SMALL VIBRATIONS OP A HOLLOW VORTEX. 
191 
regarded, then since here the variation from the circle depends on /c 3 , we may treat it 
as circular in the integration, provided we do not carry our approximation beyond k 2 . 
In this case 
\=1 — 4(L — 2)k 2 , r—2ak, R=a(l —2 Jc 2 ) 
and the energy is 
|A{L-2+(L-f)(L-PHi (L -i ) ^} 
=^M(L-2)+i(2L—1)(3L-11 )k*} 
To the lowest order this is 
= ^(L-2)(L-i) 
13. If the steady shape as just found receive a slight disturbance symmetrical 
about the straight axis, a series of waves will be propagated round the hollow. To 
prove this, and to find the time of oscillation for different modes, will be the aim of 
the remainder of this paper; and firstly I consider the case where the cross-section is 
crimped into a form given by cosn-y+N sin nv, where M.N are small com¬ 
pared with k, and functions of the time. Since they are functions of the time, the 
volume of the hollow will change, and consequently the stream function will be cyclic. 
The rate of change of volume is 
dn dn' a 2 r 2,r M cos nv + N sin nv 
^ dk^ V dv k 
(C -cf 
dv 
_^Mf 2 ’ cos nv j _ n 2 M d f 2ir cosnv , 
= ~TJ 0 (C-cf dV= ~ls du) 0 0 -c 
7ra 2 M d.B u 
kS du 
where B ;( is the coefficient of cos nv in the expansion of (C—c) 1 
Hence 
2e~ nu 
S 
and the rate of change of volume 
2tto 2 M/ . C 
kS 2 
h/i+ 
which is of the order 87rcrM(n+1)&" +1 , a quantity beyond that which we neglect. 
Hence we may employ the stream function. Let then y denote this function for the 
small motion given by £ The condition to find it is that for all values of the time t 
1 d% dv 
p dv dn' 
