144 Compressible and Non- Homogeneous Masses [CH. vn 



calculations in individual problems. Thus in the rotational problem it is 

 not possible, from a consideration of general principles, to predict whether 

 the pseudo-ellipsoidal series will initially be stable or unstable. If in any 

 problem it is unstable, cataclysmic motion will begin as soon as the first 

 point of bifurcation on the pseudo-spheroidal series is reached. This motion 

 will consist at first of an ellipsoidal elongation of the pseudo-spheroid, the 

 circular cross-sections giving place to elliptical ones, and the points of bifur- 

 cation on the pseudo-ellipsoidal series will be replaced by " dynamical points 

 of bifurcation" in this motion. In such a case, if ever it occurs, it seems to 

 be quite possible that the rotating mass may divide up into a number of 

 detached masses (instead of into only two) very much as in the tidal 

 problem. 



It will, however, be remembered that the angular momentum of the 

 pseudo-ellipsoidal series is infinite at its far end, so that much the most 

 likely event is that it increases all along the length of this series ; in this 

 case the pseudo-ellipsoidal series would initially be stable. But no such 

 general consideration can be brought forward in the case of the pear-shaped 

 series which branches off at the first point of bifurcation, and nothing justifies 

 us in predicting whether this will in general be stable or unstable. Indeed 

 it appears to be at least possible that in some problems this series may be 

 initially stable, a possibility which has been mentioned by Poincare.* 



Masses of non-uniform Composition 



146. From 140 on, we have assumed the astronomical matter to be of 

 uniform composition throughout, the pressure being a function of the density 

 only. When this restriction is removed, the discussion of equilibrium con- 

 figurations is naturally more difficult. 



Suppose that we are dealing with a mass of different types of matter 

 a, 6, c, ..., these letters referring either to chemically distinct types of matter 

 or to mixtures of such types in varying proportions. 



Consider a special problem in which the values of V T and &> are given, 

 and in which it is also given that the shells of matter occur in an assigned 

 order a,b,c, ... from the boundary inwards. 



The external boundary will of course be one of the equipotentials H = cons. 

 The surfaces of transition between the different types of matter will be sur- 

 faces at which the density changes abruptly. Thus these surfaces will coincide 

 with surfaces of constant density and hence, by equations (394), they will 

 coincide with equipotentials H = cons. 



* " Sur la Stability de 1'Equilibre des Figures Pyriformes affect^es par une Masse Fluide en 

 notation," Phil. Trans. 198 A (1901), p. 835. 



