[ n ] 
II. A Class of Functional Invariants. 
By A. B. Forsyth, M.A., F.R.S., Felloiv of Trinity College, Cambridge. 
Received March 7 ,—Read March 15, 1888. 
The investigations herein contained are indirectly connected with some results in an 
earlier memoir.* In that memou- functions called quotient-derivatives are obtained 
in the form of certain combinations of differential coefficients of a quantity y dependent 
on a single independent variable x ; and they are there shown to possess the property 
of invariance for isolated homographic transformations of the dependent and the inde¬ 
pendent variables. It is evident, however, from their form that they do not constitute 
the complete aggregate of irreducible invariants for the case of a single independent 
variable; and the deduction of this aggregate and an investigation of the relation in 
which they stand to a particular class of reciprocants were made in a subsequent 
paper.t The present memoir is a continuation of the theory of functional invariants, 
the invariants herein considered being constituted by combinations of the differential 
coefficients of a function of more than one independent variable which are such that, 
when the independent variables are transformed, each combination is reproduced save 
as to a factor depending on the transformations to which the variables are subjected. 
The transformations, in the case of which any detailed results are given, are of the 
general homographic type; and the investigations are limited to invariantive deriva¬ 
tives of a function of two independent variables only, a limitation introduced partly 
for the sake of conciseness. The characteristic properties, such as the symmetry of 
the invariants and the forms of the simultaneous linear partial differential equations 
satisfied by them, can in the case of more than two independent variables be inferred 
from the properties actually given ; but many of the deductions made are necessarily 
proper to functions of only two independent variables. 
In the matter of notation it is convenient here to state that the independent 
variables are denoted by x and y, and the dependent variable by 2 . The general 
differential coefficient + is represented by but frequently the following 
modifications for the notation of particular coefficients are made, viz. : 
2 ), q replace z^q, : 
r, s, t replace z^q, ^ 03 : 
a, h, c,d . . . Z 30 , Zjj, z^. 2 , 203 • 
9’ ■ • • ^405 ^31’ ^ 22 ’ ^13> ^04* 
* “ Invariants, Covariants, and Quotient-Derivatives associated witli Linear Differential Equations,” 
‘ Phil. Trans.,’ A, 1888, pp. 377-489. 
t “ HomograpLic Invariants and Quotient-Derivatives,” ‘ Mess, of Math.,’ vol. 17 (1888), pp. 154-192. 
25.2.89 
