386 Royal Society. 



the S3 T stems 



rfa ^-- =(00), rfaa P- =(Q1).. 



^'••=(10), ^-=(11).. 



and if *- naB **-ap 4 '**-^) + " 



then a/3 . . n a;j s,. . n . . n a/3 . . n a//3/ . . . . = l 



and U will be at once given by the equation 



U=(a/3... afi r ., ..)«/}.. U aiP ,..n.. A 9 . 



This method has been applied in the case of two variables to the 

 calculation of quadratic and cubic invariants. 



But in the case of two variables the coefficient a may be expressed 

 by a series of symbols with a single suffix, thus : 



a , a,. .. a , a, ... n , n,, .. n,,n,.. 



Now since A is of the same degree in I, m, and in I', m', the coeffi- 

 cients of all powers and products of a , a,, .. in which the degree of 

 /, m is above or below that of /', m' , will vanish in the invariant. And 

 from this and some other considerations it is shown, that not only 



mn 



n™U=0, t>n™U=0, .. |>T~ n™U=0, 



but that the coefficients of the invariant may be calculated from the 

 equations arising from equating to zero each term of the last equa- 

 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 n) have in- 

 variants of the degree m, then writing 



