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



exactly n values ; of which, n — 5 values would vanish when we supposed 

 a^, a„_i, . . . Og to become = 0, and the remaining five values would represent 



the five roots of the general equation of the fifth degree ; but such a repre- 

 sentation of the roots of that equation has been already proved to be impossible. 



[24,] Although the whole of the foregoing argument has been suggested 

 by that of Abel, and may be said to be a commentary thereon ; yet it will not 

 fail to be perceived, that there are several considerable differences between the one 

 method of proof and the other. More particularly, in establishing the cardinal 

 proposition that every radical in every irreducible expression for any one of the 

 roots of any general equation is a rational function of those roots, it has appeared 

 to the writer of this paper more satisfactory to begin by showing that the radicals 

 of highest order will have that property, if those of lower orders have it, 

 descending thus to radicals of the lowest order, and afterwards ascending again ; 

 than to attempt, as Abel has done, to prove the theorem, in the first instance, 

 for radicals of the highest order. In fact, while following this last-mentioned 

 method, Abel has been led to assume that the coefficient of the first power of 

 some highest radical can always be rendered equal to unity, by introducing 

 (generally) a new radical, which in the notation of the present paper may be 

 expressed as follows : 



:) (m)\ ^\ '■■■^Jjn) > n^"^ J L 



< a -/ 



p: < 



but although the quantity under the radical sign, in this expression, is indeed 

 free from that irrationality of the m"" order which was introduced by the radical 

 of* , it is not, in general, free from the irrationalities of the same order intro- 

 duced by the other radicals a; , ... of that order ; and consequently the new 

 radical, to which this process conducts, is in general elevated to the order m + l; 

 a circumstance which Abel does not appear to have remarked, and which 

 renders it difficult to judge of the validity of his subsequent reasoning. And 

 because the other chief obscurity in Abel's argument (in the opinion of the 



