28 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



the series of Jacobian ellipsoids coalesces with the Poincard series of 

 figures. After this point the series of Jacobian ellipsoids must, in accordance \v-itl 

 POINCARE'S doctrine of exchange of stabilities at a point of bifurcation, lose 

 stability, but the question of how it loses its stability is in a state of doubt. 

 DARWIN believed he had proved the Poincare series to be initially stable,* whereas 

 Li APOUNOFF f has maintained that this series is initially unstable. The importance 

 of this question to theories of cosmogony is, of course, great, although perhaps 

 liable to be overrated. A caution of POINCARE'S } may be borne in mind : 

 " Quelle que soit 1'hypothese [stability or instability] que doive triompher un jour, 

 je tiens a mettre toute de suite en garde contre les consequences cosmogoniques 

 qu'on pourrait en tirer. Les masses de la nature ne sont pas homogenes, et si on 

 reconnaissait que les figures pyriformes sont instables, il pourrait ne"anmoins arriver 

 qu'une masse hetdrogene flit susceptible de prendre une forme d'equilibre analogue 

 aux figures pyriformes, et qui serait stable. Le contraire pourrait d'ailleurs arriver 

 egalement." 



The present investigation was started primarily in the hope of setting this 

 question of stability at rest. I realised that to make a new series of computations 

 on the subject could be of little value, for whatever the result, there would have 

 been two- opinions on the one side to one on the other. Moreover, DARWIN has 

 stated clearly that he does not think the divergence of opinion between 

 M. LiArouNOFF and himself is one to be settled by new computations : "I feel 

 a conviction that the source of our disagreement will be found in some matter of 

 principle." I had hoped that it might be found possible to discuss the problem 

 by a purely algebraical method, involving neither laborious computations nor 

 intricate physical arguments, and that if such a discussion did not give a con- 

 vincing and satisfying answer to the question in hand, at least it might reveal the 

 source of disagreement between DARWIN and LIAPOUNOFF. The result arrived at 

 is one which, as will readily be understood from its nature, is only put forward 

 with the utmost diffidence, but it is one from which I can find absolutely no escape. 

 It is that underlying the whole question there is a complication, unsuspected equally 

 by POINCARE, DARWIN, and (in so far as I can read his writings) LIAPOUNOFF, 

 which renders nugatory the work of all these investigators on the stability of the 

 pear-shaped figure. If my method is sound, it appears, as will be explained later, 

 that it is impossible to draw any inference as to the stability of the pear from 

 computations carried only as far as the second order of small quantities. The 



* " The Stability of the Pear-shaped Figure of Equilibrium of a Rotating Mass of Liquid," ' Phil. 

 Trans.,' 200 A (1902), p. 251; also papers in 'Phil. Trans.,' 208 A (1908), p. 1, and ' Proc. Roy. Soc., 

 82 A (1909), p. 188, all combined in one paper in 'Coll. Scientific Papers,' vol. 3, p. 317. 



t "Sur mi Probleme de Tchebychef," ' Memoires de 1'Academie de St. PcStersbourg,' xvii., 3 (1905). 



I Loc. eit., p. 335. 



' Coll. Scientific Papers,' 3, p. 392. 



