Dr Muir on Resolution of Circulants. 
371 
1896 - 97 . 
1 
(4) The remaining factor in the case where n is even is obtained 
in somewhat similar fashion. Thus, when n = 8, we have 
ax' + bx 6 + cx h + dx^ + ear 8 +fx 2 + gx + h = 0 i 
and >• 
ar 6 + a ; 4 + x 2 +1 = 0 J 3 
so that, if we multiply both sides of the second equation by ax + b 
and subtract, there results 
(c - a)x 5 + (d - b)x 4 + (e - a)x 3 + (/ - b)x 2 + {g - a)x + ( h - b) — 0 , 
whence, by cyclical substitution, we obtain sufficient equations to 
produce the eliminant 
c-a 
d-b 
e-a 
f-b 
g-a 
h~b 
d-b 
e-c 
/-t 
g - c 
h-b 
a-c 
e-c 
f-d 
! 7-c 
h — d 
a-c 
b-d 
f-d 
g-e 
h — d 
a-e 
b-d 
c-e 
g-e 
h-f 
a-e 
b-f 
c-e 
d-f 
h-f 
a-g 
b-f 
c-g 
d-f 
e-g 
i 
or (by diminishin 
g each element 
of the 
1 st column by the corre- 
sponding 
element 
of the 
3rd column, each element of 
the 2 nd 
column by the corresponding element of 
the 4th column, and so 
on), 
c-e 
d-f 
e-g 
f-h 
g-a 
h-b 
d-f 
e-g 
f-h 
g-a 
h-b 
a-c 
e-g 
f~h 
g-a 
h-b 
a-c 
b-d 
f~h 
g-a 
h-b 
a-c 
b-d 
c-e 
g-a 
h-b 
a-c 
b-d 
c-e 
d-f 
h-b 
a-c 
b-d 
c-e 
d-f 
e-g 
which is again a persymmetric determinant, 
F(c-e,d -/, e - g, f - h } g - a, h - b, a-c, b - d, c - e, d -/, e - g), 
and is the factor desired. 
(5) What has been done thus far suffices to give the rational 
factors in the cases n = 3, 4, 5, the results being 
C(uj , a 2 * # 3 ) — (flq + a<£ + $ 3 ) 
a 2 $3 $3 dq 
a z - a 1 a 1 - a 2 
i 
