to the summation certain extensive classes of Series. ^29 
formula (A), we find for the above co-efficient 
02x; 
that is, by the transformation (F), writing \/~i ffir w. 
SO 
that we have 
(f) 
2x-f- 1 
1 d- A 
1 + (14-A)< 
0' 
(G) 
~ l.a ^ 
It remains to transform this latter factor into numbers of 
Bernouilli. Now, 
1 + A (1 + A)^ 
1 + 0+^)“ (1+^) |i + (14.a)2 I 
= L |i I (i+^r-i I 
2(1 + a) t^ + (l + A)“+l I 
_ _1 . > (l+A)-(l + A)-» 
-2(1+a)'*'®' (1+a)- + 1 
Now ^ ^ ^ 0 ^ ■>' = 1 by (D) : And we have therefore, for the va- 
iue of the factor in question, 
, (l + A)-(l+.A)-^ 
(1+A)“ + 1 ® 
= > + 2"^ (1 + a)L. (1 + a)-4 2* n, T.T^ 
(iq r^ i— Q , (by equation h) 
— J + (1 4 - a)i^ (I 4 - a) ^^20? 
2 ‘ 2 + a 
Again, it is demonstrated in the paper above referred to, in 
Phil. Trans. 1816, that we have universally, whatever be the 
form o? fox the value of A;, 
(1 + A)^ /(I + A) 0- =/(l + A> { A: + 0 } * 
the second member being supposed developed in powers of 0, 
and the operation denoted by f (1 -j- a) being performed on 
each power so arising : for x put 2 x^ for A;, J and — \ in suc- 
cession, and ^^*(1 -f- a), our expression becomes, 
