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

\magnification=\magstep1
\vsize=227 true mm \hsize=170 true mm
   \voffset=-0.4 true mm \hoffset=-5.4 true mm

\def\folio{\ifnum\pageno>0 \number\pageno \else\fi}

\font\Largebf=cmbx10  scaled \magstep2
\font\largerm=cmr12
\font\largeit=cmti12
\font\tensc=cmcsc10
\font\sevensc=cmcsc10 scaled 700
\newfam\scfam \def\sc{\fam\scfam\tensc}
\textfont\scfam=\tensc \scriptfont\scfam=\sevensc
\scriptscriptfont\scfam=\sevensc
\font\largesc=cmcsc10 scaled \magstep1

\def\multieqalign#1{\null\,\vcenter{\openup1\jot \mathsurround=0pt
   \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
   &&\quad \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
    \crcr#1\crcr}}\,}

\input amssym.def
\newsymbol\backprime 1038

\pageno=0

\null\vskip72pt

\centerline{\Largebf ON EQUATIONS OF THE FIFTH DEGREE}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (Transactions of the Royal Irish Academy,
   19 (1842), pp.\ 329--376.)}

\vskip36pt

\vfill

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

\vskip 12pt

\centerline{\largerm 2000}

\vskip36pt\eject

\null\vskip36pt

\noindent
{\largeit On Equations of the Fifth Degree: and especially on a
certain System of Expressions connected with those Equations,
which Professor Badano\/}\footnote*{Nuove Ricerche sulla
Risoluzione Generale delle Equazioni Algebriche del {\sc P.
Gerolamo Badano}, Carmelitano scalzo, Professore di Matematica
nella R.  Universita di Genova.  Genova, Tipografia Ponthenier,
1840.} {\largeit has lately proposed.  By {\largesc Sir William
Rowan Hamilton}, {\largerm LL.~D., P.R.I.A., F.R.A.S.}, Honorary
Member of the Royal Societies of Edinburgh and Dublin; Honorary
or Corresponding Member of the Royal or Imperial Academies of
St.~Petersburgh, Berlin, and Turin, of the American Society of
Arts and Sciences, and of other Scientific Societies at home and
abroad; Andrews' Professor of Astronomy in the University of
Dublin, and Royal Astronomer of Ireland.\par}

\vskip12pt

\centerline{Read 4th August, 1842.}

\vskip12pt

\centerline{[{\it Transactions of the Royal Irish Academy},
   vol.~xix (1842), pp.\ 329--376.]}

\bigskip

1.
{\sc Lagrange} has shown that if $\alpha$ be a given root of the
equation
$$\alpha^{n-1} + \alpha^{n-2} + \cdots + \alpha^2 + \alpha + 1
   =  0,$$
$n$ being a prime factor of $m$, and if $\mu$ denote for
abridgment the quotient
$$\mu = {1 \mathbin{.} 2 \mathbin{.} 3 \, \ldots \, m
         \over \left( 1 \mathbin{.} 2 \mathbin{.}3 \, \ldots \,
            {\displaystyle {m \over n}} \right)^n};$$
then the function
$$t = x' + \alpha x'' + \alpha^2 x''' + \ldots
          + \alpha^{m-1} x^{(m)}$$
has only $\mu$ different values, corresponding to all possible
changes of arrangement of the $m$ quantities
$x', x'',\ldots \, x^{(m)}$, which may be considered as the roots
of a given equation of the $m^{\rm th}$ degree,
$$x^m - {\sc a} x^{m-1} + {\sc b} x^{m-2}
      - {\sc c} x^{m-3} + \ldots = 0;$$
and that if the development of the $n^{\rm th}$ power of this
function~$t$ be reduced, by the help of the equation
$$\alpha^n = 1,$$
(and not by the equation $\alpha^{n-1} + \hbox{\&c.} = 0$,) to
the form
$$t^n = \xi^{(0)} + \alpha \xi' + \alpha^2 \xi'' + \ldots
       + \alpha^{n-1} \xi^{(n-1)},$$
then this power~$t^n$ itself has only
$\displaystyle {\mu \over n}$
different values, and the term~$\xi^{(0)}$ has only
$\displaystyle {\mu \over n(n - 1)}$ such values, or is a root of
an equation of the degree
$${1 \mathbin{.} 2 \mathbin{.} 3 \, \ldots \, m
         \over n (n - 1)
            \left( 1 \mathbin{.} 2 \mathbin{.}3 \, \ldots \,
            {\displaystyle {m \over n}} \right)^n},$$
of which equation the coefficients are rational functions of the
given coefficients ${\sc a}$,~${\sc b}$,~${\sc c}$, \&c.; while
$\xi', \xi'',\ldots \, \xi^{(n-1)}$
are the roots of an equation of the degree $n - 1$, of which the
coefficients can be expressed rationally in terms of $\xi^{(0)}$
and of the same original coefficients ${\sc a},\ldots$ of the
given equation in $x$.

\bigbreak

2.
For example, if there be given an equation of the sixth degree,
$$x^6 - {\sc a} x^5 + {\sc b} x^4 - {\sc c} x^3 + {\sc d} x^2
      - {\sc e} x + {\sc f} = 0,$$
of which the roots are denoted by
$x'$,~$x''$, $x'''$,
$x^{IV}$, $x^{V}$,~${\rm x}^{VI}$,
and if we form the function
$$t = x' + \alpha x'' + \alpha^2 x''' + \alpha^3 x^{IV}
       + \alpha^4 x^{V} + \alpha^5 x^{VI},$$
in which $\alpha = -1$; we shall then have
$$m = 6,\quad
  n = 2,\quad
  \mu = {720 \over 36} = 20,\quad
  {\mu \over n} = 10,\quad
  {\mu \over n (n - 1)} = 10;$$
and the function~$t$ will have twenty different values, but its
square will have only ten.  And if, by using only the equation
$\alpha^2 = 1$, and not the equation $\alpha = -1$, we reduce the
development of this square to the form
$$t^2 = \xi^{(0)} + \alpha \xi',$$
the term~$\xi^{(0)}$ will itself be a ten-valued function of the
six quantities $x',\ldots \, x^{VI}$; and $\xi'$ will be a
rational function of $\xi^{(0)}$ and ${\sc a}$, namely,
$$\xi' = {\sc a}^2 - \xi^{(0)}.$$

\bigbreak

3.
Again, if with the same meanings of $x',\ldots \, x^{VI}$, we
form $t$ by the same expression as before, but suppose $\alpha$
to be a root of the equation
$$\alpha^2 + \alpha + 1 = 0,$$
then
$$m = 6,\quad
  n = 3,\quad
  \mu = {720 \over 8} = 90,\quad
  {\mu \over n} = 30,\quad
  {\mu \over n (n - 1)} = 15;$$
so that the function~$t$ will now have 90 different values, but
its cube will have only 30; and if that cube be reduced, by the
equation $\alpha^3 = 1$, to the form
$$t^3 = \xi^{(0)} + \alpha \xi' + \alpha^2 \xi'',$$
then $\xi^{(0)}$ will be a root of an equation of the fifteenth
degree, while $\xi'$ and $\xi''$ will be the roots of a quadratic
equation, the coefficients of this last equation being rational
functions of $\xi^{(0)}$, and of the given
coefficients~${\sc a}$, \&c.

\bigbreak

4.
And if, in like manner, we consider the case
$$m = 5,\quad
  n = 5,\quad
  \mu = 120,\quad
  {\mu \over n} = 24,\quad
  {\mu \over n (n - 1)} = 6,$$
so that $x',\ldots \, x^{V}$ are the roots of a given equation
of the fifth degree
$$x^5 - {\sc a} x^4 + {\sc b} x^3
      - {\sc c} x^2 + {\sc d} x - {\sc e} = 0,$$
and
$$t = x' + \alpha x'' + \alpha^2 x''' + \alpha^3 x^{IV}
       + \alpha^4 x^{V},$$
in which $\alpha$ is a root of the equation
$$\alpha^4 + \alpha^3 + \alpha^2 + \alpha + 1 = 0,$$
then the function~$t$ has itself 120 different values, but its
fifth power has only 24; and if this fifth power be put under the
form
$$t^5 = \xi^{(0)} + \alpha \xi' + \alpha^2 \xi''
         + \alpha^3 \xi''' + \alpha^4 \xi^{IV},$$
by the help of the equation $\alpha^5 = 1$, then $\xi^{(0)}$ is a
root of an equation of the sixth degree, of which the
coefficients are rational functions of
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$,~${\sc e}$,
while $\xi'$,~$\xi''$,~$\xi'''$,~$\xi^{IV}$, are the roots of
an equation of the fourth degree, of which the coefficients are
rational functions of the same given coefficients~${\sc a}$,
\&c., and of $\xi^{(0)}$.

\bigbreak

5.
{\sc Lagrange} has shown that these principles explain the
success of the known methods for resolving quadratic, cubic, and
biquadratic equations; but that they tend to discourage the hope
of resolving any general equation above the fourth degree, by any
similar method.  And in fact it has since\footnote*{See a paper
by the present writer, ``On the Argument of Abel,'' \&c., in the
Second Part of the Eighteenth Volume of the Transactions of this
Academy.}
been shown to be impossible to express any root of any general
equation, of the fifth or any higher degree, as a function of the
coefficients of that equation, by any finite combination of
radicals and rational functions.  Yet it appears to be desirable
to examine into the validity and import of an elegant system of
radical expressions which have lately been proposed by Professor
{\sc Badano} of Genoa, for the twenty-four values of {\sc
Lagrange}'s function~$t^5$ referred to in the last article; and
to inquire whether these new expressions are adapted to assist in
the solution of equations of the fifth degree, or why they fail
to do so.

\bigbreak

6.
In order to understand more easily and more clearly the
expressions which are thus to be examined, it will be
advantageous to begin by applying the method by which they are
obtained to equations of lower degrees.  And first it is evident
that the general quadratic equation
$$x^2 - {\sc a} x + {\sc b} = 0,$$
has its roots expressed as follows:
$$x'  = \alpha + \beta,\quad
  x'' = \alpha - \beta;$$
$\alpha$ not here denoting any root of unity, but a rational
function of the coefficients of the given equation (namely
${1 \over 2} {\sc a}$), and $\beta^2$ being another rational
function of those coefficients (namely
${1 \over 4} {\sc a}^2 - {\sc b}$);
because by the general principles of article~1., when $m = 2$ and
$n = 2$, we have
$\displaystyle {\mu \over n} = 1$,
so that the function $(x' - x'')^2$
is symmetric, as indeed it is well known to be.

\bigbreak

7.
Proceeding to the cubic equation
$$x^3 - {\sc a} x^2 + {\sc b} x - {\sc c} = 0,$$
and seeking the values of the function
$$t^3 = (x' + \theta x'' + \theta^2 x''')^3,$$
in which $\theta$ is such that
$$\theta^2 + \theta + 1 = 0,$$
we know first, by the same general principles, that the number of
these values is two, because
$\displaystyle {\mu \over n} = 2$,
when $m = 3$, $n = 3$.  And because these values will not be
altered by adding any common term to the three roots
$x'$,~$x''$,~$x'''$, it is permitted to treat the sum of these
three roots as vanishing, or to assume that
$$x' + x'' + x''' = 0;$$
that is, to reduce the cubic equation to the form
$$x'^3 + p x' + q = 0.$$
In other words, the function
$$t^3 = (x_1 + \theta x_2 + \theta^2 x_3)^3,$$
in which $x_1$,~$x_2$,~$x_3$ are the three roots of the equation
with coefficients ${\sc a}$,~${\sc b}$,~${\sc c}$, will depend on
those coefficients, only by depending on $p$ and $q$, if these
two quantities be chosen such that we shall have identically
$$x^3 - {\sc a} x^2 + {\sc b} x - {\sc c}
   =  (x - {\textstyle {1 \over 3}} {\sc a})^3
       + p (x - {\textstyle {1 \over 3}} {\sc a}) + q.$$

\bigbreak

8.
This being perceived, and $x''$ and $x'''$ being seen to be the
two roots of the quadratic equation
$$x''^2 + x' x'' + x'^2 + p = 0,$$
which is obtained by dividing the cubic
$$x''^3 + p x'' - x'^3 - p x' = 0,$$
by the linear factor $x'' - x'$; we may, by the theory of
quadratics, assume the expressions
$$x''  = \alpha + \beta,\quad
  x''' = \alpha - \beta,$$
provided that we make
$$\alpha = - {\textstyle {1 \over 2}} x',\quad
  \beta^2 = - {\textstyle {3 \over 4}} x'^2 - p,$$
that is, provided that we establish the identity
$$(x'' - \alpha)^2 - \beta^2
   = x''^2 + x' x'' + x'^2 + p.$$
And, substituting for $x'$,~$x''$,~$x'''$, their values as
functions of $\alpha$ and $\beta$, and reducing by the equation
$\theta^2 + \theta + 1 = 0$, we find
$$t^3 = \{ - 3 \alpha + (\theta - \theta^2) \beta \}^3
   =  \alpha' + \beta';$$
in which
$$\alpha' = - 27 \alpha (\alpha^2 - \beta^2),\quad
  \beta'^2 = - 27 \beta^2 (9 \alpha^2 - \beta^2)^2.$$
But $\alpha$ and $\beta^2$ are rational functions of $x'$ and
$p$; and substituting their expressions as such, we find
corresponding expressions for $\alpha'$ and $\beta'^2$, namely,
$$\alpha'
   = {\textstyle {27 \over 2}} x' (x'^2 + p),\quad
  \beta'^2
   = {\textstyle {27 \over 4}} (3 x'^2 + 4p) (3 x'^2 + p)^2.$$

\bigbreak

9.
Finally, $x'$ is such that
$$x'^3 + p x' = - q;$$
and it is found on trial to be possible by this condition to
eliminate~$x'$ from the expressions for $\alpha'$ and $\beta'^2$,
obtained at the end of the last article, and so to arrive at
these other expressions, which are rational functions of $p$ and
$q$:
$$\alpha' = - {\textstyle {27 \over 2}} q,\quad
  \beta'^2 = {\textstyle {27 \over 4}} (27 q^2 + 4 p^3).$$
In this manner then it might have been discovered, what has long
been otherwise known, that the function~$t^3$ is a root of the
auxiliary quadratic equation
$$(t^3)^2 + 27 q (t^3) - 27 p^3 = 0.$$
And because the same method gives
$$(x' + \theta x'' + \theta^2 x''')
      (x' + \theta^2 x'' + \theta x''')
   =  9 \alpha^2 + 3 \beta^2 = - 3 p,$$
we should obtain the known expressions for the three roots of the
cubic equation
$$x'^3 + p x' + q = 0,$$
under the forms:
$$x' = {t \over 3} - {p \over t},\quad
  x'' = {\theta^2 t \over 3} - {\theta p \over t},\quad
  x''' = {\theta t \over 3} - {\theta^2 p \over t};$$
which are immediately verified by observing that
$$\theta^3 = 1,\quad
  \left( {t \over 3} \right)^3 - \left( {p \over t} \right)^3
   =  - q.$$
The foregoing method therefore succeeds completely for equations
of the third degree.

\bigbreak

10.
In the case of the biquadratic equation, deprived for simplicity
of its second term, namely,
$$x'^4 + p x'^2 + q x' + r = 0,$$
so that the sum of the four roots vanishes,
$$x' + x'' + x''' + x^{IV} = 0,$$
we may consider $x''$,~$x'''$,~$x^{IV}$, as roots of the
cubic equation
$$x''^3 + x' x''^2 + (x'^2 + p) x'' + x'^3 + p x' + q = 0;$$
and this may be put under the form
$$(x'' - \alpha)^3 - 3 \eta (x'' - \alpha) - 2 \epsilon = 0,$$
of which the roots (by the theory of cubic equations) may be
expressed as follows:
$$x'' = \alpha + \beta + \gamma,\quad
  x''' = \alpha + \theta \beta + \theta^2 \gamma,\quad
  x^{IV} = \alpha + \theta^2 \beta + \theta \gamma,$$
$\beta$,~$\gamma$, and $\theta$ being such as to satisfy the
conditions
$$\beta^3 + \gamma^3 = 2 \epsilon,\quad
  \beta \gamma = \eta,\quad
  \theta^2 + \theta + 1 = 0.$$
Comparing the two forms of the cubic equation in $x''$, we find
the relations
$$x' = - 3 \alpha,\quad
  x'^2 + p = 3 ( \alpha^2 - \eta ),\quad
  x'^3 + p x' + q = - \alpha^3 + 3 \alpha \eta - 2 \epsilon;$$
which give
$$\alpha
   =  - {\textstyle {1 \over 3}} x',\quad
  \eta
   =  - {\textstyle {1 \over 9}} (2 x'^2 + 3p),\quad
  \epsilon
   =  - {\textstyle {1 \over 54}} (20 x'^3 + 18 p x' + 27 q).$$
Thus, any rational function of the four roots of the given
biquadratic can be expressed rationally in terms of
$\alpha$,~$\beta$,~$\gamma$; while $\alpha$, $\beta \gamma$,
and $\beta^3 + \gamma^3$, are rational functions of
$x'$,~$p$,~$q$; and the function $x'^4 + p x'^2 + q x'$ may be
changed, wherever it occurs, to the given quantity $-r$.

\bigbreak

11.
With these preparations it is easy to express, as follows, the
function
$$(x' - x'' + x''' - x^{IV})^2,$$
which the general theorems of {\sc Lagrange}, already mentioned,
lead us to consider.  Denoting it by $4z$, we have
$$z = ( - 2 \alpha + \theta \beta + \theta^2 \gamma)^2
   =  \alpha' + \theta \beta' + \theta^2 \gamma';$$
in which
$$\alpha' = 4 \alpha^2 + 2 \beta \gamma,\quad
  \beta' = \gamma^2 - 4 \alpha \beta,\quad
  \gamma' = \beta^2 - 4 \alpha \gamma:$$
and the three values of $z$ are the three roots of the cubic
equation
$$(z - \alpha')^3 - 3 \eta' (z - \alpha') - 2 \epsilon'
   =  0;$$
in which
$$\eqalign{
\alpha'
   &= 4 \alpha^2 + 2 \eta,\cr
\eta'
   &= \beta' \gamma'
    = \eta^2 + 16 \alpha^2 \eta - 8 \alpha \epsilon,\cr
\epsilon'
   &= {\textstyle {1 \over 2}} (\beta'^3 + \gamma'^3)
    = 2 \epsilon^2 - \eta^3 - 12 \alpha \epsilon \eta
        + 48 \alpha^2 \eta^2 - 64 \alpha^3 \epsilon.\cr}$$
Substituting for $\alpha$,~$\eta$,~$\epsilon$, their values, as
functions of $x'$,~$p$,~$q$, we find
$$\eqalign{
\alpha'
   &= - {\textstyle {2 \over 3}} p;\cr
\eta'
   &= {\textstyle {1 \over 9}}
         ( - 12 x'^4 - 12 p x'^2 - 12 q x' + p^2 );\cr
\epsilon'
   &= {\textstyle {1 \over 54}}
         ( 72 p x'^4 + 72 p^2 x'^2 + 72 p q x'
               + 27 q^2 + 2 p^3);\cr}$$
and eliminating~$x'$, by the condition
$$x'^4 + p x'^2 + q x' = - r,$$
we obtain
$$\eqalign{
\eta'
   &= {\textstyle {1 \over 9}} (12 r + p^2);\cr
\epsilon'
   &= {\textstyle {1 \over 54}}
         ( - 72 pr + 27 q^2 + 2 p^3).\cr}$$
The auxiliary cubic in $z$ becomes therefore
$$(z + {\textstyle {2 \over 3}} p)^3
       - {\textstyle {1 \over 3}} (12 r + p^2)
            (z + {\textstyle {2 \over 3}} p)
       + {\textstyle {1 \over 27}} (72 pr - 27 q^2 - 2 p^3)
   =  0;$$
that is
$$z^3 + 2 p z^2 + (p^2 - 4 r) z - q^2 = 0;$$
and if its three roots be denoted by $z'$,~$z''$,~$z'''$, in an
order such that we may write
$$\eqalign{
z'
   &= {\textstyle {1 \over 4}} (x' + x'' - x''' - x^{IV})^2
    = \alpha' + \beta' + \gamma',\cr
z''
   &= {\textstyle {1 \over 4}} (x' - x'' + x''' - x^{IV})^2
    = \alpha' + \theta \beta' + \theta^2 \gamma',\cr
z'''
   &= {\textstyle {1 \over 4}} (x' - x'' - x''' + x^{IV})^2
    = \alpha' + \theta^2 \beta' + \theta \gamma',\cr}$$
we may express the four roots of the biquadratic equation under
known forms, by means of the square roots of $z'$,~$z''$,~$z'''$,
as follows:
$$\eqalign{
x'
   &=    + {\textstyle {1 \over 2}} \surd z'
         + {\textstyle {1 \over 2}} \surd z''
         + {\textstyle {1 \over 2}} \surd z''',\cr
x''
   &=    + {\textstyle {1 \over 2}} \surd z'
         - {\textstyle {1 \over 2}} \surd z''
         - {\textstyle {1 \over 2}} \surd z''',\cr
x'''
   &=    - {\textstyle {1 \over 2}} \surd z'
         + {\textstyle {1 \over 2}} \surd z''
         - {\textstyle {1 \over 2}} \surd z''',\cr
x^{IV}
   &=    - {\textstyle {1 \over 2}} \surd z'
         - {\textstyle {1 \over 2}} \surd z''
         + {\textstyle {1 \over 2}} \surd z'''.\cr}$$
It may be noticed also that the present method gives for the
product of these three square roots, the expression:
$$\eqalign{
\surd z' \mathbin{.} \surd z'' \mathbin{.} \surd z'''
   &= {\textstyle {1 \over 8}}
         (x' + x'' - x''' - x^{IV})
         (x' - x'' + x''' - x^{IV})
         (x' - x'' - x''' + x^{IV}) \cr
   &= ( - 2 \alpha + \beta + \gamma )
      ( - 2 \alpha + \theta \beta + \theta^2 \gamma )
      ( - 2 \alpha + \theta^2 \beta + \theta \gamma ) \cr
   &= - 8 \alpha^3 + 6 \alpha \eta + 2 \epsilon = - q;\cr}$$
a result which may be verified by observing that, by the
expressions given above for $\alpha'$,~$\eta'$,~$\epsilon'$, in
terms of $\alpha$,~$\eta$,~$\epsilon$, we have the relation
$$z' z'' z''' = \alpha'^3 - 3 \alpha' \eta' + 2 \epsilon'
    = ( - 8 \alpha^3 + 6 \alpha \eta + 2 \epsilon )^2.$$

