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

WRONSJKIANS (FROBENIUS, 1873)

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and then performing the multiplication in row-by-column fashion.
From this identity, by substituting l/y, for y and by noting that
dc y, _ XV(y1, Y2 )
there comes the next result, namely,
1
W(y,.. *. ', *,, ) =,/-_2W (WW(yl, y2), w(y3, y),.., W (y, 2)},
a condensation-theorem, we may note, similar to Chio's for general
determinants. (Hist., ii. p. 80.)
An interruption here takes place in order to establish the theorem
of the previous paper, namely, that if W (y1,..., n) vanish identically the y's must be connected by a linear homogeneous relation with
constant coefficients. The method of proof is gradational, the case
for X functions being made dependent on the case for X-1. Thus,
it being given that
\W(y,..., y,X) = 0,
it follows from the identity just established that
W{W(ylI Y2), W(yI, y),. W -, W(y,, yX)} 0,
and therefore by hypothesis that
c2'W(y1,y2) +c3W(y, y,) -.. +c^ W(y],yX) = 0.
This, however, through division by yi2 can be written
w c b yi te g3 fod by + c i y2,ax yJ 'ax yJ ax yL
c1yl+C2Y2 +... + CYX 0
as desired.
Returning to the condensation-theorem, Frobenius by repeated
use of it shows that
W(y1,. ).., y,) = wi( y,, y )1 w (Y i1 Y20 Y4), w(y,, y,2. )
WYI> / Y2))