102 STATICS OF EIGID BODIES 



neglected : its moment does not vanish, being equal to the sum of 

 the moments of the component forces. If, in fig. 55, AA', BB' are 

 the parallel lines of action of two opposite forces each equal to R, 



and if PAB is a line at right angles to 

 their direction, then the sum of their 

 moments about a line through P, at 

 right angles to the plane in which the 

 forces act, 



where d is the distance between the 

 line of action of the forces. A pair of 

 forces, equal in magnitude and opposite 

 in direction, but not acting in the same 



line, is called a couple. Their moment about any point P in the 

 plane containing their lines of action is independent of the position 

 of the point P, and is spoken of as the moment of the couple. 



CONDITION OF EQUILIBRIUM 



79. Since the resultant of a system of forces in a plane may be 

 either a single force or a couple, the condition for there being no 

 resultant will be that the resultant single force shall be nil, and 

 that there shall be no couple. The component of the resultant 

 force in any direction vanishes if the sum of the components 

 vanishes. Thus, in order that the resultant force may vanish, it is 

 necessary that the resolved parts in two different directions should 

 vanish. If this condition is satisfied, there can be no resultant 

 except a couple, and since the moment of a couple is the same, no 

 matter about what point the moment is taken, it appears that there 

 can be no couple if the moment about any one point is zero. Thus, 

 as a necessary and sufficient condition of equilibrium for a system 

 of forces in a plane, we have found the following : 



A system of forces in a plane will be in equilibrium if the sum 

 of the resolved parts in two directions each vanishes, and if the sum 

 of the moments about any point also vanishes. 



