/ V respecting Equations of the Fifth Degree'. "' '" '233 



it is, therefore, included under the form (II. II.), and is either six- valued or 

 three-valued, according as it does not, or as it does reduce itself to a rational 

 function of «,, by becoming a rational function of t^ ; and in neither case can it 

 become a four-valued function such as x. We must therefore suppose, that the 

 sought irrational expression b , for a root x of the general biquadratic, con- 

 tains at least one radical a/" of the third order, which, relatively to the coeffi- 

 cients a^,a^,a^,a^, must be a square-root, (and not a cube-root,) of the form 



and, relatively to the roots x^, x^, x^, x^, must admit of being expressed either 

 as a twelve-valued function, with a six-valued square, which square Is of the 

 form (II. II.) ; or else as a six- valued function, which is not itself of that form 

 (II. II.), and of which the square is three-valued. This radical a/" must there- 

 fore admit of being put under the form 



<" = K" V,, 



the factor b " being a function of the second or of a lower order, and v^ being 



one or other of the three following functions, 



v^ = x^ + x^-x^- x^, v^ = x^ + x^-x^- x^, v^ = x^ + x^-x,- X,, 



which are themselves six-valued, but have three-valued squares. And since the 

 product of the three functions «„ is symmetric, - 



v^v^v^ = 64.e^, 



(e having here the same meaning as in the second article,) we need only consider, 

 at most, two radicals of the third order, 



a,'" = 5." t;. = -/ft/' ' (a; _ 4 a. + 4 ^^,), a '" = ft/'w. = V^6;"(«,' -^a^ + ^uj; 



and may express the most general irrational function of the third order, which 

 can enter into the composition of 6 " , as follows : 



)(( (/ /; /// // /// // /// '// 



2 1 2 



