30 
Proceedings of the Royal Society of Edinburgh. [Sess. 
factors corresponding not only to the prime factors of x n — l but to all the 
composite factors as well. For example, when n is 6, not only have we 
a “I - b + c + d 4-6-1- fj a — b 4 - c — d 4 - e — f 
a+d-b—e 
b+e- c -f 
a—d+b—e 
b—e+c—f 
b + e -c -f 
c+f - d - a 
3 
b— e +c —f 
c — f + d — a 
corresponding to the prime factors 
x— 1 , .r 4- 1 , x 2 + x + 1 , x 2 -x+\ ; 
but we have 
a + c + e b + d +f 
b + d + f c + e + a , 
CL + d 
b + e 
c+f 
b + e 
c+f 
d + a 
c+f 
d + a 
e + b 
cl + 2b + 2c + d 
b + 2c + 2d + e 
c + 2d + 2e + f 
a — 2b -j- 2c — d ... 
b + 2c + 2d + e 
c + 2d + 2e +f 
d + 2e “f~ 2/ -}■ cl 
b -2c +2d - e ... 
c + 2d + 2e + f 
d + 2e +2 f + a 
e + 2/ + 2 cl + b 
5 
c — 2d+2e -f ... 
a- d 
b-e 
c~f 
b-e c-f 
c-f d- a 
d- a e-b 
a - c 
b-d 
c-e 
d-f 
b-d 
c — e 
d-f 
e — a 
c-e 
d-f 
e — a 
f-b 
d-f 
e — a 
f-b 
a — c 
corresponding to the bifid composites 
x 2 — 1 , x 3 — 1 , 
x s — 2x 2 + 2x - 1 , x 3 + 2x 2 + 2x + 1 , 
x 3 + 1 , x 4 + x 2 4- 1 ; 
and 
a + b + c 
b + c + d 
c + d + e 
d + e + f 
a-b + c 
. d-e+f 
b + c + d 
c + d + e 
d + e +f 
e+f + a 
b- c +d . . 
. e-f+a 
c + d + e 
d + e +f 
e + f + a 
f + a + b 
c—d+e . . 
f-a + b 
d + e +f 
e + f + a 
f + a + b 
a + b +c 
} 
d-e +f . . 
a-b + c 
a + b 
b + c 
c + d 
d + e 
c+f 
a-b 
... e-f 
b + c 
c + d 
d + e 
e+f 
f+a 
b-e 
. . . f-a 
c + d 
d + e 
c+f 
f+a 
a + b 
c — d 
... a-b 
d + e 
c+f 
f+a 
a + b 
b + c 
d-e 
. . . b-c 
c+f 
f+a 
a + b 
b + c 
c + d 
3 
c~f 
... c - d 
corresponding to the trifid composites 
X 4 — X 3 + X — 1 , X 4 + X 3 — X — 1 , 
x b — X 4 + X 3 — X 2 + X — 1 , X 5 + X 4 + X 3 -f X 2 + X + 1 . 
(16) The noteworthy point in regard to the determinant factors here 
is that they are without exception persymmetric, and that the existence 
of the persymmetry is not readily recognisable from an examination of 
