262 Mr G. H. Bryan, On the waves on a [June 4, 
When v/a’ is large, one root of this equation is large the other 
being small. Since by hypothesis a is not large, we find ap- 
proximately 
nk,, 
a= 2{(n?—1) via — at} Gogoddbecc00000000 (63). 
Here the corrugations on the surface die away very slowly owing 
to the great resistance offered by viscosity to changes of form of 
the liquid under its attraction and capillarity. 
18. If w=0 the value of a given by (63) is real. Moreover 
since the equation (42) does not involve any imaginary quantities 
in its expression, it is evident that if we proceed to higher 
approximations in the expansion of the various values of z or a 
in ascending powers of the small quantity a*/v no imaginaries can 
enter into them. 
Hence follows the result stated in (v) § 9, viz. that all the 
values of z are real provided that their expansions in descending 
powers of v/a’ are convergent. When this is so, the slow motions 
do not partake of the nature of waves. 
Waves on the viscous liquid surrounding a solid rotating cylinder. 
19. By introducing Bessel’s functions of the second kind the 
method of this paper may be extended to any problem relating to 
the two dimensional waves on viscous rotating liquids which, in the 
state of relative equilibrium, are bounded by concentric cylindrical 
surfaces. As an example, let us take the case when the liquid 
contains a perfectly rigid cylindrical nucleus of density o and 
radius b, the outer radius of the liquid surface being still a. 
We must now assume for the expression 
= {Ar + A'r™ + BU, (hr) + BY, (hr)je “e...... (64) 
involving four arbitrary constants. 
We find 
o ={(20—1a) Ar"+(20+02) Ar” + 2@BI, (hr) + 2@B’Y, (hr)}e “e” 
We suppose that the nucleus is rotating with angular velocity 
w, and that no slipping of the liquid takes place at the surface of 
the solid. Thus d0y/or and 0/700 both vanish when r= 6, giving 
the boundary conditions 
- Ab" + AG" 4 BJ, (hb) + BY, (hb) =0....-2.-- (66), 
nAb* —nA’b™ + hbBJ,/ (hb) + hbB’Y,/ (hb) = 0 ...(67). 
