194 Non-Spherical Masses—Dynamical Principles [ch. vii 
Since the discriminant remains invariant through all linear transforma 
tions, we have 
o- 
o 
p 
w„, 
W 12 , .. 
. w ln 
0, 6 2 , 0, ... 
- X 
W 21> 
Was, . 
• W 2n 
or b l} b 2 ,... b n = \A (175‘3). 
The condition that the configuration ©j, @ 2 , ... under discussion shall 
be one of stable equilibrium is that 3TT shall be positive for all values of 
80 1} 80 2 ,... 80 n , or again that expression (175-2) shall be positive for all values 
of <f> u fa, ... (f> n . This condition is that b u b 2 , ... b n shall all be positive. If any 
one of these quantities becomes negative, 8 W can become negative for a small 
displacement, so that the configuration has become unstable. 
The coefficients b 1} 6 2 ,... b n are called by Poincar6 “coefficients of stability.” 
A change from stability to instability occurs when any one of these coefficients 
vanishes, and the values of /x for which this occurs are, from equation (175‘3) 
given by 
A = 0 (175-4). 
Combining this with the result obtained in the last section, it appears 
that a change of stability occurs at every point of bifurcation, and at every 
point on a linear series at which /x passes through a maximum or a minimum 
value. This agrees precisely with the result obtained by other means in 
§§171 and 172. The stability or instability of the branch series at a point of 
bifurcation is most readily determined by the method already adopted in § 173; 
with the conventions there used, it appears that the branch series will be 
stable if it turns upwards from the point of bifurcation, and unstable if it 
turns downwards. 
Rotating Systems. 
176. We have so far discussed the stability of statical systems only. The 
stability of motion of a dynamical system is a much more complicated question, 
but assumes a comparatively simple form when the motion consists mainly of 
a uniform rotation. We proceed to discuss the stability of such a system. 
Let the system be referred to axes rotating in space with any velocity co 
about the axis of z in the direction from Ox to Oy. Let x, y, z be the co 
ordinates of any point referred to these axes, and let x, y, z denote their rates 
of increase. The components of velocity in space are then given by 
u — x — yco, v=y + xco, w = z (176*1), 
so that the kinetic energy T is given by 
T 2m (it 2 + j) 2 + it/ 2 ) 
= 42m (x z + y* + i 2 )-j- a>Zm(xy — yx) + |o> 2 2m (x *-\- y 2 ) ...(176"2). 
The total moment of momentum M about the e-axis is given by 
M = 2m (xv — yu ) 
= 2m (xy — yx) + co2m ( oc 2 + y 2 ) 
(176-3).