Mr. Herschel on circulating functions , &c. 14 7 
of proceeding as it would be to consider the series of natural 
numbers as consisting of several other arithmetical progres- 
sions such as 
1 . . 4 . . 7 . . 10 . . 13 , &c. 
2. .5. .8. .11. . 14. &c. 
3 .. 6 .. 9 . . 12 . . 15, &c. 
the general terms of which are respectively 3# — 2, 3a?— « 1, 
gjr, united into one, the general term of which is x. 
It is interesting then to discover some analytical artifice 
which shall obviate these inconveniences, by comprehending 
the whole extent of these and similar series in one single 
equation, whose order shall be no higher than is absolutely 
indispensible ; which shall require no preparatory investiga- 
tion to obtain it; nor the actual calculation of any superfluous 
terms for the determination of the constants in its integral ; 
and, finally, whose integration shall lead to an expression in 
functions of the index x, such that the substitution of the 
natural progression of numbers in succession for x shall pro- 
duce all the terms of the series in their order. Such an 
artifice, or train of artifices, I shall now proceed to explain. 
They turn upon a theorem familiar to every algebraist, but 
which does not seem to have been yet applied to all the uses 
of which it is susceptible. 
(3). Let us then represent by the function 
«* + / 3 * + y' + &c. 
n 
«, /3, 7, &c. being the several roots of the equation z n — 1 = 0. 
If we have occasion to denote other functions similarly com- 
posed of the roots of other equations zP — i—o, 2#— 1=0, &c. 
they will therefore be represented as follows : 
