STABILITY AND INSTABILITY 



181 



Again, if we put w = in expression (&), we obtain as the potential 

 energy in the case in which X is infinitely great in comparison with w, 





and the graph of W in this case is shown in fig. 101. There are four posi- 



tions of equilibrium, TT 3 TT 



= 0, -, TT, , 



which are respectively unstable, stable, unstable, and stable. 



The general case in which X stands in a finite ratio to w is intermediate 

 between the two extreme cases which have been considered. The graph 

 for W in the general case can be obtained by compounding the two graphs 

 already drawn. To obtain the ordinate corresponding to any value of 0, 

 we multiply the corresponding ordinates 

 in the graphs already obtained by the 

 appropriate constants, and add. The two 

 ordinates give the two terms of expres- 

 sion (5) separately : their sum gives the 

 total value of W as required. 



From this geometrical construction it 

 is clear that = remains a configura- 



u 



u 



FIG. 101 



Between these two 



0=0 



tion of unstable equilibrium. The con- 

 figuration = TT is also a configuration of 

 equilibrium, but may be either stable or unstable, 

 configurations there may be one other configuration of equilibrium, as in 

 fig. 101 ; or there may be none, as in fig. 100. Since, by 149, stable and 

 unstable configurations occur alternately, it is clear that if the configuration 

 = TT is stable, there can be no other configuration of equilibrium between 

 this and = 0, while if = TT is unstable, there must be one configuration 

 of equilibrium between = TT and = 0, and this must be stable. 



The stability or instability of the configuration = TT accordingly deter- 

 mines the nature of the solution for a given value of X. This stability or 



instability is in turn determined by the sign of at = TT. To deter- 

 mine this, let us write TT = 6 near to = TT, and neglect terms smaller 

 than 2 . We have, to this approximation, 



r 2 = a 2 cos 2 + & 2 sin 2 



7)2^ #2 



so that by equation (&), 



W = wa cos0 -\ (r 



