198 Professor Baker, On the general theory of the 



On the general theory of the stability of rotating masses of liquid. 

 By Professor H. F. Baker. 



[Read 9 February 1920.] 



I venture to make some brief provisional remarks, to which I 

 have hoped now for some years to give a more detailed examina- 

 tion, relating to the question why Sir George Darwin on the one 

 hand, and Liapounoff and Mr Jeans on the other, arrive at different 

 conclusions in regard to the stability of the so-called pear-shaped 

 figure of equilibrium of a rotating liquid mass. These point to the 

 conclusion that this is a case in which the empirical treatment of 

 the convergence of an infinite series may lead to erroneous results 

 in a concrete practical matter. 



§ 1. For a mass of homogeneous incompressible liquid rotating l| 

 as if solid about an axis, with angular momentum ju,, moment of 

 inertia I, and potential energy of gravitation — W (where W is the 

 volume integral of the product dm. dm' of tw^o elements of mass 

 divided by their mutual distance), we consider the Hamiltonian 

 function 



H=-W + Ifji^/I. 

 Let H' be the corresponding function belonging to another such 

 mass, "sufficiently" near to the former, rotating about the same 

 axis with the same angular momentum (and, for brevity, of the 

 same mass, and the same centre of mass lying on the axis), but 

 with different I and different W. We conceive that the form of 

 this second mass can be specified, relatively to that of the former, 

 by a certain number of parameters. In the actual problem the tale 

 of these parameters must be unlimited; but the methods applicable 

 when this tale is finite cannot be extended to the actual case 

 without careful examination, and in what follows we think only' 

 of a limited tale. The difference H'— H is then a function of these 

 parameters. In the case in which a change of form of the rotating 

 mass involves a dissipation of energy, a necessary and sufficient 

 condition that the form of the mass first considered should be one 

 of stable rotary equilibrium, under its own gravitation, is that 

 H' — H should be positive for all small values of the parameters 

 We adopt this as the condition. In the problem now being con- 

 sidered the first form of the mass is itself regarded as arising from 

 another (of the same mass and centre of mass), with a diff'erent 

 angular momentum, /Xq, and different I and W, say Iq and Wq, of 

 which the relative rotary equilibrium and stability have already 

 been investigated. We have then a known form of equilibrium, to 



