68 The Gravitational Potential of a Distorted Ellipsoid [CH. iv 



in which the left-hand member is now a function of x, y and z only. Suppose 

 F to consist of a series of integral algebraic terms in #, y, z, then the left-hand 

 member will consist of a series of such terms, and if this expression plus 4?rp 

 vanishes at infinity, it must also vanish at all points of space. 



Thus if F is such that equation (143) is satisfied at every point of space, 

 while equation (144) is satisfied at infinity, then equation (145) must also be 

 satisfied at every point of space. Subtracting corresponding sides of these 

 equations, we obtain as a third equation, which must be satisfied at every point 

 of space, 



= ............... (146). 



By comparison with equations (141) and (142), the last two equations are 

 seen to be equivalent to 



V 2 V i = - 



Hence it appears that if F is a series of algebraic powers satisfying equation 

 (143), and is such that equation (144) is satisfied at infinity, then the potentials 

 of the homogeneous solid bounded by the surface A,=0 will be given by equations 

 (134) and (135). 



Before leaving this result, we may notice that the limit of integration \ 

 is connected with x t y, z, the point at which the potential is evaluated, by the 

 relation 



F=0 ...... .......................... (147), 



which is true at every point of space. On differentiation we obtain 



= 



9X dx 



which is also true at every point of space. Thus equation (143) which must 

 be satisfied by F may be written in the alternative form 



f 



Jo 



^ (X) V*Fd\ - V (A.) -"-4ir/> (148). 



69. The various possible solutions of this equation, which are such that 

 the second term vanishes at infinity, determine the boundaries of solids whose 

 potentials can be written down in the form of equations (134) and (135). One 

 such solid is already known, namely the ellipsoid. For this F=f, so that 

 F=fmust be a solution of equation (148) and we must have 



............... (149). 



Of 



