148 Mr. Herschel on circulating functions , &c. 
SW- ^-t-r + fa. , V-T+/3-+&C. & 
xp q 
but when only one quantity n is under consideration, we shall, 
for convenience, omit the superior index (w) and write the 
function thus 
+ / 3 *+ &c. 
Let us also denote by P x ^ the function 
S W + b 
,(») 
+ e . S 00 + 
J 1 X X 2 1 
k 
jt • D .r— «-f 1 
omitting, in like manner, the superior index (n) and w’riting 
it P x when only one quantity n is considered. Here W'e sup- 
pose a x ,b x , See. to represent any given functions of x and 
constant quantities. The functions and P x are possessed 
then of the following properties. 
(4). S is unity whenever a? is a multiple of n: in all other 
cases, = 0. This is a well known property of the roots 
of unity. Hence, some one of the functions 
S* » Sj. _ , Sj 
3 x — n -f 1 
is necessarily unity, the rest being all zero, though when the 
numerical form of x is not specified, it is undecided which 
that one may be. Hence too it follows, that when r is a 
multiple of n> P x or 
a *- S * + b x- 1 + • • ’ • K- S x— «+I 
reduces itself to a x ; when x — 1 is a multiple of n, to b 
when x — 2 is such a multiple, to c , and so on. Thus in all 
cases P^ will reduce itself to a single term, the form of 
which will be either a x , b x , k x , in rotation, after which 
the same functions recur over again, for some one of the 
numbers 