\bigbreak

12.
In this manner, then, it might have been discovered that the four
roots $x_1$,~$x_2$,~$x_3$,~$x_4$, of the general biquadratic
equation
$$x^4 - {\sc a} x^3 + {\sc b} x^2 - {\sc c} x + {\sc d} = 0,$$
are the four values of an expression of the form
$\alpha + \beta + \gamma + \delta$,
in which, $\alpha$, $\beta^2 + \gamma^2 + \delta^2$,
$\beta \gamma \delta$, and
$\beta^2 \gamma^2 + \gamma^2 \delta^2 + \delta^2 \beta^2$,
are rational functions of the coefficients
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$,
and may be determined as such by comparison with the identical
equation
$$\eqalign{
(\alpha + \beta + \gamma + \delta - \alpha)^4
       - 2 (\beta^2 + \gamma^2 + \delta^2)
            (\alpha + \beta + \gamma + \delta - \alpha)^2
       + (\beta^2 + \gamma^2 + \delta^2)^2
   \hskip -18em \cr
   &= 8 \beta \gamma \delta
            (\alpha + \beta + \gamma + \delta - \alpha)
       + 4 (\beta^2 \gamma^2 + \gamma^2 \delta^2
            + \delta^2 \beta^2),\cr}$$
of which each member is an expression for the square of
$2 (\beta \gamma + \gamma \delta + \delta \beta)$.
It might have been perceived also that any three quantities, such
as here $\beta^2$,~$\gamma^2$,~$\delta^2$, which are the three
roots of a given cubic equation, may be considered as the three
values of an expression of the form $\alpha' + \beta' + \gamma'$,
in which, $\alpha'$, $\beta' \gamma'$, and $\beta'^3 + \gamma'^3$
are rational functions of the coefficients of that given
equation, and may have their forms determined by comparison with
the identity,
$$(\alpha' + \beta' + \gamma' - \alpha')^3
       - 3 \beta' \gamma' (\alpha' + \beta' + \gamma' - \alpha')
       - \beta'^3 - \gamma'^3
   =  0.$$
And finally that any two quantities which, as here $\beta'^3$ and
$\gamma'^3$, are the two roots of a given quadratic equation, are
also the two values of an expression of the form
$\alpha'' + \beta''$, in which $\alpha''$ and $\beta''^2$ may be
determined by comparing the given equation with the following
identical form,
$$(\alpha'' + \beta'' - \alpha'')^2 - \beta''^2 = 0.$$
Let us now endeavour to apply similar methods of expression to a
system of five arbitrary quantities, or to an equation of the
fifth degree.

\bigbreak

13.
Let, therefore, $x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$, be the five
roots of the equation
$$x^5 - {\sc a} x^4 + {\sc b} x^3 - {\sc c} x^2
      + {\sc d} x - {\sc e} = 0,
   \eqno (1)$$
and let $x'$,~$x''$,~$x'''$,~$x^{IV}$,~$x^{V}$, be the
five roots of the same equation when deprived of its second term,
or put under the form
$$x'^5 + p x'^3 + q x'^2 + r x' + s = 0,
   \eqno (2)$$
so that
$$x' + x'' + x''' + x^{IV} + x^{V} = 0,
   \eqno (3)$$
and
$$x_1 = x'  + {\textstyle {1 \over 5}} {\sc a},\quad
  x_2 = x'' + {\textstyle {1 \over 5}} {\sc a},
   \quad\hbox{\&c.}
   \eqno (4)$$
Dividing the equation of the fifth degree
$$x''^5 - x'^5 + p (x''^3 - x'^3) + q (x''^2 - x'^2)
       + r (x'' - x') = 0,
   \eqno (5)$$
by the linear factor $x'' - x'$, we obtain the biquadratic
$$x''^4 + x ' x''^3 + (x'^2 + p) x''^2
       + (x'^3 + p x' + q) x'' + x'^4 + p x'^2 + q x' + r = 0,
   \eqno (6)$$
of which the four roots are
$x''$,~$x'''$,~$x^{IV}$,~$x^{V}$.
Hence, by the theory of biquadratic equations, we may employ the
expressions:
$$x''          = \alpha + \beta + \gamma + \delta,\quad
  x'''         = \alpha + \beta - \gamma - \delta,\quad
  x^{IV}   = \alpha - \beta + \gamma - \delta,\quad
  x^{V}    = \alpha - \beta - \gamma + \delta;
   \eqno (7)$$
provided that $\alpha$,~$\beta$,~$\gamma$,~$\delta$ are such as
to satisfy, independently of $x''$, the condition:
$$\left. \eqalign{
(x'' - \alpha)^4
       - 2 (\beta^2 + \gamma^2 + \delta^2) (x'' - \alpha)^2
       - 8 \beta \gamma \delta (x'' - \alpha)
   \hskip -21em \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \beta^4 + \gamma^4 + \delta^4
       - 2 (\beta^2 \gamma^2 + \gamma^2 \delta^2
            + \delta^2 \beta^2) \cr
   &= x''^4 + x ' x''^3 + (x'^2 + p) x''^2
       + (x'^3 + p x' + q) x'' + x'^4 + p x'^2 + q x' + r;\cr}
   \right\}
   \eqno (8)$$
which decomposes itself into the four following:
$$\left. \eqalign{
   &-4 \alpha = x';\cr
   &+ 6 \alpha^2 - 2 (\beta^2 + \gamma^2 + \delta^2)
         = x'^2 + p;\cr
   &- 4 \alpha^3 + 4 \alpha (\beta^2 + \gamma^2 + \delta^2)
          - 8 \beta \gamma \delta
         = x'^3 + p x' + q;\cr
   &+ \alpha^4 - 2 \alpha^2 (\beta^2 + \gamma^2 + \delta^2)
          + 8 \alpha \beta \gamma \delta
          + (\beta^2 + \gamma^2 + \delta^2)^2
          - 4 (\beta^2 \gamma^2 + \gamma^2 \delta^2
               + \delta^2 \beta^2) \cr
   &\hskip 12em
    = x'^4 + p x'^2 + q x' + r;\cr}
   \right\}
   \eqno (9)$$
and, therefore, conducts to expressions for $\alpha$,
$\beta^2 + \gamma^2 + \delta^2$, $\beta \gamma \delta$, and
$\beta^2 \gamma^2 + \gamma^2 \delta^2 + \delta^2 \beta^2$, as
rational functions of $x'$,~$p$,~$q$,~$r$.  Again, by the theory
of cubic equations, we may write:
$$\beta^2 = \epsilon + \kappa + \lambda,\quad
  \gamma^2 = \epsilon + \theta \kappa + \theta^2 \lambda,\quad
  \delta^2 = \epsilon + \theta^2 \kappa + \theta \lambda,
   \eqno (10)$$
in which $\theta$ is a root of the equation
$$\theta^2 + \theta + 1 = 0,
   \eqno (11)$$
while $\epsilon$, $\kappa \lambda$, and $\kappa^3 + \lambda^3$
are symmetric function of $\beta^2$,~$\gamma^2$,~$\delta^2$.
Making, for abridgment,
$$\beta \gamma \delta = \eta,\quad
  \kappa \lambda = \iota,
   \eqno (12)$$
we have, by (10) and (11),
$$\kappa^3 + \lambda^3 = \eta^2 + \epsilon^3 + 3 \epsilon \iota,
   \eqno (13)$$
and
$$\beta^2 + \gamma^2 + \delta^2 = 3 \epsilon,\quad
  \beta^2 \gamma^2 + \gamma^2 \delta^2 + \delta^2 \beta^2
   =  3 (\epsilon^2 - \iota);
   \eqno (14)$$
and therefore, by (9),
$$\left. \eqalign{
   &- 4 \alpha = x';\quad
      6 (\alpha^2 - \epsilon) = x'^2 + p;\cr
   &- 4 \alpha^3 + 12 \alpha \epsilon - 8 \eta
         = x'^3 + p x' + q;\cr
   &\alpha^4 - 6 \alpha^2 \epsilon + 8 \alpha \eta
       - 3 \epsilon^2 + 12 \iota
         = x'^4 + p x'^2 + q x' + r;\cr}
   \right\}
   \eqno (15)$$
conditions which give
$$\left. \eqalign{
\alpha
   &= - {\textstyle {1 \over 4}} x';\cr
\epsilon
   &= - {\textstyle {1 \over 48}} (5 x'^2 + 8p);\cr
\eta
   &= - {\textstyle {1 \over 64}} (5 x'^3 + 4p x' + 8q);\cr
\iota
   &= + {\textstyle {1 \over 144}}
            (10 x'^4 + 11 p x'^2 + 9 q x' + p^2 + 12 r).\cr}
   \right\}
   \eqno (16)$$
Thus, $\alpha$, $\epsilon$, $\eta$ and $\iota$, on the one hand,
are rational functions of $x'$,~$p$,~$q$,~$r$; and, on the other
hand, $x'$,~$x''$,~$x'''$,~$x^{IV}$,~$x^{V}$ may be
considered as functions, although not entirely rational, of
$\alpha$,~$\epsilon$,~$\eta$,~$\iota$.  In fact, if these four
last quantities (denoted to help the memory by four Greek vowels)
be supposed to be given, and if, by extraction of a square root
and a cube root, a value of $\kappa$ be found, which satisfies
the auxiliary equation
$$\kappa^6 - (\eta^2 - \epsilon^3 + 3 \epsilon \iota) \kappa^3
       + \iota^3 = 0,
   \eqno (17)$$
and then a corresponding value of $\lambda$ by the condition
$\kappa \lambda = \iota$, we shall have $\pm \beta$ by extraction
of another square root, since
$\beta^2 = \epsilon + \kappa + \lambda$;
and may afterwards, by extraction of a third square root, either
find $\pm \gamma$ from the expression
$\gamma^2 = \epsilon + \theta \kappa + \theta^2 \lambda$,
and deduce $\delta$ from the product
$\beta \gamma \delta = \eta$,
or else find $\pm (\gamma + \delta)$
from the expression
$$(\gamma + \delta)^2 = 2 \epsilon - \kappa - \lambda
             + {2 \eta \over \beta};
   \eqno (18)$$
and may then treat $x''$, $x'''$, $x^{IV}$, $x^{V}$, as
the four values of $\alpha + \beta + \gamma + \delta$, while
$x' = - 4 \alpha$.  Hence any function whatever of the five roots
of the general equation~(1) of the fifth degree may be considered
as a function of the five quantities
${\sc a}$,~$\alpha$,~$\epsilon$,~$\eta$,~$\iota$; and if, in the
expression of that function, the values (16) be substituted for
$\alpha$,~$\epsilon$,~$\eta$,~$\iota$, so as to introduce in
their stead the quantities $x'$,~$p$,~$q$,~$r$, it is permitted
to make any simplifications of the result which can be obtained
from the relation~(2), by changing
$x'^5 + p x'^3  + q x'^2 + r x'$,
wherever it occurs, to the known quantity $-s$.

\bigbreak

14.
Consider then the twentyfour-valued function, referred to in a
former article, and suggested (as {\sc Lagrange} has shown) by
the analogy of equations of lower degrees; namely, $t^5$, in
which
$$t = x_1 + \omega x_2 + \omega^2 x_3
          + \omega^3 x_4 + \omega^4 x_5,
   \eqno (19)$$
and
$$\omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0;
   \eqno (20)$$
$\omega$ here (and not $\alpha$) denoting an imaginary fifth root
of unity, so that
$$\omega^5 = 1.
   \eqno (21)$$
Observing, that by (4) and (20), $x_1$, \&c.\ may be changed in
(19) to $x'$, \&c.; and distinguishing among themselves the 120
values of the function~$t$ by employing the notation
$$t_{abcde}
   =     \omega^5 x^{(a)}
       + \omega^4 x^{(b)}
       + \omega^3 x^{(c)}
       + \omega^2 x^{(d)}
       + \omega^1 x^{(e)},
   \eqno (22)$$
which gives, for example,
$$t_{12345}
   =     x'
       + \omega^4 x''
       + \omega^3 x'''
       + \omega^2 x^{IV}
       + \omega   x^{V};
   \eqno (23)$$
we shall have, on substituting for $x'$ its value $- 4 \alpha$,
and for $x''$,~$x'''$,~$x^{IV}$, $x^{V}$, their
values~(7), the system of twenty-four expressions following:
$$\left. \eqalign{
t_{12345}
   &= - 5 \alpha
         + {\sc b} \beta
         + {\sc c} \gamma
         + {\sc d} \delta;\cr
t_{13254}
   &= - 5 \alpha
         + {\sc b} \beta
         - {\sc c} \gamma
         - {\sc d} \delta;\cr
t_{14523}
   &= - 5 \alpha
         - {\sc b} \beta
         + {\sc c} \gamma
         - {\sc d} \delta;\cr
t_{15432}
   &= - 5 \alpha
         - {\sc b} \beta
         - {\sc c} \gamma +
         {\sc d} \delta;\cr}
   \right\}
   \eqno (24)$$
$$\left. \eqalign{
t_{12453}
   &= - 5 \alpha
         + {\sc b} \gamma
         + {\sc c} \delta
         + {\sc d} \beta ;\cr
t_{14235}
   &= - 5 \alpha
         + {\sc b} \gamma
         - {\sc c} \delta
         - {\sc d} \beta ;\cr
t_{15324}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \gamma}
         + \phantom{{\sc c} \delta}
         - \phantom{{\sc d} \beta } \cr
t_{13542}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \gamma}
         - \phantom{{\sc c} \delta}
         + \phantom{{\sc d} \beta } \cr}
   \right\}
   \eqno (25)$$
$$\left. \eqalign{
t_{12534}
   &= - 5 \alpha
         + {\sc b} \delta
         + {\sc c} \beta
         + {\sc d} \gamma;\cr
t_{15243}
   &= - \phantom{5 \alpha}
         + \phantom{{\sc b} \delta}
         - \phantom{{\sc c} \beta }
         - \phantom{{\sc d} \gamma} \cr
t_{13425}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \delta}
         + \phantom{{\sc c} \beta }
         - \phantom{{\sc d} \gamma} \cr
t_{14352}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \delta}
         - \phantom{{\sc c} \beta }
         + \phantom{{\sc d} \gamma} \cr}
   \right\}
   \eqno (26)$$
$$\left. \eqalign{
t_{12354}
   &= - 5 \alpha
         + {\sc b} \beta
         + {\sc c} \delta
         + {\sc d} \gamma;\cr
t_{13245}
   &= - \phantom{5 \alpha}
         + \phantom{{\sc b} \beta }
         - \phantom{{\sc c} \delta}
         - \phantom{{\sc d} \gamma} \cr
t_{15423}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \beta }
         + \phantom{{\sc c} \delta}
         - \phantom{{\sc d} \gamma} \cr
t_{14532}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \beta }
         - \phantom{{\sc c} \delta}
         + \phantom{{\sc d} \gamma} \cr}
   \right\}
   \eqno (27)$$
$$\left. \eqalign{
t_{12543}
   &= - 5 \alpha
         + {\sc b} \delta
         + {\sc c} \gamma
         + {\sc d} \beta ;\cr
t_{15234}
   &= - \phantom{5 \alpha}
         + \phantom{{\sc b} \delta}
         - \phantom{{\sc c} \gamma}
         - \phantom{{\sc d} \beta } \cr
t_{14325}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \delta}
         + \phantom{{\sc c} \gamma}
         - \phantom{{\sc d} \beta } \cr
t_{13452}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \delta}
         - \phantom{{\sc c} \gamma}
         + \phantom{{\sc d} \beta } \cr}
   \right\}
   \eqno (28)$$
$$\left. \eqalign{
t_{12435}
   &= - 5 \alpha
         + {\sc b} \gamma
         + {\sc c} \beta
         + {\sc d} \delta;\cr
t_{14253}
   &= - \phantom{5 \alpha}
         + \phantom{{\sc b} \gamma}
         - \phantom{{\sc c} \beta }
         - \phantom{{\sc d} \delta} \cr
t_{13524}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \gamma}
         + \phantom{{\sc c} \beta }
         - \phantom{{\sc d} \delta} \cr
t_{15342}
   &= - \phantom{5 \alpha}
         - \phantom{{\sc b} \gamma}
         - \phantom{{\sc c} \beta }
         + \phantom{{\sc d} \delta} \cr}
   \right\}
   \eqno (29)$$
in which we have made, for abridgment,
$$\left. \eqalign{
{\sc b}  &= \omega^4 + \omega^3 - \omega^2 - \omega,\cr
{\sc c}  &= \omega^4 - \omega^3 + \omega^2 - \omega,\cr
{\sc d}  &= \omega^4 - \omega^3 - \omega^2 + \omega.\cr}
   \right\}
   \eqno (30)$$
But also, by (22) and (21),
$$t_{bcdea} = \omega t_{abcde},\quad
  t_{bcdea}^5 = t_{abcde}^5;
   \eqno (31)$$
making then
$$t_{1abcd}^5 = {\sc t}_{abcd},
   \eqno (32)$$
the twenty-four values of the function~$t^5$ will be those of the
function~${\sc t}$ which arise from arranging in all possible
ways the four indices $2$,~$3$,~$4$,~$5$; that is, they are fifth
powers of the twenty-four expressions (24)~$\ldots$~(29).  It is
required, therefore, to develop these fifth powers, and to
examine into their composition.

