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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 2000.

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\centerline{\Largebf ON THE ARGUMENT OF ABEL}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Transactions of the Royal Irish Academy,
   18 (1839), pp.\ 171--259.)}

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

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\centerline{\largerm 2000}

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\centerline{\Largebf NOTE ON THE TEXT}

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The text of this edition is taken from the 18th volume of the
{\it Transactions of the Royal Irish Academy}.  A small number
of obvious typographical errors have been corrected without
comment in articles 1, 3, 5, 6, 10 and 21.

\bigbreak\bigskip

\line{\hfil David R. Wilkins}

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\line{\hfil Dublin, February 2000}

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\noindent
{\largeit On the Argument of {\largesc Abel}, respecting the
Impossibility of expressing a Root of any General Equation above
the Fourth Degree, by any finite Combination of Radicals and
Rational Functions.\par}

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\centerline{\largerm By Sir \largesc William Rowan Hamilton.}
\vskip12pt
\centerline{Read 22nd May, 1837.}
\vskip12pt
\centerline{[{\it Transactions of the Royal Irish Academy},
   vol.~xviii (1839), pp.\ 171--259.]}

\bigskip

[1.]
Let $a_1, a_2,\ldots \, a_n$ be any $n$ arbitrary quantities, or
independent variables, real or imaginary, and let
$a_1', a_2',\ldots \, a_{n'}'$ be any $n'$ radicals, such that
$$a_1'^{\alpha_1'} = f_1(a_1, \ldots \, a_n), \quad \ldots \quad
  a_{n'}'^{\alpha_{n'}'} = f_{n'}(a_1, \ldots \, a_n);$$
again, let $a_1'',\ldots \, a_{n''}''$ be $n''$ new radicals,
such that
$$\eqalign{
a_1''^{\alpha_1''}
   &= f_1'(a_1',\ldots \, a_{n'}', a_1, \ldots \, a_n),\cr
\noalign{\hbox{$\cdots\cdots\cdots$}}
a_{n''}''^{\alpha_{n''}''}
   &= f_{n''}'(a_1',\ldots \, a_{n'}', a_1, \ldots \, a_n);\cr}$$
and so on, till we arrive at a system of equations of the form
$$\eqalign{
a_1^{(m) \alpha_1^{(m)}}
   &= f_1^{(m-1)}(a_1^{(m-1)},\ldots \, a_{n^{(m-1)}}^{(m-1)},
                  a_1^{(m-2)},\ldots \, a_{n^{(m-2)}}^{(m-2)},
                  \ldots \, a_1, \ldots \, a_n),\cr
\noalign{\hbox{$\cdots\cdots\cdots$}}
a_{n^{(m)}}^{(m) \alpha_{n^{(m)}}^{(m)}}
   &= f_{n^{(m)}}^{(m-1)}(a_1^{(m-1)},\ldots \,
         a_{n^{(m-1)}}^{(m-1)},
                  a_1^{(m-2)},\ldots \, a_{n^{(m-2)}}^{(m-2)},
                  \ldots \, a_1, \ldots \, a_n),\cr}$$
the exponents $\alpha_i^{(k)}$ being all integral and prime
numbers greater than unity, and the functions $f_i^{(k-1)}$
being rational, but all being otherwise arbitrary.  Then, if we
represent by $b^{(m)}$ any rational function~$f^{(m)}$ of all the
foregoing quantities $a_i^{(k)}$,
$$b^{(m)}
    = f^{(m)}(a_1^{(m)},\ldots \, a_{n^{(m)}}^{(m)},
              a_1^{(m-1)},\ldots \, a_{n^{(m-1)}}^{(m-1)},
              \ldots \, a_1, \ldots \, a_n),$$
we may consider the quantity $b^{(m)}$ as being also an
irrational function of the $n$ original quantities,
$a_1,\ldots \, a_n$; in which latter view it may be said,
according to a phraseology proposed by {\sc Abel}, to be an
{\it irrational function of the $m^{\rm th}$ order\/}: and may be
regarded as the general type of every conceivable function of any
finite number of independent variables, which can be formed by
any finite number of additions, subtractions, multiplications,
divisions, elevations to powers, and extraction of roots of
functions; since it is obvious that any extraction of a radical
with a composite exponent, such as
${}^{\alpha_2' \alpha_1'} \!\! \surd f_1$,
may be reduced to a system
of successive extractions of radicals with prime exponents, such
as
$${}^{\alpha_1'} \!\! \surd f_1 = f_1',\quad
  {}^{\alpha_2'} \!\! \surd f_1' = f_1''.$$
Insomuch that the question, ``Whether it be possible to express a
root~$x$ of the general equation of the $n^{\rm th}$ degree,
$$x^n + a_1 x^{n-1} + \cdots + a_{n-1} x + a_n = 0,$$
in terms of the coefficients of that equation, by any finite
combination of radicals and rational functions?'', is, as
{\sc Abel} has remarked, equivalent to the question, ``Whether it
be possible to equate a root of the general equation of any given
degree to an irrational function of the coefficients of that
equation, which function shall be of any finite order $m$?'' or
to this other question: ``Is it possible to satisfy, by any
function of the form $b^{(m)}$, the equation
$$b^{(m)n} + a_1 b^{(m)n-1} + \cdots
       + a_{n-1} b^{(m)} + a_n = 0,$$
in which the exponent $n$ is given, but the coefficients
$a_1, a_2,\ldots \, a_n$ are arbitrary?''

\bigbreak

[2.]
For the cases $n=2$, $n=3$, $n=4$, this question has long since
been determined in the affirmative, by the discovery of the known
solutions of the general quadratic, cubic, and biquadratic
equations.

Thus, for $n = 2$, it has long been known that a root~$x$ of the
general quadratic equation,
$$x^2 + a_1 x + a_2 = 0,$$
can be expressed as a finite irrational function of the two
arbitrary coefficients $a_1$,~$a_2$, namely, as the following
function, which is of the first order:
$$x = b' = f'(a_1', a_1, a_2) = {-a_1 \over 2} + a_1',$$
the radical~$a_1'$ being such that
$$a_1'^2 = f_1(a_1, a_2) = {a_1^2 \over 4} - a_2;$$
insomuch that, with this form of the irrational function~$b'$,
the equation
$$b'^2 + a_1 b' + a_2 = 0$$
is satisfied, independently of the quantities $a_1$ and $a_2$,
which remain altogether arbitrary.

Again, it is well known that for $n = 3$, that is, in the case of
the general cubic equation
$$x^3 + a_1 x^2 + a_2 x + a_3 = 0,$$
a root~$x$ may be expressed as an irrational function of the
three arbitrary coefficients $a_1$,~$a_2$,~$a_3$, namely as the
following function, which is of the second order:
$$\eqalign{
x = b'' &= f''(a_1'', a_1', a_1, a_2, a_3) \cr
        &= - {a_1 \over 3} +a_1'' + {c_2 \over a_1''};\cr}$$
the radical of highest order, $a_1''$, being defined by the
equation
$$\eqalign{
a_1''^3 &= f_1'(a_1', a_1, a_2, a_3) \cr
        &= c_1 + a_1',\cr}$$
and the subordinate radical~$a_1'$ being defined by this other
equation
$$a_1'^2 = f_1(a_1, a_2, a_3) = c_1^2 - c_2^3,$$
while $c_1$ and $c_2$ denote for abridgment the following two
rational functions:
$$\eqalign{
c_1 &= - {\textstyle {1\over 54}}
         ( 2a_1^3 - 9 a_1 a_2 + 27 a_3),\cr
c_2 &= {\textstyle {1 \over 9}}
         (a_1^2 - 3 a_2);\cr}$$
so that, with this form of the irrational function~$b''$, the
equation
$$b''^3 + a_1 b''^2 + a_2 b'' + a_3 = 0$$
is satisfied, without any restriction being imposed on the three
coefficients $a_1$,~$a_2$,~$a_3$.

For $n = 4$, that is, for the case of the general biquadratic
equation
$$x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0,$$
it is known in like manner, that a root can be expressed as a
finite irrational function of the coefficients, namely as the
following function, which is of the third order:
$$x = b'''
   = f'''(a_1''', a_2''', a_1'', a_1', a_1, a_2, a_3, a_4)
   = - {a_1 \over 4} + a_1''' + a_2'''
      + {e_4 \over a_1''' a_2'''};$$
wherein
$$\eqalign{
a_1'''^2
   &= f_1''(a_1'', a_1', a_1, a_2, a_3, a_4)
      = e_3 + a_1'' + {e_2 \over a_1''},\cr
a_2'''^2
   &= f_2''(a_1'', a_1', a_1, a_2, a_3, a_4)
      = e_3 + \rho_3 a_1'' + {e_2 \over \rho_3 a_1''},\cr
a_1''^3
   &= f_1'(a_1', a_1, a_2, a_3, a_4) = e_1 + a_1',\cr
a_1'^2
   &= f_1(a_1, a_2, a_3, a_4) = e_1^2 - e_2^3;\cr}$$
$e_4$,~$e_3$,~$e_2$,~$e_1$ denoting for abridgment the following
rational functions:
$$\eqalign{
e_4 &= {\textstyle {1 \over 64}}
         ( - a_1^3 + 4 a_1 a_2 - 8 a_3),\cr
e_3 &= {\textstyle {1 \over 48}}
         (3 a_1^2 - 8 a_2),\cr
e_2 &= {\textstyle {1 \over 144}}
         ( - 3 a_1 a_3 + a_2^2 + 12 a_4),\cr
e_1 &= {\textstyle {1 \over 2}}
         ( 3 e_2 e_3 - e_3^3 + e_4^2 ) \cr
    &= {\textstyle {1 \over 3456}}
         (27 a_1^2 a_4 - 9 a_1 a_2 a_3
          + 2 a_2^3 - 72 a_2 a_4 + 27 a_3^2),\cr}$$
and $\rho_3$ being a root of the numerical equation
$$\rho_3^2 + \rho_3 + 1 = 0.$$
It is known also, that a root~$x$ of the same general biquadratic
equation may be expressed in another way, as an irrational
function of the fourth order of the same arbitrary coefficients
$a_1$,~$a_2$,~$a_3$,~$a_4$, namely the following:
$$\eqalign{
x  &= b^{\rm IV}
      = f^{\rm IV}(a_1^{IV}, a_1''', a_1'', a_1',
            a_1, a_2, a_3, a_4) \cr
   &= - {a_1 \over 4} + a_1''' + a_1^{\rm IV};\cr}$$
the radical~$a_1^{\rm IV}$ being defined by the equation
$$a_1^{{\rm IV}2}
   = f_1'''(a_1''', a_1'', a_1', a_1, a_2, a_3, a_4)
   = - a_1'''^2 + 3 e_3 + {2 e_4 \over a_1'''},$$
while $a_1'''$,~$a_1''$,~$a_1'$, and $e_4$,~$e_3$,~$e_2$,~$e_1$,
retain their recent meanings.  Insomuch that either the function
of third order $b'''$, or the function of fourth order
$b^{\rm IV}$, may be substituted for $x$ in the general
biquadratic equation; or, to express the same thing otherwise,
the two equations following:
$$b'''^4 + a_1 b'''^3 + a_2 b'''^2 + a_3 b''' + a_4 = 0,$$
and
$$b^{{\rm IV}4} + a_1 b^{{\rm IV}3} + a_2 b^{{\rm IV}2}
      + a_3 b^{\rm IV} + a_4 = 0,$$
are both identically true, in virtue merely of the forms of the
irrational functions $b'''$ and $b^{\rm IV}$, and independently
of the values of the four arbitrary coefficients
$a_1$,~$a_2$,~$a_3$,~$a_4$.

But for higher values of $n$ the question becomes more difficult;
and even for the case $n = 5$, that is, for the general equation
of the fifth degree,
$$x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 = 0,$$
the opinions of mathematicians appear to be not yet entirely
agreed respecting the possibility or impossibility of expressing
a root as a function of the coefficients by any finite
combination of radicals and rational functions: or, in other
words, respecting the possibility or impossibilty of satisfying,
by any irrational function~$b^{(m)}$ of any finite order, the
equation
$$b^{(m)5} + a_1 b^{(m)4} + a_2 b^{(m)3} + a_3 b^{(m)2}
   + a_4 b^{(m)} + a_5 = 0,$$
the five coefficients $a_1$,~$a_2$,~$a_3$,~$a_4$,~$a_5$,
remaining altogether arbitrary.  To assist in deciding opinions
upon this important question, by developing and illustrating
(with alterations) the admirable argument of {\sc Abel} against
the possibility of any such expression for a root of the general
equation of the fifth, or any higher degree; and by applying the
principles of the same argument, to show that no expression of
the same kind exists for any root of any general but lower
equation, (quadratic, cubic, or biquadratic,) essentially
distinct from those which have long been known; is the chief
object of the present paper.

\bigbreak

[3.]
In general, if we call an irrational function {\it irreducible},
when it is impossible to express that function, or any one of its
component radicals, by any smaller number of extractions of prime
roots of variables, than the number which the actual
expression of that function or radical involves; even by
introducing roots of constant quantities, or of numerical
equations, which roots are in this whole discussion considered as
being themselves constant quantities, so that they neither
influence the order of an irrational function, nor are included
among the radicals denoted by the symbols $a_1'$, \&c.; then it
is not difficult to prove that such {\it irreducible irrational
functions\/} possess several properties in common, which are
adapted to assist in deciding the question just now stated.

In the first place it may be observed, that, by an easy
preparation, the general irrational function~$b^{(m)}$ of any
order~$m$ may be put under the form
$$b^{(m)} = \sum_{\beta_i^{(m)} < \alpha_i^{(m)}} \mathbin{.}
      ( b_{\beta_1^{(m)},\ldots \,
               \beta_{n^{(m)}}^{(m)}}^{(m-1)}
         \mathbin{.} a_1^{(m)\beta_1^{(m)}} \ldots
           a_{n^{(m)}}^{(m)\beta_{n^{(m)}}^{(m)}} ),$$
in which the coefficient
$b_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m-1)}$
is a function of the order $m - 1$, or of a lower order; the
exponent $\beta_i^{(m)}$ is zero, or any positive integer less
than the prime number $\alpha_i^{(m)}$ which enters as exponent
into the equation of definition of the radical~$a_i^{(m)}$,
namely,
$$a_i^{(m) \alpha_i^{(m)}} = f_i^{(m-1)};$$
and the sign of summation extends to all the
$\alpha_1^{(m)} \mathbin{.} \alpha_2^{(m)} \ldots
         \alpha_{n^{(m)}}^{(m)}$
terms which have exponents $\beta_i^{(m)}$ subject to the
condition just now mentioned.

For, inasmuch as $b^{(m)}$ is, by supposition, a rational
function~$f^{(m)}$ of all the radicals $a_i^{(k)}$, it is, with
respect to any radical of highest order, such as $a_i^{(m)}$, a
function of the form
$$b^{(m)} = { {\sc n} (a_i^{(m)}) \over {\sc m} (a_i^{(m)}) },$$
${\sc m}$ and ${\sc n}$ being here used as signs of some
{\it whole\/} functions, or finite integral polynomes.  Now, if
we denote by $\rho_\alpha$ any root of the numerical equation
$$\rho_\alpha^{(\alpha - 1)} + \rho_\alpha^{(\alpha - 2)}
   + \rho_\alpha^{(\alpha - 3)} + \cdots + \rho_\alpha^2
   + \rho_\alpha + 1 = 0,$$
so that $\rho_\alpha$ is at the same time a root of unity,
because the last equation gives
$$\rho_\alpha^\alpha = 1;$$
and if we suppose the number $\alpha$ to be prime, so that
$$\rho_\alpha, \rho_\alpha^2 , \rho_\alpha^3,\ldots \,
   \rho_\alpha^{(\alpha - 1)}$$
are, in some arrangement or other, the $\alpha - 1$ roots of the
equation above assigned: then, the product of all the
$\alpha - 1$ whole functions following,
$${\sc m} (\rho_\alpha a)
         \mathbin{.} {\sc m} (\rho_\alpha^2 a) \ldots
         {\sc m} (\rho_\alpha^{(\alpha - 1)} a) = {\sc l} (a),$$
is not only itself a whole function of $a$, but is one which,
when multiplied by ${\sc m} (a)$, gives a product of the form
$${\sc l} (a) \mathbin{.} {\sc m} (a) = {\sc k} (a^\alpha),$$
${\sc k}$ being here (as well as ${\sc l}$) a sign of some whole
function.  If then we form the product
$${\sc m} (\rho_{\alpha_i^{(m)}} a_i^{(m)}) \mathbin{.}
      {\sc m}
         (\rho_{\alpha_i^{(m)}}^2 a_i^{(m)}) \ldots
      {\sc m}
         (\rho_{\alpha_i^{(m)}}^{\alpha_i^{(m)} - 1} a_i^{(m)})
  = {\sc l} (a_i^{(m)}),$$
and multiply, by it, both numerator and denominator of the
recently assigned expression for $b^{(m)}$, we obtain this new
expression for that general irrational function,
$$b^{(m)}
   = { {\sc l} (a_i^{(m)}) \mathbin{.} {\sc n} (a_i^{(m)})
         \over {\sc l} (a_i^{(m)})
            \mathbin{.} {\sc m} (a_i^{(m)}) }
   = { {\sc l} (a_i^{(m)}) \mathbin{.} {\sc n} (a_i^{(m)})
      \over {\sc k} (a_i^{(m)\alpha_i^{(m)}}) }
   = { {\sc l} (a_i^{(m)}) \mathbin{.} {\sc n} (a_i^{(m)})
         \over {\sc k} (f_i^{(m-1)}) }
   = {\sc i} (a_i^{(m)});$$
the characteristic~${\sc i}$ denoting here some function, which,
relatively to the radical~$a_i^{(m)}$, is whole, so that it may
be thus developed,
$$b^{(m)} = {\sc i} (a_i^{(m)})
      = {\sc i}_0 + {\sc i}_1 a_i^{(m)}
         + {\sc i}_2 a_i^{(m)2} + \cdots
         + {\sc i}_r \alpha_i^{(m)r},$$
$r$ being a finite positive integer, and the coefficients
${\sc i}_0, {\sc i}_1,\ldots \, {\sc i}_r$ being, in general,
functions of the $m^{\rm th}$ order, but not involving the
radical~$a_i^{(m)}$.  And because the definition of that radical
gives
$$a_i^{(m)h} = a_i^{(m)g} \mathbin{.} (f_i^{(m-1)})^e,$$
if
$$h = g + e \alpha_i^{(m)},$$
it is unnecessary to retain in evidence any of its powers of
which the exponents are not less than $\alpha_i^{(m)}$; we may
therefore put the development of $b^{(m)}$ under the form
$$b^{(m)} = {\sc h}_0 + {\sc h}_1 a_i^{(m)} + \cdots
   + {\sc h}_{\alpha_i^{(m)} - 1}
         (a_i^{(m)})^{\alpha_i^{(m)} - 1},$$
the coefficients ${\sc h}_0, {\sc h}_1,\ldots$ being still, in
general, functions of the $m^{\rm th}$ order, not involving the
radical $a_i^{(m)}$.  It is clear that by a repetition of this
process of transformation, the radicals
$a_1^{(m)},\ldots \, a_{n^{(m)}}^{(m)}$ 
may all be removed from the denominator of the rational function
$f^{(m)}$; and that their exponents in the transformed numerator
may all be depressed below the exponents which define those
radicals: by which means, the development above announced for the
general irrational function~$b^{(m)}$ may be obtained; wherein
the coefficient
$b_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m-1)}$
admits of being analogously developed.

For example, the function of the second order,
$$b'' = - {a_1 \over 3} + a_1'' + {c_2 \over a_1''},$$
which was above assigned as an expression for a root of the
general cubic equation, may be developed thus:
$$b'' = \sum_{\beta_1'' < 3} \mathbin{.}
         (b_{\beta_1''}' \mathbin{.} a_1''^{\beta_1''})
   = b_0' + b_1' a_1'' + b_2' a_1''^2;$$
in which
$$b_0' = - {a_1 \over 3},\quad b_1' = 1,\quad
  b_2' = {c_2 \over a_1''^3} = {c_2 \over f_1'}
       = {c_2 \over c_1 + a_1'}.$$
And this last coefficient~$b_2'$, which is itself a function of
the first order, may be developed thus:
$$b_2' = {c_2 \over c_1 + a_1'} = {\sc b}'
       = \sum_{\beta_1' < 2} \mathbin{.} ({\sc b}_{\beta_1'}
            \mathbin{.} a_1'^{\beta_1'})
       = {\sc b}_0 + {\sc b}_1 a_1';$$
in which
$${\sc b}_0 = {c_2 c_1 \over c_1^2 - a_1'^2}
      = {c_2 c_1 \over c_1^2 - f_1}
      = {c_2 c_1 \over c_2^3} = {c_1 \over c_2^2},\quad
 {\sc b}_1 = {-1 \over c_2^2}.$$

Again, the function of the third order,
$$b''' = {-a_1 \over 4} + a_1''' + a_2'''
   + {e_4 \over a_1''' a_2'''},$$
which expresses a root of the general biquadratic equation, may
be developed as follows:
$$\eqalign{
b'''
   &= \sum_{\beta_1''' < 2 \atop \beta_2''' < 2} \mathbin{.}
      (b_{\beta_1''', \beta_2'''}''
      \mathbin{.} a_1'''^{\beta_1'''}
      \mathbin{.} a_2'''^{\beta_2'''}) \cr
   &= b_{0,0}'' + b_{1,0}'' a_1'''
      + b_{0,1}'' a_2''' + b_{1,1}'' a_1''' a_2''';\cr}$$
in which
$$b_{0,0}'' = {-a_1 \over 4},\quad
  b_{1,0}'' = 1,\quad
  b_{0,1}'' = 1,$$
and
$$\eqalign{b_{1,1}''
   &= {e_4 \over a_1'''^2 \mathbin{.} a_2'''^2}
   = {e_4 \over f_1'' \mathbin{.} f_2'' }
   = {e_4 \over \displaystyle
       \left( e_3 + a_1'' + {e_2 \over a_1''} \right)
       \left( e_3 + \rho_3 a_1''
               + {e_2 \over \rho_3 a_1''} \right)} \cr
   & ={1 \over e_4} \left( e_3 + \rho_3^2 a_1''
       + {e_2 \over \rho_3^2 a_1''} \right).\cr}$$
And this last coefficient~$b_{1,1}''$, which is itself a function
of the second order, may be developed thus:
$$b_{1,1}'' = {\sc b}''
   = \sum_{\beta_1'' < 3} \mathbin{.} ({\sc b}_{\beta_1''}'
         \mathbin{.} a_1''^{\beta_1''})
   = {\sc b}_0' + {\sc b}_1' a_1'' + {\sc b}_2' a_1''^2;$$
in which
$${\sc b}_0' = {e_3 \over e_4},\quad
  {\sc b}_1' = {\rho_3^2 \over e_4},\quad
  {\sc b}_2' = {\rho_3 e_2 \over e_4 a_1''^3}
       = {\rho_3 e_2 \over e_4 (e_1 + a_1')}
       = {\rho_3 (e_1 - a_1') \over e_4 e_2^2}.$$
So that, upon the whole, these functions $b''$ and $b'''$, which
express, respectively, roots of the general cubic and biquadratic
equations, may be put under the following forms, whch involve no
radicals in denominators:
$$b'' = {-a_1 \over 3} + a_1'' + (c_1 - a_1')
          \left( {a_1'' \over c_2} \right)^2,$$
and
$$b'' = {-a_1 \over 4} + a_1''' + a_2'''
      + {1 \over e_4} \left\{ e_3 + \rho_3^2 a_1''
         + \rho_3 (e_1 - a_1')
         \left( {a_1'' \over e_2} \right)^2
         \right\} a_1''' a_2''';$$
and the functions $f_1''$,~$f_2''$, which enter into the
equations of definition of the radicals $a_1'''$,~$a_2'''$,
namely into the equations
$$a_1'''^2 = f_1'',\quad a_2'''^2 = f_2'',$$
may in like manner be expressed so as to involve no radicals in
denominators, namely thus:
$$\eqalign{
a_1'''^2
   &= e_3 + a_1'' + (e_1 - a_1')
         \left( {a_1'' \over e_2} \right)^2,\cr
a_2'''^2
   &= e_3 + \rho_3 a_1'' + \rho_3^2 (e_1 - a_1')
         \left( {a_1'' \over e_2} \right)^2.\cr}$$

It would be easy to give other instances of the same sort of
transformation, but it seems unnecessary to do so.

\bigbreak

[4.]
It is important in the next place to observe, that any term of
the foregoing general development of the general irrational
function~$b^{(m)}$, may be isolated from the rest, and expressed
separately, as follows.  Let
$b_{\gamma_1^{(m)},\ldots \, \gamma_{n^{(m)}}^{(m)}}^{(m)}$
denote a new irrational function, which is formed from $b^{(m)}$
by changing every radical such as $a_i^{(m)}$ to a corresponding
product such as
$\rho_{\alpha_i^{(m)}}^{\gamma_i^{(m)}} a_i^{(m)}$, in which
$\rho_{\alpha_i^{(m)}}$ is, as before, a root of unity; so that
$$b_{\gamma_1^{(m)},\ldots \, \gamma_{n^{(m)}}^{(m)}}^{(m)}
   = \sum_{\beta_i^{(m)} < \alpha_i^{(m)}} \mathbin{.}
      (b_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m-1)}
         \mathbin{.}
      \rho_{\alpha_1^{(m)}}^{\beta_1^{(m)} \gamma_1^{(m)}} \ldots
      \rho_{\alpha_{n^{(m)}}^{(m)}}^{\beta_{n^{(m)}}^{(m)}
            \gamma_{n^{(m)}}^{(m)}} \mathbin{.}
      a_1^{(m) \beta_1^{(m)}} \ldots
      a_{n^{(m)}}^{(m) \beta_{n^{(m)}}^{(m)}} );$$
and let any isolated term of the corresponding development of
$b^{(m)}$ or $b_{0,\ldots \,0}^{(m)}$ be denoted by the symbol
$$t_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m)}
   = b_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m-1)}
         \mathbin{.}
      a_1^{(m) \beta_1^{(m)}} \ldots
      a_{n^{(m)}}^{(m) \beta_{n^{(m)}}^{(m)}};$$
we shall then have, as the announced expression for the isolated
term, the following:
$$t_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m)}
   = {1 \over \alpha_1^{(m)} \ldots \alpha_{n^{(m)}}^{(m)}}
        \mathbin{.}
        \sum_{\gamma_i^{(m)} < \alpha_i^{(m)}} \mathbin{.}
        ( b_{\gamma_1^{(m)},\ldots \,
            \gamma_{n^{(m)}}^{(m)}}^{(m)}
         \mathbin{.}
        \rho_{\alpha_1^{(m)}}^{-\beta_1^{(m)} \gamma_1^{(m)}}
        \ldots
        \rho_{\alpha_{n^{(m)}}^{(m)}}^{-\beta_{n^{(m)}}^{(m)}
            \gamma_{n^{(m)}}^{(m)}} );$$
the sign of summation here extending to all those terms in which
every index such as $\gamma_1^{(m)}$ is equal to zero or to some
positive integer less than $\alpha_i^{(m)}$.

