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IX. 0 « circulating functions , and on the integration of a class of 
equations of finite differences into which they enter as coeffi- 
cients. By John F. W. Herschel, Esq . F. R. S. 
Read February 19, 1818. 
(1). So much has been written on the subject of recurring 
series, and the equations of differences from which they arise, 
that we can now expect little more to be added to their theory. 
This is not the case with the class of series, and their corres- 
ponding equations I propose to consider in the following 
pages, which bear a great analogy to the other, and include 
them as a particular case : I mean, series in which the same 
relation between a certain number of successive terms recurs 
periodically ; the terms so related being separated by others 
connected by relations similar in their general analytical 
form, but modified b}r a variation in the constant or variable 
coefficients which enter into the equations expressing them. 
Such series have, I believe, never yet been considered as a 
class : particular cases have very frequently occurred in the 
course of analytical investigations, and have then been treated 
by peculiar considerations of such a description as to give a 
very uninviting air to their theory, but no general view has 
hitherto been taken of their nature, and no uniform train of 
analytical artifices been exposed by whose aid they may be 
subjected to the same modes of treatment as those of the 
ordinary kind. 
(2). Let us imagine a series of quantities 
Uo>Uj,U 2 , u y &c. 
