84 



Pear-shaped Configurations of Equilibrium [CH. v 



In the special case of b = c the integral c 2 can be integrated in finite 

 terms, and equation (225) is found to assume the form 



5 - 



4 



.(226), 



where e is the eccentricity, given by e* = (a 2 c 2 )/a 2 . On numerical treatment, 

 it is found that there is only one root of this equation, namely 



e = '947741 (227). 



The corresponding values of the semi-axes are 



a=2'14!75r , b = c = '683307r (228), 



and the value of /u, is //. = '1091311. 



On comparing this with the discussion of the spheroidal series given in 

 48 51, we at once see that this first point of bifurcation is beyond the 

 point at which //, reached its maximum. It accordingly follows that all con- 

 figurations on the spheroidal series are stable up to the point at which //, 

 reaches its maximum (e = '882579; /* = '125504^), and all configurations 

 beyond this are unstable. 



Rotational Figures 



86. The determination of the first point of bifurcation on the Jacobian 

 series is a much more arduous task. The integrals cannot be evaluated in 





 Fig. 14. - 