Thus, in the case of the function of second order $b''$, which
represents, as we have seen, a root of the general cubic
equation, if we wish to obtain an isolated expression for any
term $t_{\beta_1''}''$ of its development already found, namely
the development
$$b'' = \sum_{\beta_1'' < 3} \mathbin{.}
         (b_{\beta_1''}' \mathbin{.} a_1''^{\beta_1''})
   = b_0' + b_1' a_1'' + b_2' a_1''^2
   = t_0'' + t_1'' + t_2'',$$
we have only to introduce the function
$$b_{\gamma_1''}'' = \sum_{\beta_1'' < 3} \mathbin{.}
         (b_{\beta_1''}' \mathbin{.}
         \rho_3^{\beta_1'' \gamma_1''}
            \mathbin{.} a_1''^{\beta_1''})
   = b_0' + b_1' \rho_3^{\gamma_1''} a_1''
            + b_2' \rho_3^{2\gamma_1''} a_1''^2,$$
and to employ the formula
$$t_{\beta_1''}'' = b_{\beta_1''}' \mathbin{.} a_1''^{\beta_1''}
   = {\textstyle {1 \over 3}} \mathbin{.}
      \sum_{\gamma_1'' < 3} \mathbin{.}
      ( b_{\gamma_1''}'' \mathbin{.}
         \rho_3^{- \beta_1'' \gamma_1''})
   = {\textstyle {1 \over 3}}
      ( b_0'' + \rho_3^{-\beta_1''} b_1''
         + \rho_3^{-2\beta_1''} b_2'' ).$$
In particular,
$$\eqalign{
t_0'' &= b_0' = {\textstyle {1 \over 3}}
         (b_0'' + b_1'' + b_2''),\cr
t_1'' &= b_1' a_1'' = {\textstyle {1 \over 3}}
         (b_0'' + \rho_3^{-1} b_1'' + \rho_3^{-2} b_2''),\cr
t_2'' &= b_2' a_1''^2 = {\textstyle {1 \over 3}}
         (b_0'' + \rho_3^{-2} b_1'' + \rho_3^{-4} b_2''),\cr}$$
in which
$$\eqalign{
b_0''
   &= b_0' + b_1' a_1'' + b_2' a_1''^2 \, ( = b''),\cr
b_1''
   &= b_0' + b_1' \rho_3 a_1'' + b_2' \rho_3^2 a_1''^2,\cr
b_2''
   &= b_0' + b_1' \rho_3^2 a_1'' + b_2' \rho_3^4 a_1''^2,\cr}$$
and in which it is to be remembered that
$$\rho_3^2 + \rho_3 + 1 = 0,
   \quad \hbox{and therefore}\quad
  \rho_3^3 = 1.$$

Again, if we wish to isolate any term
$t_{\beta_1''', \beta_2'''}$ of the development above assigned
for the function of third order $b'''$, which represents a root
of the general biquadratic equation, we may employ the formula
$$\eqalign{
t_{\beta_1''', \beta_2'''}'''
   &= b_{\beta_1''', \beta_2'''}''
            \mathbin{.} a_1'''^{\beta_1'''}
            \mathbin{.} a_2'''^{\beta_2'''}
      = {1 \over 2.2} \mathbin{.}
         \sum_{\gamma_1''' < 2 \atop \gamma_2''' < 2} \mathbin{.}
         ( b_{\gamma_1''', \gamma_2'''}''' \mathbin{.}
           \rho_2^{-\beta_1''' \gamma_1'''} \mathbin{.}
           \rho_2^{-\beta_2''' \gamma_2'''} ) \cr
   &= {\textstyle {1 \over 4}} \{
         b_{0,0}''' + (-1)^{-\beta_1'''} b_{1,0}'''
        + (-1)^{-\beta_2'''} b_{0,1}'''
        + (-1)^{-(\beta_1''' + \beta_2''')} b_{1,1}''' \};\cr}$$
in which we have introduced the function
$$\eqalign{
b_{\gamma_1''', \gamma_2'''}'''
   &= \sum_{\beta_1''' < 2 \atop \beta_2''' < 2} \mathbin{.}
      ( b_{\beta_1''', \beta_2'''}'' \mathbin{.}
      \rho_2^{\beta_1''' \gamma_1'''} \mathbin{.}
      \rho_2^{\beta_2''' \gamma_2'''} \mathbin{.}
      a_1'''^{\beta_1'''} \mathbin{.} a_2'''^{\beta_2'''} ) \cr
   &= b_{0,0}''
      + (-1)^{\gamma_1'''} b_{1,0}'' a_1'''
      + (-1)^{\gamma_2'''} b_{0,1}'' a_2'''
      + (-1)^{\gamma_1''' + \gamma_2'''}
            b_{1,1}'' a_1''' a_2''';\cr}$$
so that, in particular, we have the four expressions
$$\eqalign{
t_{0,0}''' &= b_{0,0}''
   = {\textstyle {1 \over 4}}
      ( b_{0,0}''' + b_{1,0}''' + b_{0,1}''' + b_{1,1}'''),\cr
t_{1,0}''' &= b_{1,0}'' a_1'''
   = {\textstyle {1 \over 4}}
      ( b_{0,0}''' - b_{1,0}''' + b_{0,1}''' - b_{1,1}'''),\cr
t_{0,1}''' &= b_{0,1}'' a_2'''
   = {\textstyle {1 \over 4}}
      ( b_{0,0}''' + b_{1,0}''' - b_{0,1}''' - b_{1,1}'''),\cr
t_{1,1}''' &= b_{1,1}'' a_1''' a_2'''
   = {\textstyle {1 \over 4}}
      ( b_{0,0}''' - b_{1,0}''' - b_{0,1}''' + b_{1,1}'''),\cr}$$
in which
$$\eqalign{
b_{0,0}''' &= b_{0,0}'' + b_{1,0}'' a_1''' + b_{0,1}'' a_2'''
               + b_{1,1}'' a_1''' a_2''',\cr
b_{1,0}''' &= b_{0,0}'' - b_{1,0}'' a_1''' + b_{0,1}'' a_2'''
               - b_{1,1}'' a_1''' a_2''',\cr
b_{0,1}''' &= b_{0,0}'' + b_{1,0}'' a_1''' - b_{0,1}'' a_2'''
               - b_{1,1}'' a_1''' a_2''',\cr
b_{1,1}''' &= b_{0,0}'' - b_{1,0}'' a_1''' - b_{0,1}'' a_2'''
               + b_{1,1}'' a_1''' a_2'''.\cr}$$

In these examples, the truth of the results is obvious; and the
general demonstration follows easily from the properties of the
roots of unity.

\bigbreak

[5.]
We have hitherto made no use of the assumed
{\it irreducibility\/} of the irrational function~$b^{(m)}$.  But
taking now this property into account, we soon perceive that the
component radicals $a_i^{(k)}$, which enter into the composition
of this irreducible function, must not be subject to, nor even
compatible with, any equations or equation of condition whatever,
except only the equations of definition, which determine those
radicals $a_i^{(k)}$, by determining their prime powers
$a_i^{(k)\alpha_i^{(k)}}$.  For the existence or possibility of
any such equation of condition in conjunction with those
equations of definition, would enable us to express at least one
of the above mentioned radicals as a rational function of others
of the same system, and of orders not higher than its own, or
even, perhaps, as a rational function of the original variables
$a_1,\ldots \, a_n$, though multiplied in general by a root of
a numerical equation; and therefore would enable us to diminish
the number of extractions of prime roots of functions, which
would be inconsistent with the irreducibility supposed.

If fact, if any such equation of condition, involving any
radical or radicals of order $k$, but none of any higher order,
were compatible with the equations of definition; then, by some
obvious preparations, such as bringing the equation of condition
to the form of zero equated to some finite polynomial function of
some radical~$a_i^{(k)}$ of the $k^{\rm th}$ order; and
rejecting, by the methods of equal roots and of the greatest
common measure, all factors of this polynome, except those which
are unequal among themselves, and are included among the factors
of that other polynome which is equated to zero in the
corresponding form of the equation of definition of the
radical~$a_i^{(k)}$; we should find that this last equation of
definition
$$a_i^{(k) \alpha_i^{(k)}} - f_i^{(k-1)} = 0$$
must be divisible, either identically, or at least for some
suitable system of values of the remaining radicals, by an
equation of condition of the form
$$a_i^{(k)g} + {\sc g}_1^{(k)} a_i^{(k)g-1} + \cdots
      + {\sc g}_{g-1}^{(k)} a_i^{(k)} + {\sc g}_g^{(k)} = 0;$$
$g$ being less than $\alpha_i^{(k)}$, and the coefficients
${\sc g}_1^{(k)},\ldots \, {\sc g}_g^{(k)}$ being functions of
orders not higher than $k$, and not involving the
radical~$a_i^{(k)}$.  Now, if we were to suppose that, for any
system of values of the remaining radicals, the coefficients
${\sc g}_1^{(k)},\ldots$ should all be $= 0$, or indeed if even
the last of those coefficients should thus vanish, we should then
have a new equation of condition, namely the following:
$$f_i^{(k-1)} = 0,$$
which would be obliged to be compatible with the equations of
definition of the remaining radicals, and would therefore either
conduct at last, by a repetition of the same analysis, to a
radical essentially vanishing, and consequently superfluous,
among those which have been supposed to enter into the
composition of the function~$b^{(m)}$; or else would bring us
back to the divisibility of an equation of definition by an
equation of condition, of the form just now assigned, and with
coefficients ${\sc g}_1^{(k)},\ldots \, {\sc g}_g^{(k)}$ which
would not all be $= 0$.  But for this purpose it would be
necessary that a relation, or system of relations, should exist
(or at least should be compatible with the remaining equations of
definition,) of the form
$${\sc g}_{g-e}^{(k)} = \nu_e a_i^{(k)e},$$
$e$ being less than $\alpha_i^{(k)}$, and $\nu_e$ being different
from zero, and being a root of a numerical equation; and because
$\alpha_i^{(k)}$ is prime, we could find integer numbers
$\lambda$ and $\mu$, which would satisfy the condition
$$\lambda \alpha_i^{(k)} - \mu e = 1;$$
so that, finally, we should have an expression for the radical
$a_i^{(k)}$, as a rational function of others of the same
system, and of orders not higher than its own, though multiplied
in general (as was above announced) by a root of a numerical
equation; namely the following expression:
$$a_i^{(k)}
   = \nu_e^\mu {\sc g}_{g - e}^{(k)-\mu} f_i^{(k-1)\lambda}.$$
And if we should suppose this last equation to be not identically
true, but only to hold good for some systems of values of the
remaining radicals, of orders not higher than $k$, we should
still obtain, at least, an equation of condition between those
remaining radicals, by raising the expression just found for
$a_i^{(k)}$ to the power $\alpha_i^{(k)}$; namely the following
equation of condition,
$$f_i^{(k-1)}
   - (\nu_e^\mu {\sc g}_{g-e}^{(k)-\mu}
         f_i^{(k-1)\lambda})^{\alpha_i^{(k)}}
   = 0,$$
which might then be treated like the former, till at last an
expression should be obtained, of the kind above announced, for
at least one of the remaining radicals.  In every case,
therefore, we should be conducted to a diminution of the number
of prime roots of variables in the expression of the
function~$b^{(m)}$, which consequently would not be irreducible.

For example, if an irrational function of the $m^{\rm th}$ order
contain any radical~$a_i^{(m)}$ of the cubic form, its exponent
$\alpha_i^{(m)}$ of the cubic form, its exponent $\alpha_i^{(m)}$
being $= 3$, and its equation of definition being of the form
$$\alpha^{(m)3}
   = f_i^{(m-1)} (a_1^{(m-1)},\ldots \,
                  a_{n^{(m-1)}}^{(m-1)},\ldots \,
                  a_1,\ldots \, a_n);$$
if also the other equations of definition permit us to
suppose that this radical {\it may\/} be equal to some rational
function of the rest, so that an equation of the form
$$a_i^{(m)} + {\sc g}_1^{(m)} = 0,$$
(in which the function~${\sc g}_1^{(m)}$ does not contain the
radical~$a_i^{(m)}$,) is {\it compatible\/} with the equation of
definition
$$a_i^{(m)3} - f_i^{(m-1)} = 0;$$
then, from the forms of these two last mentioned equations, the
latter must be {\it divisible\/} by the former, at least for some
suitable system of values of the remaining radicals: and
therefore the following relation, which does not involve the
radical~$a_i^{(m)}$, namely,
$$f_i^{(m-1)} + {\sc g}_1^{(m)3} = 0,$$
must be either identically true, in which case we may substitute
for the radical~$a_i^{(m)}$, in the proposed function of the
$m^{\rm th}$ order, the expression
$$a_i^{(m)} = - \root 3 \of {1} \mathbin{.} {\sc g}_1^{(m)};$$
or at least it must be true as an equation of condition between
the remaining radicals, and liable as such to a similar
treatment, conducting to an analogous result.

A more simple and specific example is supplied by the following
function of the second order,
$$x = - {a_1 \over 3}
      + \root 3 \of {\vphantom{\bigl(}}
         \left( c_1 + \sqrt{c_1^2 - c_2^3} \right)
      + \root 3 \of {\vphantom{\bigl(}}
         \left( c_1 - \sqrt{c_1^2 - c_2^3} \right),$$
which is not uncommonly proposed as an expression for a root~$x$
of the general cubic equation
$$x^3 + a_1 x^2 + a_2 x + a_3 = 0,$$
$c_1$  and $c_2$ being certain rational functions of
$a_1$,~$a_2$,~$a_3$, which were assigned in a former article, and
which are such that the cubic equation may be thus written:
$$\left( x + {a_1 \over 3} \right)^3
   - 3 c_2 \left( x + {a_1 \over 3} \right)
   - 2 c_1 = 0.$$
Putting this function of the second order under the form
$$x = - {a_1 \over 3} + a_1'' + a_2'',$$
in which the radicals are defined as follows,
$$a_1''^3 = c_1 + a_1',\quad
  a_2''^3 = c_1 - a_1',\quad
  a_1'^2 = c_1^2 - c_2^3,$$
we easily perceive that it is {\it permitted\/} by these
definitions to suppose that the radicals $a_1''$,~$a_2''$ are
connected so as to satisfy the following equation of condition,
$$a_1'' a_2'' = c_2;$$
and even that this supposition {\it must\/} be made, in order to
render the proposed function of the second order a root of the
cubic equation.  But the mere knowledge of the {\it
compatibility\/} of the equation of condition
$$a_2'' - {c_2 \over a_1''} = 0$$
with the equation of definition
$$a_2''^3 - (c_1 - a_1') = 0,$$
is sufficient to enable us to infer, from the forms of these two
equations, that the latter is divisible by the former, at least
for some suitable system of values of the remaining radicals
$a_1''$ and $a_1'$, consistent with their equations of
definition; and therefore that the following relation
$$c_1 - a_1' - \left( {c_2 \over a_1''} \right)^3 = 0,$$
and the expression
$$a_2'' = \root 3 \of {1} \mathbin{.} {c_2 \over a_1''},$$
are at least consistent with those equations.  In the present
example, the relation thus arrived at is found to be identically
true, and consequently the radicals $a_1'$ and $a_1''$ remain
independent of each other; but for the same reason, the radical
$a_2''$ may be changed to the expression just now given; so that
the proposed function of the second order,
$$x = {-a_1 \over 3} + a_1'' + a_2'',$$
may, by the mere {\it definitions\/} of its radicals, and even
without attending to the cubic equation which it was designed to
satisfy, be put under the form
$$x = {-a_1 \over 3} + a_1'' + \root 3 \of {1} \mathbin{.}
         {c_2 \over a_1''},$$
the number of prime roots of variables being depressed from three
to two; and consequently that proposed function was not
{\it irreducible\/} in the sense which has been already
explained.

\bigbreak

[6.]
From the foregoing properties of irrational and irreducible
functions, it follows easily that if any {\it one\/} value of
any such function~$b^{(m)}$, corresponding to any one system of
values of the radicals on which it depends, be equal to any one
root of any equation of the form
$$x^s + {\sc a}_1 x^{s-1} + \cdots
          + {\sc a}_{s-1} x + {\sc a}_s = 0,$$
in which the coefficients ${\sc a}_1,\ldots \, {\sc a}_s$ are any
rational functions of the $n$ original quantities
$a_1,\ldots \, a_n$; in such a manner that for some one system of
values of the radicals $a_1'$, \&c., the equation
$$b^{(m)s} + {\sc a}_1 b^{(m)s-1} + \cdots + {\sc a}_s = 0$$
is satisfied: then the same equation must be satisfied, also, for
{\it all\/} systems of values of those radicals, consistent with
their equations of definition.  It is an immediate consequence of
this result, that all the values of the function which has
already been denoted by the symbol
$b_{\gamma_1^{(m)},\ldots \, \gamma_{n^{(m)}}^{(m)}}^{(m)}$
must represent roots of the same equation of the $s^{\rm th}$
degree; and the same principles show that all these values of
$b_{\gamma_1^{(m)}\ldots}^{(m)}$ must be {\it unequal\/} among
themselves, and therefore must represent so many {\it different
roots\/} $x_1, x_2,\ldots$ of the same equations
$x^s + \hbox{\&c.} = 0$, if every index or exponent
$\gamma_i^{(m)}$ be restricted, as before, to denote either zero
or some positive integer number less than the corresponding
exponent $\alpha_i^{(m)}$: for if, with this restriction, any two
of the values of $b_{\gamma_1^{(m)},\ldots}^{(m)}$ could be
supposed equal, an equation of condition between the radicals
$a_1^{(m)}$, \&c. would arise, which would be inconsistent with
the supposed irreducibility of the function~$b^{(m)}$.

For example, having found that the cubic equation
$$x^3 + a_1 x^2 + a_2 x + a_3 = 0$$
is satisfied by the irrational and irreducible function~$b''$
above assigned, we can infer that the same equation is satisfied
by all the three values $b_0''$,~$b_1''$,~$b_2''$ of the
function~$b_{\gamma_1''}''$; and that these three values must be
all unequal among themselves, so that they must represent some
three unequal roots $x_1$,~$x_2$,~$x_3$, and consequently all the
three roots of the cubic equation proposed.

\bigbreak

[7.]
Combining the result of the last article with that which was
before obtained respecting the isolating of a term of a
development, we see that if any root~$x$ of any proposed
equation, of any degree $s$, in which the $s$ coefficients
${\sc a}_1,\ldots \, {\sc a}_s$
are still supposed to be rational functions of the $n$ original
quantities $a_1,\ldots \, a_n$, can be expressed as an irrational
and irreducible function~$b^{(m)}$ of those original quantities;
and if that function~$b^{(m)}$ be developed under the form above
assigned; then every term $t_{\beta_1^{(m)},\ldots}^{(m)}$ of
this development may be expressed as a rational (and indeed
linear) function of some or all of the $s$ roots
$x_1, x_2,\ldots \, x_s$ of the same proposed equation.

For example, when we have found that a root~$x$ of the cubic
equation
$$x^3 + a_1 x^2 + a_2 x + a_3 = 0$$
can be represented by the irrational and irreducible function
already mentioned,
$$x = b'' = b_0' + b_1' a_1'' + b_2' a_1''^2
   = t_0'' + t_1'' + t_2'',$$
(in which $b_1' = 1$,) we can express the separate terms of this
last development as follows,
$$\eqalign{
t_0'' &= b_0'
   = {\textstyle {1 \over 3}}
         (x_1 + x_2 + x_3),\cr
t_1'' &= b_1' a_1''
   = {\textstyle {1 \over 3}}
         (x_1 + \rho_3^{-1} x_2 + \rho_3^{-2} x_3),\cr
t_2'' &= b_2' a_1''^2
   = {\textstyle {1 \over 3}}
         (x_1 + \rho_3^{-2} x_2 + \rho_3^{-4} x_3);\cr}$$
namely, by changing $b_0''$,~$b_1''$,~$b_2''$ to
$x_1$,~$x_2$,~$x_3$ in the expressions found before for
$t_0''$,~$t_1''$,~$t_2''$.

In like manner, when a root~$x$ of the biquadratic equation
$$x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$$
is represented by the irrational function
$$\eqalign{
x  &= b''' = b_{0,0}'' + b_{1,0}''a_1''' + b_{0,1}'' a_2'''
               + b_{1,1}'' a_1''' a_2''' \cr
   &= t_{0,0}''' + t_{1,0}''' + t_{0,1}''' + t_{1,1}''',\cr}$$
in which $b_{1,0}'' = b_{0,1}'' = 1$, we easily derive, from
results obtained before, (by merely changing $b_{0,0}'''$,
$b_{0,1}'''$,~$b_{1,0}'''$,~$b_{1,1}'''$ to
$x_1$,~$x_2$,~$x_3$,~$x_4$,) the following expressions for the
four separate terms of this development:
$$\eqalign{
t_{0,0}''' &= b_{0,0}''
   = {\textstyle {1 \over 4}} (x_1 + x_2 + x_3 + x_4),\cr
t_{1,0}''' &= b_{1,0}'' a_1'''
   = {\textstyle {1 \over 4}} (x_1 + x_2 - x_3 - x_4),\cr
t_{0,1}''' &= b_{0,1}'' a_2'''
   = {\textstyle {1 \over 4}} (x_1 - x_2 + x_3 - x_4),\cr
t_{1,1}''' &= b_{1,1}'' a_1''' a_2'''
   = {\textstyle {1 \over 4}} (x_1 - x_2 - x_3 + x_4);\cr}$$
$x_1$,~$x_2$,~$x_3$,~$x_4$ being some four unequal roots, and
therefore all the four roots of the proposed biquadratic
equation.

And when that equation has a root represented in this other way,
which also has been already indicated, and in which $b_1''' = 1$,
$$x = b^{\rm IV} = {-a_1 \over 4} + a_1''' + a_1^{\rm IV}
      = b_0''' + b_1''' a_1^{\rm IV}
      = t_0^{\rm IV} + t_1^{\rm IV},$$
then each of the two terms of this development may be separately
expressed as follows,
$$\eqalign{
t_0^{\rm IV}
   &= b_1''' = {\textstyle {1 \over 2}} (x_1 + x_2),\cr
t_1^{\rm IV}
   &= b_1''' a_1^{\rm IV}
   = {\textstyle {1 \over 2}} (x_1 - x_2),\cr}$$
$x_1$ and $x_2$ being some two unequal roots of the same
biquadratic equation.

A still more simple example is supplied by the quadratic
equation,
$$x^2 + a_1 x + a_2 = 0;$$
for when we represent a root~$x$ of this equation as follows,
$$x = b' = {-a_1 \over 2} + a_1' = t_0' + t_1',$$
we have the following well-known expressions for the two terms
$t_0'$,~$t_1'$, as rational and linear functions of the roots
$x_1$,~$x_2$,
$$\eqalign{
t_0'
   &= {-a_1 \over 2} = {\textstyle {1 \over 2}} (x_1 + x_2),\cr
t_1'
   &= a_1' = {\textstyle {1 \over 2}} (x_1 - x_2).\cr}$$

In these examples, the radicals of highest order, namely $a_1'$
in $b'$,~$a_1''$ in $b''$,~$a_1'''$ and $a_2'''$ in $b'''$, and
$a_1^{\rm IV}$ in $b^{\rm IV}$, have all had the coefficients of
their first powers equal to unity; and consequently have been
themselves expressed as rational (though unsymmetric) functions
of the roots of that equation in $x$, which the function
$b^{(m)}$ satisfies; namely,
$$\eqalign{
a_1'
   &= {\textstyle {1 \over 2}}
         (x_1 - x_2),\cr
a_1''
   &= {\textstyle {1 \over 3}}
         (x_1 + \rho_3^2 x_2 + \rho_3 x_3),\cr
a_1'''
   &= {\textstyle {1 \over 4}}
         (x_1 + x_2 - x_3 - x_4),\cr
a_2'''
   &= {\textstyle {1 \over 4}}
         (x_1 - x_2 + x_3 - x_4),\cr
a_1^{\rm IV}
   &= {\textstyle {1 \over 2}}
         (x_1 - x_2);\cr}$$
the first expression being connected with the general quadratic,
the second with the general cubic, and the three last with the
general biquadratic equation.  We shall soon see that all these
results are included in one more general.

\bigbreak

[8.]
To illustrate, by a preliminary example, the reasonings to which
we are next to proceed, let it be supposed that any two of the
terms $t_{\beta_1^{(m)},\ldots}^{(m)}$ are of the forms
$$t_{2,1,3,4}'' = b_{2,1,3,4}' a_1''^2 a_2'' a_3''^3 a_4''^4,$$
and
$$t_{1,1,2,3}'' = b_{1,1,2,3}' a_1'' a_2'' a_3''^2 a_4''^3,$$
in which the radicals are defined by equations such as the
following
$$a_1''^3 = f_1',\quad
  a_2''^3 = f_2',\quad
  a_3''^5 = f_3',\quad
  a_4''^5 = f_4',$$
their exponents $\alpha_1''$,~$\alpha_2''$,~$\alpha_3''$,
$\alpha_4''$ being respectively equal to the numbers $3$,~$3$,
$5$,~$5$.  We shall then have, by raising the two terms $t''$ to
suitable powers, and attending to the equations of definition,
the following expressions:
$$\eqalign{
t_{2,1,3,4}''^{10}
   &= b_{2,1,3,4}'^{10} f_1'^6 f_2'^3 f_3'^6 f_4'^8
         a_1''^2 a_2'';\cr
t_{1,1,2,3}''^{10}
   &= b_{1,1,2,3}'^{10} f_1'^3 f_2'^3 f_3'^4 f_4'^6
         a_1'' a_2'';\cr
t_{2,1,3,4}''^6
   &= b_{2,1,3,4}'^6 f_1'^4 f_2'^2 f_3'^3 f_4'^4
         a_3''^3 a_4''^4;\cr
t_{1,1,2,3}''^6
   &= b_{1,1,2,3}'^6 f_1'^2 f_2'^2 f_3'^2 f_4'^3
         a_3''^2 a_4''^3;\cr}$$
which give
$${\sc t}_1'' = {\sc c}_1' a_1'',\quad
  {\sc t}_2'' = {\sc c}_2' a_2'',\quad
  {\sc t}_3'' = {\sc c}_3' a_3'',\quad
  {\sc t}_4'' = {\sc c}_4' a_4'',$$
if we put, for abridgment,
$$\eqalign{
{\sc t}_1'' &= t_{2,1,3,4}''^{10}  t_{1,1,2,3}''^{-10};\cr
{\sc t}_2'' &= t_{2,1,3,4}''^{-10} t_{1,1,2,3}''^{20};\cr
{\sc t}_3'' &= t_{2,1,3,4}''^{18}  t_{1,1,2,3}''^{-24};\cr
{\sc t}_4'' &= t_{2,1,3,4}''^{-12} t_{1,1,2,3}''^{18};\cr}
  \quad \eqalign{
{\sc c}_1'  &= b_{2,1,3,4}'^{10}  b_{1,1,2,3}'^{-10}
         f_1'^3 f_3'^2 f_4'^2;\cr
{\sc c}_2'  &= b_{2,1,3,4}'^{-10} b_{1,1,2,3}'^{20}
         f_2'^3 f_3'^2 f_4'^4;\cr
{\sc c}_3'  &= b_{2,1,3,4}'^{18}  b_{1,1,2,3}'^{-24}
         f_1'^4 f_2'^{-2} f_3';\cr
{\sc c}_4'  &= b_{2,1,3,4}'^{-12} b_{1,1,2,3}'^{18}
         f_1'^{-2} f_2'^2 f_4'.\cr}$$

And, with a little attention, it becomes clear that the same sort
of process may be applied to the terms
$t_{\beta_1^{(m)},\ldots}^{(m)}$ of the development of any
irreducible function~$b^{(m)}$; so that we have, in general, a
system of relations, such as the following:
$${\sc t}_1^{(m)}
      = {\sc c}_1^{(m-1)} a_1^{(m)};\quad\ldots\quad
  {\sc t}_{n^{(m)}}^{(m)}
      = {\sc c}_{n^{(m)}}^{(m-1)} a_{n^{(m)}}^{(m)};$$
in which ${\sc t}_i^{(m)}$ is the product of certain powers (with
exponents positive, or negative, or null) of the various terms
$t_{\beta_1^{(m)},\ldots}^{(m)}$; and the coefficient
${\sc c}_i^{(m-1)}$ is different from zero, but is of an order
lower than $m$.  For if any radical of the order $m$ were
supposed to be so inextricably connected, in every term, with one
or more of the remaining radicals of the same highest order, that
it could not be disentangled from them by a process of the
foregoing kind; and that thus the foregoing {\it analysis\/} of
the function~$b^{(m)}$ should be unable to conduct to separate
expressions for those radicals; it would then, reciprocally, have
been unnecessary to calculate them separately, in effecting the
{\it synthesis} of that function; which function, consequently,
would not be irreducible.  If, for example, the exponents
$\alpha_1^{(m)}$ and $\alpha_2^{(m)}$, which enter into the
equations of definition of the radicals $a_1^{(m)}$ and
$a_2^{(m)}$ should both be $= 3$, so that those radicals should
both be cube-roots of functions of lower orders; and if these two
cube-roots should enter only by their product, so that no
analysis of the foregoing kind could obtain them otherwise than
in connexion, and under the form
${\sc c}^{(m-1)} a_1^{(m)} a_2^{(m)}$; it would then have been
sufficient, in effecting the synthesis of $b^{(m)}$, to have
calculated only the cube-root of the product
$a_1^{(m)3} a_2^{(m)3} = f_1^{(m-1)} f_2^{(m-1)}
   =  f^{\backprime (m-1)}$,
instead of calculating separately the cube-roots of its two
factors $a_1^{(m)3} = f^{(m-1)}$, and $a_2^{(m)3} = f_2^{(m-1)}$:
the number of extractions of prime roots of variables might,
therefore, have been diminished in the calculation of the
function~$b^{(m)}$, which would be inconsistent with the
irreducibility of that function.

