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

SKEW DETERMINANTS (CAYLEY, 1846) 25

255

As has been pointed out elsewhere, the title of the paper is
quite misleading, the real subj ect being the construcetion of a
linear substitution for the transformation ofx2+ X2 + 3 +..
into e + e22 + e32 +... All that can be said in defence of the
inaccuracy is that skew determinants are made use of in obtaining the desired substitution. The proper place for giving an
account of the contents of the paper is thus under the heading of
'orthogonants,' if we may so name the determinaints of an
orthogonal substitution.
CAYLEY, A. (1847).
[Sur les determinants gauches. Orelle's Jourri., xxxviii. pp.
93-96; or Collected Math,. Papers, i. pp. 410-413.]
Here the title and contents agree. At the outset the former
definition is repeated, and then for a particular kind of skew
determinant, viz., those in which the condition
=rs- -Xs, (1)
is to hold even in the case where s and r are equal, " on pour
lesquels on a
Xr,s 'Xs, (r #4 s), X7,,,. = 0,, (2)
the name 'skew symmetric' ("gauche et syme'trique ") is set apart.
The reason for this is evident on the statement of the first
theorem, which is to the effect that any skew. determinant is
expressible in terms of skew symmetric determinants and those
elements of the original determinant which are not included in
the latter. "En effet," he explains,
"soit U2 le determinant gauche dont ii s'agit, cette fonction pent eftre
presentee Sons la forme
Q= EI20 + E~1X11 ~ E22k2 +.. + ~212X11IA22 +.
oiu 20 est cc que devient ~2 Si Xk1l A22,... sont re'duits 'a zero, f2, est
ce que devient le coefficient de X11 sons la me"me condition, et ainsi
de suite; c'est 'a dire, f2, est le determinant forme' par les quantite'S
Xr,,s en supposant que ces quantite's satisfassent aux conditions (2) et
en donnant 'a r,.s le valeurs 1, 2, 3,..., Ini; ~21 est le determinant
forme' pareillement en donnant 'a r, s les valeurs 2, 3,...,;22