78 



both on theoretical reasoning and on actual experiment, that 

 Mr. Jerrard's method was adequate to achieve an almost 

 equally curious and unexpected transformation, namely, the 

 reduction of the general equation 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 de- 

 pendence 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 reduc- 

 tion which Sir William Hamilton regards as very remarkable 

 in theory, and as not unimportant in practice, since it reduces 

 the solution of any proposed numerical 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 curio- 

 sity to construct and to apply. 



3. 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 ; yet, Sir William 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 inquires in what 

 sense it is true, that the general equation of the fifth degree 

 would be resolved, if, contrary to the theory of Abel, it 

 were possible to discover, as Mr. Jerrard and others have 

 sought to do, a reduction of that general equation to the 



