156 Mr. Herschel on circulating functions, &c. 
its general form) it will be sufficient to state that the final 
equation for determining "A^ will be of the form 
• "A* + . ”A,= I x ; (D) 
J H , Z H , &c. and I being certain known functions of x. 
The integrable cases of this equation (which is an ordinary 
one of finite differences, and although of the m x n ,b order 
reducible by a very obvious substitution to the m th ) are for the 
most part those in which all the functions *P , 2 P x , *”P 
are circulating functions with constant coefficients, while 
m + 1 may be of any form whatever, variable or constant. 
In these cases then the proposed circulating equation is inte~ 
grable, and they comprise nearly all which are of any real 
utility in the present state of analysis. 
The complete integral of the equation (D) involves m.n 
arbitrary constants ; and, since by means of the system (C) 
all the other unknown functions 1 A X , 2 A X may 
be expressed by linear combinations of n A x and its successive 
values, these constants will be involved in each of the several 
terms of which u x consists. But as their number exceeds 
what is necessary for expressing the complete integral of the 
proposed, whose order is only m, there must exist equations 
of relation between them to the number of — i).w, or 
else some of them must coalesce into one, by some peculiarity 
in the composition of these functions. But any such relations 
may be directly investigated by substituting the value of u x 
with all its superfluous constants in the proposed equation, 
and causing the result to vanish. It will be worth while, 
