34 ( Statics. 



directions, provided they act in the same plane, the moment of the resul- 

 tant of all these forces, taken with respect to a fixed point F, assumed at 

 pleasure in this plane, will always be equal to tht sum of the moments 

 of the forces tending to turn the system in one direction about this 

 point, minus the sum of the moments of those which tend to turn it in 

 the opposite direction. 



Fig. 20. Indeed, if we suppose that p' is the resultant of the two forces p, 

 q, p" that of p' and r, and p that of p" and <&-, if we suppose, more- 

 over, that ft represents the moment of p', and ^ that of p", then by 

 letting fall the perpendiculars FA, FE, FG, FD, FB, upon the 

 components p, q, r, w, and their resultant p, we shall have 



1. fi=p x FA + q x FE, 



2. ft' = a r X FG, 



3. px FB = !*' <* x FD-, 



adding therefore these three equations together, and suppressing 

 those quantities that cancel each other in the two members, we 

 shall have 



gxFB^pxFA + qxFE rxFG KX FD-, 

 from which it will be seen, that the moments of the two forces r, 

 v, which tend to turn the system from right to left, are of a con- 

 trary sign to that of the forces p, q, which tend to turn it from 

 left to right. 



64. If the point F were exactly in the direction of the resul- 

 tant, the moment of this force would be zero ; but, since it is equal 

 to the sum of the moments of the* forces which tend to turn the 

 system in one direction, minus the sum of the forces tending to 

 turn it in the opposite direction, we conclude that the difference 

 of these two sums of moments, taken with respect to any point 

 whatever in the direction of the resultant, is zero. 



And reciprocally, if the sum of the moments of the several forces 

 which tend to turn a system about a given point, minus the sum of 

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

 about this same point, is zero ; it must be inferred, that the resultant 

 passes through this point. 



65. As these propositions hold true, whatever be the angles 

 formed by the directions of the forces, they are applicable, when 

 these angles are infinitely small, that is, when the directions of 

 the forces are parallel. 



