224 The Configurations of Rotating Liquid Masses [ch. viii 
momentum is taken as the variable parameter, increase of angular momentum 
being represented by an upward motion. 
The spherical configuration of no rotation, or zero angular momentum, is 
represented by the point S at the bottom of the diagram. As the angular 
momentum increases, the mass first moves along the series of Maclaurin 
spheroids SM until it comes to the point of bifurcation M at which the series 
intersects the series of Jacobian ellipsoids. At this point the Maclaurin spheroids 
lose their stability, and the motion proceeds along the series of Jacobian 
ellipsoids MJ'J until the point of bifurcation J is reached. The Jacobian, 
ellipsoids lose their stability here. The second series through J is, as we 
have seen, a series of pear-shaped figures such as JP in the diagram. The 
angular momentum of these figures decreases as we proceed along the series 
from J, so that the series is unstable and the curve JP turns downwards in 
the diagram after leaving J. Thus there is no stable configuration beyond J, 
and dynamical motion of some kind must occur as soon as shrinkage has 
proceeded so far that the angular momentum is greater than that represented 
by the point J. 
Let us attempt to examine what type of dynamical motion is to be expected 
when a mass of fluid having the configuration of a Jacobian ellipsoid reaches 
the point at which secular instability sets in. 
In fig. 32 let JJ' represent the series of stable Jacobian ellipsoids in the 
neighbourhood of the point of bifurcation J. For any configuration within 
the range JJ', the third harmonic (pear-shaped) vibration is stable both 
ordinarily and secularly. Thus if any small pear-shaped vibration is set up 
J L 
** 
S 
\ 
B B"\ D 
/ 
s 
/ 
\ 
/ 
\ 
/ 
\ 
/ 
\ 
\ 
\ 
t 
‘P 
A' 
A A" 
J' 
Fig. 32. 
when the mass is in a configuration such as A, the representative point will 
oscillate backwards and forwards through some small range such as A' A A" 
until the vibration is damped by viscosity. If the vibration is set up when the 
representative point is at some point B close to J, there may still be oscillation 
through a small range, but the motion can only be stable if this range is less 
than the range B'B" in Fig. 32. For the point B" represents a secularly