1888.] 



A Class of Functional Invariants. 



431 



IV. "A Class of Functional Invariants." By A. R. Forsyth, 

 M.A., F.R.S., Fellow of Trinity College, Cambridge. 

 Received March 7, 1888. 



(Abstract.) 



The memoir is occupied with the investigation of a class of 

 functional invariants, constituted by combinations of the partial 

 differential coefficients of a function of more than one independent 

 variable. As the number of independent variables is limited to two, 

 partly for the sake of conciseness, the general definition of such an 

 invariant is that it is a function of the partial differential coefficients 

 of a dependent variable z with regard to x and y, such that when the inde- 

 pendent variables are changed to X and Y, and the same function <I> is 

 formed with regard to the new variables, the relation 



is satisfied, where 



$ =: J>*0 



• d(%, y) 



" 3(X,Y) 



The transformations for which detailed results are given are of the 

 homographic types : 



that 



J = *i , ' « 2 \ «3 (*3 + / 3 3 X + 73 Y ) 3 - 

 A . A>> ft 



7l > 72 > 73 



The characteristic properties of such invariants are — ■ 



(i.) Every invariant is explicitly free from the variables z, x, y, 



but necessarily contains p and q ; 

 (ii.) It is homogeneous in the differential coefficients of z, and is 

 of uniform and the same grade in differentiations with 

 regard to each of the independent variables ; 

 (iii.) It is symmetric or skew symmetric with regard to these 



differentiations ; 

 (iv.) It satisfies four form-equations, viz. : — 



