76 
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 p n variables, 
which have 
1-2-3 * • O n — 2) 
values, whenever p 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 n= 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 
