336 REV. T. p. KIRKMAN ON THE THEORY OF 



Let 



where PiPg- • are made by executing on the subindices of 



Pi the substitutions of Gr. 



UN 

 It is evident that # will have -y— values, if the func- 



J_i 



tions ^^^B' ' constructed on the derived groups AG, 



BGt'-, are all different algebraically. 



When all the exponents a/37- "^ ^'^^ different, the func- 

 tions ^ ^^^s' • will be all different ; for there are no two 

 substitutions alike in the series 



G, AG, EG-- 



And since no derived group DG is identical with any 

 equivalent of G, there will be as many distinct functions 

 ^, of which no one is a value of another, as there are 



groups equivalent to G. 



ITN 

 If then ^ has fewer than -^^r- values, we must have 



Ju 



either 

 or 



and the number of different exponents is < N. 

 53. a. Let us suppose that 



or that G and MG give the same function ^. 



Since the order of the exponents is the same in all the 

 terms (P) in $ and in ^,i/, and since there is no substitu- 

 tion common to G and MG. the terms (P) in ^j,j will 

 differ from those of ^ in the transposition of elements 

 which carry the same exponent ; that is, M exchanges 

 only elements carrying the same exponent. 



The derived group MG is either a derived derangement 

 of G, or the derangement by M of 



G' = MGM-^ 

 equivalent to G. 



