214 The Configurations of Rotating Liquid Masses [ch. viii 
* Phil. Trans. 190 A (1887), p. 187. 
meet the Jacobian ellipsoids. At this point of bifurcation they lose their 
stability, and since the series of Jacobian ellipsoids turns upward at this 
point, it follows that stability passes to them. 
If the mass is constrained to remain ellipsoidal there is no further point 
of bifurcation on the Jacobian series, and, as the angular momentum con 
tinually increases along this series, it follows that all configurations on it are 
stable. But it will be found later (§ 203) that when the constraint to remain 
ellipsoidal is removed, the Jacobian series loses its stability at a certain stage 
by meeting a series of non-ellipsoidal (pear-shaped) configurations. This has 
been anticipated in fig. 25. 
We have of course been concerned only with secular stability or instability. 
The conditions of ordinary stability are entirely different; for instance, as 
has been shewn by G. H. Bryan*, the Maclaurin spheroid remains “ordinarily” 
stable until its eccentricity is e = 09529. But its secular stability has dis 
appeared long before this eccentricity is reached, so that, as has already been 
noticed to be generally true, it is secular stability alone which is of interest 
in cosmogony. 
197. Before we proceed further into abstract mathematical discussion, let us 
attempt to understand the physical significance of the results so far obtained. 
Our investigation of the configurations of equilibrium of a rotating in 
compressible mass has been based upon the equations of equilibrium of the 
separate elements of the mass, but the same results could have been obtained 
from the general equations 
hw-WD 
of § 179. These again may be put in either of the forms 
8 ( W -\tfl) = 0 (a) = cons.) (197*1), 
8 ( W + £M 2 /7) = 0 (M = cons.) (197*2), 
and either equation would have given precisely the same configurations of 
equilibrium as those already obtained. 
We have found, for instance, that an incompressible mass in slow rotation 
will assume a spheroidal shape. Thus W — §<w 2 / must have a stationary value 
in this configuration, and since the configuration has been found to be stable, 
the stationary value is a minimum. Thus any displacement which causes a 
decrease in W must necessarily cause a decrease of at least equal amount in 
It is easy to find displacements which decrease W, as, for instance, by 
displacing the surface until it becomes more spherical, but this will increase 
W — |o> 2 /, and the displaced mass will oscillate stably about its position of 
equilibrium. 
Any vibration of an incompressible mass may be regarded loosely as a 
system of surface waves, and the distance from one point of zero displacement