96 ALGEBRAIC INVARIANTS
etc. Replacing x\ by y2Z2.-y-i.z2, X2 by yazi — yizs, and X3 by
y\Z2—y2Zi, we get
(№)Xi =y v Zt-y[Z„
(Zvt)X2=ytZi—y&,
(tot)XB=y& n —y&.
Our relation for a covariant K of order co now becomes
S (product of factors y f , z € , . . ., z f ) = *),
each term on the left having X+co factors with the subscript £,
etc. Apply the operator F to the left member. We obtain
a sum of terms with one determinantal factor (afiy), (apy) or
{ayz) =a x , and with X+co — 1 factors with the subscript £, etc.
The result may be modified so that the undesired factor (afiy)
shall not occur. For, it must have arisen by applying F to
a term with a factor like and hence (by the formulas
for the Xt) with a further factor z v or z$. Consider therefore
the term Ca$ n y$z n in the initial result. Then the term
— CafinynZs must occur. By operating on these with V t
we get C(a(3y)z v , —C{a^z)y v , respectively, whose sum equals
C{(Pyz)a v - (ayz)p„\ =C(/3 x a v -a x /3 v ),
as shown by expanding, according to the elements of the last
row,
«1
Pi
y 1
z 1
«2
P2
y 2
Z2
0:3
Ps
ys
Z3
<*V
Pr,
y%
z*
The modified result is therefore a sum of terms each with
one factor of type (a/3y) or a x and with X+co —1 factors with
subscript £, etc.
Applying V in succession X+co times and modifying the
result at each step as before, we obtain as a new left member
a sum of terms each with X+co factors of the types (a/3y) and
a x only. From the right member we obtain nK, where n is
a number + 0. Hence the theorem is proved.