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

JACOBIANS (CAYLEY, 1847)

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may say that When n n-ary rth-ics vanish, their Jacobian and
each of its first differential-qutiients will vanish also.
The connection of this with the problem of elimination can be
indicated in a few words. Jacobi's determinant p being of the
third degree in x,1 x2, x3, its first differential-quotients are like
f1, f2, f3 linear in x12, X22, 32, X2x3, 31 X1s2; and consequently
the resultant is at once obtained as a six-line determinant.
CAYLEY, A. (1847, February).
[On the differential equations which occur in dynamical problems.
Cacnbridge and Dub. Math. Journ., ii. pp. 210-219; or
Collected Math. Papers, i. pp. 276-284.]
This is a short exposition of Jacobi's elaborate memoir of 1844
with considerable variation in the details. The portion (~ 1)
which concerns us is of course that referring to the "fundamental
lemma." This is established in its third form, the proof, like
that originally given by Jacobi, being dependent on the theorem
DR _ x7,
Dx1,
but differing in appearance, mainly because of the use of
differentials.
BERTRAND, J. (1851, February).
[Memoire sur le determinant d'un systeme de fonctions. Journ.
(de Liouville) de Math., xvi. pp. 212-227; abstract in Comptes
Rendus.... Acad. des Sci. (Paris), xxxii. pp. 134-135.]
Recalling how Jacobi had insisted on the marked analogy
between a functional-determinant and a differential-coefficient,
Bertrand at once intimates the adoption of a new definition of
the former, which in his opinion makes the analogy still more
striking, and from which the properties of the determinant are
deducible like mere corollaries.
Save that A and 8 are used where Bertrand without distinction
uses d, the following is the definition:-If f, f2,..., f be