520 
so that 
MR. J. J. THOMSON ON THE VIBRATIONS ON A VORTEX RING, 
#=D 
J 0 (a a + ^)- r 
a 2 s 2 . a 4 s 4 
(■H593+ log ^Kl+^+^H-.. 
2 2 ^ 2 2 4 2 
f a 2 s 2 
a 4 s 4 
+ ( 92 ^l+^^2+ • • • 
Therefore 
f “ cos scb , , 
I «4= 
(a 2 + cf) 2 y 
a~ 
:D 
T1593+ log 
2 2 
V 
as. 
ta 2 s . <z% 3 
2 2 4 
+ 
a 2 s 2 . a 4 s 4 
2 3 2 2 4 2 ~^~ 
f a 2 s 2 
a 4 s 3 
+(^fs 1 + ^s, + 
Now since 
putting s=0 we find D= — 1. 
Therefore 
3 cos S(f>.d(f> 
r 
J o 
clef) 
(a 2 + <£+ a, 2 
r 
J o 
(a 2 + cf) 2 f a 
=7i 1-f 
ah 2 a 4 s 4 
' _ 2 2 ~"^2 2 4 2 ' 
+ (•11593+ log 
a 2 s 4 
as \2 ' 2 2 4 
a 2 s 4 
+fiSH-§*S,+. . 
Now approximately 
^ _ 1 C”cos 2 7uj>d<f> 
^ _ 2rfJ 0 (++</>+ 
detft dfdn 
hence A» is found, and if we substitute these values in the expressions for , 
cLt cLt 
shall be able to find the time of vibration in any particular case. 
If we suppose uk x is small, then approximately 
we 
L ~2 {^ +# { 2 ( 1<>g 2+'! 159S ) + 1 
or if we neglect 1.23 in comparison with 2 log 
2nrc l 
^= 2 b{^ + 2 nH °s +} 
Now 
'y=h mo A. ! ,'&i + 4 } (»— 1 )&„+!—(»+ 1 ) 
