respecting Equations of the Fifth Degree. 207 



general quadratic equation as a function of the two coefficients, by any finite 

 number of additions, subtractions, multiplications, divisions, elevations to powers, 

 and extractions of prime radicals, (these last extractions being supposed to be 

 reduced to the smallest possible number,) must involve the extraction of some 

 one square-root of the form 



a:=Vh\a^-4>a,), 



and must not involve the extraction of any other radical. But this square-root 

 a I is not essentially distinct from that which is usually assigned for the solution 

 of the general quadratic : it is therefore impossible to discover any new irra- 

 tional expression, finite and irreducible, for a root of that general quadratic, 

 essentially distinct from the expressions which have long been known : and the 

 only possible diffisrence between the extractions of radicals which are required 

 in any two methods of solution, if neither method require any superfluous ex- 

 traction, is that these methods may introduce different square factors into the 

 expressions of that quantity or function^, of which, in each, the square root a' 

 is to be calculated. 



[14.] Proceeding to the general cubic, 



a? + a^ x^ + a^x + a3=. 0, 



we know, first, that the three coefficients are symmetric functions of the three 

 roots, 



«. = ~ (^, + ^. + ^3)* a, = iP, ^« + ^, ^3 + ^, ajg, 03 = - X, x^ 0^3, 



so that we cannot express any one of these three arbitrary roots x^, x^, x^, as a, 

 rational function b of the three coefficients a^, a^, a^; we must therefore inquire 



whether it can be expressed as an irrational function b , involving at least 

 one radical a/ of the first order, which is to satisfy the two conditions, 



and 



«/ = F.' («.» «.' "3) ; 



the functions f^ and f/ being rational, and the prime exponent a/ being either 

 2 or 3, because it is to be equal to the number of values of the rational function 



VOL. XVIII. 2 F 



