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

276 HISTORY OF THE THEORY OF DETERMINANTS

(3) If in a zero-axial skew determinant all the minors of the 2rt"
order vanish, so also must all the minors of the (2r —l)th order.
Holding these results to be of special importance in connection
with the main subject of his paper, he devotes a section (~ 5, pp. 242 -245) to the independent consideration of them. As a basis to start
with, it is affirmed that if a zero-axial skew determinant of even order
vanishes, so likewise do all its primary minors. The proof consists
in noting that in any vanishing determinant I a,, I we have
AaaAp = AaApa,
and that as regards the special kind here in question we have further
Aaa = =A,, A, = -Aga
the result thus being at once
Aa = 0,
as desired. Following on this comes the theorem: If in a zeroaxial skew determinant * there be a coaxial minor of the 2rtl order
that does not vanish, while all those coaxial minors formable from it
by appending two additional rows and columns do vanish, then all
the minors of the (2r+1)t71 order vanish also. This is reached by
applying the preceding theorem to the given vanishing coaxial
minors of the (2r+2)th order, the result being proof of the vanishing
of a sufficient number of minors of the (2r+l)th order to enable us
to assert the vanishing of them all. Following on this again comes
a direct deduction from it, obtained by noting that since all the
minors of the (2r+1)th order vanish, so also must all those of the
(2r+2)th order, and that therefore, by attending only to those of
the latter that are coaxial, we can affirm the vanishing of all the
Pfaffians of the (r+l)th order. There is thus obtained the analogue
in Pfaffians to the well-known theorem in determinants which has
just been used at the close of the preceding proof. It may be
formulated as follows: If in any Pfaffian there be a minor of the
rth order that does not vanish, while all those minors formable from it
by appending two additional frame-lines do vanish, then all the other
* As a rule Frobenius uses' schief' not for ' gauche' but for 'gauche symetrique ':
on p. 241 the word ' alternirend ' is also applied to a skew symmetric array.