1911-12.] The Theory of Circulants from 1861 to 1880. 143 
In establishing these, the requisite differentiations are performed on 
the linear factors of the circulant. The next two are merely stated, 
namely, 
C(P,22, ... ,»») = (-If 1 " 1 
(n 4- l)(2/z + 1) 
12 
c(l,»-l, $(»-!)(»- 2),.. 
j : 2 n 1 when n is odd. 
1 0 when n is even. 
5 
The last gives the evaluation of a circulant in which the elements are the 
cosines (or sines) of the angles a ,a + b ,a-\-2b , ... ,a-\-(n — 1 )b , namely, 
{cos a - cos (a + nb)} n - {cos (a - b) - cos (a - u -1.6) 
2(1 - cos nb) 
This is reached by using the exponential expression for the cosine. 
Glaisher, J. W. L. (1877-78). 
[Sur un determinant. Assoc, frang. pour Vavancem. des sci., vi. pp. 
177-179.] 
[On the values of a class of determinants. Report . . . British Assoc. 
. . . xlvii. p. 20.] 
[On the factors of a special form of determinant. Quart. Journ. of 
Math., xv. pp. 347-356.] 
As has been already pointed out, the annexing of — x to the diagonal 
elements of the circulant C(a 1 , a 2 , . . . ,a n ), does not alter the determinant 
as regards generality. If, however, the order of the last n— 1 rows 
be reversed, thus producing a determinant equal to ( — i)*(»-i)(«T a )Q and 
symmetric with respect to the primary diagonal, the annexing of — x to 
the elements of the said diagonal produces a determinant requiring fresh 
investigation. This requirement Glaisher supplies. 
Having found that 
a - x b c 
b c - x a 
c a b — x 
a — x b c d 
b c - x d a 
c d a — x b 
d a b c — x 
- {x - (a + b + c)} 
lx‘*-(a + iob + a > 2 c) | 
( ’((X + oi^b + we) ) 
= {ic — (cl b c d) { 
’{x - (a - b + c - d)} 
. j x 2 - (a + bi + ci 2 + dfi) ) 
I •( a - bi + ci 2 - di B ) ) ’ 
and that the cofactor of x — (a + h + c + d-A e) in the next case was a quadratic 
in x 2 , he surmised the existence of a general proposition including the three, 
