1921-22.] Concomitants of Quadratic Differential Forms. 189 
foregoing invariant Ig by combinations of 1 ^, 12 ^ 3 , 14.155 expression 
of this discriminant, say □ , is 
D, 
M, 
L, 
T, 
S, 
N 
M, 
E, 
K, 
u, 
0 , 
P 
L, 
K, 
F, 
V, 
R, 
Q 
T, 
u, 
V, 
C, 
G , 
H 
S , 
0, 
R, 
G , 
B, 
.1 
p, 
Q, 
H, 
J, 
A 
In passing it may be remarked that for the line-complex 
Dpp’ -f- Fpg^ -1- Apg2 ^ 2MpiP2 + 2 DhP 4 4 - 211^2^4 , 
the other terms being absent owing to vanishing coefficients, we have 
while 
[, = o, 
F 
= 0 , 
□ = ABF 
c, 
T, 
u 
T, 
D, 
i\l 
u, 
M, 
E 
l2 = A2D2 -f- B2E2 + C2 FM- 2 ABM 2 + 2 AFT'-^ -f- 2BF [J2, 
F^AD-fBE + CF. 
32. We thus have a system of twelve members, composed of six line- 
co variants and six invariants ; there might arise a question as to whether 
these can be declared algebraically independent of one another or (what is 
the equivalent) can be declared subjected to a single relation satisfied in 
virtue of the relation 
the left-hand side of which also is covariantive under our system of equa- 
tions. The question is most simply resolved by considering the system 
when the quadratic line-complex has a canonical form, say for the 
canonical form 
DP42 + FP2‘^' -I- FPg-^ + CP4‘^ -F BP.'^ -F APg 2 + 2 NPiPg -F 2OP2P., -F 2VP3P4 , 
which includes six of the eight canonical forms due to Weiler.* For this 
form the various expressions are 
?^ = DPi2+ APg2-F2NP,Pg+ . . . 
(the omitted terms in u and in the succeeding concomitants can be written 
* Math. Aim., vol. vii, pp. 145-207. 
