152 Mr. Herschel on circulating functions, &c. 
(7). If denote any circulating function with constant 
coefficients, as 
‘*S X +6S_ I + .... *S X _„ + , 
Then, 1st, P x = P x _ „ = P x _ „ = &c. and sdly , 
/{P,. P,_,> }=/ { a ' 6 ~ *} 
For in the expression of write successively x, x — l, 
x — 2, a? — n for x, and we have (since S x _ n = S^Scc.) 
P —a S 4-6 S + kS 
X X * X—-1 1 x 
- n+l 
P —aS 4-bS + 
X—l X — I 1 X-—2 1 
•* s . 
P — £ S +b S , + 
X—2 x~z 1 X 3 1 
P *-«+i == aS ;r -« + i 
■ kS l-„ + 2 
;(A) 
and lastly P x _„ = aS x + b S_,+ • • • 4 S x _„ +1 = P x . 
Any function then, symmetrical relative to all the first mem- 
bers of the equations (A) will, by reason of the circulating 
form of their second members be also symmetrical relative to 
S,S S . , whence by the last proposition 
the truth of this becomes evident. For example 
P,- P_, ••••?,_» + , =a.b....k. 
P, + P„_, +-• P*_„ +I = « + *+ ••••*• 
(8). We are now prepared to proceed to the integration 
of circulating equations, and to determine by that means the 
general terms of such series as depend on them. The man- 
ner of reducing the law of a series of this kind into an equa- 
tion will be rendered evident by a simple instance. Suppose 
this law to oe as follows:* 
* This example is selected as having actually occurred in an enquiry of another 
nature. 
