376 Proceedings of Royal Society of Edinburgh. [sess. 
a r -a 8 + a 2 -a 3 , tlie whole determinant factor can be at once 
written, viz., 
a 7 — (Xg + (X 9 — (Xg (Xg — (Xg + (Xg — (X^ (Xg — (X^q + (X^ — (Xg (X^q — (Xj -f- (Xg — (Xg 
(x 8 — ctg + xx 3 — (x^ a g — (x 7 q + xx^ — (Xg (x 7 q — xx^ + (Xg — a g a 7 — a 2 + (Xg — xx 7 
cx 9 — ex 10 + a 4 — a 5 a 10 — a 1 + a 5 — a 6 a 4 — a 2 + a 6 — <x 7 a 2 - xx 3 + a 7 - a 8 
^10 — ^1 "h (Xg — cLq a 4 — (X 2 + <Xg — a 1 <x 2 — xx 3 4 - (x 7 — xx 8 (X 3 — a 4 4 - ci 8 — a^ 
(9) All these identities, setting forth the resolution of circulants 
into rational factors, should, of course, he demonstrable by applica- 
tion of the laws of determinants alone without any reference to the 
principles of elimination. The discovering of the factors in this 
way would not, as a rule, be an easy matter, but the establishment 
of the correctness of the resolution, when once accomplished in the 
manner of the preceding paragraphs, is not exceptionally trouble- 
some. 
As an example the case where n = 8 may be taken, the problem 
being to resolve 
«i 
«2 
a 8 
a 4 
a 5 
«6 
<x 7 
a 8 
a 8 
a 4 
CL 2 
a 3 
a 4 
% 
(Xg 
a 7 
(X 7 
a 8 
a 4 
a 2 
<x 3 
«5 
a 6 
a 6 
a 8 
a 4 
a 2 
«8 
a 5 
«5 
«6 
xx 7 
<h 
«2 
a 3 
a 4 
a 5 
CLq 
<x 7 
(Xg 
Oj 
a 2 
a 3 
a 4 
% 
a Q 
(X 7 
(Xg 
a 4 
a 2 
«2 
«8 
a 4 
«. 5 
a 6 
xx 7 
(Xg 
a 4 
into the three factors of § 4 and 
thereafter into the 
four of § 8. 
(10) The process of 
getting the first factor 
is well known. We 
have only to add to each element of the first column the correspond- 
ing element of all the other columns when the desired factor is at 
once got, and we 
have left the co-factor 
1 
<x 2 
a 3 
<x 4 
«5 
«fl 
a 7 
«8 
1 
a 4 
«2 
<x 3 
« 4 
a 5 
a 6 
xx 7 
1 
a 8 
xx 7 
«2 
(X 3 
a 4 
a h 
«6 
1 
xx 7 
a 8 
<Xj_ 
«2 
«8 
a 4 
«5 
1 
(Xg 
a 7 
(Xg 
«1 
«2 
a 3 
1 
% 
« 6 
a 7 
a 8 
a 2 
1 
« 4 
a 6 
a 6 
xx 7 
a 8 
a i 
«2 
1 
«s 
a 4 
a 5 
a 7 
a 8 
a i 
• 
