93-96] Stability 93 



Darwin's extra condition of equilibrium could only be satisfied by assigning 

 to f a special value, namely f = ()*015988e 2 , and this value gives a figure 

 whose angular momentum is greater than that of the undistorted ellipsoid. 

 Darwin accordingly announced the pear-shaped figure to be stable. 



But we shall now see that this special value for f makes it impossible to 

 carry the linear series on to third order terms at all. The condition that it 

 shall be possible to carry on the series to third order terms requires that f 

 shall have a special value, but this special value is not the one assumed by 

 Darwin ; it is a value which shews the pear-shaped figure to be unstable, as 

 we shall now see. 



95. We proceed to calculate the pear-shaped series as far as the third 

 order terms. 



An argument similar to that of 90 shews that the boundary (234) can 

 only be made a figure of equilibrium as far as third order terms by including 

 in it additional terms of degrees 5, 3 and 1. We accordingly assume for the 

 boundary of the distorted ellipsoid, 



............... (253), 



c 



where P an d Qo have the values already given in equations (232) and (236), 

 and 



so that 



R = - 2 8 



4 a 2 a 8 b s c 8 6 4 c 4 c 4 a 



2nf v + 2 (p? 2 + W + 



............ (254), 



~I 



...... (255). 



96. We can calculate the potential of this figure as far as e 3 by the 

 formulae given in the last chapter. Calculating the terms in e s in formula 

 (197) in terms of the values which have now been assigned to P, Q and R 

 we find as the value of <, 



[15jA 2 a 3 -f 2|ABa 2 /3 + 2|ACa 2 7 + |B 2 a/3 2 + ^BCa/fy + |C 2 a 7 2 ] 



If B 2 /3 3 +|BC/3 2 7 + f 



l|AB/3 3 + 

 |AB/3 7 2 4- 1|AC 7 3 ] 



