185] 
Secular Instability 
203 
acting on the particle to be — k times the velocity of the particle relative to 
the bowl. Then, so long as x, y are small, the equations of motion of the 
particle are 
x — 2 cûÿ — xPx = — kx — 
ÿ + 2 cox — apy = — ky — 
a 
99 
a 
Multiplying corresponding sides by 1, i and adding, we obtain 
d? . . , x d 
of which the solution is 
x + xy — A x e K P -f A 2 e x P, 
(x + iy) = 0, 
where 
Thus 
x -f iy = e 
= — ia) + i pH — \k 
9> _ 
- iut 
X-j = — iw 
vi 
-© 
A x e 
To refer to axes fixed in space instead of rotating with the bowl, we need 
merely multiply x + iy by e*“*. This neutralises the factor outside the curled 
brackets. We are left with two terms in A x , A 2 , representing circular motions 
of period 2 tt (-) in opposite directions in space, their amplitudes being 
proportional to e 
-hk 
[-“G) 4 ] 
respectively. The 
and to e L w J 
latter amplitude rapidly diminishes to zero; the former does the same if 
® is l ess than unity, but if w is greater than unity it increases 
indefinitely until x and y are no longer small, when our analysis no longer 
applies. Thus the initial path of the particle is a spiral of ever-increasing 
radius. 
To examine the secular stability of the upper configuration of equilibrium 
given by cos 6 = g/a> 2 a, we take a new co-ordinate z = a cos 0 , so that z is the 
distance of the particle below the centre of the bowl. Equation (185-1) now 
becomes 
W — £ <u 2 / = — mgz — | moPa? + \ maPz 2 
1 o ( 9 
*maP i z —— 
2 V «0