184 
Proceedings of the Royal Society of Edinburgh. [Sess. 
In addition to these, we have the integrals 
A , the discriminant of 2 ; 
4 
^ universal concomitant ; 
r=l 
PjPg + P 2 P 5 + P 3 P 4 , a universal concomitant, which in relations 
is always to be equal to zero ; 
so that now we have, as integrals, 
4 3 
n,n,,H 2 ; A, A,, A,; A; 
r=l s=l 
twelve in number, while there are only nine functionally independent 
integrals. There is no difficulty in verifying the three relations 
nn^ - = A{Aa + 2A,(^2 VX,) + 2A}, 
AA 5 - Aj2 = S 2 n 2 , 
which thus reduce the number of independent integrals to nine. The 
complete system will be regarded as composed of 
n,n,; A, A,; A; ^U^X,.; 
r=l S=1 
Every concomitant of the quantic can be expressed algebraically in 
terms of these. Thus there is a concomitant which has hg^ — 2fgh + ch^ 
for its leading coefficient,* and is of the second degree in x and in p ; 
it is easily proved to be 
Sa - . 
The only invariant of the form is A. The only point-covariant of 
the form is the form itself, 2. The only line-covariant of the form 
is A. The only plane-covariant of the form is II. The three covariants 
Sj , II 4 , Aj are usually called mixed concomitants. 
29. We now proceed to obtain a partiall}^ complete set of concomitants 
of the quadratic line complex alone, viz. the line-concomitants, which 
involve the line-variables only, and its pure invariants. We still have 
fifteen equations of the type 
w _ P ^ _ p ^ 
0; 
for the present purpose, these equations involve six variables P and 
twenty-one constants (with one relation) — that is, twenty-seven quantities 
in all ; and therefore a complete system of concomitants of the specified 
* See the memoir quoted in § 26 (footnote), at p. 431. 
