338 REV. T, p. KIRKMAN ON THE THEORY OP 



MG=GM 

 is a derived derangement of G, we cannot have 



54. b. Suppose^ next^ that 



or that the operation of M on the subindices of the terms 

 (P) of # gives the same algebraic result with the operation 

 thereon of R. 



This is impossible, unless the effect of M on ^, which is 

 not also an eflfect of R on ^, be limited to elements carry™ 

 ing the same exponent. 



When this is so, we have, algebraically, 

 M'-^ = MR#, 

 and if W= 1 



which has been proved impossible, if the system of expo- 

 nents fulfils conditions above prescribed, at the end of 



(53). 



55. We have therefore the following theorems — 



Theorem M, Let G be any model group made with N 

 elements of L substitutions. Let 



be the product of N different poivers of the N variables 



^ = Pl + P3+.-+Px 



be the sum of the L terms made by executing on Pj all the 



substitutions (GP) o/G. 



ilN 

 The function # has ^p- values by the permutation of 



its variables under the fixed system of exponents. And the 

 number of different functions ^ of the same degree and 

 form, of which no one is a value of another, is the number 

 of groups equivalent to G. 



Theorem N. Let G be any model group of L substitu- 

 tions. Let 



