^^2 The Rev. Thos. P. Kirk man on 



and the charges brought by the standard G^ against certain 

 of its equivalents in what precedes, instead of being, that 

 each had under S a value algebraically identical with O^y 

 might as truly have been, that none had a value not a 

 value of (Btj, or lacked any value of (3a- But I am not sure 

 that the wider form of the charges made would better have 

 fixed the ideas of the student Certainly it would have made 

 no difference in the steps of demonstration or in the result 

 concerning the number and definitions of functions to be 

 found. 



15. No allusion to groups was made in the Question 

 proposed by the Academy of Paris, in 1858, for the com- 

 petition of i860. 



" Quels peuvent etre les nombres de valeurs des fonctions 

 bien definies, qui contiennent un nombre donn^ de lettres, et 

 comment peut on former les fonctions pour lesquelles il 

 existe un nombre donne de valeurs ? " 



I soon discovered, on making in 1859 my first acquaint- 

 ance with groups and their functions, that the groups were 

 themselves the functions. All possible finite functions of n 



letters with their -jr values, k being the order of any group. 



G of ;^ elements, are before us when every G is written out with 



its -77 — 1 derivates under every S in turn. If G is maximum, 



fC 



n/=kQ = l^Q, as in article 4. If G of order k is not maxi- 

 mum, it is a subgroup £- of some maximum G', whose Q 

 columns under S, each of L products, have for equal sub- 



n f 

 divisions the -^ values of the subgroup g, one subdivision 



being ^. And these -^ values can be dictated from those 



Q columns of the maximum G', which G' can be written 



{kily. i)=L), 



G' = ^ + a-ig + a^g + . . + a^g. 

 Thus it was clear to me that, to attempt to answer 



