MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
IlG 
37. Hence we have the following theorem ; 
To express any simultaneous concomitant of 
{r, s, tXq, - pf, 
{a, b, c, dXq, — p>f, 
{e,f h, h, iXq, — p)^ 
in terms of the quantities u^, u^, u^, . . . , which are the irreducible solutions of 
A^f = 0, the equation characteristic of all these concomitants, it is sufficient to take the 
coefficient Cq of the highest power of q in to construct a new function Fq, which is the 
same combination of the coefficiejits of 
{u^, u^, ufXT — 
(— —pf^ 
as Cq is of the coefficie7its of the binary qualities, and to divide Fq by ... -m)_^ 
where m is the degree of ^ in q and — p, and I, V, I ",... are the degrees of Cg in the 
coefficients of A„_o, A;;/_ 2 , A,;//_ 2 ,... respectively. 
The theorem is illustrated by one or two examples which have already occurred in 
the reduction of Qj, Qg, Qy. Thus, for Hg = rt — s^, we have only one quantic entering 
into its composition, viz., Aq ; so that n = 2, / = 2 ; I' = 0 = I" ... , and m = 0 
hence, 
up • 2 • 3 Hg = U .2 — uf, 
that is, 
Hg {in^u<^ 
Again 
— Hg^ = r [qid — qc) — %s{p>c — qb) + t {pb — qa) 
= q{~ cr 2bs — at) + p {dr — 2cs + bt) ; 
so that two qualities enter. Thus, we have, for Hg^, 
n — 2, I = 1 •, 
n = 3, /' = 1 ; I" = r = ... = 0; 
m = 1 . 
Hence 
- 1 + 3 • ^ Hgi = -u^{- uf) + 2wpi3 -Uof- U^) 
— upp, + 2upi- + upi^ 
as before ; and so for others. 
