SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 173 



same value of A l in terms of J 4 J5 4 (7 4 Z) 4 , and so on. The successive 

 values of A are 



A, = A\ 3 2 . 4 2 - B, (2 2 . 3 2 + 2 2 . 4 2 + 3 2 . 4 2 ) + &amp;lt;7 4 (2 2 + 3 2 + 4 2 ) - D 4 , 



^^J^ 2 ^ 2 ^ 2 ^ 2 -^^ 22 - 82 - 4 ^ 22 - 32 - 5 ^ 22 - 42 - 5 ^ 32 - 42 - 52 ) 

 + C 6 (2 2 . 3 2 + 2 2 . 4 2 + 2 2 .5 2 + 3 2 .4 2 + 3 2 .5 2 + 4 2 .5 2 ) 

 - D b (2 2 + 3 2 + 4 2 + 5 2 ) + E 6 , &c., 



the law of which is readily noticed. The last of these values, 

 which is that which we wish to determine, contains the quantities 

 A, B, C, D, E, &c., with an infinite index, and these quantities 

 are known ; they are the same as those which enter into equa 

 tions (a). 



Dividing the ultimate value of A : by the infinite product 



2 2 .3 2 .4 2 .5 2 .6 2 ...&c., 

 we have 



&quot; D (.2*. 3&quot;. 4&quot; + 2&quot;. 3&quot;. 5 a + 3&quot;. 4&quot;. 5&quot; + &C 7 

 E .S .^.o 1 + ^~4\ff + &C ) + &C 



The numerical coefficients are the sums of the products which 

 could be formed by different combinations of the fractions 



1 i i i i Ac 



I 2 2&quot; 3&quot; 5 2 6*&quot; 



after having removed the first fraction p. If we represent 



the respective sums of products by P lf Q x , R^ S lt T I} ... &c., and 

 if we employ the first of equations (e) and the first of equa 

 tions (6), we have, to express the value of the first coefficient a, 

 the equation 



2 2 .3 2 .4 2 .5 2 ... 



CQ l - DR V + ES l - &c., 



