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

SKEW DETERMINANTS (SCHEIBNER, 1859) 287
lished. Naturally following on this comes the proof (p. 156)
that each of the non-coaxial primary minors is the product of
two Pfaffians, the result being written in the form
Ap = (p+l,..., 2m, 0,..., p-l)(q+l,..., 2m, 0)..., q-1),
where the suffixes of the elements of the original determinant
are 0, 1,..., 2m. On a later page (p. 158) it is shown that a
similar proposition holds when the original determinant is of
even order, namely,
Ap = (-1)(o,,,..., 2+)(+,...,p-,+,..., q-1).
Cayley's theorem regarding a "bordered" skew symmetric
determinant thus appears broken up into two parts.
The paper concludes with the suggestions that a skew symmetric determinant should be called a Wechseldeterminante, that
its square root should be called a Halbdeterminante, and that the
latter should be denoted by
a01 a02 a3.... aoP
aC12 a13.... alP
C23.... 2p
CCp - 1,p
an expression which would thus be an alternative for (0,1,2,..., p)
and which would vanish for even values of p.
SOUILLART, C. (1860, Sept.).
[Note sur la question 405 et sur une composition de carres..
Nozv. Annales de Math., xix. pp. 320-322.]
Souillart's subject is the skew determinant
a b c d
-b a -d c
-c d a -b
-d -c b a,
and his observations are (1) that it is equal to
(a2 + b2 + c2 + d)2,