164 Compressible and Non- Homogeneous Masses [CH. vn 



equilibrium of approximately spheroidal or ellipsoidal shape, these remaining 

 stable until the two masses concerned reached a certain critical distance 

 from one another, after which dynamical motion was found to occur. And 

 the general features of this dynamical motion were broadly the same for each 

 model in both of the two problems. 



In the rotational problem the situation is very different. So long as the 

 rotation is slow the figures of equilibrium for every model are necessarily 

 spheroidal in shape, but for more rapid rotations the shapes of the figures of 

 equilibrium have been found to vary greatly. In the incompressible model, 

 we found a sequence of figures, spheroidal, ellipsoidal, pear-shaped, ending 

 with fission into two detached masses. In Roche's model, on the other 

 hand, we found a pseudo-spheroidal series which ended abruptly by matter 

 being thrown off from the equator. 



The incompressible model and Roche's model, may be regarded as limits 

 of homogeneity and non-homogeneity. The composite model considered in 

 154 provided a continuous transition between these two extremes. In this 

 we had a nucleus of volume V N and an atmosphere of volume V A , and were 

 able to determine the motion for all values of the ratio V N /V A . The limiting 

 value V N /V A = oo gives of course the incompressible model, while the limiting 

 value V N /V A = gives Roche's model. 



These same two models may, however, be regarded as fixing the limits of 

 compressibility and non-compressibility, and when they are regarded in this 

 light the composite model does not provide a gradual transition from one to 

 the other. A convenient sequence of figures of varying compressibility is 

 provided by masses obeying the pressure-density law 



(423), 



where 7 varies from one mass to another. The value 7 = oo provides a 

 completely incompressible mass, while the value 7 = 1| provides, as we have 

 already seen ( 149), a model in which the mass is entirely concentrated at 

 the centre, as in Roche's model. 



Thus a general study of figures obeying the law (423) for values of 7 

 from H to oo will provide a continuous transition from Roche's model to the 

 incompressible model, through a series of figures of continually varying com- 

 pressibility. To such a study we now proceed, limiting ourselves, for reasons 

 already explained, to the rotational problem. 



