173, 174] Adiabatic Model 175 



Then we may assume* 



^ 



T 



J 



Inserting these values into equation (465) and equating Coefficients, we 

 obtain 



e (L_ 



' - n * j - - j 



.(468), 



and four similar equations ; there are also three equations such as 



_ . 



2-rr (p - a) abc 



In these equations the six coefficients c n , c^, ... C 23 are linear functions of 

 the six coefficients L, M, N, I, m, n only, but the coefficients, d 1} d 2 , d s are 

 linear functions of L, M, N, I, m, n, p, q and r. 



It follows that the six equations (468) form a set of linear equations for 

 the determination of L, M, N, I, m, n. The solution of these equations, if 

 written down directly in analytical form, would be too complicated to convey 

 any definite meaning to the mind. Fortunately there is an approximate 

 solution of a very simple type, namely "f* 



(approximation A) ...... (470). 



To understand the meaning of this approximation, we may notice that it 

 satisfies equations (468) if c u , c w ,...c aa are neglected. Thus the approxi- 

 mation is arrived at by neglecting terms of degree 4 in AP;(1), and is 

 therefore equivalent to treating the boundary q = 1 as ellipsoidal when 

 calculating its gravitational potential. 



174. To obtain some idea of the amount of error involved in this approxi- 

 mate solution, I have worked out exactly the true solution in two special 

 cases. 



It will be remembered that the configurations for an incompressible mass 

 consist of spheroids, ellipsoids and pear-shaped figures. The corresponding 

 configurations for the compressible mass are derived from the foregoing by 

 distortion and consist of pseudo-spheroids, pseudo-ellipsoids and pear-shaped 



* The numerical multiplier 4 is introduced in order to facilitate comparison with the cor- 

 responding analysis in Chapter V. 



t Bakerian Lecture for 1917 (not yet published). 



