218 ON THE DETERMINATION OF THE ATTRACTIONS 



In order to resolve the system (41), let us here consider the 

 general type of the equations therein contained, viz. 



_ d*^ } cos 6 8 _ r d^ r / \,. y+1 



dB^ r + (r l) sin 6 s . r ' d0 8 _ r + (sin G\. r T 



Now if we reflect on the nature of the results obtained in 

 a preceding part of this paper, it will not be difficult to see that 

 - r is of the form 



where p is a rational and entire function of ^ = cos# 8 _ r , and i a 

 whole number. 



By substituting this value in the general type and making 



X wl = - (*'+* -2) ..................... (43) 



we readily obtain 



To satisfy this equation, let us assume 



Then by substituting in the above and equating separately the 

 coefficients of the various powers of ^, we have in the first place 

 from the highest 



\^ = -e(e + r-l) ..................... (44), 



and afterwards generally 



e i It . e i 2t 1 . 

 <+' ~ " 2t + 2 x 2e + r - 2t - 3 *' 



But the equation (43) may evidently be made to coincide with 

 (44), by writing i (r} for , and i (r+1} for e, since then both will be 

 comprised in 



-"' !*'"' ) + '- -2) .................. (45). 



Hence we readily get for the general solution of the system 

 (41), 



