PHILOSOPHICAL TRANSACTIONS. 
I. On the Instability of the Pear-shaped Figure of Equilibrium of a 
Rotating Mass of Liquid. 
By J. H. Jeans, M.A., F.R.S. 
Received March 21,—Read April 6, 1916. 
1. The main result obtained in this paper is sufficiently indicated by its title. In a 
recent paper* I showed that the stability of the pear-shaped figure could only be decided 
on after the figure itself had been calculated as far as terms involving the cube of the 
parameter e, which is used to measure the deviation of the pear-shaped figure from 
the critical Jacobian ellipsoid. In the present paper these third-order terms are 
calculated, and the pear-shaped figure is definitely shown to be unstable. 
A statement of the results obtained, and a discussion of their bearing on the wider 
question of which this problem forms a part, will be found at the end of the paper 
(§§23-27). 
2. The discussion lias to begin with a determination of the potential of a distorted 
ellipsoid, carried as far as the third order of the small quantities involved. With a 
view to shortening very lengthy computations, it is convenient to arrange the algebraic 
solution in a form somewhat different from that previously given. The solution now 
given can readily be extended to any order of small quantities, and appears to lead to 
the most concise series of computations for terms of all degrees above the second. 
Potential of a Distorted Ellipsoid carried to the Terms of Third and 
Higher Orders. 
3. As before, the undisturbed ellipsoid (which will ultimately be supposed to be the 
critical Jacobian) is taken to be the surface X = 0 in the family of surfaces 
- ^ 2 , v 2 , ^ 
— a 2 + A b 2 +A c 2 + X 
( 1 ) 
while the disturbed ellipsoid whose potential we require (which will ultimately be 
* “On the Potential of Ellipsoidal Bodies, and the Figures of Equilibrium of Rotating Liquid Masses,” 
‘Phil. Trans.,’ A, vol. 215, p. 27. 
VOL. CCXVII. — A 549. 
B 
[Published July 11, 1916. 