\bigbreak

15.
For this purpose it is convenient first to consider those parts
of any one such power, which are common to the three other powers
of the same group, (24) or (35), \&c., and, therefore, to
introduce the consideration of six new functions, determined by
the following definition:
$${\sc v}_{abc}
   =  {\textstyle {1 \over 4}}
         (    {\sc t}_{2abc} + {\sc t}_{a2cb}
            + {\sc t}_{bc2a} + {\sc t}_{cba2} );
   \eqno (33)$$
which gives, for example,
$$\left. \eqalign{
{\sc v}_{345}
   &= (-5 \alpha)^5
         + 60 (-5 \alpha)^2 {\sc b} {\sc c} {\sc d}
               \beta \gamma \delta \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 10 \{ (- 5 \alpha)^3 + 2 {\sc b} {\sc c} {\sc d}
               \beta \gamma \delta \}
            (    {\sc b}^2 \beta^2
               + {\sc c}^2 \gamma^2
               + {\sc d}^2 \delta^2 ) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 5 (-5 \alpha)
            ( {\sc b}^4 \beta^4
               + {\sc c}^4 \gamma^4
               + {\sc d}^4 \delta^4
               + 6 {\sc b}^2 {\sc c}^2 \beta^2  \gamma^2
               + 6 {\sc c}^2 {\sc d}^2 \gamma^2 \delta^2
               + 6 {\sc d}^2 {\sc b}^2 \delta^2 \beta^2 );\cr}
   \right\}
   \eqno (34)$$
this being (as is evident on inspection) the part common to the
four functions
${\sc t}_{2345}$, ${\sc t}_{3254}$,
${\sc t}_{4523}$, ${\sc t}_{5432}$,
or to the fifth powers of the four expressions in the group~(24).
By changing $\beta$,~$\gamma$,~$\delta$, first to
$\gamma$,~$\delta$,~$\beta$, and afterwards to
$\delta$,~$\beta$,~$\gamma$, the expression (34) for
${\sc v}_{345}$ will be changed successively to those for
${\sc v}_{453}$ and ${\sc v}_{534}$, which, therefore, it is
unnecessary to write; and
${\sc v}_{354}$,~${\sc v}_{543}$,~${\sc v}_{435}$,
may be formed, respectively,
from ${\sc v}_{345}$,~${\sc v}_{453}$,~${\sc v}_{534}$,
by interchanging $\gamma$ and $\delta$.  Or, after substituting
in (34) for $\beta^2$,~$\gamma^2$,~$\delta^2$, their values~(10),
and writing $\eta$ for $\beta \gamma \delta$, it will only be
necessary to multiply $\kappa$ by $\theta$, and $\lambda$ by
$\theta^2$, wherever they occur, in order to change
${\sc v}_{345}$ to ${\sc v}_{453}$; and to repeat this process,
in order to change ${\sc v}_{453}$ to ${\sc v}_{534}$: while
${\sc v}_{345}$,~${\sc v}_{453}$,~${\sc v}_{534}$
will be changed, respectively, to
${\sc v}_{354}$,~${\sc v}_{543}$,~${\sc v}_{435}$,
by interchanging $\theta$ and $\theta^2$, or $\kappa$ and
$\lambda$.

\bigbreak

16.
In this manner it is not difficult to perceive that we may write
$$\left. \eqalign{
  {\sc v}_{345} &= g + h + i,\cr
  {\sc v}_{453} &= g + \theta   h + \theta^2 i,\cr
  {\sc v}_{534} &= g + \theta^2 h + \theta   i,\cr}
   \right\}
   \eqno (35)$$
and
$$\left. \eqalign{
  {\sc v}_{354} &= g' + h' + i',\cr
  {\sc v}_{543} &= g' + \theta   h' + \theta^2 i',\cr
  {\sc v}_{435} &= g' + \theta^2 h' + \theta   i',\cr}
   \right\}
   \eqno (36)$$
in which,
$$\left. \eqalign{
g = g'
   &= (-5 \alpha)^5
         + 60 (- 5\alpha)^2 \eta {\sc b} {\sc c} {\sc d}
         + 10 \{ (-5 \alpha)^3
                  + 2 \eta {\sc b} {\sc c} {\sc d} \}
            \epsilon ({\sc b}^2 + {\sc c}^2 + {\sc d}^2) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 5 (-5 \alpha) \epsilon^2
            ( {\sc b}^4 + {\sc c}^4 + {\sc d}^4
               + 6 {\sc c}^2 {\sc d}^2 + 6 {\sc d}^2 {\sc b}^2
               + 6 {\sc b}^2 {\sc c}^2 ) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 10 (-5 \alpha) \iota
            ( {\sc b}^4 + {\sc c}^4 + {\sc d}^4
               - 3 {\sc c}^2 {\sc d}^2
               - 3 {\sc d}^2 {\sc b}^2
               - 3 {\sc b}^2 {\sc c}^2 );\cr}
   \right\}
   \eqno (37)$$
$$\left. \multieqalign{
  h  &= k \kappa + l \lambda^2, &
  i  &= \kappa' \lambda + l' \kappa^2;\cr
  h' &= k \lambda + l \kappa^2, &
  i' &= k' \kappa + l' \lambda^2;\cr}
   \right\}
   \eqno (38)$$
$$\left. \eqalign{
k  &= 10 \{ (-5 \alpha)^3 + 2 \eta {\sc b} {\sc c} {\sc d} \}
            (             {\sc b}^2
               + \theta   {\sc c}^2
               + \theta^2 {\sc d}^2 ) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 10 (-5 \alpha) \epsilon
            ( {\sc b}^4 + \theta {\sc c}^4
                  + \theta^2 {\sc d}^4
                  - 3 {\sc c}^2 {\sc d}^2
                  - 3 \theta {\sc d}^2 {\sc b}^2
                  - 3 \theta^2 {\sc b}^2 {\sc c}^2 );\cr
l  &= 5 (-5 \alpha)
            ( {\sc b}^4 + \theta {\sc c}^4 + \theta^2 {\sc d}^4
                + 6 {\sc c}^2 {\sc d}^2
                + 6 \theta {\sc d}^2 {\sc b}^2
                + 6 \theta^2 {\sc b}^2 {\sc c}^2 );\cr}
   \right\}
   \eqno (39)$$
and $k'$, $l'$ are formed from $k$,~$l$, by interchanging
$\theta$ and $\theta^2$.  Hence also, by the same properties of
$\epsilon$,~$\eta$,~$\iota$, which were employed in deducing
these equations, we have:
$$\left. \eqalign{
h h'
   &= k^2 \iota + l^2 \iota^2
       + kl ( \eta^2 - \epsilon^3 + 3 \epsilon \iota );\cr
h^3 + h'^3
   &= 2 (3 k^2 - l^2 \iota) l \iota^2
       + (k^2 + 3 l^2 \iota) k
            ( \eta^2 - \epsilon^3 + 3 \epsilon \iota )
       + l^3
            ( \eta^2 - \epsilon^3 + 3 \epsilon \iota )^2;\cr}
   \right\}
   \eqno (40)$$
and $i i'$, $i^3 + i'^3$ have corresponding expressions, obtained
by accenting $k$ and $l$.

\bigbreak

17.
If then we make
$$g  = {\sc h}_1 + \surd {\sc h}_2,\quad
  g' = {\sc h}_1 - \surd {\sc h}_2;
   \eqno (41)$$
$$h^3 + h'^3 = 2 {\sc h}_3;\quad
  h^3 - h'^3 = 2 \surd {\sc h}_4;
   \eqno (42)$$
$$i'^3 + i^3 = 2 {\sc h}_5,\quad
  i'^3 - i^3 = 2 \surd {\sc h}_6;
   \eqno (43)$$
we see that the six functions~${\sc v}$ may be expressed by the
help of square-roots and cube-roots, in terms of these six
quantities~${\sc h}$, by means of the following formul{\ae}:
$$\left. \eqalign{
{\sc v}_{345}
   &= {\sc h}_1 + \surd {\sc h}_2
       + \root 3 \of {{\sc h}_3 + \surd {\sc h}_4}
       + \root 3 \of {{\sc h}_5 - \surd {\sc h}_6};\cr
{\sc v}_{453}
   &= {\sc h}_1 + \surd {\sc h}_2
       + \theta   \root 3 \of {{\sc h}_3 + \surd {\sc h}_4}
       + \theta^2 \root 3 \of {{\sc h}_5 - \surd {\sc h}_6};\cr
{\sc v}_{534}
   &= {\sc h}_1 + \surd {\sc h}_2
       + \theta^2 \root 3 \of {{\sc h}_3 + \surd {\sc h}_4}
       + \theta   \root 3 \of {{\sc h}_5 - \surd {\sc h}_6};\cr}
   \right\}
   \eqno ({\rm a})$$
and
$$\left. \eqalign{
{\sc v}_{354}
   &= {\sc h}_1 - \surd {\sc h}_2
       + \root 3 \of {{\sc h}_3 - \surd {\sc h}_4}
       + \root 3 \of {{\sc h}_5 + \surd {\sc h}_6};\cr
{\sc v}_{543}
   &= {\sc h}_1 - \surd {\sc h}_2
       + \theta   \root 3 \of {{\sc h}_3 - \surd {\sc h}_4}
       + \theta^2 \root 3 \of {{\sc h}_5 + \surd {\sc h}_6};\cr
{\sc v}_{435}
   &= {\sc h}_1 - \surd {\sc h}_2
       + \theta^2 \root 3 \of {{\sc h}_3 - \surd {\sc h}_4}
       + \theta   \root 3 \of {{\sc h}_5 + \surd {\sc h}_6}:\cr}
   \right\}
   \eqno ({\rm b})$$
which have accordingly, with some slight differences of notation,
been assigned by Professor {\sc Badano}, as among the results of
his method of treating equations of the fifth degree.  We see
too, that the six quantities ${\sc h}_1,\ldots \, {\sc h}_6$, (of
which indeed the second, namely ${\sc h}_2$, vanishes), are
rational functions of $\alpha$,~$\epsilon$,~$\eta$,~$\iota$; and
therefore, by article~13., of $x'$,~$p$,~$q$,~$r$.  But it is
necessary to examine whether it be true, as Professor
{\sc Badano} appears to think (guided in part, as he himself
states, by the analogy of equations of lower degrees), that these
quantities~${\sc h}$ are all rational functions of the
coefficients $p$,~$q$,~$r$,~$s$, of the equation (2) of the fifth
degree; or, in other words, to examine whether it be possible to
eliminate from the expressions of those six quantities~${\sc h}$,
the unknown root~$x'$ of that equation, by its means, in the same
way as it was found possible, in articles 11. and 9. of the
present paper, to eliminate from the correspondent expressions,
the roots of the biquadratic and cubic equations which it was
there proposed to resolve.  For, if it shall be found that any
one of the six quantities ${\sc h}_1,\ldots \, {\sc h}_6$, which
enter into the formul{\ae} (a) and (b), depends essentially, and
not merely in appearance, on the unknown root~$x'$; so as to
change its value when that root is changed to another, such as
$x''$, which satisfies the same equation~(2): it will then be
seen that these formul{\ae}, although true, give no assistance
towards the general solution of the equation of the fifth degree.

\bigbreak

18.
The auxiliary quantities
$\omega$,~${\sc b}$,~${\sc c}$,~${\sc d}$,
being such that, by their definitions (20) and (30),
$$\left. \eqalign{
-1 + {\sc b} + {\sc c} + {\sc d} &= 4 \omega^4,\cr
-1 + {\sc b} - {\sc c} - {\sc d} &= 4 \omega^3,\cr
-1 - {\sc b} + {\sc c} - {\sc d} &= 4 \omega^2,\cr
-1 - {\sc b} - {\sc c} + {\sc d} &= 4 \omega,\cr}
   \right\}
   \eqno (44)$$
while $\omega$,~$\omega^2$,~$\omega^3$,~$\omega^4$ are the four
imaginary fifth roots of unity, we shall have, by the theory of
biquadratics already explained, the following identical equation:
$$\left. \eqalign{
\{ & (x + 1)^2 - ({\sc b}^2 + {\sc c}^2 + {\sc d}^2) \}^2
         - 8 {\sc b} {\sc c} {\sc d} (x + 1)
         - 4 ( {\sc b}^2 {\sc c}^2
         + {\sc c}^2 {\sc d}^2
         + {\sc d}^2 {\sc b}^2) \cr
   &\qquad = \{ (x + 1)^2 + 5 \}^2 + 40 (x + 1) + 180,\cr}
   \right\}
   \eqno (45)$$
the second member being equivalent to
$$x^4+ 4 x^3 + 4^2 x^2 + 4^3 x + 4^4;$$
we find, therefore, that
$${\sc b}^2 + {\sc c}^2 + {\sc d}^2 = -5;\quad
  {\sc b} {\sc c} {\sc d} = -5;\quad
  {\sc b}^2 {\sc c}^2
         + {\sc c}^2 {\sc d}^2
         + {\sc d}^2 {\sc b}^2 = -45;
   \eqno (46)$$
and, consequently,
$${\sc b}^4 + {\sc c}^4 + {\sc d}^4 = 115.
   \eqno (47)$$
Hence, by (37), the common value of $g$ and $g'$, considered as a
function of $\alpha$,~$\epsilon$,~$\eta$,~$\iota$, is:
$$g = g' = 125 ( - 25 \alpha^5 + 50 \alpha^3 \epsilon
       - 60 \alpha^2 \eta + 31 \alpha \epsilon^2
       - 100 \alpha \iota + 4 \epsilon \eta );
   \eqno (48)$$
and if in this we substitute, for the quantities
$\alpha$,~$\epsilon$,~$\eta$,~$\iota$, their values (16), or
otherwise eliminate those quantities by the relations~(15), and
attend to the definitions~(41) of the quantities ${\sc h}_1$ and
${\sc h}_2$, we find:
$${\sc h}_1 = {125 \over 12}
         ( 25 x'^5 + 25 p x'^3 + 25 q x'^2 + 25 r x' + pq );
   \eqno (49)$$
and, as was said already,
$${\sc h}_2 = 0.
   \eqno (50)$$
It is therefore true, of {\it these\/} two quantities~${\sc h}$,
that they are independent of the root~$x'$ of the proposed
equation of the fifth degree, or remain unchanged when that root
is changed to another, such as $x''$, which satisfies the same
equation: since it is possible to eliminate $x'$ from the
expression (49) by means of the proposed equation (2), and so to
obtain ${\sc h}_1$ as a rational function of the coefficients of
that equation, namely,
$${\sc h}_1 = {125 \over 12} ( pq - 25 s ).
   \eqno (51)$$
Indeed, it was evident {\it \`{a} priori\/} that ${\sc h}_1$ must
be found to be equal to {\it some\/} rational function of those
four coefficients, $p$,~$q$,~$r$,~$s$, or some symmetric function
of the five roots of the equation~(2); because it is, by its
definition, the sixth part of the sum of the six
functions~${\sc v}$, and, therefore, the twenty-fourth part of
the sum of the twenty-four different values of the
function~${\sc t}$; or finally the mean of all the different
values which the function~$t^5$ can receive, by all possible
changes of arrangement of the five roots, $x',\ldots \, x^{V}$,
or $x_1,\ldots \, x_5$, among themselves.  The evanescence of
${\sc h}_2$ shows farther, that, in the arrangement assigned
above, the sum of the three first of the six functions~${\sc v}$,
or the sum of the twelve first of the twenty-four
functions~${\sc t}$, is equal to the sum of the other three, or
of the other twelve of these functions.  But we shall find that
it would be erroneous to conclude, from the analogy of these
results, even when combined with the corresponding results for
equations of inferior degrees, that the other four
quantities~${\sc h}$, which enter into the formul{\ae} (a) and
(b), can likewise be expressed as rational functions of the
coefficients of the equation of the fifth degree.

\bigbreak

19.
The auxiliary quantities ${\sc b}^2$, ${\sc c}^2$, ${\sc d}^2$,
being seen, by (46), to be the three roots $z_1$,~$z_2$,~$z_3$,
of the cubic equation
$$z^3 + 5 z^2 - 45 z - 25 = 0,
   \eqno (52)$$
which decomposes itself into one of the first and another of the
second degree, namely,
$$z - 5 = 0,\quad z^2 + 10 z + 5 = 0;
   \eqno (53)$$
we see that one of the three quantities ${\sc b}$, ${\sc c}$,
${\sc d}$, must be real, and $= \pm \surd 5$, while the other two
must be imaginary.  And on referring to the definitions~(30), and
remembering that $\omega$ is an imaginary fifth root of unity, so
that $\omega^4$ and $\omega^3$ are the reciprocals of $\omega$
and $\omega^2$, we easily perceive that the real one of the three
is ${\sc d}$, and that the following expressions hold good:
$${\sc b}^2 = - 5 - 2 {\sc d};\quad
  {\sc c}^2 = -5 + 2 {\sc d};\quad
  {\sc d}^2 = 5;
   \eqno (54)$$
with which we may combine, whenever it may be necessary or
useful, the relation
$${\sc b} {\sc c} = - {\sc d}.
   \eqno (55)$$
If then we make, for abridgment,
$$\zeta = (\theta - \theta^2) {\sc d}
   =  (\theta - \theta^2)
         (\omega^4 - \omega^3 - \omega^2 - \omega),
   \eqno (56)$$
$\theta$ being still the same imaginary cubic root of unity as
before, so that
$$\zeta^2 = - 15;
   \eqno (57)$$
we shall have, in (39),
$$\left. \eqalign{
   & {\sc d}^2 + \theta   {\sc b}^2 + \theta^2 {\sc c}^2
      = 10 - 2 \zeta,\cr
   & {\sc d}^4 + \theta   {\sc b}^4 + \theta^2 {\sc c}^4
      = - 20 + 20 \zeta,\cr
   & {\sc b}^2 {\sc c}^2 + \theta   {\sc c}^2 {\sc d}^2
               + \theta^2 {\sc d}^2 {\sc b}^2
         = 30 + 10 \zeta;\cr}
   \right\}
   \eqno (58)$$
and, consequently (because ${\sc b} {\sc c} {\sc d} = -5$),
$$\left. \eqalign{
\theta k
   &= - 100 (5 - \zeta) (25 \alpha^3 + 2 \eta)
          + 500 (11 + \zeta) \alpha \epsilon;\cr
\theta l
   &= - 2000 (2 + \zeta) \alpha;\cr}
   \right\}
   \eqno (59)$$
while $\theta^2 k'$ and $\theta^2 l'$ are formed from $\theta k$
and $\theta l$, by changing the signs of $\zeta$.  It is easy,
therefore, to see, by the remarks already made, and by the
definitions (42) and (43), that the quantities
${\sc h}_3$,~${\sc h}_4$,~${\sc h}_5$,~${\sc h}_6$, when
expressed as rational functions of
$\alpha$,~$\epsilon$,~$\eta$,~$\iota$, or of
$x'$,~$p$,~$q$,~$r$, will not involve either of the two imaginary
roots of unity, $\theta$ and $\omega$, except so far as they may
involve the combination~$\zeta$ of those roots, or the radical
$\sqrt{-15}$; and that ${\sc h}_5$ will be formed from
${\sc h}_3$, and ${\sc h}_6$ from ${\sc h}_4$, by changing the
sign of this radical.  We shall now proceed to study, in
particular, the composition of the quantity~${\sc h}_4$; because,
although this quantity, when expressed by means of
$x'$,~$p$,~$q$,~$r$, is of the thirtieth dimension relatively to
$x'$, ($p$, $q$, and $r$ being considered as of the second,
third, and fourth dimensions, respectively), while ${\sc h}_3$
rises no higher than the fifteenth dimension; yet we shall find
it possible to decompose ${\sc h}_4$ into two factors, of which
one is the twelfth dimension, and has a very simple meaning,
being the product of the squares of the differences of the four
roots $x''$,~$x'''$,~$x^{IV}$,~$x^{V}$; while the other
factor of ${\sc h}_4$ is an exact square, of a function of the
ninth dimension.  We shall even see it to be possible to
decompose this last function into three factors, which are each
as low as the third dimension, and are rational functions of the
five roots of the original equation of the fifth degree; whereas
it does not appear that ${\sc h}_3$, when regarded as a function
of the same five roots, can be decomposed into more than three
rational factors, nor that any of these can be depressed below
the fifth dimension.

\bigbreak

20.
Confining ourselves then for the present to the consideration of
${\sc h}_4$, we have, by (42) and (38), the following expression
for the square-root of that quantity:
$$\surd {\sc h}_4
   =  {\textstyle {1 \over 2}}
         ( \kappa^3 - \lambda^3 )
         \{ k^3 - 3 k l^2 \kappa \lambda
             - l^3 (\kappa^3 + \lambda^3) \};
   \eqno (60)$$
and, therefore, by (59), and by the same relations between
$\kappa$, $\lambda$, and $\epsilon$,~$\eta$,~$\iota$, which were
used in deducing the formul{\ae} of the sixteenth article, we
obtain the following expression for the quantity~${\sc h}_4$
itself, considered as a function of
$\alpha$,~$\epsilon$,~$\eta$,~$\iota$:
$${\sc h}_4 = 2^{10} 5^{18} \{
            ( \eta^2 - \epsilon^3 + 3 \epsilon \iota )^2
          - 4 \iota^3 \} {\sc l}^2;
   \eqno (61)$$
in which we have made, for abridgment,
$${\sc l} = \mu^3 - 3 \iota \mu \nu^2
            + ( \eta^2 - \epsilon^3 + 3 \epsilon \iota ) \nu^3,
   \eqno (62)$$
and
$$\mu = (-5 + \zeta) (5 \alpha^3 + {\textstyle {2 \over 5}} \eta)
       + (11 + \zeta) \alpha \epsilon,\quad
  \nu = 4 (2 + \zeta) \alpha.
   \eqno (63)$$
Now, without yet entering on the actual process of substituting,
in the expression (61), the values (16) for
$\alpha$,~$\epsilon$,~$\eta$,~$\iota$;
or of otherwise eliminating those four quantities by means of the
equations (15), in order to express ${\sc h}_4$ as a function of
$x'$,~$p$,~$q$,~$r$, from which $x'$ is afterwards to be
eliminated, as far as possible, by the equation of the fifth
degree; we see that, in agreement with the remarks made in the
last article, this expression (61) contains (besides its
numerical coefficient) one factor, namely,
$$( \eta^2 - \epsilon^3 + 3 \epsilon \iota )^2 - 4 \iota^3
   =  (\kappa^3 - \lambda^3)^2,
   \eqno (64)$$
which is of the twelfth dimension; and another, namely,
${\sc l}^2$, which is indeed itself of the eighteenth, but is the
square of a function~(62), which is only of the ninth dimension:
because $\alpha$,~$\epsilon$,~$\eta$,~$\iota$, are to be
considered as being respectively of the first, second, third, and
fourth dimensions; and, therefore, $\mu$ is to be regarded as
being of the third, and $\nu$ of the first dimension.

