400 SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
(3.) Furthermore, if we observe that 
i-(l+Af+^={l-(l + A)^} + (l + A)"{l-(l + A)n, 
we shall have, by applying’ each of these operative symbols to F(A)y'(0), 
{l-(l + Ay+nF(A)/(0) = {l-(l + A)^}F(A)/(0) + {l-(l + A)nF(A)/(^+0); (7.) 
and therefore 
{l-(H-Ar}F(A)/(^-|-0)-{l-(H-A)nF(A)/(jo+0) = {(l + A)^-(l + A)nF(A)/(0); (8.) 
(4.) In the particular case where F(A) = ^, these become 
and 
(9.) 
(10.) 
(5.) Designating by S(x“) the sum of the wth powers of the natural numbers from 
1 to X inclusive, or putting 
S(^«)_i»_j_2»-i- 
it is demonstrated (Examples, § 6, Ex. 23.) that 
and again, in § 8, Exp. 1 1 of the same work, that 
■ S ( a ”)^(1 + A ). ^ -^ ^^' ^ 0 ” ( 12 .) 
(6.) Furthermore, it will be necessary to recall in what follows, the notation and 
conventions of ‘circulating functions,’ as explained in my paper on that subject, 
published in the Philosophical Transactions for 1818, vol. cviii. p. 144. Denoting 
by the sum of the j^th powers of the .sth roots of unity divided by s, or the function 
y»'+/3'+y'+&c.), 
where «, (B, y, &c. are those roots, any funetion of the form 
will circulate in its successive values as x increases by units from 0 : being expressed 
by when .r is a multiple of .9; by B^,, when a? — 1 is such a multiple, and so on. It 
A^, B^, &c. be simply constant, the function may be termed di periodic one, since it 
assumes in periodic and constantly recurring succession the values A, B, C — N, A, B, 
&c. ad wjlnitum. If a' be a specified number, as 2, 3, &c., we shall not the less use 
the notations 2^, 3^, &c. to express the respective quantities ^(a^+f^^), 3 (o^'^+^'^+ 7 ^)j 
&c., where a, (B, &c. are the corresponding roots of unity. And we shall accordingly 
