104 Pear-shaped Configurations of Equilibrium [CH. v 



These are the conditions that the surface (294) shall be a possible figure 

 of equilibrium under a rotation o>. 



The points of bifurcation and points at which o> 2 reaches a turning point 

 will be determined by the Hessian of this system of equations*, namely 



da, 



~- ri -<)' 



= (300). 



If it were possible to calculate the 6's in terms of the a's and solve equa- 

 tions (299) and (300) in the most general case, we should obtain a complete 

 knowledge of all the linear series and their points of bifurcation. As this is 

 not possible, we start from a known configuration and trace out configurations 

 by following the different series. 



106. The simplest configuration is the circular one, for which 



a v a 2 (t s = =0. 



With these values all the 6's vanish, and equations (299) are satisfied for 

 all values of o>. Thus there is a linear series of circular configurations, along 

 which o> increases from zero upwards, and this is obviously the two-dimen- 

 sional analogue of the series of Maclaurin's spheroids. 



To search for points of bifurcation on this series we examine configura- 

 tions in which a 1? a 2 , ... are all small. Neglecting squares of these small 

 quantities equation (293) becomes 



fr = a 2 4- aj (?? + a 2 ^- 1 ) + a 2 (rf + a 4 ?r 2 ) + 

 so that, by comparison with equation (288), 



and, from (291), 



Vi = Trp [a, (f + 77) + a 2 (f 2 + T; 2 ) + J a 3 ( f + rf) + . . . - ft? } + cons. 



Comparing with equation (297), the value of b n is seen to be a n /n. Thus 

 the determinant in equation (300) reduces to its leading diagonal, and the 

 points of bifurcation on the circular series are given by 



As a typical solution we have 



27T P ~ l n 



Cf. 22, p. 24. 



