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

JACOBIANS (DONKIN, 1854) 247
DONKIN, W. F. (1854, February).
[Demonstration of a theorem of Jacobi's relative to functional
determinants. Cambridge and Dub. Math. Journ., ix. pp.
161-163.]
The theorem or identity in question is that of the year 1844.
The functions being, u2,.., u, and the independent variables
X1, x2,..., x), Donkin says that the functional determinant
may be represented by
Dax1 31 t%' 'x2 n
32 32 32
ax1 3x2 axn
ax ' ' ' a
it being understood that each symbol of differentiation is
it being understood that each symbol of differentiation is
operative only upon that one of the functions which has the
same suffix as the upper D of the symbol. As a consequence of
this he considers that the non-zero member of the identity
sought to be established would be
x^1 a2 *3Z *...U2 3 (A)
2 2....
3 32 an
ax3 3^ " ' 3ax,
Dx1 Dx2 2x
where the upper D's of the first row being now without a suffix
are supposed to be no longer restricted in their effect. As,
however, the unrestricted symbol 3/Dxi is held to be equivalent to
32 D+3+,,n
axi Dxi aOx
the determinant operating on u2u3... un has the first element of
each column equal to the sum of all the other elements of the