88, 89] Stability 87 



STABILITY OF THE PEAR-SHAPED FIGURES 



89. It is accordingly clear that in the tidal and double-star problems 

 there are no stable configurations beyond the spheroidal and ellipsoidal 

 figures already specified; the pear-shaped configurations are in every case 

 unstable, the mass having become unstable before these configurations are 

 reached. In the rotational problem, on the other hand, the series out of 

 which the pear-shaped series bifurcates is itself stable up to the point of 

 bifurcation, so that the pear-shaped figures, as already explained in 21, 

 may be either stable or unstable. 



The criterion of stability for these pear-shaped figures has already been 

 given in 21 ; if on passing along the series from the point of bifurcation, 

 the angular momentum is found initially to increase, then the figures are 

 stable ; if on the other hand it is found initially to decrease, then the figures 

 are unstable. As far as first order terms, it is obvious that the angular mo- 

 mentum will be the same as at the point of bifurcation, so that to apply this 

 criterion, we must proceed as far as second order terms in our determination 

 of the series. 



This problem has formed the subject of a series of classical papers by 

 Poincare, Darwin and Liapounoff. The general problem was first opened by 

 Poincare's memoir in Vol. 7 of the Acta Mathematica (1885), to which 

 reference has already been made. The criterion of stability was not accurately 

 .stated here, and the necessary modification was announced by Schwarz- 

 schild* in 1896. The accuracy of Schwarzschild's criterion of stability was 

 admitted by Poincare in a paper published in 1901f; in this same paper 

 Poincare developed a method of carrying ellipsoidal harmonic potentials as 

 far as the second order terms, and reduced the criterion of stability to an 

 algebraic form, without however undertaking the necessary computations. 

 At this stage the problem was taken up by Darwin, who, after preparing the 

 ground by preliminary investigations!, published in 1902 a paper, "The 

 Stability of the Pear-Shaped Figure of Equilibrium of a Rotating Mass of 

 Fluid." In the paper the equation of the pear-shaped figure was found as 

 far as terms of the second order ; and its moment of momentum calculated. 

 This was found to increase on passing along the series, so that the pear- 

 shaped figure was announced to be stable. 



* K. Schwarzschild, Mnnchener Inaug. Dissert. (1896). 



t "Sur la Stabilite de 1'Equilibre des Figures Pyriformes affectees par une Masse Fluide en 

 rotation." Phil. Trans. 197 A (1901), p. 333. 



J "Ellipsoidal Harmonic Analysis." Phil. Trans. 197 A (1901), p. 461; "On the Pear- 

 shaped figure of Equilibrium of a Kotating Mass of Liquid." Phil. Trans. 198 A (1901), p. 301. 



Phil. Trans. 200 A (1902), p. 251; see 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. in, 

 p. 317. 



