164 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Forsyth, A. R. (1884). 
[On certain symmetric products involving prime roots of unity. 
Messenger of Math., xiv, pp. 40-56.] 
Forsyth begins in effect by equating two forms of the eliminant of 
x n -pp? 1 ' 1 +p 2 X n ~‘ 1 -... = (»- a 1 )(x - a 2 ) ... (x - a n ) = 0 ) 
X s - 1 = (x- coffa - w 2 ) . . . (x - (O s ) = 0 ) 
obtaining naturally 
X = (a s 
n 
a; =wi 
( y~P l*” _1 +...) = (- l)“ ls+1 ' . (1 - «!)(! - a?) ... (1 - a?) , 
and when s = 3 
where 
C(cr,r,v) = (l-al)(l-al). . . (l-a 3 n ) 
°-= 1 -P3+P6-P9 + • • • 
T= -Pi+Pt-Pi+Pio- • • • 
v = P 2 ~P>5 +P 8 -Pll+ • • • 
Three paragraphs (pp. 43-46) are then devoted to the evaluation of 
circulants. The procedure adopted is essentially the same as Minozzi’s 
of 1878. As illustrative examples of its effectiveness the five-line and 
seven-line circulants are taken, the result in the former case agreeing 
with Glaisher’s, and in the latter being new. 
Muir, T. (1884). 
[Note on the final expansion of circulants. Messenger of Math., xiv, 
pp. 169-175.] 
The mode of expansion referred to is that used by Minozzi and Forsyth, 
the main object being to correct, if necessary, the latter’s expansion of 
C (a 0 ,a 1 ,a 2 , . . . a Q ), or, say, C(0, 1, 2, 3, 4, 5, 6). It is first pointed 
out that the procedure is considerably shortened by using an additional 
property, namely, that when 
Aa%a?a$a s 3 ala£a% 
is a term of the expansion, then also is 
a term, where m, n, p, q, r, s are numbers less than 7, and such that 
m , 2m , 3 m , 4m , 5m , 6m = ; m >n ,p , q ,r , s (mod. 7). 
Thus, having got the term —2lalala 2 a 3 in the expansion of C(0, 1,2, 3, 4, 
5, 6), we operate on the suffixes 1,2, 3 by multiplying in succession by 
the numbers 2, 3, 4, 5, 6 and casting out the sevens, the result being 
