328 The Rev. Thos. P. Kirkman 



on 



another, from the above groups G^, G^d, &c., in our table 

 under S, which are turned by that 2 into the functions 



The above proof that no two of G^'s / charges fell on 

 the same group, is equally valid concerning the t charges of 

 ^bGdd . • . The reader may ask, why not valid concerning 

 the ^—m charges of G^, or the t—m-\-v charges of G^^, &c. 

 The reason is, that when there is no G, there can be none of 

 the reductions of the form {MM. p. 342, in the equation 

 below {e) ) 



in which we are to remember that (Bd^i = (B^ , and that I ^+1 

 contains Oj, the product 616^. 



13. We have now, retaining our table (A„) cleared of 

 outmarkings, to choose another S, which will determine 

 another index-group I^+i. The processes for determining 

 the number of distinct functions that our table (A„) will give 

 under its new 2, are in all things the same as above in art. 8, 

 •&c. ; and thus we can obtain all possible many-valued 

 functions constructive on (A^) under every different S. 



If we then proceed so to deal with every other table of 

 maximum equivalent transitive groups of n elements, we 

 shall obtain, without omission or repetition, the entire 

 number with all their values, of functions possible of n letters 

 that cannot be obtained as products of smaller functions. 

 There are always, for n>^^ several, and very soon many, 

 maximum transitive groups of various orders <^« ! 



We have next to consider the number of terms in our 

 won functions. This is determined in M,M. p. 343, by the 

 comparison of (B^^ with G^. 



It may also be shown as follows : — 



Since Jm+i, common to Q^ and It+i, is 



Jm+i =1 + Gi + 02 + . . . + Qmi (t^) 



