64 STATICS OF SYSTEMS OF PARTICLES 



Now if the whole system of particles is in equilibrium, it follows 

 that each particle separately must be in equilibrium. It follows 

 from 33, that 



(a) the sum of the components in any direction, of all the forces 

 acting on any single particle, must vanish; 



and from the theorem just proved in 48, that 



(b) the sum of the moments about any line, of all the forces 

 acting on any single particle, must vanish. 



If, however, the sum of the components of the forces acting on 

 each particle vanishes, it follows by addition that the sum of the 

 components of all the forces acting on all the particles must vanish. 

 The sum of the components of the internal forces, however, van- 

 ishes by itself, for the internal forces consist of pairs of actions 

 and reactions, and the two components in any direction of such a 

 pair of forces are equal and opposite. 



Since the total sum vanishes, and the sum of the components 

 of internal forces vanishes, it follows that the sum of the com- 

 ponents of external forces vanishes. 



A similar proposition is true of the moments of the external 

 forces. The sum of the moments about any line L of all the internal 

 forces is nil, for the moments of an action and reaction are equal 

 and opposite. The sum of the moments of all the forces, internal 

 and external, is zero, for each sum of the moments of the forces 

 acting on each particle is zero separately. Thus the sum of the 

 moments of the external forces is zero. 



Thus we have proved the following theorems : 



When a system of particles is in equilibrium under the action 

 of any system of external forces, 



(a) the sum of the components of all these forces in any direction 

 is zero ; 



(b) the sum of the moments of all these forces about any line is zero. 

 Speaking loosely, we may say that these theorems express that 



there is no tendency to advance in any direction or to turn about 

 any line. 



