206.] 



CONDITIONS OF EQUILIBRIUM. 



123 



and Rr into a resultant vector H 1 by geometric addition, we 

 have found the elements of reduction R, H' for the origin O'. 



204. If the new origin O' had been selected on the line / of 

 the original resultant, no new couple (R, r) would have been 

 introduced, and H would not have been changed. But when- 

 ever the line of action / of the resultant is changed, the vector 

 of the resultant couple H is changed. 



By increasing the distance r between / and /' the moment Rr 

 of the additional couple is increased. The effect of combining 

 this additional couple Rr with H is, in general, to vary both the 

 magnitude of the resulting couple H' and the angle <f> it makes 

 with the direction of the resultant R. It can be shown that the 

 line /' of the new resultant can always be selected so as to 

 reduce the angle <f> to zero. The line / for which $ = o, i.e. for 

 which the vector H of the resultant couple is parallel to the 

 resultant force R, is called the central axis of the given system 

 of forces. We proceed to show how it can be found. 



205. Let the vector H be resolved at O into a component 



J~f Q = H cos $ along /, and a component 

 angles to /(Fig. 67). In the plane pass- 

 ing through / at right angles to H ly it 

 is always possible to find a line / par- 

 allel to / at a distance r Q from /, such 

 as to make Rr Q = H v 



The line / so determined is the cen- 

 tral axis. For, if this line be taken as 

 the line of the resultant R, the addi- 

 tional couple Rr Q destroys the compo- 

 nent ff v so that the resulting couple 

 HQ has its vector parallel to R. 



206. As the direction of the vector 

 H is always changed in passing from 



at right 



Fig. 67. 



line to line, there can be but one central axis for a given system 

 of forces. 



