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

42 HISTORY'OF THE THEORY OF DETERMINANTS

correspondant 'a une permutation quelconque de ces nombres, la fonction,
~(l 2. n)
(oti I de'signe la somme de tons les termes qu'on obtient en permutant
ces nombres d'une manie're quelconque) est ce qu'on nomme DMerrninant."
It is readily seen that this is much more genera] than any definition in use up to that time, and that it agrees with the ordinary
definition only when the f unction (1 2... n) takes the particular
form X. P - X.,, or XI.X2... X,,. Further, it is not the same
generalisation as we are familiar with from Cauchy's great
mremoir of 1812, where determinants are viewed as a special
class of alternatirtg symmbetric functions. This is shown quite
clearly by the only other case brought forward by Cayley,
viz., the case where the f unction (1 2... 'a) is given the form X12.34 Other examples, not given by Cayley, are~4 a3 i.e.2 a123 -a132 - a213 ~a 9L31 + a312-a.21'
1~~2 *q** a12 - 13 -a23 a3.2
a23 a23 a32 a 13 a 31 a12 a21
There is also in the same paper a more direct contribution
to the theory of general determinants, viz., the theorem afterwards 'associated with Cayley's -name, and which,-to use later
phraseology,-gives the expression for a determinant in terms of
its own devertebrated coaxial mninors and its primary diagonal
elemenuts. In the actual wording of the description of the theorem
it is, not unnaturally, applied to a skew determinant only; but
there is clearly nothing in the nature of the case to confine it to
this special form. The description is"En effet, soit S2 le dtterminant gauche dont il s'agit, cette fonction
pent eftre presentee sons la forme
2 = 20 + S2 1X\ + 22A22 ~. + 21?A11X22 +.
oS2~ est ce que devient 2 si Al, 'Xn re'dnits 'a zero 2 est
011 0 ~ ~ ~ ~ 1'22'.sn
ce que devient le eo~3fflicient de A,, sous la me'me condition, et ainsi de
suite; c'est 'a-dire: 2~ est le determinant f orme6 par les quantite's A,, en
snpposant que ces quantite's satisfassent anx condition (2), et en
donnant At r les valeurs 1, 2, n. 21 est le determinant forme'
pareillement en donnaut 'a r, s les valeurs 2, 3,...,n91; 22 s'obtient eni
donnant A r,) s les valeurs 1, 3,..., m1, et ainsi de suite: cela est aise'
de voir si l'on range les quantite's i,., en forme de carre'"