223 
203, 204] Pear-shaped Configurations 
harmonic analysis which had been used by Poincare, Darwin and Liapounoff. 
My results agreed with those previously given by Darwin up to a certain 
distance, and where they began to disagree I was able to discover* a quite 
simple error which invalidated Darwin’s discussion of the problem. On carrying 
the discussion of the figure as far as the third order of small quantitiesf, 
I confirmed the conclusion already reached by Liapounoff, that the pear-shaped 
series was initially unstable, and was further able to shew that the correction 
of Darwin’s error made the difference between stability and instability in his 
final result. When Darwin’s error had been corrected in this way, the three 
investigations of Darwin, Liapounoff and myself agreed in finding the pear- 
shaped figure to be unstable. In 1920 H. F. BakerJ expressed doubts 
(unsubstantiated by detailed calculation) as to the accuracy of my treatment 
of certain series, but withdrew his criticism five years later§, and expressed 
himself as satisfied that the expansion in series which I had used in my papers 
could be placed on a sure foundation. 
There seems, then, to be little room for doubt that the series of pear-shaped 
configurations is initially unstable. 
Sequence of Configurations of Rotating Liquid. 
204. The equilibrium configurations of a mass of homogeneous rotating 
liquid accordingly fall into linear series which may be arranged diagrammati- 
cally as shewn in fig. 25, and also on the left of fig. 31 (the right-hand half 
anticipates results which will be obtained later). In this diagram the angular 
* l.c. p. 76. 
+ Phil. Trans. 217 A (1916), p. 1, and Problems of Cosmogony and Stellar Dynamics, p. 87. 
% Proc. Cam.b. Phil. Soc. xx. (1920), p. 198. § Ibid. xxm. (1925), p. 1.