152 Proceedings of the Royal Society of Edinburgh. [Sess. 
of which an equivalent is 
II ( cq + a 2 w r + a 3 (j)l + . . . + a n w ” -1 
r = l ' 
where oo r is any root of the equation 
x n + 1 = 0 . 
His theorem is thus to the effect that a circuiant oj the (2m) #/l order is 
expressible as the product of a circuiant and a skew circuiant each of the 
m** order ; and he gives an alternative proof of it dependent on the 
ordinary factorial expressions for the determinants in question. It is noted 
also that on account of the relation between the roots of the equations 
the circuiant of 
x 2n+l -1-1 = 0, x 
’ ^4 ’ 
,2n+l 
1 = 0 
n , a 
n > ^2n+l 
is equal to the skew circuiant of 
or, say, to 
Q>\ j ^2 > ^3 ’ ’ • • ' > ^2 n ) & Zn+\ j 
C , $2 ’ ‘ ’ , ®2»+l)* 
Glaisher, J. W. L. (1880). 
[On some algebraical expressions which are unaltered by certain sub- 
stitutions. Messenger of Math., x, pp. 60-63.] 
In effect the main result of the paper is that if s stands for oq + a 2 + • . . 
+ a n , then 
C(g - a Y , s - a 2 , . . . , s - a n ) = ( - 1 ) n ~\n - l)C(cq , a 2 , . . . , a n ) , 
and consequently that 
C(g - oq , s - a 2 , . . . , s - a n ) _ ^ , a 2 , , a n ) ^ 
(g - cq) + (g -- a 2 ) + . . . + (g - a n ) cq + a 2 + . . . + a n ’ 
a simple example of the latter being 
1 
c + a 
a + b 
1 
b 
c 
1 
b + c 
c + a 
=(-i) 2 
1 
a 
b 
1 
CL + b 
b + c 
1 
c 
a 
or, in non-determinant rotation, 
(b + c) 2 + (c + a) 2 + (a + b ) 2 - (c -f a)(a + b) — (a + b)(b + c) - (b + c)(c + a) 
= a 2 + b 2 + c 2 — be — ca — ab . 
The result of making one or two other readily suggested substitutions for 
a ± , a 2 , , a n in C(aq , a 2 , . . . , a ) is also noted. 
