157 
1915-16.] The Theory of Circulants from 1880 to 1900. 
a bed e f 
— c a b e f —d 
-b - c a f - d - e 
- e - d f a - c - b 
-d f e b a - e 
f e d c b a 
C \a +f , b + e , c + d) . C '(a -f ,b - e , c - d) , 
where the determinants on the left are no longer circulants but resemble 
circulants in having the elements of the first row repeated in each of the 
other rows. Both are centro-symmetric ; and in addition the former of the 
two has the array of its first three rows made up of the array of C(a, b, c ) 
and the array of ( — l)^ 3-1 ) 3 C(d^ e, /), while in the other arrays of C\a, b, c) 
and ( — l)K3-i)3qfc e,j) are similarly used. 
Muir, T. (1881). 
[On the resolution of a certain determinant into quadratic factors. 
Messenger of Math., xi, pp. 105-108.] 
The theorem here established is that the determinant whose every 
element is the sum of the corresponding elements of the circulants 
,a 2 ,...,a n ), ( - . C(6 X ,b 2 , ... , b n ) 
is divisible by 
( Cl j + (Ofl&2 4 “ ... 4 * 0) n + CO 1 Cl 2 + ... + CO n ^ 1 Ct n j 
— (£q + wb 2 + ... 4 - co n 1 b n )(b 1 4- oi~ 1 b 2 + . . . + c o~ n+1 b n ) , 
co being one of the imaginary n th roots of unity. 
When n is 5 the determinant in question is 
a l 
4- 
h 
a 2 
+ 
4- 
h 
a 4 
4- 
h 
a b 
+ 
h 
«5 
4- 
h 
cq 
4- 
h 
a 2 
4- 
a s 
4- 
h 
« 4 
4- 
K 
a 4 
4- 
h 
«5 
4- 
h 
a i 
4- 
h 
a 2 
4- 
h 
« 3 
4- 
h 
a z 
4- 
\ 
«4 
4- 
«5 
4- 
a i 
4- 
h 
a 2 
4- 
^3 
a 2 
4- 
h 
a z 
4- 
h 
«4 
4- 
h 
«5 
4- 
h 
a i 
4- 
and may be viewed as the eliminant of a set of linear homogeneous equations 
in x 1 , x 2 , x s , x A , x 5 . Using the multipliers co°, co 1 , co 2 , co s , co 4 with these 
equations and performing addition we obtain 
cq 4- co -1 a 2 4- ... 4- co _4 a 5 _ _ x, + wx 5 + co 2 .r 4 + co 3 # 3 + cfix 2 
b 1 + co& 2 4- ... 4- oo 4 /q aq + o)X 2 4- co 2 £ 3 + co 3 aq 4- co 4 aq ’ 
and therefore on putting co -1 for co 
<q + co« 2 4- ... 4- co 4 a 5 x 1 4- co 4 aq 4- co 3 aq 4- co 2 aq 4- ux 2 
4" co l b 2 4- ... 4" oo 4 /q x 4 4- co 4 aq 4* c o 3 aq 4~ co 2 aq 4" coaq 
