24 Mr Basset, On the Stability of [Nov. 28, 



is, however, no plain and well-defined statement that the condition 

 in question is intended to apply to a viscous and not to a friction- 

 less liquid ; and from a careful perusal of the whole passage, I fail 

 to discover any expressions, except one or two of a very vague kind, 

 which indicate that the authors are contemplating the case of a 

 viscous liquid. It is important to recollect that the conditions of 

 stability of a viscous liquid depend upon totally different principles 

 from those of a frictionless liquid, and I propose in the present 

 paper to carefully examine this question. 



Stability of Viscous Figures of Equilibrium. 



2. When a mass of liquid is rotating as a rigid body about a 

 fixed axis under the influence of its own attraction, it possesses 

 potential energy which may be determined as follows. In the 

 first place let the mass of liquid be supposed to be spherical, and 

 let 1^0 t>e the work which must be done in order to remove every 

 element of the liquid to infinity against the attraction of its parts ; 

 in the next place let W be the corresponding work which must be 

 done when the liquid has its actual form ; then if JJ be the poten- 

 tial energy 



U=Wo-W (1). 



From this result we see that the potential energy in the sphe- 

 rical form is zero, as ought to be the case, for in this form the 

 liquid is incapable of doing any work. 



The total energy E is equal to 



E=^T<o'+ U (2), 



where / is the moment of inertia, and co the angular velocity of the 

 liquid about the fixed axis of rotation ; also since 



/ft) = h, 



where h is the constant angular momentum, the value of ^ may be 

 written 



E = yi'/I+U (3). 



The first term of E of course represents kinetic energy, but it 

 may also be regarded as potential energy ; for the conditions of 

 relative equilibrium will be unchanged, if the rotation is annulled 

 and replaced by a repulsive force ft)V or AV//^ acting upon every 

 element. If the liquid be supposed to be distributed in the form 

 of an infinitely thin annulus of indefinitely large radius, whose 

 centre coincides with the axis of rotation and whose plane is 

 perpendicular to it, the term ^1^/1 will be indefinitely small; 

 whence the value of this term in the actual state may be regarded 

 as the work which must be done in order to bring the liquid from 

 the annular into the actual form. 



