202 Non-Spherical Masses—Dynamical Principles [ch. yii 
a constant angular velocity co, while a small particle, such as a shot, is free 
to roll about inside. When the shot is in a position at an angle 6 from the 
lowest point of the bowl, the potential energy W is equal to — mga cos 6 . 
This has only one minimum, at 6 = 0, so that if the bowl is smooth, the 
particle will fall to the lowest point of the bowl and remain there; the rota 
tion of the bowl is entirely irrelevant, and the equilibrium in this position is 
“ordinarily” stable. 
To examine the question of secular instability, we have to consider the 
variations not of W but of W — £<w 2 / (formula 184rl). We have 
W — ^g) 2 / = — mga cos 6 — £ra&> 2 a 2 sin 2 6 (185T). 
Possible configurations of equilibrium occur when 
^(W-WD= o, 
and this is when 6 = 0, or when 
cos 6 = Ar (185-2). 
The root 6 = 0, of course, represents the position of equilibrium at the lowest 
point of the bowl, while the second root represents equilibrium in which 
the particle rotates with the bowl, at such a height up the face of the 
bowl that the tangential component of centrifugal force mo> 2 a sin 6 cos 6 is 
precisely equal to the tangential component of gravity mg sin 6 . The second 
configuration only exists when &> 2 is greater than gfa. If o> 2 has a value less 
than g/a, equation (185‘2) cannot be satisfied, since cos 6 is necessarily less 
than unity. 
To examine the secular stability of the position of equilibrium at the 
lowest point of the bowl, we expand W— $co 2 I in the neighbourhood of this 
position in powers of displacements from the position, and obtain 
W — | &)*/ = — mga + \rnft 1 (ga — <w 2 a 2 ). 
Comparing this with the general formula (184-3), we see that the coefficient 
of stability corresponding to the single variable co-ordinate 6 has the value 
m (ga — o) 2 a 2 ). This is positive so long as co 2 is less than g/a, and for such 
values of <w 2 the equilibrium at the lowest point of the bowl is secularly 
stable. As soon as to 2 exceeds this critical value, the coefficient of stability 
m (ga — &> 2 a 2 ) becomes negative, and equilibrium at the lowest point becomes 
secularly unstable. The particle now slips away from this configuration, the 
value of 6 oscillating about a mean value which recedes further and further 
from the original equilibrium value 6 = 0 . 
The initial path of the particle is readily calculated. Let us take hori 
zontal rectangular co-ordinates Ox, Oy passing through the lowest point of 
the bowl, and rotating with the bowl. Let us assume the frictional force