OF ELLIPSOIDS OF VARIABLE DENSITIES. 219 



H-D JM f t <>} {i^ - I'M -1} Jrtu (,) 



- j 



______ ._._ 4 _ 



) +7 5} ^ ' 



where /-t = cos # g _ r , and i (r) represents any positive integer what- 

 ever, provided i (r) is never greater than i (r+l} . 



Though we have thus the solution of every equation in the 

 system (41), yet that of the first maybe obtained under a simpler 

 form by writing therein for X^ its value i (2} * deduced from (45). 

 We shall then immediately perceive that it is satisfied by 



cos 

 In consequence of the formula (45), the equation (42) becomes 



n_. , 



1 2 ' /^ 2\ * ~7 ~~" \ 2 ' 



1 2 ' /^ 2\ * ~7 \ 2 ' i 



dp 2 p(l-p*) dp { p 2 1-p 



which is satisfied by making k = \ (i (8) +2a))(i (8) -}- 2co+n 1), 

 and 



- 2 X 2/ (8) + 2&) + s - 2 . 2i (8} + 2&) + ^ - 4 2W _ 4 __ 



CC ' 



where co represents any whole positive number. 



Having thus determined all the factors of $, it now only 

 remains to deduce the corresponding value of H. But H the 

 particular value satisfying the differential equation in H, will be 

 had from (/> by simply making therein 

 ,. _ 



since in the present case we have generally a r ' = a. 



Hence, it is clear that the proper values of 1? 2 , # 3 , &c. to 

 be here employed are all constant, and consequently the factor 