In the cases of the irreducible functions $b'$,~$b''$,~$b'''$,
$b^{\rm IV}$, which have been above assigned, as representing
roots of the general quadratic, cubic, and biquadratic equations,
the theorem of the present article is seen at once to hold good;
because in these the radicals of highest order are themselves
terms of the developments in question, the coefficients of their
first powers being already equal to unity.  Thus in the
development of $b'$, we have $a_1' = t_1'$; in $b''$, we have
$a_1'' = t_1''$; in $b'''$, we have $a_1''' = t_{1,0}'''$, and
$a_2''' = t_{0,1}'''$; and in $b^{\rm IV}$, we have
$a_1^{\rm IV} = t_1^{\rm IV}$.

\bigbreak

[9.]
By raising to the proper powers the general expressions of the
form
$${\sc t}_i^{(m)} = {\sc c}_i^{(m-1)} a_i^{(m)},$$
we obtain a system of $n^{(m)}$ equations of this other form
$${\sc t}_i^{(m) \alpha_i^{(m)}}
   = {\sc c}_i^{(m-1)\alpha_i^{(m)}} f_i^{(m-1)}
   = f_i^{\backprime (m-1)},$$
$f_i^{\backprime (m-1)}$ being some new irrational function, of
an order lower than $m$; and by combining the same expressions
with those which define the various terms
$t_{\beta_1^{(m)},\ldots}^{(m)}$, the number of which terms we
shall denote by the symbol $t^{(m)}$, we obtain another system of
$t^{(m)}$ equations, of which the following is a type,
$${\sc u}_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m-1)}
   = b_{\beta_1^{(m)},\ldots \,
      \beta_{n^{(m)}}^{(m)}}^{\backprime (m-1)},$$
if we put, for abridgment,
$${\sc u}_{\beta_1^{(m)},\ldots}^{(m-1)}
   = t_{\beta_1^{(m)},\ldots}^{(m)} \mathbin{.}
         {\sc t}_1^{(m)-\beta_1^{(m)}} \cdots
         {\sc t}_{n^{(m)}}^{(m)-\beta_{n^{(m)}}^{(m)}},$$
and
$$b_{\beta_1^{(m)},\ldots}^{\backprime (m-1)}
   = b_{\beta_1^{(m)},\ldots}^{(m-1)} \mathbin{.}
         {\sc c}_1^{(m-1)-\beta_1^{(m)}} \cdots
         {\sc c}_{n^{(m)}}^{(m-1)-\beta_{n^{(m)}}^{(m)}}.$$
In this manner we obtain in general $n^{(m)} + t^{(m)}$
equations, in each of which the product of certain powers, (with
positive, negative, or null exponents,) of the $t^{(m)}$ terms of
the development of the irrational function~$b^{(m)}$, is equated
to some other irrational function, $f^{\backprime(m-1)}$ or
$b^{\backprime (m-1)}$, of an order lower than $m$.  Indeed, it
is to be observed, that since these various equations are
obtained by an elimination of the $n^{(m)}$ radicals of highest
order, between their $n^{(m)}$ equations of definition and the
$t^{(m)}$ expressions for the $t^{(m)}$ terms of the development
of $b^{(m)}$, they cannot be equivalent to more than $t^{(m)}$
distinct relations.  But, among them, they must involve
explicitly all the radicals of lower orders, which enter into the
composition of the irreducible function~$b^{(m)}$.  For if any
radical~$a_i^{(k)}$, of order lower than $m$, were wanting in all
the $n^{(m)} + t^{(m)}$ functions of the forms
$$f_i^{\backprime (m-1)}
   \quad\hbox{and}\quad
  b_{\beta_1^{(m)},\ldots}^{\backprime (m-1)},$$
we might then employ instead of the old system of radicals
$a_1^{(m)},\ldots$ of the order $m$, a new and equally numerous
system of radicals $a_1^{\backprime (m)},\ldots$ according to the
following type,
$$a_i^{\backprime (m)} = {\sc t}_i^{(m)}
   = {}^{\alpha_i^{(m)}} \!\! \sqrt{ f_i^{\backprime (m-1)} };$$
and might then express all the $t^{(m)}$ terms of $b^{(m)}$, by
means of these new radicals, according to the formula
$$t_{\beta_1^{(m)},\ldots}^{(m)}
   = b_{\beta_1^{(m)},\ldots}^{\backprime (m-1)} \mathbin{.}
         a_1^{\backprime (m) \beta_1^{(m)}} \, \ldots \,
         a_{n^{(m)}}^{\backprime (m) \beta_{n^{(m)}}^{(m)}},$$
which would not involve the radical~$a_i^{(k)}$; so that in this
way the number of extractions of prime roots of variables might
be diminished, which would be inconsistent with the
irreducibility of $b^{(m)}$.

The results of the present article may be exemplified in the case
of any one of the functions $b'$,~$b''$,~$b'''$,~$b^{\rm IV}$,
which have already been considered.  Thus, in the case of the
function~$b''$, which represents a root of the general cubic
equation, we have
$${\sc t}_1'' = t_1'',\quad
  {\sc c}_1'' = 1,\quad
  f_1^{\backprime\prime} = f_1',\quad
  \beta_{\beta_1''}^{\backprime\prime} = b_{\beta_1''}',\quad
  {\sc u}_{\beta_1''}'
   = t_{\beta_1''}'' \mathbin{.} t_1''^{-\beta_1''},$$
and the $n^{(m)} + t^{(m)} = 1 + 3 = 4$ following relations hold
good:
$$t_1''^3 = f_1',\quad
  t_0'' = b_0',\quad
  1 = b_1',\quad
  t_2'' t_1''^{-2} = b_2';$$
of which indeed the third is identically true, and the second
does not involve $a_1'$, because
$\displaystyle b_0' = - {a_1 \over 3}$;
but both the first and fourth of these relations involve that
radical~$a_1'$, because $f_1' = c_1 + a_1'$, and
$\displaystyle b_2' = {c_1 - a_1' \over c_2^2}$.

\bigbreak

[10.]
Since each of the $t^{(m)}$ terms of the development of $b^{(n)}$
can be expressed as a rational function of the $s$ roots
$x_1,\ldots \, x_s$ of that equation of the $s^{\rm th}$ degree
which $b^{(m)}$ is supposed to satisfy; it follows that every
rational function of these $t^{(m)}$ terms must be likewise a
rational function of those $s$ roots, and must admit, as such, of
some finite number~$r$ of values, corresponding to all possible
changes of arrangement of the same $s$ roots amongst themselves.
The same term or function must, for the same reason, be itself a
root of an equation of the $r^{\rm th}$ degree, of which the
coefficients are symmetrical functions of the $s$ roots,
$x_1,\ldots \, x_s$, and therefore are rational functions of the
$s$ coefficients ${\sc a}_1,\ldots \, {\sc a}_s$, and ultimately
of the $n$ original quantities $a_1,\ldots \, a_n$; while the
$r - 1$ other roots of this new equation are the $r - 1$ other
  values of the same function of $x_1,\ldots \,x_s$,
corresponding to the changes of arrangement just now mentioned.
Hence, every one of the $n^{(m)} + t^{(m)}$ functions
${\sc t}_i^{(m) \alpha_i^{(m)}}$ and
${\sc u}_{\beta_1^{(m)},\ldots}^{(m-1)}$, and therefore also
every one of the $n^{(m)} + t^{(m)}$
functions~$f_i^{\backprime (m-1)}$ and
$b_{\beta_1^{(m)},\ldots}^{\backprime (m-1)}$, to which they
are respectively equal, and which have been shown to contain,
among them, all the radicals of orders lower than $m$, must be a
root of some such new equation, although the degree $r$ will not
in general be the same for all.  Treating these new equations and
functions, and the radicals of the order $m - 1$, as the equation
$x^s + \hbox{\&c.} = 0$, the function~$b^{(m)}$, and the radicals
of the order~$m$ have already been treated; we obtain a new
system of relations, analogous to those already found, and
capable of being thus denoted:
$$\eqalign{
{\sc t}_i^{(m-1)}
   &= {\sc c}_i^{(m-2)} a_i^{(m-1)};\cr
{\sc t}_i^{(m-1) \alpha_i^{(m-1)}}
   &= f_i^{\backprime (m-2)};\cr
{\sc u}_{\beta_1^{(m-1)},\ldots}^{(m-1)}
   &= b_{\beta_1^{(m-1)},\ldots}^{\backprime (m-2)}.\cr}$$
And so proceeding, we come at last to a system of the form,
$${\sc t}_1' = {\sc c}_1 a_1', \quad \ldots \, \quad
  {\sc t}_{n'}' = {\sc c}_{n'} a_{n'}';$$
in which the coefficient~${\sc c}_i$ is different from zero, and
is a rational function of the $n$ original quantities
$a_1,\ldots \, a_n$; while ${\sc t}_i'$ is a rational function of
the $s$ roots $x_1,\ldots \, x_s$ of that equation of the
$s^{\rm th}$ degree in $x$ which it has been suppose that
$b^{(m)}$ satisfies.  We have therefore the expression
$$a_i' = {{\sc t}_1' \over {\sc c}_i};$$
which enables us to consider every radical~$a_i'$ of the first
order, as a rational function~${\sc f}_i'$ of the $s$ roots
$x_1,\ldots \, x_s$, and of the $n$ original quantities
$a_1,\ldots \, a_n$: so that we may write
$$a_i' = {\sc f}_i'(x_1,\ldots \, x_s, a_1,\ldots \, a_n).$$
But before arriving at the last mentioned system of relations,
another system of the form
$${\sc t}_1'' = {\sc c}_1' a_1'',
   \quad\ldots\quad
  {\sc t}_{n''}'' = {\sc c}_{n''}' a_{n''}''$$
must have been found, in which the coefficient~${\sc c}_i'$ is
different from zero, and is a rational function of
$a_1',\ldots \, a_{n'}'$ and of $a_1,\dots, a_n$, while
${\sc t}_i''$ is a rational function of $x_1,\ldots \, x_s$; we
have therefore the expression
$$a_i'' = {{\sc t}_i'' \over {\sc c}_i'},$$
and we see that every radical of the second order also is equal
to a rational function of $x_1,\ldots \, x_s$ and of
$a_1,\ldots \, a_n$: so that we may write
$$a_i'' = {\sc f}_i''(x_1,\ldots \, x_s, a_1,\ldots \, a_n).$$

And re-ascending thus, through orders higher and higher, we find,
finally, by similar reasonings, that every one of the
$n' + n'' + \cdots + + n^{(k)} + \cdots + n^{(m)}$ radicals which
enter into the composition of the irrational and irreducible
function~$b^{(m)}$, such as the radical~$a_i^{(k)}$, must be
expressible as a rational function~${\sc f}_i^{(k)}$ of the roots
$x_1,\ldots \,x_s$, and of the original quantities
$a_1,\ldots \,a_n$: so that we have a complete system of
expressions, for all these radicals, which are included in the
general formula
$$a_i^{(k)}
   = {\sc f}_i^{(k)}(x_1,\ldots \, x_s, a_1,\ldots \, a_n).$$

Thus, in the case of the cubic equation and the function~$b''$,
when we have arrived at the relation
$$t_1''^3 = f_1',$$
in which
$$t_1'' = {\textstyle {1 \over 3}}
            (x_1 + \rho_3^2 x_2 + \rho_3 x_3),
   \quad \hbox{and}\quad
  f_1' = c_1 + a_1',$$
we find that the rational function
$$t_1''^3 = {\textstyle {1 \over 27}}
            (x_1 + \rho_3^2 x_2 + \rho_3 x_3)^3$$
admits only of {\it two\/} different values, in whatever way the
arrangement of the three roots $x_1$,~$x_2$,~$x_3$ may be
changed; it must therefore be itself a root of a quadratic
equation, in which the coefficients are symmetric functions of
those three roots, and consequently rational functions of
$a_1$,~$a_2$,~$a_3$; namely, the equation
$$\eqalign{
0  &= (t_1''^3)^2 - {\textstyle {1 \over 27}} \{
            (x_1 + \rho_3^2 x_2 + \rho_3 x_3)^3
          + (x_1 + \rho_3^2 x_3 + \rho_3 x_2)^3 \} (t_1''^3) \cr
   &\mathrel{\phantom{=}}
      + {\textstyle {1 \over 729}}
            (x_1 + \rho_3^2 x_2 + \rho_3 x_3)^3
            (x_1 + \rho_3^2 x_3 + \rho_3 x_2)^3 \cr
   &= (t_1''^3)^2 + {\textstyle {1 \over 27}}
            (2 a_1^3 - 9 a_1 a_2 + 27 a_3) (t_1''^3)
          +  \left( {a_1^2 - 3 a_2 \over 9} \right)^3.\cr}$$
The same quadratic equation must therefore be satisfied when we
substitute for $t_1''^3$ the function~$x_1 + a_1'$ to which it is
equal, and in which $a_1'$ is a square root; it must therefore be
satisfied by {\it both\/} values of the function~$c_1 \pm a_1'$,
because the radical~$a_1'$ must be subject to no condition except
that by which its square is determined; therefore, this radical
$a_1'$ must be equal to the semi-difference of two unequal roots
of the same quadratic equation; that is, to the semi-difference
of the two values of the rational function~$t_1''^3$; which
semi-difference {\it is itself a rational function of\/}
$x_1$,~$x_2$,~$x_3$, namely
$$\eqalign{
a_1' &= {\textstyle {1 \over 54}}
            \{   (x_1 + \rho_3^2 x_2 + \rho_3 x_3)^3
               - (x_1 + \rho_3^2 x_3 + \rho_3 x_2)^3 \} \cr
     &= {\textstyle {1 \over 18}}
            (\rho_3^2 - \rho_3) (x_1 - x_2) (x_1 - x_3)
                                (x_2 - x_3)
      = {\sc f}_1'(x_1, x_2, x_3).\cr}$$

The same conclusion would have been obtained, though in a
somewhat less simple way, if we had employed the relation
$$t_2'' t_1''^{-2} = b_2',$$
in which
$$t_2'' t_1''^{-2}
   = { 3(x_1 + \rho_3^2 x_3 + \rho_3 x_2) \over
        (x_1 + \rho_3^2 x_2 + \rho_3 x_3)^2 },\quad
  b_2' = {c_1 - a_1' \over c_2^2}.$$

\bigbreak

[11.]
In general, let $p$ be the number of values which the rational
function~${\sc f}_i^{(k)}$ can receive, by altering in all
possible ways the arrangements of the $s$ roots
$x_1,\ldots \, x_s$, these roots being still treated as arbitrary
and independent quantities, (so that $p$ is equal either to the
product $1 \mathbin{.} 2 \mathbin{.} 3 \, \ldots\, s$, or to some
submultiple of that product); we shall then have an
{\it identical\/} equation of the form
$${\sc f}_i^{(k)p} + {\sc d}_1 {\sc f}_i^{(k)p-1} + \cdots
      + {\sc d}_{p-1} {\sc f}_i^{(k)}
      + {\sc d}_p = 0,$$
in which the coefficients ${\sc d}_1,\ldots \, {\sc d}_p$ are
rational functions of $a_1,\ldots \, a_n$; and therefore at least
{\it one} value of the radical~$a_i^{(k)}$ must satisfy the
equation
$$a_i^{(k)p} + {\sc d}_1 a_i^{(k)p-1} + \cdots
      + {\sc d}_{p-1} a_i^{(k)}
      + {\sc d}_p = 0.$$
But in order to do this, it is necessary, for reasons already
explained, that {\it all\/} values of the same radical
$a_i^{(k)}$, obtained by multiplying itself and all its
subordinate radicals of the same functional system by any powers
of the corresponding roots of unity, should satisfy the same
equation; and therefore that the number~$q$ of these values of
the radical~$a_i^{(k)}$ should {\it not exceed\/} the degree $p$
of that equation, or the number of the values of the rational
function~${\sc f}_i^{(k)}$.

Again, since we have denoted by $q$ the number of values of the
radical, we must suppose that it satisfies identically an
equation of the form
$$a_i^{(k)q} + {\sc e}_1 a^{(k)q-1} + \cdots
      + {\sc e}_{q-1} a_i^{(k)}
      + {\sc e}_q = 0,$$
the coefficients ${\sc e}_1,\ldots \, {\sc e}_q$ being rational
functions of $a_1,\ldots \, a_n$; and therefore that at least one
value of the function~${\sc f}_i^{(k)}$ satisfies the equation
$${\sc f}_i^{(k)q} + {\sc e}_1
         \mathbin{.} {\sc f}_i^{(k)q-1} + \cdots
       + {\sc e}_{q-1} {\sc f}_i^{(k)}
   = 0.$$
Suppose now that the $s$ roots $x_1,\ldots \, x_s$ of the
original equation in $x$,
$$x^s + {\sc a}_1 x^{s-1} + \cdots
       + {\sc a}_{s-1} x + {\sc a}_s = 0,$$
are really unconnected by any relation among themselves, a
supposition which requires that $s$ should not be greater than
$n$, since ${\sc a}_1,\ldots \, {\sc a}_s$ are rational functions
of $a_1,\ldots \,a_n$; suppose also that $a_1,\ldots \, a_n$ can
be expressed, reciprocally, as rational functions of
${\sc a}_1,\ldots \, {\sc a}_s$, a supposition which requires,
reciprocally, that $n$ should not be greater than $s$, because
the {\it original quantities\/} $a_1,\ldots \, a_n$ {\it are, in
this whole discussion, considered as independent of each other}.
With these suppositions, which involve the equality $s= n$, we
may consider the $n$ quantities $a_1,\ldots \, a_n$, and
therefore also the $q$ coefficients
${\sc e}_1,\ldots \, {\sc e}_q$, as being symmetric functions of
the $n$ roots $x_1,\ldots \, x_n$ of the equation
$$x^n + {\sc a}_1 x^{n-1} + \cdots
       + {\sc a}_{n-1} x + {\sc a}_n = 0;$$
we may also consider ${\sc f}_i^{(k)}$ as being a rational but
unsymmetric function of the same $n$ arbitrary roots, so that we
may write
$$a_i^k ={\sc f}_i^{(k)}(x_1,\ldots \, x_n);$$
and since the truth of the equation
$${\sc f}^{(k)q} + {\sc e}_1 {\sc f}_i^{(k)q-1} + \cdots
       + {\sc e}_q = 0$$
must depend only on the {\it forms of the functions}, and not on
the {\it values of the quantities\/} which it involves, (those
values being altogether arbitrary,) we may alter in any manner
the arrangement of those $n$ arbitrary quantities
$x_1,\ldots \, x_n$, and the equation must still hold good.  But
by such changes of arrangement, the symmetric coefficients
${\sc e}_1,\ldots \, {\sc e}_q$ remain unchanged, while the
rational but unsymmetric function ${\sc f}_i^{(k)}$ takes, in
succession, all those $p$ values of which it was before supposed
to be capable; thse $p$ unequal values therefore must all be
roots of the same equation of the $q^{\rm th}$ degree, and
consequently $q$ must {\it not be less\/} than $p$.  And since it
has been shown that the former of these two last mentioned
numbers must {\it not exceed\/} the latter, it follows that they
must be {\it equal\/} to each other, so that we have the relation
$$q = p:$$
that is, the radical~$a_i^{(k)}$ and the rational function
${\sc f}_i^{(k)}$ must be exactly {\it coextensive in
multiplicity of value}.

For example, when, in considering the irreducible irrational
expression $b''$ for a root of the general cubic, we are
conducted to the relation assigned in the last article,
$$a_1' = {\sc f}_1'(x_1, x_2, x_3) = {\textstyle {1 \over 18}}
   (\rho_3^3 - \rho_3) (x_1 - x_2) (x_1 - x_3) (x_2 - x_3);$$
we can then at pleasure infer, either that the radical~$a_i'$
must admit (as a radical) of two and only two values, if we have
previously perceived that the rational function~${\sc f}_1'$
admits (as a rational function) of two values, and only two,
corresponding to changes of arrangement of the three roots
$x_1$,~$x_2$,~$x_3$, namely, the two following values, which
differ by their signs,
$$\pm {\textstyle {1 \over 18}}
   (\rho_3^3 - \rho_3) (x_1 - x_2) (x_1 - x_3) (x_2 - x_3);$$
or else we may infer that the function~${\sc f}_1'$ admits thus
of two values and two only, for all changes of arrangement of
$x_1$,~$x_2$,~$x_3$, if we have perceived that the radical~$a_1'$
(as being given by its square,
$$a_1'^2 = f_1 = c_1^2 - c_2^3,$$
which square is rational,) admits, itself, of the two values
$\pm a_1'$ which differ in their signs.

\bigbreak

[12.]
The conditions assumed in the last article are all fulfilled,
when we suppose the coefficients ${\sc a}_1$ \&c.\ to coincide
with the $n$ original quantities $a_1$, \&c., that is, when we
return to the equation originally proposed;
$$x^n + a_1 x^{n-1} + \cdots + a_{n-1} x + a_n = 0,$$
which is the general equation of the $n^{\rm th}$ degree: so that
we have, for any radical~$a_i^{(k)}$, which enters into the
composition of any irrational and irreducible function
representing any root of any such equation, an expression of the
form
$$a_i^{(k)} ={\sc f}_i^{(k)}(x_1,\ldots \, x_n);$$
the radical and the rational function being coextensive in
multiplicity of value.  We are, therefore, conducted thus to the
following important theorem, to which {\sc Abel} first was led,
by reasonings somewhat different from the foregoing: namely, that
``if a root~$x$ of the general equation of any particular degree
$n$ can be expressed as an irreducible irrational function
$b^{(m)}$ of the $n$ arbitrary coefficients of that equation,
then every radical~$a_i^{(k)}$, which enters into the composition
of that function~$b^{(m)}$, must admit of being expressed as a
rational, though unsymmetric function~${\sc f}_i^{(k)}$ of the
$n$ arbitrary roots of the same general equation; and this
rational but unsymmetric function~${\sc f}_i^{(k)}$ must admit of
receiving exactly the same variety of values, through changes of
arrangement of the $n$ roots on which it depends, as that which
the radical~$a_i^{(k)}$ can receive, through multiplications of
itself and of all its subordinate functional radicals by any
powers of the corresponding roots of unity.''

Examples of the truth of this theorem have already been given, by
anticipation, in the seventh and tenth articles of this Essay; to
which we may add, that the radicals $a_i''$ and $a_i'$, in the
expressions given above for a root of the general biquadratic,
admit of being thus expressed:
$$\eqalign{
a_1'' &= {\textstyle {1 \over 48}}
         \{   (x_1 + x_2 - x_3 - x_4)^2
            + \rho_3^2 (x_1 - x_2 + x_3 - x_4)^2
            + \rho_3   (x_1 - x_2 - x_3 + x_4)^2 \} \cr
      &= {\textstyle {1 \over 12}}
         \{   x_1 x_2 + x_3 x_4
            + \rho_3^2 (x_1 x_3 + x_2 x_4)
            + \rho_3   (x_1 x_4 + x_2 x_3) \}; \cr
a_1'  &= {\textstyle {1 \over 3456}}
         \{   x_1 x_2 + x_3 x_4
            + \rho_3^2 (x_1 x_3 + x_2 x_4)
            + \rho_3   (x_1 x_4 + x_2 x_3) \}^3 \cr
    &\mathrel{\phantom{=}}
         - {\textstyle {1 \over 3456}}
         \{   x_1 x_2 + x_3 x_4
            + \rho_3^2 (x_1 x_4 + x_2 x_3)
            + \rho_3   (x_1 x_3 + x_2 x_4) \}^3 \cr
      &= {\textstyle {1 \over 1152}} (\rho_3^2 - \rho_3)
            (x_1 - x_2) (x_1 - x_3) (x_1 - x_4)
            (x_2 - x_3) (x_2 - x_4) (x_3 - x_4).\cr}$$

But before we proceed to apply this theorem to prove, in a manner
similar to that of {\sc Abel}, the impossibility of obtaining any
finite expression, irrational and irreducible, for a root of the
general equation of the fifth degree, it will be instructive to
apply it, in a new way, (according to the announcement made in
the second article,) to equations of lower degrees; so as to
draw, from those lower equations, a class of illustrations quite
different from those which have been heretofore adduced: namely,
by showing, {\it \`{a} priori}, with the help of the same general
theorem, that no new finite function, irrational and irreducible,
can be found, essentially distinct in its radicals from those
which have long since been discovered, for expressing any root of
any such lower but general equation, quadratic, cubic or
biquadratic, in terms of the coefficients of that equation.

