AND DIAGRAMS OP FORCES. 165 



Note. It is often convenient to turn one of the figures round in its own 

 plane 90. Corresponding lines are then parallel to each other, and this is 

 sometimes more convenient in comparing the diagrams by the eye. 



Since every polygon in the one figure has three or more sides, every 

 point in the other figure must have three or more lines meeting in it. Since 

 every line in the one figure has two, and only two, extremities, every line in 

 the other figure must be a side of two, and only two, polygons. If either of 

 these figures be taken to represent the pieces of a frame, the other will repre- 

 sent a system of forces such that, these forces being applied as tensions or 

 pressures along the corresponding pieces of the frame, every point of the frame 

 will be in equilibrium. 



The simplest example is that of a triangular frame without weight, ABC, 

 jointed at the angles, and acted on by three forces, P, Q, R, applied at the 

 angles. The directions of these three forces must meet in a point, if the frame 

 is in equilibrium. We shall denote the lines of the figure by capital letters, 

 and those of the reciprocal figure by the corresponding small letters ; we shall 

 denote points by the lines which meet in them, and polygons by the lines 

 which bound them. 



Here, then, are three lines, A, B, C, forming a triangle, and three other 

 lines, P, Q, R, drawn from the angles and meeting in a point. Of these 

 forces let that along P be given. Draw the first line p of the reciprocal 

 diagram parallel to P, and of a length representing, on any convenient scale, 

 the force along P. The forces along P, Q, R are in equilibrium, therefore, if 

 from one 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 



