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permutation of those ” 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, d@ 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 
