3i6 The Rev. Thos. P. Kirk man on 



2. The question — what are all the possible groups of n 

 elements ? — is exceedingly difficult. 



The question — how many distinct functions, with their 

 values, are constructive on the groups when given? — is trivial 

 in comparison with the former. 



When I have said what I think needful for the second 

 aim, I have no doubt that my new theorem, although it is 

 nothing but rigorous algebra, will have for the reader, who 

 loves analysis, a charm of magical surprise. 



3. The data for the discussion of a definite case of the 

 problem of many-valued functions are three. 



First, any system of Q equivalent maximum groups 



[M.M. p. 280, 283, 334) of n elements x-^x^x^ x^. For 



these elements let i 2 3 ... ;^ stand in what follows. 



Second, any chosen system S of ^^ exponents, not all 

 alike {M.M, 57, p. 341), viz. : — 



« = ^« +^3 +^y + +/^ = « + /^ + ^+ +p-n; (i) 



which is to be read a a's, b ]3's, c y's, &c., or a ^times, j3 ^times, 

 7 rtimes, &c., repeated. 



Third, an Index group I, determined, as we shall see, 

 by the system S. 



4. A maximum group G has, and can have, no derived 

 derangement GP = PG {M.M. 8, p. 281) by a substitution P. 

 If L be the order of G, i.e., the number of its substitutions, 

 and Q be that of its equivalents {M.M. p. 280) 



G, AGA-i, BGB-\ &c., 



G is also defined as maximum by the condition 



QL = «! 



For example of an equivalent (;/ = 4) 



1234 = G and 1234 = G' 

 2341 3142 

 3412 4321 

 4123 2413 



