77 



through of this principle, that in order to prove that a function 

 has K values, it is not enough to show that it is invariable for 

 a certain number of substitutions. It is required also that it 

 shall be proved to be variable for all other substitutions. 



There is an infinite number of functions which answer the 

 definition of M. Mathieu, and are invariable for the substitu- 

 tions which he considers, which yet have not the required 

 number of values. 



This question of groups, contrary to the very frequent 

 custom of the Academy for a length of years, has been with- 

 drawn from competition, after being proposed once only, viz., 

 for 1860. I can only say that I regret this, and I wish that 

 the Academy had given one chance more to the improved 

 notation which they have done me the honour to commend. 



It is to be hoped that a question which has been long under 

 consideration, with no great result, and which, in my humble 

 opinion, has somewhat been retarded by the too great generality 

 of Cauchy's notation of substitutions, may soon receive a satis- 

 factory solution. 



The numerals in what follows refer to the articles of my 

 memoir, which has been so peacefully buried, I hope alive, 

 though without name or record, in the " dark unfathomed, 

 caves'^ of waste paper in the French Institute, 



Theorems on Groups. 



" Def. Let A^, A^ A,, be any arrangements of the same N 

 elements 1 2 3 . . N. 

 The operation : — 



executes on A^, the substitution ^, by putting for the element 



a in A^„ that which stands over a in the substitution. The 



result is the permutation A/, or, what is the same thing, the 



A A 

 product of the two substitutions, —^ .-fU in that order, where 



