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

238 HISTORY OF THE THEORY OF DETERMINANTS
functions of x1, x,..., x,, and the latter receive n distinct sets
of increments
Axl AX2.... x
A2X1 A2.... A2,,
AnX1 A Z1 *..v. A xnVn
with the result that the corresponding increments of the functions
are
Af Alf, l....
Afi A2J2.... A2f,,
Af. A.f.... A.
then the limiting value of the ratio of the determinant of the
second array to the determinant of the first array as the elements
of the latter array are indefinitely diminished is called the
determinant of the n functions. Since in the circumstances
mentioned
Akf = AfiAki + fA ~' + -A
Dx, ax2 ax,
for all values of Ic and i not greater than n, it follows from
the multiplication-theorem that the aforesaid limiting value is
equal to
afi 'af, 3./
Df1 Df1.... n,....
Dlx ~.2 1 ' ' Xn;
Jafn Df Df
3Z1 2x2 x 3
and, this determinant being independent of the increments given
to the independent variables, it is held that the definition is
legitimised. It might have been added that the name assigned
to the limiting value is also thereby justified.
The more important of Jacobi's results, eight or nine in number,
are then re-established, precedence being given to those regard