The theory of determinants in the historical order of development, by Sir Thomas Muir.

CIRCULANTS (SCOTT, 1878)

381

The last is an extension of Wolstenholme's result of 1867, giving
the evaluation of a circulant in which the elements are the cosines
(or sines) of the angles a, a+b, a+2b,..., a+(n-l)b,
namely,
{cos a-cos (a nb) n- {cos(at-b)-cos(a+ -l. b)}
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. franc. pour l'avancem. 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,, a2,..., ), 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 (- 1)~('-l)(1-2)C 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 = - x-(a+b+c)}
b c-x a a+wb+-2c)
c a b-x a.(+ b+w)
a-x b c cd = {x-(a +b+-c-c+d)}
b c-x d a x x-(a-b+c-d)}
c d a-x b x2j-(c+bi+ +d23)
c - Cl Ct b fx2-(a~b~+ci2+di3)}
d a b c-x ( (ac-bi+ci2-di3)f,
and that the cofactor of x-(a+b+c+d+e) in the next case was a
quadratic in x2, he surmised the existence of a general proposition
including the three, and stated his surmise at the meetings of the