102 MR., A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
so that Aq is an invariant of index 2 ; and similarly for A'g), 
= 6 A;, 
so that A'q is an invariant of index 2 , 
It is desirable to modify the forms of X and Y, so as to express them explicitly in 
terms of the quantities .... When the values of A, B, C. tq, u\, are 
substituted in X, and it is multiplied by — u-^, it takes the form 
qh' — 2pqs + 2 {qqr — {p>q + p q)'sqypt}, 
which may be denoted by ; and when exactly the same operations are applied to 
Y, it takes the form 
qh' — ‘Zpq's + 'p>h + 2 {qqr — [p)q + p'q) s' pp't'}, 
which may be denoted by ^'q. 
It is now easy to verify that 
= 6^0 ; 
so that ^0 is an invariant of index 2 ; and similarly that 
0 ~ 0 ’ 
0 ~ 0 j 
so that Q is an invariant of index 2 . 
24. The general result of the preceding investigation can be enunciated as follows :— 
Every simultaneous invariant of tivo functions z and z' of two indep)endent variables, 
■which involves no di^erential coejfeients of order higher than the second, can he 
exqmessed in terms of the five Irreducible invariants J {ofi index 1 ) and Aq, A'q, 
'^ 0 ’ ^^0 of index 2 ) where 
J = qxf — p'cq, 
Aq = q^r — ^Ipcqs + pH, 
A'o = cqV - 2p'q's' + p'H', 
^0 = (f'*' — +2rt' + 2 [qq'r — {pf +/(/)« + pqht}, 
gt'o = q'V — 2p'q's + p'H + 2 [qqV — {pq' + p'q) s' + 
and' p, q, r, s, t; p, q , r , s', t'; have their ordinary significations as pjarticd differential 
coefficients of z and ofi z'*. 
* It is easy to see that the invariant Aq, formed for 2 + As' is 
Aq + AHq + A'xI’q + A'^A'q. 
This remark is practically due to Professor Cayley. 
