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

216- HISTORY OF THE THEORY OF DETERMINANTS
current, and it is seen that the expansion in terms of the
elements of the last row and their cofactors leads at once to.
Allegret's result.
BRtIOSCHI, F. (1857).
[Solution de la question 350 (Wrouski). lNouv. Artraces de Math.,.
xvi. pp. 248-249.1
The problem having been set to find what Wrouski called the
"Aleph "functions * of the roots x1, X2'..., Xn~ of the equation
ao ~n cqalx71 + ct2xn-2 +... +at, = 0
in terms of the coefficients, Brioschi begins by saying that the~
rth of the said functions, being the complete homogeneous
function of degree r, is the coefficient of z" in the product
(I + XJZ+X12z2~ +. )(+X2z;+X2 2Z2~..). (+X~,Z+Xn 2Z2-[
(l- X1z)( - x2z)... (lxnZ)' r sy
He thus has
and therefore by differentiation
But having Also by a well-known theorem
q'z-) X + +2 +
-Xi + r-2z+ xi 3Z2~..
+ X2 + X2Z + X23Z~...
-81~S2z+SIZ2+...
* WRoNSKI, H. Inrodu~2ctionh 'a la Philosophie des 11iath~mcatique8
(pp. 65,..)vi~+ 270 pp., Paris, 181 1.