208 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



Theorem III. A couple may be replaced by another in the 

 same plane having equal moment. 



Let the couple be P^P^ and 

 the arm be A S (Fig. 53). At C 

 on AS produced and at S apply 

 four equal and opposite forces Q 

 of such magnitude that 



ft 



B 



AB 



The resultant of the parallel 

 forces, P 1? Q B , is equal to P l plus 

 Q 3 applied at S on account of 

 the above equation. This is 

 counterbalanced by the forces P 2 

 Fig. 53. and Q applied at B, leaving the 



couple Q^Qz of moment 



f\ 7?/T ~p A 7? 



\cJ JD\J === JL * J* Ify 



equivalent to the original couple. 



A force -couple is determined therefore by its plane and moment, 

 and may be represented by a free vector perpendicular to its plane 

 and of length equal to the moment. 



Theorem IV. Composition of Couples. Suppose the two couples 

 are in different planes. By turning each in its own plane bring all 



the four forces into directions 

 perpendicular to the intersection 

 of their planes, and then by 

 varying one of the couples cause 

 them to have the same arm AS. 

 The forces Qjfi applied at A 

 compound by the parallelogram 

 into E v P 2 and Q 2 applied at 

 S compound into E% equal and 

 opposite to E v The arm of all 

 these couples is the same, there- 

 rig. 54. fore their moments are propor- 

 tional to P, Q and E. The vectors 



representing the moments are perpendicular to AS and to P, Q and E 

 respectively, thus they form the sides and diagonal of a parallelogram 

 similar to that of P, Q, E. Therefore couples are compounded by 

 compounding their moments by the law of addition of vectors. 



