381 
1896-97.] Dr Muir on Resolution of Circulants. 
elements of the other columns, and each element of the 2nd 
column by the corresponding element of the 1st column, we have 
(«! + 2 a 2 + 2 a 3 + 2 a 4 ) 
OL i CLc ) CXt 4 
a 2 — a 4 - a 4 a 2 - a 3 
CC 3 CL 4 ^2 ^3 ^9 
the determinant in which is exactly the same as the second of the 
two factors we started with. The theorem is thus established, and 
a comparison with the result of § 10 leads us to the curious 
identity 
a x 
“«3 
a 2 
-a 4 
cl 3 cc 4 
— 
a 2 
-a 4 
a x 
-a 4 
& 2 “ CLq 
a 3 
-a 4 
a 2 
-a 3 
CC-^ "* CLc) 
a 3 - a 4 a 2 — a 3 a x — a 2 a 2 — a x a 3 — a 2 
cu 
a 4 a, 
& 3 Ct-^ CL% Cl 2 Cl-^ 
a 3 — a 4 a 2 — cl 3 a x — a 2 
CL 3 CL 4 Cfc) 
Cf 3 CL 4 
(14) Taking now an even-ordered circulant, — say, as before, the 
circulant of the 8th order, 
0 (^ 1 , u 2 , ct 3 , cc 4 , , <x 4 , u 3 , u 2 ) 
we see again, by reason of the centro-symmetry, that it is resolv- 
able into 
a x 
+ a 2 
a 2 + a 3 
u 3 -f- u 4 
U 4 
+ « 5 
a x 
-a 2 
a 2 - 
a 3 
a 3 
-a 4 
a 4 
-a s 
a 2 
+ a 3 
(%i + ct 4 
a 2 + a b 
a 3 
+ ^4 
a 2 
-a 3 
a x - 
a 4 
a 2 
-a 5 
a 3 
-a 4 
a 3 
+ u 4 
a 2 + a 5 
<%i + u 4 
a 2 
+ a 3 
a 3 
-a 4 
a 2 ~ 
a 5 
a l 
-a 4 
a 2 
-a 3 
a 4 
+ a 5 
a 3 + a 4 
(X 2 -p Ct 3 
CL^ 
+ a 2 
a 4 
- a 5 
a 3 — 
a 4 
a 2 
-a 3 
a x 
— a 2 
From each of these a known linear factor is separable, the result 
being 
{a x 4 - 2 a 2 + 2 a 3 + 2 a 4 + a 5 ) (a x - 2 a 2 + 2 a 3 - 2 a 4 + a 5 ) 
a Y + a 2 a 2 + a 3 a 3 + a 4 1 
• 
CL-^ Che) CLcy (Xg CLi ^ 1 
a 2 + a 3 a x + a 4 a 2 + a b 1 
a 2 — a 3 a Y - a 4 a 2 - a 5 1 
a 3 + a 4 a 2 + a 5 a x + a 4 1 
a 3 - a 4 a 2 — a 5 a x — a 4 - 1 
a 4 + a b a 3 + a 4 a 2 + a 3 1 
CL 4 6 ^^ 6 )^^ CL 9 1 
These two determinants, however, are clearly transformable into 
u x ~ a 3 a 2 4 - cc 3 — 04 
a 2 -a 4 a x + a 4 - a 2 
a 3 ~ a 5 a 2 Jr °'b~ a i 
a 4 a ?> -\- a 4 - a 2 - a 5 
a 5 a 2 + a b - a x — a 4 
a 4 a x + a 4 — a 2 - a 3 
a x - a 3 a 2 - a 3 -\-a l 
a 2 - a 4 a 1 — a 4 + a 2 
et 3 ^2 ^5 h ot 3 
a 4 a 3 -a 4 J r a 2 - a 5 
a b a 2 - a b J r a l - a 4 
a 4 a 1 -a 4 + a 2 - a 3 
