true position in the plan of the point

C of the elevations, and the distance C P will be its horizontal distance from B. Since, now, its vertical distance can easily be obtained from either front or side elevation, a new diagram can now be easily constructed which shall contain the proper dimensions to obtain a correct side view of this elbow. Proceed, then, to construct diagrams shown in Fig. 337. making C M equal to C M, Fig. 336, M B equal to C P of the plan, Fig. 336; a line connecting the points C and B will represent the center line of the intermediate portion of the pipe and give its true relation to the vertical portion whose center line is represented by B H, Fig. 337. By drawing the outlines of the pipe at the required distance on either side of the center lines B H and B C, a correct side view of the miter is obtained. Since, as has been referred to above, the upper portion of the pipe appears vertical in one view and inclined in the other (see Fig. 336), a correct side view of the upper elbow is more difficult to be obtained. While different methods may be devised for obtaining it, the following is perhaps the simplest: As the upper section of the pipe, as shown by Fig. 336, is of indefinite length, any point may be assumed, as D, from which to take measurement for obtaining the angle of the upper elbow. Since the true length of the line C B of either elevation has already been obtained and given in Fig. 337, and since the true length of the part C D can be derived from the side view of Fig. 336, it is necessary only to obtain the true distance between the points D and B of the elevations to obtain the proper angle at the point C. By dropping a vertical line from the point D to a horizontal line drawn from the point C in the side view, Fig. 336, the horizontal distance between C and D may be obtained. By tranferring this distance, o C, to the plan of the front view, and locating its distance from C, as indicated by D, this point will give the true position of the point D in the plan, and the line D P will give the true horizontal distance between the points D and B. In Fig. 338 let the distance 0 C be equal to the line D P of Fig. 336. At point 0 erect a perpendicular, 0 D, making the distance O D equal to o D of the side elevation, Fig. 336. From the point C drop a perpendicular, C B, making that distance equal to the vertical hight between the points C and B, as measured on line C M of the front view; a diagonal line connecting the points D and B will readily be seen to give the true distance between the points bearing those letters in Fig. 336. Proceed now to construct the triangle shown in Fig. 339, making C B equal to C B of Fig. 337. From C as a center, with a radius equal to C D as obtained from the Bide view in Fig. 336, draw a small are, which intersed with the arc drawn from the point B, with a radius equal to B D as obtained in Fig. 338; this will give the correct angle of the upper elbow at C. A complete view of the miter may be obtained by further adding outlines of the pipe at equal distances on either side of the center lines, and connecting their angles, as shown by the line bf. Having now obtained two correct side views of the two elbows, the problem of obtaining the patterns for the same can be solved by the regular method.