254 Mr G. H. Bryan, On the waves on a [June 4, 
R,, R, being real functions of 7 Then the value of yw in (8) can 
be written 
= = ” exp — (a, + 0,)t. {o"/a" + (R, +4R,)} 
Beal ] n 2 2 Z = it < 
=A,e" {(r"/a"+ BR, + RV expe \n8 —a,t+tan™ Eaton 
where A, is a constant = A/a’. 
If we had taken 
=e er (Ale BS (hn) eee (35), 
we should have obtained instead of (33) the corresponding equa- 
tion in 2 
2oa" 3 2 a =(( 2 7 
|: (2 — = :) -- nk, 4n(n —1) ‘ (20 de 2eh eS ee 
+ 4n (n—1)*2. Baa ta aie =10ta5. eee (36), 
which may be obtained from (83) by writing — wu for au. 
The complex roots of this equation are conjugate to those of 
(33), and this is also true of the values of a h’. Moreover taking 
a’ =a, — wa, we are led to the corresponding solution 
v = Ang exp = (a, 77 1a) t. (n/a se (R, ae h,)} 
os 9b (7 mm 2 AD a, alt ( iwi = rial 
=Aje {(r"/a"+ RB)’ + RY exp —t pe a,t + tan ZS wea 
hia ate gue ea eRe (37), 
and combining (34), (87) we get the real wave motion 
be —ayt n/n 2 3 sin ae 1 Le 
ya dee + RY+ RY |nd aft tan" pi a} 
win Seta ORNs SRE (38). 
This represents a system of waves travelling round the cylinder 
with angular velocity «,/n relatively to the rotating mass or 
o+4,/n relatively to axes fixed in space, and, if a, 1s positive, 
gradually dying away, the modulus of decay being I/a,. If a, is 
negative they increase indefinitely with the time till the motion is 
so large that the squares of the relative velocities of the fluid can 
be no longer neglected, consequently the circular form of the 
cylinder is unstable. 
The particles of fluid which in the steady motion form a con- 
centric circle of radius 7 will be disturbed in waves of which 
the phase is a function of 7, and therefore changes as we proceed 
from the free surface of the cylinder inwards. 
