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

SKEW DETERMINANTS (BALTZER, 1857)

283

Baltzer's proof that the rational integral function H, which is
the square root of A, changes signs when two suffixes, r and s,
are interchanged is a simplification of Brioschi's, the operation
and even the notion of differentiation being dispensed with.
The function resulting from the change being H' he concludes
like Brioschi that
H2 = H'2;
also the aggregate of the terms in H which contain as. being
acsB, say, he infers as Brioschi does that B cannot be affected by
the change, and that therefore a,.sB will be altered into at,,B
or -arsB. Here, however, he brings the demonstration quickly
to a satisfactory end by saying that since some of the terms
*of H' are thus seen to differ in sign only from the corresponding
terms of H, the equation H= H'2 shows all of them must so
differ; and this is what was to be proved.
Jacobi's notation for the function H is then introduced, the
formal intimation being that (1,2,3,.., n) is used to denote
the aggregate whose fiJst term is ca2a34,..., an, c and hose
square is A. The other value of /A is thus of course representable by (2,1,3,.., n), or (2,3,..., n,1), or.... As this implies
also that
A.= ~(2,3,..., r-1, r+1,..., n)
we have now the means, so far as symbolism is concerned, of
removing the ambiguity from the various terms of the identity
JA = al2N/A'22~A,,+ 3 33 +., *lna/A'n.
As for the knowledge necessary to use the symbolism aright,
Baltzer's dictum is that the sign taken to precede
(2,3,..., r-l, r+1,......,3. )
in substituting for JA',, must be such that the equation
J/A',,./JA's = A,.
will be satisfied; and this he proves will take place when the
sign-factor of (2,3,..., r-1, r+ 1,..., n) is (-1)r. By hypothesis, he says, the left-hand side
-. (- 1)+(2,3,...,r-l,r+1..., ).(-)(2,3,..., s-1,s+1,..., )
= (- 1)_+s(2,3,..., r- l, I+ 1,.., ~)(2,3,..., s- l, s+ 1,.., I),