502 
MR. J. J. THOMSON ON THE VIBRATIONS OF A VORTEX RING, 
lienee 
*.= (“,+7 log «,) + ((«+/.)»+ O*.(13) 
except when and 
^ 0= wie^ -^(( a + p) 3 +C 3 ) f .(14) 
If we substitute these values for the jit’s the equation 
-Jma^t, = <l> 
gives 
lo S K i) f 8a3=<1> 
or, since k l is approximately e 2 /4« 2 , we get if we substitue this value for k } and ve 2 oj 
for m 
1 $ ,2 a ^ . . 
w -i“^ lo g7 =,I> .(15) 
The second term on the left-hand side of this equation being small compared with 
the first, we get as a rougher approximation 
..(1^) 
The equation 
& i + £( (n—l)% l+l —(n +1 } = a n 
gives on substitution 
.(17) 
Substituting for 0. in equation (11) we find 
.< 18 > 
This agrees to the degree of approximation we are working to with the value for 
the velocity of translation of a circular vortex found by Sir W. Thomson and given in 
Professor Tait’s translation of Helmholtz’s paper on “Vortex Motion” (Phil. Mag., 
June, 1867). 
The value of <£ given by equation (16) is also the same as that obtained by Sir W. 
Thomson. 
Substituting for the ft’s in equation (9) we find 
maa lt - 
K da' dpj[_ 
1 log (»ifM 
(^)W ((a+p)2+ ^. 
-^ 8 +2)logg^f ^/((a+ P WY 
=A 
