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permutation of those n roots among themselves, as it has 

 values, when considered as a radical, arising from the intro- 

 duction of factors which are roots of unity. And in pro- 

 ceeding to apply this general principle to equations of the 

 fifth degree, the same illustrious mathematician employed 

 certain properties of functions of five variables, which may 

 be condensed into the two following theorems: that, if a 

 rational function of five independent variables have a prime 

 power symmetric, without being symmetric itself, it must be 

 the square root of the product of the ten squares of diffe- 

 rences of the five variables, or at least that square root mul- 

 tiplied by some symmetric function ; and that, if a rational 

 function of the same variables have, itself, more than two 

 values, its square, its cube, apd its fifth power have, each, 

 more than two values also. Sir W. H. conceived that the 

 reflections into which he had been led, were adapted to re- 

 move some obscurities and doubts which might remain upon 

 the mind of a reader of Abel's argument ; he hoped also 

 that he had thrown light upon this argument in a new way, 

 by employing its premisses to deduce, a priori^ the known 

 solutions of quadratic, cubic, and biquadratic equations, 

 and to show that no new solutions of such equations, with 

 radicals essentially different from those at present used, re- 

 main to be discovered : but whether or no he had himself 

 been useful in this way, he considered Abel's result as esta- 

 blished : namely, that it is impossible to express a root of 

 the general equation of the fifth degree, in terms of the co- 

 efficients of that equation, by any finite combination of radi- 

 cals and rational functions. 



2. What appeared to him the fallacy in Mr. Jerrard's very 

 ingenious attempt to accomplish this impossible object, had 

 been already laid before the British Association at Bristol, 

 and was to appear in the forthcoming volume of the reports of 

 that Association. Meanwhile, Sir William Hamilton was 

 anxious to state to the Academy his full conviction, founded 



