I55-] STABILITY. 93 



tend to bring the rod back to its position of equilibrium. In 

 this case the original position of the rod is said to be a position 

 of stable equilibrium. The effect of the earth's magnetism on 

 the needle of a compass offers a familiar example. 



If, however, in the original position the forces tend to diminish 

 the distance A 1 A 2 (Fig. 43), the couple arising after displace- 

 ment tends to increase the displacement and thus to remove the 

 rod still farther from equilibrium. The original position in this 

 case is said to be one of unstable equilibrium. The weight rest- 

 ing on a vertical post and the reaction of the support on which 

 the rod stands may be taken as an illustration. 



Finally, a third case would arise if the forces F lt F 2 , being 

 still equal and opposite, were applied at one and the same point 

 of the rod. The forces would then remain in equilibrium after 

 any displacement of the rod ; such equilibrium is called neutral 

 or astatic. 



154. These different cases of equilibrium can be distinguished 

 by the algebraic sign of the product A^A^ F 1 = A 2 A l F 2 , which 

 is negative for stable equilibrium, since A^A^ and F 1 have 

 opposite sense (Fig. 42), positive for unstable equilibrium 

 (Fig. 43), and indeterminate (since A^^o) for neutral equi- 

 librium. 



It is to be noticed that these considerations will hold whether 

 the rotation of angle </> take place in the positive or negative 

 sense. But they hold only within certain limits for the angle of 

 rotation. Thus, in the example illustrated by Figs. 42 and 43, 

 when <f> reaches the value TT, the nature of the equilibrium is- 

 changed. 



155. Strictly speaking, investigations of stability are not 

 purely statical, but require a kinetic examination of the subse- 

 quent motion. However, the principles of statics are sufficient 

 to determine the nature of the equilibrium for infinitesimal dis- 

 placements, i.e. when only the initial motion of the body is- 

 considered. 



