1^ 



In the theorem H which follows, I have exactly enumerated 

 a very large and well defined family of grouped groups, of 

 which only a small portion had been before detected, without 

 attempt at enumeration. 



When the exponents in the denominator of the derivant Q 

 are all unity, the groups contain only substitutions of the two 

 first species : in other cases there are substitutions of the third 

 species also. 



I hope that my theorems on the connection between groups 

 and the functions constructible on them, will be deemed of 

 some importance. 



I remember well the embarrassment I felt in asking myself 

 the simple questions; 1st. What is the group to which the 

 3-valued function ab-\-cd belongs? 2nd. What is the general 

 theory of the connection between the group and the function ? 



I know not where an answer can be found to these inquiries, 

 except in my own Memoir. 



The most important addition that has been made in France 

 to our knowledge of this subject since the days of Cauchy, is 

 a Memoir in Liouville's Journal, January, 1860. 



And I beg here to express my admiration of the demonstra- 

 tion there given of the existence of functions of jf variables, 

 -which have 



1.2-3 • • (p''—2) 

 values, whenever j? is a prime number. 



It appears to me that the use there made of the impossible 

 subindices invented by Galois, is a very high analytical 

 achievment. 



I had succeeded in both finding and completely enumerating 

 these groups for yz:= 1, before I had the pleasure of reading 

 the demonstration of the more general theorem. 



Proof of the confusion of ideas that has reigned may be 

 found in the Paper referred to in the " Journal of Liouville." 

 The corollary at page 20 is not demonstrated, neither is the 

 enunciation true; and there appears to be an oversight all 



