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

80 HISTORY OF THE THEORY OF DETERMINANTS

The first matter of real interest is reached on p. 1, where the
following theorem is given: 'Soient s la re'sultante de 'ordre i
forme'e avec les termes dz tableaua
ao ali... ci_i
ao a1.. * abo bl... bil
10 11 ~ ~
et s" resulttante de 'ordre i-1 formee avec les termies compris
dans le tableau
S(+aoobl) S(aob2)... S(+~a9bi)
S(~aoci) S(+ o~OA).. S(.aoc,_i)
S (~a o?) S (a02).. S(~aoi-).
s(+s(_+- c )... s(+_c,~i-l> i
La riesultante s" sera egale a s, au facteur pres ao2, en sorte
qu'on aura s"- =o-2 S."
This is one form of the theorem afterwards well known as
effecting the transformation of any determinant into one of the
next lower order. It may be viewed as a case of Hermite's
result of the year 1849.
On p. 17 particular cases cease to be considered, and the
multiplication of an array of i rows and 2i columns by a similar
array is taken up, with a result in accordance with that arrived
at by Binet and Cauchy in 1812. From this result, by specialisation, the ordinary multiplication-theorem is then deduced, and
with it (Chio's " theoreme ix.") the first part of the memoir closes.
The second part, which begins on p. 23, concerns the solving
of a set of 2n equations of a type which will be sufficiently
specified by giving the set where n = 3, namely,
x +y +z = da
xS +y1 +Z = d2
- + +2 + y2+ Z = dj
X5 + Y + 5+5 = g6
The connection of this with what precedes consists in the fact,
arrived at by Sylvester in his solution of the problem of the
canonisation of the quintic, that A, a, ~ are then the roots of
the equation in w