Mr. W. II. Prcecc on the Parallelogram of Forces. 429 



(3) The moment of a couple is the proauc of the magnitude 

 of either force into the arm of the couple. (It is the numerical 

 measure of its importance.) 



Axioms. 



(1) Any system of forces may be replaced by their resultant. 



(2) Two equal and opposite forces acting on different points 

 of a rigid body, so as to balance each other, are upon the same 

 straight line. 



(3) Two equal and opposite couples acting at the same point 

 of the same rigid body, balance each other. 



(This is a Cor. to Definition 3 ; for the two couples have the 

 same moments, but of different signs.) 



1. Let the two forces P, Q act upon the point A ; it is required 

 to find the direction of their resultant. 



Take A C, A D respectively equal 

 in magnitude and direction to the 

 forces P, Q. Through C draw C B 

 parallel to AD, and through D 

 draw D B parallel to A C, meeting 

 CBinB. Join A B. ThenACBD 

 is a parallelogram, and A B is its 

 diagonal. 



At B, rigidly connected with A, apply a force P L equal and 

 opposite to P, and also a force Q x equal and opposite to Q. 



The system is in equilibrium ; for at the points A, B we have 

 the couple (P, P : ) acting in one direction, and also the couple 

 (Q> Qi) acting in the other direction; and these couples are 

 equal, for the moment of (P, Pj) is B D x C c, and the moment of 

 (Qj Qi) is ADxD*/; and these two products are evidently 

 equal, for they are each equal to the area of the parallelogram 

 AD B C. Hence they balance each other, and the system is in 

 equilibrium. 



Now the forces P and Q have a resultant which acts between 

 them ; we may therefore replace them by their resultant without 

 disturbing the equilibrium : call it R. 



The forces P 2 and Q, have also a resultant which acts between 

 them ; we may also replace them by their resultant, which we 

 will call R 1 . 



But these two systems of forces are equal and opposite ; and 

 since they balance each other, their resultants must be equal and 

 opposite and also balance each other ; and therefore, by axiom 2 } 

 the resultants must be in the same straight line. 



Hence the resultant of the forces P and Q acting at A must 

 be along the diagonal A B of the parallelogram A C B D whose 

 sides are equivalent to the forces P and Q. 



