Composition and Decomposition of Forces. 37 



or 



qXDB=pxDA + rX DC. 



Moreover, as the point F, taken arbitrarily, may be higher or 44 ' 

 lower, as we please, the point D has not been supposed to be in 

 one point of the direction of the resultant rather than in another ; 

 it follows, therefore, that the moments of several parallel forces, 

 taken with respect to any point whatever in the direction of the resul- 

 tant, are such, that the sum of the moments of the forces which tend 

 to turn the system in one direction, is always equal to the sum of the 

 moments of those which tend to turn it in the opposite direction. 



68, Therefore by taking with contrary signs the moments of 

 the forces which tend to turn the system in opposite directions, 

 and by taking also with contrary signs the forces which act in 

 opposite directions, we may infer as a general conclusion j 



1. That the resultant of any number whatever of parallel forces 

 is always equal to the sum of all these forces ; 



2. That this resultant, which is parallel to the component forces, 

 passes through a series of points each of which has this property, that 

 the sum of the moments, taken with respect to this point, is zero. 



The above propositions are of the greatest importance. We 

 shall see soon with what facility they enable us to find the centre 

 of gravity of bodies. We proceed now to the consideration of 

 forces the directions of which are inclined to each other. 



69. Let there be any number of forces p, q, r, &c., all exert- Fig. 22. 

 ed in the same plane, let the force p, acting according to AE, be 

 represented by AE, and let the force q, acting according to BG, 



be represented by BG, and the force r, acting according to CL be 

 represented by CL. Through a point F, taken arbitrarily in the 

 plane of these forces, suppose two straight lines FC', FB", mak- 

 ing any angle with each other (and for the sake of greater sim- 

 plicity, let this angle be a right angle) ; and let us imagine the 

 forces p, q, r, or AE, BG, CL, decomposed each into two others, 

 one of which shall be parallel to FC, and the other to FB", and 

 which will consequently be represented, each by the correspond- 

 ing side of the parallelogram, the diagonal of which represents 4^ 

 the given force. 



It is clear from what has been said, that the forces AV, BH, 66}68t 

 CK, being parallel, will have for their resultant a single force 

 DN, parallel to AV, BH, &c., the value of which will be 



