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

 and this vanishes when X = X' through the factor 



/+.+*. ...(169). 



1 +v 1+ v 



It must also vanish when X = 0, and v given by equation (162) must also 

 vanish when X = 0. These conditions are most easily satisfied by making 



w = w =w"=... = w (n ) = ... = 0, when X = ......... (170). 



Thus equations (164) (166) will contain a complete solution of the 

 original equation if w, w', etc. are all chosen so as to vanish when X = 0, 

 while 6 is given by equation (168). 



72. We now turn to equation (156). It is convenient to transform to 

 new variables f , TJ, f, X connected with the old variables x, y, z, X by the 

 relations 



Differentiation with respect to the new variable X is given by 



9 9 a* a a * 



Expressed in terms of these new coordinates, equation (156) becomes 

 du 



In this we may put/+ </> = 0, or, from equation (169), 



/- U 



T+i+ 



and the equation reduces to 



Theoretically, this equation determines u/(l+v), so that a solution of 

 this equation combined with the solution for v obtained in 71 will yield a 

 complete solution of the problem. For our present purpose it is convenient 

 to examine solutions in powers of a parameter e. 



SOLUTION IN POWERS OF A PARAMETER 



73. Equation (174) may readily be solved in powers of a parameter e 

 on assuming a solution of the form 



u 



^ - = effi + e-g* + etg* + (17o). 



1 + v 



