MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
11 
Substituting now in X', in Jq^, and for Ug the values as given for Uj and Ug in the 
last two equations (so as to express all the aggregates of coefficients proper to the 
rank 3 in terms of Pg, P^, Pg and to leave the residue of terms—if there be such a 
residue—as a function of u^, u.,, Ug, u^) and gathering together like terms, we find 
P " 
•2^8 
P 
8 
X' + 27<Pg - I ^ + I Voi = 27u./Pg 4- h t; + 9..,P,P 
_ 
2v., 
Hence we have 
Z — I 3 Qe 1 
and it follows that the first educt of U 3 (= Q^g) when reduced hij means of Ug is 
functionally equivalent to the invariant Qjg. 
33. In the preceding investigation the Jacobian of the function Aq and any other 
function of the series in § 16 entered. The following formulm, interesting in 
themselves, are of use in a verification that Z actually satisfies all the differential 
equations which are characteristic of an invariant:— 
For m > 2 , 
and, for = 2 , 
and, for m > 2 , 
and, for m = 2 , 
SAm-o . ,, hA,„_3 
= m(m-l)(q f-’- + A,„_. 
Ai 
3A,| 
Ao 
0 A,„ _ 2 
dp 
0A„ 
dq - 
^‘2 '!A - 
op 
0A(, 
0? - 
— m {m 
— m {m 
2pq 
— 
1 ) (pfp + A ..,-8 
0A,„_3 
Cq 
- 1)P 
h 
j 
and, for all values of m, 
