MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL TNYARTANTS. 
117 
38. We might, if we pleased, carry the theorem further, for every simultaneous con¬ 
comitant satisfies the two characteristic equations 0, = 0, and is therefore 
expressible in terms of the simultaneous irreducible solutions of these two equations. 
Such irreducible solutions are necessarily functional combinations of Wj, u^, u^, . . . 
such as satisfy 0 ; and, as in the earlier cases of §§ 14, 17, it is easy to show 
that these irreducible solutions within the rank three are equivalent to the set 
= Aq, 
— Aj, 
{u^Urj -j- = Jqj, (dropping a factor 6) 
(wgWg — Urj^) u-^~^ ■= Hj, 
{u^u^ — Su^u^Ug -J- 2^7^) = Q, 
which are f, Ayy, <^, .9-, A, Q respectively in Gordan’s notatiou. Thus, every simul¬ 
taneous concomitant within the rank three can be expressed algebraically—though 
not necessarily rationally—in terms of these six quantities. 
The actual expression can be obtained by a development of the method adopted in 
the preceding theorem. It is first necessary to replace the covariant by its value in 
terms of iq, ii^, u^, . . . ; then to substitute by means of the equations 
Ui. - Aq, 
Ug — Aj, 
u^Uf.Ug = -f (miJqi + 
Ur^i^ui = U'lhi^Q + (tqJoi (iqJoi + ^qA^), 
for u^, ^q, Wg, lig, u~, Ug. The result, we know, must appear as a function of Aq, Hq, A^, 
Jq^, Hj, Q ; and, therefore, the terms involving Ug will disappear, and the factors 
will cancel. 
For example, in the case of Hq^, = p (Gordan) = (Salmon), we have, dropping 
the factor 12 and using L^, which is | Hq^ 
= — u^Ug — 2ugUrj • 
= ^ {ni + 
2 (7^iJoi + ^^3Al) -f- q yy {W;^^A.q^H^ -f- (hqJp^ q-%Ai)^} 
A, 
Ag A;^ 
_ /Aj -pj I Aq „ Jqj“ 
so that 
l^AqA, = + Jq,^. 