\bigbreak

[13.]
Beginning then with the general quadratic,
$$x^2 + a_1 x + a_2 = 0,$$
let us endeavour to investigate, {\it \`{a} priori}, with the
help of the foregoing theorem, all possible forms of irrational
and irreducible functions $b^{(m)}$, which can express a root~$x$
of this quadratic, in terms of the two arbitrary coefficients
$a_1$,~$a_2$, so as to satisfy identically, or independently of
the values of those two coefficients, the equation
$$b^{(m)2} + a_1 b^{(m)} + a_2 = 0.$$
The two roots of the proposed quadratic being denoted by the
symbols $x_1$ and $x_2$, we know that the two coefficients $a_1$
and $a_2$ are equal to the following symmetric functions,
$$a_1 = - (x_1 + x_2),\quad a_2 = x_1 x_2;$$
we cannot therefore suppose either root to be a rational function
$b$ of these coefficients, because an unsymmetric function of two
arbitrary quantities cannot be equal to a symmetric function of
the same; and consequently we must suppose that the exponent $m$
of the order of the sought function~$b^{(m)}$ is greater than
$0$.  The expression $b^{(m)}$ for $x$ must therefore involve at
least one radical~$a_1'$, which must itself admit of being
expressed as a rational but unsymmetric function of the two roots
$x_1$,~$x_2$,
$$a_1' = {\sc f}_1'(x_1, x_2),$$
and of which some prime power can be expressed as a rational
function of the two coefficients $a_1$,~$a_2$,
$$a_1'^{\alpha_1'} = f(a_1, a_2),$$
the exponent $\alpha_1'$ being equal to the number of the values
$${\sc f}_1'(x_1, x_2),\quad {\sc f}_1'(x_2, x_1),$$
of the unsymmetric function~${\sc f}_1'$, and consequently being
$= 2$; so that the radical~$a_1'$ must be a square root, and must
have two values differing in sign, which may be thus expressed:
$$+a_1' = {\sc f}_1'(x_1, x_2),\quad
  -a_1' = {\sc f}_1'(x_2, x_1).$$
But, in general, whatever rational function may be denoted by
${\sc f}$, the quotients
$${{\sc f}(x_1, x_2) + {\sc f}(x_2, x_1) \over 2}
   \quad\hbox{and}\quad
  {{\sc f}(x_1, x_2) - {\sc f}(x_2, x_1) \over 2(x_1 - x_2)}$$
are some symmetric functions, $a$ and $b$; so that we may put
generally
$${\sc f}(x_1, x_2) = a + b(x_1 - x_2),\quad
  {\sc f}(x_2, x_1) = a - b(x_1 - x_2);$$
therefore, since we have, at present,
$${\sc f}_1'(x_2, x_1) = - {\sc f}_1'(x_1, x_2),$$
the function~${\sc f}_1'$ must be of the form
$${\sc f}_1'(x_1,x_2) = b(x_1 - x_2),$$
the multiplier $b$ being symmetric.  At the same time,
$$a_1' = b(x_1 - x_2),$$
and therefore the function~$f_1$ is of the form
$$f_1(a_1, a_2) = a_1'^2 
   = b^2 (x_1 - x_2)^2 = b^2 (a_1^2 - 4 a_2),$$
so that the radical~$a_1'$ may be thus expressed,
$$a_1' = \sqrt{b^2(a_1^2 -4 a_2)},$$
in which, $b$ is some rational function of the coefficients
$a_1$,~$a_2$.  No other radical~$a_2'$ of the first order can
enter into the sought irreducible expression for $x$; because the
same reasoning would show that any such new radical ought to be
reducible to the form
$$a_2' = c(x_1 - x_2) = {c \over b} a_1',$$
$c$ being some new symmetric function of the roots, and
consequently some new rational function of the coefficients; so
that, after calculating the radical~$a_1'$, it would be
unnecessary to effect any new extraction of prime roots for the
purpose of calculating $a_2'$, which latter radical would
therefore be superfluous.  Nor can any radical~$a_1''$ of higher
order enter, because such radical would have $2 \alpha_1''$
values, $\alpha_1''$ being greater than $1$, while any rational
function~${\sc f}_1''$, of two arbitrary quantities $x_1$,~$x_2$,
can receive only two values, through any changes of their
arrangement.  The exponent $m$, of the order of the sought
irreducible function~$b^{(m)}$, must therefore be $= 1$, and this
function itself must be of the form
$$b' = b_0 + b_1 a_1',$$
$b_0$ and $b_1$ being rational functions of $a_1$,~$a_2$, or
symmetric functions of the two roots $x_1$,~$x_2$, which roots
must admit of being separately expressed as follows:
$$x_1 = b_0 + b_1 a_1',\quad
  x_2 = b_0 - b_1 a_1',$$
{\it if any expression of the sought kind can be found for either
of them}.  It is, therefore, necessary and sufficient for the
existence of such an expression, that the two following
quantities,
$$b_0 = {x_1 + x_2 \over 2},\quad
  b_1 = {x_1 - x_2 \over 2 a_1'},$$
should admit of being expressed as rational functions of
$a_1$,~$a_2$; and this condition is satisfied, since the
foregoing relations give
$$b_0 = - {a_1 \over 2},\quad
  b_1 = {1 \over 2b}.$$

We find, therefore, as the sought irrational and irreducible
expression, and as the only possible expression of that kind, (or
at least as one with which all others must essentially coincide,)
for a root~$x$ of the general quadratic, the following:
$$x = b'
   = {-a_1 \over 2}
       + {1 \over 2b} \sqrt{ b^2 (a_1^2 - 4 a_2) };$$
$b$ still denoting any arbitrary rational function of the two
arbitrary coefficients $a_1$,~$a_2$, or any numerical constant,
(such as the number ${1 \over 2}$, which was the value of the
quantity $b$ in the formul{\ae} of the preceding articles,) and
the two separate roots $x_1$,~$x_2$, being obtained by taking
separately the two signs of the radical.  And thus we see
{\it \`{a} priori}, that {\it every\/} method, for
calculating a root~$x$ of the general quadratic equation as a
function of the two coefficients, by any finite number of
additions, subtractions, multiplications, divisions, elevations
to powers, and extractions of prime radicals, (these last
extractions being supposed to be reduced to the smallest possible
number,) {\it must\/} involve the extraction of some one
square-root of the form
$$a_1' = \sqrt{ b^2 ( a_1^2 - 4a_2 ) },$$
and must {\it not\/} involve the extraction of any other radical.
But this square-root $a_1'$ is not essentially distinct from that
which is usually assigned for the solution of the general
quadratic: it is therefore impossible to discover any {\it new\/}
irrational expression, finite and irreducible, for a root of that
general quadratic, essentially distinct from the expressions
which have long been known: and the only possible difference
between the extractions of radicals which are required in any two
methods of solution, if neither method require any superfluous
extraction, is that these methods may introduce different square
factors into the expressions of that quantity or function~$f_1$,
of which, in each, the square root $a_1'$ is to be calculated.

\bigbreak

[14.]
Proceeding to the general cubic,
$$x^3 + a_1 x^2 + a_2 x + a_3 = 0,$$
we know, first, that the three coefficients are symmetric
functions of the three roots,
$$a_1 = - (x_1 + x_2 + x_3),\quad
  a_2 =   x_1 x_2 + x_1 x_3 + x_2 x_3,\quad
  a_3 = - x_1 x_2 x_3,$$
so that we cannot express any one of these three arbitrary roots
$x_1$,~$x_2$,~$x_3$, as a rational function~$b$ of the three
coefficients $a_1$,~$a_2$,~$a_3$; we must therefore inquire
whether it can be expressed as an irrational function~$b^{(m)}$,
involving at least one radical~$a_1'$ of the first order, which
is to satisfy the two conditions,
$$a_1'^{\alpha_1'} = f_1(a_1, a_2, a_3),$$
and
$$a_1' = {\sc f}_1'(x_1, x_2, x_3);$$
the functions $f_1$ and ${\sc f}_1'$ being rational, and the
prime exponent $\alpha_1'$ being either $2$ or $3$, because it is
to be equal to the number of values of the rational
function~${\sc f}_1'$ obtained by changing in all possible ways
the arrangement of the three roots $x_1$,~$x_2$,~$x_3$, and
therefore must be a divisor of the product $1 \mathbin{.} 2
\mathbin{.} 3 = 6$.

Now by the properties of rational functions of three variables,
(of which an investigation shall soon be given, but which it is
convenient merely to enunciate here, that the course of the main
argument may not be too much interrupted,) no three-valued
function of three arbitrary quantities $x_1$,~$x_2$,~$x_3$, can
have a symmetric cube; and the only two-valued functions, which
have symmetric squares, are of the form
$$b(x_1 - x_2)(x_1 - x_3)(x_2 - x_3),$$
$b$ being a symmetric but otherwise arbitrary multiplier.  We
must therefore suppose, that the radical~$a_1'$ is a square-root,
and that it may be thus represented:
$$\eqalign{
a_1' &= {\sc f}_1'(x_1, x_2, x_3)
        = b(x_1 - x_2)(x_1 - x_3)(x_2 - x_3) \cr
   &= \surd
      \{ b^2 (x_1 - x_2)^2 (x_1 - x_3)^2 (x_2 - x_3)^2 \} \cr
   &= \surd
      \{ b^2 (a_1^2 a_2^2 - 4 a_1^3 a_3 - 4 a_2^3
            + 18 a_1 a_2 a_3 - 27 a_3^2) \} \cr
   &= \sqrt{ - 108 b^2 (c_1^2 - c_2^3) },\cr}$$
$b$ being here rational with respect to $a_1$,~$a_2$,~$a_3$, as
also are $c_1$ and $c_2$, which last have the same meanings here
as in the second article; so that the function~$f_1$ is of the
form,
$$f_1 (a_1, a_2, a_3) = - 108 b^2 (c_1^2 - c_2^3).$$

No other radical of the first order,~$a_2'$, can enter into the
sought irreducible expression $b^{(m)}$; because the same
reasoning would give
$$a_2' = c(x_1 - x_2)(x_1 - x_3)(x_2 - x_3)
   = {c \over b} a_1',$$
$c$ being rational with respect to $a_1$,~$a_2$,~$a_3$, so that
the radical~$a_2'$ would be superfluous.  On the other hand, no
expression of the form $b_0 + b_1 a_1'$ can represent the
three-valued function~$x$; we must therefore suppose that if the
sought expression $b^{(m)}$ exist at all, it is, at lowest, of
the second order, and involves at least one radical~$a_1''$, such
that
$$a_1''^{\alpha_1''} = \, ( \, f_1' = \, ) \, b_0 + b_1 a_1',$$
and
$$a_1'' = {\sc f}_1''(x_1, x_2, x_3);$$
the rational function~${\sc f}_1''$ admitting of $2\alpha_1''$
values, and consequently the exponent $\alpha_1''$ being $= 3$,
(since it cannot be $= 2$, because no function of three variables
has exactly four values,) so that we must suppose the radical
$a_1''$ to be a cube-root, of the form
$$a_1'' = \root 3 \of {b_0 + b_1 a_1'},$$
$b_0$ and $b_1$ being rational with respect to
$a_1$,~$a_2$,~$a_3$.  But in order that a six-valued rational
function~${\sc f}_1''$, of three arbitrary quantities
$x_1$,~$x_2$,~$x_3$, should have a two-valued cube, it must be of
the form
$${\sc f}_1'' (x_1, x_2, x_3)
   = (p_0 + p_1 a_1')(x_1 + \rho_3^2 x_2 + \rho_3 x_3);$$
in which $p_0$ and $p_1$ are symmetric, $a_1'$ has the form
recently assigned, and $\rho_3$ is a root of the numerical
equation
$$\rho_3^2 + \rho_3 + 1 = 0;$$
we must therefore suppose that
$$a_1'' = (p_0 + p_1 a_1')(x_1 + \rho_3^2 x_2 + \rho_3 x_3),$$
and
$$b_0 + b_1 a_1'
   = 27(p_0 + p_1 a_1')^3 \left\{ c_1
         + {\textstyle {1 \over 18}} (\rho_3^2 - \rho_3)
            {a_1' \over b} \right\},$$
$c_1$ retaining here its recent meaning; so that the radical
$a_1''$ may be considered as the cube-root of this last
expression.  If any other radical~$a_2''$ of the second order
could enter into the composition of $b^{(m)}$, it ought, for the
same reasons, to be either of the form
$$a_2''  = (q_0 + q_1 a_1')(x_1 +\rho_3^2 x_2 + \rho_3 x_3),$$
or else of the form
$$a_2''  = (q_0 + q_1 a_1')(x_1 +\rho_3 x_2 + \rho_3^2 x_3),$$
$\rho_3$ being here the same root of the numerical equation
$\rho_3^2 +\rho_3 + 1 = 0$, as in the expression for
$\alpha_1''$; we should therefore have either the relation
$$a_2'' = {q_0 + q_1 a_1' \over p_0 + p_1 a_1'} a_1'',$$
or else the relation
$$a_2'' = {9 c_2 (p_0 + p_1 a_1)(q_0 + q_1 a_1') \over a_1''},$$
$c_2$ retaining its recent meaning; so that in each case it would
be superfluous to perform any new extraction of a cube-root or
other radical in order to calculate $a_2''$, after $a_1'$ and
$a_1''$ had been calculated; and consequently no such other
radical~$a_2''$ of the second order can enter into the
composition of the irreducible function~$b^{(m)}$.  If then that
function be itself of the second order, it must be capable of
being put under the form
$$b'' = b_0' + b_1' a_1'' + b_2' a_1''^2,$$
$b_0'$,~$b_1'$,~$b_2'$ being functions of the forms
$$b_0' = (b_0')_0 + (b_0')_1 a_1',\quad
  b_1' = (b_1')_0 + (b_1')_1 a_1',\quad
  b_2' = (b_2')_0 + (b_2')_1 a_1',$$
in which the radicals $a_1'$ and $a_1''$ have the forms lately
found, and $(b_0')_0,\ldots \, (b_2')_1$ are rational functions
of $a_1$,~$a_2$,~$a_3$.  And on the same supposition, the three
roots $x_1$,~$x_2$,~$x_3$, of that equation must, in some
arrangement or other, be represented by the three expressions,
$$\eqalign{
x_\alpha &= b_0''
    = b_0' + b_1' a_1'' + b_2' a_1''^2,\cr
x_\beta  &= b_1''
   = b_0' + \rho_3   b_1' a_1'' + \rho_3^2 b_2' a_1''^2,\cr
x_\gamma &= b_2''
   = b_0' + \rho_3^2 b_1' a_1'' + \rho_3   b_2' a_1''^2,\cr}$$
$\rho_3$ retaining here its recent value: which expressions
reciprocally will be true, if the following relations,
$$\eqalign{
b_0'
   &= {\textstyle {1 \over 3}}
         (x_\alpha + x_\beta + x_\gamma),\cr
b_1' a_1''
   &= {\textstyle {1 \over 3}}
         (x_\alpha + \rho_3^2 x_\beta + \rho_3   x_\gamma),\cr
b_2' a_1''^2
   &= {\textstyle {1 \over 3}}
         (x_\alpha + \rho_3   x_\beta + \rho_3^2 x_\gamma),\cr}$$
can be made to hold good, by any suitable arrangement of the
roots $x_\alpha$,~$x_\beta$,~$x_\gamma$, and by any suitable
selection of those rational functions of $a_1$,~$a_2$,~$a_3$,
which have hitherto been left undetermined.  Now, for this
purpose it is necessary and sufficient that the arrangement of
the roots $x_\alpha$,~$x_\beta$,~$x_\gamma$, should coincide with
one or other of the three following arrangements, namely
$x_1$,~$x_2$,~$x_3$, or $x_2$,~$x_3$,~$x_1$, or
$x_3$,~$x_1$,~$x_2$; the value of $3 b_1' (p_0 + p_1 a_1')$
being, in the first case, unity, in the second case,~$\rho_3$;
and, in the third case,~$\rho_3^2$; while, in every case, the
value of $b_0'$ is to be
$\displaystyle {-a_1 \over 3}$,
and that of
$b_1' b_2' (b_0 + b_1 a_1')$ is to be $c_2$.  All these
suppositions are compatible with the conditions assigned before;
nor is there any essential difference between the three cases of
arrangement just now mentioned, since the passage from any one to
any other may be made (as we have seen) by merely multiplying the
coefficient~$b_1'$, which admits of an arbitrary multiplier, by
an imaginary cube-root of unity.  We have, therefore, the
following irrational and irreducible expression for the root~$x$
of the general cubic, as a function of the second order,
$$x = b'' = {-a_1 \over 3} + {a_1'' \over 3 (p_0 + p_1 a_1')}
      + { 3 c_2 (p_0 + p_1 a_1') \over a_1''};$$
in which it is to be remembered that
$$a_1''^3 = 27 (p_0 + p_1 a_1')^3 \left\{ c_1
         + {\textstyle {1 \over 18}} (\rho_3^2 - \rho_3)
            {a_1' \over b} \right\},$$
and that
$$a_1'^2 = - 108 b^2 (c_1^2 - c_2^3);$$
$c_1$ and $c_2$ having the determined values above referred to,
namely
$$c_1 = - {\textstyle {1 \over 54}}
         (2a_1^3 - 9 a_1 a_2 + 27 a_3),\quad
  c_2 = {\textstyle {1 \over 9}}
         (a_1^2 - 3 a_2),$$
and $\rho_3$ being an imaginary cube-root of unity, but $b$ and
$p_0$,~$p_1$, being any arbitrary rational functions of
$a_1$,~$a_2$,~$a_3$, or even any arbitrary numeric constants;
except that $b$ must be different from $0$, and that $p_0$,~$p_1$
must not both together vanish.  (In the formul{\ae} of the
earlier articles of this essay, these three last quantities had
the following particular values,
$$b = {\textstyle {1 \over 18}} (\rho_3^2 - \rho_3),\quad
  p_0 = {\textstyle {1 \over 3}},\quad
  p_1 = 0.)$$
By substituting for the cubic radical~$a_1''$ the three unequal
values $a_1''$,~$\rho_3 a_1''$,~$\rho_3^2 a_1''$, in the general
expression, just now found, for $x$, we obtain separate and
unequal expressions for the three separate roots
$x_1$,~$x_2$,~$x_3$; these roots, and every rational function of
them, may consequently be expressed as rational functions of the
two radicals $a_1'$ and $a_1''$; and therefore it is unnecessary
and improper, in the present research, to introduce any other
radical.  But these two radicals $a_1'$ and $a_1''$ are not
essentially distinct from those which enter into the usual
formul{\ae} for the solution of a cubic equation: it is therefore
impossible to discover any {\it new} irrational expression,
finite and irreducible, for a root of the general cubic,
{\it essentially distinct\/} from those which have long been
known; and the only possible difference, with respect to the
extracting of radicals, between any two methods of solution which
both are free from all superfluous extractions, consists in the
introduction of different square factors into that quantity or
function~$f_1$, of which, in each, the square root $a_1'$ is to
be calculated; or in the introduction of different cubic factors
into that other quantity or function~$f_1'$, of which, in each
method, it is requisite to calculate the cube-root $a_1''$.  It
is proper, however, to remember the remarks which have been made,
in a foregoing article, respecting the {\it reducibility\/} of a
certain expression, involving {\it two\/} cubic radicals $a_1''$
and $a_2''$, which is not uncommonly assigned for a root of the
cubic equation.

\bigbreak

[15.]
But it is necessary to demonstrate some properties of rational
functions of three variables, which have been employed in the
foregoing investigation.  And because it will be necessary to
investigate afterwards some analogous properties of functions of
four and five arbitrary quantities, it may be conducive to
clearness and uniformity that we should begin with a few remarks
respecting functions which involve two variables only.

Let ${\sc f}(x_\alpha, x_\beta)$ denote any arbitrary rational
function of two arbitrary quantities $x_1$,~$x_2$, arranged in
either of their only two possible arrangements; so that the
function~${\sc f}$ admits of the two following values
$${\sc f}(x_1, x_2) \quad\hbox{and}\quad {\sc f}(x_2, x_1),$$
which for conciseness may be thus denoted,
$$(1,2) \quad\hbox{and}\quad (2,1).$$
These different {\it values\/} of the proposed function~${\sc f}$
may also be considered as being themselves two different
{\it functions\/} of the same two quantities $x_1$,~$x_2$ taken
in some determined order; and may, in this view, be denoted thus,
$${\sc f}_1(x_1, x_2) \quad\hbox{and}\quad {\sc f}_2(x_1, x_2),$$
or, more concisely,
$$(1,2)_1 \quad\hbox{and}\quad (1,2)_2:$$
they may also, on account of the mode in which they are formed
from one common type ${\sc f}(x_\alpha, x_\beta)$ be said to be
{\it syntypical functions}.  For example, the two values,
$$a x_1 + a x_2
   = (1,2) = {\sc f}(x_1, x_2) = {\sc f}_1(x_1, x_2) = (1,2)_1,$$
and
$$a x_2 + a x_1
   = (2,1) = {\sc f}(x_2, x_1) = {\sc f}_2(x_1, x_2) = (1,2)_2,$$
of the function~$a x_\alpha + b x_\beta$, may be considered as
being two different but {\it syntypical\/} functions of the two
variables $x_1$ and $x_2$.  And again, in the same sense, the
functions
$\displaystyle {x_1^2 \over x_2}$
and
$\displaystyle {x_2^2 \over x_1}$
are syntypical.

Now although, in general, two such syntypical functions,
${\sc f}_1$ and ${\sc f}_2$, are unconnected by any relation
among themselves, on account of the independence of the two
arbitrary quantities $x_1$ and $x_2$; yet, for some
{\it particular forms\/} of the original or typical
function~${\sc f}_1$, they may become connected by some such
relation, without any restriction being thereby imposed on those
two arbitrary quantities.  But all such relations may easily be
investigated, with the help of the two general forms obtained in
the thirteenth article, namely,
$${\sc f}_1 = a + b (x_1 - x_2),\quad
  {\sc f}_2 = a - b (x_1 - x_2),$$
in which $a$ and $b$ are symmetric.  For example, we see from
these forms that the two syntypical functions ${\sc f}_1$ and
${\sc f}_2$ become equal, when they reduce themselves to the
symmetric term or function~$a$, but not in any other case; and
that their squares are equal without their being equal
themselves, if they are of the forms $\pm b(x_1 - x_2)$, but not
otherwise.  We see too, that we cannot suppose
${\sc f}_2 = \rho_3 {\sc f}_1$, without making $a$ and $b$ both
vanish; and therefore that two syntypical functions of two
arbitrary quantities cannot have equal cubes, if they be
themselves unequal.

\bigbreak

[16.]
After these preliminary remarks respecting functions of two
variables, let us now pass to functions of three; and accordingly
let ${\sc f}(x_\alpha, x_\beta, x_\gamma)$, or more concisely
$(\alpha, \beta, \gamma)$, denote any arbitrary rational function
of any three arbitrary and independent quantities
$x_1$,~$x_2$,~$x_3$, arranged in any arbitrary order.  It is
clear that this function~${\sc f}$ has in general six different
values, namely,
$$(1,2,3),\quad
  (2,3,1),\quad
  (3,1,2),\quad
  (2,1,3),\quad
  (3,2,1),\quad
  (1,3,2),$$
or, in a more developed notation,
$${\sc f}(x_1, x_2, x_3), \quad\ldots\quad
      {\sc f}(x_1, x_3, x_2),$$
corresponding to the six different possible arrangements of the
three quantities on which it is supposed to depend; and that
these six {\it values\/} of the function~${\sc f}$ may also be
considered as six different but {\it syntypical functions\/} of
the same three arbitrary quantities $x_1$,~$x_2$,~$x_3$, taken in
some determined order; which functions may be thus denoted,
$${\sc f}_1(x_1, x_2, x_3), \, \ldots \, \,
      {\sc f}_6(x_1, x_2, x_3),$$
or, more concisely,
$$(1,2,3)_1, \, \ldots \, \, (1,2,3)_6.$$
For example, the six following values,
$$\eqalign{
a x_1 + b x_2 + c x_3 &= (1,2,3) = {\sc f}(x_1, x_2, x_3),\cr
a x_2 + b x_3 + c x_1 &= (2,3,1) = {\sc f}(x_2, x_3, x_1),\cr
a x_3 + b x_1 + c x_2 &= (3,1,2) = {\sc f}(x_3, x_1, x_2),\cr
a x_2 + b x_1 + c x_3 &= (2,1,3) = {\sc f}(x_2, x_1, x_3),\cr
a x_3 + b x_2 + c x_1 &= (3,2,1) = {\sc f}(x_3, x_2, x_1),\cr
a x_1 + b x_3 + c x_2 &= (1,3,2) = {\sc f}(x_1, x_3, x_2),\cr}$$
of the original or typical function
$$a x_\alpha + b x_\beta + c x_\gamma
   = {\sc f}(x_\alpha, x_\beta, x_\gamma),$$
may be considered as being six syntypical functions,
${\sc f}_1$,~${\sc f}_2$, ${\sc f}_3$,
${\sc f}_4$, ${\sc f}_5$,~${\sc f}_6$,
of the three quantities $x_1$,~$x_2$,~$x_3$.  Such also are the
six following,
$${x_1 \over x_2} + x_3,\quad
  {x_2 \over x_3} + x_1,\quad
  {x_3 \over x_1} + x_2,\quad
  {x_2 \over x_1} + x_3,\quad
  {x_3 \over x_2} + x_1,\quad
  {x_1 \over x_3} + x_2,$$
which are the values of the function
$\displaystyle {x_\alpha \over x_\beta} + x_\gamma$.

Now, in general, six such syntypical functions of three arbitrary
quantities are all unequal among themselves; nor can any ratio or
other relation between them be assigned, (except that very
relation which constitutes them syntypical,) so long as the form
of the function~${\sc f}$, although it has been supposed to be
rational, remains otherwise entirely undetermined.  But, for some
{\it particular forms\/} of this original or typical
function~${\sc f}(x_\alpha, x_\beta, x_\gamma)$, relations may
arise between the six syntypical functions
${\sc f}_1,\ldots \, {\sc f}_6$,
without any restriction being thereby imposed on the three
arbitrary quantities $x_1$,~$x_2$,~$x_3$; for example, the
function~${\sc f}$ may be partially or wholly symmetric, and then
the functions ${\sc f}_1,\ldots \, {\sc f}_6$ will, some or all,
be equal.  And we are now to study the chief {\it functional
conditions}, under which relations of this kind can arise.  More
precisely, we are to examine what are the conditions under which
the number of the values of a rational function~${\sc f}$ of
three variables, or of the square or cube of that function, can
reduce itself below the number six, in consequence of two or more
of the six syntypical functions ${\sc f}_1,\ldots \, {\sc f}_6$,
or of their squares or cubes, which are themselves syntypical,
becoming equal to each other.  And for this purpose we must first
inquire into the conditions requisite in order that any
{\it two\/} syntypical functions, or that any two values of
${\sc f}$, may be equal.

