88 Pear-shaped Configurations of Equilibrium [CH. v 



/ 



Darwin's investigation had not been long published when doubt was 



cast on the accuracy of his conclusions. A paper appeared in 1905 by 

 Liapounoff* in which he stated that he could prove that the pear-shaped 

 figure was unstable. Liapounoflfs method was very different from that of 

 Darwin, and a large part of his investigation appeared in the Russian 

 language; owing perhaps to these circumstances neither investigator was 

 able to announ.ce the exact spot in which the error of the other lay, and the 

 problem remained an open one. The method of treatment given in the 

 present chapter will, it is hoped, shew the source of the divergence of the 

 results obtained by these two investigators. 



90. As far as the first order of small quantities, the pear-shaped figure 

 has already ( 83) been found to be 



S + g + |- 1 + eP ' = (230), 



where 



P -f (*? + frf + 1? + *) (231), 



so that 



The potential of this figure can be found by the method already given in 

 70 of Chapter IV. As regards the internal and boundary potentials, the 

 terms in e will be of degrees 3 and 1, those in e 2 of degrees 4, 2, 0, those in 

 e* will be of degrees 5, 3, 1, and so on. It is at once clear that the general 

 equation of equilibrium 



...... (233) 



cannot be satisfied as far as e z , for terms in e* of degree 4 in x, y, z occur on 

 the left of this equation, and have no balancing terms on the right. To 

 satisfy the equation of equilibrium, it is found to be necessary to add terms 

 in e 2 of degrees 4, 2 and to the left of equation (230), and such terms then 

 appear also on the right of equation (233). 



91. Thus, to calculate the pear-shaped figure as far as second order 

 terms, we assume the boundary of the figure to be 



where 



Q = J [L? + Mif + N? + 2^ 2 2 + 2m? 2 J 2 + 2wf V + 2 (pf 2 + qjf + r? 8 ) + s] 



......... (235), 



* "Sur un Probteme de Tchebychef." Mtmoires de VAcademie de St P2tersbour(] , xvn, 3 

 (1905). 