\bigbreak

21.
Again, on examining the factor (64), we see that it is the square
of another function, namely $\kappa^3 - \lambda^3$, which is
itself of the sixth dimension, and is rational with respect to
$x''$,~$x'''$,~$x^{IV}$,~$x^{V}$, though not with respect
to $\alpha$,~$\epsilon$,~$\eta$,~$\iota$, nor with respect to
$x'$,~$p$,~$q$,~$r$.  This function $\kappa^3 - \lambda^3$ may
even be decomposed into six linear factors; for first, we have,
by (11),
$$\kappa^3 - \lambda^3
   =  (\kappa - \lambda)
      (\kappa - \theta   \lambda)
      (\kappa - \theta^2 \lambda);
   \eqno (65)$$
and secondly, by (10),
$$3 \kappa
   = \beta^2 + \theta^2 \gamma^2 + \theta   \delta^2,\quad
  3 \lambda
   = \beta^2 + \theta   \gamma^2 + \theta^2 \delta^2,
   \eqno (66)$$
expressions which give
$$\left. \eqalign{
\kappa - \lambda
   &= {\textstyle {1 \over 3}} (\theta - \theta^2)
         (\delta^2 - \gamma^2),\cr
\kappa - \theta   \lambda
   &= {\textstyle {1 \over 3}} (1 - \theta)
         (\beta^2 - \delta^2),\cr
\kappa - \theta^2 \lambda
   &= {\textstyle {1 \over 3}} (\theta^2 - 1)
         (\gamma^2 - \beta^2);\cr}
   \right\}
   \eqno (67)$$
but also, by (7),
$$\left. \eqalign{
\delta^2 - \gamma^2
   &= {\textstyle {1 \over 4}}
         (x'' - x''') (x^{V} - x^{IV}),\cr
\beta^2 - \delta^2
   &= {\textstyle {1 \over 4}}
         (x'' - x^{IV}) (x''' - x^{V}),\cr
\gamma^2 - \beta^2
   &= {\textstyle {1 \over 4}}
         (x'' - x^{V}) (x^{IV} - x''');\cr}
   \right\}
   \eqno (68)$$
and
$$(\theta - \theta^2) (1 - \theta) (\theta^2 - 1)
   =  (1 - \theta)^3 = - 3 (\theta - \theta^2);
   \eqno (69)$$
therefore,
$$\kappa^3 - \lambda^3
   = - 2^{-6} 3^{-2} (\theta - \theta^2)
         (x'' - x''')
         (x'' - x^{IV})
         (x'' - x^{V})
         (x''' - x^{IV})
         (x''' - x^{V})
         (x^{IV} - x^{V}).
   \eqno (70)$$
Thus, then, the square of the product of these six linear factors
(70), and of the numerical coefficients annexed, is equal to the
function~(64), of the twelfth dimension, which itself entered as
a factor into the expression~(61) for ${\sc h}_4$; and we see
that this square is free from the imaginary radical~$\theta$,
because, by (11),
$$(\theta - \theta^2)^2 = - 3;
   \eqno (71)$$
and that it is a symmetric function of the four roots,
$x''$,~$x'''$,~$x^{IV}$,~$x^{V}$, being proportional to
the product of the squares of their differences, as was stated in
article~19.: so that this square (though not its root) may be
expressed, in virtue of the biquadratic equation~(6), as a
rational function of $x'$,~$p$,~$q$,~$r$; which followed also
from its being expressible rationally, by (64), in terms of
$\epsilon$,~$\eta$,~$\iota$.

\bigbreak

22.
Introducing now, in the expression~(64), here referred to, the
values~(16), or the relations~(15), we find, after reductions:
$$\left. \eqalign{
\kappa^3 + \lambda^3
   &= \eta^2 - \epsilon^3 + 3 \epsilon \iota \cr
   &= - 2^{-6} 3^{-3} \{ 25 x'^6 + 75 p x'^4
               + (48 p^2 + 45 r) x'^2 + 27 pq x' \cr
   &\mathrel{\phantom{=}} \mathord{}
       - 2 p^3 + 72 pr - 27 q^2 \};\cr}
   \right\}
   \eqno (72)$$
$$\left. \eqalign{
(\kappa^3 + \lambda^3)^2
   &= (\eta^2 - \epsilon^3 + 3 \epsilon \iota)^2 \cr
   &= 2^{-12} 3^{-6} \{ 625 x'^{12} + 3750 p x'^{10}
       + (8025 p^2 + 2250 r) x'^8 + 1350 p q x'^7 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (7100 p^3 + 10350 pr - 1350 q^2) x'^6
       + 4050 p^2 q x'^5 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (2004 p^4 + 15120 p^2 r - 4050 p q^2 + 2025 r^2) x'^4 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (2592 p^3 q + 2430 pqr) x'^3 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (-192 p^5 + 6732 p^3 r - 1863 p^2 q^2 + 6480 p r^2
             - 2430 q^2 r) x'^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (-108 p^4 q + 3888 p^2 qr - 1458 p q^3) x' \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 4 p^6 - 288 p^4 r + 108 p^3 q^2 + 5184 p^2 r^2
             - 3888 p q^2 r + 729 q^4 \};\cr}
   \right\}
   \eqno (73)$$
$$\left. \eqalign{
4 \kappa^3 \lambda^3
   &= 4 \iota^3 \cr
   &= 2^{-10} 3^{-6} \{ 1000 x'^{12} + 3300 p x'^{10}
             + 2700 q x'^9 + (3930 p^2 + 3600 r) x'^8 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 5940 p q x'^7 + (1991 p^3 + 7920 pr + 2430 q^2) x'^6
             + (3807 p^2 q + 6480 qr) x'^5 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (393 p^4 + 5076 p^2 r + 2673 p q^2 + 4320 r^2) x'^4 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (594 p^3 q + 7128 pqr + 729 q^3) x'^3 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (33 p^5 + 792 p^3 r + 243 p^2 q^2
                + 4752 p r^2 + 2916 q^2 r) x'^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (27 p^4 q + 648 p^2 qr + 3888 q r^2) x'
       + p^6 + 36 p^4 r + 432 p^2 r^2 + 1728 r^3 \};\cr}
   \right\}
   \eqno (74)$$
and, finally,
$$\left. \eqalign{
(\kappa^3 - \lambda^3)^2
   &= (\eta^2 - \epsilon^3 + 3 \epsilon \iota)^2 - 4 \iota^3 \cr
   &= - 2^{-12} 3^{-3} \{ 125 x'^{12} + 350 p x'^{10}
             + 400 q x'^9 + (285 p^2 + 450 r) x'^8 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 830 p q x'^7 + (32 p^3 + 790 pr + 410 q^2) x'^6
             + (414 p^2 q + 960 qr) x'^5 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (-16 p^4 + 192 p^2 r + 546 p q^2 + 565 r^2) x'^4 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (-8 p^3 q + 966 pqr + 108 q^3) x'^3 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (12 p^5 - 132 p^3 r + 105 p^2 q^2
             + 464 p r^2 + 522 q^2 r) x'^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (8 p^4 q - 48 p^2 qr + 54 p q^3 + 576 q r^2) x' \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 16 p^4 r - 4 p^3 q^2 - 128 p^2 r^2 + 144 p q^2 r
             + 256 r^3 - 27 q^4 \}.\cr}
   \right\}
   \eqno (75)$$

\bigbreak

23.
This last result may be verified, or rather proved anew, and at
the same time put under another form, which we shall find to be
useful, by a process such as the following.  The biquadratic
equation (6), of which the roots are
$x''$,~$x'''$,~$x^{IV}$,~$x^{V}$, shows that, whatever
$x$ may be,
$$\left. \eqalign{
(x - x'') (x - x''') (x - x^{IV}) (x - x^{V})
   \hskip -12em \cr
   &= x^4 + x' x^3 + x'^2 x^2 + x'^3 x + x'^4
       + p (x^2 + x' x + x'^2) + q (x + x') + r;\cr}
   \right\}
   \eqno (76)$$
and, therefore, that
$$(x' - x'') (x' - x''') (x' - x^{IV}) (x' - x^{V})
   =  5 x'^4 + 3 p x'^2 + 2 q x' + r.
   \eqno (77)$$
If we then multiply the expression (75) by the square of this
last function~(77), we ought to obtain a symmetric function of
all the five roots of the equation of the fifth degree, namely,
the product of the ten squares of their differences, multiplied
indeed by a numerical coefficient, namely, $- 2^{-12} 3^{-3}$, as
appears from (70) and (71): and consequently an expression for
this product itself, that is for
$$\left. \eqalign{
{\sc p}  &= (x' - x'')^2 (x' - x''')^2
         (x' - x^{IV})^2 (x' - x^{V})^2 (x'' - x''')^2
   \hskip -12em \cr
   &\mathrel{\phantom{=}} \mathord{}
      \times (x'' - x^{IV})^2 (x'' - x^{V})^2
      (x''' - x^{IV})^2 (x''' - x^{V})^2
      (x^{IV} - x^{V})^2,\cr}
   \right\}
   \eqno (78)$$
must be obtained by multiplying the factor
$125 x'^{12} + \hbox{\&c.}$ which is within the brackets in (75),
by the square of $5 x'^4 + 3 p x'^2 + 2 q x' + r$,
and then reducing by the condition that
$x'^5 + p x'^3 + q x'^2 + r x' = - s$.  Accordingly this process
gives:
$$\left. \eqalign{
{\sc p}  &= 3125 s^4 - 3750 p q s^3 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (108 p^5 - 900 p^3 r + 825 p^2 q^2
             + 2000 p r^2 + 2250 q^2 r) s^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       - (72 p^4 q r - 16 p^3 q^3 - 560 p^2 q r^2 + 630 p q^3 r
             + 1600 q r^3 - 108 q^5) s \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 16 p^4 r^3 - 4 p^3 q^2 r^2 - 128 p^2 r^4 + 144 p q^2 r^3
             + 256 r^5 - 27 q^4 r^2;\cr}
   \right\}
   \eqno (79)$$
an expression for the product of the squares of the differences
of the five roots of an equation of the fifth degree, which
agrees with known results.  And we see that with this meaning of
${\sc p}$, we may write:
$$(\kappa^3 - \lambda^3)^2
   = - 2^{-12} 3^{-3} {\sc p}
            (5 x'^4 + 3 p x'^2 + 2 q x' + r)^{-2}.
   \eqno (80)$$
The expression (61) for ${\sc h}_4$ becomes, therefore:
$${\sc h}_4 = - 2^{-2} 3^{-3} 5^{18} {\sc p}
         \left(
            {\mu^3 - 3 \iota \mu \nu^2
                + (\eta^2 - \epsilon^3 + 3 \epsilon \iota) \nu^3
               \over 5 x'^4 + 3 p x'^2 + 2 q x' + r}
         \right)^2;
   \eqno (81)$$
$\mu$ and $\nu$ having the meanings defined by (63).

\bigbreak

24.
With respect now to the factor~${\sc l}$, which enters by its
square into the expression~(61), and is the numerator of the
fraction which is squared in the form~(81), we have, by (62),
(63), and (57),
$$\left. \eqalign{
{\sc l}  &=  {\textstyle {4 \over 5}} ( 15625 \alpha^9
             + 24375 \alpha^7 \epsilon + 3750 \alpha^6 \eta
             - 16125 \alpha^5 \epsilon^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
          + 1500 \alpha^5 \iota + 3900 \alpha^4 \epsilon \eta
             + 7605 \alpha^3 \epsilon^3
             - 8820 \alpha^3 \epsilon \iota
             - 6260 \alpha^3 \eta^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
          - 1290 \alpha^2 \epsilon^2 \eta
             + 120 \alpha^2 \eta \iota
             + 156 \alpha \epsilon \eta^2
             + 8 \eta^3 ) \cr
   &\mathrel{\phantom{=}} \mathord{}
          + {\textstyle {12 \over 25}} \zeta (
               15625 ( \alpha^9 - \alpha^7 \epsilon)
             + 3750 \alpha^6 \eta - 125 \alpha^5 \epsilon^2
             + 15500 \alpha^5 \iota
             - 2500 \alpha^4 \epsilon \eta \cr
   &\mathrel{\phantom{=}} \mathord{}
             + 1125 \alpha^3 \epsilon^3
             - 4500 \alpha^3 \epsilon \iota
             - 100 \alpha^3 \eta^2
             - 10 \alpha^2 \epsilon^2 \eta
             + 1240 \alpha^2 \eta \iota
             - 100 \alpha \epsilon \eta^2
             + 8 \eta^3);\cr}
   \right\}
   \eqno (82)$$
and when we substitute for $\alpha$, $\epsilon$, $\eta$,
$\iota$, their values~(16), we find, after reductions, a result
which may be thus written:
$$2^6 5^2 {\sc l} = 5 {\sc l}' - \zeta {\sc l}'';
   \eqno (83)$$
if we make, for abridgment,
$$\left. \eqalign{
{\sc l}' &= 25 p x'^7 + 275 q x'^6 + (135 p^2 - 350 r) x'^5
             + 210 p q x'^4 \cr
   &\mathrel{\phantom{=}} \mathord{}
             + (141 p^3 - 500 pr + 385 q^2) x'^3 \cr
   &\mathrel{\phantom{=}} \mathord{}
          + (93 p^2 q - 20 q r) x'^2 + 20 p q^2 x' - 4 q^3; \cr
{\sc l}'' &= 1750 x'^9 + 2825 p x'^7 + 2100 q x'^6
             + (1120 p^2 + 1825 r) x'^5 + 1615 p q x'^4 \cr
   &\mathrel{\phantom{=}} \mathord{}
          + (39 p^3 + 1060 pr + 500 q^2) x'^3
             + (109 p^2 q + 620 qr) x'^2
             + 68 p q^2 x' + 12 q^3.\cr}
   \right\}
   \eqno (84)$$
With these meanings of ${\sc l}'$ and ${\sc l}''$, the
quantity~${\sc h}_4$, considered as a rational function of
$x'$,~$p$,~$q$,~$r$, may therefore be thus expressed:
$${\sc h}_4 = - 2^{-14} 3^{-3} 5^{14} {\sc p}
         \left(
            {5 {\sc l}' - \zeta {\sc l}''
               \over 5 x'^4 + 3 p x'^2 + 2 q x' + r}
         \right)^2;
   \eqno (85)$$
${\sc p}$ being still the quantity~(79), and $\zeta$ being still
$= \sqrt{-15}$.

\bigbreak

25.
Depressing, next, as far as possible, the degrees of the powers
of $x'$, by means of the equation~(2) of the fifth degree which
$x'$ must satisfy, we find:
$$\left. \eqalign{
{\sc l}'
   &= {\sc l}_0' + {\sc l}_1'  x' + {\sc l}_2'  x'^2
         + {\sc l}_3'  x'^3 + {\sc l}_4'  x'^4;\cr
{\sc l}'' &= {\sc l}_0'' + {\sc l}_1'' x' + {\sc l}_2'' x'^2
         + {\sc l}_3'' x'^3 + {\sc l}_4'' x'^4;\cr}
   \right\}
   \eqno (86)$$
in which the coefficients are thus composed:
$$\left. \eqalign{
{\sc l}_0'  &= - 110 p^2 s - 4 q^3 + 350 rs,\cr
{\sc l}_1'  &= - 110 p^2 r + 20 p q^2 - 275 q s + 350 r^2,\cr
{\sc l}_2'  &= - 17 p^2 q - 25 ps - 55 qr,\cr
{\sc l}_3'  &= + 31 p^3 - 175 pr + 110 q^2,\cr
{\sc l}_4'  &= - 90 pq;\cr}
   \right\}
   \eqno (87)$$
and
$$\left. \eqalign{
{\sc l}_0'' &= - 45 p^2 s + 12 q^3 - 75 rs;\cr
{\sc l}_1'' &= - 45 p^2 r + 68 p q^2 - 350 ps - 75 r^2;\cr
{\sc l}_2'' &= + 64 p^2 q - 1075 ps + 195 qr;\cr
{\sc l}_3'' &= - 6 p^3 - 90 pr + 150 q^2;\cr
{\sc l}_4'' &= + 190 pq - 1750s.\cr}
   \right\}
   \eqno (88)$$
But because, after the completion of all these transformations
and reductions, it is seen that the five quantities
$$5 {\sc l}_0' - \zeta {\sc l}_0'',\quad
  5 {\sc l}_1' - \zeta {\sc l}_1'',\quad
  5 {\sc l}_2' - \zeta {\sc l}_2'',\quad
  5 {\sc l}_3' - \zeta {\sc l}_3'',\quad
  5 {\sc l}_4' - \zeta {\sc l}_4'',
   \eqno (89)$$
which become the coefficients of
$x'^0$,~$x'^1$,~$x'^2$,~$x'^3$,~$x'^4$,
in the numerator $5 {\sc l}' - \zeta {\sc l}''$ of the fraction
to be squared in the formula~(85), are not proportional to the
five other quantities
$$r,\quad 2q,\quad 3p,\quad 0,\quad 5,
   \eqno (90)$$
which are the coefficients of the same five powers of $x'$ in the
denominator of the same fraction, it may be considered as already
evident, at this stage of the investigation, that the root~$x'$
enters, not apparently, but also really, into the composition of
the quantity~${\sc h}_4$.

