550 REPORT— 1901. 



constants defined by the formula 



x(x + 1) . . . (\ + « - 1) = X"" + <"' r-' + . . . C, ^ 



Then the necessary and sufficient condition that .v be the first singular point on 

 L as we go from a to infinity is that the inequality 



O a " a 



holds independently of a, however small e may be, for an infinite number of values 

 of n ; while the inequality 



holds for all sufficiently large values of v where we take a first and then e 

 sufficiently small. 



3. Poinoare's Pear-sliaped Flqnre of EquUihrinm of Potatinq Liquid. 



By G. H. Darwin, F.R.S. 



Ellipsoidal harmonic analysis has usually been presented in such a form as to 

 make numerical calculation almost impossible, but the author believes that he has 

 succeeded in removing this defect in a paper for the ' Philosophical Transactions,' 

 now in the press. By aid of the methods of that paper the limit of stability 

 of Jacobi's ellipsoid becomes calculable. According to the principles established 

 by M. Poincar^, stability ceases when we arrive at a stage where a coefficient of 

 stability vanishes, and where there is interchange of stabilities between two 

 coalescent series of figures. The figure which coalesces with the Jacobian at 

 this point is the pear-shaped figure sketched by Poincar^. No attempt is made 

 in this paper to indicate the methods pursued, but results will merely be given.' 



If o) denotes the angular velocity of an ellipsoid of liquid, and p the density, it 



is well known that bifurcation of the Maclaurin ellipsoid occurs when-^— = 'ISTl, 



2t7P 



and when a number jx to which the moment of momentum is proportional is 

 •30375.2 



One of the equatorial axes then begins to elongate, and the other to shorten, as 

 the angular velocity diminishes and the moment of momentum increases. These 

 ellipsoidal figures with three unequal axes are the Jacobian ellipsoids. 



The problem to be solved is to find when a coefficient of stability in the 

 Jacobian series first vanishes, and to determine the nature of the figure which 

 coalesces with the Jacobian. 



If the phraseology of spherical harmonic analysis be adopted, it is found con- 

 venient to take as the principal axis of quasi-symmetry for the ellipsoidal 

 liarmonics the longest axis of the Jacobian ellipsoid. Then it appears that the 

 iirst to vanish of the coefficients of stability is that corresponding to the third 

 zonal harmonic. 



The following short table gives the leading facts concerning the Jacobian 

 ellipsoids as far as just beyond their instability. The last line in the table gives 

 the corresponding facts as to the critical Jacobian, which is a figure of bifurcation. 

 The axes of the ellipsoids a, b, c are given in such a form that their product (the is 



' A paper giving the details of the investigation was presented to the Royal 

 Society in October 1901. 



' See Proo. U.S., vol. xli. p. 319. 