\bigbreak

[17.]
If any two such values be denoted by the symbols
$${\sc f}(x_{\alpha_1}, x_{\beta_1}, x_{\gamma_1}),
   \quad\hbox{and}\quad
  {\sc f}(x_{\alpha_2}, x_{\beta_2}, x_{\gamma_2}),$$
or, more concisely, by the following,
$$(\alpha_1, \beta_1, \gamma_1)
   \quad\hbox{and}\quad
  (\alpha_2, \beta_2, \gamma_2),$$
it is clear that in passing from the one to the other, and
therefore in passing from some one arrangement to some other of
the three indices $\alpha$,~$\beta$,~$\gamma$, (which must
themselves coincide, in some arrangement or other, with the
numbers $1$,~$2$,~$3$,) we must have changed some index, such as
$\alpha$, to some other, such as $\beta$, which must also have
been changed, itself, either to $\alpha$ or to $\gamma$; this
latter index $\gamma$ remaining in the first case unaltered, but
being changed to $\alpha$ in the second case.  And, in whatever
order the indices $\alpha_1$,~$\beta_1$,~$\gamma_1$ may have
coincided with $\alpha$,~$\beta$,~$\gamma$, it is obvious that
the function
$${\sc f}(x_{\alpha_1}, x_{\beta_1}, x_{\gamma_1})
   \quad\hbox{or}\quad
  (\alpha_1, \beta_1, \gamma_1)$$
must coincide with the syntypical function
$${\sc f}_i(x_\alpha, x_\beta, x_\gamma)
   \quad\hbox{or}\quad
  (\alpha, \beta, \gamma)_i,$$
for some suitable index~$i$, belonging to the system
$1$,~$2$, $3$, $4$, $5$,~$6$; the equation
$$(\alpha_1, \beta_1, \gamma_1)
   = (\alpha_2, \beta_2, \gamma_2),$$
is therefore equal to one or other of the two following, namely,
either
$$\hbox{1st, } \ldots \,
   (\alpha, \beta, \gamma)_i = (\beta, \alpha, \gamma)_i,$$
or
$$\hbox{2nd, } \ldots \,
   (\alpha, \beta, \gamma)_i = (\beta, \gamma, \alpha)_i.$$
In the first case, the function~${\sc f}_i$ is symmetric with
respect to the two quantities $x_\alpha$,~$x_\beta$, and
therefore involves them only by involving their sum and product,
which may be thus expressed,
$$x_\alpha + x_\beta = - a_1 - x_\gamma,\quad
  x_\alpha x_\beta = a_2 + a_1 x_\gamma + x_\gamma^2,$$
$a_1$ and $a_2$ being symmetric functions of the three quantities
$x_1$,~$x_2$,~$x_3$, namely, the following,
$$a_1 = - (x_1 + x_2 + x_3),\quad
  a_2 = x_1 x_2 + x_1 x_3 + x_2 x_3;$$
so that if we put, for abridgment,
$$a_3 = - x_1 x_2 x_3,$$
the three quantities $x_1$,~$x_2$,~$x_3$ will be the three roots
of the cubic equation
$$x^3 + a_1 x^2 + a_2 x + a_3 = 0.$$
In this case, therefore, we may consider ${\sc f}_i$ as being a
rational function of the root~$x_\gamma$ alone, which function
will however involve, in general, the coefficients $a_1$ and
$a_2$; and we may put
$${\sc f}_i(x_\alpha, x_\beta, x_\gamma)
   = {\phi(x_\gamma) \over \chi(x_\gamma)}
   = {\chi(x_\alpha) \mathbin{.} \chi(x_\beta)
            \mathbin{.} \phi(x_\gamma)
         \over
         \chi(x_\alpha) \mathbin{.} \chi(x_\beta)
            \mathbin{.} \chi(x_\gamma)}
   = \psi(x_\gamma),$$
$\phi$,~$\chi$ and $\psi$ denoting here some rational and whole
functions of $x_\gamma$, which may however involve rationally the
coefficients of the foregoing cubic equation.  And since it is
unnecessary, on account of that equation, to retain in evidence
the cube or any higher powers of $x_\gamma$, we may write simply
$${\sc f}_i(x_\alpha, x_\beta, x_\gamma)
  = a + b x_\gamma + c x_\gamma^2,$$
$a$,~$b$,~$c$ being here symmetric functions of the three
quantities $x_1$,~$x_2$,~$x_3$: so that, in this case, the six
syntypical functions, or values of the function~${\sc f}$, reduce
themselves to the three following
$$a + b x_1 + c x_1^2,\quad
  a + b x_2 + c x_2^2,\quad
  a + b x_3 + c x_3^2.$$
Nor can these three reduce themselves to any smaller number,
without their all becoming equal and symmetric, by the vanishing
of $b$ and $c$.

In the second case, the form of ${\sc f}_i$ being such that
$$(\alpha, \beta, \gamma)_i = (\beta, \gamma, \alpha)_i,$$
it must also be such that
$$(\beta, \gamma, \alpha)_i = (\gamma, \alpha, \beta)_i;$$
for the same reason we must have
$$(\beta, \alpha, \gamma)_i = (\alpha, \gamma, \beta)_i
   = (\gamma, \beta, \alpha)_i,$$
so that the function changes when any two of the three indices
are interchanged, but returns to its former value when any two
are interchanged again; from which it results that the two
following combinations
$$(\alpha, \beta, \gamma)_i + (\beta, \alpha, \gamma)_i
   \quad\hbox{and}\quad
{ (\alpha, \beta, \gamma)_i - (\beta, \alpha, \gamma)_i \over
   (x_\alpha - x_\beta)
   (x_\alpha - x_\gamma)
   (x_\beta - x_\gamma) }$$
remain unchanged, after all interchanges of the indices, and are
therefore symmetric functions, such as $2a$ and $2b$, of the
three quantities $x_1$,~$x_2$,~$x_3$: so that we may write
$${\sc f}_i(x_\alpha, x_\beta, x_\gamma)
   = (\alpha, \beta, \gamma)_i
   = a + b (x_\alpha - x_\beta) (x_\alpha - x_\gamma)
           (x_\beta - x_\gamma);$$
and consequently the six syntypical functions, or values of the
function~${\sc f}$, reduce themselves in this case to the two
following,
$$a \pm b (x_1 - x_2) (x_1 - x_3) (x_2 - x_3),$$
in which $a$ and $b$ are symmetric.  It is evident that any
farther diminution of the number of values of ${\sc f}$,
conducts, in this case also, to the one-valued or symmetric
function~$a$.

   Combining the foregoing results, we see that if an unsymmetric
rational function of three arbitrary quantities have fewer than
six values, it must be reducible either to the two-valued form
$$a + b (x_1 - x_2) (x_1 - x_3) (x_2 - x_3),$$
or to the three-valued form
$$a + bx + c x^2.$$

\medskip

[18.]
It is possible, however, that some analogous but different
reduction may cause either---I. the square, or II. the cube of a
function~${\sc f}$ of three variables, to have a smaller number
of values than the function~${\sc f}$ itself.  But, for this
purpose, it is necessary that we should now have a relation of
one or other of the two forms following, namely, either
$${\rm I. } \ldots \quad
   (\alpha_2, \beta_2, \gamma_2)
   = - (\alpha_1, \beta_1, \gamma_1)$$
or
$${\rm II. } \ldots \quad
   (\alpha_2, \beta_2, \gamma_2)
   = \rho_3 (\alpha_1, \beta_1, \gamma_1)$$
$(\rho_3$ denoting, as above, an imaginary cube root of unity,)
instead of the old functional relation
$(\alpha_2, \beta_2, \gamma_2) = (\alpha_1, \beta_1, \gamma_1)$.
And as we found ourselves permitted, before, to change that old
relation to one or other of these two,
$$\hbox{1st,}\quad
   (\beta, \alpha, \gamma)_i = (\alpha, \beta, \gamma)_i;\quad
  \hbox{2nd,}\quad
   (\beta, \gamma, \alpha)_i = (\alpha, \beta, \gamma)_i;$$
so are we now allowed to change the two new relations to the
four following:
$$\eqalign{
\hbox{I.~1, } \ldots \quad
   (\beta, \alpha, \gamma)_i
   &=      - (\alpha, \beta, \gamma)_i;\cr
\hbox{II.~1, } \ldots \quad
   (\beta, \alpha, \gamma)_i
   &= \rho_3 (\alpha, \beta, \gamma)_i;\cr}
      \quad
\eqalign{
\hbox{I.~2, } \ldots \quad
   (\beta, \gamma, \alpha)_i
   &=      - (\alpha, \beta, \gamma)_i;\cr
\hbox{II.~2, } \ldots \quad
   (\beta, \gamma, \alpha)_i
   &= \rho_3 (\alpha, \beta, \gamma)_i;\cr}$$
the relation (I.) admitting of being changed to one or other of
the two marked (I.~1) and (I.~2); and the
relation (II.) admitting, in like manner, of being changed either
to (II.~1) or to (II.~2).  But the relations
(I.~2) and (II.~1) conduct only to evanescent
functions, because (I.~2) gives
$$(\gamma, \alpha, \beta)_i
   = - (\beta, \gamma, \alpha)_i
   = + (\alpha, \beta, \gamma)_i,\quad
  (\alpha, \beta, \gamma)_i
   = - (\gamma, \alpha, \beta)_i
   = - (\alpha, \beta, \gamma)_i,$$
and (II.~1) gives
$$(\alpha, \beta, \gamma)_i
   = \rho_3   (\beta, \alpha, \gamma)_i
   = \rho_3^2 (\alpha, \beta, \gamma)_i:$$
we may therefore confine our attention to the other two
relations.  Of these (I.~1) requires that the function
$\displaystyle
   {(\alpha, \beta, \gamma)_i \over (x_\alpha - x_\beta)}$
should not change its value when $x_\alpha$ and $x_\beta$ are
interchanged, and consequently, by what was shown above, that it
should be reducible to the form $a + b x_\gamma + c x_\gamma^2$;
in this case, therefore, we have the expression,
$$(\alpha, \beta, \gamma)_i
   = {\sc f}_i(x_\alpha, x_\beta, x_\gamma)
   = (x_\alpha - x_\beta)(a + b x_\gamma + c x_\gamma^2),$$
the coefficients $a$,~$b$,~$c$, being symmetric functions of
$x_1$,~$x_2$,~$x_3$.  Accordingly the square of this function
${\sc f}_i$ admits in general of three values only, while the
function is itself in general six-valued; because the square of
the factor $x_\alpha - x_\beta$, but not that factor itself, can
be expressed as a rational function of $x_\gamma$, and of the
quantities $a_1$,~$a_2$,~$a_3$, which are symmetric relatively to
$x_1$,~$x_2$,~$x_3$.  It may even happen that the function
itself shall have only two values, and that its square shall be
symmetric, namely, by the factor $a + b x_\gamma + c x_\gamma^2$
being reducible to the form
$b (x_\alpha - x_\gamma) (x_\beta - x_\gamma)$, in which the
coefficient~$b$ is some new symmetric function; but the results
of the last article enable us to see that the functions thus
obtained, namely, those of the form
$$b   (x_\alpha - x_\beta)
      (x_\alpha - x_\gamma)
      (x_\beta - x_\gamma),$$
or more simply of the form
$$b (x_1 - x_2) (x_1 - x_3) (x_2 - x_3),$$
are the only two-valued functions of three variables which have
symmetric squares: they enable us also to see easily that the
square of a three-valued function of three variables is always
itself three-valued.  It remains, then, only to consider the
relations (II.~2); which requires that the function
$${ (\alpha, \beta, \gamma)_i \over
      x_\alpha + \rho_3^2 x_\beta + \rho_3 x_\gamma }$$
should be of the two-valued form
$a + b   (x_\alpha - x_\beta)
         (x_\alpha - x_\gamma)
         (x_\beta - x_\gamma)$;
because, if we denote it by $\phi(x_\alpha, x_\beta, x_\gamma)$,
we have
$$\phi(x_\alpha, x_\beta, x_\gamma)
   = \phi(x_\beta, x_\gamma, x_\alpha)
   = \phi(x_\gamma, x_\alpha, x_\beta),$$
and
$$\phi(x_\beta, x_\alpha, x_\gamma)
   = \phi(x_\alpha, x_\gamma, x_\beta)
   = \phi(x_\gamma, x_\beta, x_\alpha);$$
we have, therefore, in this case,
$$\eqalign{
(\alpha, \beta, \gamma)_i
   &= {\sc f}_i(x_\alpha, x_\beta, x_\gamma) \cr
   &= \{ a + b (x_\alpha - x_\beta) (x_\alpha - x_\gamma)
             (x_\beta - x_\gamma) \}
      (x_\alpha + \rho_3^2 x_\beta + \rho_3 x_\gamma),\cr}$$
$a$ and $b$ being symmetric coefficients, which must not both
together vanish; and accordingly we find, {\it \`{a} posteriori},
that whereas this function~${\sc f}_i$ has always itself six
values, its cube has only two.  The foregoing analysis shows at
the same time, that if an unsymmetric function of three variables
have fewer than six values, its cube cannot have fewer values
than itself; and accordingly it is easy to see that the cubes of
those two-valued and three-valued functions, which were assigned
in the last article, are themselves two-valued and three-valued.
In fact, the passage from any one to any other of the values of
any such (two-valued or three-valued) function, may be performed
by interchanging some two of the three quantities
$x_1$,~$x_2$,~$x_3$; and if such interchange could have the
effect of multiplying the function by an imaginary cube-root of
unity, $\rho_3$, another interchange of the same two quantities
would multiply again by the same factor $\rho_3$; and therefore
the two interchanges combined would multiply by $\rho_3^2$, which
is a factor different from unity, although any two such
successive interchanges of any two quantities $x_\alpha$,
$x_\beta$, ought to make no change in the function.  If, then, a
rational function of three arbitrary quantities have a symmetric
cube, it must be itself symmetric.

The form of that six-valued function of three variables which has
a two-valued cube, may also be thus deduced, from the functional
relation (II.~2).  Omitting for simplicity, the lower
index $i$, which is not essential to the reasoning, we find,
by that relation,
$$(\beta, \gamma, \alpha)
   = \rho_3   (\alpha, \beta, \gamma);\quad
  (\gamma, \alpha, \beta)
   = \rho_3^2 (\alpha, \beta, \gamma);$$
$$(\gamma, \beta, \alpha)
   = \rho_3   (\alpha, \gamma, \beta);\quad
  (\beta, \alpha, \gamma)
   = \rho_3^2 (\alpha, \gamma, \beta);$$
so that
$$(\alpha, \beta, \gamma)
      \mathbin{.} (\alpha, \gamma, \beta)
  = (\beta, \gamma, \alpha) (\beta, \alpha, \gamma)
  = (\gamma, \alpha, \beta) (\gamma, \beta, \alpha)
  = e,$$
this product $e$ being some symmetric function; at the same time,
the sum
$(\alpha, \beta, \gamma) + (\alpha, \gamma, \beta)$
is a three-valued function~$y_\alpha$, which may be put under the
form
$$y_\alpha = a+ b x_\alpha + c x_\alpha^2,$$
$a$,~$b$, and $c$ being symmetric, and $b$ and $c$ being obliged
not both to vanish.  Attending therefore to that cubic equation
of which $x_\alpha$,~$x_\beta$,~$x_\gamma$ are the roots, we have
$$y_\alpha^2 = a^{(2)} +b^{(2)} x_\alpha + c^{(2)} x_\alpha^2,$$
$a^{(2)}$,~$b^{(2)}$, and $c^{(2)}$ denoting here some symmetric
functions, and $c$,~$c^{(2)}$ being obliged not both to vanish;
and consequently, by eliminating $x_\alpha^2$, we obtain an
equation of the form
$$(b c^{(2)} - c b^{(2)}) x_\alpha
   = c a^{(2)} - a c^{(2)} + c^{(2)} y_\alpha - c y_\alpha^2,$$
in which the coefficients of $y_\alpha$ and $y_\alpha^2$ cannot
both vanish, and in which therefore the coefficient of $x_\alpha$
cannot vanish, because the three-valued function~$y_\alpha$ must
not be a root of any equation with symmetric coefficients,
below the third degree; we have therefore an expression of the
form
$$x_\alpha = p + q y_\alpha + r y_\alpha^2,$$
in which $p$,~$q$,~$r$ are symmetric, and $q$ and $r$ do not both
vanish.  But
$$y_\alpha
   = (\alpha, \beta, \gamma) + (\alpha, \gamma, \beta)
   = (\alpha, \beta, \gamma)
       + {e \over (\alpha, \beta, \gamma)};$$
and the cube of $(\alpha, \beta, \gamma)$ is a two-valued
function; therefore
$$x_\alpha = p' + q' (\alpha, \beta, \gamma)
                + r' (\alpha, \beta, \gamma)^2,$$
the functions $p'$,~$q'$,~$r'$ being either symmetric or
two-valued, and consequently undergoing no change, when we pass
successively from the first to the second, or from the second to
the third, of the three functions
$(\alpha, \beta, \gamma)$,
$(\beta, \gamma, \alpha)$,
$(\gamma, \alpha, \beta)$,
by changing at each passage $x_\alpha$ to $x_\beta$, $x_\beta$ to
$x_\gamma$, and $x_\gamma$ to $x_\alpha$; and we have seen that
these three last-mentioned functions bear to each other the same
ratios as the three cube-roots of unity
$1$,~$\rho_3$,~$\rho_3^2$; we have therefore
$$x_\beta  = p' + q' \rho_3   (\alpha, \beta, \gamma)
                + r' \rho_3^2 (\alpha, \beta, \gamma)^2,\quad
  x_\gamma = p' + q' \rho_3^2 (\alpha, \beta, \gamma)
                + r' \rho_3   (\alpha, \beta, \gamma)^2;$$
and thus, finally, the six-valued function which has a two-valued
cube is found anew to be expressible as follows,
$$(\alpha, \beta, \gamma)
   = {1 \over 3 q'}
         (x_\alpha + \rho_3^2 x_\beta + \rho_3 x_\gamma);$$
in which the coefficient $\displaystyle {1 \over 3 q'}$ is a
two-valued function, of the form
$${1 \over 3q'} = a + b (x_1 - x_2) (x_1 - x_3) (x_2 - x_3),$$
$a$,~$b$, denoting here some new symmetric functions.

The theorems obtained incidentally in this last discussion supply
us also with another mode of proving that the cube of a
three-valued function of three arbitrary quantities must be
itself three-valued: for if we should suppose
$y_\beta = \rho_3 y_\alpha$, and consequently
$y_\gamma = \rho_3 y_\beta = \rho_3^2 y_\alpha$, in which
$y_\alpha = a + b x_\alpha + c x_\alpha^2$, and $b$ and $c$ do
not both vanish, we should then have relations of the forms
$$\eqalign{
x_\alpha
   &= p + q y_\alpha + r y_\alpha^2,\cr
x_\beta
   &= p + q \rho_3   y_\alpha + r \rho_3^2 y_\alpha^2,\cr
x_\gamma
   &= p + q \rho_3^2 y_\alpha + r \rho_3   y_\alpha^2;\cr}$$
but these would require that we should have the equation
$$x_\alpha + \rho_3^2 x_\beta + \rho_3 x_\gamma
   = 3q \mathbin{.} y_\alpha,$$
a condition which it is impossible to fulfil, because the first
member has six values, and the second only three.

\bigbreak

[19.]
The discussion of the forms of functions of four variables may
now be conducted more briefly, than would have been consistent
with clearness, if we had not already treated so fully of
functions in which the number of variables is less than four.

Let $x_1$,~$x_2$,~$x_3$,~$x_4$ be any four arbitrary quantities,
or roots of the general biquadratic,
$$x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0;$$
and let ${\sc f}(x_1, x_2, x_3, x_4)$, or, more concisely
$(1,2,3,4)$, denote any rational function of them.  By altering
the arrangement of these four roots, we shall in general obtain
twenty-four different but {\it syntypical\/} functions; of which
each, according to the analogy of the foregoing notation, may be
denoted by any one of the four following symbols:
$$(\alpha, \beta, \gamma, \delta)
   = {\sc f}(x_\alpha, x_\beta, x_\gamma, x_\delta)
   = (1,2,3,4)_i = {\sc f}_i(x_1, x_2, x_3, x_4).$$
In passing from any one to any other of these twenty-four
syntypical functions ${\sc f}_1,\ldots \, {\sc f}_{24}$, by a
change of arrangement of the four roots, some one of these roots,
such as the first in order, must be changed to some other, such
as the second; and this second must at the same time be changed
either to the first or to a different root, such as the third;
while, in the former case, the third and fourth roots may either
be interchanged among themselves or not; and, in the latter case,
the third root may be changed either to the first or to the
fourth.  We have therefore four and only four distinct sorts of
changes of arrangement, which may be typified by the passages
from the function $(\alpha, \beta, \gamma, \delta)$ to the four
following:
$$\hbox{I.   } \ldots \, (\beta, \alpha, \gamma, \delta);\quad
  \hbox{II.  } \ldots \, (\beta, \alpha, \delta, \gamma);\quad
  \hbox{III. } \ldots \, (\beta, \gamma, \alpha, \delta);\quad
  \hbox{IV.  } \ldots \, (\beta, \gamma, \delta, \alpha);$$
and may be denoted by the four characteristics
$$\nabla_1,\quad \nabla_2, \quad, \nabla_3, \quad \nabla_4;$$
or more fully by the following,
$$\nablaperm{1}{a,b},\quad \nablaperm{2}{a,b},\quad
  \nablaperm{3}{a,b,c},\quad \nablaperm{4}{a,b,c};$$
$\nablaperm{1}{a,b}$ implying, when prefixed to any function
$(\alpha, \beta, \gamma, \delta)$, that we are to interchange the
$a^{\rm th}$ and $b^{\rm th}$ of the roots on which it depends;
$\nablaperm{2}{a,b}$, that we are to interchange among
themselves, not only the $a^{\rm th}$ and $b^{\rm th}$, but also
that $c^{\rm th}$ and $d^{\rm th}$; $\nablaperm{3}{a,b,c}$, that
we are to interchange the $a^{\rm th}$ to the $b^{\rm th}$, the
$b^{\rm th}$ to the $c^{\rm th}$, and the $c^{\rm th}$ to the
$a^{\rm th}$; namely, by  putting that which had been
$b^{\rm th}$ in the place of that which had been $a^{\rm th}$,
and so on; and finally $\nablaperm{4}{a,b,c}$, that the
$a^{\rm th}$ is to be changed to the $b^{\rm th}$, the
$b^{\rm th}$ to the $c^{\rm th}$, the $c^{\rm th}$ to the
$d^{\rm th}$, and the $d^{\rm th}$ to the $a^{\rm th}$; so that
we have, in this notation,
$$\eqalign{
\hbox{I.   } \ldots \,
      \nablaperm{1}{1,2} (\alpha, \beta, \gamma, \delta)
   &= (\beta, \alpha, \gamma, \delta);\cr
\hbox{II.  } \ldots \,
      \nablaperm{2}{1,2} (\alpha, \beta, \gamma, \delta)
   &= (\beta, \alpha, \delta, \gamma);\cr
\hbox{III. } \ldots \,
      \nablaperm{3}{1,2,3} (\alpha, \beta, \gamma, \delta)
   &= (\beta, \gamma, \alpha, \delta);\cr
\hbox{IV.  } \ldots \,
      \nablaperm{4}{1,2,3} (\alpha, \beta, \gamma, \delta)
   &= (\beta, \gamma, \delta, \alpha).\cr}$$
The first sort of change may be called, altering in a {\it simple
binary cycle\/}; the second, in a {\it double binary cycle\/};
the third, in a {\it ternary\/}; and the fourth, in a
{\it quaternary\/} cycle.  And every possible equation,
$$(\alpha_2, \beta_2, \gamma_2, \delta_2)
   = (\alpha_1, \beta_1, \gamma_1, \delta_1),$$
between any two of the twenty-four syntypical functions
${\sc f}_i$ may be denoted by one or other of the four following
symbolic forms, in each of which the two members may be conceived
to be prefixed to a function such as
$(\alpha_1, \beta_1, \gamma_1, \delta_1)$:
$$\hbox{I.   } \ldots \, \nablaperm{1}{a,b} = 1;\quad
  \hbox{II.  } \ldots \, \nablaperm{2}{a,b} = 1;\quad
  \hbox{III. } \ldots \, \nablaperm{3}{a,b,c} = 1;\quad
  \hbox{IV.  } \ldots \, \nablaperm{4}{a,b,c} = 1;$$
or, without any loss of generality, by one of the four following,
in each of which the two members are conceived to be prefixed to
a function such as
$(\alpha, \beta, \gamma, \delta)_i$:
$$\hbox{I.   } \ldots \, \nablaperm{1}{1,2} = 1;\quad
  \hbox{II.  } \ldots \, \nablaperm{2}{1,2} = 1;\quad
  \hbox{III. } \ldots \, \nablaperm{3}{1,2,3} = 1;\quad
  \hbox{IV.  } \ldots \, \nablaperm{4}{1,2,3} = 1;$$
the I${}^{\rm st}$ and II${}^{\rm nd}$ suppositions conducting to
twelve-valued functions, the III${}^{\rm rd}$ to an eight-valued,
and the IV${}^{\rm th}$ to a six-valued function; while every
possible pair of equations between any three of the same
twenty-four syntypical functions, if it be not included in a
single equation of this last set, may be put under one or other
of the six following forms:
$$\vbox{\halign{\hfil\hbox{#}\enspace $\ldots$\enspace
   &\hfil $#$&\quad $#$&\qquad
   \hfil\hbox{#}\enspace $\ldots$\enspace
   &\hfil $#$&\quad $#$\cr
   (I. I.)
      & \nablaperm{1}{1,2} = 1,
      & \nablaperm{1}{1,3} = 1; &
   (I. I.)${}'$
      & \nablaperm{1}{1,2} = 1,
      & \nablaperm{1}{3,4} = 1; \cr
\noalign{\vskip3pt}
   (I. II.)
      & \nablaperm{1}{1,2} = 1,
      & \nablaperm{2}{1,3} = 1; &
   (I. III.)  & \nablaperm{1}{1,2} = 1,
      & \nablaperm{3}{2,3,4} = 1; \cr
\noalign{\vskip3pt}
   (II. II.)
      & \nablaperm{2}{1,2} = 1,
      & \nablaperm{2}{1,3} = 1; &
   (II. III.)
      & \nablaperm{2}{1,2} = 1,
      & \nablaperm{3}{1,2,3} = 1; \cr}}$$
which conduct respectively to functions with four, six, three,
one, six and two values; nor can any form of condition,
essentially distinct from all the ten last mentioned, be obtained
by supposing any three or more equations to exist between the
twenty-four functions~${\sc f}_i$.

