1902.] Figure of Equilibrium of a Botating Mass of Liquid. 18o 



The apparently rapid convergence of the series seemed to render- 

 such a reversal of the result almost incredible. In order, however, to 

 feel yet more sure, I made a rough estimate of the contribution of the 

 eighth zonal harmonic, and found that it would only amount to xTs^h 

 part of that critical total which would just show the pear to be 

 unstable. 



Since the convergency of the series is obviously rapid, I regard it as 

 proved, but by something short of absolute algebraic proof, that the 

 pear is stable. 



The numbers obtained in the course of the work afford the means of 

 giving a second approximation to the form of the pear, and the result 

 is shown in figures, drawn with the largest value of e, which seemed 

 consistent with a fair degree of approximation. 



I originally called the figure " pear-shaped " because M. Poincare's 

 conjectural sketch in the ' Acta Mathematica ' was very like a pear. 

 In the first approximation, shown in my former paper, the resemblance 

 to a pear was not striking, and it needs some imagination to see the 

 pear-shape in the new figures ; but a distinctive name is so convenient 

 that we may as well continue to call it by that name. 



The effects of the new terms are almost entirely concentrated at the 

 two ends. They tend to augment the protuberance of the stalk end, 

 and to diminish the depression at the blunt end so much as nearly to 

 fill it up. Over the greater part of the figure the depressions and 

 protuberances are less conspicuous than they were. 



I think it is hardly too much to say that in a well-developed " pear " 

 the Jacobian ellipsoid has nearly regained its primitive figure, but that 

 it has shot forth a protuberance at one end. A consideration of the 

 figures and of a conjectural extension of them almost reminds one of 

 some such phenomenon as the protrusion of a filament of protoplasm 

 from a mass of living matter. Notwithstanding the warning of 

 M. Poincare as to the danger of applying these results to hetero- 

 geneous masses and thence to cosmogony, I cannot restrain myself 

 from joining him in seeing in this almost life-like process a counter- 

 part to at least one form of the birth of double stars, planets, and 

 satellites. 



VOL. LXXI. 



V 



