116 FUNDAMENTAL THEORY OF COUPLES. 



Draw AM perpendicular to CD and AN to BC; then 

 because the moments are equal 



P AN AB 

 therefore _ = _ = __ ;: 



therefore the resultant of P and Q acts in A C. Similarly 

 the resultant of P and Q at C acts in CA, so that the 

 resultants balance each other., 



Hence the two couples balance each other ; and, there- 

 fore, two like couples of equal moment in the same, plane 

 are equivalent. 



If the forces of the couples were parallel we must ' 

 suppose the couples like, and take a third couple unlike 

 and of equal moment, with its forces not parallel to the 

 forces of the two couples. 



Then this couple will balance each of the two couples, 

 and therefore the two couples are equivalent. 



Next suppose two unlike couples of equal moment in 

 parallel planes. 



We may always replace them by two couples of equal 

 moment having equal and parallel arms and forces. 



Let any plane be drawn cutting the forces in the points 

 A, B, (7, D (ng. 35). 



AB is equal and parallel to CD, and therefore ABCD 

 is a parallelogram of which the diagonals bisect each other 

 at their point of intersection 0. 



The resultant of P at A and P at G is a parallel force 

 2P at ' y and of P at B and P at D is an equal parallel 

 but opposite force 2P at 0, hence the system is in equi- 

 librium. 



Therefore two like couples of equal moment in parallel 

 planes are equivalent. 



Hence the resultant of any number of couples in the 

 same or parallel planes is a couple, in a parallel plane of 

 moment equal to the algebraical sum of the moments of 

 the couples. 



