CHAP, n.] DIFFERENT POINTS OF APPLICATION. 21 



Two forces therefore which form a couple cannot be replaced 

 by a single force. Their resultant is an indefinitely small force 

 situated in any position in the plane of the forces, at an infinite 

 distance. 



21. Conditions of Equilibrium. If then, similarly to Art. 

 4, any number of forces lying in the same plane and having 

 different points of application, are in equilibrium, the force 

 polygon always closes. 



For this reason, as already repeatedly seen in the practical 

 applications of our last chapter, the force polygon formed by 

 the exterior forces must always close. 



But inversely, if the force polygon closes, it does not follow 

 that the forces are in equilibrium a couple may result. 



To determine whether this is the case inspect also the " equi- 

 librium polygon." If this also closes [i.e., if S and S n inter- 

 sect] the forces are in equilibrium. If this does not close [i.e., 

 if S and S n are parallel] there is no single resultant, but the 

 forces can be replaced by a couple, and this couple, as we have 

 seen, may have any position in the plane. 



Thus if we suppose in Fig. 11, PI. 3, P t and P 2 decomposed 

 into their components S , S 1? and S^ S 2 , the compressive strains 

 in S x at c and d are equal and opposite [see ()]. We have 

 then S and S 2 remaining, which again form a couple which 

 must have the same action as the first. 



Hence we see that one couple can be replaced by another with- 

 out changing the action of the forces. 



It is easy to determine a simple relation between any two 

 couples. 



If from c we lay off c a equal to o 1, and c o equal to Co, we 

 have o a parallel to C 1 or S l5 and therefore to c d. Join a d and 

 o d. The triangles c d a and c d o having a common base c d 

 and their vertices o and a in a line parallel to c d, are equal in 

 area. The side c a of one is known, and the opposite apex lies 

 in the line of the force P 2 . Its area is then c a P x multiplied 

 by half of the perpendicular distance of P x from P 2 , and is 

 therefore' completely determined. So also for the other trian- 

 gle, one side of which o c is one force of the new couple, and 

 the. opposite apex of which lies in the other force S 2 . 



IIenc<Jk^& couple can be turned at will in its plane of action, 

 and the intensity and direction of its forces can be changed at 

 will if tho area of the triangle the base of which is one of the 



