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Proceedings oj the Royal Society 
Hence (50), with i= 1, becomes, to a first approximation, 
• • • < 58 >- 
This and (52) used to find the two unknowns X and q 2 give 
X = jr, an( ^ S 2 = 3,m 2 a 2 , 
for a first approximation. How, with i = 1, (51) becomes 
X = ~(l - - 
and therefore re/n/is infinitely small. Hence (52) gives for a second 
approximation 
q2 = 3m 2 a 2 (l+^ . . . (59 ) ; 
and we have 
1 _ 2 1 ( 1 _&n\ 
q 2 X 3 m 2 a 2 \ 3o>/ 
. (60). 
Using now (57), (59), (60), and (56) in (50), we find to a second 
approximation 
= Im 2 a 2 flog— + i + -1159 N ) . . (61). 
2 \ ma 4 J 
whence 
- n 1 
O) 
Compare this result with (43) above. The fact that, as in (43), 
- n is positive in (61), shows that in this case also the direction in 
which the disturbance travels round the cylinder is retrograde (or 
opposite to that of the translation of fluid in the undisturbed vortex) ; 
and, as is to be expected, the values of —n are approximately 
equal in the two cases, when ma is small enough ; but it is smaller 
by a relatively small difference in (60) than in (43), as is also to be 
expected. 
The case of ma small and i> 1 has a particularly simple approxi- 
mate solution for the smallest g-root of the transcendental (50). With 
any value of i instead of unity we still have (58), as a first approxima- 
tion for q small. Eliminating g 2 /m 2 a 2 between this and (52) we still 
find X = i ; but instead of n = 0 by (51), we now have n — (i - l)o>. 
Thus is proved the solution for waves of deformation of sectional 
figure travelling round a cylindrical vortex, announced thirteen years 
ago without proof in my first article respecting Yortex Motion.* 
* “ Vortex Atoms,” Proe. Eoy. Soc. Edin. , Feb. 18, 1867. 
