55-58] The Double-Star Problem 53 



the slightly elliptical orbit r = a (1 e cos 6). And when frictional forces 

 are introduced, as for instance by making the masses fluid, or by supposing 

 the solid masses covered by shallow oceans, the instability is one of orbital 

 motion only and not one of the configurations of the masses. *_When the 

 secondary is supposed wholly fluid so that k' 2 is variable, the fluidity of the 

 mass modifies the stage at which instability sets in, but introduces no new 

 instability of its own. The mechanism by which this instability is set up is 

 that which has been studied by Darwin under the name of "Tidal Friction"*; 

 it produces a secular increase or decrease in the mean radius of the orbit. 



It is important to notice that the case of MjM f = oo , in which M' is of 

 infinitesimal mass, is not, from our present point of view identical with the 

 case of p = oc in the diagram shewn in fig. 7, in which M ' is supposed 

 to disappear altogether. The former problem is one having one more degree 

 of freedom than the latter, and this one degree of freedom happens to be 

 secularly unstable for all finite values of r. In the latter problem, in which 

 the system is supposed to reduce to a single rotating body, the angular 

 momentum increases steadily from to oo on passing along the path SBJ in 

 fig. 7, so that the configurations on this path are all stable so long as the 

 mass is constrained to remain ellipsoidal. 



57. A special problem arises when the rotation of the secondary is not 

 affected by forces exerted on it by the primary. The primary, which is the 

 body whose configurations and stability we are specially considering, may 

 now be a small satellite rotating round a massive planet, which our choice 

 of terms compels us to call the secondary. The term M'k' z <*> in equation 

 (101) may now be replaced by M'k'*w', where a/ is the angular velocity of 

 the secondary and neither this nor k' 2 is subject to variation. The angular 

 momentum is accordingly 



MM' 

 N\=Mk*a>+ --.o^ + cons ................ (104). 



A similar problem occurs when the secondary is treated as a point so 

 that A/ 2 = 0. This leads back to Roche's problem discussed in 51. The 

 moment of momentum is again given by equation (104) in which the final 

 constant now vanishes. Let us investigate the stability of systems in which 

 the moment of momentum is given by equation (104). 



58. A case of special interest arises when the primary M is infinitesimal. 

 The value of M now becomes infinitesimal also, but N\/M remains finite, 

 being given by 



~ 



* Sir G. Darwin. Coll. Works, Vol. n. 



