respecting Equations of the Fifth Degree. 231 



such as a^' of the same order enter into the expression of the irreducible 



function b ; because this other radical would be obliged to be of one or other 

 of the two forms following, namely either 



"a" = (^0 + Ix a/) («^i + P3 ^2 + P3 U3), 



or else 



< = (?o + S'l a/) («i + ft ^2 + />/ U3) , 



Pj being the same cube-root of unity in these expressions, as in the expression for 

 a," ; and the product of the two last trinomial factors is symmetric, 



(w, + /)3* Wj + /)3 M3) (m, + />3 W2 + ft' W3) = 144 e^ ; 



a/ 



so that either the quotient — or the product aj' a," would' be a two-valued 



function, which would be known when a,' had been calculated, without any new 

 extraction of radicals. At the same time, if we observe that 



we see that the three values w,, u^, u^ of the three-valued function x^x^-\- x Xg 



can be expressed as rational functions of the radicals a," and a/, or as irrational 

 functions of the second order of the coefficients a,, a^, a^, O4 of the proposed 

 biquadratic equation, namely the following, 



' ^ i ^^ Po + Piai ^ «! > 



_ 1 $ j_ P3fflr , 144^2(^0 + piai' ) > 



«.-7^«2i- p^ + p^a,' + p3«." y 



so that if the biquadratic equation can be resolved at all, by any finite combina- 

 tion of radicals and rational functions, the solution must begin by calculating a 

 square-root a/ and a cube-root a/', which are in all essential respects the same 

 as those required for resolving that other equation of which Uy, u^, u^ are roots, 

 namely the following cubic equation : 



u^ — a^u^ -{- (^Oi a.j — 4a^)u-{- (4 a^ — o,^) o^ — Oj^ = ; 



VOL. XVIII. 2 I 



