ROYAL IRISH ACADEMY, *May* 22*nd*, 1837

SIR WILLIAM HAMILTON laid before the Academy an account of some investigations in which he had recently been engaged, respecting Equations of the Fifth Degree. They related chiefly to three points: first the argument of Abel against the possibility of generally and algebraically resolving such equations; second, the researches of Mr. Jerrard; and third, the conceivable reduction, in a new way, of the original problem to a more simple form.

1. The argument of Abel consisted of two principal parts; one
independent of the degree of the equation, and the other dependent on
that degree. The general principle was first laid down, by him, that
whatever may be the degree *n* of any general algebraic equation, if
it be possible to express a root of that equation, in terms of the
coefficients, by any finite combination of rational functions, and of
radicals with prime exponents, then every radical in such an
expression, when reduced to its most simple form, must be equal to a
rational (though not a symmetric) function of the *n* roots of the
original equation; and must, when considered as such a function, have
exactly as many values, arising from the permutations of those *n*
roots among themselves, as it has values, when considered as a
radical, arising from the introduction of factors which are roots of
unity. And in proceeding 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 differences 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 W. H. 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 premisses to deduce, *à 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 no 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.

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 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 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 unimportant
in practice, since it reduces the solution of any proposed numerical
equation of the fifth degree, even without 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.

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 binomial form, or to the extraction of a fifth root of an expression in general imaginary? And he conceives, that the propriety of considering such extraction as an admitted instrument of calculation in elementary algebra, is ultimately founded on this: that the two real equations

into which the imaginary equation

resolves itself, may be transformed into two others which are of the forms

so that each of these two new equations expresses one given real number as a known rational function of one sought real number. But notwithstanding the interest which attaches to these two particular forms of rational functions, and generally to the analogous forms which present themselves in separating the real and imaginary parts of a radical of the

it ought now to be the object of those who interest themselves in the improvement of this part of algebra, to inquire, whether the dependence of the two real numbers

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