CHAPTER VII 



THE MOTION OF COMPRESSIBLE AND NON-HOMOGENEOUS 



MASSES 



GENERAL THEORY 



138. So far we have discussed only the behaviour of masses of perfectly 

 homogeneous and incompressible matter. In so doing we have followed 

 the classical line of development, based upon the researches of Maclaurin, 

 Jacobi, Poincar6 and Darwin. Astronomical matter must however be highly 

 compressible and far from homogeneous, so that the question of how far we 

 are justified in attributing to real astronomical matter the behaviour which 

 is found to occur in ideal incompressible masses is obviously one of great 

 importance. In the present chapter we shall develop a general theory of the 

 configurations of equilibrium of compressible masses, and shall in particular 

 attempt to examine in a general way the effect of compressibility in intro- 

 ducing departures from the motion predicted by the incompressible model 

 which we have so far had under consideration. 



139. If p is the pressure at any point x, y, z of a mass rotating with 

 angular velocity about the axis of z, the equations of equilibrium will be 



< 386 >> 



(387), 



in which V is the potential of the whole gravitational field of force, including 

 tidal forces if any are present. Thus we may write 



V=V M +V T ........................... (389), 



where V M is the gravitational potential of the rotating mass under con- 

 sideration, and V T is the potential of the tidal field. 



Writing 



H= F+ Jo> 2 <V + ?/ 2 ) ........................ (390), 



