386 MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



" Equation adjointe " of LAGUANGE ; they are all transformable by a substitution 

 similar to that which transforms the original dependent variable, viz., multiplication 

 by some power of dz/dx ; and they possess the property that all combinations of them, 

 similar to those by which they are constructed, are expressible explicitly in terms of 

 variables of the set. From this set of dependent variables there are deduced functions 

 of them and their differential coefficients possessing the invariantive property ; and, 

 again, from the aggregate all composite functions are excluded. These functions are 

 entitled identical covariants. 



Finally there is obtained a third class of functions possessing the invariantive 

 property, and involving in their expressions the dependent variables and the 

 coefficients of the differential equation ; and those functions are excluded from the 

 aggregate, which can be algebraically compounded by means of functions occurring 

 earlier in the class, of invariants, and of identical covariants. These functions are 

 entitled mixed covariants. 



For purposes of simplicity and of distinction between these classes of functions 

 there is an advantage in considering, as the ground form, a differential quantic 

 (being the sinister of the differential equation) rather than the differential equation 

 itself; for, in the case of identical covariants of order equal to and greater than that 

 of the equation satisfied by the variable in question, they can by means of the 

 equation be changed into mixed covariants. It is necessary to mention this both 

 here and later when the functions occur ; but, beyond this mention, further notice is 

 not taken of the possible fusion of the two classes of functions. 



10. The general aggregate of concomitants of the differential equation is taken as 

 including these three classes of functions, and later in the memoir it is shown to be 

 complete ; and the expression of every function is only implicitly general, that is, it is 

 given in connexion with the canonical form of the equation. A few of the prim- 

 invariants of lowest index are given for a semi-canonical form, but these are the only 

 exceptions. Again, my aim has been the investigation of invariautive forms from the 

 purely algebraical or functional point of view, and not from the geometrical ; I have 

 nowhere in this part adverted to SYLVESTER'S Reciprocants. The identity of some 

 classes of the latter with HALPHEN'S Differential Invariants is known*, and thus the 

 three species of covariantive functions constituted by Differential Invariants, Recipro- 

 cants, and Invariants of Differential Quantics have known points of connexion. 

 The discovery of further relations between them would be of great interest and 

 value. 



* SYLVESTER, " On the Method of Reciprocants as containing an exhaustive theory of the Singularities 

 of Curves," Nature,' vol. 33, 1886, p. 227. 



