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

WRONTSKIANS (FROBENIUS, 1873)

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complementaries, we obtain by taking N in succession equal to
0, 1, 2,..., it- I the set of equations
YIZI +g21Z2 +... +YnZn = 0,
91,iZI +Y2,1Z2 +-... +YniZn = 0,
Yl,n —2Zi +Y2,n-s2Z2+. +. +Ynn- Zn = 0J
Yl,n-IZ1 +Y2,n-1Z2+ +?Yn,n2-1iZn
in which, as we may note for ourselves, the definition of the z's is
involved. If, for shortness' sake, we put
8sg for yz,aZ + 2,Z Z2,p +.+ YaznP
w ae thus know that
= -,0 = 3 '0,() 0n-),0 0,
and, 1.
But by differentiating s,- 1,0 once, X -,0 twice, and so on, it is readily
shown that more is known in regard to sg., that, indeed,
"~;F = if a+8 -Putting therefore a = 0, 3 = 0, 1, 2,, n-i, we have
zjyg +Zxy2 +. +Znyn = 0,
ZJ,1Yl/ +Z2,1Y2 +. +Zn,1Y = 0,
Zl,n-2Yi+Z",n2Y2+. +Znn —2Yn 0,
Zi, n-Y+Z2,nIY2.+.. +zn, lz-IY2?n
and are face to face with the interesting fact that what we have
reached is almost exactly producible from the previous set of
equations by the interchange of y and z. Important conclusions
follow of course with ease. In the first place, the determinant of
the new set being W(z,, Zn) and not being zero, it is known
that the z's are not connected by a linear homogeneous relation
with constant coefficients. Next, it is seen that the solution of the
set is
y- IW z-)