178 Compressible and Non- Homogeneous Masses [OH. vn 



But to obtain all the compressible solutions adjacent to a given incom- 

 pressible solution, we must retain Ao> 2 . In particular, the retention of A&> 2 

 will be necessary in searching for points of bifurcation in the compressible 

 problem. A point of bifurcation in the compressible problem will be adjacent 

 to the corresponding point of bifurcation in the incompressible problem, but 

 will not in general have the same rotation. 



Thus in searching for the points of bifurcation in the incompressible 

 problem, we retain the so far undetermined quantity A&> 2 in our equations. 

 The three equations (475) (477) determine three relations between p, q t r 

 and Aw 2 , but to determine these four quantities fully a further equation is 

 needed, this equation of course expressing the condition for a point of 

 bifurcation. 



177. Let us confine our attention to the particular point of bifurcation 

 at which the pseudo-spheroidal figure gives place to a pseudo-ellipsoidal 

 figure. The corresponding point in the incompressible problem is the point 

 of bifurcation at which the Jacobian ellipsoids join the Maclaurin spheroids. 

 At this point equations (69) and (70) of 37 are both satisfied, as well as 

 the equations of equilibrium (65) (67). Combining equations (69) and 

 (70), we obtain as the equation determining the position of this point of 

 bifurcation, 



a*J AA = <?J c (481), 



or, using the equation of equilibrium (67), 



a*bcJ AA = 0abc (482). 



The actual values of a, b, c, 6 and &> 2 are those already given in 174. 



In the compressible problem, the equation of the boundary has been 

 taken to be 



where F< stands for the fourth-degree terms La?/ a* + . . . , which have already 

 been determined. This may be put in the form 



where 



etc ...................... (484). 



The condition determining a point of bifurcation in the incompressible 

 problem is readily seen to be 



a*b'cJ' AA = 6'a'b'c, 

 where 0', J' AA refer to an ellipsoid of semi-axes a, b', c'. 



