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

JACOBIANS (TRZASKA, 1871)

267

TRZASKA, W. (1871).
[O pewnem zastosowaniu wyznacznik6w funkcyjnych. Pamietnik
Towarzystwa Nauk Scislych w Paryzu, i. pp. 113-121.*]
The theorem here dealt with is an extension of one noted by
Bertrand in 1864. In order to suggest the mode of proof it may be
conveniently formulated as follows: The necessary and sufficient
condition that thefunctions ul, U2..., un of the independent variables
x1, x2,..., x, shall satisfy p relations independent of these variables
is that the leading minor of the (n -p)t" order in the n-by-i array
aul Oul aul aul an,
Du2 Du2 Du2 3nU1 DU
ax, ax ' x ax.... Ox
x1 x2 fl-P X11-1)~1
aun-P 'a Un-p aun-p aull-P?Un —p
1ax ax ' Dax Dx' axx
1 2 n~~~f-p fl-P+1 fln
Du, Dufi 'u,, au,, au,,
ox ax ''~ ax,_ ax,,,, ~~ ax,
x1 x2... Dx Dx I... x,,
shall not vanish, but that all the minors of the (n-p +1)tlL order
obtained by bordering the said leading minor shall vanish.
On putting p = 1 and m = n+h we have the case given by
Bertrand, and on further putting h = 0 we reach the fundamental
case dealt with by Jacobi.
MINCHIN G. M. (1871).
[Elementary demonstration of a fundamental theorem. Quart.
Journ. of Math., xii. pp. 172-175.]
The theorem in question is that of which the conclusion is
a(Y1 IY2. Yn) aD(xI, x2..1 X,)
D(x, xI)., X,,) D(Y1 Y2., y,,)
and the demonstration is based on the fact that the Jacobian is the
-determinant of the set of equations which give the increments
*Or Baraniecki's text-book, chap. xii. ~118.