SMALL VIBRATIONS OF A HOLLOW VORTEX. 
185 
The series connected with X», Y* to complete the solution for a given M«, N„ are 
found from 
0= — 2a 3 UX'(3&-f 4 cos v-\-6k cos 2v)YL n cos nv 
-\-k(\k-\- cos v+f k cos 2t))(e+2a 3 UM s cos nv) 
+ (2k) (1 -\-k cos v)%X. n cos nv 
with a corresponding equation for Y n in which e=0. 
We need only consider for the first approximation the principal terms, which give 
Xv /(C-c)=^ ) ^+^(M, cos »«+N. sin nv) f? 
Since the circulation is to vanish, 
e+2a 3 MJJ:=0 
_2a 2 U 1 f 
x_ y'2*v(c-c)t 
—M — 
+ (M„ cos nv-\- N„ sin nv) 
R'J 
11. We are now in a position to determine the first term in the expression, giving 
the form of the hollow, viz., that part which will destroy the term in cos 2v in the 
value of the surface velocity. U denoting this velocity we have 
re_(M 4 J i i 
cdS 2 |_ \ bw buJ \ bv bv 
Now at the mean section bxp/bv= 0, and is therefore at least of the first order of 
small quantities near the circle u=u 0 . Hence 
where 
TT ( 0 — 
a 2 S ]_ bu bu\ 
TT , (C 0 -c) 8 W 
1 a 2 S 0 \ bu d 
a? TJi 0 0 
——= a+/3 cos v-\-y cos 2v 
2 Yo 
and a, /3, y are the values given in (20) when h-\- £ is substituted for k in the functions 
in u. 
Now £ is of the form M cos 2v, hence 
2« 2 U n 
M 
X= 
y/C2h)f{G-c) 
Rq I fh o 
-H7+W C0S 2V 
Xt o ix 2 
] B 
MDCCCLXXXIV. 
