374 Proceedings of Royal Society of Edinburgh. [sess. 
necessitates that C(a l , a 2 , . . . . , a 8 ) should he resolvable also into 
four factors, and further, that there also must he two of the 1st 
degree, one of the 2nd, and one of the 4th. For the sake of clear- 
ness it may he well to go through the reasoning again, say for the 
case of the factor x^+l : — 
The equations concerned are 
a^x 1 + a 2 cc 6 + a 3 x 5 + . . . . +a 8 = 0 
a 4 +i =o V 
and, since in the first equation - 1 may he substituted for a- 4 , it 
follows that the highest power of x left after the substitutions have 
been made must he the 3rd. That is to say, the final equation 
must he of the form 
A 1 a 3 + A 2 x 2 + A 3 x + A 4 = 0 . 
The cyclical substitution will then give three other similar 
equations, and the eliminant obtained will he a determinant of the 
4th order, having each of its elements of the first degree. 
The actual complete result is 
C(oq , a 2 , a 3 j . • . j u 8 ) = (oq 4- a 2 4- oq 4- ... 4- cq) 
* (a 1 -a 2 + a 3 - ... — a 8 ) 
■ 
. a 7 — a 5 + a 3 — cq a 8 — cr 6 + u 4 - a 2 
a 8 — (Zq 4- oq — ct 9 cq — a 7 4- cq — ct 3 
a b ~ a i a e> ~ a 2 a i~ a s a s ~ a i 
a 6 — a 2 a 7 — a 3 a s — oq a 1 — a b 
a 7 — a 3 a 8 — cq a 1 — a 5 a 2 — a 6 
ct 8 ' cq cq cq cq ct 7 , 
where it has again to he observed that the determinants are 
persymmetric. 
The results for the cases from n = 2 to n= 10 may he tabulated 
as follows 
: — • 
Factors of 
Factors of 
x n -l 
C(&2 ) • • • • 3 etn) 
n — 2 
X — 1 
a x + a 2 
X + 1 
cq - cq 
‘ _____ 
