170, i7i] Adiabatic Model 171 



in which of course 



n = y. + 10,2 (a-a + y a) 



= p F;(l)-(p -er)#-|-> 2 (^ + .v 2 ) ............ (451). 



Equation (450) contains the solution for all compressible^ masses, and so 

 must include the solution for the incompressible problem in which p Q <r 

 vanishes. 



In this solution of the incompressible problem, the figure is known to be 

 ellipsoidal, so that P = 0, while F^(l) becomes the potential of a homo- 

 geneous ellipsoid of unit density, and so is given by 



Vi(I) = -7rabc(J A x* + J B f + J c z 2 -J) ............ (452). 



Thus, omitting all terms which disappear when the mass is incompressible, 

 equation (450) reduces to 



= - 7rp abc (J A x* + J B y 2 + J c z* - J) + &> 2 (x* + y 2 ) + C. . .(453). 



The term in c (p <r) on the left-hand has been retained because it is 

 obvious that the equation can only be satisfied by supposing c (p a) to 

 remain finite when (p a) vanishes. We know that in any case c must 

 become infinite when the mass is incompressible, for the value of dp /dp 

 then is infinite. 



In the general problem, let us put 



. 

 Trace 



this equation defining 6. Then equation (453) becomes 



+ f) ......... (455), 



and on equating coefficients of 2 , y z and 2 2 , we find 



7, 



ZTTpQdbc a 5 

 w 2 



27r/o a6c 



Jc 



.(456). 



These are the conditions that an ellipsoid of semi-axes a, b, c shall be a 

 figure of equilibrium for a mass of uniform density p rotating with angular 

 velocity o>. It is at once seen that they are identical with the three 

 equations (65) (67) which were found to determine the solution of this 

 problem in 36. 



