64, 65] 



The Double-Star Problem 



63 



On the other hand, if the big body is regarded as a point or rigid sphere, 

 tidal friction cannot operate, and the problem now becomes identical with 

 Roche's problem already discussed. The value of R in limiting stability 

 when p = is no longer oo , but 2*455 r. Thus tidal friction in the primary 

 can increase the value of R from this value to infinity. 



Darwin describes a system as " partially stable " when it is stable except 

 for the tidal friction arising from the tides in the primary. And he remarks 

 that, inasmuch as tidal friction is a slowly acting cause of instability, partial 

 stability of this kind is from the point of view of cosmical evolution of even 

 greater interest than the full secular stability of the system. 



Again, a slightly different problem occurs when the big body is supposed 

 to be an ellipsoid petrified in its configuration of equilibrium, so that the 

 masses are both in equilibrium but tidal friction cannot act on the primary. 



Darwin believes .that the limit of partial stability of a series of configura- 

 tions, such as that represented in fig. 8, can be found by discovering the 

 value at which 



W = (Mi 



. ' M + M'- 



is a minimum, this value M' representing all that part of the moment of 

 momentum which is liable to variation when tides cannot be raised in M'. 

 A slight modification of the argument of 64 will shew that the configura- 

 tion of closest approach cannot be even "partially stable." It. accordingly 

 appears that the configuration of limiting " partial stability " must lie at a 

 point intermediate between and R in fig. 8. Darwin calculates some 

 configurations of " partial stability," and gives the following table of values 

 for R, the closest approach consistent with partial stability, and for the axes 

 of the primary and secondary when in this critical position, a being the mean 

 radius of the combined mass : 



It appears that changes in the ratio of the masses produce surprisingly 

 little change in the critical value of R/a. Whatever the ratio of the masses 



