1921-22.] Concomitants of Quadratic Differential Forms. 
Next, we have a line-co variant x, obtained by forming either 
187 
___ dv dv dv dv ^ dv dv 
or (what is the equivalent except as to an arithmetical factor) 
dll dw du dw dll dio dii dw dii d'W du dw 
^ 
it is quadratic in the variables, and of the fourth degree in the coefficients 
D, M, . . ., A. 
Next, we have a line-covariant y, obtained by forming either 
dll dx du dx dll dx du dx du dx du dx 
Fp„ ep, ^3‘‘"SP5 ^■''aFg W-i 
or (what is the equivalent except as to an arithmetical factor) 
_ dv dw dv dw dv dw dv dw dv div dv dw 
W,'^d¥, d¥\'^W^ dP^'^W^ 
this also is quadratic in the variables, and it is of the fifth degree in the 
coefficients. 
Lastly, as regards line-covariants, there is a line-co variant 2 ;, obtained 
by forming eithei* 
_ du di/ da dij du dy du dy du dy du dy 
Fp^'^&p, ^"’"aPg W^'^dP^ Ws'^dP^ atq’ 
or the same combination of v and x, or the combination 
d'lv dw ^ d'W dw ^ div dw 
dP^ dP 
dP, dP, dP, ap^ 
this line-covariant also is quadratic in the variables, and it is of the sixth 
degree in the coefficients. 
31. As regards pure invariants, they now can be obtained simply. 
The operator 
a2 
a2 
02 
dp-^dpQ dpodp^ ^Ih^Pc, 
is invariantive ; and tv, v, w, x, y, z are covariants. Hence 
Ii = V^, 
lo = VV , 
13 = , 
1 4 = , 
F = v?/, 
L. = , 
are six invariants of degrees one, two, three, four, five, and six respectively. 
