RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 5 



extremity of p we draw q parallel to Q, and from the other extremity r parallel 

 to R, so as to form a triangle pqr, then q and r will represent on the same scale 

 the forces along Q and R. 



To determine whether these forces are tensions or pressures, make a point 

 travel along p in the direction in which the force in P acts on the point of con- 

 course of PQR, 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 corres- 

 ponding 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. Hence, if we draw a triangle, having p 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 indicates 

 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 cor- 

 responds 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 qca. 



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



VOL. XXVI. PART I. B 



