162-165] 



Roche's Model 



163 



than that of the corresponding branch of this equipotential, dynamical 

 motion must have occurred before the components approach to within a 

 distance R of one another, and this motion will consist of matter streaming 

 out of the conical end of the critical equipotential. 



164. Other features of interest in the double-star problem are revealed 

 by a study of the composite model in which the nuclei are homogeneous 

 masses of finite size. The different forms which the two nuclei can assume 

 are precisely those which appear in the double-star problem of Darwin 

 already discussed in Chapter III ( 60 65). The boundaries of these 

 masses are equipotential surfaces, and are surrounded by other equipotentials, 

 any closed one of which may form the boundary of a possible atmosphere. 



For instance in fig. 84, the thick curves form the boundaries determined 



Fig. 34. 



by Darwin for the closest stable approach of equal masses (cf. fig. 13, p. 64). 

 The thin curves surrounding the nuclei are external equipotentials, and the 

 atmospheres of the stars may be bounded by any one of these. 



Darwin's figures were at the closest distance which was consistent with 

 stability for homogeneous masses, but it is at once apparent that the 

 boundaries of the atmospheres may be closer than this. They may be in 

 actual contact without stability being violated, or the two atmospheres may 

 be merged into a single atmosphere which will now be bounded by a single 

 closed equipotential surrounding both stars. Thus our investigation suggests 

 that heterogeneity will in general lessen the distance of closest approach 

 found by Darwin for the incompressible mass. 



165. The models just considered may be regarded as marking the limit 

 of non-homogeneity in one direction, the limit in the other direction being 

 provided by the perfectly homogeneous model studied in Chapters III to VI. 



In both the tidal and double-star problems, the motion of the non- 

 homogeneous models has been found to be very similar in its broad outlines 

 to that already discovered for the perfectly homogeneous model. In each 

 problem we found in both models a single series of configurations of 



112 



