66 The Gravitational Potential of a Distorted Ellipsoid [en. iv 



so as to give the potential of a distorted ellipsoid as far as second order terms, 

 but on attempting to apply his method it is found that the second order terms 

 in themselves are inadequate to determine the stability or instability of the 

 branch series ; a knowledge of third order terms is demanded. 



As Poincare's method does not seem to admit of extension, or at least of 

 easy extension, to the calculation of third order terms, it is found necessary 

 to develop some other method of writing down the third order terms required. 

 Such a method is now given. In the present chapter we confine ourselves 

 entirely to this problem in potential theory ; the determination of the con- 

 figurations of equilibrium being reserved for Chapter V. 



GENERAL THEORY 



67. In our discussion of ellipsoidal configurations of equilibrium, the 

 ellipsoid was supposed to be the surface X = in the family of surfaces 



If we write 



...... 



then the potential at the point x, y, z of the solid ellipsoid of density p is, as 

 in equations (54) and (55), given by 



F = (\)/dX .......................... (132), 



<- [V(X)/<fX 



Jo 



where the lower limit X in equation (132) is the positive root of the equation 



/=o. 



Suppose the family of surfaces determined by equation (130) to be dis- 

 torted so that their equation becomes F^,F being a function of #, y, z and 

 X, and let the distortion be such that the surface X = oo remains at infinity. 



We require to find the potential of a homogeneous mass bounded by the 

 surface X = in the distorted family of surfaces. Let us examine under 

 what conditions it is possible for the external and internal potentials to be 

 given by* 



f 



(134), 



,..(135), 



I. 



* The forms of these equations are of course suggested by analogy with equations (132) and 

 (133). For a direct derivation of equations (134) and (135) see Phil. Trans. 215 A, p. 29. 



