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 kad 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 
