' 
125] AS DEPENDING ON THE SYMBOLIC EQUATION 0 n = 1. 125 
satisfies the equation 8 r = 1, where r is less that n, then that r must be a sub 
multiple of n; it follows that when n is a prime number, the group is of necessity 
of the form 
1, a, a 2 ,... a n_1 , (a n = l); 
and the same may be (but is not necessarily) the case, when n is a composite 
number. But whether n be prime or composite, the group, assumed to be of the 
form in question, is in every respect analogous to the system of the roots of the 
ordinary binomial equation x n — 1 = 0; thus, when n is prime, all the roots (except 
the root 1) are prime roots; but when n is composite, there are only as many prime 
roots as there are numbers less than n and prime to it, &c. 
The distinction between the theory of the symbolic equation 8 n = 1, and that of 
the ordinary equation x n — 1 = 0, presents itself in the very simplest case, n — 4. For, 
consider the group 
1, % /3, 7> 
which are a system of roots of the symbolic equation 
0 4 = 1. 
There is, it is clear, at least one root /3, such that /3 2 = 1; 
represent the group thus, 
1, a, /3, afi, (/3 2 = 1); 
we may therefore 
then multiplying each term by a as further factor, we have for the group 1, a 2 , a/3, 
a 2 /3, so that a 2 must be equal either to /3 or else to 1. In the former case the 
group is 
1, a, a 2 , a 3 , (a 4 =l), 
which is analogous to the system of roots of the, ordinary equation ic 4 — 1 = 0. For 
the sake of comparison with what follows, I remark, that, representing the last- 
mentioned group by 
1, /3, 7. 
we have the table 
1, 
a, 
/3, 
7 
1 
1 
a 
/3 
7 
a 
a 
/3 
7 
1 
/3 
/3 
7 
1 
a 
7 
7 
1 
a 
/3