412 
SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
and therefore by equation ( 27 .): 
R=m^. ^ - A ^^{^x-\-n-\-n.O) 
x{x-\-n — s-\-n,0) 
-l-w?^_ 2 ,.&c.+ &e., 
which resolves itself, by the transformation of equation (9.), into two sets of terms, 
R'+R", viz. 
^+«.0)+ &c. . . (43.) 
and 
K"=m^. 
1-(1 + Af^ 
P^ + S'-r 
Xin+n.0)+m, 
l-(l+Af^+«" 
-%(w— .s+w.0)4- &c. • • (44.; 
(30.) If, in pursuance of the process followed in the development of Y, we put 
X{x)=7n^.x{x+n) +m^_,.x{x-\-n — s) kc. {t terms), 
(denoting also by Xo(a?), Xi(.r) &c., what this expression becomes when for z we put 
successively x, x — 1, x—2, &c.), we shall have 
R'=: ” X(a;+^z.O) 
1 [x 
=;tXW. 
."fcfiA0.X'W + 
( x{x + n) {x + 2n) 
( 1.2.3 
x{x + n) 
1.2 
,wA0 
1.2 
■&c. (45.) 
The whole assemblage of such terms, giving z all its values, from x to a; — 
therefore will constitute a circulating function explicit in x, and which we shall de- 
note by Z'. 
As regards R", since s and t are given numerically, it constitutes a periodic function 
with constant coefficients, to obtain which we have only to consider that, supposing 
any one of the exponents p^+ 5 '^ to be represented by 
&c., 
we shall have by equation (16.), 
l-(H-A) 
PT, + <l'r 
1-(1+A)‘ 
.w> 
1-(1+A)^ 
.w^_,+ &c. 
(31.) In the particular case in which all the functions &c. are constant, 
we may consider them as being themselves the coefficients of a periodic function, 
such that 
Xi — Xo-'l^h + Xi • • • • .%»-! 
so that if we should meet with such expressions as Xm+i-^ &c., they are to be taken 
as equivalent to Xw Xi^ a mode of regarding a series of arbitrary constants 
occurring in a certain order which will tend greatly to simplify and add clearness to 
what follows. Now we have, generally Xi being constant. 
