1919-20.] Determinants connected with Circulants. 31 
the forms in which they are first obtained. For example, if the factor 
corresponding to & 4 — x z -\-x — 1 be wanted, we take the equations 
ax h + 5x 4 + cx 3 + dx 2 + ex +f— 0 i 
x^ - x z + x - 1=0] 
and seek their eliminant. Using the second equation to free the first 
from the terms in x 5 and x 4 , we obtain 
(c + b + a)x 3 + ( d - a)x 2 + (e — b)x + (f+b + a) = 0 , 
whence three other equations are got by cyclical substitution, and then 
from the set of four the result 
c + b + a d - a e-b f+ b + a 
d + c + b e — b f — c a + c + b 
e+d+c f-c a-d b+d+c 
f+e+d a-d b-e c+e+d . 
The persymmetry latent in this has to be brought into evidence by 
performing the operations 
col 4 + col 3 + col 2 
col 3 + col 2 + colj 
col 2 + eolj . 
(17) When, however, the persymmetric form is known, we can readily 
utilise our knowlege to evolve it directly from the circulant itself, and 
thereby simultaneously obtain the cofactor. Thus, performing on 
the operations 
we obtain 
a b c d e f 
b c d e f a 
c d e f a b 
d e f a b c 
e f a b c d 
f a b c d e 
colj + col 2 + col 3 , . . . , col 4 + col 5 + col 6 
a + b + c 
b + c + d 
c + d + 6 
d + e +f 
e 
f 
b + c + d 
c + d + 6 
d + e +f 
e +f + a 
f 
a : 
c + d + 6 
d + & + f 
e +f + a 
f+a + b 
a 
b 
d + e +f 
e +f+a 
f + a + b 
a + b + c 
b 
c 
e+f + a 
f + a + b 
a + b + c 
b + c + d 
c 
d 
f + a + b 
a + b +c 
b + c + d 
c + d + e 
d 
e 
the last two rows of which, by the performance of the operations 
