1921-22.] Concomitants of Quadratic Differential Forms. 191 
These equations can be regarded in either of two ways. We can ignore 
the relation 
0-2 + 0-4 + o-g = 0 ; 
and then there is no syzygy (linear or other) between u, v, w, x, y, z. 
We could also resolve the equations so as to express , cr^ , o-g linearly 
in terms of u, v, w, x ,y, z, and substitute the results in the foregoing 
relation ; owing to the symmetry, the coefficients would be invariants. 
It is preferable to choose the latter way. 
We may therefore summarise the results in a declaration that the 
required system contains 
(i) six invariants, independent of one another ; 
(ii) six line-covariants, connected by a single linear relation ; 
(iii) the permanent relation P^Pg -f- P 2 P 5 + P 3 P 4 = 0. 
An Algebraically Complete System of Concomitants: 
Differential Invariants. 
33. It is now possible to construct, with comparative ease, a complete 
aggregate of independent solutions of the full system of equations in § 25, 
which are fifteen in number. The quantities that occur in those equations 
are 
10 , the coefficients in the original quadratic form ; 
21 , the coefficients in the associated line-co variant ; 
14, the number of variables (point, line, plane) ; 
that is, 45 in all. Consequently, a complete aggregate of independent 
solutions of the system of equations must contain thirty members. 
We have already constructed a couple of limited aggregates. In the 
first of them, we had the concomitants which involved none of the 
twenty-one coefficients of the line-covariant ; they were nine in number, 
and were denoted by 
S,2,; n,n,; A, A,; A; 
of which, in particular, A is a quadratic line-covariant, also having twenty- 
one coefficients constructed out of the ten coefficients of the original 
quadratic form X. In the second of them, we had the associated line- 
covariant i^(P), and five other line-covariants v(P), 'Ip(P), «t(P), y(P), ^(P), 
the six being connected by a single relation in virtue of the equation 
