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.