166 RECIPROCAL FIGURES, FRAMES, 



concourse of PQK, and let the point travel in the same direction round the 

 polygon pqr. Then, the direction in which the point travels along any side 

 of the polygon will be the direction in which the force acts along the corre- 

 sponding piece of the frame on the point of concourse. If it acts from the 

 point of concourse, the force is a tension ; if towards it, it is a pressure. 



The other extremity of P meets B and C, and the forces along these 

 three pieces are in equilibrium. Henoe, if we draw a triangle, having /> for 

 one side and lines parallel to B and C for the others, the sides of this triangle 

 will represent the three forces. 



Such a triangle may be described on either side of p, the two together 

 would form a parallelogram of forces ; but the theory of reciprocal figures indi- 

 cates that only one of these triangles forms part of the diagram of forces. 



The rule for such cases is as follows : Of the two extremities of p, one 

 corresponds to the closed figure PRB, and the other to the closed figure PQC, 

 these being the polygons of which P is a side in the first figure. 



We must, therefore, draw b parallel to B from the intersection of p and r, 

 and not from the other extremity, and we must draw c parallel to C from 

 the intersection of p and q. 



We have now a second triangle, pbc, corresponding to the forces acting at 

 the point of concourse of P, B, C. To determine whether these forces are 

 tensions or pressures, we must make a point travel round pbc, so that its 

 course along p is in the opposite direction to its course round pqr, because the 

 piece P acts on the points PBC and PQR with equal and opposite forces. 



If we now consider the equilibrium of the point of concourse of QC and A, 

 we shall find that we have determined two of these forces by the lines q and c, 

 and that the third force must be represented by the line a which completes the 

 triangle '/'. 



We have now constructed a complete diagram of forces, in which each 

 force is represented by a single line, and in which the equilibrium of the forces 

 meeting at any point is expressed visibly by the corresponding lines in the 

 other figure forming a closed polygon. 



There are in this figure six lines, having four points of concourse, and 

 forming four triangles. To determine the direction of the force along a given 

 line at any point of concourse, we must make a point travel round the cor- 

 responding polygon in the other figure in a direction which is positive with 

 respect to that polygon. For this purpose it is desirable to name the polygons 



