Factors of Circulants. 
1918-19.] 
47 
diagonals, and by observing that the consecutive numbers as subscripts 
alternate in position from the first to the last, second, second last, etc. 
From I, II, IV, we have 
3(A 1 -A 4 ) - a 18 +Ct 27 + a 45 
3(A 2 0 + A 3 0 7 + A & 0 4 ) = a is + a 27 0 3 + a 45 0 6 
3( A 2 0 8 + A g 0 2 + A 5 0 5 ) ■= a 18 + a 27 0 6 + a 45 ^ 3 j 
and since 1 4- d 3 + d 6 = 0, we have 
C(a 18 , a 27 , a 45 ) = 27 (A, - A 4 )(A 2 0 + A 3 0 7 + A 5 0 4 )(A 2 0 8 + A 3 0 2 + A 6 0«) 
= 27( A x - A 4 )(A 2 2 + A 3 2 + A 5 2 - A 2 A 3 - A 2 A 5 - A 3 A 5 ). 
If we write I, II, III, IV as follows : — 
( Al - A 4 ) + (A 2 - A 4 )(0 + 0 3 ) + (A 8 - A 4 )(0 2 + f) + (A 5 - A 4 )(0 4 + 05) = a 18 
( A, - A 4 ) + ( A 2 - A 4 )(0 2 + 0 7 ) + (A 3 - A 4 )(0 4 + 05) + (A 5 - A 4 )(0 + 08) = a 27 
(A x - A 4 ) + ( A 2 - A 4 )(0 3 + 08) + (A 3 - A 4 )(03 + 00) + (A 5 - A 4 )(0» + 00) = a 36 
( A x - A 4 ) + ( A 2 - A 4 )(0 4 + 05) + (A 3 - A 4 )(0 + 08) + (A s - A 4 )(0 2 + 0 7 ) = a 45 , 
then multiply the first, second, and fourth by (d + d 8 ), (d 2 + d 7 ) , (d 4 + d 5 ), 
respectively, and add, we have 
6(A 2 - A 4 ) - 3(A 3 - A 4 ) - 3(A 5 - A 4 ) = a 18 (0 + 0 8 ) + a 27 (0 2 + 0 7 ) + a 45 (0 4 + 0 5 ) 
- 3(A 2 - A 4 ) + 6(A 3 - A 4 ) - 3( A 5 - A 4 ) = a 18 (0 2 + 0 7 ) + a 27 (0 4 + 05) + a 45 (0 + 0 8 ) 
- 3(A 2 - A 4 ) - 3( A 3 - A 4 ) + 6(A 5 - A 4 ) = a 18 (0 4 4- 05) + a 27 (0 + 08) + a 45 (0 2 + 0 7 ). 
The product of these gives 
27(2 A 2 - A 3 - A 5 )( - A 2 + 2 A 3 - A 5 )( - A 2 - A 3 + 2A 5 ) 
— a 18 2 (^ a 27 ~ ^ a 45 — a is) a 27 2 (^ a 45 ~ ^ a 18 — a 2?) *h a 45"(^ a 18 — ^ a 27 - a 4 5 ) ~ ^ a 18 a 27 a 45 * 
Syracuse University, 
Syracuse, N.Y., 
October 1918 . 
( Issued separately May 16 , 1919 .) 
