370 
Proceedings of Royal Society of Edinburgh. [sess. 
and that when n is even we have two, viz., 
fl&l + $2 • • • + 
$2 o • « J • 
(3) The remaining factor, in the case where n is odd, is readily 
obtained hy using along with the given linear equation the equa- 
tion 
instead of x n -~ 1 = 0. Thus, when n — 5, we have 
ax 4 + bx 3 + ex 2 + dx + e = 0 
and 
a 4 + x 3 + x 2 + x + 1 = 0 
whence it follows that 
(b - a)x 3 + (c - a)x 2 + (d - a)x + (e - a) = 0, 
and hy cyclical substitution 
(c - b)x 3 + (d - + (e - &)& + (a - 6) = 0, 
((7 - c)x 3 + (e - c)# 2 + (a - c)cc + (& — c) = 0 , 
(e - d)x 3 + (a - c?)a; 2 + (b - <7)x + (c - d) = 0 , 
from which four equations we have the eliminant 
b- 
a 
c 
- a 
d- 
a 
e-a 
c - 
b 
d 
-b 
e - 
b 
a-b 
d- 
c 
e 
- c 
a - 
c 
b - c 
e - 
d 
a 
-d 
b- 
d 
c-d 
or (hy diminishing each element of the first three columns hy the 
corresponding element in the column immediately following), 
b — c c-d d -e e-a 
c — d d -e e — a a-b 
d -e e — a a-b b - c 
e—a a-b b - c c—d 
— a persy mmetric determinant, ordinarily written 
P(5 - c , c — d , d-e , e-a , a-b , b - c, c-d)* 
* For further details regarding the co-factor of a-^ + a^ . . . + a n in the 
case where n is odd, a paper on “ Circulants of Odd Order ” may be consulted 
in the Quart. Journ. of Math., xviii. pp. 261-265. 
