376 Sir William Rowan Hamilton on Equations of the Fifth Degree. 



function If'. Indeed, it is not pretended that a full account has been given, in the 

 present paper, of the reasons which Professor Badano has assigned for believing 

 that the twenty-four quantities which have been called h are all symmetric* func- 

 tions of the five roots of the equation of the fifth degree ; and that those quanti- 

 ties are connected by certain relations among themselves, which would, if valid, 

 conduct to the following expression for resolving an equation of that degree, ana- 

 logous to the known radical expressions for the solution of less elevated equations : 



<* = Ki + a/Kj -f V Kg -f \/k4 + V K3 — V'k4 

 + V{Vi,-\- -/Kg + A/i74rVI^-|- Vk, - V'kJ 

 + \/{k, 4- /Kg + 0V K, -f a/ Kg + e^V'lE^^T-T^} 



+ v/{K3 + -v/k« + e^v'K, + 1/K3 + e^^^;:=wT,}. 



But it has been shown, in the foregoing articles, that at least some of the relations 

 here referred to, between the twenty-four quantities h, do not in general exist ; 

 since we have not, for example, the relation of equality between h^ and Hg, which 

 would be required, in order to justify the substitution of a single symbol K4 for 

 these two quantities. It has also been shown that each of these two unequal 

 quantities, h^ and Hg, in general changes its value, when the arrangement of the 

 five roots of the original equation is changed in a suitable manner : and that h,, 

 •H,3, H,9, are also unequal, and change their values, at least in the example above 

 chosen. And thus it appears, to the writer of the present paper, that the inves- 

 tigations now submitted to the Academy, by establishing (as in his opinion they 

 do) the failure of this new and elegant attempt of an ingenious Italian analyst, 

 have thrown some additional light on the impossibility (though otherwise proved 

 before) of resolving the general equation of the fifth degree by any finite combi- 

 nation of radicals and rational functions. 



* " Dunque le H sono quantita costanti sotto la sostituzione di qualunque radice dell' equa- 

 zione." To show that the constancy, thus asserted, does not exist, has been the chief object pro- 

 posed in the present paper ; to which the writer has had opportunities of making some additions, 

 since it was first communicated to the Academy. 



