106 Pear-shaped Configurations of Equilibrium [CH. v 



108. To search for points of bifurcation on this series, we have to examine 

 configurations for which all the a's are small except a 2 . We readily find that 

 b n must be of the form 



- + terms linear in a n+2 , a 



- n+4 , 



Hence when m<n, db n /da m = 0, and when m = n> 



da n n (1 



Thus in equation (300) all terms below the leading diagonal vanish ; the 

 determinant reduces to the product of the terms in its leading diagonal, and 

 the equation for points of bifurcation reduces to the separate equations 



- 2aa 2 ) 



= 1- 



,4,5,...) (305). 



Simplified with the help of equations (303) and (304), this equation is 

 found to reduce to 



This equation is readily solved by graphical methods. In fig. 17 the 

 curve which is concave to the axis of a. is the parabola y = J (1 - a 2 ), while 



Fig. 17. 



the remaining curves are the graphs of 



U 



for the values n = 3, 4, . . . . As we pass along the elliptic series, starting from 

 the point of bifurcation with the circular series, we may suppose that we pass 

 along the axis OP in fig. 17 from a = to a = 1. 



