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

SKEW DETERMINANTS (NATANT, 1867)

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introduced as an equivalent for 'zero-axial skew.' When the skew
determinant has not a zero diagonal, he calls it 'improperly symmetral' (uneigentlich symmetral).
VELTMANN, W. (1871): MERTENS, F. (1876).
[Beitrage zur Theorie der Determinanten. Zeitschrift f. Math. u.
Phys., xvi. pp. 516-525.]
[Ueber die Determinanten, deren correspondirende Elemente a,
and a,, entgegengesetzt gleich sind. Crelle's Journ., lxxxii.
pp. 207-211.]
The two parts of the fundamental proposition regarding zeroaxial skew determinants are here established from a direct
consideration of the terms of the final development of a general
determinant as set forth by Cauchy in 1841. This, as we have
seen, had already been done by Cayley himself in 1860: Veltmann
and Mertens, however, arrange their matter with greater care
and in much fuller detail.
CLIFFORD, W. K. (1873?).
[On Pfaffians. Math. Papers, pp. 535-537.]
In ~ 1 of this posthumously printed note Clifford defines a Pfaffian
by means of what, following Hankel, he calls 'alternate units.' *
Such a definition was of course to be expected in view of the
analogous definition of a determinant involved in the work of Grassmann (1844) and Cauchy (1847). Forming a linear function of
the binary products of four alternate units, I., 12, 3, 14, say the
function
t12t112 + aq13t13 + a14144 + '(23t23 -+ a24t214 + at34t3t4,
and raising it to the second power, we obtain
2 (a12Ca4-a13a24 + a14a23) 1112t1t4,
in which the Pfaffian of the second degree appears as a factor.
It is the general result including this which Clifford uses, his
* Cauchy's 'clefs alg6briques' (1847): Sylvester's 'polar umbrae' (1862):
Hankel's ' alternirende Einheiten' (1867).
M.D. III. S