248 Mr G. H. Bryan, On the waves on a [June 4, 
(7) On the waves on a viscous rotating cylinder, an illustration 
of the influence of viscosity on the stability of rotating quid. By 
G. H. Bryan, B.A., St Peter's College. 
1. In his important memoir on the stability of rotating liquid*, 
M. Poincaré has proved that under certain circumstances Mac- 
laurin’s spheroid, if formed of a perfect liquid, may be stable even 
though the total kinetic and potential energy for given angular 
momentum is not a minimum. If however there be any viscosity 
in the liquid such figures of relative equilibrium must be un- 
stable. 
Since then the conditions of stability of rotating liquid are 
affected by the presence or absence of viscosity, I thought it might 
be of interest to give a hydrodynamical investigation of the waves 
on viscous rotating liquid, in some simple case which admits of 
mathematical solution, with the view of showing more clearly 
what is the effect of viscosity on the stability of the relative 
equilibrium. 
Now this can be done if the motion is two dimensional, the 
surface of the liquid being compelled to remain cylindrical by 
constraints which do not otherwise affect the motion. Without 
some such constraints the cylinder would obviously be highly un- 
stable. In the following investigation the cylinder is supposed to 
be oscillating about steady rotation in the circular form, this being 
apparently the only case in which the solution leads to intelligible 
results. In the simple sub-case when the liquid is not rotating 
the problem becomes exactly analogous, in two dimensions, to 
Prof. H. Lamb’s investigation for the oscillations about the spherical 
form T. 
2. We suppose the cylinder when undisturbed to be of radius 
a and to rotate about its axis with angular velocity w. The liquid 
is supposed self-attracting, of density p, the kinematical coefficient 
of viscosity being represented by v and the surface tension by 7. 
Let the motion be referred to rectangular axes of «, y rotating 
about the axis of the cylinder with angular velocity #. When 
the liquid is steadily rotating the coordinates (#, y) of any fluid 
particle referred to these axes will remain constant. Hence when 
the liquid is slightly disturbed the rates of change of these co- 
ordinates will be small. Let them be denoted by u,v Then the 
total component velocities of the particle parallel to the instan- 
taneous positions of the axes are u— @y and v+ eae. 
* «Sur l’équilibre d’une masse fluide animée d’une mouvement de rotation.” 
Acta mathematica, Vol. vu. p. 259. 
+ “On the Oscillations of a Viscous Spheroid.” Proc. London Math. Soc. 
Vol. x11. p, 51. 
