82 



Pear-shaped Configurations of Equilibrium [OH. v 



be before instability can set in through this distortion. It follows that the 

 ellipsoid will first become unstable through a distortion in which the number 

 of furrows is the fewest possible, namely one, and this is the third zonal har- 

 monic distortion.-- 



83. The value of P at the first point of bifurcation on every ellipsoidal 

 series must accordingly be of the form 



and the corresponding value of P is 



*2 \ 



.(213). 



We obtain at once from equation (203), 

 x 



so that we may write 



a t ) 



.(215). 



Equation (208) can now be satisfied, and on equating coefficients we 

 obtain 



0,0=0 



.(216). 



If we introduce new functions of a, b, c defined by 



- r xdx = r ^ dx 



.then equations (216) are found to assume the form 

 ZCL 



(217), 



* 26 2 



2c 2 l " a 



~ 9^2 c ~ 9A"2 C i + 9^ ( 3 2 + C,) - 



9/-2 i A A 2 ' 9/}2 ; 

 ~(i j o i_A^n L,U j Q 



