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

246 HISTORY OF THE THEORY OF DETERMINANTS

functions Of Yl, Y2,... l Y n being greater than m, and the y's
be functions of xl r2,.... 1, then
DQk 'a rn),,. {Dj(iU'Ua~ 2... Un) D(yj9, YP2..Yp"r)}
where all, a2,.., a l and Yi'1 72'... I7nare fixed sets of
rn integers chosen, from 1, 2,..., n, and 3102 / -2,/3,, as any
such set whatever. Taking then what he considers to be a
particular case of this, namely,
D(Yal Ya2...y ar) __y~DYal Ya2...Yj ' D(X91 X2... SJ
he points out that if the sets al, a2,..., a,,, and Yi' 72', 7
be identical the determinant on the left is equal to 1, and that
on the other hand if even one of the y's be not included among
the a's the determinant will have a column of zeros and
Itherefore be zero itself. The generalisation aimed at thus is,
that If y1, y2,... I y,, be fu~nctions of x1, x2,.. x11, and in be
less than n, then
~~I, a(YY2. ). (XP, XP,... X = 1 or
1X2 XP11) D ( Y~ Y
accor,~ding as the a set of m integers chosen from, 1, 2,..., n i~s
identical or not with the y set. The illustrative example taken
is the case where n = 2, and the mnode of stating it is that
{(~~ ~ DY~)D~i Dx _ Dx -) = 1 or
according as a, /3 =p, q or not, it being understood that i, j is in
succession
2, 3;...; 2, n