30 General Dynamical Principles [CH. n 



for a system at rest ; they will occur in pairs of the form X 2 = p icr, and 

 these will. lead to roots for X of the form 



\= qip, 



so that the complete time factor for an oscillation is found to be of the form 

 Ae^ cos (pt e) + Be~& cos (pt - 77). 



If q is different from zero for any vibration, the amplitude of this vibration 

 will continually increase owing to the presence of the factors e qt , and the 

 system will be unstable. Thus the condition for stability is that q shall be 

 zero for every vibration, and this in turn requires that all the roots in X 2 

 shall be real and negative a condition which is the same in form as that for 

 the stability of a non-rotating system. 



A transition from stability to instability occurs whenever one of the 

 roots in X 2 vanishes. Putting X = in equations (39) we find that these 

 equations reduce to 



6^ = 0, M 2 = 0, etc. 



and the condition for a change from stability to instability is seen to be that 

 one of the coefficients b l} 6 2 shall vanish. These coefficients are seen to 

 be precisely the Poiricare " coefficients of stability " calculated for a system of 

 potential energy W i&> 2 /. 



28. Multiply equations (32) by 0,, # 2 ... and add corresponding sides. 

 We obtain 



~(^+TT-!o> 2 /) = M + M+ ............ '. ..... (40). 



The same result is readily obtained from the equations of energy and 

 angular momentum. The equation of energy is 



2 + ...+a>G .................. (41) 



or, using equations (27) and (30), 



which is identical with equation (40). 



When F l = F 2 = . . . = 0, so that no forces act except the couple G necessary 

 to maintain the rotation constant, the equation has the integral 



T B + W-^1 = constant ........................ (42). 



For equilibrium, W \u?I must, as we have seen, be stationary. Con- 

 sider first what kind of equilibrium obtains when W \u?I is an absolute 

 minimum. When any small displacement of the system occurs, W Jo> 2 / is 

 necessarily increased, so that the constant value of T R + W - J&> 2 / is greater 



