i6o Mr. Herschel on circulating functions , 
CL* + ft* + y* + & C. 
n 
the direct integration of each term will give 
u r + 0-rr + &c - } + c 
but the peculiar values of a, / 3 ,&c. (the roots of unity) render 
this expression useless. Neither is it of any service to regard 
as the general term of a recurring series whose equation 
is S , = S 
X +W X 
for the general expression of the sum of such a series 
becomes illusory when applied to this particular case. We 
shall be successful, however, if we regard u x as originating 
from an equation of differences A u = S„ , or u „ — m = S , 
and treat it as a particular case of the equation integrated in 
(9). Thus if we assume 
U X = + 9 A X . S M + • • ■ 
the system of equations (C) becomes 
*A .S 
X — -f- I 
2 A X+I — ‘A^ = 1 
3 A 
X+ I 
‘A =0 
4 A®+i — 3 A a 
These give 
'A x+1 - n A x =o 
i A I A ^ 1,1 I A , H A 1 A a I A 
A t— A *+i» A x— A X+1 — A X + 2» A *~ A ^+3’ 
5 A„, = 'A 
x + n — i 
and finally 
'A I+ „ - 'A t = 1 
But, since a particular value of u x suffices to determine the 
general one, we need only seek a particular value of ‘A : 
