92 



Pear-shaped Configurations of Equilibrium [OH. v 



negligible, will be represented by all points inside a certain rectangle ABCD 

 surrounding the point in fig. 1 6. They do not fall into linear series, as it 

 was assumed by Poincare and Darwin that they would. 



Jacobian 



P 



Fig. 16. 



That the two linear series will lose their identity and give place to a 

 two-dimensional area seems to be predicted by Poincare's analysis of which 

 an account has already been given in 22, 23 of Chap. II. For the 

 condition that a point of bifurcation shall occur at 0, namely A = 0, is also 

 the condition that the direction of the linear series shall be indeterminate 

 at 0, or, what is the same thing, that the two linear series shall become 

 merged into an area as they approach the point of bifurcation. 



Thus it now appears that an expansion as far as # is not adequate to 

 reveal the direction in which the second linear series turns on starting out 

 from the point of bifurcation 0. The difficulty is introduced by the artificial 

 method of expansion in powers of the parameter e ; the linear series are in 

 reality completely determinate, but an expansion as far as e 2 only does not 

 suffice to determine them. A precisely similar complication occurs in con- 

 sidering the direction in which lines of force start out from a point of equi- 

 librium in an electrostatic field*. 



94. Sir G. Darwin seems to have carried out his investigation under 

 the impression that there would be a unique configuration of equilibrium 

 when the calculations were carried as far as e 2 , and this led him to introduce 

 a spurious condition of equilibrium, the effect of which was to limit him to 

 one of the doubly infinite series we have discovered f. In point of fact, 



* Phil. Trans. 215 A, p. 74. 



t For details, see Phil. Trans. 215 A, p. 76. 



