British Association. 117 



have a prime power symmetric, without being symmetric itself, it 

 must be the square root of the product of the ten squares of differ- 

 ences of the five variables, or at least that square root multiplied by 

 some symmetric function ; and that if a rational function of the same 

 variables have itself more than two values, its square, its cube, and its 

 fifth power have each more than two values also. Sir William Ha- 

 milton conceived that the reflections into which he had been led were 

 adapted to remove 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 

 premises 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, 

 remain to be discovered : but whether or not he had himself been 

 useful in this way, he considered Abel's result as established : namely, 

 that it is impossible to express a root of the general equation of the 

 fifth degree, in terms of the coefficients of that equation, by any finite 

 combination of radicals and rational functions. 



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 his full conviction, founded 

 both on theoretical reasoning and on actual experiment, that Mr. Jer- 

 rard's method was adequate to achieve an almost equally curious and 

 unexpected transformation, namely, the reduction of the general equa- 

 tion of the fifth degree, with five coefficients, real or imaginary, to a 

 trinomial form ; and therefore ultimately to that very simple state in 

 which the sum of an unknown number (real or imaginary) and of its 

 own fifth power is equalled to a known (real or imaginary) number. 

 In this manner the general dependence of the modulus and amplitude 

 of a root of the general equation of the fifth degree on the five moduli 

 and five amplitudes of the five coefficients of that equation, is reduced 

 to the dependence of the modulus and amplitude of a new (real or 

 imaginary) number on the one modulus and one amplitude of the sum 

 of that number and its own fifth power ; a reduction which Sir William 

 Hamilton regards as very remarkable in theory, and as not unim- 

 portant in practice, since it reduces the solution of any proposed nu- 

 merical equation of the fifth degree even with imaginary coefficients, 

 to the employment, without tentation, of the known logarithmic tables, 

 and of two new tables of double entry, which he has had the curiosity 

 to construct and to apply. 



It appears possible enough that this transformation, deduced from 

 Mr. Jerrard's principles, conducts to the simplest of all forms under 

 which the general equation of the fifth degree can be put j yet Sir 

 M^illiam Hamilton thinks that algebraists ought not absolutely to 

 despair of discovering some new transformation which shall conduct 

 to a method of solution more analogous to the known ways of resolving 

 equations of lower degrees, though not like them dependent entirely 

 upon radicals. He inquired in what sense it is true that the general 



