204 Sir William R. Hamilton on the Argument of Abel, 



discovered, for expressing any root of any such lower but general equation, 

 quadratic, cubic, or biquadratic, in terms of the coefficients of that equation. 



[13.] Beginning then with the general quadratic, 



0^ + a, X + Oj = 0, 

 let us endeavour to investigate, a priori, with the help of the foregoing theorem, 



all possible forms of irrational and irreducible functions b , which can express a 

 root X of this quadratic, in terms of the two arbitrary coefficients a^, a^, so as to 

 satisfy identically, or independently of the values of those two coefficients, the 

 equation 



jW 4- a. 6^"^ + a, = 0. 



The two roots of the proposed quadratic being denoted by the symbols x^ and x^, 

 we know that the two coefficients o^ and a^ are equal to the following symmetric 

 functions, 



«. = - (^. + ^J' <^. = ^x ^. ; 



we cannot therefore suppose either root to be a rational function b of these co- 

 efficients, because an unsymmetric function of two arbitrary quantities cannot be 

 equal to a symmetric function of the same ; and consequently we must suppose 



that the exponent m of the order of the sought function b is greater than 0. 



The expression 6 "" for a; must therefore involve at least one radical a/, which 

 must itself admit of being expressed as a rational but unsymmetric function of 

 the two roots x^, x^, 



< = f/ (^1, a:^), 



and of which some prime power can be expressed as a rational function of the 

 two coefficients a,, a^, 



the exponent a^ being equal to the number of the values 



F, (x,, x^), Fi (Xj, X,), 



