42 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Theorem V.* — Corresponding to every rational factor of x n — 1 there 
is a rational factor of the circulant of the nth order. 
Thus, corresponding to the factors x+1 and x— 1, the factors of the 
circulant are cq — a 2 + a 3 . . . — ( — 1 ) n a n , and a 1 + a 2 + . . . + a n , respectively. 
The factors of the circulant corresponding to binomial factors of x n — 1 can 
be expressed as circulants or skew circulants, while those corresponding to 
multinomial factors can be expressed as persymmetric determinants. 
Theorem VI. j* — The circulant C(a , a , . . . ,a ,b ,b , . . . , b) whose 
elements in the first row are p as followed by n — p b’s is equal to zero 
when p and n—p are not prime to each other, and is equal to 
{a — b) n ~\p .a-\-n —p . b) when p is prime to n —p. 
A simple proof of this theorem depending upon the properties of the 
roots of unity might be given. 
2. The principal object of this paper is to exhibit the rational and real 
factors of certain forms of circulants. This will be done, for the most 
part, by considering the factors (cq + a. 2 0 + a 3 0 2 + . . . +<x n d n_1 ) themselves, 
making use of the properties of the roots of unity. 
3. It may be observed in the first place that every circulant 
can be factored into real linear and quadratic factors. For, since 
0 k +0 n ~\k = 1 , 2 , . . . n — 1) is real, the product (a l + a 2 0 + a 3 0 2 . . . + qJPr 1 ) 
(cq + a 2 d n_1 + a 3 0 n ~ 2 -fi . . . + a n 0) is a real quadratic expression. 
It may be observed next that if, in Theorem III, a m+l = a m+2 = a m+3 = . . . 
= a 2m = 0, then 
C(cq , cl 2 j • * * > ,0,0,... 0) 2m = C(cq , a 2 ? • • (<q , ^ > • ■ • ’ (I) 
4. The circulant C, of order n=r.s, where a h = a kr+h ' r - — -j, 
has for its value 
Q __ G r ( a i > » • • » ^l+s^l.r) • Cfig + U x+r + • . . a i+s^l .r j Sa, 2 ’ SGt, 3 ’ ' ’ ’ SCl r) 
(i a 1 -\-a 1+r + . . . +a 1+s -zi. r ) 
The theorem and method of proof may be illustrated by taking the case 
where n = 12. We have n = S’4<, or n — 26, and starting with the former 
we have the following relations between the elements : 
a., = a. = a Q — a. 
* Muir, “ On the Resolution of Circulants into Rational Factors,” Proc. Roy. Soc. Edin., 
xxi, 1896, pp. 369-382. 
t Muir, “A Special Circulant considered by Catalan,” Proc. Roy. Soc. Edin ., xxiv, 
Part 6, 1903. 
