154 Mr. Herschel on circulating functions , &c. 
(9). Given the circulating equation 
+ 1 P x • + 2 P X • U x-z + mp x- U x _ m = m + 1 P , 
where 1 P^, 2 P r , &c. ar<? any circulating functions, with either 
constant or variable coefficients, the period of their circulation 
being the same in each, and equal to n ; to integrate it 
Assume 1 A x . + "A, . S^.-f ”A r . S r _„ + , 
Then we have 
U x-l— n \-l' S * + IA *_ i* S x— 1 + ” _ ‘ A x— I- S *— a+i 
K-* s ,+ n \-f s_, + ”- 2 a 1 _ 2 . s m+i 
and so on ; and supposing 
,p ,= ‘^' S .+V S « + Vs M+J .ta. 
we shall have by (5), 
• *P, = VV* ■ S. + * 6 . • 'A^, . S._, + 
'^x • ”~‘ A x— r ®x — n+i 
similarly, 
«_,• 2 P, = V ” _ ' A x- 2 - S x + %• W-x- V. + 
a ^-’ ,-2A x-x-S,_„ +1 ,&C.=&C. 
The equation then will become by substitution 
° = S 4‘ A * + V A *_, + V ' A x_x + • • ■ • 
+ S ;_,{* A 1 ,+ ‘V'A^. + ’i/A^, + . . . . 
m h n ~ m + 2 A 
* A x—m 
+, M 
+ S J « +I f A -+ i/ A X— I + ** 2A X — 2 + 

n has here been supposed greater than m. If the contrary 
