406 SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
in which, for/’(0) writing ip(5— 1 it becomes 
Similarly, 
]_fl4A'l'* 1 — fl + Avf 1 
A t 4 -a‘. 0)= ^ 1 +-y •0)+<p(-y+5.0)4-(p(A’+l-|-5.0)j, 
and so on ; the general form assumed by our equation (9.), on applying this process, 
being 
= ^^^f/{0)+/(P+0)+/(3jO+0).... +/(«-]. ;,+0)| . . (33.) 
Supposing, then, for brevity, we denote the combination 
put 
1-(1 + A)‘ 
A 
by V, and that we 
'v|/i(A') = (p(5— 1) ; %//2(5)=<?(5— 1)+?)(.0 ; •vf/3('y)=9(-5- l) + '?(^)+?’(-s+l) ; &C., 
then we shall have, finally, 
S,.=- — I 
+^'fx-2-V'4'2(-y+0.5) 
+ &C.; (34.) 
=X+Y, 
where X represents the non-periodical part, a function of x, and Y the periodical, 
whose coefficients are constant. 
(21.) For the actual evaluation of these functions, all we have to do is to develope 
the operative characteristic in powers of A, and the attached functions in powers of 
0, and to apply them term by term to each other. As regards the function X, we 
find, by so doing, 
X= - y.<P(^+ 6’- 1) — j 
, \x{x + s){x + 2s) ^ x{x + s) . I f{x + s- 
+ 1 rxs .AO — .^.AO J. ^ 
\x{x-\-s ) . . (x-l-3s) 
- 1 ) 
+ &c., (35.) 
in which it will be recollected that 
A0=1; A0^=],AW=2; A0'=1, A"0^=6; AW=6; &c. &c. 
(22.) In like manner, denoting by '^{s) in general, any of the functions 
