1915-16.] The Theory of Circulants from 1880 to 1900. 169 
where, by diminishing in the determinant on the right the second and 
third rows by ax -{-by times the first, there results 
1 1 1 
ay - ax -by . bx — ax - by 
bx — ax -by ay — ax — by 
In other words, 
(a 2 -ab + b 2 )(x 2 - xy + y 2 ) = P 2 - PQ + Q 2 , 
if P, Q be taken equal to bx — ax—by, ay — ax— by respectively. 
Muir, T. (1896). 
[On the resolution of circulants into rational factors. Proc. Roy. Soc. 
Edin., xxi, pp. 369-382.] 
Viewing the circulant C(a x , a 2 , . . . , a n ) as the eliminant of the 
equations 
apc^~ x + a 2 x n ~ 2 + . . . +a n = 0 i 
s5 n -i=or 
the author concludes that corresponding to every rational factor of x n — 1 
there must be a rational factor of the circulant ; and his object is to 
determine such factors and to present them, when found, in the most 
convenient forms. The means employed for the purpose is elimination. 
Thus, to begin with, Catalan’s factor corresponding to 
(; x n - l)/(x - 1) 
is obtained by deducing from the equations 
ax 4 + bx 3 + cx 2 + tfo + e •-= 0 * 
cc 4 + a 3 + a; 2 + a + 1 = 0 J 
the cubic 
(b - a)x 3 + (c - a)x 2 + (d- a)x + (e - a) = 0 , 
thence by cyclical substitution three other equations, and finally 
b 
- a 
e - a 
d 
- a 
e - 
a 
c 
-b 
d-b 
e 
-b 
a - 
b 
d 
- c 
e - c 
a 
— c 
b- 
c 
e 
-d 
a — d 
b 
-d 
c - 
d 
which being inherently persymmetric is more conveniently written 
P(6 — c,c-d,d — e,e-a,a-b,b — c,c — d). 
The process is then extended to find the factor corresponding to 
(x 2m -l)/(x 2 -l); 