\bigbreak

26.
The foregoing calculations have been laborious, but they have
been made and verified with care, and it is believed that the
results may be relied on.  Yet an additional light will be thrown
upon the question, by carrying somewhat farther the analysis of
the quantity or function~${\sc h}_4$, and especially of the
factor~${\sc l}$; which, though itself of the ninth dimension
relatively to the roots of the equation of the fifth degree, is
yet, according to a remark made in the nineteenth article,
susceptible of being decomposed into three less complicated
factors; each of these last being rational with respect to the
same five roots, and being only of the third dimension.  In fact,
we have, by (62), and by (11), (12), (13),
$${\sc l} = (\mu + \kappa \nu + \lambda \nu)
         (\mu + \theta   \kappa \nu + \theta^2 \lambda \nu)
         (\mu + \theta^2 \kappa \nu + \theta   \lambda \nu);
   \eqno (91)$$
that is, by (10),
$${\sc l} = (\mu - \epsilon \nu + \beta^2  \nu)
         (\mu - \epsilon \nu + \gamma^2 \nu)
         (\mu - \epsilon \nu + \delta^2 \nu);
   \eqno (92)$$
in which, by the same equations, and by (63) and (57),
$$\left. \eqalign{
\mu - \epsilon \nu
   &= (-5 + \zeta) ( 5 \alpha^3
             + {\textstyle {2 \over 5}} \beta \gamma \delta )
          + ( 1 - \zeta ) \alpha (\beta^2
          + \gamma^2 + \delta^2); \cr
\nu &= (8 + 4 \zeta) \alpha;\quad
\zeta = \sqrt{-15}.\cr}
   \right\}
   \eqno (93)$$
Thus, ${\sc l}$ is seen to be composed of three factors,
$${\sc l} = {\sc m}_1 {\sc m}_2 {\sc m}_3,
   \eqno (94)$$
$${\sc m}_1 = \mu - \epsilon \nu + \beta^2  \nu,\quad
  {\sc m}_2 = \mu - \epsilon \nu + \gamma^2 \nu,\quad
  {\sc m}_3 = \mu - \epsilon \nu + \delta^2 \nu,
   \eqno (95)$$
of which each is a rational, integral, and homogeneous function,
of the third dimension, of the four quantities
$\alpha$,~$\beta$,~$\gamma$,~$\delta$, and, therefore, by (7), of
the four roots $x''$,~$x'''$,~$x^{IV}$,~$x^{V}$, of the
biquadratic equation~(6); or finally, by~(4), of the five roots
$x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$, of the original equation~(1)
of the fifth degree: because we have
$$x'' = x_2 - {\textstyle {1 \over 5}}
            (x_1 + x_2 + x_3 + x_4 + x_5),
   \quad\hbox{\&c.};
   \eqno (96)$$
or because
$$\left. \eqalign{
20 \alpha &= x_2 + x_3 + x_4 + x_5 - 4 x_1,\cr
4 \beta   &= x_2 + x_3 - x_4 - x_5,\cr
4 \gamma  &= x_2 - x_3 + x_4 - x_5,\cr
4 \delta  &= x_2 - x_3 - x_4 + x_5.\cr}
   \right\}
   \eqno (97)$$
And the first of these three factors of ${\sc l}$ may be
expressed by the following equation:
$$100 {\sc m}_1 = 5 {\sc m}_1' - \zeta {\sc m}_1'';
   \eqno (98)$$
in which,
$$\left. \eqalign{
{\sc m}_1' &= 4 x_1^3 - 3 x_1^2 (x_2 + x_3 + x_4 + x_5)
          - 2x_1 (x_2^2 + x_3^2 + x_4^2 + x_5^2) \cr
   &\mathrel{\phantom{=}} \mathord{}
          - 2 x_1 (x_2 x_3 + x_4 x_5)
          + 6 x_1 (x_2 + x_3) (x_4 + x_5) \cr
   &\mathrel{\phantom{=}} \mathord{}
          + 2 \{ x_2 x_3 (x_2 + x_3) + x_4 x_5 (x_4 + x_5) \}
          - 3 \{ x_2 x_3 (x_4 + x_5) + x_4 x_5 (x_2 + x_3) \};\cr}
   \right\}
   \eqno (99)$$
and
$$\left. \eqalign{
{\sc m}_1'' &= 4 x_1^3 - 3 x_1^2 (x_2 + x_3 + x_4 + x_5)
          + 2x_1 (x_2^2 + x_3^2 + x_4^2 + x_5^2) \cr
   &\mathrel{\phantom{=}} \mathord{}
          + 14 x_1 (x_2 x_3 + x_4 x_5)
          - 6 x_1 (x_2 + x_3) (x_4 + x_5) \cr
   &\mathrel{\phantom{=}} \mathord{}
          - 3 \{ x_2 x_3 (x_2 + x_3) + x_4 x_5 (x_4 + x_5) \}
          -   \{ x_2 x_3 (x_4 + x_5) + x_4 x_5 (x_2 + x_3) \} \cr
   &\mathrel{\phantom{=}} \mathord{}
          - \{ x_2^3 + x_3^3 + x_4^3 + x_5^3
             - 2 (x_2^2 + x_3^2) (x_4 + x_5)
             - 2 (x_4^2 + x_5^2) (x_2 + x_3) \};\cr}
   \right\}
   \eqno (100)$$
while the second factor,~${\sc m}_2$, can be formed from
${\sc m}_1$ by merely interchanging $x_3$ and $x_4$; and the
third factor ${\sc m}_3$ from ${\sc m}_2$, by interchanging $x_4$
and $x_5$.

\bigbreak

27.
If, now, we substitute the expression~(94) for the numerator of
the fraction which is to be squared in the formula~(81), and
transform also in like manner the denominator of the same
fraction, by introducing the five original roots
$x_1,\ldots \, x_5$, through the equations~(77) and (4), we find:
$${\sc h}_4
   = - {2^{-2} 3^{-3} 5^{18} {\sc p} {\sc m}_1^2
         {\sc m}_2^2 {\sc m}_3^2
               \over (x_1 - x_2)^2 (x_1 - x_3)^2
                     (x_1 - x_4)^2 (x_1 - x_5)^2};
   \eqno (101)$$
and we see that this quantity cannot be a symmetric function of
those five roots, unless the product of the three factors
${\sc m}_1$,~${\sc m}_2$,~${\sc m}_3$
be divisible by the product of the four differences
$x_1 - x_2,\ldots \, x_1 - x_5$.  But this would require that at
least some of those three factors~${\sc m}$ should be divisible
by one of these four differences, for example by $x_1 - x_2$;
which is not found to be true.  Indeed, if any one of these
factors, for example,~${\sc m}_1$, were supposed to be divisible
by any one difference, such as $x_1 - x_2$, it is easy to see,
from its form, that it ought to be divisible also by each of the
three other differences; because, in ${\sc m}_1$, we may
interchange $x_2$ and $x_3$, or $x_4$ and $x_5$, or may
interchange $x_2$ and $x_4$, or $x_2$ and $x_5$, if we also
interchange $x_3$ and $x_5$, or $x_3$ and $x_4$; but a rational
and integral function of the third dimension cannot have four
different linear divisors, without being identically equal to
zero, which does not happen here.  The same sort of reasoning may
be applied to the expressions~(95), combined with (93), for the
three factors ${\sc m}_1$,~${\sc m}_2$,~${\sc m}_3$, considered
as functions, of the third dimension, of
$\alpha$,~$\beta$,~$\gamma$,~$\delta$; because if any one of
these functions could be divisible by any one of the four
following linear divisors,
$$\left. \eqalign{
x_1 - x_2 &= -5 \alpha - (  \beta  + \gamma + \delta),\cr
x_1 - x_3 &= -5 \alpha - (  \beta  - \gamma - \delta),\cr
x_1 - x_4 &= -5 \alpha - (- \beta  + \gamma - \delta),\cr
x_1 - x_5 &= -5 \alpha - (- \beta  - \gamma + \delta),\cr}
   \right\}
   \eqno (102)$$
it ought from its form to be divisible by all of them, which is
immediately seen to be impossible.  The conclusion of the
twenty-fifth article is, therefore, confirmed anew; and we see,
at the same time, by the theory of biquadratic equations, and by
the meanings of $\epsilon$,~$\eta$,~$\iota$, that the denominator
of the fraction which is to be squared, in the form (81) for
${\sc h}_4$, may be expressed as follows:
$$\left. \eqalign{
5 x'^4 + 3 p x'^2 + 2 q x' + r
   &= (x_1 - x_2) (x_1 - x_3) (x_1 - x_4) (x_1 - x_5) \cr
   &= (5 \alpha)^4 - 6 \epsilon (5 \alpha)^2
          + 8 \eta (5 \alpha) - 3 (\epsilon^2 - 4 \iota);\cr}
   \right\}
   \eqno (103)$$
a result which may be otherwise proved by means of the
relations~(15).

\bigbreak

28.
The investigations in the preceding articles, respecting
equations of the fifth degree, have been based on analogous
investigations made previously with respect to biquadratic
equations; because it was the theory of the equations
last-mentioned which suggested to Professor {\sc Badano} the
formul{\ae} marked (a) and (b) in the seventeenth article of this
paper.  But if those formul{\ae} had been suggested in any other
way, or if they should be assumed as true by definition, and
employed as such to fix the meanings of the quantities~${\sc h}$
which they involve; then, we might seek the values and
composition of these quantities ${\sc h}_1,\ldots \, {\sc h}_6$,
by means of the following converse formul{\ae}, which (with a
slightly abridged notation) have been given by the same author:
$$\left. \eqalign{
{\sc h}_1 + \surd {\sc h}_2
   &= {\textstyle {1 \over 3}}
         ({\sc v}_{345} + {\sc v}_{453} + {\sc v}_{534});\cr
{\sc h}_3 + \surd {\sc h}_4
   &= {\textstyle {1 \over 27}}
         (             {\sc v}_{345}
            + \theta^2 {\sc v}_{453}
            + \theta   {\sc v}_{534})^3;\cr
{\sc h}_5 - \surd {\sc h}_6
   &= {\textstyle {1 \over 27}}
         (             {\sc v}_{345}
            + \theta   {\sc v}_{453}
            + \theta^2 {\sc v}_{534})^3;\cr}
   \right\}
   \eqno ({\rm c})$$
and
$$\left. \eqalign{
{\sc h}_1 - \surd {\sc h}_2
   &= {\textstyle {1 \over 3}}
         ({\sc v}_{354} + {\sc v}_{543} + {\sc v}_{435});\cr
{\sc h}_3 - \surd {\sc h}_4
   &= {\textstyle {1 \over 27}}
         (             {\sc v}_{354}
            + \theta^2 {\sc v}_{543}
            + \theta   {\sc v}_{435})^3;\cr
{\sc h}_5 + \surd {\sc h}_6
   &= {\textstyle {1 \over 27}}
         (             {\sc v}_{354}
            + \theta   {\sc v}_{543}
            + \theta^2 {\sc v}_{435})^3.\cr}
   \right\}
   \eqno ({\rm d})$$
Let us, therefore, employ this other method to investigate the
composition of ${\sc h}_4$, by means of the equation
$$54 \surd {\sc h}_4
   =  (             {\sc v}_{345}
         + \theta^2 {\sc v}_{453}
         + \theta   {\sc v}_{534} )^3
    - (             {\sc v}_{354}
         + \theta^2 {\sc v}_{543}
         + \theta   {\sc v}_{435} )^3;
   \eqno (104)$$
determining still the six functions~${\sc v}$ by the
definition~(33), so that each shall still be the mean of four of
the twenty-four functions~${\sc t}$; and assigning still to these
last functions the significations~(32), or treating them as the
fifth powers of twenty-four different values of {\sc Lagrange}'s
function~$t$, which has itself 120 values: but expressing now
these values of $t$ by the notation
$$t_{abcde}
   =     \omega^5 x_a
       + \omega^4 x_b
       + \omega^3 x_c
       + \omega^2 x_d
       + \omega   x_e,
   \eqno (105)$$
which differs from the notation~(22) only by having lower instead
of upper indices of $x$; and is designed to signify that we now
employ (for the sake of a greater directness and a more evident
generality) the five arbitrary roots $x_1$,~\&c., of the original
equation~(1), between which roots no relation is supposed to
subsist, instead of the other roots $x'$,~\&c., of the
equation~(2), which equation was supposed to have been so
prepared that the sum of its roots should be zero.

\bigbreak

29.
Resuming, then, the calculations on this plan, and making for
abridgment
$${\sc a} = x_a + x_b + x_c + x_d + x_e,
   \eqno (106)$$
so that $-{\sc a}$ is still the coefficient of the fourth power
of $x$ in the equation of the fifth degree; making also
$${\sc w}_{abcde}
   =  x_a^4 x_b + 2 x_a^3 x_d^2 + 4 x_a^3 x_c x_e
       + 6 x_a^2 x_b^2 x_e + 12 x_a^2 x_b x_c x_d,
   \eqno (107)$$
and
$${\sc x}_{bcde}
   =  5 (     {\sc w}_{abcde}
            + {\sc w}_{bcdea} + {\sc w}_{cdeab}
            + {\sc w}_{deabc} + {\sc w}_{eabcd});
   \eqno (108)$$
we find (because $\omega^5 = 1$), for the fifth power of the
combination (105) of the five roots~$x$, the expression:
$$t_{abcde}^5
   =  {\sc a}^5
       + (\omega^4 - 1) {\sc x}_{bcde}
       + (\omega^3 - 1) {\sc x}_{cebd}
       + (\omega   - 1) {\sc x}_{edcb}
       + (\omega^2 - 1) {\sc x}_{dbec};
   \eqno (109)$$
and therefore, for the six functions~${\sc v}$, with the same
meanings of those functions as before, the formula:
$$\left. \eqalign{
{\sc v}_{cde}
   &= {\textstyle {1 \over 4}}
         ( t_{12cde}^5 + t_{1c2ed}^5
               + t_{1de2c}^5 + t_{1edc2}^5 ) \cr
   &= {\sc a}^5 + (\omega + \omega^4 - 2) {\sc y}_{cde}
       + (\omega^2 + \omega^3 - 2) {\sc y}_{dce};\cr}
   \right\}
   \eqno (110)$$
in which,
$$4 {\sc y}_{cde}
   = {\sc x}_{2cde} + {\sc x}_{c2ed}
         + {\sc x}_{de2c} + {\sc x}_{edc2}.
   \eqno (111)$$
If then we make
$$\left. \multieqalign{
{\sc y}_{345} &= {\sc y}_5' + {\sc y}_5'', &
{\sc y}_{435} &= {\sc y}_5' - {\sc y}_5'', \cr
{\sc y}_{453} &= {\sc y}_3' + {\sc y}_3'', &
{\sc y}_{543} &= {\sc y}_3' - {\sc y}_3'', \cr
{\sc y}_{534} &= {\sc y}_4' + {\sc y}_4'', &
{\sc y}_{354} &= {\sc y}_4' - {\sc y}_4''; \cr}
   \right\}
   \eqno (112)$$
we shall have, by (20) and (30), the following system of
expressions for the functions~${\sc v}$:
$$\left. \eqalign{
  {\sc v}_{345}
   &= {\sc a}^5 - 5 {\sc y}_5' + {\sc d} {\sc y}_5'';\cr
  {\sc v}_{453}
   &= {\sc a}^5 - 5 {\sc y}_3' + {\sc d} {\sc y}_3'';\cr
  {\sc v}_{534}
   &= {\sc a}^5 - 5 {\sc y}_4' + {\sc d} {\sc y}_4'';\cr}
   \right\}
   \eqno (113)$$
and
$$\left. \eqalign{
  {\sc v}_{354}
   &= {\sc a}^5 - 5 {\sc y}_4' - {\sc d} {\sc y}_4'';\cr
  {\sc v}_{543}
   &= {\sc a}^5 - 5 {\sc y}_3' - {\sc d} {\sc y}_3'';\cr
  {\sc v}_{435}
   &= {\sc a}^5 - 5 {\sc y}_5' - {\sc d} {\sc y}_5'';\cr}
   \right\}
   \eqno (114)$$
${\sc d}$ being still
$= \omega^4 - \omega^3 - \omega^2 + \omega$,
so that ${\sc d}^2$ is still $= 5$.  We have also the equation:
$$\left. \eqalign{
   &\mathrel{\phantom{=}} \mathord{} \mathbin{\phantom{+}}
         {\sc x}_{2345}
       + {\sc x}_{3254}
       + {\sc x}_{4523}
       + {\sc x}_{5432} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {\sc x}_{2453}
       + {\sc x}_{4235}
       + {\sc x}_{5324}
       + {\sc x}_{3542} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {\sc x}_{2534}
       + {\sc x}_{5243}
       + {\sc x}_{3425}
       + {\sc x}_{4352} \cr
   &= \mathord{} \mathbin{\phantom{+}}
         {\sc x}_{2354}
       + {\sc x}_{3245}
       + {\sc x}_{5423}
       + {\sc x}_{4532} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {\sc x}_{2543}
       + {\sc x}_{5234}
       + {\sc x}_{4325}
       + {\sc x}_{3452} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {\sc x}_{2435}
       + {\sc x}_{4253}
       + {\sc x}_{3524}
       + {\sc x}_{5342};\cr}
   \right\}
   \eqno (115)$$
because the first member may be converted into the second by
interchanging any two of the four roots
$x_2$,~$x_3$,~$x_4$,~$x_5$, on which (and on $x_1$), the
functions~${\sc x}$ depend, and therefore the difference of these
two members must be equal to zero; since, being at highest of the
fifth dimension, it cannot otherwise be divisible by the function
$$\varpi =  (x_2 - x_3) (x_2 - x_4) (x_2 - x_5)
            (x_3 - x_4) (x_3 - x_5) (x_4 - x_5),
   \eqno (116)$$
which is the product of the six differences of the four roots
just mentioned, and is itself of the sixth dimension.  We may
therefore combine with the expressions (113) and (114) the
relations:
$${\sc y}_{345} + {\sc y}_{453} + {\sc y}_{534}
   =  {\sc y}_{354} + {\sc y}_{543} + {\sc y}_{435};
   \eqno (117)$$
and
$${\sc y}_3'' + {\sc y}_4'' + {\sc y}_5'' = 0.
   \eqno (118)$$

\bigbreak

30.
With these preparations for the study of the functions~${\sc v}$,
or of any combination of those functions, let us consider in
particular the first of the three following factors of the
expression~(104) for $54 \surd {\sc h}_4$:
$$\left. \eqalign{
{\sc v}_{345} - {\sc v}_{354}
       + \theta^2 ({\sc v}_{453} - {\sc v}_{543})
       + \theta   ({\sc v}_{534} - {\sc v}_{435});\cr
{\sc v}_{345} - {\sc v}_{543}
       + \theta^2 ({\sc v}_{453} - {\sc v}_{435})
       + \theta   ({\sc v}_{534} - {\sc v}_{354});\cr
{\sc v}_{345} - {\sc v}_{435}
       + \theta^2 ({\sc v}_{453} - {\sc v}_{354})
       + \theta   ({\sc v}_{534} - {\sc v}_{543});\cr}
   \right\}
   \eqno (119)$$
$\theta$ being still an imaginary cube-root of unity.  We find:
$$\left. \eqalign{
{\sc v}_{345} - {\sc v}_{354}
   &=   5 ( {\sc y}_4' - {\sc y}_5') - {\sc d} {\sc y}_3'';\cr
{\sc v}_{534} - {\sc v}_{435}
   &= - 5 ( {\sc y}_4' - {\sc y}_5') - {\sc d} {\sc y}_3'';\cr
{\sc v}_{453} - {\sc v}_{543}
   &= 2 {\sc d} {\sc y}_3'';\cr}
   \right\}
   \eqno (120)$$
expressions which show immediately that
$${\sc v}_{345} + {\sc v}_{453} + {\sc v}_{534}
   =  {\sc v}_{354} + {\sc v}_{543} + {\sc v}_{435},
   \eqno (121)$$
and, therefore, by (c) and (d), that
$${\sc h}_2 = 0,$$
as was otherwise found before.  Also,
$$2 \theta^2 - \theta - 1
   =  (\theta - 1) (2\theta + 1)
   =  - (1 - \theta) (\theta - \theta^2);
   \eqno (122)$$
and, consequently, by (120), the first of the three factors (119)
is equivalent to the product of the two following:
$$1 - \theta,\quad
  5 ({\sc y}_4' - {\sc y}_5') - \zeta {\sc y}_3'';
   \eqno (123)$$
in which, as before,
$$\eta = (\theta - \theta^2){\sc d} = \sqrt{-15}.$$
But, by (112) and (117),
$$2 ({\sc y}_4' - {\sc y}_5')
   =  {\sc y}_{534} - {\sc y}_{435}
         - ( {\sc y}_{345} - {\sc y}_{354} )
   =  2 ( {\sc y}_{534} - {\sc y}_{435} )
         + {\sc y}_{453} - {\sc y}_{543},
   \eqno (124)$$
and
$$2 {\sc y}_3'' = {\sc y}_{453} - {\sc y}_{543};
   \eqno (125)$$
so that the first factor~(119) may be put under the form:
$${\textstyle {1 \over 2}} (1 - \theta)
   \{ 10 ( {\sc y}_{534} - {\sc y}_{435} )
          + (5 - \zeta) ( {\sc y}_{453} - {\sc y}_{543} ) \}.
   \eqno (126)$$
Besides, by (111), the three differences
$${\sc y}_{cde} - {\sc y}_{ced},\quad
  {\sc y}_{cde} - {\sc y}_{edc},\quad
  {\sc y}_{cde} - {\sc y}_{dce},
   \eqno (127)$$
are divisible, respectively, by the three products
$$(x_2 - x_c) (x_d - x_e),\quad
  (x_2 - x_d) (x_e - x_c),\quad
  (x_2 - x_e) (x_c - x_d);
   \eqno (128)$$
and, therefore, the factor~(126) is divisible by the product
$$(x_2 - x_3) (x_4 - x_5),
   \eqno (129)$$
the quotient of this division being a rational and integral and
homogeneous function of the five roots~$x$, which is no higher
than the third dimension, and which it is not difficult to
calculate.

