STABILITY AND INSTABILITY 183 



fig. 102, having a horizontal tangent and point of inflection at P. 

 On one side the potential energy is less than at P, on the other 

 side it is greater. 



Let Q, Q' be two adjacent configurations on these two sides 

 of P. If the system is placed at Q, it must move so that its poten- 

 tial energy decreases, and therefore moves away from P. If it is 

 placed at Q r , for the same reason it must move 

 at first towards P, but it will move beyond P 

 and will then continue to move away from P, 

 for it cannot come to rest until its potential 

 energy is again equal to that at Q', and this - " 

 cannot happen in the neighborhood of P. Thus 

 if the system starts from any configuration in the neighborhood 

 of P, it will ultimately be moving away from P. In other words, 

 the equilibrium is unstable. 



Thus if = at P, the equilibrium is, in general, unstable. 



. 

 An exception has to be made when - = 0; for then we have 



z 



This case may be treated as in 148, and we find that the 



/ &W\ 

 equilibrium is stable or -unstable according as / I is positive 



, . \ 0X1 /ft 



or negative. x 



152. Higher degrees of singularity may be treated in the same 

 way, and we easily obtain the following general rules : 



If the first differential coefficient which does not vanish is of odd 

 order, the equilibrium is unstable. 



If the first differential coefficient which does not vanish is of even 

 order, the equilibrium is stable or unstable according as this differ- 

 ential coefficient is positive or negative. 



It is possible for all the differential coefficients to vanish, in 

 which case the problem is best treated by other methods. 



