Mr. Herschel on circulating functions, &c. 149 
x , x — i, x — 2, .... . ,r — ft -f 1 
is necessarily a multiple of n. 
If the coefficients a x , b x , See. be all constant, or if 
P . S 4 - 6 . S 4 -. . . .k. S 
X X 1 X—~l * X— »4-I 
and we give to a? the several values o, 1, 2, 3, 4, &c. to 
infinity, in succession, the first n values of P x will be in their 
order , , 
a, b, c, k 
after which the same set of quantities will be reproduced in 
the same order by continuing the substitution, and so on to 
infinity. The function P x may be called in this case a circu- 
lating function, and the same name (with less propriety how- 
ever) may be extended to the case when the coefficients are 
variable. The system of coefficients a, 6, c, k may 
be called a period. Hence if the terms of a series be in rota- 
tion, a , 6, c, . . . k, a, b, Se c. the general term will be truly 
represented by P^, or a -j- 6 -j- . . . . k S x _ M + , , and if 
they coincide in rotation with the values of n functions a x , 
b , See. thus : 
a Q , b v c 2 ,,.. \__ x , a n , b n+l . Sec. 
the general term will be 
a x' S * + b x' S x— 1 + • • • • h- S * — n + 1 
(5). Any function, however complicated, of x, S x , .... 
S, , is reducible to the form P^ 
Let ?(*)=/{*, S^, S j _ i> ....S i _ B+i | 
be the proposed function. If then x be of the form mn, or a 
multiple of n, we have S x = 1, and the rest zero, therefore 
in this case 
