CHAPTER V 



PEAR-SHAPED CONFIGURATIONS OF EQUILIBRIUM 



78. In Chapter III we discovered the existence of a number of series 

 of ellipsoidal configurations of equilibrium. We were able to examine the 

 stability of these configurations subject to the restriction that they were con- 

 strained to remain ellipsoidal. When it was possible for them to be dis- 

 torted from the ellipsoidal shape, it was not found feasible to examine their 

 general stability because we had no means of writing down the gravitational 

 potential of a distorted ellipsoid. 



The investigation of Chapter IV has now provided us with a formula for 

 the potential of a distorted ellipsoid, and we can proceed to search for con- 

 figurations of equilibrium which are of the shape of distorted ellipsoids. In 

 this way we discover the points of bifurcation on the ellipsoidal series already 

 discussed, and so obtain a complete knowledge of the stability of these 

 series. 



Let us take the equation of the general distorted ellipsoid to be 



. + 8i + $- 1 + P - (201). 



As far as first powers of e, the internal potential of the solid whose boundary 

 is given by equation (201) is 



Vi = -*paJbcr^d\ (202), 



Jo A 



where 



(203), 



and the potential at the boundary, V b , is given by the same formula. 



GENERAL THEORY. 



79. Let us apply this to the general double-star problem discussed in 

 50. So long as we are concerned only with the search for configurations of 

 equilibrium, this problem, as we have seen ( 52), includes the rotational and 

 tidal problems as special cases, although the problems become separate when 

 questions of stability are discussed. 



The condition that the surface (201) shall be a figure of equilibrium for 

 the primary mass in the double-star problem is (cf. 51) that 



iyS. ||2 '2 \ 



f) = -^abce(- + ^ + ^-l + eP a ] 



(204) 



