183 
1921 - 22 .] Concomitants of Quadratic Differential Forms. 
from taking the function 
2(Xi + AX/', X 2 + XX/, X 3 + XX 3 ", X 4 + AX/) 
for any arbitrary quantity X and picking out the coefficients of X^, X^, X*^. 
Also E = 0 is the equation of a quadric surface on which the point 
X^ , . . ., X4 lies; is the condition that the points Xj , . . X^ and 
X4", . . ., X4" are conjugate with respect to the surface X^O; and X2 = 0 
is the equation of a quadric surface on which the point X^", . . X^" lies. 
We are accustomed to the tangential equation of a quadric; so we 
have a plane-covariant 
11 = (All, ^22 5 ^33’ -^44’ ^23’ -^315 ^12’ ^14? ^^24 > ^34 ^ ^1 J ^2’ ^3’ ^ 4 )“" 
where the coefficients A^^ have the significance assigned in § 22. And then 
we have 
ni = ia„n, = 
obtained by selecting the coefficients of and in the function 
n(Ui-t-/xUi , Ug-i-ftUg , U3 4 - ft 113^^, Ui-i-/xUf ). 
Also 11 = 0 is the condition that the plane Ui , . . U4 touches the quadric ; 
rii = 0 is the condition that the planes Ui , . . U4 and Uj", . . U4" are 
conjugate with respect to the quadric; and is the condition that the 
plane Ui", . . U4" touches the quadric. 
Next, there is a line- co variant A. We write 
B, = ad- P , 
\i — hd — iiP , 
Q, = cd -iP , 
d = be -f - , 
Q = ca- cp - , 
i = ah-lP, 
li = hn — gm , 
with the relation 
and we have 
g = df~ mn , 
P = gl-a7l, 
\i = dg - nl , 
q — am - Ih , 
\ = dh — Im , 
r = hm - hi , 
I 
II 
S = bn -fm . , 
\ = hf - bg , 
t=fn- cm , 
II 
1 
0 =// - Im , 
V = gm - fl , 
n-t- 04 -v = 0, 
A = dVp + -f 2IP4P3 -t- 2 tP 4 Pi -F 2 SP1P5 -}- 2 nP^Pg 
+ ePp -H 2kP2P3 + 2UP0P4 -f- 20P2P5 + 2pP2Pg 
+ f PgS + 2 VP 3 P, + 2 rP^Pj + 2 qPsPj 
+ cP^2 + 2gP,P5 + 2hP^P3 
+ bP 52 + 2 jP 3 P 3 
+ aP/. 
And, with A, we have 
A, = p^A, A,==1VA. 
