450 Proceedings of the Royal Society 
Whether in Case II. or Sub-case II. we see that the disturbance 
consists of an undulation travelling round the cylinder with angular 
velocity 
or of two such undulations superimposed on one another, travelling 
round the cylinder with angular velocities greater than and (alge- 
braically) less than the angular velocity of the mass of the liquid at 
its free surfaces by equal differences. The propagation of the wave 
of greater velocity is in the same direction as that in which the 
liquid revolves ; the propagation of the other is in the contrary 
direction when N>i 2 (as it certainly is in some cases). 
If the free surface be started in motion with one or other of the two 
principal angular velocities (34), or linear velocities aw 
and the liquid be then left to itself, it will perform the simple har- 
monic undulatory movement represented by (6), (26), (23). But if 
the free surface be displaced to the corrugated form (30) and then 
left free either at rest or with any other distribution of normal 
velocity than either of those, the corrugation will, as it were, split 
into two sets of waves travelling with the two different velocities 
The case i = 0 is clearly exceptional, and can present no un- 
dulations travelling round the cylinder. It will be considered 
later. 
The case i — 1 is particularly important and 
evaluate H for it remark that 
I -fmr) = I ' 0 (mr) I 
and 1 1 (mr) = T 0 (mr) ) 
interesting. 
To 
(36). 
How the general solution of (24) is 
-( 
tv — ( E + D log 
m 2 r 2 7ft4 r 4 
1 + ~^ + ¥a* + &c - 
+ D (»V V „V S2 + &e . 
• (36), 
) J 
where E and D are constants Hence according to our notation 
