202 Sir "William R. Hamilton on the Argument of Abel, 



For example, when, in considering the irreducible irrational expression b" for a 

 root of the general cubic, we are conducted to the relation assigned in the last 

 article, 



«■' = F.' (^.' ^2' ^3) = Vff {p\-p^) (^ -^2) (^ -^3) (^2-^3) ; 



we can then at pleasure infer, either that the radical 0/ must admit (as a 

 radical) of two and only two values, if we have previously perceived that the 

 rational function f/ admits (as a rational function) of two values, and only two, 

 corresponding to changes of arrangement of the three roots x^, x^, x^, namely, 

 the two following values, which differ by their signs, 



± ^5 (p'-Ps) (^ -^2) (^ -^3) (^2-^3) ; 



or else we may infer that the function f/ admits thus of two values and two only, 

 for all changes of arrangement of x^, x.^, x^, if we have perceived that the radical 

 a/ (as being given by its square, 



which square is rational,) admits, itself, of the two values ± a/ which differ in 

 their signs. 



[12.] The conditions assumed in the last article are all fulfilled, when we 

 suppose the coefficients a^ &c. to coincide with the 71 original quantities a^ &c., 

 that is, when we return to the equation originally proposed ; 



a:" + a, a;""' + . . . + o„_i a; + a„ = 0, 



which is the general equation of the n'^ degree : so that we have, for any radical 

 a , which enters into the composition of any irrational and irreducible function 

 representing any root of any such equation, an expression of the form 



(*) (*) / \ 



the radical and the rational function being coextensive in multiplicity of value. 

 We are, therefore, conducted thus to the following important theorem, to which 

 Abel first was led, by reasonings somewhat different from the foregoing : 



