84 STATICS. [140. 



140. The given system of forces will be in equilibrium if the 

 following two conditions of equilibrium are fulfilled : 



Jt = o, H=o. 



It will be noticed that the moment Fp of the couple intro- 

 duced by transferring the force F to the point O is the moment 

 of the force Fwith respect to this point O. 



Hence, a plane system of forces is in equilibrium if (a) its 

 resultant is zero, and (b) the algebraic sum of the moments of all 

 its forces is zero with respect to any point in its plane. 



141. It is evident that the magnitude and direction of the 

 resultant R do not depend on the selection of the origin O. 

 But the position of this resultant and the magnitude of the 

 resulting couple // will in general differ for different points 

 selected as origin. Indeed, the origin can be so taken as to 

 make the couple H vanish (unless the resultant R be zero) ; 

 that is, the whole system can be reduced to a single resultant. 



To do this (see Art. 135), it is only necessary, after determin- 

 ing R and H for some point O, to transfer R to a parallel line 

 at such a distance r from its original position as to make the 

 moment Rr of the couple introduced by the transfer equal and 

 opposite to the moment ^Fp ; i.e. we must take (Art. 135) 



The line along which this single resultant acts is called the 

 central axis of the given system of forces. 



142. For a purely analytical reduction of a plane system of 

 forces the system is referred to rectangular axes Ox, Oy, arbi- 

 trarily assumed in the plane (Fig. 37). Every force /MS resolved 

 at its point of application P (x, y) into two components X, Y, 

 parallel to the axes, so that 



F=/ 7 sin, 



