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



Let us now assume for w, w', w" ... expansions of the form 



w = ew + e* w 2 + e?w 3 -f . . . , 



w ' = ewi + e*Wz + e 3 w s ' + . . . . 

 Then the value of 6 given by equation (168) becomes 



+' 2M" + 3/X'" + . . .) 



Equating coefficients of powers of e in equations (164) (166) we now 

 obtain 



STir-' + lr) =-^X (184), 



A dx d\J 



I & + S =-V^-4A^ [f 



and similar equations. 



74. Let us now introduce an operator D denned by 



On differentiation with respect to X, we find 



W _ I (P_ Id 2 Id 2 



so that 3I)/3X is simply V 2 transformed into f , rj, coordinates. 



Transformed into f, 77, ? coordinates, equations (184) etc. become (cf. 

 equation (172)) 



4^! =- l~ Ul (189), 



C7A< (7A, 



We have already found that u 1 = P ) a function of f , 77, f on ty> so that 

 equation (189) has the integral 



