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

240 HISTORY OF THE THEORY OF DETERMINANTS

and the corresponding theorem when the functions are only
impnlicitly given, that is to say, by means of connecting equations
F(X, X2..., Ax2,., X f..., f) =,
F,,(x,,2*..., x,,,, f,2,.., A)= O.
The mode of treatment will be readily guessed from what has
gone before.
The same cannot, however, be confidently affirmed in connection with the theorem which expresses the functional determinant as a single product. This is found grouped under the
heading " Diverses formes que l'on peut donner a un determinant"
(fonctionnel), the said forms being obtainable on varying the
systems of increments assigned to the variables. In the first
example, the array of increments of the x's is taken to be
AAx 0.... 0
0 A22.... 0
~. o... *...
0 0.... Ax
which necessitates the other array being
fl AA __2 AA. At AAf
Xpi ax e VX
D x cn 2 a ' x
M A2X9 2 22. 2
DxX22 ax2 aD x2
'f af2 aff
nnxdn the a nxn
and the ratio of the determinants of the arrays to be
E fl La a. -. )
x1 2 nx'
To this there is added "C'est l'expression donnee comme
definition par M. Jacobi,"-a remark, however, equally applicable
when, as at the outset, the increments of the x's were taken in