25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 197 



Then all the arguments which occur in the values of - , r, u, and v 



likewise occur in those of - , r 1} u l} and v l} but the coefficients of the 



*'i 



corresponding terms will differ by a quantity which contains y 2 as a factor. 



Also there will be additional terms in the value of - , r 1} u 1} and v 1} 



r i 



with arguments of the form 



where k does not vanish, and these will also contain y 2 as a factor in 

 every term. 



Hence , r r l} u u^ , and v v, will contain y 2 as a factor in 



r 1 r 



every term. 



Also z = 0, and therefore (z z,) 2 = z, 2 , which will also contain y 2 as a 

 factor in every term. 



/I IV 



Hence ( will be of the order of y* at least, while 



\r, r] 



- ~ ~J (( u ~ i) s + ( v ~ rf + ( z ~ z ')' - ( r ~ r i) 2 } 



'/ 



will be of the order of y 4 at least. 



Therefore, by the same principle as before, the value of can 



contain no constant term of the order of y 2 . 



That is, (7=0 generally; and as this holds good for every value of e' 

 we must have 



<? = 0, <?, = <), <7 2 = 0, &c. 



CASE III. 



Next, let the values x, y, z belong to the solution in which y vanishes 

 and e is finite, while a;,, y lt z 1 belong to the general case in which e 1 and 

 y are both finite, the value of e being now changed to e 1 while nt + e, 

 and therefore also n, retains the same value as before. 



