156 
Proceedings of the Royal Society of Edinburgh. [Sess. 
a+d b+e 
c +f 
a-d 
b - e c -f 
= 
c+f a+d 
b + e 
f-c 
a—d b—e 
b + e c+f 
a + d 
e — b 
f—c a—d 
= C (a + d ,b + e, 
c+f) • 
C (a- 
- d,b - 
It is next found that when n is odd there is no difficulty in deducing 
Glaisher’s theorem from Scott’s. We have only to change the columns 
of C (a-\-d , b + e, c+/) into rows, after the signs of the second row and 
second column of C '(a — d, b — e, c—j), and then perform row-by-row 
multiplication, there being a series of identities like 
(a + d , c+f ,b + e\a- d , e-b , c-f) = a 2 + 2ce - d 2 - 2bf 
to be relied on. 
Lastly, attention is drawn to the fact that there is a second way of 
expressing a circulant of the (4n + 2)^ order as a circulant of the (2n4-iy 71 
order, namely, by resolving it in accordance with Scott’s theorem, 
substituting for the skew circulant an ordinary circulant, and then 
performing row-by-row multiplication. For example, 
C (a ,b , c , d , e ,f) 
where 
a+f b + e c+d 
c + d a -f- f b + e 
b+e c+d a+f 
L M N 
N L M 
M N L , 
a -f 
e-b 
c-d 
c — d 
a-f 
e-b 
e-b 
c-d 
a-f 
L = a 2 + e 2 + c 2 - b 2 - d 2, -/ 2 . = 2a 2 - 2& 2 say 
M = 2ae - 2 \bd 4- 2 ah - 2 ad 
H = 2 ae - 2 bd — 2 ab + 2 ad , 
the 2 implying summation of the terms got by changing 
a , e , c into e , c , a , 
b,d,f into d,f,b. 
As companion results to Scott’s are given (§§ 9, 10) the identities 
and 
a b c d e f 
c a b e f d 
b c a f d e 
e d f a c b 
d f e b a c 
f e d c b a 
— C (a +f , b + e , c + d ) . Q>(a — f , b — e , c d) , 
