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

ALTERNANTS (BRIOSCHI, 1857)

187r

n -row, - row2,,-,3
n1 row2 - rOW2l-4,
n - rown-2- row,.
BRIOSOHI, F. (1857, Oct.).
[Sullo svillippo di un determinante. Ainmali di Mat., i. pp.
9-11; or Opere miat., i. pp. 273-275.]
Brioschi enunciates -without proof the proposition that the
even-ordered determinant
1 1 1 1 1 1
- a, (xl -, a)2 x - a' (x 2 - (12)- a, (x 2 -
1 1 1 1 1 1
-2 a1 (X2 -a,1)2 X2 -a~2 (X2 - a2)2. -. a,, (X2 -a,,)2
1. 1 1 1 1 1
XZn -Q (X2 -a 1)2 X2 a- a2 (X2n - (1)2).. X- aq, (X2n- at )2
which is seen to be a function of 2n x's and qi a's, is equal to
114 (a,, a2 an) (XIL,) IIZ1 I X2 ~ X2.)n
where (x) = (x - x,)(x -X2)... (x - xI). He then obtains
similar expressions for the principal minors, namely, (1) for the
cofactor of any element in an odd-numbered column, and (2)
for the cofactor of any element in an even-numbered column, his
procedure being to express the minor in question in terms of
determinants like the original but of the order 2n -2 and then
to make the substitutions which are thus rendered possible.
PROUHET, E. (1857, Nov.).
[Questions 410, 411. Nouv. Annales de Mlath., (1) xvi. pp. 403,
404; xvii. pp. 187-190.]
By reason of the existence of the identity

2-'' cos'a = cos sa + s cos (s - 2)a~ + 4s(s - 1) cos (s - 4) a +...