MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
103 
Theory of Eduction. 
25. It has already appeared from § 4 that the operator 
operating on z jiroduces an invariant of index unity. But for the purposes of this 
operation z may be regarded merely as an unchanging quantity, and, therefore, it 
may be replaced by an absolute invariant (of index zero); and, when the operator acts 
upon an absolute invariant, there results a new invariant, of the next higher rank in 
the differential coefficient of the variable and of index unity. 
We can, however, make the operator an absolute invariant, for the index of Ag is 2; 
and, therefore, 
is an absolute invariantive operator vffiich, when it operates on an absolute invariant, 
generates a new absolute invariant of next higher rank. 
The operator can evidently be applied any number of times in succession, so that, 
if I be an absolute invariant. 
is an absolute invariant for all values of the index r. 
Similarly, the result of operating upon any absolute invariant with the operator 
^ dx Ej 
is to give a relative invariant of index unity ; and, if we are considering simultaneous 
invariants in two variables z and z , then 
are absolute invariantive operators, which, when applied to absolute invariants, 
produce absolute invariants. 
26. Thus, in the case of a single dependent variable, we have 
B-QgAg 
C = 
