112 Pear-shaped Configurations of Equilibrium [CH. v 



entirely at our disposal, and we shall accordingly take A, = 0. We shall, as a 

 matter of numerical convenience, replace the parameter 6 by a new parameter 

 0'such that 10000 2 = <9' 2 . 



As far as terms in #', the pear-shaped figure determined by equation (333) 

 is now found to be 



0'2 1 10?) = 1. 



The corresponding pear-shaped figure in three-dimensions was 



and we see that the two figures agree as closely as possible if we take 0' = e. 

 Thus our new choice of parameter results in 6' having the same meaning as 

 e has in the three-dimensional problem. 



The second order solution now assumes the definite form 

 r z = 1 + |r 2 cos 2</> + 20' . 10 " ' (r 3 cos 3</> - - l r cos 0) 



+ 10- 3 0' 2 (^ cos 40 - ^r 2 cos 20 + 6 ^ 4 7B ) . . .(336), 

 while the value of &> 2 is given by 



~ = 0-3750 (1 + 0-0513<9 /2 ) . . .(337). 



Zirp 



We may notice that this rate of increase of o> 2 is closely analogous to that 

 in the three-dimensional problem 



= 014200 (1 + 0-052270 2 ). 



On calculating the moment of inertia of the curve defined by equation 

 (336), we find 



which compares with 



Mk* (1 - 0-09378e 2 ) 



in the three-dimensional problem. 



Calculating the moment of momentum in the cylindrical problem, we find 

 M&m = Mk<?a> (l - 0-1423(9 /2 ). 



This shews that Mk 2 a) diminishes as we proceed along the two-dimensional 

 pear-shaped series, and therefore that the series is initially unstable. 



114. The agreement between the two-dimensional and three-dimensional 

 problems has so far been so marked that it may be hoped that it will persist 

 into those regions in which the three-dimensional figure cannot be calculated. 

 On the assumption that this is the case, we may infer the advanced stages of 

 the three-dimensional problem from those of the two-dimensional problem. 