\bigbreak

31.
In this manner we are led to establish an equation of the form:
$${\sc v}_{345} - {\sc v}_{354}
       + \theta^2 ( {\sc v}_{453} - {\sc v}_{543} )
       + \theta   ( {\sc v}_{534} - {\sc v}_{435} )
   =  (1 - \theta) (x_2 - x_3) (x_4 - x_5) {\sc n}_1;
   \eqno (130)$$
in which if we make
$$2 {\sc n}_1 = 10 {\sc n}_1' + (5 - \zeta) {\sc n}_1'',
   \eqno (131)$$
we have
$${\sc n}_1'
   = {{\sc y}_{534} - {\sc y}_{435}
         \over (x_2 - x_3) (x_4 - x_5)},\quad
  {\sc n}_1''
   = {{\sc y}_{453} - {\sc y}_{543}
         \over (x_2 - x_3) (x_4 - x_5)}.
   \eqno (132)$$
Effecting the calculations indicated by these last formul{\ae},
we find
$${\sc n}_1'
   = {\textstyle {5 \over 4}} ({\sc m}_1'' - {\sc m}_1'),\quad
  {\sc n}_1''
   = - {\textstyle {5 \over 2}} {\sc m}_1'',
   \eqno (133)$$
${\sc m}_1'$and ${\sc m}_1''$ being determined by the equations
(99) and (100); and, therefore,  with the meaning~(98) of
${\sc m}_1$, we find the relation:
$${\sc n}_1 = - 125 {\sc m}_1.
   \eqno (134)$$
Thus, the first of the three factors (119) may be put under the
form:
$$ - 125 (1 - \theta)(x_2 - x_3) (x_4 - x_5) {\sc m}_1;
   \eqno (135)$$
in deducing which, it is to be observed, that the first
term,~$x_a^4 x_b$, of the formula~(107) for ${\sc w}_{abcde}$,
gives, by (108), the five following terms of ${\sc x}_{bcde}$:
$$5 x_a^4 x_b + 5 x_b^4 x_c + 5 x_c^4 x_d
      + 5 x_d^4 x_e + 5 x_e^4 x_a;
   \eqno (136)$$
and these five terms of ${\sc x}$ give, respectively, by (111),
the five following parts of ${\sc y}_{cde}$:
$$\left. \eqalign{
   &{\textstyle {5 \over 4}}
      x_1^4 ( x_2 + x_c + x_d + x_e ),\cr
   &{\textstyle {5 \over 4}}
      ( x_2^4 x_c + x_c^4 x_2 + x_d^4 x_e + x_e^4 x_d ),\cr
   &{\textstyle {5 \over 4}}
      ( x_c^4 x_d + x_2^4 x_e + x_e^4 x_2 + x_d^4 x_c ),\cr
   &{\textstyle {5 \over 4}}
      ( x_4^4 x_e + x_e^4 x_d + x_2^4 x_c + x_c^4 x_2 ),\cr
   &{\textstyle {5 \over 4}}
      ( x_e^4 + x_d^4 + x_c^4 + x_2^4 ) x_1;\cr}
   \right\}
   \eqno (137)$$
which are to be combined with the other parts of ${\sc y}$,
derived, in like manner, through ${\sc x}$, from the other terms
of ${\sc w}$, and to be submitted to the processes indicated by
the formul{\ae} (132), in order to deduce the values (133) of
${\sc n}_1'$ and ${\sc n}_1''$, and thence, by (131) and (98),
the relation (134) between ${\sc n}_1$ and ${\sc m}_1$, which
conducts, by (130), to the expression~(135).  For example, the
first and last of the five parts (137) of ${\sc y}$, contribute
nothing to either of the two quotients (132), because those parts
are symmetric relatively to $x_c$,~$x_d$,~$x_e$; but the second
part~(137) contributes
$$- {\textstyle {5 \over 4}}
      ( x_2^3 + x_2^2 x_d + x_2 x_d^2 + x_d^3 + x_e^3
          + x_e^2 x_c + x_e x_c^2 + x_c^3 ),
   \eqno (138)$$
to the quotient
$${{\sc y}_{cde} - {\sc y}_{edc} \over (x_2 - x_d) (x_e - x_c)},
   \eqno (139)$$
and
$$+ {\textstyle {5 \over 4}}
      ( x_2^3 + x_2^2 x_e + x_2 x_e^2 + x_e^3 + x_c^3
          + x_c^2 x_d + x_c x_d^2 + x_d^3 ),
   \eqno (140)$$
to the quotient
$${{\sc y}_{cde} - {\sc y}_{dce} \over (x_2 - x_e) (x_c - x_d)};
   \eqno (141)$$
this second part~(137) of ${\sc y}$ contributes therefore, by
(132),
$$- {\textstyle {5 \over 4}}
      ( x_2^3 + x_2^2 x_3 + x_2 x_3^2 + x_3^3 + x_4^3
          + x_4^2 x_5 + x_4 x_5^2 + x_5^3 ),
   \eqno (142)$$
to the quotient~${\sc n}_1'$, and the same quantity with its sign
changed to the quotient~${\sc n}_1''$: and the other parts of the
same two quotients are determined in a similar manner.

\bigbreak

32.
The two other factors~(119) may respectively be expressed as
follows:
$$- 125 (1 - \theta^2) (x_2 - x_4) (x_3 - x_5) {\sc m}_2,
   \eqno (143)$$
and
$$- 125 (\theta - \theta^2) (x_2 - x_5) (x_3 - x_4) {\sc m}_3;
   \eqno (144)$$
in which, ${\sc m}_2$ and ${\sc m}_3$ are formed from
${\sc m}_1$, as in the twenty-sixth article; because the second
factor~(119) may be formed from the first, by interchanging $x_3$
and $x_4$, and multiplying by $- \theta^2$; and the third factor
may be formed from the second, by interchanging $x_4$ and $x_5$,
and multiplying again by $- \theta^2$.  If then we multiply the
three expressions (135) (143) (144) for the three factors (119)
together, and divide by three, we find:
$$18 \surd {\sc h}_4
   = - 5^9 (\theta - \theta^2) \varpi
            {\sc m}_1 {\sc m}_2 {\sc m}_3;
   \eqno (145)$$
$\varpi$ denoting here the product (116) of the six differences
of the four roots $x_2,\ldots \, x_5$.  The expression (101) for
${\sc h}_4$ itself is therefore reproduced under the form:
$${\sc h}_4
   = - 2^{-2} 3^{-3} 5^{18} \varpi^2
         {\sc m}_1^2 {\sc m}_2^2 {\sc m}_3^2;
   \eqno (146)$$
and the conclusions of former articles are thus confirmed anew,
by a method which is entirely different, in its conception and in
its processes of calculation, from those which were employed
before.

\bigbreak

33.
It may not, however, be useless to calculate, for some particular
equation of the fifth degree, the numerical values of some of the
most important quantities above considered, and so to illustrate
and exemplify some of the chief formul{\ae} already established.
Consider therefore the equation:
$$x^5 - 5 x^3 + 4 x = 0;
   \eqno (147)$$
of which the roots may be arranged in the order:
$$x_1 = 2,\quad
  x_2 = 1,\quad
  x_3 = 0,\quad
  x_4 = -1,\quad
  x_5 = -2;
   \eqno (148)$$
and may (because their sum is zero) be also written thus:
$$x'         = 2,\quad
  x''        = 1,\quad
  x'''       = 0,\quad
  x^{IV} = -1,\quad
  x^{V}  = -2.
   \eqno (149)$$
Employing the notation~(32), in combination with (22) or with
(105), we have now:
$$\left. \eqalign{
{\sc t}_{2345}
   &= (2 + \omega^4 - \omega^2 - 2\omega)^5;\cr
{\sc t}_{3254}
   &= (2 + \omega^3 - 2\omega^2 - \omega)^5;\cr
{\sc t}_{4523}
   &= (2 - \omega^4 - 2\omega^3 + \omega^2)^5;\cr
{\sc t}_{5432}
   &= (2 - 2\omega^4 - \omega^3 + \omega)^5.\cr}
   \right\}
   \eqno (150)$$
But $\omega^5 = 1$; therefore,
$${\sc t}_{5432} = (-2 - \omega^4 + \omega^2 + 2 \omega)^5,
   \eqno (151)$$
and
$${\sc t}_{2345} + {\sc t}_{5432} = 0.
   \eqno (152)$$
Again,
$${\sc t}_{3254} = (1 - \omega^2)^5 (2 - \omega)^5,\quad
  {\sc t}_{4523} = (1 - \omega^3)^5 (2 - \omega^4)^5;
   \eqno (153)$$
and if we make
$$(2 - \omega)^5 = {\sc e} - {\sc o},\quad
  (2 + \omega)^5 = {\sc e} + {\sc o},
   \eqno (154)$$
we shall have
$${\sc e} = 32 + 80 \omega^2 + 10 \omega^4,\quad
  {\sc o} = 80 \omega + 40 \omega^3 + \omega^5;
   \eqno (155)$$
also,
$$(1 - \omega^2)^5
   =  - 5 \omega^2 (1 - \omega^2) (1 - \omega^2 + \omega^4);
   \eqno (156)$$
we find, therefore, by easy calculations,
$$\left. \eqalign{
(1 - \omega^2)^5 {\sc e}
   &= 300 + 430 \omega - 110 \omega^2
         - 540 \omega^3 - 80 \omega^4,\cr
(1 - \omega^2)^5 {\sc o}
   &= 600 + 190 \omega - 405 \omega^2
         - 395 \omega^3 + 10 \omega^4;\cr}
   \right\}
   \eqno (157)$$
and by subtracting the latter of these two products from the
former, and afterwards changing $\omega$ to its reciprocal, we
obtain:
$$\left. \eqalign{
{\sc t}_{3254}
   &= -300 + 240 \omega   + 295 \omega^2
         - 145 \omega^3 - 90 \omega^4,\cr
{\sc t}_{4523}
   &= -300 + 240 \omega^4 + 295 \omega^3
         - 145 \omega^2 - 90 \omega.\cr}
   \right\}
   \eqno (158)$$
We have, therefore, by (20),
$${\sc t}_{3254} + {\sc t}_{4523} = - 750;
   \eqno (159)$$
and, consequently, by (33) and (152),
$${\sc v}_{345} = - {375 \over 2}.
   \eqno (160)$$

\bigbreak

34.
In like manner, to compute, in this example, the second of the
functions~${\sc v}$, we have
$$\left. \eqalign{
{\sc t}_{2453}
   &= (2 + \omega^4 - \omega^3 - 2 \omega^2)^5
      = - {\sc t}_{3542};\cr
{\sc t}_{4235}
   &= (1 - \omega)^5 (2 + \omega^3)^5,\quad
{\sc t}_{5324}
    = (1 - \omega^4)^5 (2 + \omega^2)^5;\cr}
   \right\}
   \eqno (161)$$
adding then the two products~(157) together, and afterwards
changing $\omega $ to $\omega^3$ and $\omega^2$ successively, we
find, by (154):
$$\left. \eqalign{
{\sc t}_{4235}
   &= 900 + 620 \omega^3 - 515 \omega
         - 935 \omega^4 - 70 \omega^2,\cr
{\sc t}_{5324}
   &= 900 + 620 \omega^2 - 515 \omega^4
         - 935 \omega   - 70 \omega^3;\cr}
   \right\}
   \eqno (162)$$
but, by (20), (30), and (54),
$$2 (\omega   + \omega^4) = - 1 + {\sc d},\quad
  2 (\omega^2 + \omega^3) = - 1 - {\sc d},\quad
  {\sc d}^2 = 5;
   \eqno (163)$$
therefore,
$${\sc t}_{2453} + {\sc t}_{3542} = 0,\quad
  {\sc t}_{4235} + {\sc t}_{5324} = 2250 - 1000 {\sc d};
   \eqno (164)$$
and
$${\sc v}_{453} = {\textstyle {1 \over 2}} (1125 - 500 {\sc d}).
   \eqno (165)$$

\bigbreak

35.
To compute the third of the functions~${\sc v}$, we have, in the
present question, the relations:
$${\sc t}_{2534} = - {\sc t}_{3254},\quad
  {\sc t}_{5243} = - {\sc t}_{4235},\quad
  {\sc t}_{3425} = - {\sc t}_{5324},\quad
  {\sc t}_{4352} = - {\sc t}_{4523};
   \eqno (166)$$
and, therefore, by (159) and (164),
$${\sc v}_{534} = - 375 + 250 {\sc d}.
   \eqno (167)$$
For the fourth function~${\sc v}$, we have, by processes entirely
similar to the foregoing:
$$\left. \eqalign{
{\sc t}_{2354} &= - (1 - \omega^3)^5 (2 + \omega^4)^5,\quad
{\sc t}_{4532}  = - (1 - \omega^2)^5 (2 + \omega)^5,\cr
{\sc t}_{2354} &+ {\sc t}_{4532} = - 2250 - 1000 {\sc d};\cr}
   \right\}
   \eqno (168)$$
$$\left. \eqalign{
{\sc t}_{3245} &= - (1 - \omega^4)^5 (2 - \omega^2)^5,\quad
{\sc t}_{5423}  = - (1 - \omega  )^5 (2 - \omega^3)^5,\cr
{\sc t}_{3245} &+ {\sc t}_{5423} = + 750;\cr}
   \right\}
   \eqno (169)$$
$${\sc v}_{354} = - 375 - 250 {\sc d}.
   \eqno (170)$$
For the fifth function~${\sc v}$, we have the relations:
$${\sc t}_{2543} = - {\sc t}_{2354};\quad
  {\sc t}_{5234} = - {\sc t}_{4325};\quad
  {\sc t}_{3452} = - {\sc t}_{4532};
   \eqno (171)$$
and, therefore, by (168),
$${\sc v}_{543} = {\textstyle {1 \over 2}} (1125 + 500 {\sc d}).
   \eqno (172)$$
Finally, for the sixth function~${\sc v}$, we have
$${\sc t}_{2435} = - {\sc t}_{5423};\quad
  {\sc t}_{4253} = - {\sc t}_{3524}:\quad
  {\sc t}_{5342} = - {\sc t}_{3245};
   \eqno (173)$$
and, therefore, by (169),
$${\sc v}_{435} = - {375 \over 2}.
   \eqno (174)$$
The three first values of ${\sc v}$ may therefore be thus
collected:
$${2 \over 125} {\sc v}_{345} = - 3;\quad
  {2 \over 125} {\sc v}_{453} = 9 - 4 {\sc d};\quad
  {2 \over 125} {\sc v}_{534} = -6 + 4 {\sc d};
   \eqno (175)$$
and the three last values, in an inverted order, may in like
manner be expressed by the equations:
$${2 \over 125} {\sc v}_{435} = - 3;\quad
  {2 \over 125} {\sc v}_{543} = 9 + 4 {\sc d};\quad
  {2 \over 125} {\sc v}_{354} = -6 - 4 {\sc d}.
   \eqno (176)$$

\bigbreak

36.
It is evident that these six values of ${\sc v}$ are of the forms
(113) and (114), and that they verify, in the present case, the
general relation (121).  They show also, by (c) and (d) of
article~28., that not only ${\sc h}_2$, but ${\sc h}_1$, vanishes
in this example; the common value of the two sums~(121), of the
three first and three last values of ${\sc v}$, being zero.
Accordingly, if we compare the particular equation (147) with the
general forms (1) and (2), we find the following values of the
coefficients (${\sc b}$, ${\sc c}$, ${\sc d}$, ${\sc e}$, not
having here their recent meanings):
$${\sc a} = 0,\quad
  {\sc b} = - 5,\quad
  {\sc c} = 0,\quad
  {\sc d} = 4,\quad
  {\sc e} = 0,
   \eqno (177)$$
and
$$p = - 5,\quad
  q = 0,\quad
  r = 4,\quad
  s = 0;
   \eqno (178)$$
and therefore the formula~(51) gives here
$${\sc h}_1 = 0.
   \eqno (179)$$
We find also, with the same meanings of $\theta$ and $\zeta$ as
in former articles:
$$\left. \eqalign{
{2 \over 125}
         (             {\sc v}_{345}
            + \theta^2 {\sc v}_{453}
            + \theta   {\sc v}_{534} )
   &= 3 (4 \theta^2 - \theta) + 4 \zeta;\cr
{2 \theta^2 \over 125}
         (             {\sc v}_{354}
            + \theta^2 {\sc v}_{543}
            + \theta   {\sc v}_{435} )
   &= 3 (4 \theta - \theta^2) + 4 \zeta;\cr}
   \right\}
   \eqno (180)$$
and, therefore, by ({\rm c}) and ({\rm d}),
$$\left. \eqalign{
2^3 3^3 5^{-9} ({\sc h}_3 + \surd {\sc h}_4)
   &= \{ 3 (4 \theta^2 - \theta) + 4 \zeta \}^3,\cr
2^3 3^3 5^{-9} ({\sc h}_3 - \surd {\sc h}_4)
   &= \{ 3 (4 \theta - \theta^2) + 4 \zeta \}^3;\cr}
   \right\}
   \eqno (181)$$
equations which give, by (11) and (57):
$$\surd {\sc h}_4
   = 2^{-2} 5^{10} (\theta - \theta^2) (23 + 3 \zeta);
   \eqno (182)$$
and
$${\sc h}_4 = - 2^{-3} 3^1 5^{20} (197 + 69 \zeta).
   \eqno (183)$$
Let us now compare these last numerical results with the general
formul{\ae} found by other methods in earlier articles of this
paper.

\bigbreak

37.
The method of the thirteenth article gives, in the present
article,
$$\left. \eqalign{
   &\alpha = - {\textstyle {1 \over 2}},\quad
    \beta  = 1,\quad
    \gamma = {\textstyle {1 \over 2}},\quad
    \delta = 0,\quad
    \epsilon = {\textstyle {5 \over 12}},\quad
    \eta   = 0,\cr
   &\kappa = {4 + \theta^2 \over 12},\quad
    \lambda = {4 + \theta \over 12},\quad
    \iota = \kappa \lambda = {13 \over 144},\cr
   &\kappa^3 + \lambda^3 = {35 \over 864},\quad
    {\textstyle {1 \over 2}} (\kappa^3 - \lambda^3)
      = - 2^{-5} 3^{-1} (\theta - \theta^2);\cr}
   \right\}
   \eqno (184)$$
and therefore, by (59),
$$\left. \eqalign{
   &{3 \theta k \over 250} = 5 (1 - \zeta),\quad
    {3 \theta l \over 250} = 12 (2 + \zeta),\cr
   &k^3 - 3 k l^2 \kappa \lambda - l^3 (\kappa^3 + \lambda^3)
      = - 2^3 3^1 5^{10} (23 + 3 \zeta);\cr}
   \right\}
   \eqno (185)$$
and, accordingly, if we multiply the last expressions (184) by
the last expression (185), we are led, by the general formula
(60), to the same result for $\surd {\sc h}_4$, and therefore for
${\sc h}_4$, as was obtained in the last article by an entirely
different method.  The general formula (60) may also, in virtue
of the equations (13), (59), (62), (63), (70), (116), and (4), be
written thus:
$$18 \surd {\sc h}_4 = - 5^9 (\theta - \theta^2) \varpi {\sc l};
   \eqno (186)$$
which agrees, by (94), with the general result (145), and in
which we have now
$$\varpi = 1 \mathbin{.} 2 \mathbin{.} 3
         \mathbin{.} 1 \mathbin{.} 2 \mathbin{.} 1
   =  12;
   \eqno (187)$$
while ${\sc l}$ may be calculated by the definitions (62) and
(63), which give, at present, by the values~(184) for $\alpha$,
$\epsilon$, $\eta$, $\iota$,
$$\mu = {\textstyle {5 \over 6}} (1 - \zeta),\quad
  \nu = - 2 (2 + \zeta),
   \eqno (188)$$
and
$${\sc l} = - {\textstyle {15 \over 8}} (23 + 3 \zeta):
   \eqno (189)$$
and thus we arrive again at the same value of $\surd {\sc h}_4$
as before.  The same value of ${\sc l}$ may be obtained in other
ways, by other formul{\ae} of this paper; for example, by those
of the 24th and 25th articles, which give, in the present
question,
$${\sc l}'  = - 2^3 3^1 5^2 23;\quad
  {\sc l}'' = + 2^3 3^2 5^3.
   \eqno (190)$$
We may also decompose ${\sc l}$ into three factors ${\sc m}$,
which are here:
$${\sc m}_1 = - {\textstyle {1 \over 2}} (3 + 4 \zeta);\quad
  {\sc m}_2 =   {\textstyle {1 \over 2}} (3 -   \zeta);\quad
  {\sc m}_3 =   {\textstyle {5 \over 2}};
   \eqno (191)$$
and which conduct still to the same result.

