1896 - 97 .] Dr Muir on Resolution of Circulants. 379 
or, for the sake of symmetry afterwards, 
C(<x 5 , (Xq , a * , a l , a 2 > % » ^4) 0 
The co-factor of a 5 + a 6 + a 7 + a x + a 2 4 - a 3 + a 4 we know to be 
CCrj Qq 
a 6 a 5 
a 7 
-«6 
^7 
$2 
a 3 cl 2 
a 4 - a 3 
(Lr 0 - a 4 
a 6 
— a 5 
$7 6^0 
a x - a 7 
a . 2 - cq 
a 3 ~ a 2 
(^2 ^3 
«5 
-a 4 
a 6 ~ a 5 
j 
6^2 ^7 
a 2 - cq 
^3 6^2 
« 4 
-«s 
a 5 -a i 
a 6 — cq 
a 7 -a 6 
cq - a 7 
Ctcy Ct^ 
% 
-«2 
Ct^ 6^3 
cq - cq 
a 6 ~ a 5 
6^2 £^7 
- eq 
a 3 — 
— ^3 
CL g Cq 
which, on 
putting a 7 , 
a 5 = *^2 > 
^3 » ^4 
becomes 
0 
6^3 
«2‘ 
-«3 
6^2 ^2 
(X.) (^2 
C 3 q ^2 
CO 
e 
1 
rH 
• 
« 3 - 
-a 4 
6?2 ^3 
«2 - a 2 
cq - cq 
CLi) 
Qj ^ 6^3 
• 
— a 4 
6^3 
Cq $ 2 
Cbcy (^2 
« 4 ‘ 
”«8 
• 
6^3 (^2 
cq cq 
6^2 ““ " CL>J 
« 2 - 
«8‘ 
-a 2 
• 
cq - cq 
&2 ^3 
<C^2 ^2 
a 2 - 
- cq 
^3 — ^2 
(^2 ^3 
• 
and this, by Cayley’s theorem regarding zero-axial skew determin- 
ants of even order, is equal to 
cl 3 cl 4 cl.) cLq cl 7 * ct.-y ll 2 rq cl 3 0 ^ ^ 
^2 ^3 ^2 
(Xcy (^2 
0^2 $3 
6^3 a 
(12) As an example of an even-ordered circulant, let us take the 
case of n = 8. The two linear factors having been removed, the 
remaining factor we know from § 9 to be 
a 2 
~ a 8 
a 3 
- a 4 
« 4 
~a 2 
% 
a 3 
a 6 
- «4 
a 7 
a 5 
®i 
— a 7 
«2 
~ a 8 
a 3 
~ a i 
a 4 
~a 2 
a b 
“«8 
a 6 
-a 4 
00 
— a Q 
«1 
- a 7 
a 2 
~a 8 
a 3 
— a 4 
a 4 
— cl 2 
«6 
— a 3 
a 7 
-«5 
a 8 
a 1 
- a 7 
a 2 
“«8 
«3 
- a 4 
a 4 
-«2 
a 6 
-a 4 
a 7 
“«5 
a s 
-« 6 
a 4 
“« 7 
«o 
~«8 
% 
— a 4 
a b 
-«8 
«6 
-a 4 
a 7 
-a 5 
a 8 
-a 6 
«1 
- a- 
i 
a 2 
— a 8 
and this, on putting a 8 , a 7 , a 6 = a 2 , a 3 , a 4 becomes 
