1880 to 1900. 
155 
1915-16.] The Theory of Circulants from 
( - l)£ (3+ } , 3. C(a , b , c , d , e ,f) = 
a 
-b 
c 
-d 
e 
-f 
e 
-f 
a 
-b 
c 
-d 
c 
-d 
e 
a 
. -b 
f 
- a 
b 
- c 
d 
- e 
d 
- e 
f 
- a 
b 
- c 
b 
- c 
d 
- e 
f 
- a 
The second form is got from the first by deleting the negative signs, 
reversing the order of the rows and then the order of the columns, the 
result being 
( “ I)** 8-1 )? • C(a , 6 , e , d , e , /) = 
a f e d c b 
chafed 
e d c b a f 
b a f e d c 
d c b a f e 
f e d c b a 
From these two results by multiplication there is obtained 
P 
R 
Q • 
Q 
P 
R . 
( 
) 2 
R 
Q 
P . 
(-1) 3 {C(« 
cs 
a, 
C5> 
II 
. -P 
-R 
-Q 
• -Q 
-P 
-R 
. -R 
-Q 
-P 
and therefore 
C( 
a , b , c } d , e , f) 
= 
C(P,Q,B), 
where 
P = 
a 2 - 
- bf + ce-d 2 +ec 
-fb 
= a- 
2 + 2 ce-d 2 - 
•26/, 
Q = 
ae - 
- bd + c 2 — db + ea 
- f 2 
= c 2 
+ 2 ea -f 2 - 
2db, 
R = 
ac - 
-W + ca-df+e 2 - 
- fd : 
= e 2 
+ 2 ac -b 2 - 
2 fd. 
In the second place, Scott’s theorem regarding a circulant of the (2n) th 
order is shown to be immediately deducible from Zehfuss’ theorem of 1862, 
the reason being that by reversing the order of the last n rows and there- 
after the order of the last n columns the circulant becomes centro- 
symmetric. For example, 
a b c f e d 
f a b e d c 
e f a d c b 
b c d a f e 
c d e b a f 
d e f c b a 
C(a , 6 , c , d , e ,/) 
