37 8 Proceedings of Royal Society of Edinburgh. 
[sess. 
«8 
~a 6 
Cl 4 
~ a 6 
«5 
— a 7 
00 
52 
1 
a 7 
00 
e 
-a 2 
a 4 
— a 6 
a 5 
~ a 7 
«6 
-a 8 
a- 
«8 
— C< 2 
cq 
— a 3 
a 5 
- a 7 
a 6 
-a s 
« 7 
— cq 
a 8 
-«2 
cq 
«2 
“«4 
a 6 
-a s 
a 7 
— 
«8 
- a 2 
a x 
“«S 
a 2 
- « 4 
«s 
— a 5 
a 7 
~ a i 
a s 
— a 2 
cq 
~«3 
a 2 
-«4 
-«5 
-a 6 
a s 
— a 2 
a i 
— a 3 
« 2 
- « 4 
H 
“«5 
« 4 
“«6 
a 6 
— a 7 
which is exactly the same as the determinant of § 4. 
Finally, by increasing each element of the 1st column by the 
corresponding element of the 5th, and each element of the 2nd 
column by the corresponding element of the 6th ; and thereafter 
diminishing each element of the 6th row by the corresponding 
element of the 4th, each element of the 5th row by the corre- 
sponding element of the 3rd, and so on, we have 
a* 
i 
— a 1 -f- $ 8 
“«6 
« 8 
— C&2 + <q 
“«6 
a 5 
-a 7 
a 6 
1 
a 
00 
Cq 
- cq 
a 8 
- a. 
a s 
— Cq 4* cq 
“«6 
cq 
- a 3 + a 5 
- a 7 
-«s 
a 7 
- a 1 
-a 2 
cq 
- a. 
• 
• 
a^ 
- cq 
a 6 
— a 2 
a 7 
~ a 3 
a 8 
-a 
• 
• 
a 6 
-«2 
a 7 
— a 3 
a 8 
- a 4 
cq 
- a. 
• 
• 
a 7 
“«8 
a 8 
-« 4 
cq 
-a 5 
a 2 
— a { 
• 
• 
a 8 
- « 4 
a 1 
— « 5 
a 2 
~ a 6 
a 8 
- a, 
which clearly resolves into the two determinants of § 8, viz., 
P (a 7 - oq + a 3 — a 5 , a 8 — a 2 + a 4 - a 6 , a 1 - a 3 + a 5 — a *) , 
IP ( a ^ a i , ctg a ,-) , . . . . a 8 a 4 , a 7 oq a 3 ck 7 ) • 
(11) We come now to a very interesting property of circulants 
of the form 
C(^b > ^2 ’ ^3 > • • • • j 1 ^2) » 
that is to say, circulants whose first row, after the element in the 
place (1,1) has been deleted, is the same when read backwards as 
when read forwards. This property is to the effect that the co- 
factor of 
a 7 -f- a.-) -f- a.^ + a 8 - 1 - a 2 
in the case of a circulant of odd order, and the co-factor of 
( a j -j- a.i -f- a 8 -f- . . . - 1 - a 2 -t- (oq — ^2 P ^3 — a 8 — oq) 
in the case of a circulant of even order are complete squares. 
As an example of the former case let us take the circulant of the 
7th order, viz., 
C (a i , a 2 , a 2 , a 4 , oq , a§ , $7) 
