72 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
respectively. The transformed independent variables are denoted by X and Y; and 
quantities bearing to them the same relation as the foregoing bear to x and y are 
denoted by P, Q, . . . . And in three different instances it has been necessary, 
for the sake of uniformity of notation for similar successions of quantities, to use 
different symbols for the same quantity occurring in different successions; these are 
= -L (§§ 3. 12). = - Ai (§§ 12, 13), Mia = A (§§ 16, 17). 
The general results of the memoir may be stated as follows :— 
Every invariant is explicitly free from the variables themselves, viz., the dependent 
and the two [iii\ independent variables ; it is homogeneous in the differential coefficients 
of the dependent variable ; it is of uniform grade in differentiations with regard to 
each of the dependent variables, and it is either symmetric or skew symmetric with 
regard to such differentiations. 
It satisfies six [m^ + iii\ linear partial differential equations, all of the first order, of 
which four \yn^'\ are characteristic equations and determine the form of the invariant, 
and the remaining two are index equations and are identically satisfied when the 
form is known and the index is derived by inspection from the form. 
Every invariant involves the two [in] differential coefficients of the first order. 
The following results relative to irreducible invariants derived from a single 
dependent variable z are given :—The invariants can be ranged in sets, each set being 
proper'"' to a particular rank. There ]S no invariant proper to the rank 1 ; there is 
one proper to the rank 2 ; there are three invariants proper to the rank 3 ; and, 
for a value of n greater than 3, there are n-\- 1 invariants proper to the rank n, which 
can be chosen so as to be linear in the differential coefficients of order n. Every 
invariant can be expressed in terms of these irreducible invariants ; and the expres¬ 
sion involves invariants of rank no higher than the order of the highest differential 
coefficient which occurs in that invariant. 
In the case of irreducible invariants, involving differential coefficients of tivo 
dependent variables, it is shown that there is a single one proper to the rank 1, and 
that there are four proper to the rank 2. 
Some eductive operators are given ; and in one case the eclucts are discussed so as 
to select those of the invariants thus obtained which are evidently reducible. Some 
general results analogous to reversor operations are derived. 
Finally, it is shown how the theory of binary forms can be partly connected with 
the theory of functional invariants ; for functional invariants are expressible in terms 
of the simultaneous concomitants of a certain set of quantities, viewed as binary 
quantics of successive orders in q and — qi as variables. 
[Note added December 5, 1888.t—The invariants in the present memoir are distinct 
* An invariant is said to be proper to the rank n when the highest differential coefficient of z 
occurring in it is of order n. 
t This addition is due to a desire which has been expressed that some indication should be given of 
the difference between tlie functions considered in the present memoir and invariautive functions of 
