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Academy, on the 25th of April last, offering to them this 
solution of their prize-question, on condition that they would 
award to me some kind of extra medal, after satisfying 
themselves that my work is what it pretends to be, the whole 
of that theory of which, in their proposals of I 808 , they 
desired to obtain only some notable extension. 
He received a formal reply, dated May 1 1 , informing him 
that the matter was referred to the Committee of the competi- 
tion of I860: he has since heard nothing more. 
The prize-question for 1860 was the following: — 
“ Quels peuvent etre les nombres de valeurs des fonctions 
bien dehnies qui contiennent un nombre donne de lettres, et 
comment peut-on former les fonctions pour lesquelles il existe 
un nombre donne de valeurs ?” 
A complete answer to this question, which is more than 
the Academy required in its proposals in 1858, must supply a 
clear and direct method of forming every possible transitive 
r- valued function of n letters not divisible by, nor containing, 
asymmetric function, with an enumeration of all its equiva- 
lent functions, that is, functions of like form and degree, of 
which no one is a value of another, and also with a demon- 
stration that no other such r-valued functions of n letters are 
possible. Such a complete answer is given in the Memoir 
of which this Paper is the abstract. 
To expect here definite results in general formulae for 
every value of n, is about as reasonable a thing as to ask for 
a complete exposition of the theory of numbers in general 
formulae. I have the honour to present the results demanded, 
both positive and negative, for values of n below n — - 11 , and 
to shew that all further results can be obtained with certainty 
by my method up to any value of n. It is a great satisfaction 
to me to be able to say, that the positive results, i.e., all the 
groups and functions thus far found, fall under the definite 
and general though defectively arranged theorems, of the 
treatise which I sent to Paris, and of the supplement, which 
