1896-97.] Dr Muir on Resolution of Circulants. 
375 
vi = 3 
i— i 
i 
m 4 + < x 2 + m 3 
X 2 + X + 1 
P ( — ^3 j a 3 — % $ CL\ — ttg) 
71 = 4 
03-1 
a x + a 2 + a 3 + a 4 
03+1 
a 1 - a 2 + a s - a 4 
X 2 + 1 
P (a 3 -a 1 , a 4 -a 2 , a 1 -a 3 ) 
n — 5 
03-1 
M 4 + M 2 + • • • + M§ 
03 4 + 03 3 + . . . + 1 
P(^2 ^3 3 % ^4 5 ® e ® 5 ^5 CC^ 3 ^2 9 • • • 5 ^3 $4) 
71 = 6 
03-1 
flq + # 2 + • " . + 
03 + 1 
- a 2 + • • • ~ a 6 
X 2 + 03 + 1 
P(a 3 -a 4 + a 6 - a lf a A - ct 5 + a 1 - a 2) a 5 - a 6 + a 2 - a 3 ) 
03 2 - 03 + 1 
P( - a 3 - a 4 + a 6 + a x , - a 4 - a 5 + a 4 + a 2 , - a 5 - a 6 + a 2 + a 3 ) 
71 = 7 
03—1 
M 4 + cl 2 + . . . + m 7 
03 6 + 03 5 + . „ . +1 
P (^2 ^3 2 ^3 ^4 3 ® • ® ® 5 ^0 $7 5 $7 ££4 9 $4 6^2 ? ® 9 9 9 
8 
II 
00 
03-1 
$ 4 + d 2 + . . . + Mg 
03 + 1 
d/j d^o "i" ® ® ® CCq 
03 2 + 1 
P (m 7 -m 5 + m 3 -m 1} a 8 - m 6 + m 4 -m 2 , a 4 - a 7 + a 5 - a 3 ) 
03 4 +l 
P(ttg m 4 , m 8 m 2 , » , , , cc 8 m 4 , m 4 m ^ tt 3 — ® 7 ) 
?i = 9 
03-1 
eq + cs 2 + . , . + a 9 
X 2 + 03 + 1 
P(a 9 - a 2 + a 6 - a 4 + a 3 - «i , oq -a 8 + a 7 -a 5 + a 4 -a 2 , a 2 - a 9 + « 8 - m 6 +m 5 - a 3 ) 
03 6 + 03 3 + 1 
P(a 4 — CI 7 , - « 8 > • • • ) a 9 ~ a 3 > n 4 — a 4 , . . . , m 5 — a 8 ) 
' ii ~ 1 0 
03-1 
a i + <^2 + • • *4* a io 
03+1 
$4 ^2 "P • • • ^10 
03 4 + 03 3 + . . . + 1 
P ( CC 7 Mg + M 2 M 3 , CKg Ci/g + — G3 4 , » t, . . , — A3 4 + C ? 8 — < 33 ^ J 
03 4 — 03 3 + . . . + 1 
P(a 7 + a 8 -a 2 - a 3 , a 8 + a 9 -a 3 -a 4 . m 3 + m 4 - m 8 - m 3 ) 
It is well worthy of notice that all the persymmetric deter- 
minants here given would he sufficiently specified by writing only 
the first element , the others being obtained at once therefrom by 
the cyclical substitution. For example, when n— 10, the factor of 
C(a 19 •••, a 10 ) corresponding to the factors x^ + x^ + x 2 + x + 1 of 
cc 10 — 1 is known to be a persymmetric determinant of the 4th order; 
and, consequently, if it be further known that the first element is 
