1915-16.] The Theory of Circulants from 1880 to 1900. 167 
for the elements of the column. The third theorem is that if the linear 
factors of a circulant, say of the fifth order, be a, /3, y, . . . , and the 
complementary minors of the elements of the first row be A , — B , C , — . .. 
then 
C(a/3yS , a/3ye , a/38e , ay8e , /3y8e) = 5 5 ABCDE . 
This is proved by using the preceding theorem five times over in the 
form 
A + ojE + w 2 D 4- co 3 C + w 4 B = /3ySe , 
A + w 2 E + o) 4 D + coC + to 3 B = ySea , 
and thence obtaining the linear factors of 
C(a/3yS , a/3ye , a/38e , aySe , /3ySe) 
in the form 
5A , 5B , 5C, 5D, 5E. 
The fourth theorem is that if the elements of the first row of a circulant 
of the n th order be multiplied by co n , co a ~ 1 , ..., co respectively, the elements 
of the second row by co 11-1 , co n ~ 2 , . . ., co, co n respectively, the elements of the 
third row by oo n ~ 2 , a> n " 3 , . . .,cd, co n , of 1 ' 1 respectively, and so on, where co is any 
n th root of unity, the circulant is unaltered in value . Here for proof we 
have only to multiply the rows in order by co°, co 1 , co 2 , co s , co 4 , and then divide 
the columns by co 5 , co 4 , co 3 , co 2 , co. The fifth theorem concerns skew circulants, 
being the analogue of the same author’s theorem of 1881 regarding ordinary 
circulants. 
In order to throw light on the law of the coefficients of the final 
expansion of C(a, b, c, d, e,) the determinant 
da 
b/3 
cy 
d8 
ee 
e P 
ay 
b8 
ce 
da 
dy 
e8 
ae 
ba 
de 
ea 
a/3 
by 
be 
Ca 
d(3 
ey 
aS 
is framed by attaching to each element of C(<x, b, c, d,e) the corresponding 
element of ( — l) 3+2+1 C(a, /3, y, S, e). This, which degenerates into the 
ordinary five-line circulant on putting a — b = c — d = e — 1 or a = /3 = y = S = e 
= 1, is found to be equal to 
2a 5 . adySe - 2a 3 ^e . 2a 3 yS + 2a 2 6 2 cf . 2a 2 /3y 2 
o o o o o - lOabcde . a/?ySe 
+ 2a 5 . abcde - 2a 3 /3e . 2 a 3 cd + 2a 2 /3 2 d . 2a 2 5c 2 
where all the coefficients of the ordinary circulant except the last are 
in a manner “ sifted ” into their unit constituents. 
