52 Ellipsoidal Configurations of Equilibrium [OH. in 



where k is the radius of gyration of the primary. Adding this to the similar 

 expression for the secondary, we obtain for the total moment of momentum 

 M of the system 



............... (100), 



1 _ 2 



or, replacing R by its value (M + JM v )*o> , 



M-(J4-JT^).'+-^^- a -7* ...... (101). 



(M + 1/0* 



56. When the primary M is infinitely massive compared with the 

 secondary M' t the total moment of momentum M has the value M = Mk-w, 

 and the variation of M is precisely that of a freely rotating mass ; it increases 

 steadily from M = at S to M = oo at / in fig. 7. 



For finite values of the ratio M/M' the value of given M by equation 

 (101) becomes infinite when o> = i.e. at the two ends of the linear series 

 of configurations similar to those shewn in fig. 7. 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 r which divided 

 stable from unstable configurations in fig. 7 must now be replaced by another 

 curved line passing through 8. 



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 & 2 and A/ 2 are constants 

 the limit of this range is easily found from equation (101) by making M = 0. 

 The limit of stability is found to be given by 



...(102) 



+ Y 



or, in terms of R (using equation (89)), 



.................. (103) 



and the range of stability is again seen to be infinitesimal i.e. limited to 

 very great values of R when M/M' is infinite. 



The result shews that there can only be secular stability of a large and 

 small mass rotating round one another when the smaller mass is at a very 

 great distance from the larger*. We are dealing, it must be noticed, with 

 secular stability only ; the question means nothing except when dissipative 

 forces are present. When there are no dissipative forces, as for instance 

 if both bodies are perfectly rigid, a circular orbit of no matter what radius is 

 thoroughly stable, the orbit r = a giving place when slightly disturbed to 



* Cf. Sir G. Darwin. Coll. Works, m. p; 442. 



