1 72 Sir William R. Hamilton on the Argument of Abel, 



(™) . 



we may consider this quantity b as being also an irrational function of the n 



original quantities, a„ •••«„; in which latter view it may be said, according to 

 a phraseology proposed by Abel, to be an irrational Junction of the to'* order : 

 and may be regarded as the general type of every conceivable function of any 

 finite number of independent variables, which can be formed by any finite num- 

 ber of additions, subtractions, multiplications, divisions, elevations to powers, and 

 extractions of roots of functions ; since it is obvious that any extraction of a 



radical with a composite exponent, such as ^ f, may be reduced to a system 

 of successive extractions of radicals with prime exponents, such as 



Insomuch that the question, " Whether it be possible to express a root a; of the 

 general equation of the n"" degree, 



^ + a, j; + • • • + a„-i *• + a„ = 0, 



in terms of the coefficients of that equation, by any finite combination of radicals 

 and rational functions ?" is, as Abel has remarked, equivalent to the question, 

 " Whether it be possible to equate a root of the general equation of any given 

 degree to an irrational function of the coefficients of that equation, which func- 

 tion shall be of any finite order m ?" or to this other question : " Is it possible to 



(") 

 satisfy, by any function of the form b , the equation 



(m)n (m)n—l (m) 



b J^a,b + • • • + «„_i i> + «„ = 0. 



in which the exponent n is given, but the coefficients a,, a^, ... a are arbi- 

 trary ?" 



[2.] For the cases w = 2, « = 3, w = 4, this question has long since been 

 determined in the affirmative, by the discovery of the known solutions of the 

 general quadratic, cubic, and biquadratic equations. 



Thus, for w =: 2, it has long been known that a root x of the general qua- 

 dratic equation, 



x'^-\-a^a;-\-a^z=0. 



