In Fig. 684 is shown a pictorial representation of a fork, or crotch, consisting of three branches of equal size and taper; all uniting so as to form one round pipe.

In the plan. Fig. 685, ABC represents the base of article or size of the huge pipe and B D E C G one of the tapering branches. The other branches are partly shown in plan by A G C S T and A U V B G.

Fig. 684.   Perspective View of Three Pronged Fork with Tapering Branches.

Fig. 684. - Perspective View of Three-Pronged Fork with Tapering Branches.

In the elevation the branch is shown by J K L M N and the half profile of small end by K R L.

An inspection of the engraving will show that the perimeter of the larger end of the branch must be divided into three parts, two of which form the joints or connections with the branches on either side of it while the third part must form one-third of the base or circumference of the large pipe with which it is to be united. In the elevation P M represents the plane of the base or upper end of the round pipe of which A B C is the profile or plan, and J O is assumed as the hight of the central point at which all the branches meet. From J of the elevation or G of the plan to either of the three points A, B or C any suitable curve may be chosen as the profile upon which to make a joint or miter between adjacent branches. As J O is equal to G A or G C, a quarter circle is assumed as the most suitable curve; therefore from O as a center describe the quarter circle P J of the elevation, corresponding with A G of the plan. In order to complete the elevation of the branch J K L M N, it will be necessary to obtain the elevation of the miter line G C. Therefore divide P J into any convenient number of equal parts, as shown by the small figures, and from the points thus obtained carry lines to the right parallel with P M. From G, on G C, set off spaces equal to the distances from the points in P O to the line O J, as shown, and from the points thus obtained in G C erect perpendiculars cutting lines of similar number drawn from P J. A line traced through these points of intersection, as shown by J N, will give the miter line in elevation corresponding with G C of the plan. Divide C H of the plan into the same number of equal parts as P J of the elevation, and from the points thus obtained erect perpendiculars rutting N M. Divide K R L, the profile of the smaller end of the branch, into the same number of equal parte as the larger end - that is, as many as are found in J N M - and from the points of division drop lines perpendicular to K L, cutting the same. Connect points in K L with those in JNM by solid and dotted lines in the manner shown in the drawing. Upon all of these lines it will be necessary to construct see tions in order to obtain the true distances as if measured upon the surface of the branch. As each of the branch pipes consists of symmetrical halves when divided by the line G F of the plan half sections only need be constructed, all projections being measured from the dividing plane represented by G F in the plan and shown in elevation by J KLMN.

In Fig. 686 are shown the sections having for their bases the solid lines of the elevation, which are constructed in the following manner: Upon any horizontal line, as P Q, set off from P the lengths of the several solid lines of the elevation, as indicated by the small figures corresponding with those in JNM. At P, which corresponds with all the points in K L of the elevation, erect a perpendicular, P H, upon which set off the hights of the points in K R L, as 2' 2, 3' 3, etc., shown by P 2, P 3, etc. At each of the points near Q erect a perpendicular, which make equal in hight to the length of line drawn from the point of corresponding number in G C H of the plan to the line G H. Thus make 9' 9, 10' 10, etc., equal to 9" a, 10" b, etc., of the plan. From the points 9, 10, etc., draw solid lines to the points in H P, connecting points correspondingly connected by the solid lines of the elevation. The sections having for their bases the dotted lines of the elevation are shown in Fig. 687, and are constructed in exactly the same manner. Upon Y Z, set off from Y the lengths of the dotted lines of the elevation, numbering the points near to correspond with those in J N M of the elevation.

The perpendiculars erected from these points are tin-same as those similarly located in Fig. 686, and the perpendicular X Y is a duplicate of H P of Fig. 686. Prom the points 9, 10, 12. etc., draw dotted lines to points in X Y. connecting points correspondingly connected by dotted lines of the elevation.

To describe the pattern shown in Fig. 688 proceed as follows: Draw any line, as J K, in length equal to J K of elevation. Fig. 685. With K of pattern as center, and K 2 of profile as radius, describe a small arc (2), which cut with one struck from J of pattern as center, and 8' 2 of Fig. 687 as radius, thus establishing point 2 of pattern. With point 2 of pattern as center, and 9 2 of Fig. 686 as radius, describe another small arc (9), which intersect with one struck from J of pattern as center, and J 9' of elevation as-radius, thus establishing the point 9 of pattern. Proceed in this manner until the remaining points are located, all as clearly indicated by the solid and dotted lines in Fig. 688. By drawing lines through the points thus obtained the half pattern shown by K Q LMNJ is the result. The other half, as shown by

K Q' L' M' N' J, can be obtained in a similar manner, or by duplication.

Fig. 685,   Plan and Elevation of Three Pronged Fork.

Fig. 685, - Plan and Elevation of Three-Pronged Fork.

Fig. 686.   Diagram of Sections Upon Solid Lines in J K L M N of Fig. 685.

Fig. 686. - Diagram of Sections Upon Solid Lines in J K L M N of Fig. 685.

Fig. 687.   Diagram of Sections Upon Dotted Lines in J K L M N of Fig. 685.

Fig. 687. - Diagram of Sections Upon Dotted Lines in J K L M N of Fig. 685.

Figj. 688.   Pattern of Tapering Branch.

Figj. 688. - Pattern of Tapering Branch.