216 The Configurations of Rotating Liquid Masses [oh. viii 
third zonal harmonic. The onset of instability is thus marked by a point of 
bifurcation, the second series through which is a series of the pear-shaped 
figures obtained by imposing a third zonal harmonic displacement upon an 
ellipsoid. On following the Jacobian series beyond this point of bifurcation, 
Poincar6 found that a whole succession of further points of bifurcation were 
encountered, representing instabilities which entered respectively through 
harmonics of orders 4, 5, 6, All this is in entire agreement with the 
general principles just discussed. 
Poincar6 discussed the problem of stability in terms of ellipsoidal harmonic 
analysis, and in this he was followed by Darwin* and Schwarzschildj\ 
I have found*, however, that both this and more difficult questions to follow 
are more easily treated by using ordinary Cartesian co-ordinates x, y, z; we 
shall accordingly use these in the present discussion of the problem. 
199. In discussing ellipsoidal configurations, it proved convenient to 
regard the standard ellipsoid 
which formed the boundary of the rotating mass, as the special member 
A = 0 of the family of surfaces 
but are derived from ellipsoids by distortion. We shall take the boundary of 
the distorted surface to be 
where e is a small parameter which measures the amplitude of the distortion, 
while P 0 determines its distribution, and we shall find it convenient to regard 
* Coll. Works, hi. Papers 10, 11, 12 and 13. 
t Neue Ann. d. Sternwarte, München, m. (1897), p. 275, and Inaug. Dissert. München. 1896, 
t Phil. Trans. Roy. Soc. 215 A (1915), p. 27. 
entered through a displacement which was everywhere proportional to the 
Pear-shaped Configurations of Equilibrium. 
(199-1), 
We have now to consider surfaces which are not themselves ellipsoidal 
(199-3), 
this surface as the particular surface A. = 0 of the family of surfaces 
(199-4). 
To deduce the value of P from P 0 , we write P 0 in the form 
and the value of P is taken to be