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

SKEW DETERMINANTS (TANNER, 1878)

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are terms of [1234] and {123} respectively. The determinant
[123] had already presented itself to Bellavitis in 1857 (Hist., ii.
p. 96).
The two kinds of functions are not without a certain relationship,
as may be seen on expanding any one of them in terms of the
elements of the last row and their cofactors; for example,
[1234] = {123 y4 —124})y3+134)y2 —234}yl,
{1234 [123]a -[124] +[134de- d
{1234) = [123] -[124]- + [134] — [2341
ax4 D3 DX2 ax,
The relation which hose of the first kithatd bear to Pfaffians is considerably closer, it being understood that by Pfaffians in the title
of the paper is meant a particular type of Pfaffian, namely, Pfaffians
whose elements are of the form
Dys Dyr
_- -, or say rs.
Xe?)Xs '
Thus, expanding [1234] in terms of the minors of its first two rows
and their complementaries, we obtain
12 34- 13 24 + 14 23 + 23. 14- 24 -13 + 34 * 12,
i.e. 2{12.34-13.24+14 23};
so that, using a different notation for Pfaffians, we have the identity
k X D = 2! Y D2 Y_ Y y3 _Y3 D_ aY1
9x1 Dx2 Dx3 4 2 4
Y1 Y2 Y3 Y4 DY3 _y2 DY4 _Y2
3a a a a x9 ax3 3x2 3x4
ax1 9X2 9X3 x4 aY4 Dy3
Y1 Y2 Y3 Y4 Dx3 Dx4,
there being for [123456], [12345678],.... similar Pfaffian expressions with the coefficients 3!, 4!,.... respectively.
As may be expected from this relationship, Tanner readily discovers
analogues to the simpler properties of Pfaffians; and, in view of his
previous paper of the same year, it is equally natural to find him
formulating the 'complementaries' of these analogues. He also