A little attention will not fail to evince the justice of this
enumeration of the conditions under which a rational function of
four arbitrary variables can have fewer than twenty-four values:
yet it may not be useless to remark, as connected with this
inquiry, that, in virtue of the notation here employed, the
supposition $\nablaperm{1}{a,b} = 1$ involves the supposition
$\nablaperm{1}{b,a} = 1$; the supposition
$\nablaperm{2}{a,b} = 1$ involves the suppositions
$\nablaperm{2}{b,a} = 1$, $\nablaperm{2}{c,d} = 1$,
$\nablaperm{2}{d,c} = 1$; $\nablaperm{3}{a,b,c} = 1$ involves
$\nablaperm{3}{b,c,a} = 1$, $\nablaperm{3}{c,a,b} = 1$,
$\nablaperm{3}{a,c,b} = 1$, $\nablaperm{3}{c,b,a} = 1$,
$\nablaperm{3}{b,a,c} = 1$; $\nablaperm{4}{a,b,c} = 1$
involves $\nablaperm{4}{b,c,d} = 1$, $\nablaperm{4}{c,d,a} = 1$,
$\nablaperm{4}{d,a,b} = 1$, $\nablaperm{2}{a,c} = 1$,
$\nablaperm{4}{a,d,c} = 1$, $\nablaperm{4}{d,c,b} = 1$,
$\nablaperm{4}{c,b,a} = 1$, $\nablaperm{4}{b,a,d} = 1$;
while the system $\nablaperm{1}{a,b} = 1$,
$\nablaperm{2}{a,b} = 1$, is equivalent to the system
$\nablaperm{1}{a,b} = 1$, $\nablaperm{1}{c,d} = 1$;
$\nablaperm{1}{a,b} = 1$, $\nablaperm{3}{a,b,c} = 1$, to
$\nablaperm{1}{a,b} = 1$, $\nablaperm{1}{a,c} = 1$;
$\nablaperm{1}{a,b} = 1$, $\nablaperm{4}{a,b,c} = 1$, to
$\nablaperm{1}{a,b} = 1$, $\nablaperm{3}{b,c,d} = 1$;
$\nablaperm{1}{a,b} = 1$, $\nablaperm{4}{a,c,b} = 1$, to
$\nablaperm{1}{a,b} = 1$, $\nablaperm{2}{a,c} = 1$;
$\nablaperm{2}{a,b} = 1$, $\nablaperm{4}{a,b,c} = 1$, to
$\nablaperm{2}{a,b} = 1$, $\nablaperm{1}{b,d} = 1$;
$\nablaperm{3}{a,b,c} = 1$, $\nablaperm{3}{b,c,d} = 1$, to
$\nablaperm{2}{a,b} = 1$, $\nablaperm{3}{a,b,c} = 1$;
$\nablaperm{3}{a,b,c} = 1$, $\nablaperm{4}{a,b,c} = 1$, to
$\nablaperm{3}{a,b,c} = 1$, $\nablaperm{1}{c,d} = 1$;
and $\nablaperm{4}{a,b,c} = 1$, $\nablaperm{4}{a,c,b} = 1$, to
$\nablaperm{3}{a,b,c} = 1$, $\nablaperm{4}{a,b,c} = 1$: analogous
equivalences also holding good for other systems of analogous
conditions.

Let us now consider more closely the effects of the ten different
suppositions (I.),$\ldots$ (II.~III.).

In the case (I.), the function~${\sc f}$ is symmetric relatively
to some two roots $x_\alpha$,~$x_\beta$, and may be put under the
form of a rational function of two others, $x_\gamma$,
$x_\delta$, or simply of their difference,
$$\hbox{(I.)} \, \ldots \,
F = \phi(x_\gamma - x_\delta);$$
it being understood that this function~$\phi$ may involve the
coefficients $a_1$,~$a_2$,~$a_3$,~$a_4$, which are symmetric
relatively to $x_1$,~$x_2$,~$x_3$,~$x_4$: because it is in
general possible to determine rationally any two roots
$x_\gamma$,~$x_\delta$, of an equation of any given degree, when
their difference $x_\gamma - x_\delta$ is given.

In the case (II.), we may interchange some two roots,
$x_\alpha$,~$x_\beta$, if we at the same time interchange the two
others; and the function may be put under the form
$$\hbox{(II.)} \, \ldots \,
F = \phi( x_\alpha + x_\beta - x_\gamma - x_\delta,
          \overline{x_\alpha - x_\beta} \mathbin{.}
          \overline{x_\gamma - x_\delta} );$$
because any rational  function of the four roots may be
considered as a rational function of the four combinations
$$x_\alpha + x_\beta,\quad
  x_\alpha - x_\beta,\quad
  x_\gamma + x_\delta,\quad
  x_\gamma - x_\delta,$$
or of the four following,
$$x_\alpha + x_\beta + x_\gamma + x_\delta,\quad
  x_\alpha + x_\beta - x_\gamma - x_\delta,\quad
  x_\alpha - x_\beta,\quad
  \overline{x_\alpha - x_\beta}
      \mathbin{.} \overline{x_\gamma - x_\delta};$$
of which the first may be omitted, as symmetric, and the third as
being here obliged to enter only by its square, which square
$(x_\alpha - x_\beta)^2$ is expressible as a rational function of
$x_\alpha + x_\beta - x_\gamma - x_\delta$, involving also the
symmetric coefficients $a_1$,~$a_2$,~$a_3$, which are allowed to
enter in any manner into $\phi$.

In the case (III.), some three roots
$x_\alpha$,~$x_\beta$,~$x_\gamma$, may all be interchanged, the
fourth root remaining unaltered; and, on account of what has been
shown respecting functions of three variables, we may write
$$\hbox{(III.)} \, \ldots \,
F = \phi(x_\delta)
      +  (x_\alpha - x_\beta)
         (x_\alpha - x_\gamma)
         (x_\beta - x_\gamma)
         \psi(x_\delta),$$
the function~$\psi$ (as well as $\phi$) being rational.

In the case (IV.), we may change $x_\alpha$ to $x_\beta$, if we
at the same time change $x_\beta$ to $x_\gamma$, $x_\gamma$ to
$x_\delta$, and $x_\delta$ to $x_\alpha$; and the
function~${\sc f}$ is of the form
$$\hbox{(IV.)} \, \ldots \,
F = \phi( \overline{x_\alpha - x_\beta + x_\gamma - x_\delta}
         \mathbin{.} \overline{x_\alpha - x_\gamma}
         \mathbin{.} \overline{x_\beta - x_\delta} );$$
because the condition $\nablaperm{4}{1,2,3} = 1$ involves the
condition $\nablaperm{2}{1,3} = 1$, and consequently the present
function~${\sc f}$ must be rational relatively to the two
combinations
$$x_\alpha + x_\gamma - x_\beta - x_\delta
   \quad\hbox{and}\quad
   \overline{x_\alpha - x_\gamma} \mathbin{.}
      \overline{x_\beta - x_\delta};$$
or relatively to the two following,
$$x_\alpha - x_\beta + x_\gamma - x_\delta
   \quad\hbox{and}\quad
   \overline{x_\alpha - x_\beta + x_\gamma - x_\delta}
         \mathbin{.} \overline{x_\alpha - x_\gamma}
         \mathbin{.} \overline{x_\beta - x_\delta};$$
but of these two last-mentioned combinations, the former alone
changes, and it changes in its sign alone, when the operation
$\nablaperm{4}{1,2,3}$ performed, so that it can enter only by
its square; which square
$( x_\alpha - x_\beta + x_\gamma - x_\delta )^2$
can be expressed as a rational function of the product
$$( x_\alpha - x_\beta + x_\gamma - x_\delta )
   ( x_\alpha - x_\gamma ) ( x_\beta - x_\delta ),$$
and of those symmetric coefficients which may enter in any manner
into $\phi$.

By similar reasonings it appears, that in the six other cases
(I.~I.) $\ldots$ (II.~III.), we have, respectively, the six
following forms for ${\sc f}$:
$$\hbox{(I.~I.)} \, \ldots \,
F = \phi(x_\delta)
   = a + b x_\delta + c x_\delta^2 + d x_\delta^3;$$
$$\hbox{(I.~I.)${}'$} \, \ldots \,
F = \phi(x_\alpha + x_\beta - x_\gamma - x_\delta);$$
$$\hbox{(I.~II.)} \, \ldots \,
F = \phi(x_\alpha x_\beta + x_\gamma x_\delta)
   = a + b(x_\alpha x_\beta + x_\gamma x_\delta)
       + c(x_\alpha x_\beta + x_\gamma x_\delta)^2;$$
$$\hbox{(I.~III.)} \, \ldots \,
F = a;$$
$$\hbox{(II.~II.)} \, \ldots \,
F = \phi( \overline{x_\alpha - x_\beta} \mathbin{.}
          \overline{x_\gamma - x_\delta} );$$
$$\hbox{(II.~III.)} \, \ldots \,
F = a + b (x_\alpha - x_\beta) (x_\alpha - x_\gamma)
          (x_\alpha - x_\delta) (x_\beta - x_\gamma)
          (x_\beta - x_\delta) (x_\gamma - x_\delta).$$
To one or other of the ten forms last determined, may therefore
be reduced every rational function of four arbitrary quantities,
which has fewer than twenty-four values.  And although the
functions (I.~I.)${}'$ and (II.~II.) are six-valued, as well as
the function (IV.), yet these three functions are all in general
distinct from one another; the function (IV.) being one which
does not change its value, when the four roots
$x_\alpha$,~$x_\beta$,~$x_\gamma$,~$x_\delta$ are all changed in
some one {\it quaternary cycle}, but the function
(I.~I.)${}'$ being one which allows {\it either or both\/} of
{\it some two pairs\/} $x_\alpha$,~$x_\beta$ and
$x_\gamma$,~$x_\delta$ to have its two roots interchanged, and
the function (II.~II.) being characterized by its allowing
{\it any two roots\/} to be interchanged, if {\it the other two
roots\/} be interchanged at the same time.  It may be useful also
to observe, that the three-valued function (I.~II.) belongs, as a
particular case, to each of these three six-valued forms, and may
easily be deduced from the form (I.~I.)${}'$, as follows:
$${\sc f} = \psi(x_\alpha + x_\beta - x_\gamma - x_\delta)
   = \psi(x_\gamma + x_\delta - x_\alpha - x_\beta)
   = \chi \overline{(x_\alpha + x_\beta - x_\gamma - x_\delta)^2}
   = \phi (x_\alpha x_\beta + x_\gamma x_\delta).$$

Attending next to conditions of the forms
$$\nabla = -1,\quad \nabla= \rho_3,$$
instead of attending only to conditions of the form
$$\nabla = 1,$$
we discover the forms which a rational function of four arbitrary
variables must have, in order that its square or cube may have
fewer values than itself; which functional forms are the
following:

The general twenty-four-valued function~${\sc f}$ will have its
square twelve-valued, if it be either of the form
$${\sc f}
   = (x_\alpha - x_\beta)
         \mathbin{.} \psi(x_\gamma - x_\delta),$$
or of this other form
$${\sc f} = (x_\alpha - x_\beta) \mathbin{.}
      \psi( x_\alpha + x_\beta - x_\gamma - x_\delta,
           \overline{x_\alpha - x_\beta} \mathbin{.}
           \overline{x_\gamma - x_\delta} ).$$
The same general or twenty-four-valued function will have an
eight-valued cube, if it be of the form
$${\sc f} = \{ \phi(x_\delta)
         + (x_\alpha - x_\beta) (x_\alpha - x_\gamma)
           (x_\beta - x_\gamma) \psi(x_\delta) \}
      (x_\alpha + \rho_3^2 x_\beta + \rho_3 x_\gamma),$$
$\rho_3$ being, as before, an imaginary cube-root of unity.  The
twelve-valued function (I.) will have a six-valued square, if it
be reducible to the form
$${\sc f} = (x_\gamma - x_\delta) \mathbin{.}
      \psi (x_\alpha + x_\beta - x_\gamma - x_\delta).$$
The twelve-valued function (II.) will have a six-valued square,
if it be of the form
$${\sc f}
   = (x_\alpha + x_\beta - x_\gamma - x_\delta)
         \mathbin{.} \psi( \overline{ x_\alpha - x_\beta }
         \mathbin{.} \overline{ x_\gamma - x_\delta } ),$$
or of the form
$${\sc f}
   = (x_\alpha - x_\beta) (x_\gamma - x_\delta)
         \mathbin{.}
         \psi(x_\alpha + x_\beta - x_\gamma - x_\delta),$$
or of the form
$${\sc f}
   = (x_\alpha + x_\beta - x_\gamma - x_\delta)
      \mathbin{.}
      \psi( \overline{ x_\alpha + x_\beta - x_\gamma - x_\delta }
      \mathbin{.} \overline{ x_\alpha - x_\beta }
      \mathbin{.} \overline{ x_\gamma - x_\delta } ).$$
The eight-valued function (III.) will have its square
four-valued, if it be of the form
$${\sc f} = (x_\alpha - x_\beta) (x_\alpha - x_\gamma)
      (x_\beta - x_\gamma) \psi( x_\delta ).$$
The six-valued functions (IV.), (I.~I.)${}'$, (II.~II.), will
have their squares three-valued, if they be reducible,
respectively, to the forms,
$${\sc f} = ( x_\alpha - x_\beta + x_\gamma - x_\delta )
      ( x_\alpha - x_\gamma ) ( x_\beta - x_\delta ) \mathbin{.}
      \psi( x_\alpha x_\gamma + x_\beta x_\delta ),$$
$${\sc f}
   = ( x_\alpha + x_\beta - x_\gamma - x_\delta )
         \mathbin{.}
         \psi( x_\alpha x_\beta + x_\gamma x_\delta ),$$
$${\sc f}
   = ( x_\alpha - x_\beta ) ( x_\gamma - x_\delta )
         \mathbin{.}
         \psi( x_\alpha x_\beta + x_\gamma x_\delta );$$
and the last-mentioned six-valued function, (II.~II.), will have
its cube two-valued, if it be reducible to the form
$$\eqalign{
F &= \{ a + b (x_\alpha - x_\beta) (x_\alpha - x_\gamma)
          (x_\alpha - x_\delta) (x_\beta - x_\gamma)
          (x_\beta - x_\delta) (x_\gamma - x_\delta) \} \cr
  &\mathrel{\phantom{=}}
      \times
         \{
            x_\alpha x_\beta + x_\gamma x_\delta
          + \rho_3^2 (x_\alpha x_\gamma + x_\beta x_\delta)
          + \rho_3   (x_\alpha x_\delta + x_\beta x_\gamma)
         \},\cr}$$
$\rho_3$ being still an imaginary cube-root of unity.  And the
square of the two-valued function (II.~III.) will be symmetric,
if it be of the form
$${\sc f} =   b (x_\alpha - x_\beta) (x_\alpha - x_\gamma)
          (x_\alpha - x_\delta) (x_\beta - x_\gamma)
          (x_\beta - x_\delta) (x_\gamma - x_\delta).$$
But there exists no other case of reduction essentially distinct
from these, in which the number of values of the square or cube
of a rational function of four independent variables is less than
the number of values of that function itself.  Some steps,
indeed, have been for brevity omitted, which would be requisite
for the full statement of a formal demonstration of all the
foregoing theorems; but these omitted steps will easily occur to
any one, who has considered with attention the investigation of
the properties of rational functions of three variables, given in
the two preceding articles.

\bigbreak

[20.]
The foregoing theorem respecting functions of four variables
being admitted, let us now proceed to apply them to the
{\it \`{a} priori\/} investigation of all possible expressions,
finite and irreducible, of the form $b^{(m)}$, for a root~$x$ of
the general biquadratic equation already often referred to,
namely,
$$x^4 + a_1 x^3 +a_2 x^2 + a_3 x + a_4 = 0.$$

It is evident in the first place that we cannot express any such
root~$x$ as a rational function of the coefficients
$a_1$,~$a_2$,~$a_3$,~$a_4$, because these are symmetric
relatively to the four roots $x_1$,~$x_2$,~$x_3$,~$x_4$, and a
symmetric function of four arbitrary and independent quantities
cannot be equal to an unsymmetric function of them; we must
therefore suppose that $m$ in $b^{(m)}$ is greater than $0$, or,
in other words, that the function~$b^{(m)}$ is irrational, with
respect to the quantities $a_1$,~$a_2$,~$a_3$,~$a_4$, if any
expression of the required kind can be found at all for $x$.  On
the other hand, the general theorem of {\sc Abel} shows that if
any such expression exist, it must be composed of some finite
combination of quadratic and cubic radicals, together with
rational functions; because $2$ and $3$ are the only prime
divisors of the product
$24 = 1 \mathbin{.} 2 \mathbin{.} 3 \mathbin{.} 4$.
And the first and only radical of the first order in $b^{(m)}$,
must be a square-root, of the form
$$\eqalign{
a_1' &= b (x_1 - x_2) (x_1 - x_3) (x_1 - x_4)
          (x_2 - x_3) (x_2 - x_4) (x_3 - x_4) \cr
      &= \sqrt{ -442368 \mathbin{.} b^2
            \mathbin{.} (e_1^2 - e_2^3) }
       = \sqrt{ -2^{14} \mathbin{.} 3^3
            \mathbin{.} b^2 \mathbin{.} (e_1^2 - e_2^3) },\cr}$$
$b$ being some symmetric function of $x_1$,~$x_2$,~$x_3$,~$x_4$,
and $e_1$,~$e_2$ having the same meanings here as in the
second article; because no rational and unsymmetric function of
four arbitrary quantities $x_1$,~$x_2$,~$x_3$,~$x_4$, has a prime
power symmetric, except either this function~$a_1'$, or else some
other function~$a_2'$ which may be deduced from it by a
multiplication such as the following,
$\displaystyle a_2' = {c \over b} a_1'$.
But a two-valued expression of the form $f_1' = b_0 + b_1 a_1'$
cannot represent a four-valued function, such as $x$; we must
therefore suppose that the sought expression $b^{(m)}$ contains a
radical~$a_1''$ of the second order, and this must be a
cube-root, of the form
$$a_1'' = (p_0 + p_1 a_1')(u_1 + \rho_3^2 u_2 + \rho_3 u_3)
   = \root 3 \of{} ( b_0 + b_1 a_1' );$$
in which $\rho_3$ is, as before, an imaginary cube-root of
unity; $p_0$,~$p_1$, $b_0$,~$b_1$ are symmetric relatively to
$x_1$,~$x_2$,~$x_3$,~$x_4$, or rational relatively to
$a_1$,~$a_2$,~$a_3$,~$a_4$;
$$u_1 = x_1 x_2 + x_3 x_4,\quad
  u_2 = x_1 x_3 + x_2 x_4,\quad
  u_3 = x_1 x_4 + x_2 x_3;$$
and
$$b_0 + b_1 a_1' = 1728 ( p_0 + p_1 a_1')^3
      \left\{ e_1
         + {1 \over 1152} (\rho_3^2 - \rho_3) {a_1' \over b}
         \right\},$$
the rational function~$e_1$, and the radical~$a_1'$ retaining
their recent meanings: because no rational function~${\sc f}_1''$
of four independent variables $x_1$,~$x_2$,~$x_3$,~$x_4$, which
cannot be reduced to the form thus assigned for $a_1''$, can have
itself $2 \alpha_1''$ values, $\alpha_1''$ being a prime number
greater than $1$, if the number of values of the prime power
${\sc f}_1''^{\alpha_1''}$ be only $2$.  Nor can any other
radical such as $a_2''$ of the same order enter into the
expression of the irreducible function~$b^{(m)}$; because this
other radical would be obliged to be of one or other of the two
forms following, namely either
$$a_2'' = (q_0 + q_1 a_1')(u_1 + \rho_3^2 u_2 + \rho_3   u_3),$$
or else
$$a_2'' = (q_0 + q_1 a_1')(u_1 + \rho_3 u_2   + \rho_3^2 u_3),$$
$\rho_3$ being the same cube-root of unity in these expressions,
as in the expression for $a_1''$; and the product of the two last
trinomial factors is symmetric,
$$(u_1 + \rho_3^2 u_2 + \rho_3   u_3)
  (u_1 + \rho_3   u_2 + \rho_3^2 u_3) = 144 e_2;$$
so that either the quotient
$\displaystyle {a_2'' \over a_1''}$ or the product $a_2'' a_1''$
would be a two-valued function, which would be known when $a_1'$
had been calculated, without any new extraction of radicals.  At
the same time, if we observe that
$$u_1 + u_2 + u_3 = a_2,$$
we see that the three values $u_1$,~$u_2$,~$u_3$ of the
three-valued function $x_\alpha x_\beta + x_\gamma x_\delta$ can
be expressed as rational functions of the radicals $a_1''$ and
$a_1'$, or as irrational functions of the second order of the
coefficients $a_1$,~$a_2$,~$a_3$,~$a_4$ of the proposed
biquadratic equation, namely the following,
$$\eqalign{
u_1 = {1 \over 3} \left\{
         a_2 + {a_1'' \over p_0 + p_1 a_1'}
             + {144 e_2 (p_0 + p_1 a') \over a_1''}
         \right\},\cr
u_2 = {1 \over 3} \left\{
         a_2 + {\rho_3   a_1'' \over p_0 + p_1 a_1'}
             + {144 e_2 (p_0 + p_1 a') \over \rho_3   a_1''}
         \right\},\cr
u_3 = {1 \over 3} \left\{
         a_2 + {\rho_3^2 a_1'' \over p_0 + p_1 a_1'}
             + {144 e_2 (p_0 + p_1 a') \over \rho_3^2 a_1''}
         \right\};\cr}$$
so that if the biquadratic equation can be resolved at all, by
any finite combination of radicals and rational functions, the
solution must begin by calculating a square-root $a_1'$ and a
cube-root $a_1''$, which are in all essential respects the same
as those required for resolving that other equation of which
$u_1$,~$u_2$,~$u_3$ are roots, namely the following cubic
equation:
$$u^3 - a_2 u^2 + (a_1 a_3 - 4 a_3) u
      + (4 a_2 - a_1^2) a_4 - a_3^2 = 0;$$
which may also be thus written,
$$(u - {\textstyle {1 \over 3}} a_2)^2
   - 48 e_2 (u - {\textstyle {1 \over 3}} a_2)
   - 128 e_1 = 0.$$
Reciprocally if $u_1$,~$u_2$,~$u_3$ be known, by the solution of
this cubic equation, or in any other way, we can calculate $a_1'$
and $a_1''$, without any new extraction of radicals, since if we
put, for abridgment,
$$\eqalign{
t_1 &= u_2 - u_3 = (x_1 - x_2) (x_3 - x_4),\cr
t_2 &= u_1 - u_3 = (x_1 - x_3) (x_2 - x_4),\cr
t_3 &= u_1 - u_2 = (x_1 - x_4) (x_2 - x_3),\cr}$$
we have
$$a_1' = b t_1 t_2 t_3,$$
and
$$a_1''
   = (p_0 + p_1 b t_1 t_2 t_3)
         (u_1 + \rho_3^2 u_2 + \rho_3 u_3).$$
Again, it is important to observe, that if any one of the three
quantities $t_1$,~$t_2$,~$t_3$, such as $t_1$, be given, the
other two $t_2$,~$t_3$, and also $u_1$,~$u_2$,~$u_3$, can be
deduced from it, without any new extraction; because, in general,
the difference of any two roots of a cubic equation is sufficient
to determine rationally all the three roots of that equation: it
must therefore be possible to express the radicals $a_1'$ and
$a_1''$ as rational functions of $t_1$; and accordingly we find
$$a_1' = b t_1 (144 e_2 - t_1^2),$$
and
$$a_1'' = \{ p_0 + p_1 b t_1 (144 e_2 - t_1^2) \} \left(
            {\rho_3^2 - \rho_3 \over 2} t_1
            + {576 e_1 \over 48 e_2 - t_1^2}
            \right);$$
while $t_1$ may be reciprocally be expressed as follows,
$$t_1 = u_2 - u_3
      = {\textstyle {1 \over 3}} (\rho_3 - \rho_3^2) \left\{
         {a_1'' \over p_0 + p_1 a_1'}
         - {144 e_2 (p_0 + p_1 a_1') \over a_1''} \right\}.$$
Hence the most general irrational function of the second order,
$$f_1'' = b_0' + b_1' a_1'' + b_2' a_1''^2,$$
which can enter into the composition of $b^{(m)}$, and in which
$b_0'$,~$b_1'$,~$b_2'$ are functions of the first order, and of
the forms
$$(b_0')_0 + (b_0')_1 a_1',\quad
  (b_1')_0 + (b_1')_1 a_1',\quad
  (b_2')_0 + (b_2')_1 a_1',$$
may be considered as a rational function of $t_1$,
$$f_1'' = \phi(t_1)
   = \phi( \overline{x_1 - x_2}
            \mathbin{.} \overline{x_3 - x_4} );$$
it is, therefore, included under the form (II.~II.), and is
either six-valued or three-valued, according as it does not, or
as it does reduce itself to a rational function of $u_1$, by
becoming a rational function of $t_1^2$, and in neither case can
it become a four-valued function such as $x$.  We must therefore
suppose, that the sought irrational expression $b^{(m)}$, for a
root~$x$ of the general biquadratic, contains at least one
radical~$a_1'''$ of the third order, which, relatively to the
coefficients $a_1$,~$a_2$,~$a_3$,~$a_4$, must be a square-root,
(and not a cube-root,) of the form
$$a_1''' = \sqrt{ b_0' + b_1' a_1'' + b_2' a_1''^2 };$$
and, relatively to the roots $x_1$,~$x_2$,~$x_3$,~$x_4$, must
admit of being expressed either as a twelve-valued function, with
a six-valued square, which square is of the form (II.~II.); or
else as a six-valued function, which is not itself of that form
(II.~II.), and of which the square is three-valued.  This radical
$a_1'''$ must therefore admit of being put under the form
$$a_1''' = b_\alpha'' v_\alpha,$$
the factor $b_\alpha''$ being a function of the second or of a
lower order, and $v_\alpha$ being one or other of the three
following functions,
$$v_1 = x_1 + x_2 - x_3 - x_4,\quad
  v_2 = x_1 + x_3 - x_2 - x_4,\quad
  v_3 = x_1 + x_4 - x_2 - x_3,$$
which are themselves six-valued, but have three-valued squares.
And since the product of the three functions $v_\alpha$ is
symmetric,
$$v_1 v_2 v_3 = 64 \mathbin{.} e_4,$$
($e_4$ having here the same meaning as in the second article,) we
need only consider, at most, two radicals of the third order,
$$a_1''' = b_1'' v_1
   = \sqrt{ b_1''^2 (a_1^2 - 4 a_2 + 4 u_1) },\quad
  a_2''' = b_2'' v_2
   = \sqrt{ b_2''^2 (a_1^2 - 4 a_2 + 4 u_2) };$$
and may express the most general irrational function of the third
order, which can enter into the composition of $b^{(m)}$, as
follows:
$$f_1''' = b_{0,0}'' + b_{1,0}'' a_1''' + b_{0,1}'' a_2'''
            + b_{1,1}'' a_1''' a_2''';$$
the coefficients of this expression being functions of the second
or of lower orders,  If we suppress entierely one of the two last
radicals, such as $a_2'''$, without introducing any higher
radical~$a_1^{\rm IV}$, we shall indeed obtain a simplified
expression, but cannot thereby represent any root, such as
$x_\alpha$, of the proposed biquadratic equation; for if we could
do this, we should then have a system of two expressions for two
different roots, $x_\alpha$,~$x_\beta$, of the forms
$$x_\alpha = b_0'' + b_1'' a_1''',\quad
  x_\beta  = b_0'' - b_1'' a_1''',$$
which would give
$$b_0'' = {\textstyle {1 \over 2}} (x_\alpha + x_\beta);$$
but this last rational function, although six-valued, cannot be
put under the form (II.~II.), and therefore cannot be equal to
any function of the second order, such as $b_0''$.  Retaining
therefore both the radicals, $a_1'''$,~$a_2'''$, we have next to
observe, that if the function~$f_1'''$ can coincide with the
sought function~$b^{(m)}$, so as to represent some one root
$x_\alpha$ of the proposed biquadratic equation, it must give a
system of expressions for all the four roots
$x_\alpha$,~$x_\beta$,~$x_\gamma$,~$x_\delta$, in some
arrangement or other, by merely changing the signs of those two
radicals of the third order; namely the following system,
$$\eqalign{
x_\alpha &= b_{0,0}'' + b_{1,0}'' a_1''' + b_{0,1}'' a_2'''
            + b_{1,1}'' a_1''' a_2''',\cr
x_\beta  &= b_{0,0}'' + b_{1,0}'' a_1''' - b_{0,1}'' a_2'''
            - b_{1,1}'' a_1''' a_2''',\cr
x_\gamma &= b_{0,0}'' - b_{1,0}'' a_1''' + b_{0,1}'' a_2'''
            - b_{1,1}'' a_1''' a_2''',\cr
x_\delta &= b_{0,0}'' - b_{1,0}'' a_1''' - b_{0,1}'' a_2'''
            + b_{1,1}'' a_1''' a_2''';\cr}$$
which four expressions for the four roots conduct to the four
following relations,
$$\eqalign{
b_{0,0}''
   &= {\textstyle {1 \over 4}}
            (x_\alpha + x_\beta + x_\gamma + x_\delta),\cr
b_{1,0}'' a_1'''
   &= {\textstyle {1 \over 4}}
            (x_\alpha + x_\beta - x_\gamma - x_\delta),\cr
b_{0,1}'' a_2'''
   &= {\textstyle {1 \over 4}}
            (x_\alpha - x_\beta + x_\gamma - x_\delta),\cr
b_{1,1}'' a_1''' a_2'''
   &= {\textstyle {1 \over 4}}
            (x_\alpha - x_\beta - x_\gamma + x_\delta).\cr}$$

