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

35 2 HISTORY OF THE THEORY OF DETERMINANTS

Jacobi's theorem of 1835 regarding Bezout's condensed eliminant
suggests the similar theorem regarding the bigradient eliminant,*
namely, if w be a common root of the equations
aox" +al x"1- +... = 0, b Xl + bx1-l..+. 0,
then the signed primary minors associated with any row of
(a,..., a1,,),
(bo,... b, )n,are proportional to
w +-l1 w +n-2,., w, 1.
In dealing with the highest-common-factor of A and B and with
the subject of elimination, Baltzer profits far less than he ought
to have done from the work of Trudi, whom indeed he does not
mention.
In conclusion, it is just worth noting that he now unreservedly
adopts the name 'discriminant,' but, strangely enough, introduces
it at first (~ 11. 19) not in its quite general sense, but in connection
with the bigradient and other forms of the resultant of f(x) and.f'(x).
ISE, E. (1873): JANNI, V. (1874).
[Sul grado della risultante. Giornale di Mat., xi. p. 253.]
[Sul grado dell' eliminante del sistema di due equazioni. Giornale
di Mat., xii. p. 27.]
The bigradient form of eliminant is here used in the establishing
of the proposition that if the coefficients ar, b., be functions of the
rt1 degree in one and the same variable y, the eliminant is of the (mn)tl
degree in the same variable. Janni's proof, though not quite so good
as it might have been, is the more interesting. The eliminant being
ao al a2 a. ao al a a
bo bl b2
b b, b2.. bo b 2,
* Gordan (1870), in quoting the two from Baltzer, says that mn of the primary
minors of the former eliminant are secondary minors of the latter. (Math. Annalen,;iii. p. 356.)