442 PROF. F. JENKIN ON THE PRACTICAL APPLICATION OF RECIPROCAL 



reaction of the support, the push due to the compressed member A, and the pull 



due to the extended member C. The directions 

 of the three triplets of forces are shown by 

 arrows. 



In the reciprocal figure lines parallel to each 

 triplet of forces must make a triangle, and the 

 whole figure must consist of six lines. Begin 

 with the joint 1, and draw one triangle mnq with the sides 

 P, A, C, such that they are parallel to P, A, C in fig. 1 ; while the 

 line P is equal on any convenient scale to the known force due to 

 the upward reaction at the point of support. Then PAC, fig. 2, 

 is the simple polygon of forces acting at 1. 



Next beginning at point m, and using the line A to express 

 the upward push at 2 (fig. 1) draw the triangle mno, which re- 

 presents the polygon of forces at joint 2. We next find that mqo 

 represents the polygon of forces at joint 3, and that the fig. 2 

 is the reciprocal of fig. 1. 



The reciprocal figure is thus built up of a series of the well-known polygons 

 of forces acting at each joint, but so arranged that the line representing the elas- 

 tic force exerted by each member does not require to be drawn twice, but forms 

 part of two polygons, in which, however, it represents forces acting in opposite 

 directions. 



Thus the line mq in fig. 2 forms part of the triangle qnm, in which it repre- 

 sents a pull from m to q, acting at point 1, and it also forms part of the triangle 

 mqo, in which it represents a pull from q to m, acting at joint 3. It is in 

 choosing the form of the component polygons, and in their arrangement, that a 

 certain discrimination is required. 



To aid in the construction of the figures, we may observe that the lines repre- 

 senting the external forces acting on a rigid frame in equilibrium must in the 

 reciprocal figure form a closed polygon, and when these lines are parallel, as when 

 weights only are applied, this polygon becomes infinitely thin, and is represented 

 by a single straight line, subdivided into parts proportional to the forces. Thus. 

 in figs. 1 and 2, the three vertical forces are represented by the lines no, oq, and 

 qn, which represent respectively the solid weight, acting from n to o, the up- 

 ward reaction P acting from o to q and P from q to n. This line (on), subdivided 

 in the ratio of the loads, may be conveniently termed the line of loads, and 

 reappears in all reciprocal figures of framework under parallel forces. 



The reciprocal figures corresponding to the ordinary Warren girder will now 

 be described in Plate XVII. fig. 1. The load is supposed to be applied at the 

 bottom joints, and will just be assumed as equal to 10 tons at each of the eight 

 joints. These weights are represented by short vertical lines in fig. 1. 



