518 
MR. J. J. THOMSON ON THE VIBRATION S ON A VORTEX RING, 
columnar vortex, which by formula 61 of his paper “ On the Vibrations of a Columnar 
Vortex,” is 
277 l 
/ /co27T 3 6 
log A. 
° 2ke 
■1159 
The results would agree approximately if Ij2e were indefinitely great compared 
with n, but it obliges us to neglect the factor log n, and thus the agreement instead 
of getting better as it ought gets worse as n increases. The way I determined 
A n is only suitable when uk / is small, as it is only allowable to expand cos ncj) 
as 1 
2 
when n<f) remains small within the limits of integration. 
I have therefore 
endeavoured to determine A n in a way which shall not be open to these objections. 
With the notation of the paper since 
A„=V' 
7T.I o 
cos n<f).cl(f> 
(a 3 + p 3 -+- £’ 3 —2 ap cos </>) 2 
we may easily prove that if a—p be small compared with a, that 
\ = (n— 1)A „_ 1 —(n — 3) A,, 
and 
a 
a 
/ d Ai—i 
dAJ 
\ da 
da t 
> 
+ 
1— J 
dA n 
1 — 
V da 
da 
— (?/ -J- 3 ) A a — (n -j-1) A n 
If we make these substitutions we find from equations (10) and (12) 
{%+ \{{n — 1 )8L» +1 — {n + 1 )&*_i) } 
c ^=±maa H { < gl l — 
Hence we have only to find an expression for 
7r((a + p) 3 + £ 3 f 
p 2ir cos n<f> 
j,(^ + ^ S in A) 
Now as is very small this integral will be very large, and as the large part 
arises when <f> is small or nearly 2n, we may write as the approximate value of 
2 G 50 cos ncf)d(f> 
V+fJ 
4 f 00 cos 2 n<f>.d<f) 
JoW+W 