\bigbreak

38.
An equation of the fifth degree, which, like that here assumed as
an example, has all its roots unequal, may have those roots
arranged in 120 different ways; and any one of these arrangements
may be taken as the basis of a verification such as that
contained in the last five articles.  But we have seen that no
such change of arrangement will affect the value of either
${\sc h}_1$ or ${\sc h}_2$; and with respect to ${\sc h}_4$ which
has been more particularly under our consideration in this paper,
it is not difficult to perceive that an interchange of any two of
the four last roots ($x_2$,~$x_3$,~$x_4$,~$x_5$, or
$x''$,~$x'''$,~$x^{IV}$,~$x^{V}$), of the proposed
equation of the fifth degree, will merely change the sign of the
square-root, $\surd {\sc h}_4$, in the foregoing formul{\ae},
without making any change in the value of ${\sc h}_4$ itself,
which has been shown to depend on the first root ($x_1$ or $x'$)
alone.  It will, however, be instructive to exemplify this
last-mentioned dependence, by applying the foregoing general
processes to the case of the equation of the fifth degree~(147),
the two first roots being made to change places with each other,
in such a manner that the order shall now be chosen as follows:
$$x_1 = 1,\quad
  x_2 = 2,\quad
  x_3 = 0,\quad
  x_4 = -1,\quad
  x_5 = -2,
   \eqno (192)$$
or (since the sum of all five vanishes),
$$x' = 1,\quad
  x'' = 2,\quad
  x''' = 0,\quad
  x^{IV} = - 1,\quad
  x^{V} = - 2.
   \eqno (193)$$
We find, for this new case, by calculations of the same sort as
in recent articles of this paper, the following new system of
equations for the values of the six functions~${\sc v}$:
$$\left. \eqalign{
{\textstyle {2 \over 125}} {\sc v}_{345} = 12 + 4 {\sc d};\quad
{\textstyle {2 \over 125}} {\sc v}_{453} = -9 - 4 {\sc d};\quad
{\textstyle {2 \over 125}} {\sc v}_{534} = -3;\cr
{\textstyle {2 \over 125}} {\sc v}_{435} = 12 - 4 {\sc d};\quad
{\textstyle {2 \over 125}} {\sc v}_{543} = -9 + 4 {\sc d};\quad
{\textstyle {2 \over 125}} {\sc v}_{354} = -3;\cr}
   \right\}
   \eqno (194)$$
in which, ${\sc d}$ has again the meaning assigned by (30): and,
consequently,
$$\left. \eqalign{
{2 \theta^2 \over 125}
         (             {\sc v}_{345}
            + \theta^2 {\sc v}_{453}
            + \theta   {\sc v}_{534} )
   &= 3 (5 \theta^2 - 2 \theta) - 4 \zeta;\cr
{\textstyle {2 \over 125}}
         (             {\sc v}_{354}
            + \theta^2 {\sc v}_{543}
            + \theta   {\sc v}_{435} )
   &= 3 (5 \theta - 2 \theta^2) - 4 \zeta;\cr}
   \right\}
   \eqno (195)$$
$$\left. \eqalign{
2^4 3^3 5^{-9} \surd {\sc h}_4
   &= \{ 5 (5 \theta^2 - 2 \theta) - 4 \zeta \}^3
       - \{ 3 (5 \theta - 2 \theta^2) - 4 \zeta \}^3;\cr
\surd {\sc h}_4
   &= 2^{-3} 5^9 7 (\theta - \theta^2) (55 - 6 \zeta);\cr}
   \right\}
   \eqno (196)$$
and
$${\sc h}_4 = - 2^{-6} 3^1 5^{19} 7^2 ( 497 - 132 \zeta):
   \eqno (197)$$
results which differ from those obtained with the former
arrangement of the five roots of the proposed equation~(147), but
of which the agreement with the general formul{\ae} of the
present paper may be evinced by processes similar to those of the
last article.

\bigbreak

39.
As a last example, if the arrangement of the same five roots be
$$x_1 = 0,\quad
  x_2 = 1,\quad
  x_3 = 2,\quad
  x_4 = -1,\quad
  x_5 = -2,
   \eqno (198)$$
we then find easily that all the six quantities~${\sc v}$ vanish,
and, therefore, that we have, with this arrangement,
$$\surd {\sc h}_4 = 0,\quad {\sc h}_4 = 0.
   \eqno (199)$$
All these results respecting the numerical values of ${\sc h}_4$,
for different arrangements of the roots of the proposed
equation~(147), are included in the common expression:
$${\sc h}_4
   = - 2^{-4} 3^3 5^{18}
         \left(
            {5 (72 x' + 5 x'^3) - 2 \zeta (38 x' - 17 x'^3)
               \over 5 x'^4 - 15 x'^2 + 4}
         \right)^2;
   \eqno (200)$$
which results from the formula~(85), combined with (79) and (86)
(87) (88): and thus we have a new confirmation of the correctness
of the foregoing calculations.

\bigbreak

40.
It is then proved, in several different ways, that the
quantity~${\sc h}_4$, in the formul{\ae} which have been marked
in this paper (a), (b), (c), (d), and which have been proposed by
Professor {\sc Badano} for the solution of the general equation
of the fifth degree, is not a symmetric function of the five
roots of that equation.  And since it has been shown that the
expression of this quantity~${\sc h}_4$ contains in general the
imaginary radical~$\zeta$ or $\sqrt{-15}$, which changes sign in
passing to the expression of the analogous quantity~${\sc h}_6$,
we see that these two quantities, ${\sc h}_4$ and ${\sc h}_6$,
are not generally equal to each other, as Professor {\sc Badano},
in a supplement to his essay, appears to think they must be.
They are, on the contrary, found to be in general the two unequal
roots of a quadratic equation, namely,
$${\sc h}_4^2 + {\sc q} {\sc h}_4 + {\sc r}^2 = 0,
   \eqno (201)$$
in which
$${\sc q} = - ({\sc h}_4 + {\sc h}_6)
   =  2^{-13} 3^{-3} 5^{15} \varpi^2
            (5 {\sc l}'^2 - 3 {\sc l}''^2),
   \eqno (202)$$
and
$${\sc r} = \surd {\sc h}_4 \mathbin{.} \surd {\sc h}_6
   =  - 2^{-14} 3^{-3} 5^{15} \varpi^2
            (5 {\sc l}'^2 - 3 {\sc l}''^2),
   \eqno (203)$$
$\varpi$, ${\sc l}'$, and ${\sc l}''$, having the signification
already assigned; and the values of the coefficients ${\sc q}$
and ${\sc r}$ depend essentially, in general, on the choice of
the root~$x'$, although they can always be expressed as rational
functions of that root.

\bigbreak

41.
It does not appear to be necessary to write here the analogous
calculations, which show that the two remaining quantities
${\sc h}_3$ and ${\sc h}_5$, which enter into the same
formul{\ae} (a), (b), (c), (d), are not, in general, symmetric
functions of the five roots of the proposed equation of the fifth
degree, nor equal to each other, but roots of a quadratic
equation, of the same kind with that considered in the last
article.  But it may be remarked, in illustration of this general
result, that for the particular equation of the fifth degree
which has been marked (147) we find, with the arrangement~(148)
of the five roots, the values:
$${\sc h}_3 = 2^{-3} 3^{-2} 5^9 (1809 - 914 \zeta),\quad
  {\sc h}_5 = 2^{-3} 3^{-2} 5^9 (1809 + 914 \zeta);
   \eqno (204)$$
with the arrangement (192),
$${\sc h}_3 = 2^{-2} 3^{-2} 5^9 (1269 + 781 \zeta),\quad
  {\sc h}_5 = 2^{-2} 3^{-2} 5^9 (1269 - 781 \zeta);
   \eqno (205)$$
and, with the arrangement~(198),
$${\sc h}_3 = 0,\quad {\sc h}_5 = 0.
   \eqno (206)$$
The general decomposition of these quantities ${\sc h}_3$ and
${\sc h}_5$, into factors of the fifth dimension, referred to in
a former article, results easily from the equations of definition
(42) and (43), which give:
$$\left. \eqalign{
  2 {\sc h}_3 &= (h + h') (h + \theta  h') (h + \theta^2 h');\cr
  2 {\sc h}_5 &= (i + i') (i + \theta  i') (i + \theta^2 i').\cr}
   \right\}
   \eqno (207)$$
And the same equations, when combined with (40) and (38), show
that the combinations
$${\sc h}_3^2 - {\sc h}_4 = h^3 h'^3,\quad
  {\sc h}_5^2 - {\sc h}_6 = i^3 i'^3,
   \eqno (208)$$
are exact cubes of rational functions of the five roots of the
equation of the fifth degree, which functions are each of the
tenth dimension relatively to those five roots, and are symmetric
relatively to four of them; while each of these functions $h h'$
and $i i'$, decomposes itself into two factors, which are also
rational functions of the five roots, and are no higher than the
fifth dimension.

\bigbreak

42.
In the foregoing articles, we have considered only those six
quantities~${\sc h}$ which were connected with the composition of
the six functions~${\sc v}$, determined by the definition~(33).
But if we establish the expressions,
$$\left. \eqalign{
{\sc t}_{2cde}
   &= {\sc v}_{cde} + {\sc v}_{cde}'
          + {\sc v}_{cde}''
          + {\sc v}_{cde}''',\cr
{\sc t}_{c2ed}
   &= {\sc v}_{cde}
          + \phantom{{\sc v}_{cde}'}
          - \phantom{{\sc v}_{cde}''}
          - \phantom{{\sc v}_{cde}'''} \cr
{\sc t}_{de2c}
   &= {\sc v}_{cde}
          - \phantom{{\sc v}_{cde}'}
          + \phantom{{\sc v}_{cde}''}
          - \phantom{{\sc v}_{cde}'''} \cr
{\sc t}_{edc2}
   &= {\sc v}_{cde}
          - \phantom{{\sc v}_{cde}'}
          - \phantom{{\sc v}_{cde}''}
          + \phantom{{\sc v}_{cde}'''} \cr}
   \right\}
   \eqno (209)$$
which include the definition~(33), and give,
$$\left. \eqalign{
{\sc v}_{cde}'
   &= {\textstyle {1 \over 4}}
            ( {\sc t}_{2cde} + {\sc t}_{c2ed}
          - {\sc t}_{de2c}
          - {\sc t}_{edc2} ),\cr
{\sc v}_{cde}''
   &= {\textstyle {1 \over 4}}
            ( {\sc t}_{2cde}
          - \phantom{{\sc t}_{c2ed}}
          + \phantom{{\sc t}_{de2c}}
          - \phantom{{\sc t}_{edc2}} ),\cr
{\sc v}_{cde}'''
   &= {\textstyle {1 \over 4}}
            ( {\sc t}_{2cde}
          - \phantom{{\sc t}_{c2ed}}
          - \phantom{{\sc t}_{de2c}}
          + \phantom{{\sc t}_{edc2}} ),\cr}
   \right\}
   \eqno (210)$$
we are conducted to expressions for the squares of the three
functions ${\sc v}'$, ${\sc v}''$, ${\sc v}'''$, which are
entirely analogous to those marked (a) and (b), and have
accordingly been assigned under such forms by Professor
{\sc Badano}, involving eighteen new quantities
${\sc h}_7,\ldots \, {\sc h}_{24}$; which quantities, however,
are not found to be symmetric functions of the five roots of the
equation of the fifth degree, though they are symmetric relative
to four of them.

\bigbreak

43.
In making the investigations which conduct to this result, it is
convenient to establish the following definitions, analogous to,
and in combination with, that marked (111):
$$\left. \eqalign{
4 {\sc y}_{cde}'
   &= {\sc x}_{2cde} + {\sc x}_{c2ed}
         - {\sc x}_{de2c} - {\sc x}_{edc2},\cr
4 {\sc y}_{cde}''
   &= {\sc x}_{2cde}
          - \phantom{{\sc x}_{c2ed}}
          + \phantom{{\sc x}_{de2c}}
          - \phantom{{\sc x}_{edc2}},\cr
4 {\sc y}_{cde}'''
   &= {\sc x}_{2cde}
          - \phantom{{\sc x}_{c2ed}}
          - \phantom{{\sc x}_{de2c}}
          + \phantom{{\sc x}_{edc2}};\cr}
   \right\}
   \eqno (211)$$
for thus we obtain,
$$\left. \eqalign{
{\sc x}_{2cde}
   &= {\sc y}_{cde} + {\sc y}_{cde}'
          + {\sc y}_{cde}''
          + {\sc y}_{cde}''',\cr
{\sc x}_{c2ed}
   &= {\sc y}_{cde}
          + \phantom{{\sc y}_{cde}'}
          - \phantom{{\sc y}_{cde}''}
          - \phantom{{\sc y}_{cde}'''},\cr
{\sc x}_{de2c}
   &= {\sc y}_{cde}
          - \phantom{{\sc y}_{cde}'}
          + \phantom{{\sc y}_{cde}''}
          - \phantom{{\sc y}_{cde}'''},\cr
{\sc x}_{edc2}
   &= {\sc y}_{cde}
          - \phantom{{\sc y}_{cde}'}
          - \phantom{{\sc y}_{cde}''}
          + \phantom{{\sc y}_{cde}'''};\cr}
   \right\}
   \eqno (212)$$
$$\left. \eqalign{
{\sc v}_{cde}'
   &= (\omega^4 - \omega) {\sc y}_{cde}'
          + (\omega^3 - \omega^2) {\sc y}_{dce}'',\cr
{\sc v}_{cde}''
   &= (\omega^4 - \omega) {\sc y}_{cde}''
          - (\omega^3 - \omega^2) {\sc y}_{dce}',\cr
{\sc v}_{cde}''
   &= (\omega^4 + \omega - 2) {\sc y}_{cde}'''
          - (\omega^3 + \omega^2 - 2) {\sc y}_{dce}'''.\cr}
   \right\}
   \eqno (213)$$
Introducing also the following notations, analogous to (112),
$$\left. \multieqalign{
{\sc y}_{345}'
   &= {\sc y}_5^{\backprime\prime}
          + {\sc y}_5^{\backprime\prime\prime}, &
{\sc y}_{435}''
   &= {\sc y}_5^{\backprime\prime}
          - {\sc y}_5^{\backprime\prime\prime}, \cr
{\sc y}_{453}'
   &= {\sc y}_3^{\backprime\prime}
          + {\sc y}_3^{\backprime\prime\prime}, &
{\sc y}_{543}''
   &= {\sc y}_3^{\backprime\prime}
          - {\sc y}_3^{\backprime\prime\prime}, \cr
{\sc y}_{534}'
   &= {\sc y}_4^{\backprime\prime}
          + {\sc y}_4^{\backprime\prime\prime}, &
{\sc y}_{354}''
   &= {\sc y}_4^{\backprime\prime}
          - {\sc y}_4^{\backprime\prime\prime}; \cr}
   \right\}
   \eqno (214)$$
$$\left. \multieqalign{
{\sc y}_{345}''
   &= {\sc y}_5^{\backprime\backprime\prime}
          + {\sc y}_5^{\backprime\backprime\prime\prime}, &
{\sc y}_{435}'
   &= {\sc y}_5^{\backprime\backprime\prime}
          - {\sc y}_5^{\backprime\backprime\prime\prime}, \cr
{\sc y}_{453}''
   &= {\sc y}_3^{\backprime\backprime\prime}
          + {\sc y}_3^{\backprime\backprime\prime\prime}, &
{\sc y}_{543}'
   &= {\sc y}_3^{\backprime\backprime\prime}
          - {\sc y}_3^{\backprime\backprime\prime\prime}, \cr
{\sc y}_{534}''
   &= {\sc y}_4^{\backprime\backprime\prime}
          + {\sc y}_4^{\backprime\backprime\prime\prime}, &
{\sc y}_{354}'
   &= {\sc y}_4^{\backprime\backprime\prime}
          - {\sc y}_4^{\backprime\backprime\prime\prime}; \cr}
   \right\}
   \eqno (215)$$
and
$$\left. \multieqalign{
{\sc y}_{345}'''
   &= {\sc y}_5^{\backprime\backprime\backprime\prime}
    + {\sc y}_5^{\backprime\backprime\backprime\prime\prime}, &
{\sc y}_{435}'''
   &= {\sc y}_5^{\backprime\backprime\backprime\prime}
    - {\sc y}_5^{\backprime\backprime\backprime\prime\prime}, \cr
{\sc y}_{453}'''
   &= {\sc y}_3^{\backprime\backprime\backprime\prime}
    + {\sc y}_3^{\backprime\backprime\backprime\prime\prime}, &
{\sc y}_{543}'''
   &= {\sc y}_3^{\backprime\backprime\backprime\prime}
    - {\sc y}_3^{\backprime\backprime\backprime\prime\prime}, \cr
{\sc y}_{534}'''
   &= {\sc y}_4^{\backprime\backprime\backprime\prime}
    + {\sc y}_4^{\backprime\backprime\backprime\prime\prime}, &
{\sc y}_{354}'''
   &= {\sc y}_4^{\backprime\backprime\backprime\prime}
    - {\sc y}_4^{\backprime\backprime\backprime\prime\prime};
         \cr}
   \right\}
   \eqno (216)$$
we find, by (30), results analogous to (113) and (114), namely,
$$\left. \multieqalign{
{\sc v}_{345}'
   &= {\sc b} {\sc y}_5^{\backprime\prime}
          + {\sc c} {\sc y}_5^{\backprime\prime\prime}, &
{\sc v}_{435}'
   &= {\sc b} {\sc y}_5^{\backprime\backprime\prime}
    - {\sc c} {\sc y}_5^{\backprime\backprime\prime\prime}, \cr
{\sc v}_{453}'
   &= {\sc b} {\sc y}_3^{\backprime\prime}
    + {\sc c} {\sc y}_3^{\backprime\prime\prime}, &
{\sc v}_{543}'
   &= {\sc b} {\sc y}_3^{\backprime\backprime\prime}
    - {\sc c} {\sc y}_3^{\backprime\backprime\prime\prime}, \cr
{\sc v}_{534}'
   &= {\sc b} {\sc y}_4^{\backprime\prime}
    + {\sc c} {\sc y}_4^{\backprime\prime\prime}, &
{\sc v}_{354}'
   &= {\sc b} {\sc y}_4^{\backprime\backprime\prime}
    - {\sc c} {\sc y}_4^{\backprime\backprime\prime\prime}; \cr}
   \right\}
   \eqno (217)$$
$$\left. \multieqalign{
{\sc v}_{345}''
   &= {\sc c} {\sc y}_5^{\backprime\backprime\prime}
    + {\sc b} {\sc y}_5^{\backprime\backprime\prime\prime}, &
{\sc v}_{435}''
   &= {\sc c} {\sc y}_5^{\backprime\prime}
    - {\sc b} {\sc y}_5^{\backprime\prime\prime}, \cr
{\sc v}_{453}''
   &= {\sc c} {\sc y}_3^{\backprime\backprime\prime}
    + {\sc b} {\sc y}_3^{\backprime\backprime\prime\prime}, &
{\sc v}_{543}''
   &= {\sc c} {\sc y}_3^{\backprime\prime}
    - {\sc b} {\sc y}_3^{\backprime\prime\prime}, \cr
{\sc v}_{534}''
   &= {\sc c} {\sc y}_4^{\backprime\backprime\prime}
    + {\sc b} {\sc y}_4^{\backprime\backprime\prime\prime}, &
{\sc v}_{354}''
   &= {\sc c} {\sc y}_4^{\backprime\prime}
    - {\sc b} {\sc y}_4^{\backprime\prime\prime}; \cr}
   \right\}
   \eqno (218)$$
and
$$\left. \multieqalign{
{\sc v}_{345}'''
   &= {\sc d} {\sc y}_5^{\backprime\backprime\backprime\prime}
    - 5 {\sc y}_5^{\backprime\backprime\backprime\prime\prime}, &
{\sc v}_{435}'''
   &= {\sc d} {\sc y}_5^{\backprime\backprime\backprime\prime}
    + 5 {\sc y}_5^{\backprime\backprime\backprime\prime\prime},
         \cr
{\sc v}_{453}'''
   &= {\sc d} {\sc y}_3^{\backprime\backprime\backprime\prime}
    - 5 {\sc y}_3^{\backprime\backprime\backprime\prime\prime}, &
{\sc v}_{543}'''
   &= {\sc d} {\sc y}_3^{\backprime\backprime\backprime\prime}
    + 5 {\sc y}_3^{\backprime\backprime\backprime\prime\prime},
         \cr
{\sc v}_{534}'''
   &= {\sc d} {\sc y}_4^{\backprime\backprime\backprime\prime}
    - 5 {\sc y}_4^{\backprime\backprime\backprime\prime\prime}, &
{\sc v}_{354}'''
   &= {\sc d} {\sc y}_4^{\backprime\backprime\backprime\prime}
    + 5 {\sc y}_4^{\backprime\backprime\backprime\prime\prime}.
         \cr}
   \right\}
   \eqno (219)$$
And squaring the eighteen expressions (217) (218) (219), we
obtain others, for the eighteen functions ${\sc v}'^2$,
${\sc v}''^2$, ${\sc v}'''^2$, which depend indeed on eighteen
others of the forms~${\sc y}$, determined by the definitions
(211) (214) (215) (216), but which are free, by (54) and (55),
from the imaginary fifth root of unity,~$\omega$, except so far
as that root enters by means of the combination~${\sc d}$, of
which the square is $= 5$.

