148 Compressible and Non-Homogeneous Masses [CH. vn 



When either V T or o> 2 are allowed to vary, we obtain a linear series of 

 configurations by picking out the appropriate equipotential surface from 

 each set. When V T and <*> 2 both vanish, the equipotentials are spheres and 

 the boundary is therefore spherical ; as we pass along the linear series the 

 boundary will depart more and more from the spherical shape. 



One property of Roche's model may be noticed at once. There can be no 

 points of bifurcation or turning points on any linear series. For when V T 

 and a) are given, the value of 11 is uniquely determined by equation (400) 

 and hence the boundary is uniquely determined. But the condition for a 

 point of bifurcation or a turning point is that there shall be two adjacent 

 configurations of equilibrium, and hence (by 141) two different boundaries, 

 possible for the same value of V T and o>. 



It follows that all possible configurations for a Roche's model lie on one 

 linear series, and this may in every case be supposed to originate in the 

 spherical configuration for which V T and o> both vanish. As we proceed 

 along this series, the different boundaries are equipotentials which differ 

 more and more from spheres, until finally it may happen that the equi- 

 potential which forms the boundary coincides with one which marks a 

 transition from closed to open equipotentials. On moving one step further 

 along the linear series we shall find that there is no closed equipotential 

 capable of containing the whole mass. There is therefore no equilibrium 

 configuration consistent with values of w 2 and V T beyond a certain limit, and 

 as soon as this limit is exceeded, a cataclysm of some kind must occur. 



151. The transition from an open equipotential to a closed one must 

 necessarily be through one which intersects itself, and therefore through an 

 equipotential on which a point of equilibrium occurs. Such a point is deter- 

 mined by the equations 



aft = an = an = 



dx dy dz 



Since H is necessarily constant over the surface of every equipotential, 

 including the boundary, this condition may be put in the alternative form 



or, again, from equations (391) (393), 



2 = 0. 



dn 



We are now back to the point of view of 147 ; the series terminates 

 as soon as dp/dn vanishes at any point of the boundary. But we have 

 now seen that this will occur at a point at which the equipotential which 



