OF ELLIPSOIDS OF VARIABLE DENSITIES. 221 



The differential equation which serves to determine If when 

 we introduce a instead of h as independent variable, may in the 

 present case be written under the form 



- 2) a' 2 - (i + 2w) (i + 2a> + n - 1 ) a 2 } H, 



and the particular integral here required is that which vanishes 

 when h is infinite. Moreover it is easy to prove, by expanding 

 in series, that this particular integral is 



provided we make the variable r to which A w refers vanish 

 after all the operations have been effected. 



But the constant k' may be determined by comparing the 

 coefficient of the highest power of a in the expansion of the last 

 formula with the like coefficient in that of the expression (46), 

 and thus we have 



/I/ ii-VO I *- / _ _ _ 



2. 4. 6. ..2ft) 

 Hence we readily get for the equivalent of (47), 



rr T>nn n4-2^+2ft)-l.w + 2/+2ft)+l...w+2i+4ft)-3 



= P0 i e 2 ...e,_ 1 x 2 . 4. 6.. .2ft, 



... x Ka i+2w (- l) w a i A w a 2r f^daa 1 ^'^ 1 (a 2 - a' 2 ) 



In certain cases the value of Fjust obtained will be found 

 more convenient than the foregoing one (47). Suppose for 

 instance we represent the value of V when h = 0, or a = a' by F . 

 Then we shall hence get 



2 . 4. 6.. .2ft) 