Reciprocally, if these four last conditions can be satisfied, by
any suitable arrangement of the four roots, and by any suitable
choice of those coefficients or functions which have hitherto
been left undetermined, we shall have the four expressions just
now mentioned, for the four roots of the general biquadratic, as
the four values of an irrational and irreducible function
$b'''$, of the third order,  Now, these four conditions are
satisfied when we suppose
$$x_\alpha = x_1,\quad
  x_\beta  = x_2,\quad
  x_\gamma = x_3,\quad
  x_\delta = x_4;$$
$$b_{0,0}'' = {-a_1 \over 4};\quad
  b_{1,0}'' = {1 \over 4 b_1''};\quad
  b_{0,1}'' = {1 \over 4 b_2''};$$
and finally,
$$b_{1,1}'' = {16 e_4 \over b_1'' b_2'' v_1^2 v_2^2};$$
but not by any suppositions essentially distinct from these.  It
is therefore possible to express the four roots of the general
biquadratic equation, as the four values of an irrational and
irreducible expression of the third order $b'''$, namely as the
following:
$$\eqalign{
x_1 &= b_{0,0}'''
      = {-a_1 \over 4} + {a_1''' \over 4 b_1''}
         + {a_2''' \over 4 b_2''}
         + {16 b_1'' b_2'' e_4 \over a_1''' a_2'''};\cr
x_2 &= b_{0,1}'''
      = {-a_1 \over 4} + {a_1''' \over 4 b_1''}
         - {a_2''' \over 4 b_2''}
         - {16 b_1'' b_2'' e_4 \over a_1''' a_2'''};\cr
x_3 &= b_{1,0}'''
      = {-a_1 \over 4} - {a_1''' \over 4 b_1''}
         + {a_2''' \over 4 b_2''}
         - {16 b_1'' b_2'' e_4 \over a_1''' a_2'''};\cr
x_4 &= b_{1,1}'''
      = {-a_1 \over 4} - {a_1''' \over 4 b_1''}
         - {a_2''' \over 4 b_2''}
         + {16 b_1'' b_2'' e_4 \over a_1''' a_2'''};\cr}$$
and there exists no system of expressions, essentially distinct
from these, which can express the same four roots, without the
introduction of some radical, such as $a_1^{\rm IV}$, of an order
higher than the third.  We must, however, remember that these
expressions involve several arbitrary symmetric functions of
$x_1$,~$x_2$,~$x_3$,~$x_4$, or arbitrary rational functions of
$a_1$,~$a_2$,~$a_3$,~$a_4$, which enter into the composition of
the radicals $a_1'$,~$a_1''$,~$a_1'''$,~$a_2'''$, though only in
the way of multiplying a function by an exact square or cube
before the square-root or cube-root is extracted: namely, the
quantity $b$ in $a_1'$; $p_0$ and $p_1$ in $a_1''$; and, in the
radicals $a_1'''$,~$a_2'''$, twelve other arbitrary quantities,
introduced by the functions $b_1''$,~$b_2''$, which latter
functions may be thus developed,
$$\eqalign{
b_1'' = r_{0,0} + r_{0,1} a_1'
         + ( r_{1,0} + r_{1,1} a_1' ) a_1''
         + ( r_{2,0} + r_{2,1} a_1' ) a_1''^2,\cr
b_2'' = r_{0,0}^\backprime + r_{0,1}^\backprime a_1'
         + ( r_{1,0}^\backprime
            + r_{1,1}^\backprime a_1' ) a_1''
         + ( r_{2,0}^\backprime
            + r_{2,1}^\backprime a_1' ) a_1''^2.\cr}$$
In the earlier articles of this Essay, these fifteen arbitrary
quantities had the following particular values,
$$b = {\rho_3^2 - \rho_3 \over 1152};\quad
  p_0 = {\textstyle {1 \over 12}};\quad
  p_1 = 0;$$
$$r_{0,0} = {\textstyle {1 \over 4}};\quad
  r_{0,1} = r_{1,0} = r_{1,1} = r_{2,0} = r_{2,1} = 0;$$
$$r_{0,0}^\backprime = {\textstyle {1 \over 4}};\quad
  r_{0,1}^\backprime = r_{1,0}^\backprime = r_{1,1}^\backprime
      = r_{2,0}^\backprime = r_{2,1}^\backprime = 0.$$
Apparent differences between two systems of expressions of the
third order, for the four roots of a biquadratic equation, may
also arise from differences in the arrangement of those four
roots.

Analogous reasonings, the details of which will easily suggest
themselves to those who have studied the foregoing discussion,
show that if we retain only one radical of the third order
$a_1'''$, but introduce a radical of the fourth order
$a_1^{\rm IV}$, for the purpose of obtaining the only other sort
of irrational and irreducible expression $b^{(m)} = b^{\rm IV}$,
which can represent a root of the same general biquadratic
equation, we must then suppose this new radical~$a_1^{\rm IV}$
to be a square-root, of the form
$$a_1^{\rm IV} = p'''(x_1 - x_2)
   = \sqrt{ p'''^2 \left( - {v_1^2 \over 4} + 12 e_3
                          + {32 e_4 \over v_1} \right) };$$
$p'''$ being a function of the third or of a lower order, which
in the earlier articles of this Essay had the particular value
${1 \over 2}$; which $v_1$ has the meaning recently assigned, and
$e_3$,~$e_4$ have those which were stated in the second article;
we must also employ the expressions
$$\eqalign{
x_1 &= b_0''' + b_1''' a_1^{\rm IV}
      = {-a_1 \over 4} + {v_1 \over 4}
         + {a_1^{\rm IV} \over 2 p'''},\cr
x_2 &= b_0''' - b_1''' a_1^{\rm IV}
      = {-a_1 \over 4} + {v_1 \over 4}
         - {a_1^{\rm IV} \over 2 p'''},\cr}$$
and
$$x_3 = {-a_1 \over 4} - {v_1 \over 4}
         + {p''' t_1 \over 2 a_1^{\rm IV}},\quad
  x_4 = {-a_1 \over 4} - {v_1 \over 4}
         - {p''' t_1 \over 2 a_1^{\rm IV}},$$
$t_1$ retaining here its recent meaning; or, at least, we must
make suppositions, and must employ expressions, not differing
essentially from these.

But all the radicals $a_1'$,~$a_1''$, $a_1'''$,
$a_2'''$,~$a_1^{\rm IV}$, introduced in the present article,
agree in all essential respects with those which have been long
employed, for the calculation of the roots of the general
biquadratic equation; it is, therefore, impossible to discover
any new expression for any one of those four roots, which, after
being cleared from all superfluous extractions of radicals, shall
differ essentially, in the extractions that remain, from the
expressions that have been long discovered.  And the only
important difference, with respect to these extractions of
radicals, between any two general methods for resolving
biquadratic equations, if both be free from all superfluous
extractions, is, that after calculating first, in both methods, a
square-root $a_1'$, and a cube-root $a_1''$, (operations which
are equivalent to those required for the solution of an auxiliary
cubic equation,) we may afterwards either calculate two
{\it simultaneous\/} square-roots $a_1'''$,~$a_2'''$, as in the
method of {\sc Euler}, or else two {\it successive\/}
square-roots $a_1'''$,~$a_1^{\rm IV}$, as in the method of
{\sc Ferrari} or {\sc Des Cartes}: for, in the view in which they
are here considered, the methods of these two last-mentioned
mathematicians do not essentially differ from each other.

\bigbreak

[21.]
It is not necessary, for the purposes of the inquiry into the
possibility or impossibility of representing, by any expression
of the form $b^{(m)}$, a root~$x$ of the general equation of the
fifth degree,
$$x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 = 0,$$
to investigate all possible forms of rational functions of five
variables, which have fewer than 120 values; but it is necessary
to discover all those forms which have five or fewer values.
Now, if the rational function
$${\sc f}(x_1, x_2, x_3, x_4, x_5)$$
have fewer than six values, when the five arbitrary roots
$x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$, of the above-mentioned general
equation are interchanged in all possible ways, it must, by a
still stronger reason, have fewer than six values, when only the
first four roots $x_1$,~$x_2$,~$x_3$,~$x_4$, are interchanged in
any manner, the fifth root~$x_5$ remaining unchanged.

Hence, by the properties of functions of four variables, the
function~${\sc f}$ must be reducible to one of the four following
forms, corresponding to those which, in the nineteenth article,
were marked (I.~III), (II.~III), (I.~II.), and (I.~I.):
$$\eqalign{
\hbox{(a)} \quad
   &\phi(x_5);\cr
\hbox{(b)} \quad
   &\phi(x_5,           \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_4}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_4}
            \mathbin{.} \overline{x_3 - x_4} );\cr
\hbox{(c)} \quad
   &\phi(x_5, x_1 x_2 + x_3 x_4);\cr
\hbox{(d)} \quad
   &\phi(x_5, x_4);\cr}$$
or at least to some form not essentially distinct from these.  In
making this reduction, the following principle is employed, that
any symmetric function of $x_1$,~$x_2$,~$x_3$,~$x_4$, is a
rational function of $x_5$ and of the five coefficients
$a_1$,~$a_2$,~$a_3$,~$a_4$,~$a_5$; which latter coefficients are
tacitly supposed to be capable of entering in any manner into the
rational functions $\phi$.

It may also be useful to remark, before going farther, that the
four forms here referred to, of functions of four variables, with
four or fewer values, may be deduced anew as follows.  Retaining
the abridged notation $(\alpha, \beta, \gamma, \delta)$, we see
immediately that if the six syntypical functions
$$(1,2,3,4),\quad
  (2,3,1,4),\quad
  (3,1,2,4),\quad
  (1,3,2,4),\quad
  (3,2,1,4),\quad
  (2,1,3,4)$$
be not unequal among themselves, they must either all be equal,
in which case we have the four-valued form $\phi(x_4)$ or
(I.~I.), or else must distribute themselves into two distinct
groups of three, or into three distinct groups of two equal
functions.  But if we suppose
$$(1,2,3,4) = (2,3,1,4) = (3,1,2,4),$$
in order to get the reduction to two groups, the functions
$(1,2,3,4)$ and $(2,1,3,4)$ being not yet supposed to be equal;
and then require that the six following values of
$(\alpha, \beta, \gamma, \delta)$,
$$(1,2,3,4),\quad
  (2,1,3,4),\quad
  (1,2,4,3),\quad
  (2,1,4,3),\quad
  (1,3,4,2),\quad
  (3,1,4,2),$$
shall not be all unequal; we must either make some supposition,
such as
$$(1,2,3,4) = (1,2,4,3),$$
which conducts to the
one-valued form (I.~III.), or else must make some supposition,
such as
$$(1,2,3,4) = (2,1,4,3),$$
which conducts to the two-valued
form (II.~III.).  And if we suppose
$$(1,2,3,4) = (2,1,3,4),$$
in order to reduce the six functions
$(1,2,3,4),\, \ldots \, \, (2,1,3,4)$ to three distinct groups,
the functions $(1,2,3,4)$ and $(2,3,1,4)$ being supposed unequal;
and then require that of the six following values,
$$(1,2,3,4),\quad
  (2,3,1,4),\quad
  (3,1,2,4),\quad
  (1,2,4,3),\quad
  (2,4,1,3),\quad
  (4,1,2,3),$$
there shall be fewer than five unequal; we must either suppose
$$(2,3,1,4) = (4,1,2,3),$$ in which case we are conducted to the
three-valued form (I.~II.); or else must suppose
$$(2,3,1,4) = (2,4,1,3),$$
which conducts again to the four-valued function (I.~I.), by
giving
$(1,2,3,4) = \phi(x_3)$.

Now of the four forms (a), (b), (c), (d), the form (a) is
five-valued, and therefore admissible in the present inquiry; but
the form (b) is, in general, ten-valued; the form (c) has, in
general fifteen values; and the form (d) has twenty.  If, then,
we are to reduce the functions (b) (c) (d) within that limit of
number of values to which we are at present confining ourselves,
we must restrict them by some new conditions, of which the
following are sufficient types:
$$\eqalign{
\hbox{(b)${}'$} \quad
   &\phi(x_5,           \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_4}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_4}
            \mathbin{.} \overline{x_3 - x_4} ) \cr
   &\quad =
    \phi(x_5,         - \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_4}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_4}
            \mathbin{.} \overline{x_3 - x_4} );\cr
\hbox{(b)${}''$} \quad
   &\phi(x_5,           \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_4}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_4}
            \mathbin{.} \overline{x_3 - x_4} ) \cr
   &\quad =
    \phi(x_4,           \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_5}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_5}
            \mathbin{.} \overline{x_3 - x_5} );\cr
\hbox{(b)${}'''$} \quad
   &\phi(x_5,           \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_4}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_4}
            \mathbin{.} \overline{x_3 - x_4} ) \cr
   &\quad =
    \phi(x_4,         - \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_5}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_5}
            \mathbin{.} \overline{x_3 - x_5} );\cr
\hbox{(c)${}'$} \quad
   &\phi(x_5, x_1 x_2 + x_3 x_4)
      = \phi(x_5, x_1 x_3 + x_2 x_4);\cr
\hbox{(c)${}''$} \quad
   &\phi(x_5, x_1 x_2 + x_3 x_4)
      = \phi(x_4, x_1 x_2 + x_3 x_5);\cr
\hbox{(c)${}'''$} \quad
   &\phi(x_5, x_1 x_2 + x_3 x_4)
      = \phi(x_4, x_1 x_3 + x_2 x_5);\cr
\hbox{(d)${}'$} \quad
   &\phi(x_5, x_4) = \phi(x_5, x_3);\cr
\hbox{(d)${}''$} \quad
   &\phi(x_5, x_4) = \phi(x_4, x_3);\cr
\hbox{(d)${}'''$} \quad
   &\phi(x_5, x_4) = \phi(x_2, x_3).\cr}$$
(To suppose $\phi(x_5, x_4) = \phi(x_4, x_5)$, would indeed
reduce the number of values of the function (d) from twenty to
ten, but a new reduction would be required, in order to depress
that number below six, and thus we should still be obliged to
employ one of the three conditions (d)${}'$ (d)${}''$
(d)${}'''$.)  Of these twelve different conditions
(b)${}'$ $\ldots$ (d)${}'''$, some one of which we must employ,
(or at least some condition not essentially different from it,)
the three marked (b)${}'$ (c)${}'$ (d)${}'$ are easily seen to
reduce respectively the three functions (b) (c) (d) to the
five-valued form (a); they are therefore admissible, but they
give no new information.  The supposition (b)${}''$ conducts us
to equate the function (b) to the following,
$$\phi(x_3, \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_5}
            \mathbin{.} \overline{x_1 - x_4}
            \mathbin{.} \overline{x_2 - x_5}
            \mathbin{.} \overline{x_2 - x_4}
            \mathbin{.} \overline{x_5 - x_4}),$$
because it allows us to interchange $x_5$ and $x_3$, inasmuch as
$x_3$ may previously be put in the place of $x_4$ and $x_4$ in
the place of $x_3$, by interchanging at the same time $x_1$ and
$x_2$,---a double interchange which does not alter the product
$\overline{x_1 - x_3} \, \ldots \, \overline{x_3 - x_4}$,
since it only changes simultaneously the signs of the two factors
$x_1 - x_2$ and $x_3 - x_4$; or because, if we denote the
function (b) by the symbol $(1,2,3,4,5)$, we have
$$(1,2,3,4,5) = (2,1,4,3,5),$$
and also by (b)${}''$,
$$(1,2,3,4,5) = (1,2,3,5,4),$$
so that we must have
$$(1,2,3,4,5) = (2,1,4,5,3) = (1,2,5,4,3);$$
but also the condition (b)${}''$ gives
$$(1,2,5,4,3) = (1,2,5,3,4);$$
we must therefore suppose
$$(1,2,3,5,4) = (1,2,5,3,4),$$
that is
$$\eqalign{ &\phi(x_4,  \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_5}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_5}
            \mathbin{.} \overline{x_3 - x_5} ) \cr
   &\quad
   = \phi(x_4,        - \overline{x_1 - x_2}
            \mathbin{.} \overline{x_1 - x_3}
            \mathbin{.} \overline{x_1 - x_5}
            \mathbin{.} \overline{x_2 - x_3}
            \mathbin{.} \overline{x_2 - x_5}
            \mathbin{.} \overline{x_3 - x_5} ),\cr}$$
which is an equation of the form (b)${}'$, and reduces the
function (b) to the form (a), and ultimately to a symmetric
function~$a$, because $x_5$ and $x_4$ may be interchanged.
The supposition (b)${}'''$ conducts to a two-valued function,
which changes value when any two of the five roots are
interchanged, so that the sum
$(1,2,3,4,5) + (1,2,3,5,4)$, and the quotient
$${ (1,2,3,4,5) - (1,2,3,5,4) \over
      (x_1 - x_2) (x_1 - x_3) \ldots (x_4 - x_5) },$$
are some symmetric functions, which may be called $2a$ and $2b$;
we have therefore, in this case, a function of the form,
$$\hbox{(e)} \quad
a + b \,          \overline{x_1 - x_2}
      \mathbin{.} \overline{x_1 - x_3}
      \mathbin{.} \overline{x_1 - x_4}
      \mathbin{.} \overline{x_1 - x_5}
      \mathbin{.} \overline{x_2 - x_3}
      \mathbin{.} \overline{x_2 - x_4}
      \mathbin{.} \overline{x_2 - x_5}
      \mathbin{.} \overline{x_3 - x_4}
      \mathbin{.} \overline{x_3 - x_5}
      \mathbin{.} \overline{x_4 - x_5},$$
in which $a$ and $b$ are symmetric.  The remaining suppositions,
(c)${}''$, (c)${}'''$, (d)${}''$, (d)${}'''$, are easily seen to
conduct only to symmetric functions; for instance, (c)${}''$
gives
$$\eqalign{
\phi(x_5, x_1 x_2 + x_3 x_4)
   &= \phi(x_4, x_3 x_5 + x_2 x_1)
    = \phi(x_1, x_3 x_5 + x_2 x_4) \cr
   &= \phi(x_1, x_2 x_4 + x_3 x_5)
    = \phi(x_5, x_2 x_4 + x_3 x_1)
    = \phi(x_5, x_1 x_3 + x_2 x_4),\cr}$$
so that the condition (c)${}'$ is satisfied, and at the same time
$x_5$ is interchangeable with $x_4$.  And it is easy to see that
the five-valued function~$\phi(x_\alpha)$ may be put under the
form
$$\hbox{(f)} \quad
b_0 + b_1 x_\alpha + b_2 x_\alpha^2 + b_3 x_\alpha^3
      + b_4 x_\alpha^4;$$
the coefficients $b_0, b_1, b_2, b_3, b_4$ being symmetric.  It
is clear also that neither this five-valued function (f), nor the
two-valued function (e), admits of any reduction in respect to
number of values, without becoming altogether symmetric.  There
are, therefore, no unsymmetric and rational functions of five
independent variables, with fewer than six values, except only
the two-valued function (e), and the five-valued function (f).

Suppose now that we have the equation
$$a_1' = {\sc f}_1'(x_1, x_2, x_3, x_4, x_5),$$
${\sc f}_1'$ being a rational but unsymmetric function; and that
$$a_1'^{\alpha_1'} = f_1(a_1, a_2, a_3, a_4, a_5),$$
the exponent $\alpha_1'$ being prime, and the function~$f_1$
being rational relatively to $a_1,\ldots \, a_5$, and
therefore symmetric relatively to $x_1,\ldots \, x_5$.  With
these suppositions, the function~${\sc f}_1'$ must, by the
principles of a former article, have exactly $\alpha_1'$ values,
corresponding to changes of arrangement of the five arbitrary
quantities $x_1,\ldots \, x_5$; the exponent $\alpha_1'$ must
therefore be a prime divisor of the product 120
($= 1 \mathbin{.} 2 \mathbin{.} 3 \mathbin{.} 4 \mathbin{.} 5)$;
that is, it must be $2$, or $3$, or $5$.  But we have seen that
no rational function of five variables has exactly three values;
and if we supposed it to have five values, so as to put, (by what
has been already shewn,)
$$a_1' = b_0 + b_1 x_\alpha + b_2 x_\alpha^2 + b_3 x_\alpha^3
         + b_4 x_\alpha^4,$$
we should then have three other equations of the forms
$$\eqalign{
a_1'^2 &= b_0^{(2)} + b_1^{(2)} x_{\alpha}  
                    + b_2^{(2)} x_{\alpha}^2
                    + b_3^{(2)} x_{\alpha}^3
                    + b_4^{(2)} x_{\alpha}^4,\cr
a_1'^3 &= b_0^{(3)} + b_1^{(3)} x_{\alpha}  
                    + b_2^{(3)} x_{\alpha}^2
                    + b_3^{(3)} x_{\alpha}^3
                    + b_4^{(3)} x_{\alpha}^4,\cr
a_1'^4 &= b_0^{(4)} + b_1^{(4)} x_{\alpha}  
                    + b_2^{(4)} x_{\alpha}^2
                    + b_3^{(4)} x_{\alpha}^3
                    + b_4^{(4)} x_{\alpha}^4,\cr}$$
the coefficients being all symmetric, and being determined
through the elimination of all higher powers of $x_\alpha$ than
the fourth, by means of the equations
$$\eqalign{
x_\alpha^5 + a_1 x_\alpha^4
           + a_2 x_\alpha^3
           + a_3 x_\alpha^2
           + a_4 x_\alpha
% CORRECTION!
           + a_5
   &= 0,\cr
x_\alpha^6 + a_1 x_\alpha^5
           + a_2 x_\alpha^4
           + a_3 x_\alpha^3
           + a_4 x_\alpha^2
           + a_5 x_\alpha
   &= 0,\quad\hbox{\&c.};\cr}$$
and it would always be possible to find symmetric multipliers
$c_1$,~$c_2$,~$c_3$,~$c_4$, which would not all be equal to $0$,
and would be such that
$$\eqalign{
c_1 b_2 + c_2 b_2^{(2)} + c_3 b_2^{(3)} + c_4 b_2^{(4)}
   &= 0,\cr
c_1 b_3 + c_2 b_3^{(2)} + c_3 b_3^{(3)} + c_4 b_3^{(4)}
   &= 0,\cr
c_1 b_4 + c_2 b_4^{(2)} + c_3 b_4^{(3)} + c_4 b_4^{(4)}
   &= 0;\cr}$$
in this manner then we should obtain an equation of the form
$$c_1 a_1' + c_2 a_1'^2 + c_3 a_1'^3 + c_4 a_1'^4
   = c_1 b_0 + c_2 b_0^{(2)} + c_3 b_0^{(3)} + c_4 b_0^{(4)}
     + (c_1 b_1 + c_2 b_1^{(2)} + c_3 b_1^{(3)} + c_4 b_1^{(4)})
         x_\alpha,$$
in which it is impossible that the coefficient of $x_\alpha$
should vanish, because the five unequal values of $a_1'$ could
not all satisfy one common equation, of the fourth or of a lower
degree; we should therefore have an expression for $x_\alpha$ of
the form
$$x_\alpha = d_0 + d_1 a_1'
                 + d_2 a_1'^2
                 + d_3 a_1'^3
                 + d_4 a_1'^4,$$
the coefficients $d_0,\ldots\, d_4$ being symmetric; and for the
same reason we should have also
$$\eqalign{
x_\beta    &= d_0 + d_1 \rho_5   a_1'
                  + d_2 \rho_5^2 a_1'^2
                  + d_3 \rho_5^3 a_1'^3
                  + d_4 \rho_5^4 a_1'^4,\cr
x_\gamma   &= d_0 + d_1 \rho_5^2 a_1'
                  + d_2 \rho_5^4 a_1'^2
                  + d_3 \rho_5   a_1'^3
                  + d_4 \rho_5^3 a_1'^4,\cr
x_\delta   &= d_0 + d_1 \rho_5^3 a_1'
                  + d_2 \rho_5   a_1'^2
                  + d_3 \rho_5^4 a_1'^3
                  + d_4 \rho_5^2 a_1'^4,\cr
x_\epsilon &= d_0 + d_1 \rho_5^4 a_1'
                  + d_2 \rho_5^3 a_1'^2
                  + d_3 \rho_5^2 a_1'^3
                  + d_4 \rho_5   a_1'^4,\cr}$$
$x_\alpha$,~$x_\beta$,~$x_\gamma$,~$x_\delta$,~$x_\epsilon$
denoting, in some arrangement or other, the five roots
$x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$, and
$\rho_5$,~$\rho_5^2$,~$\rho_5^3$,~$\rho_5^4$
being the four imaginary fifth-roots of unity; consequently we
should have
$$5 d_1 a_1' = x_\alpha + \rho_5^4 x_\beta + \rho_5^3 x_\gamma
               + \rho_5^2 x_\delta + \rho_5 x_\epsilon;$$
a result which is absurd, the second member of the equation
having 120 values, while the first member has only five.  We must
therefore suppose that the exponent $\alpha_1'$ is $=2$, and
consequently must adopt the expression
$$a_1'
   = b (x_1 - x_2) (x_1 - x_3) (x_1 - x_4) (x_1 - x_5)
       (x_2 - x_3) (x_2 - x_4) (x_2 - x_5) (x_3 - x_4)
       (x_3 - x_5) (x_4 - x_5),$$
the factor $b$ being symmetric.  This, therefore, is the only
rational and unsymmetric function of five arbitrary quantities,
which has a prime power (namely its square) symmetric.

