Factors of Circulants. 
45 
1918-19.] 
which, since 0 6 = — 1 , becomes (a 4 — a 7 ) + (a 2 — a 4 )0, and similarly for the other 
factors, {{a 1 — a 1 ) — (a 2 — a^0}, etc. ; and the product of all the factors is 
_ {(<*1 ~ a if + K ~ a 4 ) 8 } { K - 2a 3 + Q 6 - (a 2 - a 4 ) 6 } (K + 4a s + a r) 2 ~ ( a 2 + 5a i) 2 } 
{(«].- 2 a 3 + a 7 ) 2 -(a 2 -a 4 ) 2 } ( ' 
For the case of (76) take n = 7 and the relations are a 3 = a 5 = a 7 = a 9 = a 11 
= a 13 , a 4 = a 6 = a 10 = a 12 = a u . The factors are : {(a 1 — a 3 + a 4 — a 8 ) + (a 2 — a 4 )0}, 
1^ — a 3 — a 4 + a 8 ) — (a 2 — a 4 )d}, etc. 
The product of all the factors gives 
r = {( ^-^3 + a 4 -a 8 ) 7 -(^2- a 4) 7 }{( a i-%~ a 4 ± a s) 7 + ( a 2- a 4) 7 }{( a l + 6 %) 2 -( a 2 + 5 °h + ff 8 ) 2 } /7 7/\ 
(a 4 - a 3 ) 2 - (a 2 + a 8 — 2 a 4 ) 2 ' 
If in (7a) a n+1 ^a 3 , and in (76) a n:fl = a 4 , both reduce to 
q i ( a i - a s) n ~ ( a 2 ~ a ^ n }{ ( a i - a ?T + ( a 2, - a d n } { («1 + W - 1 . a 3 ) 2 - (fl 2 + n - 1 . a 4 ) 2 } 
(^1 — a ^)' A ~ { a 2 ~ a ^f‘ 
^ +”-" 1 ' - («2 +^“1 • « 4 ) 2 } • • . ( 8 ) 
If in (8) a 4 = a 2 , it becomes 
C=,(o 1 -a 3 ) 2n 1 .a 3 ) 2 -(«(« 2 ) 2 } . . (8') 
If in (S') a 3 = a 1} then 
0(^2 , CKg , (^2 , . . . , ^ 2)2 n — Q (w> 1) . . (8) 
as is obvious from the determinant itself. 
If in (7a) a 4 =a 2 , it becomes 
C = (a 4 ^n+ i) w (^i 2a 3 (^1 ”1~ ^ 2 . g , 3 -f- a w _|_x) 2 (%a 2 ) 2 ) ■ (7a ) 
which, if a 3 = a 1 , becomes 
C = ( - l) w “ 2 («i-«n+i) 2n_2 {( ? ^- 1 • a \ + a n+xf ~ (na 2 ) 2 } . . (7a'") 
If in (76) a 4 = a 2 , it becomes 
C = {(a 1 -a 3 ) 2 -(a 2 -a n+1 ) 2 } w " 1 {(a 1 + ?z - 1 . a 3 ) 2 - (n - 1 • a 2 + a n+1 ) 2 }. (76") 
6. The circulant of order 2n—l, where 
has for value 
(t 2k - a 2k+2 I / Z. _ T O w \ 
C = (a 4 - a 2 ) 2n (n - 1 . a l + ?i . a 2 ) 
(9) 
This may be proved in a similar manner to the others by using the 
properties of the roots of unity, but a very simple proof using determinants 
is as follows : — 
Starting with the circulant C(a 4 , a 2 , a 4 , . . . , a 4 ) and performing on it the 
following operations: co^-colg , col 2 -col 4 , . . . col^-col^ , col 2re - col 2w+1 ; 
