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

SKEW DETERMINANTS (BALTZER, 1857)

2S8

thus showing, as he says (1) that the ratios on the left are independent of r, and (2) that, when the sign of one of the roots has
been fixed, the others are known (" dass durch das Zeichen einer
unter diesen Wurzeln die Zeichen der iibrigen Wurzeln bestimmt
sind)."
Turning now to the section specially set apart for the consideration of skew determinants, we find that it opens with
Cayley's theorem regarding a zero-axial determinant of even
order, the requirement being, as here worded, to prove that such
a determinant is the square of a rational integral function of
the elements. The proof is essentially the same as Spottiswoode's and Brioschi's, and differs from Cayley's merely in
that it does not begin with a determinant of a more general
form than is necessary,-a point which it is desirable to
insist upon, as Baltzer ignores the fact, and then does not
hesitate to say in a footnote that Cayley's proof "leaves
manifold doubts unrelieved." In fact the theorem which
Cayley proves is, that if c zero-cxial skew determinant of
odd order be 'botrdered' the resulting determinant is the product of two Pfaffians: whereas what the three others prove,
is the particular case of this in which the skewness extends
to the bordering elements.
The development with which the proof begins Baltzer writes
in the form
A = an11A1 - af, 1aisA',
rs
where A' is the cofactor of a,., in An, and r and s have the
values 2, 3,..., n. He then uses the fact that All is a zeroaxial skew determinant of odd order, and that therefore by a
preceding result
A' = A' = l xA'
rs sr rr ss S
so that there is obtained
a = ECa a1 C/A, A';
1's
and since in this aggregate the values possible for r are exactly
those possible for s, he concludes (without knowing the signs of