44 
Proceedings of the Royal Society of Edinburgh. [Sess. 
If in (2) we put a 1 = a h r+1 (h = 1, 2, . . . , s— 1), then 
C = 0 (3) 
That is, the C of onr example vanishes if in (2') a t = a 4 = a 7 = a 10 ; in 
(2") = a 5 = u 9 ; in (2 /// ) a 1 = a s = a 5 = a 7 = a 9 = a n ] in (2 iy ) <x 1 = a 7 . 
If in (2) a 2 = a 3 = . . . =a r , then we have 
C = GT ( a i > W • • • «l+i=I , r) («i + <*l+r + ■ • • + gj+i=i . r ~ + a l+r + • • • + . r + T - 1 . S . gg) 
( tt i + a 1+r + . . . + ai+izr! . r ) r 
If in (4) a 2 ;= 0, we have 
C = C r (a 1 , « 1+r , . . . i. r ) .... (5) 
In the case where s = 2, (4) becomes 
0 _ C r ( a i , QH+r)(«i + a 1+r - 2a 2 ) yF " 1 (a 1 + g 1+r + 2 , r - 1 . a 2 ) 
(«! + «!+,)" 
= («! - «i+r) r (% + <^i+r - 2a 2 Y~\a l + a 1+r + 2 . r - 1 . a 2 ) . . (4') 
which when a 2 = 0 is 
C = (a\-al +r ) r (5') 
The case of (2) when s = 2 takes the form 
C(«1 j 0*2 j • • • j j 5 • • "5 ®'r)’2 l r == (gj ^r+l) • t/(a 4 "I - ^r+l ) 2tt 2 j 26? 3 , . • . 2 Qy) 
and if in this a 2 = a 3 = a 4 = . . . = a r , we have 
C(®j j ^2 ’ • * • a 2 3 %+r j ®2 » ^2 ‘ • ' ^2)2^ 
= (^1 ®i+y* ^^2)^ ($ 4 “1“ "1“ 2 ,T 1 , Ct 2 ) • (6) 
which when a 2 = a 1+r is an example of Theorem VI. 
5. The circulant C of order 2 n, (n even), where 
{ ~ a M+z l (* = 2. 3, 
I a 2Jc-i — a 2k+i j 
,n - 1), 
except that a n+1 =i=a 3 has for value 
p _ { ( a \ a n+i) n 4- (a 2 a ^ n }{( a i 2(^3 + (<^ 2 Q h) w }(( < h"t^ 2.a 3 + a w+1 )^ (fl 2 + 9fl—l.<% 4 ) 2 } \ 
{(a 1 -2a 3 + a n+1 ) 2 -(a 2 -a 4 )} ' ' 
The circulant C of order 2 n, (n odd), where 
j a 2k a 2k+2 ) _ 
I ^2k-\ = ^2fc+l j 
2 3 
o, 
1 ). 
except that a w+1 =/=a 4 has for value 
{(gj gg g 4 + g«+i) n + (g 2 ^4) W }{(^i CL%-¥ (l^ (a 2 — g 4 ) n } { (gj + ?£ hgg) 2 (tt 2 + 2.(Z 4 + C^ n+1 ) 2 | 
K^— a. 3 ) 2 — (& 2 — 2 a 4 .+ g«+i) 2 } 
( 76 ) 
The method of proof may be illustrated by using for (7a) the same 
case as was used in art. 4. The relations between the elements are 
a z = a 5 = a 9 = a n , a 4 = a 6 = a 8 = a 10 = a 12 , and the first factor is 
a 4 + a 2 0 + a, 7 6 G + a 3 (0 2 + 0 4 + 6 8 + 6 10 ) + g 4 (0 3 + 0 5 + Q 7 + 6» + 6 U ), 
