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 peut 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 I 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. 
