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



would be unnecessary to effect any new extraction of prime roots for the pur- 

 pose of calculating a^, which latter radical would therefore be superfluous. Nor 

 can any radical a," of higher order enter, because such radical would have 2 a^" 

 values, a/' being greater than 1, while any rational function f/', of two arbitrary 

 quantities x,, x^, can receive only two values, through any changes of their 

 arrangement. The exponent m, of the order of the sought irreducible function 

 b^'"-', must therefore be = 1, and this function itself must be of the form 



b' — b^ + b^ a/, 



bg and 6, being rational functions of a,, o^, or symmetric functions of the two 

 roots x^, Xj, which roots must admit of being separately expressed as follows : 



if any expression of the sought kind can be found for either of them. It is, 

 therefore, necessary and sufficient for the existence of such an expression, that 

 the two following quantities, 



.^1 + ^2 I ^1 - ^2 



should admit of being expressed as rational functions of a^, a^; and this con- 

 dition is satisfied, since the foregoing relations give 



We find, therefore, as the sought irrational and irreducible expression, 

 and as the only possible expression of that kind, (or at least as one with which 

 all others must essentially coincide,) for a root x of the general quadratic, the 

 following : 



'ft 



b still denoting any arbitrary rational function of the two arbitrary coefficients 

 a , a^, or any numerical constant, (such as the number t, which was the value 

 of this quantity b in the formulae of the preceding articles,) and the two separate 

 roots x^, a!j, being obtained by taking separately the two signs of the radical. 

 And thus we see d priori, that every method, for calculating a root x of the 



