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

232 HISTORY OF THE THEORY OF DETERMINANTS

Since, however, (i, k)f1 = - (I, i)f1 and (i, i)f = 0, we can apply to
the latter the general proposition that if aik be any quantities
whatever such that aik =- aki, aii = 0, and Hi stand for
aao, + al, + aai,ax axl axn
then
aH MH1 aHn
J 1.+, + =
Dx Da^ ax1
and so reach the first of our aims as desired. There only then
remains to show that the lemma holds in the case of two
variables, and this is unnecessary because it is then identical
with the familiar proposition
Df1 - Df1
ax y ay ax
In addition to this gradational proof Jacobi gives one of a
different kind. Since Ai, he says, does not involve differentialcoefficients with respect to xi, it follows that A and -DA
cannot involve differential-coefficients taken twice with respect
to any one variable. Further, second differential-coefficients
taken with respect to different variables xi, X, cannot occur
anywhere save int
DA~ DA.
x~ 't'aA,
axi ax,
All we have got to show therefore is that the cofactor of
3Ai aAk
in -i+ - vanishes. To do this we express Ai in terms of the
elements of one column and their cofactors, say
Ai= f + a.. +... +,fn
U1D+Ua+, +anDxk
D-
* The reason for this, of course, is that - + i =0.
aXi DOiVk
It would have been well to make clear here that every term of the final
expansion of DAxi contains one and only one second differential-coefficient.