\bigbreak

44.
If, now, we write like Professor {\sc Badano} (who uses, indeed,
as has been stated already, a notation slightly different),
$$\left. \eqalign{
{\sc v}_{453}'''^2
   &= {\sc h}_{19} + \surd {\sc h}_{20}
          + \root 3 \of { {\sc h}_{21}
                + \surd {\sc h}_{22} }
	  + \root 3 \of { {\sc h}_{23}
                - \surd {\sc h}_{24} };\cr
{\sc v}_{534}'''^2
   &= {\sc h}_{19} + \surd {\sc h}_{20}
          + \theta   \root 3 \of { {\sc h}_{21}
                + \surd {\sc h}_{22} }
	  + \theta^2 \root 3 \of { {\sc h}_{23}
                - \surd {\sc h}_{24} };\cr
{\sc v}_{345}'''^2
   &= {\sc h}_{19} + \surd {\sc h}_{20}
          + \theta^2 \root 3 \of { {\sc h}_{21}
                + \surd {\sc h}_{22} }
	  + \theta   \root 3 \of { {\sc h}_{23}
                - \surd {\sc h}_{24} };\cr}
   \right\}
   \eqno ({\rm a}''')$$
and
$$\left. \eqalign{
{\sc v}_{543}'''^2
   &= {\sc h}_{19} - \surd {\sc h}_{20}
          + \root 3 \of { {\sc h}_{21}
                - \surd {\sc h}_{22} }
	  + \root 3 \of { {\sc h}_{23}
                + \surd {\sc h}_{24} };\cr
{\sc v}_{435}'''^2
   &= {\sc h}_{19} - \surd {\sc h}_{20}
          + \theta   \root 3 \of { {\sc h}_{21}
                - \surd {\sc h}_{22} }
	  + \theta^2 \root 3 \of { {\sc h}_{23}
                + \surd {\sc h}_{24} };\cr
{\sc v}_{354}'''^2
   &= {\sc h}_{19} - \surd {\sc h}_{20}
          + \theta^2 \root 3 \of { {\sc h}_{21}
                - \surd {\sc h}_{22} }
	  + \theta   \root 3 \of { {\sc h}_{23}
                + \surd {\sc h}_{24} };\cr}
   \right\}
   \eqno ({\rm b}''')$$
together with twelve other expressions similar to these, and to
those already marked (a) and (b), but involving the functions
${\sc v}'$ and ${\sc v}''$; we shall have, as the same author has
remarked, a system of converse formul{\ae}, analogous to (c) and
(d), for the determination of the values of the eighteen
quantities ${\sc h}_7,\ldots \, {\sc h}_{24}$.  Among these, we
shall content ourselves with here examining one of the most
simple, namely the following:
$${\sc h}_{19} = {\textstyle {1 \over 6}}
         (   {\sc v}_{345}'''^2
           + {\sc v}_{453}'''^2
           + {\sc v}_{534}'''^2
           + {\sc v}_{354}'''^2
           + {\sc v}_{543}'''^2
           + {\sc v}_{435}'''^2 );
   \eqno (220)$$
for the purpose of showing, by an example, that this quantity is
not independent of the arrangement of the five roots of the
original equation of the fifth degree.

\bigbreak

45.
Resuming with this view, the equation (147), and the arrangement
of the roots (148), we find the following system of the
twenty-four values of the function~${\sc x}_{bcde}$:
$$\left. \multieqalign{
{\sc x}_{2345} &= -  500; &
{\sc x}_{3254} &= -   90; &
{\sc x}_{4523} &=    240; &
{\sc x}_{5432} &=    500; \cr
{\sc x}_{2453} &=   1165; &
{\sc x}_{4235} &= -  935; &
{\sc x}_{5324} &= -  515; &
{\sc x}_{3542} &= - 1165; \cr
{\sc x}_{2534} &=     90; &
{\sc x}_{5243} &=    935; &
{\sc x}_{3425} &=    515; &
{\sc x}_{4352} &= -  240; \cr}
   \right\}
   \eqno (221)$$
$$\left. \multieqalign{
{\sc x}_{2354} &= -  620; &
{\sc x}_{3245} &= -  295; &
{\sc x}_{5423} &=    145; &
{\sc x}_{4532} &=     70; \cr
{\sc x}_{2543} &=    620; &
{\sc x}_{5234} &= -  720; &
{\sc x}_{4325} &=    720; &
{\sc x}_{3452} &= -   70; \cr
{\sc x}_{2435} &= -  145; &
{\sc x}_{4253} &=    375; &
{\sc x}_{3524} &= -  375; &
{\sc x}_{5342} &=    295; \cr}
   \right\}
   \eqno (222)$$
which give, by (211),
$$\left. \multieqalign{
4 {\sc y}_{345}''' &= -  150; &
4 {\sc y}_{453}''' &=   1450; &
4 {\sc y}_{534}''' &= - 1600; \cr
4 {\sc y}_{435}''' &=    150; &
4 {\sc y}_{543}''' &=    550; &
4 {\sc y}_{354}''' &= -  400; \cr}
   \right\}
   \eqno (223)$$
and, therefore, by (216),
$$\left. \multieqalign{
8 {\sc y}_5^{\backprime\backprime\backprime\prime}
   &=      0; &
8 {\sc y}_3^{\backprime\backprime\backprime\prime}
   &=   2000; &
8 {\sc y}_4^{\backprime\backprime\backprime\prime}
   &= - 2000; \cr
8 {\sc y}_5^{\backprime\backprime\backprime\prime\prime}
   &= -  300; &
8 {\sc y}_3^{\backprime\backprime\backprime\prime\prime}
   &=    900; &
8 {\sc y}_4^{\backprime\backprime\backprime\prime\prime}
   &= - 1200; \cr}
   \right\}
   \eqno (224)$$
whence, by (219),
$$\left. \multieqalign{
{\textstyle {2 \over 125}} {\sc v}_{345}'''
   &=    3; &
{\textstyle {2 \over 125}} {\sc v}_{453}'''
   &= -  9 + 4 {\sc d}; &
{\textstyle {2 \over 125}} {\sc v}_{534}'''
   &=   12 - 4 {\sc d}; \cr
{\textstyle {2 \over 125}} {\sc v}_{435}'''
   &= -  3; &
{\textstyle {2 \over 125}} {\sc v}_{543}'''
   &=    9 + 4 {\sc d}; &
{\textstyle {2 \over 125}} {\sc v}_{354}'''
   &= - 12 - 4 {\sc d}; \cr}
   \right\}
   \eqno (225)$$
and the squares of these six second members are
$$9,\quad 161 \mp 72 {\sc d},\quad 224 \mp 96 {\sc d},
   \eqno (226)$$
so that we have, by (220), with this arrangement of the five
roots of the equation (147),
$${\sc h}_{19} = 2^{-1} 3^{-1} 5^6 197.
   \eqno (227)$$
But with the arrangement (192), we find, by similar calculations,
$$\left. \multieqalign{
{\textstyle {2 \over 125}} {\sc v}_{345}'''
   &=    6 + 4 {\sc d}; &
{\textstyle {2 \over 125}} {\sc v}_{453}'''
   &= -  9 - 4 {\sc d}; &
{\textstyle {2 \over 125}} {\sc v}_{534}'''
   &= -  3; \cr
{\textstyle {2 \over 125}} {\sc v}_{435}'''
   &= -  6 + 4 {\sc d}; &
{\textstyle {2 \over 125}} {\sc v}_{543}'''
   &=    9 - 4 {\sc d}; &
{\textstyle {2 \over 125}} {\sc v}_{354}'''
   &= +  3; \cr}
   \right\}
   \eqno (228)$$
of which the squares are
$$116 \pm 48 {\sc d},\quad 161 \pm 72 {\sc d},\quad 9;
   \eqno (229)$$
and we have now
$${\sc h}_{19} = 2^{-1} 3^{-1} 5^6 11^1 13,
   \eqno (230)$$
a value different from that marked (227).  And, finally, with the
arrangement of the roots (198), we find instead of the quantities
(225) or (228), the following:
$$\mp 18 - 8 {\sc d},\quad \pm 6,\quad 0,
   \eqno (231)$$
of which the squares are
$$644 \pm 288 {\sc d},\quad 36,\quad 0,
   \eqno (232)$$
and give still another value for the quantity~${\sc h}$ now under
consideration, namely,
$${\sc h}_{19} = 2^1 3^{-1} 5^7 17.
   \eqno (233)$$

\bigbreak

46.
The twelve other expressions which have been referred to, as
being analogous to (a) and (b), are of the forms:
$$\eqalignno{
{\sc v}_{345}'^2
   &= {\sc h}_7 + \surd {\sc h}_8
       + \root 3 \of {{\sc h}_9
             + \surd {\sc h}_{10}}
       + \root 3 \of {{\sc h}_{11}
             - \surd {\sc h}_{12}}; &({\rm a}')\cr
{\sc v}_{354}'^2
   &= {\sc h}_7 - \surd {\sc h}_8
       + \root 3 \of {{\sc h}_9
             - \surd {\sc h}_{10}}
       + \root 3 \of {{\sc h}_{11}
             + \surd {\sc h}_{12}}; &({\rm b}')\cr
{\sc v}_{534}''^2
   &= {\sc h}_{13} + \surd {\sc h}_{14}
       + \root 3 \of {{\sc h}_{15}
             + \surd {\sc h}_{16}}
       + \root 3 \of {{\sc h}_{17}
             - \surd {\sc h}_{18}}; &({\rm a}'')\cr
{\sc v}_{435}''^2
   &= {\sc h}_{13} - \surd {\sc h}_{14}
       + \root 3 \of {{\sc h}_{15}
             - \surd {\sc h}_{16}}
       + \root 3 \of {{\sc h}_{17}
             + \surd {\sc h}_{18}}; &({\rm b}'')\cr}$$
and they give, as the simplest of the expressions deduced from
them, the two following, which are analogous to that marked
(220):
$$\eqalignno{
{\sc h}_7    &= {\textstyle {1 \over 6}}
         (   {\sc v}_{345}'^2
           + {\sc v}_{453}'^2
           + {\sc v}_{534}'^2
           + {\sc v}_{354}'^2
           + {\sc v}_{543}'^2
           + {\sc v}_{435}'^2 );
   &(234)\cr
{\sc h}_{13} &= {\textstyle {1 \over 6}}
         (   {\sc v}_{345}''^2
           + {\sc v}_{453}''^2
           + {\sc v}_{534}''^2
           + {\sc v}_{354}''^2
           + {\sc v}_{543}''^2
           + {\sc v}_{435}''^2 ).
   &(235)\cr}$$
For the case of the equation (147), and the arrangement of roots
(148), we find the numerical values:
$$\left. \multieqalign{
{\textstyle {2 \over 5}} {\sc v}_{345}'
   &= - 126 {\sc b} -   7 {\sc c}; &
{\textstyle {2 \over 5}} {\sc v}_{453}'
   &=   202 {\sc b} -  11 {\sc c}; &
{\textstyle {2 \over 5}} {\sc v}_{534}'
   &=    25 {\sc b} +  50 {\sc c}; \cr
{\textstyle {2 \over 5}} {\sc v}_{435}''
   &= - 126 {\sc c} +   7 {\sc b}; &
{\textstyle {2 \over 5}} {\sc v}_{543}''
   &=   202 {\sc c} +  11 {\sc b}; &
{\textstyle {2 \over 5}} {\sc v}_{354}''
   &=    25 {\sc c} -  50 {\sc b}; \cr}
   \right\}
   \eqno (236)$$
$$\left. \multieqalign{
{\textstyle {2 \over 5}} {\sc v}_{435}'
   &= -  18 {\sc b} +  47 {\sc c}; &
{\textstyle {2 \over 5}} {\sc v}_{543}'
   &=   100 {\sc b} - 175 {\sc c}; &
{\textstyle {2 \over 5}} {\sc v}_{354}'
   &= -  61 {\sc b} -  52 {\sc c}; \cr
{\textstyle {2 \over 5}} {\sc v}_{345}''
   &= -  18 {\sc c} -  47 {\sc b}; &
{\textstyle {2 \over 5}} {\sc v}_{453}''
   &=   100 {\sc c} + 175 {\sc b}; &
{\textstyle {2 \over 5}} {\sc v}_{534}''
   &= -  61 {\sc c} +  52 {\sc b}; \cr}
   \right\}
   \eqno (237)$$
which may be obtained, either by the method of article~43.,
combined with the values (221) (222) of the twenty-four
functions~${\sc x}$; or by the formul{\ae} (210), combined with
the following table:
$$\left. \multieqalign{
{\textstyle {2 \over 5}} {\sc t}_{2345}
   &= - 175 {\sc b} - 25 {\sc c}; &
{\textstyle {2 \over 5}} {\sc t}_{2435}
   &= - 150 - 11 {\sc b} - 77 {\sc c}; \cr
{\textstyle {2 \over 5}} {\sc t}_{2453}
   &= + 377 {\sc b} + 89 {\sc c}; &
{\textstyle {2 \over 5}} {\sc t}_{2543}
   &=   450 + 111 {\sc b} + 27 {\sc c} + 200 {\sc d}; \cr
{\textstyle {2 \over 5}} {\sc t}_{2534}
   &=   150 + 77 {\sc b} - 11 {\sc c}; &
{\textstyle {2 \over 5}} {\sc t}_{2354}
   &= - 450 - 111 {\sc b} - 27 {\sc c} - 200 {\sc d}; \cr}
   \right\}
   \eqno (238)$$
and with the condition, that, if we write for abridgment,
$${\sc t}_{bcde}
   =       {\sc t}_{bcde}^{(0)}
       + {\sc b} {\sc t}_{bcde}'
       + {\sc c} {\sc t}_{bcde}''
       + {\sc d} {\sc t}_{bcde}''',
   \eqno (239)$$
we have in general the relations,
$$\left. \eqalign{
{\sc t}_{edcb}
   &=      {\sc t}_{bcde}^{(0)}
       - {\sc b} {\sc t}_{bcde}'
       - {\sc c} {\sc t}_{bcde}''
       + {\sc d} {\sc t}_{bcde}''';\cr
{\sc t}_{cebd}
   &=      {\sc t}_{bcde}^{(0)}
       + {\sc c} {\sc t}_{bcde}'
       - {\sc b} {\sc t}_{bcde}''
       - {\sc d} {\sc t}_{bcde}'''.\cr}
   \right\}
   \eqno (240)$$
And hence, for the same equation of the fifth degree, and the
same arrangement of the roots, we find, by (54) and (55):
$$\left. \eqalign{
  {\sc h}_7
   &= - 2^{-2} 3^{-1} 5^4 ( 10975 + 706 {\sc d} );\cr
  {\sc h}_{13}
   &= - 2^{-2} 3^{-1} 5^4 ( 10975 - 706 {\sc d} ).\cr}
   \right\}
   \eqno (241)$$
But, for the same equation (147), with the arrangement of the
roots (192), we find, by similar calculations, the values:
$$\left. \eqalign{
  {\sc h}_7
   &= - 2^{-2} 3^{-1} 5^4 ( 10975 - 1472 {\sc d} );\cr
  {\sc h}_{13}
   &= - 2^{-2} 3^{-1} 5^4 ( 10975 + 1472 {\sc d} );\cr}
   \right\}
   \eqno (242)$$
and with the arrangement (198),
$$\left. \eqalign{
  {\sc h}_7
   &= - 2^{-2} 3^{-1} 5^4 ( 10975 + 3832 {\sc d} );\cr
  {\sc h}_{13}
   &= - 2^{-2} 3^{-1} 5^4 ( 10975 - 3832 {\sc d} ).\cr}
   \right\}
   \eqno (243)$$
We see, therefore, that in this example, the difference of the
two quantities ${\sc h}_7$ and ${\sc h}_{13}$ is neither equal to
zero, nor independent of the arrangement of the five roots of the
equation of the fifth degree.  However, it may be noticed that in
the same example, the sum of the same two quantities ${\sc h}_7$
and ${\sc h}_{13}$ has not been altered by altering the
arrangement of the roots; and in fact, by the method of the 43rd
article, we find the formula:
$$\left. \eqalign{
- {\textstyle {48 \over 5}} ({\sc h}_7 + {\sc h}_{13})
   &= \mathbin{\phantom{+}}
         ({\sc x}_{2345} - {\sc x}_{5432})^2
       + ({\sc x}_{2453} - {\sc x}_{3542})^2
       + ({\sc x}_{2534} - {\sc x}_{4352})^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + ({\sc x}_{3254} - {\sc x}_{4523})^2
       + ({\sc x}_{4235} - {\sc x}_{5324})^2
       + ({\sc x}_{5243} - {\sc x}_{3425})^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + ({\sc x}_{2354} - {\sc x}_{4532})^2
       + ({\sc x}_{2543} - {\sc x}_{3452})^2
       + ({\sc x}_{2435} - {\sc x}_{5342})^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + ({\sc x}_{3245} - {\sc x}_{5423})^2
       + ({\sc x}_{5234} - {\sc x}_{4325})^2
       + ({\sc x}_{4253} - {\sc x}_{3524})^2;\cr}
   \right\}
   \eqno (244)$$
of which the second member is in general a symmetric function of
the five roots, and gives, in the case of the equation (147), by
(221) and (222), the following numerical value, agreeing with
recent results,
$${\sc h}_7 + {\sc h}_{13} = - 2^{-1} 3^{-1} 5^6 439.
   \eqno (245)$$

\bigbreak

47.
It seems useless to add to the length of this communication, by
entering into any additional details of calculation: since the
foregoing investigations will probably be thought to have
sufficiently established the inadequacy of Professor
{\sc Badano}'s method\footnote*{Professor {\sc Badano}'s rule is,
to substitute, in each~${\sc h}$, for each power of $x'$, the
fifth part of the sum of the corresponding powers of the five
roots, $x',\ldots \, x^{V}$; and he proposes to extend the same
method to equations of all higher degrees.}
for the general solution of equations of the fifth degree,
notwithstanding the elegance of those systems of radicals which
have been proposed by that author for the expression of the
twenty-four values of {\sc Lagrange}'s function~$t^5$.  Indeed,
it is not pretended that a full account has been given, in the
present paper, of the reasons which Professor {\sc Badano} has
assigned for believing that the twenty-four quantities which have
been called ${\sc h}$ are all symmetric\footnote\dag{``Dunque le
${\sc h}$ sono quantit\`{a} costanti sotto la sostituzione di
qualcunque radice dell'equaz\-ione.''  To show that the
constancy, thus asserted, does not exist, has been the chief
object proposed in the present paper; to which the writer has had
opportunities of making some additions, since it was first
communicated to the Academy.}
functions of the five roots of the equation of the fifth degree;
and that those quantities are connected by certain relations
among themselves, which would, if valid, conduct to the following
expression for resolving an equation of that degree, analogous to
the known radical expressions for the solution of less elevated
equations:
$$\eqalign{
t^5
   &= {\sc k}_1 + \surd {\sc k}_2
       + \root 3 \of { {\sc k}_3 + \surd {\sc k}_4}
       + \root 3 \of { {\sc k}_3 - \surd {\sc k}_4} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \surd \{ {\sc k}_5 + \surd {\sc k}_6
       + \root 3 \of { {\sc k}_7 + \surd {\sc k}_8}
       + \root 3 \of { {\sc k}_7 - \surd {\sc k}_8} \} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \surd \{ {\sc k}_5 + \surd {\sc k}_6
       + \theta   \root 3 \of { {\sc k}_7 + \surd {\sc k}_8}
       + \theta^2 \root 3 \of { {\sc k}_7 - \surd {\sc k}_8} \}
            \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \surd \{ {\sc k}_5 + \surd {\sc k}_6
       + \theta^2 \root 3 \of { {\sc k}_7 + \surd {\sc k}_8}
       + \theta   \root 3 \of { {\sc k}_7 - \surd {\sc k}_8} \}.
            \cr}$$
But it has been shown, in the foregoing articles, that at least
some of the relations here referred to, between the twenty-four
quantities~${\sc h}$, do not in general exist; since we have not,
for example, the relation of equality between ${\sc h}_4$ and
${\sc h}_6$, which would be required, in order to justify the
substitution of a single symbol~${\sc k}_4$ for these two
quantities.  It has also been shown that each of these two
unequal quantities, ${\sc h}_4$ and ${\sc h}_6$, in general
changes its value, when the arrangement of the five roots of the
original equation is changed in a suitable manner: and  that
${\sc h}_7$, ${\sc h}_{13}$, ${\sc h}_{19}$, are also unequal,
and change their values, at least in the example above chosen.
And thus it appears, to the writer of the present paper, that the
investigations now submitted to the Academy, by establishing (as
in his opinion they do) the failure of this new and elegant
attempt of an ingenious Italian analyst, have thrown some
additional light on the impossibility (though otherwise proved
before) of resolving the general equation of the fifth degree by
any finite combination of radicals and rational functions.

\bye