25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 201 



(1) Three terms of the first class; 



(2) One term of the first and two of the second class ; 



or (3) Three terms of the second class, one of which at least involves e" 

 as a factor. 



Such a term formed in the first of these ways would be divisible by 

 (y yj 3 and therefore by (y 2 y^) 3 , since it can only involve even powers of 

 y and y^ 



Such a term formed in the second of these ways would be divisible 

 by e 2 (y y^ and therefore by e 2 (y 2 y/) or by (y 2 y/) 2 . 



Also such a term formed in the third of these ways would be divisible 

 by e 4 or by (/-y, 5 ) 2 - 



Hence, by the same principle as before, the value of --- must be 



i\ r 



divisible by (y 2 y^) 2 . 



That is 

 is divisible by (y 2 y^) 2 ; or 



is divisible by (y 2 y^) 5 . 



Now divide by y' ji, and then put 7i 2 = y 2 ; 



N 

 therefore 2F ^ y- - 2 Gy" = 0, 



F M 



G = N' 



which is the last of the relations announced above. 



The results obtained in Cases III. and IV. may be rendered more 

 general in the following manner : 



Let P denote the constant term in the reciprocal of the Moon's radius 

 vector, considered as a function of e" and y 2 . 



Then, taking e 2 , e*, and y 2 to be related as in Case III., we have, by 

 the same reasoning as before, 



dP dP 



= , , 2 . (e^ e 2 ) + -V-, 2 v . y 2 + terms of higher dimensions in e* e 2 and y 2 . 

 a (e ) a (y) 



A. 26 



