MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
79 
A function of the i^artial differential coefficients of z ivith regard to x and to y is 
called cm invariant if when the independent variables are changed to X and Y and the 
same function $ of the new variables is formed, the equation 
is satisfied, inhere 
4 ) = 
_ d (x, y) 
■"0(x/Yy 
5 . The following properties of irreducible invariants are easily obtained :— 
(i.) An invariant does not contain the dependent variable, nor either of the 
independent variables. 
(ii.) An invariant is homogeneous in tire differential coefficients. 
(hi.) An invariant is of uniform grade,equal to its index m, in differentiation 
with regard to x\ and of uniform grade, also equal to its index m, in 
differentiation with regard to y. 
(iv.) An invariant is either symmetric or skew symmetric in differentiation with 
regard to the independent variables. 
All these properties hold of Ag, the index of which is easily seen to be 2; it is a 
symmetric invariant, that is, it is unchanged if x and y be interchanged. 
The index of a symmetric invariant is an even integer; the index of a skew 
symmetric invariant is an odd integer. 
These properties hold for functions which ai-e invariants for any general transforma¬ 
tion, and not merely for the homographic transformations to be adopted; but the 
forms of possible functions, as well as the value of J, will be determined by the 
character of the transformation. And, in particular, for the homographic transforma¬ 
tion it is easy to prove that 
J = 
7n 72. 73 i 
(«3 + /^3^ + 73Y) 
6 . The method adopted for the determination of the forms of invariants will be to 
obtain the partial differential equations satisfied by them ; these equations can be ob¬ 
tained, as in a similar case,t by using the principle of complete infinitesimal variation. 
For this purpose it will be necessary to have the formulae expressing the relations 
between differential coefficients of ^ when the variables are transformed. This relation 
is given in the following proposition, the transformations being supposed any wdiatever. 
The special application to the homographic transformation will afterwards be made. 
* The grade of a term is the sum of the orders of differentiation with regard to one variable of the 
factors ; thus, the ic-grade of Aq is 2 ; the i/-grade is 2. 
t “ Homographic Invariants and Quotient-Derivatives,” ‘ Mess, of Math.,’ vol. 17 (1888), pp. 154-192. 
