respecting Eqitations of the Fifth Degree. 237 



and 



_ — a, Vi^ p'" f 1 _ —a, v, p"'ti 



^, retaining here its recent meaning ; or, at least, we must make suppositions, 

 and must employ expressions, not differing essentially from these. 



But all the radicals, a,', a,", a,'", a^", a'^, introduced in the present article, 

 agree in all essential respects with those which have been long employed, for 

 the calculation of the roots of the general biquadratic equation ; it is, therefore, 

 impossible to discover any new expression for any one of those four roots, which, 

 after being cleared from all superfluous extractions of radicals, shall differ essen- 

 tially, in the extractions that remain, from the expressions that have been long 

 discovered. And the only important difference, with respect to these extractions 

 of radicals, between any two general methods for resolving biquadratic equations, 

 if both be free from all superfluous extractions, is, that after calculating first, in 

 both methods, a square-root a,', and a cube-root a,", (operations which are equi- 

 valent to those required for the solution of an auxiliary cubic equation, ) we may 

 afterwards either calculate two simultaneous square-roots a/", a^", as in the 

 method of Euler, or else two successive square-roots a/", a/'', as in the method 

 of Ferrari or Des Cartes : — for, in the view in which they are here consi- 

 dered, the methods of these two last-mentioned mathematicians do not essentially 

 differ from each other. 



[21.] It is not necessary, for the purposes of the inquiry into the possibility 



or impossibility of representing, by any expression of the form &'•'", a root x of 

 the general equation of the fifth degree, 



x^-\-a^x^-\- a^s^-^-a^x^-Y a^x-\-a^^= 0, 



to investigate all possible forms of rational functions of five variables, which have 

 fewer than 120 values ; but it is necessary to discover all those forms which have 

 five or fewer values. Now, if the rational function 



F (Xj, x^, ajj, x^, x^) 



have fewer than six values, when the five arbitrary roots a;,, x.^, x.^, x^, x^, of the 



