SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 175 



t&amp;gt; 2 c 3 d 4 e 5 , &c, and consequently for those of bcde, c., results entirely 

 analogous to that which we have found above for the value of 

 the first coefficient a^. 



211. If now we represent by P 2 , Q,, P z , S 2 , &c., the quantities 



1+1+1+1. 



I 2 3* 4* 5* 



1*. 3 2 I 2 . 4 2 I 2 . 5 2 3 2 . 



&c., 

 which are formed by combinations of the fractions 1 , 1 , 1 , 



2 , ^5 ... &c. to infinity, omitting ^ the second of these fractions 

 we have, to determine the value of b 2 , the equation 



, - &c. 



Representing in general by P n Q n R n S n ... the sums of the 

 products which can be made by combining all the fractions 



p &amp;gt; 2* &amp;gt; g2 &amp;gt; f , -^2 &quot;- to infinity, after omitting the fraction 1 



only; we have in general to determine the quantities a lt 6 2 , c 

 d 4 , e s ..., &c., the following equations: 



A-BP l +CQ l -DB l 



., , 



^- . O . -T . O ... 



A - P 2 + CQ 2 - DR + ES - &c. = 2i ,- &quot; 2 ? 



4 -^= 



l a .2 2 .3*.5.6.. ~ 

 &c. 



