1896-97.] Dr Muir on Resolution of Circulants. 
369 
On the Resolution of Circulants into Rational Factors. 
By Thomas Muir, LL.D. 
(Read December 7, 1896.) 
(1) If we think of the circulant C (aq , a 2 , . . . , , a n _ x , a n ) as 
the result of the elimination of x from the equations 
a^x n ~ x + a 2 x n ~ 2 + .... + a n _ x x + a n = 0 
x n = 1 
it is readily apparent that, corresponding to every rational factor 
of x n — 1, there must be a rational factor of the circulant. Thus 
the circulant of the 6th order 
a 4 
a 2 
«8 
a 4 
a 5 
a ( 
a 6 
a 3 
a 4 
a. 
a b 
a 6 
flq 
a 2 
a 3 
a 
a 4 
a 5 
a 4 
a 2 
a. 
a 3 
a 5 
a 6 
a 4 
a, 
a 2 
«8 
a 4 
a 5 
% 
a 
must have four rational factors, viz., the factor 
flq + Cq> d" $3 + $4 + Ct^ + CIq 
corresponding to the solution x=\ of the equation £ 6 =1, the 
factor 
CL i — $2 + ttg — & 4 + CL b — CLq 
corresponding to the solution x = - 1, and two other factors corre- 
sponding to the partial equations 
a 2 + ^+1 = 0, £ 2 -£+l=0. 
The main object of this paper is the determination of such 
factors, and the presentation of them, when found, in the most 
suitable forms. 
(2) It is clear, at the outset, that when n is odd we have always 
one linear factor, viz., 
oq + $2 F .... + 0. n J 
