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



71. Let us consider equation (155) first. On substituting for ^ (X), this 

 equation reduces to 



We have, however, 



A.0 



so that the equation becomes 



n A /^ x dv dv\) d\ /4iA 

 V 2 w+/V 2 ?; + 4(S T5 - + ^-H = - -r- ...... (157). 

 \ Adx 3X/J A \AA=O 



This must be satisfied for all values of X, and therefore, in particular, for 

 the value X = 0. Hence we must have 



v = when X = ........................ (158). 



It will be remembered that the boundary of the distorted ellipsoid is 

 supposed to be 



and equation (158) shews that < A=0 reduces to MA,=O- Thus different values 

 of M A=0 determine different boundaries, and if w x=0 can be made perfectly 

 general we can. solve the potential problem for the most general boundary 

 possible. 



Since v must vanish when X = 0, equation (157) becomes 



= ............ (160) 



in which X 7 is- momentarily used to denote the value of X given by the 

 equation f+ (f> = 0. 



The most general way of satisfying this equation is to make 



x dv 



where <r may be any function of x, y, z and X which vanishes when X = X' 

 and also when X = 0. 



Regarding this as an equation for v, let us try a solution 



v = w +fw' +f 2 w" + ... +f w + (162). 



