86 STATICS. [144. 



144. If R be not zero, R and H can be combined into a single 

 resultant R' equal and parallel to R at the distance H/R from 

 it (see Art 141). The equation of the line of this single result- 

 ant R'y i.e. the central axis of the system of forces, is found by 

 considering that it makes the angle a with the axis of x and that 

 its distance from the origin is 



H/R = 2 (x Y- 

 Hence its equation is 



f - 2F-17 2AT- 2 (x Y-yX) = o. 

 If R = o, the system is equivalent to the couple 



unless H itself be also zero, in which case the system is in 

 equilibrium. 



145. The same results can be obtained by a transformation 



of co-ordinates. Let R = V(S^T) 2 -f (2F) 2 and H= 2 (x Y-yX ) 

 be the resultant force and couple for a point O as origin. If 

 some other point O', whose co-ordinates with respect to O are 

 ( , 77, be taken as new origin and x\ y 1 be the co-ordinates of the 

 point of application P of the force F for parallel axes through 

 O\ the resultant R remains the same while the resulting couple 

 becomes 



Hence this new couple will vanish whenever the origin O'(%, rj) 

 is taken on the straight line whose equation referred to the 

 original axes is 



77-77=0. 



This equation of the central axis agrees with the equation found 

 in Art. 144; it represents the line of action of the single 

 resultant to which the system can be reduced. 



