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



that they tend to discourage the hope of resolving any general equation above 

 the fourth degree, by any similar method. And in fact it has since* been shown 

 to be impossible to express any root of any general equation, of the fifth or any 

 higher degree, as a function of the coefficients of that equation, by any finite 

 combination of radicals and rational functions. Yet it appears to be desirable to 

 examine into the validity and import of an elegant system of radical expressions 

 which have lately been proposed by Professor Badano of Genoa, for the twenty- 

 four values of Lagrange's function f referred to in the last article; and to in- 

 quire whether these new expressions are adapted to assist in the solution of equa- 

 tions of the fifth degree, or why they fail to do so. 



6. In order to understand more easily and more clearly the expressions which 

 are thus to be examined, it will be advantageous to begin by applying the method 

 by which they are obtained to equations of lower degrees. And first it is evident 

 that the general quadratic equation, 



A-* — A^ -f- B = 0, 



has its roots expressed as follows : 



x' = a-\-^, x" =ia — ^', 



a not here denoting any root of unity, but a rational function of the coefficients 

 of the given equation (namely t}a), and /3^ being another rational function of 

 those coefficients (namely j^A^ — b) ; because by the general principles of article 



1., when m = 2 and n := 2, we have - = 1, so that the function (x' — x"y is 



n 



symmetric, as Indeed it is well known to be. 



7. Proceeding to the cubic equation 



X^ — AX'^ -\- BX — C = 0, 



and seeking the values of the function 



f = (or' + ex" -f e' x"'y, 



in which 6 is such that 



e^ + + 1 = 0, 



* See a paper by the present writer, " On the Argument of Abel," &c., in the Second Part of 

 the Eighteenth Volume of the Transactions of this Academy. 



