MK. A. li. i''0K8YTH ON A CLASS OS SUNCTIONAL INVARIANTS. 
89 
Now, for the first set of terms 
UqU^ — '^< 7 ^ = {<f {ac — h^) — fq {ad — he) + {hd — c") j 
where Hi is the Hessian of Aj considered as a groimd-forrn in <i and — p \ and the 
third set of terms is 
“sT = -<h„5. 
For the middle set of terms it is easily found, by the results already proved, tliat 
the terms within the bracket can be expressed in the form 
ip [q {r'^ d — 3?‘5c + (2s^ + rt) h — sta] + p {t~a — Msh + (26'“ -j- rt) c — md] ] 
= Lj , 
say ; so that we have 
Q, = H,+ 2U-4Hob 
14. Considering now the question as to whether each of the functions thus obtained 
will satisfy (i.) and (ii.), for one and the same numerical vaiue of the index X ^^^’O^ahly 
associated with it, we may proceed as follows. Writing the equations in the form 
aj=-ikf . (i'.), 
fli/=3X/.(ii'.), 
we have the following result : — 
/ = 
%f = 
Hl/ = 
X = 
A„ 
6A(j 
6Ao 
2 
Ai : 
9Ai 
OAi 
3 
Jqi 
l2J„i 
12.101 
4 
L., 
i 
1 
1-2L. 
121.. 
4 
I 
Hoi 1 
9Hoi 
9Hoi 
3 
Ho 1 
'-Ho 
6Ho 
2 
Hi 
12Hi 
12Hi 
4 
MUCCCLXXXIX.-A. 
N 
