165, 166] Adiabatic Model 165 



THE CONFIGURATIONS OF ROTATING COMPRESSIBLE MASSES 



166. It will be convenient to write the pressure-density 4aw (423) in 

 the form 



c 



'O' 

 7' 



We may notice that this law includes as special cases Laplace's law 

 [p = \ G (p 2 " 2 )] an d also the law obeyed by a gas in adiabatic or convective 

 equilibrium [p Q = 0]. 



We find at once that < (p), defined by equation (395), is given by 



so that the general equation of equilibrium (396) becomes 



-^ pv~ l = ft + C ........................ (424), 



in which, as before, 



Operating with V 2 we obtain at once as the differential equation which 

 must be satisfied by p, 



' V 2 /3 y- 1 = -47r/3 + 2o> 2 ..................... (425). 



Taking the point of maximum density p as origin, it will be possible to 

 expand p in the form 



P = po- pz- p s - p,- ........................... (426), 



where p 2 , p 3 , p ... are functions of x, y, z, of degrees 2, 3, 4 ... respectively. 

 The value of p 2 is 



the differential coefficients being evaluated at the origin. Since the origin 

 is supposed to be the point of maximum density, p 2 must be negative for all 

 values of x, y and z. Changing axes, it must be possible to put p 2 in the 

 form 



where a denotes the density at the boundary of the mass. 



If we further put 



p 9 + p* +... = - e (p - a) P OJ 



