GROUPS AND Many-valued functions. 337 



Suppose, first, that 



MG = G'M. 

 There is in $j,/ a term which differs tactically from 



'V = xlxlxl • 'X\, 

 by the exchange of certain elements 



for elements 



which carry the same exponents as before. 



The same function ^M) both in the tactical and in the 

 algebraic sense, is constructed on the derived MG and on 

 the derangement G'M; and G'M differs from G' only in 

 the exchange of certain vertical rows carrying the expo- 

 nent d for others carrying the same 6, and of certain 

 vertical rows carrying the exponent yu. for others carrying 

 fjb, &c. Consequently the same algebraical function is 

 constructed on G' and on G, if the same is constructed on 

 G and on MG which is no derived derangement of G. 



Hence if G be a maximum group which has no derived 

 derangement, and if the system of exponents be such that 

 no group equivalent to G gives the same function ^ with 

 G, we cannot have 



Suppose, secondly, that MG is a derived derangement 

 of G. 



The function ^j^j will differ tactically from ^ only in the 

 exchange of certain vertical rows of ^ for others which 

 carry the same exponents as before. This change leaves 



Pi = MPi 

 still true algebraically, where MPi is the result of operating 

 with M on the subindices of Pi . 



If then the system of exponents be such that no group 

 equivalent to G gives the same function ^ with G, and 

 that Pi changes in algebraic value by the operation MP^ 

 on its subindices, when 



SER. III. VOL. I. XX 



