MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
85 
Every Irreducible Invariant must involve Zq^ and 
For every irreducible invariant f satisfies the five equations ; if then it be indepen¬ 
dent of we have df/ dz^-^ = 0. Since there is no term in it which involves Zq^, and 
since there is a single term involving Zq^ in AQf= 0, viz., z^i {b/jdz^Q), we must there¬ 
fore have bf jdz^Q, i.e., the function must be independent of z^^. From = 0, it then 
follows that bf jdz^Q and both vanish; from A^f = 0, it then follows that 
df /dzii and O/’/O^ooboth vanish, and, therefore, that f involves no difierential coefficients 
of the second order. Proceeding in this way to the successive orders, it appears that f 
involves no differential coefficients whatever ; so that it cannot be an invariant, other 
than a constant or 2 . 
12. Proceeding now to the consideration of invariants which involve difierential 
coefficients of the third order as the highest, and denoting them for convenience by 
a, h, c, d {= z^Q, z^^, z^i, z^q, respectively), we have, as the subsidiary equations 
necessary for the construction of tlie general solution of A^f = 0, the set 
dp dq dr ds dt da db dc dd 
q 0 2s t 0 Zb 2c d. 0 " 
To deduce that general solution, eight independent integrals of the subsidiary set 
must be obtained ; bearing in mind the character of the invariants (§11) ultimately to 
be arrived at, we take these integrals in the form 
Wi = q, 
== t, 
Wg = qs — pt, 
= (fr — 2pqs + pH, 
% ~ ^^3 
u^ = pd — qc, 
Urj ;= p^d — 2pqc -{- q^h, 
Uq = qP d — 3p~qc -fi 3pqHj — qhc. 
Any solution of the equation Ag/’ = 0 can be expressed as a functional combination 
of u-^, . . ., itg; thus 
rt — S'' = * ■ — 3 
7 7 o 'Hrl* 
hd — c^= — ^ 
{ad — hcY — 4 {ac — h^) [hd — c^) = 
(upig — UpirjY — 4 (upiri — U^) {UpiQ — UjH 
and so on. 
