GROUPS AND MANY-VALUED FUNCTIONS. 339 



/yja /y)B /vjV , . /y>0 



(^^^, /3=y, ■ ■ O'^o, 

 be a product wTiicli changes value algebraically by the 

 operation MP^ on its subindices, when MG is a derived 

 derangement of G, and such a jyroduct that no group equi- 

 valent to G gives the same function 



with G. 



UN 



The function $ has —^ values by the permutation of 



the variables under the fixed system of exponents, which 



UN 



values are formed on G and on its -^ i derived groups. 



Theorem P. Let F be any function symmetric or not of 

 the N variables XyX^x-'Xj^, which does not comprise the 

 term P^ above described, which gives ^ = Pi + P2+ • •, by 



JJN 



the group G, having -^ values. 



Let 



Pi + Fi=Si, P^ + F^^Sa, &c. 

 be what 



P + F=S 

 becomes by the substitutions of G. 

 The function 



{$} = Si + S,+ S3+--+S^ 



has —:f— values. 



56. It is of importance that we should demonstrate also 

 the following 



Theorem Q.. If tioo equivalent groups G and G' give by 

 any system of exponents the same function ^, this function 



UN 



^ has not -y— values. 



Li 



Let us suppose that the given system of exponents is 

 written over the elements in G and in G', and that 



G = „G' = „IlGE-i 

 where the symbol = « affirms algebraic identity. 



