142 Compressible and Non-Homogeneous Masses [OH. vn 



Since V + fa 2 (a? + y 2 ) is constant over the boundary in a configuration 

 of equilibrium, the value of V T becomes 



For a problem in which V T , o> and the shape of the boundary are given, 

 the value of 



is known at every point inside the boundary. It follows that 



'dv i . 



must be given at all points inside the boundary except for a constant, and 

 this determines dV/dn at all points of the boundary, except for a constant. 

 But 



[fdV 



II to 



is given, being equal to 4vr times the total mass of the rotating body, so that 

 d V/dn is uniquely determined at every point of the boundary. 



It follows from equation (396) that under the conditions now contem- 

 plated, p and dp/dn are determined at every point of the boundary, and 

 from this and equation (398) it is easy to see that the solution for p is 

 unique.* 



It follows that configurations of equilibrium may be specified by their 

 boundaries alone, but a more important result also follows. When V T and o> 2 

 are given and the boundary is given, there will be an endless number of 

 possible vibrations in which the internal particles move, while those at the 

 boundary remain in position. The result just obtained shews that none of 

 these can ever be of zero frequency, so that no points of bifurcation can 

 occur, and the internal vibrations, if stable in the initial configuration of the 

 mass, must always remain stable. 



From the circumstance that configurations of equilibrium may be specified 

 by their boundaries alone, it will be clear that the various configurations 

 must fall into linear series much in the same way as in the incompressible 

 problem. The configuration for no rotation and no tidal forces will of 

 course be spherical. 



142. In the rotational problem there will obviously be a series, analogous 

 to the Maclaurin spheroids, in which the boundary is a figure of revolution. 



* It is difficult to construct a rigorous proof, for complications of a mathematical nature 

 arise. See Proc. Roy. Soc. 93 (1917), p. 416. 



