1918-19.] 
Factors of Circulants. 
43 
The factor 
a 4 + a 2 6 + a 8 0 2 4- a 4 0 3 + a b 0 4 + a 6 0 5 + a 7 6 Q + a 8 0 7 + a 9 0 8 + a lo 0 9 4- a n 0 10 + a l2 0 u 
becomes 
(a 4 + a 4 # 3 + a 7 <9 6 + a lo 0 9 ) + a 2 (0 + <9 4 4 0 7 + 6 10 ) + a 3 (<9 2 4 0 5 + 0 8 + 0 11 ), 
which, on account of 1 -f $ 3 + d 6 + d 9 = 0, reduces to 
1 . (a 1 + afl 8 + a 7 0 6 + a 3 0 ( 
The other factors follow : 
Corresponding to - 0 
0 2 
2. (a 4 + a 4 0 9 + a 7 0° + a ]0 # 3 ) 
3. ( a 4 4- a 4 $^ + a 7 + a 49 6® ) 
4. (a 4 + a 4 + a 7 4- ci^q ) + 4a 2 $ 3 + 4a 3 $ 4 , , — 0 2 
5. ( a x + a 4 0 9 + a 7 0 6 + a 10 <9 3 ) , , 6 3 
6. (a 3 + a 4 (9 3 + a 7 0 e 4 a lo 0 9 ) - 0 3 
7 . (a 3 + CL^ 4 d 7 4 CL ^ o ) 4“ 4a 2 $ 4 4 4<2g$ 3 ,, 0 4 
8. (a x + a 4 0 Q + a 7 + a 10 6 Q ) ,, - 0 4 
9. (a 4 4 a 4 <9 3 + a 7 6 Q + a lo 0 9 ) ,, 6 b 
10. „ - 0 b 
11. (ffj 4 a 4 0 6 4 a 7 4 a 4(3 $ 6 ) 0 6 
12. (a 4 "f- a 4 4 cl 7 4 dj o ) 4 4(^2 4 4a 3 , , — 
The product of the first, third, fifth factors is 
C(a 3 , a 4 , a 7 , a 10 ) 
( O' I + 4* <^4 4 ^io) 
The product of the second, sixth, eleventh, as well as of the eighth, 
ninth, tenth, gives the same result. 
The product of the fourth, seventh, twelfth gives 
C(a 4 4 a 4 4- a 7 4- a 10 , 4a 2 , 4a 3 ). 
Therefore 
0 J C 3 ( a i > « 4 » « 7 , «io) • C(a 4 + a 4 4- a 7 4 a 30 , 4a 2 , 4a 3 ) 
(a 3 "l - a 4 4” a 7 4" a 10 ) 3 
Taking r = 4 and s = 3 so that the relations are a 2 = a 6 = a 10 , a 3 = a 7 = <x n , 
a 4 = (X 8 = a i2’ we fiave 
q _ C 4 (a 3 , % , a 9 ) . C(a 1 + a 5 4 a 9 , 3a 2 > 3a 3 , 3a 4 ) ,g"\ 
( a i 4- a 6 4 a 9 ) 4 
With r — 2 and s = 6 the relations are a 2 = a i = a 6 = a 8 = a 1{) = <x 12 , and 
we have 
q = C 2 (a 1 , a 3 , a b , a 7 , a 9 , a n ) . C(a 4 + a 3 4- a 5 + a 7 + a 9 + a u , 6 a) 
(a 4 + a 3 + a 5 + a^ + a 9 + a 3 4 ) 2 
With r = 6ands = 2 the relations are a 2 = a 8 , a 3 = a 9 , a^ = a 10 , a 5 = a 11 , 
a Q = a 12 , and we have 
q _ C 6 (a 4 , a 7 ) . C(a 3 + a 7 , 2a 2 , 2 a 3 , 2a 4 , 2a 5 , 2a fi ) 
(a 4 4 - a 7 ) 6 
= (a 4 <^' 7 )^ • C(a 4 + a 7 , 2a 2 , 2 a 3 , 2a 4 , 2a 5 , 2a 6 ) . . . {2 ) 
