AND THE ACTION OP TWO VORTICES IN A PERFECT FLUID. 
521 
Substituting we find 
log 2K 1 -\-(n 2 —l)(n-\-l) log n-\-l — (n — l) log n— 1) 
OjL LITCb 
m 
= say 
d[3ii TtlCl n / 0 1 n 1 r» 
W=W {w ’log 2^-log 2^) 
m 
sa 7 
Thus 
or the time of vibration 
Now if n be not very large 
d 3 /3„ , ( m \ 2 r 0 
W+(wjAA-=o 
=2 72 
f—n 2 log 2 ac x = — w 2 log 
a 
and the time of vibration 
^=(72 3 —1) log 2 * 7 =— (% 2 — 1) log - 
= 27t j^~n\/n 2 —1 log- 
/2a 3 & e 
If w be large 
f=n 2 log 2^+w 3 log n= —n 2 log 
Cb 
g=n 2 log 2 ^* 7 = —7i 3 log — 
a 
ne 
and the time of vibration 
_ /toe 3 0 , a 
=2 7 u n h s 
or if l be the wave length == 27 rct/n 
a* ne 
= 2iT 
j^TT^COe^ , l 
i ~ P log live 
which agrees approximately with Sir William Thomson’s result. 
3x2 
