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.