104 MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
as the three irreducible invariants proper to the rank three, and they form the com¬ 
plete system of irreducible absolute invariants within this rank. Hence 
A = 
are absolute invariants proper to the rank four. But it is not to be inferred that 
A, B, C, A', B', C' constitute the complete system of irreducible absolute invariants 
within the rank four. 
Again, in the operator the quantities p and q are first differential coefficients of an 
unchanging quantity 2 ; they can be replaced by first differential coefficients of any 
absolute invariant I, and then 
A 
\dy dx dx dt/J 
is an absolute invariantive operator, the operation of which on absolute invariants 
produces other absolute invariants. Hence 
E = A. 
0 B a\ 
u 
dx 
dx dy) 
4 /SC 
d 
0C 0\ 
dx 
dx dy] 
a/SA 
d 
0A 0 ^ 
\9y 
dx 
dx dy j 
C, 
- W W A, 
-Tv; B, 
are absolute invariants proper to the rank four, and they are of the second degree in 
the differential coefficients of the fourth order. Among the six quantities A', B', C', 
D, E, F there is the relation, 
A'D + B'E -f C'F = 0 ; 
so that only five of them can be independent. 
27 . In any higher rank n let I, J, K, . . . be the invariants, absolute and irreducible, 
proper to that rank; let I' denote the absolute invariant educed from I by the 
operator 
and I,„ the absolute invariant educed from 
I by the operator 
A 
di/ dx 
dx dy) ’ 
