93 



* will have -^^^ values by the permutation of the N variables, 



JLi 



which can be formed upon G and its ^ 1 derived groups. 



And the number of different functions O, all of the same 

 algebraic degree, of which no one is a value of another, is the 

 number of groups equivalent to G. 



Theo. P. Let G be any group of L substitutions. 



Let 



i Xl 1-^ X;i • Xy 



Where 



be a term such that it changes in its algebraic value by the 

 operation MP performed on its subindices, where MG is 

 any derived derangement of G, and such that no group 

 equivalent to G gives the same algebraic function, 

 $ = r, + R+- + P^ 



with G. The function O has — — values, which are con- 



XJ 



structed on G and on its — — — 1 derived groups. 



Theo. Q. If two equivalent groups G and G^ of L substi- 

 tutions give for a certain system of exponents of XiX-z" x^ the 



same algebraic function <I>i, <I>i has not ^^-^— values. 



Theo. R. The number of groups equivalent to a group G 

 being given, and any system of exponents of Xi x-> • • Xy being- 

 selected, there is always an assignable number of algebraic 

 functions, 



= Pi + P2+-- + P^ 

 of the degree determined by that system, such that each 



7r N 

 function has '--— values, and that no function is a value of 



another. 



1 . There are three equivalent functions, and no more, of the 

 6th degree, made with four letters, which have each six values. 



