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

ORTHOGONANTS (GRAVELAA', 1875)

295

Salmon, seeking to shorten Cauchy's procedure of 1829, had used as
a substitute for Sturm's series of derived functions the series
L,(x), Lnl(x)).... L(), 1.
To this Gravelaar objects that, though the substituted series has
one fundamental property in common with Sturm's series, it does
not possess a further essential property, namely, the impossibility
of two consecutive members of it simultaneously vanishing. In
effect the objection means that Salmon did not, like his predecessor
Cauchy, consider the case where the equation under discussion has
equal roots.
The proof offered (~~ 10, 11, pp. 306-310) is of less interest than
the preparatory theorems on which it depends, namely, (1) that the
r"t differential coefficient of L,,(x) is, save for the arithmetical factor
(- l)rr, the sum of the coaxial minors of the (n-r)th order: (2) that
the necessary and sufficient condition for the equation L,,(x) = 0
having a real root a repeated m times is that all the minors of the
(n —m + 1)t" order vanish when x is put equal to a; (3) that if L,(x)
contain the factor (x-a)"1, then all the primary minors contain the
factor (x —a)m-1.
No proof of the first theorem is given, but it is readily derivable
from Trudi's form of Jacobi's theorem regarding the complete
differential of a determinant. In support of the second theorem,
but really proving little more than the converse, it is pointed out
that if L,(x) contain the factor (x-a)"', the putting of x equal to a
must cause all the differential coefficients of L,(x) up to and
including the (m-Il)t" to vanish; that therefore, from the first
theorem, with the help of a property of axisymmetric determinants,
all the coaxial minors of the (nz-m -l)t`l and higher orders must
vanish; and that thence, with the help of another property of
axisymmetric determinants, the other minors of the (n —m+l)th
and higher orders must vanish also. As for the third theorem, it
is first shown to be true of the primary minors that are coaxial,
and thereafter of the others by considering a two-line minor of the
adjugate of L,(x) and by assuming that a secondary coaxial minor
is divisible by (x-a)"'-1.