74 



The following theorems form part of a Memoir presented 

 to the Imperial Academy of Paris, in competition for their 

 prize medal which was offered in 1858, to be awarded in 1860. 

 The subject proposed is — 



" Quels peuvent etre les nombres de valeurs des fonctions 

 bien definies qui contiennent un nombre donne de lettres, et 

 comment pent on former les fonctions pour lesquelles il existe 

 un nombre donne de valeurs ? " 



No prize has been awarded, and the report of the Referees 

 gives no details of the Memoirs presented. Comptes Rendus, 

 Mar., 1861. 



The Memoir which I had the honour to present to the 

 Academy contains numerous results, which 1 believe to be 

 new, and which at least are so far important that they are 

 contributions, I hope, useful and pregnant, to a direct answer 

 to the prize question. 



I may be permitted to mention what appears to me to be 

 more or less important, as well as new, in my results. 



1. The enumeration of groups of forms already known, but 

 not enumerated. 



2. The definition and enumeration of large classes of 

 groups which were before, so far as I can learn, unknown. 



3. The discovery of the third and principal species of 

 substitutions in grouped groups. 



4. The determination of the number of equivalent m-valued 

 functions that can be constructed on a given group ; i.e., of 



functions of the same degree, of which none is among the m 

 values of another. 



The mathematician will remark that there is a little con- 

 fusion of ideas in the writings of the most recent French 

 investigators, about the relation of groups and functions. 



