Functions formed on Groups. 5 3 



under aaa/3/3/3, 

 1 function of 2 terms and of 10 values ; 

 1 function of 6 terms and of 60 values ; 

 1 function of 12 terms and of 60 values : 



under oa/3/3/3/3, 

 1 function of 6 terms and of 60 values ; 

 1 function of 6 terms and of 10 values; 

 1 function of 3 terms and of 1 5 values' : 



under a/3,3/3/3/3. 

 1 function of 6 terms and of 1 value. 

 In all, 143 different functions. 



I believe that no difficulty, which has not been overcome 

 in the pages preceding, will be found in giving an equally 

 good account of all the many-valued functions constructible 

 on any tribe of maximum groups of n letters ; and this by 

 inspection of the paradigm of any one only of the equiva- 

 lent groups, whether transitive or not. 



P.S. — I must beg leave to withdraw my unguarded promise in 

 the last page of " Functions given by Groups." E.G., the 6-valued 

 function of Serret (KG. 16) has no value which can be so written, 

 that when exponents are effaced it shall be a single group. 



In my 87th year I can safely predict that this will be my last 

 contribution to the theory of Groups and their Functions. My 

 last word on the more difficult, but not less (over 34 years ago) 

 fully worked out, Theory of the Polyedra, is to be found (I have 

 a few copies to give) in Vol. XLIII. of the Proceedings of the 

 Literary and Philosophical Society of Liverpool. Vide'm C.R. the 

 Prize Questions, proposed at Paris in 1858, for i860 and 1861. 



Concerning the opinion of any living reader, upon what was 

 30 years ago by the R.S. permitted to appear of my Polyedra, I 

 have to say exactly what I said in P.G., p. 19, about groups and 

 their functions. 



