207 



tion above written, operated upon by the symbol >. This always 

 gives a system of linear equations for finally determining the coeffi- 

 cients. It does not appear possible fully to explain this method 

 without entering into more details than can be given in an abstract. 



From the general equations for so determining the coefficients, the 

 number and degrees of the distinct invariants belonging to any given 

 function may theoretically be determined ; and this has been done 

 for the simplest case, viz. quadratic functions. But the expressions 

 for higher degrees appear so complicated that an answer to this im- 

 portant question can hardly be expected from this method, in any 

 case not already known. 



The view of invariants here taken has suggested a series of other 

 functions of which invariants form the last term. These functions, 

 which I propose to call Variants, may be thus expressed. If 

 functions of the degrees r, s, . . (r, s, . . being less than ri) have in- 

 variants of the degree m, then writing 



,. d -. d 

 d= Jr ' rfro 



n 

 of which the last is a simple invariant, since, omitting the factor 



O a =ao y a =a, .. "b ao=an. 

 n 



With the exact relation between these functions and covariants I 

 am not at present acquainted. 



VOL. VII. 



