336 The Rev. Thos. P, Kirkman on 



out by d, and we deal with I^+i compared with G^^j just as 

 we dealt with it (inyo) compared with G^. We have next 

 to mark out, by d'^ (t-m) times, other groups, each of which 

 gives a function that has a value identical with G a- The 

 same group may be more than once marked by d^ and more 

 than once by d^ ; it matters not for how many reasons we 

 reject it as useless under S. We have now done with G^^, 

 and Oaa is one of our sought functions. 



We take next any group Gaaa that is not marked out 

 either by d or d'^, and repeat the same process till there is 

 no group in our table (A„) that is not marked out, except 



that we have handled and done with. 



If there are R of these so done with, our table (A„) has 

 given us R distinct Q-valued functions (5^^, &c., all under 

 the same 2, which will differ in the number of their terms,, 

 (article 13). 



When we have thus used every S under which r 

 groups in (Hw) have Q values, and have dealt in like manner 

 with every table (BJ (C„) &c., of which there are always 

 more than one for ;2>5, of equivalent maximum transitive 

 groups, we have found, and can dictate with all its values^ 

 every possible finite many-valued function of 11 letters (as 

 shown in article 15), which is not a mere product of simpler 

 functions. 



The cases of S under which Q-r groups in our tables 

 (B,0 &c., give functions of fewer than Q values, are all 

 simple, and the Q-r are easily laid aside. The proof will 

 appear in practice upon definite values of n^ the number of 

 letters to be handled. 



I have tacitly assumed in what precedes, that there is no 

 ^-valued whole and rational function of n letters that cannot 

 be formed by writing a certain S over a group and its k- i 

 derivates. Instead of piling up words to prove this nega- 

 tive, I content myself with promising that, being given 



