% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf INVESTIGATIONS RESPECTING EQUATIONS}

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\centerline{\Largebf OF THE FIFTH DEGREE}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   1 (1841), pp.\ 76--80.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 2000}

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\noindent
\centerline{\largeit Investigations respecting Equations
      of the Fifth Degree.}

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\centerline{\largerm By Sir \largesc William Rowan Hamilton.}
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\centerline{Communicated May 22nd, 1837.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~i (1841), pp.\ 76--80.]}

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{\sc 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, {\it \`{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 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
{\it 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
$$\eqalign{
x^5 - 10 x^3 y^2 + 5 x y^4 &= a,\cr
5 x^4 y - 10 x^2 y^3 + y^5 &= b,\cr}$$
into which the imaginary equation
$$(x + \sqrt{-1}y)^5 = a + \sqrt{-1} b$$
resolves itself, may be transformed into two others which are of
the forms
$$\rho^5 = r,
   \quad\hbox{and}\quad
  {5 \tau - 10 \tau^3 + \tau^5 \over
         1 - 10 \tau^2 + 5 \tau^4} = t,$$
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 $n^{\rm th}$ degree; Sir
William Hamilton does not conceive that they both possess so
eminent a prerogative of simplicity as to entitle the inverses of
them alone to be admitted among the instruments of elementary
algebra, to the exclusion of the inverses of all other real and
rational functions of single real variables.  And he thinks, that
since Mr.\ Jerrard has succeeded in reducing the general equation
of the fifth degree, with five imaginary coefficients, to the
trinomial form above described, which resolves itself into the
two real equations following,
$$\eqalign{
x^5 - 10 x^3 y^2 + 5 x y^4 + x &= a,\cr
5 x^4 y - 10 x^2 y^3 + y^5 + y &= b,\cr}$$
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 $x$ and $y$, in these two last
equations, on the two real numbers $a$ and $b$, cannot be
expressed by the help of the real inverses of some new real and
rational or even transcendental functions of single real
variables; or, (to express the same thing in a practical, or in a
geometrical form,) to inquire whether the two sought real numbers
cannot be calculated by a finite number of tables of single
entry, or constructed by the help of a finite number of curves:
although the argument of Abel excludes all hope that this can be
accomplished, if we confine ourselves to those particular forms
of rational functions which are connected with the extraction of
radicals.

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