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

JACOBIANS (JACOBI, 1844-1845)

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and thus know as above that
A = el,... -a.x
where al, a,... involve no differential-coefficients taken with
respect to xi or with respect to x,.
The observation made in the course of the first proof that
A, A1,..., An are themselves functional determinants leads
Jacobi to the conception of "partial functional determinants"
on the analogy of partial differential-quotients. The fundamental lemma then becomes viewable as the analogue of
'a_ af= O
Dx y
or, in Jacobi's words, "gravissimam manifestat analogiam
determinantium functionalium et quotientium differentialium
partialium."
Apparently this recalls to Jacobi another analogy of the same
kind, which he had omitted to draw attention to in his paper
of 1830, when the first two cases of the lemma had been
originally enunciated by him. The proposition involving the
said analogy he now generalises thus:-If f, f1, f,.., f, be
expressible as series the terms of which involve only powers of
the variables x, xl, x2,..., xl, the functional determinant does
qot involve a term in x-lx-1 x2-1.... x,-1. In support of
it he has only to point out that the functional determinant is
equal to
3(fA)+,(,/Al) + (.fAn)
ax ax1 Dax
and that the development of the 7cth term of this expansion
cannot contain a term in 1/xk 1.
After referring to a possible application of the lemma in
connection with definite multiple integrals, Jacobi concludes ~ 2
by returning to the lemma itself and throwing it into a third
form originally announced in 1841 (De determ. funct. ~ 9).