PHILOSOPHICAL TRANSACTIONS. 
I. Further Consideration of the Stability of the Pear-shaped Figure of 
a Rotating Mass of Liquid. 
By Sir G. H. Darwin, K.C.B., F.R.S., Plumian Professor of Astronomy, and 
Fellow of Trinity College, Cambridge. 
Received October 29,—Read December 12 , 1907. 
Introduction. 
In vol. 17, No. 3 (1905), of the ‘ Memoirs of the Imperial Academy of St. Petersburg,’ 
M. Liarounoff has published an abstract of his work on figures of equilibrium of 
rotating liquid under the title “ Sur un Probleme de Tchebychef.” In this paper he 
explains how he has obtained a rigorous solution for the figure and stability of the 
pear-shaped figure, and he pronounces it to be unstable. In my paper in the 
‘ Philosophical TransactionsI had arrived at an opposite conclusion. 
The stability or instability depends, in fact, on the sign of a certain function which 
M. Liapounoff calls A, and which I denote A 0 + X (B/) 2 /C/, where A 0 is equal to 
21 s [1 (u 2 )' + 2£. } ] —^cr.! + X \i, s]. 
M. Liapounoff tells us that, after having seen my conclusion he repeated all his 
computations and confirmed his former result. He attributes the disagreement 
between us to the fact that I have only computed portion of an infinite series, and 
only used approximate forms for the elliptic integrals in the several terms. He 
believes that the inclusion of the neglected residue of the infinite series would lead to 
an opposite conclusion. 
In my computation the function ^3 [1 (cr 2 ) 2 + 2 £ 4 ] —^cr 4 is decisively negative, and 
being numerically greater than X{(B/) 2 /C/ + [7, s]}, which is positive, the sum of the 
two is negative. The inclusion of the neglected residue undoubtedly tends to make 
this whole function positive, but after making the revision, explained in the present 
paper, it remains incredible, to me at least, that the neglected residue can amount to 
the total needed to invert the sign. 
It may be worth mentioning that in revising my work I notice that 
As [-§- (<u ) 2 + 2 £ t ] —Leu owes its negative sign to the term —-§- 0 - 4 . This term arises 
from the energy of the double layer, called |DD. It comes from the portion of the 
term f-7rp 2 je 3 (1 — \e) dcr, which gives rise to a term in e 4 with a negative sign. This 
* Series A, vol. 200, pp. 251-314. 
VOL. CCVIII.-A 427. 
B 
4.3.08 
