1 18 On the Algebraic Resolution of Equations of the Fifth Degree. 



But it remains to consider uv. 



5. From the equation (e) there spring at once 



/(0/[^)=S2+(*+* 4 )m+(* 8 + * 8 )» i 



f(L*)f(l 3 ) =@2 + (** + t> + (t + ft>. 



Now the product of the two first members of these two equa- 

 tions, or the function 



must manifestly be symmetrical with respect to u and v. It is, 

 in effect, expressible by 



(S2) 2 



+ (t + t 2 -r-t 3 + £ 4 )£2(«+t-) 

 + (*+* 4 )(**+ ft 8 ) (**+««) 



+ {(* + * 4 ) a + <** + l?)*}!». 



And if from this expression we eliminate ifi + v 2 by means of 

 the identical equation 



u 2 ~\- v 2 = (u -\- v) 2 — 2uv, 

 we shall immediately obtain 



Mmf{<?)m=L + \uv; .... (e,) 



in which L is rational and even integral relatively to A lf A 2 , . . A 5 

 (arts. 4, 5), and X is a numerical constant. 

 6. According-, then, to M. Hermite's theory, 



ought — on the hypothesis that the equation in x admits generally 

 of a finite algebraical solution — to involve no radical higher than 

 a cubic. His conclusion, however, appears to me to be quite 

 untenable. It follows, indeed, from what has been demonstrated 

 in my ( Essay/ that the fifth power of the function in question, 



[/WP IM)Y [/(< 3 )] 5 UWY, 



depends directly on an Abelian equation, and therefore involves 

 in its solution quadratic and cubic radicals only. But although 

 /W /(*") f( iS ) /vO i s > as we ^ as it s fifth power, six-valued, we 

 cannot, with the aid of Lagrange's theory of homogeneous 

 functions, establish a rational communication between the two 

 functions 



m m /(< 3 ) m> l/wp imi b um s imr, 



as I pointed out to Mr. Cayley in the Philosophical Magazine 

 for May 18G1. 



7. Setting aside, therefore, the objection raised from the theory 

 of Lagrange, we may now see clearly the way in which quintic 

 radicals enter into the expression for uv. 



