198 Non-Spherical Masses—Dynamical Principles [ch. yii 
putting F x = F 2 = ... = 0, and assuming each proportional to the same 
time-factor e M . The equations then reduce to 
(aj\ 2 + bj) 6 X + &)X/3 12 0 2 + (o\/3 is 0 s + ... = 0, 
®Xy3 gl $i + (uX./ 6«2^2 + • • • + (u- g X 2 + b s ) 0 8 -\- ... — 0 
Eliminating the 0’s, we find as the equation giving X, 
a 1 X 2 + 6 1 , <wX/3 12 , co\(3 13 , 
(oXfS^i , a 2 X -i- b 2 , ( 0 XS 23 , ... — 0 • 
(OXfin , G)\(3<)2 > «3 X ”f* &3 , . . . 
.(180-7). 
Since /3 rg = — this equation is unchanged when the sign of X is changed. 
The equation is therefore an equation in X 2 , just as when the system is at rest. 
But the roots in X 2 are not necessarily all real as they are for a system at rest; 
in general they will occur in pairs of the form X 2 = p ± ia, and these will lead 
to roots for X of the form 
X = ± q ± ip, 
so that the complete time-factor for an oscillation is found to be of the form 
Ae qt cos (pt — e) + Be~ qt cos (pt — tj). 
If q is different from zero for any vibration, the amplitude of this vibration 
must 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. 
Stability and Instability. 
181. Multiplying the general equations of motion (178*2) by 8 1 , 6 2 ,... and 
adding corresponding sides, we obtain 
§ t (T R + W- iu’I) = Fj 1 + Fj,+ (18M). 
When F 1 — F 2 = ... = 0, so that no forces act except the couple 0 necessary 
to maintain the rotation constant, the equation has the integral 
T R + W— ^&> 2 / = constant (181-2). 
For equilibrium we have seen that W — ^<u 2 / must be stationary. Consider 
first what kind of equilibrium obtains when W — %o>*I is an absolute minimum. 
When any small displacement of the system occurs, W — 2 J is necessarily 
increased, so that the constant value of T R + W — %(o 2 l is greater than its value 
when at rest in the equilibrium configuration by a small amount c. Through 
out the subsequent motion the value of T R can never increase beyond c, so 
that the motion is absolutely stable. This argument cannot however be 
reversed, and the system is not necessarily unstable if W— ^cd 2 / is not an 
absolute minimum.