256 Mr G. H. Bryan, On the waves on a [June 4, 
This shows that if the liquid be rotating there cannot be 
a system of gradually diminishing waves whose position relative 
to the rotating mass remains fixed. 
(iii) Nor can it have a pair of conjugate complex roots, except 
when w= 0. 
Since the roots of (36) are conjugate to those of (33) such a 
pair, if it existed, would have to be roots of both these equations, 
and therefore also of the two equations (39), (40). 
Hence we cannot have two waves travelling with equal relative 
velocities in opposite directions, or combining to form relatively 
stationary oscillatory motions of the liquid. 
(iv) In the cases when n=0 or n=1, if we remove extraneous 
factors the equation in z reduces to 
ae Jz We ep Ie Jz ='0 eae (41) 
where n=0 or 1 respectively. Now whatever be the value of n, 
the roots of this equation are all real and positive. For if we put 
z equal, in turn, to zero, and the successive roots of the equation 
(39), we find the left-hand side is alternately positive and nega- 
tive, hence the equation has one root between each of these values 
of z. We may similarly show that its roots are also separated by 
those of (40). 
When n= 0 the motion is symmetrical about the axis of the 
cylinder, and the fluid particles rotate round this axis with small 
angular velocities relative to the moving axes, which are functions 
of the distance from the centre. 
When n=1, we get unsymmetrical types of slow motion of the 
liquid within the cylinder, in which the form of the surface 
remains unaltered. 
In either case the motions gradually die away, and are un- 
affected by the rotation. 
(v) When #=0 the equation becomes 
(22 + nk, at/v® — 4m (n —1) 2} (2-"? ST, Je — 2a MPT, sf 2) 
+4n(n—1)'2.2°@ PPT, 2 =0...(42), 
in which all the coefficients are real. Putting 2 equal in turn to 
the various roots of (41) we find by the results just obtained in (iv) 
that the sign of the left hand is alternately positive and negative, 
hence the equation has at least one real root between each of 
these values of z. It can therefore at most have two conjugate 
complex roots, and it may have all its roots real, We shall see 
