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

JACOBIANS (GILBERT, 1869) 265
and the case where n is 3 is similarly made to rest on the cases n = 1,
n = 2. The work of verification, however, increases seriously with
each step.
IMSCHENETSKY, V. G. (1869).
[Sur l'integration des equations aux derivees partielles du premier
ordre. Archiv d. Math. u. Phys., 1. pp. 278-474.]
The third section of the first chapter (pp. 284-287) of Imschenetsky's memoir is devoted to a full and carefully expressed proof of
the theorem regarding the vanishing of the Jacobian. What is of
interest in it is the mode in which the Jacobian is shown by the
translator J. Hoiiel to be expressible as a single product. The
functions of x,..., x being ul,.., u, there is first deduced
by elimination
ul = f1(1. * * * *, ),
=2 = 2(1', X2.. ' Xn),
u3 = 0 3(1 U2 X3, 3.., n
in = ~0n(Ui, u2, *. * n-l, X),
and then unity in the form
1.....
' 5, 1
Du2. 11....e
n
is multiplied in row-by-column fashion by.(.,....,,)/a(xl,.... X),
the result obtained being
am, a^^ a^i
a~1 2
ax1 D3x2 'x3
3P2 a2
* x2 2X3
a03
*X3

I.