222 The Configurations of Rotating Liquid Masses [oh. vm 
This problem formed the subject of a series of classical papers by Poincaré, 
Darwin and Liapounoff*. The general problem was first broached by Poincaré’s 
memoir in Vol. 7 of the Acta Mathematica (1885), to which reference has 
already been made. The criterion of stability was inaccurately stated there, 
and the necessary modification was announced by Schwarzschild "f* in 1896. 
The accuracy of Schwarzschild’s criterion of stability was admitted by Poincaré 
in a paper published in 1901 % ; in this paper Poincaré also 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, by 
a method which was substantially identical with that which Poincaré had 
developed the previous year; and the moment of momentum of the con 
figuration was calculated. This was found to increase on passing along the 
series, so that the pear-shaped figure was announced to be stable. 
Darwin’s investigation had not long been published when doubt was cast 
on the accuracy of his conclusions. In 1905 Liapounoff published a paper IT 
in which he stated that he could prove that the pear-shaped figure was 
unstable. Liapounoff’s method was very different from that of Darwin, and 
a large part of his investigation appeared in the Russian language ; owing 
perhaps to these circumstances, neither investigator was able to announce the 
exact spot in which the error of the other lay, and the problem remained an 
open one for some years. 
In 1915** I approached the problem from a new angle, using ordinary 
Cartesian co-ordinates, in the way already explained, in place of the ellipsoidal * * * § ** 
* Poincaré’s papers on this subject occupy 122 pages in the Acta Mathematica, and 41 pages in 
the Phil. Trans. Darwin’s papers occupy 247 pages in the Phil. Trans, or 237 pages in his Collected 
Works. Liapounoff’s papers occupy 750 pages in the Memoirs of the St Petersburg Academy. 
My own two papers referred to later occupy 86 pages in the Phil. Trans. Considerations of space 
make it impossible to give more than the barest outlines of this great mass of mathematical re 
search. A fuller account of my own investigation will be found in my Problems of Cosmogony 
and Stellar Dynamics, pp. 87—102. 
f K. Schwarzschild, Miinchener Inaug. Dissert. (1896). 
+ “Sur la Stabilité de l’Equilibre des Figures Pyriformes affectées par une Masse Fluide en 
rotation,” Phil. Trans. 197 A (1901), p. 333. 
§ “Ellipsoidal Harmonie Analysis,” Phil. Trans. 197 A (1901), p. 461; “On the Pear-shaped 
Figure of Equilibrium of a Rotating 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. lxxxii A (1909), p. 188, all combined in one paper in Collected Scientific Papers, 
hi. p. 317. 
11 “ Sur un Problème de Tchebychef,” Mémoires de l’Académie de St Pétersbourg, xvii. 3 (1905), 
and other papers published by the Academy. 
** Phil. Trans. 215 A (1914), p. 27.