respecting Equations of the Fifth Degree. 229 



F = (x„ - a;^ + a;^ - x^) {x^ - x^) (x^ -x^).^ (x^ x^ + x^ x^), 



P = (^a + ^/3 - ^y — ^5) • -f (^a ■^13 + ^y^^)» 



v = {x^ — x^){x^-x^).^{x^x^-\-x^Xs); 



and the last-mentioned six-valued function, (II. II.), will have its cube two- 

 valued, if it be reducible to the form 



T-{a-\-b{x^- x^) {x^ - x^) {x^ - x^) {x^ - x^)(x^ - x^) {x^ - x^)} 

 X{x^x^ + x^x^ + P3' (^„ ^y + a^^ a;^) + ft (^„ ^5 + ^^ Xy) }, 



^3 being still an imaginary cube-root of unity. And the square of the two- 

 valued function (II. III.) will be symmetric, if it be of the form 



F = S (x„ - x^) {x^ - x^) {x^ - x^) (Xp — x^) (Xp - x^) (x^ - x^) . 



But there exists no other case of reduction essentially distinct from these, in 

 which the number of values of the square or cube of a rational function of four 

 independent variables is less than the number of values of that function itself. 

 Some steps, indeed, have been for brevity omitted, which would be requisite 

 for the full statement of a formal demonstration of all the foregoing theorems ; 

 but these omitted steps will easily occur to any one, who has considered with 

 attention the investigation of the properties of rational functions of three 

 variables, given in the two preceding articles. 



[20.] The foregoing theorems respecting functions of four variables being 

 admitted, let us now proceed to apply them to the d priori Investigation of all 



possible expressions, finite and irreducible, of the form b , for a root .r of the 

 general biquadratic equation already often referred to, namely, 



X* ■\- a^ x^ -\- a^ 3^ -{- a^ X -{- a^ := . 



It Is evident in the first place that we cannot express any such root x as a 

 rational function of the coefficients a,, a^, a^, a^, because these are symmetric 

 relatively to the four roots iF„ x^, x^, x^, and a symmetric function of four arbi- 

 trary and independent quantities cannot be equal to an unsymmetric function of 



them ; we must therefore suppose that m in b is ' greater than 0, or, in other 



