246 Sir William R. Hamilton on the Argument of Abel, 



or other ; and therefore, if we observe that f/'* has been concluded to be a 

 function of the two-valued form, we find ourselves obliged to suppose that the 

 five roots may be expresse d as follows, (if the supposition under inquiry be 

 correct,) 



x^ - e,' + e,' F/' + ej f/'^ + e,' f/- + e,' f/'", 



^^ = «o' + P, e; F." + f/ el F/'^ + /)/ el f/'^ + p,^ ei F/'^ 



^ = eo' + ft^e/F/' + p/e,' F/'^ + p^e/ <'\9'< ^r, 



^s = e: + p,' el F/' + p, el v^ + p/ el f/'=' + p/ e,' f/'*, 



^e = ^o' + P.' el F/' + p/ el F."^ + p/ e/ F/- + /,, e; F/'S 



Co', ... ^5' being either symmetric or two-valued ; but these expressions conduct 

 to the absurd result, 



5 el F," = x„ -J- p/ <r^ -1- /)3^ ^y 4- /)/ or^ -t- p, x^, 



in which the first member has only ten, while the second member has 120 

 values. We are therefore obliged to reject as inadmissible the supposition 



*i — J\-> 



and we find that no rational function of five arbitrary variables can have any 

 prime power two-valued, if its own values be more numerous than two. 



[22.] There is now no difficulty in proving, after the manner of Abel, that 

 it is impossible to represent a root of the general equation of the fifth degree, 

 as a function of the coefficients of that equation, by any expression of the form 



h " ; that is, by any finite combination of radicals and rational functions. 



For, in the first place, since the coefficients a,, . . . 05 are symmetric functions 

 of the roots x^, . . .x^, it is clear that we cannot express any one of the latter 

 as a rational function of the former ; m in S , must therefore be greater than 



; and the expression h if it exist at all, must involve at least one radical of 

 the first order, a/, which must admit of being expressed as a rational but unsym- 



metric function f/ of the five roots, but must have a prime power f/"' symme- 



