In Fig. 626, DC K L shows a portion of the larger pipe, of which M P N O is the section; H G B A a portion of the smaller pipe, of which E J F I is the section; and A B C D the elevation of the transition piece necessary to form a connection between the two pipes at the angle H A L. The drawing also shows that the ends of the two pipes to be joined are square, or cut off at right angles, so that the lower base of A B C D is a perfect circle whose diameter is D C (or M N) and the upper base is a circle whose diameter is A B (or E F). and also that the side A D is vertical. In the choice of a method of dividing the surface of the piece ABCD into triangles, either the elevation or the plan can be made, use of for that purpose, according to convenience. In the demonstration here given the elevation has been used by way of variety, all as shown in Fig. 627. Proceed then to divide the plan of the upper base into any convenient number of equal spaces, as shown, and drop a line from each point at right angles to A B, cutting A B, and numbering each point to correspond with the number upon the plan. In like manner divide the plan of the lower base into the same number of equal spaces, and erect a perpendicular line from each, cutting the line D C, and numbering the points of intersection in the same order, or to correspond with the points in the upper base, all as shown. Connect the points in D C with points of similar number in A B by solid lines, also connect points in D C with points of the next higher number in A B by dotted lines, which will result in a triangulation suitable for the purpose.

Fig. 626. - Elevation of a Transition Piece Joining Two Round Pipes of Unequal Diameter at an Angle.

Fig. 627. - Triangulation of Transition Piece Shown in Fig. 626.

The next step will be to construct sections through the piece upon all of the lines upon the elevation (both solid and dotted), which operations are shown in Figs. 628 and 629, and which may be done in the following manner: Upon any horizontal line, as T S of Fig. 628, erect a perpendicular, as T U. Upon T S set off from T the several distances of the points in the lower base from the center line 1 7 of the plan, as measured upon the vertical lines (Fig. 627), all as indicated by the small figures. Upon T U set off the lengths of the solid lines of the elevation, numbering each point thus obtained to correspond with its line in the elevation. From each of the points upon T U draw horizontal lines to the right, making each in length equal to the distance of points of corresponding number in the plan of the upper base from the center line 1 7, as measured upon the lines at right angles to line A B, thus obtaining the points 2', 3', 4', etc. Now connect these points with points of corresponding number in the base line T S by means of solid lines, as shown.

Fig. 628. - Diagram of Sections Taken on Solid Lines of Fig. 627.

Fig. 629. - Diagram of Sections Taken on Dotted Lines of Fig. 627.

In constructing the sections upon the dotted lines of the elevation, shown in Fig. 629, the same course is to be pursued as that employed in Fig. 628. The base line W V is a duplicate of T S. Upon the perpendicular line erected at W set off the lengths of the several dotted lines of the elevation, numbering each point thus obtained to correspond with the number at the top of its line in the elevation. From each point draw a horizontal line to the right as before, which make equal in length to the similar lines in Fig. 628, numbering each point as shown by the small figures 2', 3', 4', etc. Now connect each of these points with the point of next lower number in the base line V W by a dotted line.

Fig. 631. - Perspective View of Model.

Having obtained all the necessary measurements, the pattern for one-half the envelope of A B C D may be developed in the following manner: Draw any line, as A D in Fig. 630, which make equal in length to A D of the elevation. "With A as a center, and a radius equal to 1 2 of the plan of the upper base, Fig. 627, strike a small arc, which intersect with another struck from D as a center, with a radius equal to the dotted line 1 2' of the diagram, Fig. 629, thus establishing the position of the point 2 in the upper line of the pattern. From D as a center, with a radius equal to 1 2 of the plan of the lower base, Fig. 627, strike a small arc, which intersect with another struck from point 2, just obtained, with a radius equal to the solid line 2 2' of the diagram, Fig. 628, thus fixing the position of point 2 in the lower line of the pattern. So continue, using the lengths of the dotted lines in the diagram, Fig. 629, in connection with the lengths of the spaces in the plan of the upper base, to develop upper line of the pattern, and the lengths of the solid lines in the diagram, Fig. 628, in connection with the lengths of the spaces in the plan of the lower base, to develop the lower line of the pattern, using each combination alternately until the pattern is complete. As each new point of the pattern is determined it should be numbered, and the solid or dotted line used in obtaining the same may be drawn across the pattern if desired, merely as a means of noting progress, but these lines are not necessary, as each point is simply used as a center from which to find the next point beyond.

Fig. 630. - Pattern of Transition Piece Shown in Fig. 627.

Sometimes, in order to more thoroughly understand the method employed in such an operation as the foregoing, it is desirable to construct a small model, which can be made from cardboard or thin metal, the details of which are clearly shown in Fig. 681. The pieces forming the upper and lower bases of it should be duplicates of the half plans of the upper and lower bases shown in Fig. 627. having the lines there shown drawn upon them, and the piece forming the back is a duplicate of the plane figure A B C D. These three parts may be cut in one piece, after which a right angle bend on the lines A B and C D will bring the two bases into correct relative position. Five quadrilateral figures corresponding to those shown in Fig. G28 may now be cut and fastened in position, according to their numbers, between the two bases of the model. Threads or wires can be so placed as to correspond in position with the dotted lines shown in the elevation, Fig. 627, to complete the model. The model is only useful before the pattern is developed to assist in showing the shapes and order or rotation of the various triangles; and one constructed to the dimensions of any problem which may occur to the student at the outset of his study of triangulation will serve to assist his imagination in all subsequent operations.