124 



STATICS. 



[207. 



It appears from the construction of the central axis given in 

 Art. 205, that the vector of the resulting couple for this axis / 

 is HQ = HCOS$ ; it is, therefore, less than for any other line. 



It is instructive to observe how the vector H increases and 

 changes its direction as we pass from the central axis / to any 

 parallel line /. 



The transformation from / to / requires the introduction of a 

 couple (R, r Q ) whose vector Rr Q (Fig. 68) is at right angles to 

 the plane (/ , /) and combines with //" to form the resulting 



couple H for /. As the distance r Q 

 of / from / is increased, both the 

 magnitude of H and the angle </> it 

 makes with / increase until, for an 

 infinite r Q , the angle $ becomes a 

 right angle. 



-R 

 I- 



207. It is evident that since i 

 = ff cos (f>, the product RH cos is a 

 constant quantity for a given system 

 of forces. It has been called the 

 invariant of the system. 



If the elements of reduction for the 

 central axis (R, H Q ) be given, those 

 for any parallel line / at the distance r Q from the central axis are 

 determined by the equations 



Fig. 68. 



208. To sum up the results of the preceding articles, it has 

 been shown that any system of forces acting on a rigid body can 

 be reduced, in an infinite mimber of ways, to a resultant R in 

 combination with a couple H. For all these reductions the mag- 

 nitude, direction, and sense of the resultant R are the same, but 

 the vector H of the couple changes according to the position 

 assumed for the line of R. There is one, and only one, position 

 of R, called the central axis of the system, for which the vector 



