1888. ] viscous rotating cylinder. 253 
This may be regarded as an equation in h or in a whose roots 
are the values of either of these quantities corresponding to the 
various free waves on the cylinder. 
2 
If we write 22/00 = Geter renee sees (32), 
n+1 
and multiply throughout by z 2, it reduces to an equation in z, 
viz. 
2 4 — 2 ae 
(< =F 2 = :) ar nl, =—4n (nm = e| GRIN Z ai Och eat) 2) 
+4n(n—1J2.¢ CTP? Jf /z=0......(33). 
oats Tn 2 z is always divisible by z2*””, and we are justified in re- 
moving this factor from the equation in z on the grounds that 
the value z= 0 is in general inadmissible, as we shall now show. 
7. For z=0 gives h=0, a=0. Also when h is indefinitely 
diminished J, (ha) becomes ultimately = h"a"/{2"T' (n+ 1)}. We 
thus obtain from (28) 
B=—2°T (n+ 1)A/h’, 
and therefore ar = 0. 
Hence there is no motion of the fluid—this is as we should 
expect. If however we suppose that the surface is deformed 
so that the normal displacement is ce’ we find by (24) that the 
normal surface traction necessary to maintain this displacement 
and acting outwards is fe” where 
i 
=cak,, 
and therefore the displacement will not continue without the 
application of force unless k, =0. But in this case 0 is a root of 
the divided equation (33). Thus the factor 2”*” is irrelevant. 
S. The equation (33) has an infinite number of roots, in 
general complex, and corresponding to each of these values of z 
we find a different function yy given by (8), (28) which satisfies all 
the conditions of the problem but which is complex. From each 
of these we may however form an expression ~ corresponding to a 
real wave motion of the liquid. Assuming z and therefore a and 
h complex let us write a = a,+ ca,, and express 
2n (n —1) J, (hr)/[{h’a* — 2n (n — 1)} J, (ha) — 2haJ,,,, (ha)] 
in the form R,++&,, 
18—2 
