258 
■ ME. W. H. L. ETJSSELL ON THE CALCIJLHS OE SYMBOLS. 
From whence, by equating the coefficients of (f), 
‘P2n-\('^ )H-^(’J’ + 2w)=:(P2n_i(7r + 2) + ^(7r ), 
(£"*5;— l)^2„_i(7r)= (g"”*;— l)^(7r), 
whence 
Again, 
i. 
_ / \ S^”'dir — 1/1/ \ 
<P2n-M=—a W- 
e^dv — 1 
<P2n-2{'^) + ^(tT + — 1 )<P2n-M = ?> 2 n- 2 (’J' + 2) + ?>2„_i(7r + 1)^, 
<P2n-2{-^)=-^ ^ 
e d-TT — its dir — 1 J 
1 1 
, d d 
1 1 
i — 1 
We shall now investigate another form of this expansion, by which we shall be able 
to obtain a remarkable expression for the general term. We shall express the unknown 
functions by a notation slightly differing from that which we have just employed. The 
reason for so doing will be easily seen by the reader. 
Let 
Then 
(f’+f«(^))-*‘=f“"+?”+‘fi"’(^+2)+f”9?>(>r+2)+ . . . 
• • • 
?.'"+'>(y)=?.<-'(x+2)+0(ir), 
or 
+ 1 ) (^) g 2 ^ ^(n)^ _ _ 
Wherefore, solving this equation in finite differences, we have 
2£-"”^ 0(7r). 
9'”+'^= 2) + + 1)^T, 
Again, 
whence 
Hence 
= 2£-"4} 2£-"4}^(7r) 
dr d d T 2 
where, however, a proper coiTection must be added after each performance of the sjTnbol 2. 
and similarly. 
