230 The Configurations of Rotating Liquid Masses [oh. viii 
if both masses shrink, the rates of shrinkage, and consequent rates of increase 
in density, will in all probability be quite different for the two masses. We 
can, however, construct an artificial problem in which the density, if supposed 
uniform to begin with, remains uniform throughout the shrinkage, or in which 
the two densities, if not supposed equal to begin with, change so as always to 
retain the same ratio. The physical conditions are now represented by an 
increase in the absolute densities, while the moment of momentum remains 
constant and, exactly as in § 196, these conditions may equally be represented 
by supposing both densities to remain constant while the moment of momen 
tum increases. 
209. In the more general problem in which the secondary is not regarded 
merely as a point, the moment of momentum of the primary about the centre 
of gravity of the system is 
where k is the radius of gyration of the primary. Adding the similar ex 
pression for the secondary, we obtain for the total moment of momentum M 
of the system 
210. When the primary M is infinitely massive compared with the 
secondary M\ the total moment of momentum M has the value M = il№&>, 
and the variation of M is precisely that of a freely rotating mass; it increases 
steadily from M = 0 to M = oo as we pass along the series SBJ in fig. 33. 
For finite values of the ratio M/M', the value of M given by equation 
(209T) becomes infinite when <o = 0, i.e. at the two ends of the linear series 
of configurations shewn in fig. 33. Thus on leaving S, M decreases until a 
minimum is reached, and all configurations beyond this minimum will be 
unstable. Thus the curved line BR" T" which divided stable from unstable 
configurations in fig. 33 must now be replaced by another curved line 
passing through S. 
It accordingly appears that when M/M' is large the linear series becomes 
unstable very near to S, the range of stability vanishing altogether when 
M/M' is infinite. If both masses are rigid, so that k 2 and k' 2 are constants, the 
limit of this range is easily found from equation (209T) by making SM = 0. 
The limit of stability is found to be given by 
or, replacing R by its value y^(M+ M')^(o 
.(209-2). 
(2101),