Let us next inquire whether it be possible to find any
unsymmetric but rational function,
$$a_1'' = {\sc f}_1''(x_1, x_2, x_3, x_4, x_5),$$
which, having itself more than two values, shall have a prime
power two-valued,
$$a_1''^{\alpha_1''} = f_1'
   = a + b (x_1 - x_2) \ldots (x_4 - x_5).$$
If so, the function~${\sc f}_1''$ must have exactly $2\alpha_1''$
values, and consequently the prime exponent $\alpha_1''$ must be
either three or five, because it must be a divisor of 120, and
cannot be $=2$, since no rational function of five arbitrary
quantities has exactly four values: so that $a_1''$ or
${\sc f}_1''$ must be either a cube-root of a fifth-root of the
two-valued function~$f_1'$.  And the six or ten values of
${\sc f}_1''$ must admit of being expressed as follows:
$$(1,2,3,4,5)_i;\quad
  \rho_{\alpha_1''} (1,2,3,4,5)_i;
         \enspace\ldots \enspace
  \rho_{\alpha_1''}^{\alpha_1'' - 1} (1,2,3,4,5)_i;$$
$$(1,2,3,4,5)_k;\quad
  \rho_{\alpha_1''}^{\backprime} (1,2,3,4,5)_k;
         \enspace\ldots \enspace
  \rho_{\alpha_1''}^{\backprime \alpha_1'' - 1} (1,2,3,4,5)_k;$$
in which $\rho_{\alpha_1''}$ and $\rho_{\alpha_1''}^\backprime$
are imaginary cube-roots or fifth-roots of unity, according as
$\alpha_1''$ is $3$ or $5$; while $(1,2,3,4,5)_i$ and
$(1,2,3,4,5)_k$ are some two different values of the function
${\sc f}_1''$, which may be called ${\sc f}_1''$ and
${\sc f}_1^{\backprime\prime\prime}$, and correspond to different
arrangements of $x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$, being also
such that
$$\eqalign{
{\sc f}_1^{\prime\prime \alpha_1''}
      = (1,2,3,4,5)_i^{\alpha_1''}
   &= a + b (x_1 - x_2) \ldots (x_4 - x_5),\cr
{\sc f}_1^{\backprime\prime\prime \alpha_1''}
      = (1,2,3,4,5)_k^{\alpha_1''}
   &= a - b (x_1 - x_2) \ldots (x_4 - x_5).\cr}$$
These last equations show that the cube or fifth power (according
as $a_1''$ is $3$ or $5$) of the product of
$(1,2,3,4,5)_i$ and $(1,2,3,4,5)_k$ is symmetric, and
consequently, by what was lately proved, that this product itself
is symmetric; so that we may write
$${\sc f}_1''  \mathbin{.} {\sc f}_1^{\backprime\prime\prime}
   = (1,2,3,4,5)_i \mathbin{.} (1,2,3,4,5)_k = c,$$
and therefore
$$\nabla (1,2,3,4,5)_i \mathbin{.} \nabla (1,2,3,4,5)_k = c,$$
$\nabla$ being here the characteristic of any arbitrary change of
arrangement of the five roots, which change, however, is to
operate similarly on the two functions to which the symbol is
prefixed.  (For example, if we suppose
$$(1,2,3,4,5)_i = (1,2,3,5,4),\quad
  (1,2,3,4,5)_k = (1,2,4,3,5),$$
and if we employ $\nabla$ to indicate that change which consists
in altering the first to the second, the second to the third, the
third to the fourth, the fourth to the fifth, and the fifth to
the first of the five roots in any one arrangement, we shall
have, in the present notation,
$$\nabla (1,2,3,4,5)_i = (2,3,5,4,1),\quad
  \nabla (1,2,3,4,5)_k = (2,4,3,5,1);$$
and similarly in other cases.)  Supposing then that $\nabla$
denotes the change of arrangement of the five roots which is made
in passing from that value of the function~${\sc f}_1''$ which is
$= (1,2,3,4,5)_i$ to that other value of the same function which
is $= \rho_{\alpha_1''} (1,2,3,4,5)_i$, we see that the same
change performed on $(1,2,3,4,5)_k$ must multiply this latter
value not by $\rho_\alpha''$ but by $\rho_{\alpha_1''}^{-1}$;
which factor is, however, of the form
$\rho_{\alpha_1''}^\backprime$, so that we may denote the
$2 \alpha_1''$ values of ${\sc f}_1''$ as
follows:
$$(1,2,3,4,5)_i; \quad \nabla (1,2,3,4,5)_i;
         \quad\ldots\quad
  \nabla^{\alpha_1'' - 1} (1,2,3,4,5)_i;$$
$$(1,2,3,4,5)_k; \quad \nabla (1,2,3,4,5)_k;
         \quad\ldots\quad
  \nabla^{\alpha_1'' - 1} (1,2,3,4,5)_k.$$
We see, at the same time, that the sum of the two functions
$(1,2,3,4,5)_i$ and $(1,2,3,4,5)_k$ admits of {\it at least\/}
$\alpha_1''$ different values, namely,
$$\nabla^0 \{ (1,2,3,4,5)_i + (1,2,3,4,5)_k \}
   = {\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime},$$
$$\nabla^1 \{ (1,2,3,4,5)_i + (1,2,3,4,5)_k \}
   = \rho_{\alpha_1''}   {\sc f}_1''
      + \rho_{\alpha_1''}^{-1}
         {\sc f}_1^{\backprime\prime\prime},$$
$$\cdots\cdots\cdots$$
$$\nabla^{\alpha_1'' - 1} \{ (1,2,3,4,5)_i + (1,2,3,4,5)_k \}
   = \rho_{\alpha_1''}^{\alpha_1'' - 1}   {\sc f}_1''
      + \rho_{\alpha_1''}^{-(\alpha_1'' - 1)}
         {\sc f}_1^{\backprime\prime\prime},$$
On the other hand, this sum
${\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime}$
cannot admit of {\it more\/} than $\alpha_1''$ values, because it
must satisfy an equation of the degree $\alpha_1''$, with
symmetric coefficients; which results from the two relations
$${\sc f}_1^{\prime\prime \alpha_1''}
      + {\sc f}_1^{\backprime\prime\prime \alpha_1''}
   = 2a,\quad
{\sc f}_1'' {\sc f}_1^{\backprime\prime\prime} = c,$$
and is either the cubic equation
$$({\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime})^3
      - 3c ({\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime})
      - 2a = 0,$$
or the equation of the fifth degree
$$({\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime})^5
      - 5c   ({\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime})^3
      + 5c^2 ({\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime})
      - 2a = 0,$$
according as $\alpha_1''$ is $3$ or $5$.  We must therefore
suppose that the function
${\sc f}_1'' + {\sc f}_{\backprime\prime\prime}$ has
{\it exactly\/} $\alpha_1''$ values, and consequently that
$\alpha_1''$ is $5$ and not $3$, because no rational function of
five independent variables has exactly three values.  And from
the form and properties of the only five-valued function of five
variables, we must suppose farther, that
$${\sc f}_1'' + {\sc f}_1^{\backprime\prime\prime}
   = {\sc f}_1'' + {c \over {\sc f}_1''}
   = b_0 + b_1 x_\alpha
         + b_2 x_\alpha^2
         + b_3 x_\alpha^3
         + b_4 x_\alpha^4,$$
$x_\alpha$ being some one of the five roots $x_1,\ldots \,x_5$,
and the coefficients $b_0,\ldots \, b_4$ being symmetric; and
that conversely the root~$x_\alpha$ may be thus expressed,
$$x_\alpha
   = d_0
       + d_1
         \left( {\sc f}_1'' + {c \over {\sc f}_1''} \right)
       + d_2
         \left( {\sc f}_1'' + {c \over {\sc f}_1''} \right)^2
       + \cdots
       + d_4
         \left( {\sc f}_1'' + {c \over {\sc f}_1''} \right)^4,$$
the coefficients $d_0,\ldots \, d_4$ being symmetric.  We must
also suppose that by changing ${\sc f}_1''$, successively, to
$\rho_5 {\sc f}_1''$,~$\rho_5^2 {\sc f}_1''$,~$\rho_5^3
      {\sc f}_1''$,~$\rho_5^4 {\sc f}_1''$,
we shall obtain successively, expressions for the other four
roots $x_\beta$,~$x_\gamma$,~$x_\delta$,~$x_\epsilon$, in some
arrangement or other; and therefore, if we observe that
${\sc f}_1''^5$ has been concluded to be a function of the
two-valued form, we find ourselves obliged to suppose that the
five roots may be expressed as follows, (if the supposition under
inquiry be correct,)
$$\eqalign{
x_\alpha
   &= e_0' + e_1' {\sc f}_1''
           + e_2' {\sc f}_1''^2
           + e_3' {\sc f}_1''^3
           + e_4' {\sc f}_1''^4,\cr
x_\beta
   &= e_0' + \rho_5   e_1' {\sc f}_1''
           + \rho_5^2 e_2' {\sc f}_1''^2
           + \rho_5^3 e_3' {\sc f}_1''^3
           + \rho_5^4 e_4' {\sc f}_1''^4,\cr
x_\gamma
   &= e_0' + \rho_5^2 e_1' {\sc f}_1''
           + \rho_5^4 e_2' {\sc f}_1''^2
           + \rho_5   e_3' {\sc f}_1''^3
           + \rho_5^3 e_4' {\sc f}_1''^4,\cr
x_\delta
   &= e_0' + \rho_5^3 e_1' {\sc f}_1''
           + \rho_5   e_2' {\sc f}_1''^2
           + \rho_5^4 e_3' {\sc f}_1''^3
           + \rho_5^2 e_4' {\sc f}_1''^4,\cr
x_\epsilon
   &= e_0' + \rho_5^4 e_1' {\sc f}_1''
           + \rho_5^3 e_2' {\sc f}_1''^2
           + \rho_5^2 e_3' {\sc f}_1''^3
           + \rho_5   e_4' {\sc f}_1''^4,\cr}$$
$e_0',\ldots \, e_5'$ being either symmetric or two-valued; but
these expressions conduct to the absurd result,
$$5 e_1' {\sc f}_1'' = x_\alpha + \rho_5^4 x_\beta
                          + \rho_5^3 x_\gamma
                          + \rho_5^2 x_\delta
                          + \rho_5   x_\epsilon,$$
in which the first member has only ten, while the second member
has 120 values.  We are therefore obliged to reject as
inadmissible the supposition
$${\sc f}_1''^{\alpha_1''} = f_1';$$
and we find that no rational function of five arbitrary variables
can have any prime power two-valued, if its own values be more
numerous than two.

\bigbreak

[22.]
There is now no difficulty in proving, after the manner of {\sc
Abel}, that it is impossible to represent a root of the general
equation of the fifth degree, as a function of the coefficients
of that equation, by any expression of the form $b^{(m)}$; that
is, by any finite combination of radicals and rational functions.

For, in the first place, since the coefficients
$a_1,\ldots \, a_5$ are symmetric functions of the roots
$x_1,\ldots \, x_5$, it is clear that we cannot express any one
of the latter as a rational function of the former; $m$ in
$b^{(m)}$, must therefore be greater than $0$; and the
expression~$b^{(m)}$ if it exist at all, must involve at least
one radical of the first order, $a_1'$, which must admit of being
expressed as a rational but unsymmetric function~${\sc f}_1'$ of
the five roots, but must have a prime power ${\sc
f}_1'^{\alpha_1'}$ symmetric, and consequently must be a
square-root, of the form deduced in the last article, namely,
$$a_1' = b (x_1 - x_2) \ldots (x_4 - x_5),$$
the factor $b$ being symmetric.  And because any other radical of
the same order, $a_2'$, might be deduced from $a_1'$ by a
multiplication such as the following,
$\displaystyle a_2' = {c \over b} a_1'$,
we see that no such other radical~$a_2'$, of the first order, can
enter into the expression $b^{(m)}$, when that expression is
cleared of all superfluous functional radicals.  On the other
hand, a two-valued expression such as
$$f_1' = b_0 + b_1 a_1'$$
cannot represent the five-valued function~$x$; if then the sought
expression $x = b^{(m)}$ exist at all, it must involve some
radical of the second order, $a_1''$, and this radical must admit
of being expressed as a rational function of the five roots,
which function is to have, itself, more than two values, but to
have some prime power ${\sc f}_1''^{\alpha_1''}$, two-valued.
And since it has been proved that no such function~${\sc f}_1''$
exists, it follows that no function of the form $b^{(m)}$ can
represent the sought root~$x$ of the general equation of the
fifth degree.  If then that general equation admit of being
resolved at all, it must be by some process distinct from any
finite combination of the operations of adding, subtracting,
multiplying, dividing, elevating to powers, and extracting roots
of functions.

\bigbreak

[23.]
It is, therefore impossible to satisfy the equation
$$b^{(m)5} + a_1 b^{(m)4} + a_2 b^{(m)3}
      + a_3 b^{(m)2} + a_4 b^{(m)} + a_5 = 0,$$
by any finite irrational function~$b^{(m)}$; the five
coefficients $a_1$,~$a_2$,~$a_3$,~$a_4$,~$a_5$ being supposed to
remain arbitrary and independent.  And, by still stronger reason,
it is impossible to satisfy the equation
$$b^{(m) n} + a_1 b^{(m) n - 1} + \cdots
      + a_{n-1} b^{(m)} + a_n = 0,$$
if $n$ be greater than five, and $a_1,\ldots \, a_n$ arbitrary.
For if we could do this, then the irrational function~$b^{(m)}$
would, by the principles already established, have exactly $n$
values; of which $n - 5$ values would vanish when we supposed
$a_n, a_{n-1},\ldots \, a_6$ to become $= 0$, and the remaining
five values would represent the five roots of the general
equation of the fifth degree; but such a representation of the
roots of that equation has been already proved to be impossible.

[24.]
Although the whole of the foregoing has been suggested by that of
{\sc Abel}, and may be said to be a commentary thereon; yet it
will not fail to be perceived, that there are several
considerable differences between the one method of proof and the
other.  More particularly, in establishing the cardinal
proposition that every radical in every irreducible expression
for any one of the roots of any general equation is a rational
function of those roots, it has appeared to the writer of this
paper more satisfactory to begin by showing that the radicals of
highest order will have that property, if those of lower orders
have it, descending thus to radicals of the lowest order, and
afterwards ascending again; than to attempt, as {\sc Abel} has
done, to prove the theorem, in the first instance, for radicals
of the highest order.  In fact, while following this
last-mentioned method, {\sc Abel} has been led to assume that the
coefficient of the first power of some highest radical can always
be rendered equal to unity, by introducing (generally) a new
radical, which in the notation of the present paper may be
expressed as follows:
$$\root \alpha_k^{(m)} \of { \left\{
      \sum_{\beta_i^{(m)} < \alpha_i^{(m)}
            \atop \beta_k^{(m)} = 1}
         \mathbin{.}
      ( b_{\beta_1^{(m)},\ldots \, \beta_{n^{(m)}}^{(m)}}^{(m-1)}
         \mathbin{.}
         a_1^{(m) \beta_1^{(m)}} \ldots
         a_{n^{(m)}}^{(m) \beta_{n^{(m)}}^{(m)}} )
      \right\}^{\alpha_k^{(m)}} };$$
but although the quantity under the radical sign, in this
expression is indeed free from that irrationality of the
$m^{\rm th}$ order which was introduced by the radical
$a_k^{(m)}$, it is not, in general, free from the irrationalities
of the same order introduced by the other radicals
$a_1^{(m)},\ldots$ of that order; and consquently the new
radical, to which this process conducts, is in general elevated
to the order $m + 1$; a circumstance which {\sc Abel} does not
appear to have remarked, and which renders it difficult to judge
of the validity of his subsequent reasoning.  And because the
other chief obscurity in {\sc Abel}'s argument (in the opinion of
the present writer) is connected with the proof of the theorem,
that a rational function of five independent variables cannot
have five values, and five only, unless it be symmetric
relatively to four of its five elements; it has been thought
advantageous, in the present paper, as preliminary to the
discussion of the forms of functions of five arbitrary
quantities, to establish certain auxiliary theorems respecting
functions of fewer variables; which have served also to determine
{\it \`{a} priori\/} all possible solutions (by radicals and
rational functions) of all general algebraic equations below the
fifth degree.

\bigbreak

[25.]
However, it may be proper to state briefly here the simple and
elegant reasoning by which {\sc Abel}, after {\sc Cauchy}, has
proved that if a function of five variables have fewer than five
values, it must be either two-valued or symmetric.  Let the
function be for brevity denoted by
$(\alpha, \beta, \gamma, \delta, \epsilon)$; and let $\nabla$ and
$\nabla^\backprime$ denote such changes, that
$$\eqalign{
(\beta, \gamma, \delta, \epsilon, \alpha)
   &= \nabla (\alpha, \beta, \gamma, \delta, \epsilon),\cr
(\beta, \epsilon, \alpha, \gamma, \delta)
   &= \nabla^\backprime
         (\alpha, \beta, \gamma, \delta, \epsilon).\cr}$$
These two changes are such that we have the two symbolic
equations
$$\nabla^5 = 1,\quad \nabla^{\backprime 5} = 1;$$
but also, by supposition, some two of the five functions
$$\nabla^0 (\alpha, \beta, \gamma, \delta, \epsilon),\quad
  \ldots \quad
  \nabla^4 (\alpha, \beta, \gamma, \delta, \epsilon)$$
are equal among themselves, and so are some two of the five
functions
$$\nabla^{\backprime 0}
         (\alpha, \beta, \gamma, \delta, \epsilon),\quad
  \ldots \quad
  \nabla^{\backprime 4}
         (\alpha, \beta, \gamma, \delta, \epsilon);$$
we have therefore two equations of the forms
$$\nabla^r = 1,\quad \nabla^{\backprime {r^\backprime}} = 1,$$
in which $r$ and $r^\backprime$ are each greater than $0$, but
less than $5$; and by combining these equations with the others
just now found, we obtain
$$\nabla = 1,\quad \nabla^\backprime = 1:$$
that is
$$(\beta, \gamma, \delta, \epsilon, \alpha)
   = (\alpha, \beta, \gamma, \delta, \epsilon),
      \quad\hbox{and}\quad
(\beta, \epsilon, \alpha, \gamma, \delta)
   = (\alpha, \beta, \gamma, \delta, \epsilon).$$
Hence
$$(\gamma, \alpha, \beta, \delta, \epsilon)
   = (\beta, \gamma, \delta, \epsilon, \alpha)
   = (\alpha, \beta, \gamma, \delta, \epsilon);$$
and in like manner,
$$(\alpha, \gamma, \delta, \beta, \epsilon)
   = (\alpha, \beta, \gamma, \delta, \epsilon)
   = (\gamma, \alpha, \beta, \delta, \epsilon);$$
we may therefore interchange the first and second of the five
elements of the function, if we at the same time interchange
either the second and third, or the third and fourth; and a
similar reasoning shows that we may interchange any two, if we
at the same time interchange any two others.  An even number of
such interchanges leaves therefore the function unaltered; but
every alteration of arrangement of the five elements may be made
by either an odd or an even number of such interchanges: the
function, therefore, is either two-valued or symmetric; it having
been supposed to have fewer than five values.  Indeed, this is
only a particular case of a more general theorem of {\sc Cauchy},
which is deduced in a similar way: namely, that if the number of
values of a rational function of $n$ arbitrary quantities be less
than the greatest prime number which is itself not greater than
$n$, the number of values of that function must then be either
two or one.

\bigbreak

[26.]
It is a necessary consequence of the foregoing argument, that
there must be a fallacy in the very ingenious process by which
Mr.~{\sc Jerrard} has attempted to reduce the general equation of
the fifth degree to the solvible form of {\sc De Moivre}, namely,
$$x^5 - 5b x^3 + 5 b^2 x - 2e = 0,$$
of which a root may be expressed as follows,
$$x = \root 5 \of {} \{ e + \sqrt {e^2 - b^5} \}
      + {b \over \root 5 \of {} \{ e + \sqrt {e^2 - b^5} \}}:$$
because this process of reduction would, if valid, conduct to a
finite (though complicated) expression for a root~$x$ of the
general equation of the fifth degree,
$$x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 = 0,$$
with five arbitrary coefficients, real or imaginary, as a
function of those five coefficients, through the previous
resolution of certain auxiliary equations below the fifth degree,
namely, a cubic, two quadratics, another cubic, and a
biquadratic, besides linear equations and {\sc De Moivre}'s
solvible form; and therefore ultimately through the extraction of
a finite number of radicals, namely, a square-root, a cube-root,
three square-roots, a cube-root, a square-root, a cube-root,
three square-roots, and a fifth-root.  Accordingly, the fallacy
of this process of reduction has been pointed out by the writer
of the present paper, in an ``Inquiry into the Validity of a
Method recently proposed by {\sc George B. Jerrard}, Esq., for
transforming and resolving Equations of Elevated Degrees:''
undertaken at the request of the British Association for the
Advancement of Science, and published in their Sixth Report.  But
the same Inquiry has confirmed the adequacy of
Mr.~{\sc Jerrard}'s method to accomplish an almost equally
curious and unexpected transformation, namely, the reduction of
the general equation of the fifth degree to the trinomial form
$$x^5 + Dx + E = 0;$$
and therefore ultimately to this very simple form
$$x^5 + x = e;$$
in which, however, it is essential to observe that $e$ will in
general be imaginary even when the original coefficients are
real.  If then we make, in this last form,
$$x = \rho (\cos \theta + \sqrt{-1} \sin \theta),$$
and
$$e = r (\cos v + \sqrt{-1} \sin v),$$
we can, with the help of Mr.~{\sc Jerrard}'s method, reduce the
general equation of the fifth degree, with five arbitrary and
imaginary coefficients, to the system of the two following
equations, which involve only real quantities:
$$\rho^5 \cos 5 \theta + \rho \cos \theta = r \cos v;\quad
  \rho^5 \sin 5 \theta + \rho \sin \theta = r \sin v;$$
in arriving at which system, the quantities $r$ and $v$ are
determined, without tentation, by a finite number of rational
combinations, and of extractions of square-roots and cube-roots
of imaginaries, which can be performed by the help of the usual
logarithmic tables; and $\rho$ and $\theta$ may afterwards be
found from $r$ and $v$, by {\it two new tables of double entry},
which the writer of the present paper has had the curiosity to
construct and to apply.

\bigbreak

[27.]
In general, if we change $x$ to $x + \sqrt{-1} y$, and
$a_i$ to $a_i + \sqrt{-1} b_i$, the equation of the fifth degree
becomes
$$(x + \sqrt{-1} y)^5
      + (a_i + \sqrt{-1}) (x + \sqrt{-1}y)^4 + \cdots
      + a_5 + \sqrt{-1} b_5 = 0,$$
and resolves itself into the two following:
$$\eqalign{
\hbox{I.}\quad
   &x^5 - 10 x^3 y^2 + 5 x y^4
      + a_1 (x^4 - 6 x^2 y^2 + y^4) - b_1 (4 x^3 y - 4 x y^3) \cr
   &\quad \mathord{}
      + a_2 (x^3 - 3 x y^2) - b_2(3 x^2 y - y^3)
      + a_3 (x^2 - y^2) - 2 b_3 xy
      + a_4 x - b_4 y + a_5 = 0;\cr}$$
and
$$\eqalign{
\hbox{II.}\quad
   &5 x^4 y - 10 x^2 y^3 + y^5
      + a_1 (4 x^3 y - 4 x y^3) + b_1 (x^4 - 6 x^2 y^2 + y^4) \cr
   &\quad \mathord{}
      + a_2 (3 x^2 y - y^3) + b_2 (x^3 - 3 x y^2)
      + 2 a_3 xy + b^3 (x^2 - y^2)
      + a_4 y + b_4 x + b_5 = 0;\cr}$$
in which all the quantities are real: and the problem of
resolving the general equation with imaginary coefficients is
really equivalent to the problem of resolving this last system;
that is, to the problem of deducing, from it, {\it two real
functions\/} ($x$ and $y$) of {\sc ten} {\it arbitrary real
quantities\/}
$a_1,\ldots\, a_5, b_1,\ldots\, b_5$.
Mr.~{\sc Jerrard} has therefore accomplished a very remarkable
simplification of this general problem, since he has reduced it
to the problem of discovering {\it two real functions of\/}
{\sc two} {\it arbitrary real quantities}, by showing that,
without any real loss of generality, it is permitted to suppose
$$a_1 = a_2 = a_3 = b_1 = b_2 = b_3 = b_4 = 0,$$
and
$$a_4 = 1,$$
$a_5$ and $b_5$ alone remaining arbitrary: though he has failed
(as the argument developed in this paper might have shown
beforehand that he must necessarily fail) in his endeavour to
calculate the latter two, or the former ten functions, through
any finite number of extractions of square-roots, cube-roots, and
fifth-roots of expressions of the form $a + \sqrt{-1} b$.

\bigbreak

[28.]
But when we come to consider in what sense it is true that we are
in possession of methods for extracting, without tentation, such
roots of such imaginary expressions; and therefore in what sense
we are permitted to postulate the extraction of such radicals, or
the determination of both $x$ and $y$, in an imaginary equation
of the form
$$x + \sqrt{-1} y = \root \alpha \of {a + \sqrt{-1} b},$$
as an instrument of calculation in algebra; we find that this
depends ultimately on our being able to reduce all such
extractions to the employment of {\it tables of single entry\/}:
or, in more theoretical language, to {\it real functions of
single real variables}.  In fact, the equation last-mentioned
gives
$$(x + \sqrt{-1} y)^\alpha = a + \sqrt{-1} b,$$
that is, it gives the system of the two following:
$$x^\alpha - {\alpha(\alpha - 1) \over 1 \mathbin{.} 2}
         x^{\alpha - 2} y^2
   + \hbox{\&c.} = a,\quad
  \alpha x^{\alpha - 1} y
   - {\alpha(\alpha - 1)(\alpha - 2)
         \over 1 \mathbin{.} 2 \mathbin{.} 3}
         x^{\alpha - 3} y^3
   + \hbox{\&c.} = b;$$
which, again, give
$$(x^2 + y^2)^\alpha = a^2 + b^2,$$
and
$${\displaystyle \alpha {y \over x}
   - {\alpha(\alpha - 1)(\alpha - 2)
         \over 1 \mathbin{.} 2 \mathbin{.} 3}
      \left( {y \over x} \right)^3 + \cdots \over
   \displaystyle
   1 - {\alpha(\alpha - 1) \over 1 \mathbin{.} 2}
         \left( {y \over x} \right)^2 + \cdots}
   = {b \over a}.$$
If then we put
$$\phi_1(\rho) = \rho^\alpha,$$
and
$$\phi_2(\tau) = {\displaystyle \alpha \tau
   - {\alpha(\alpha - 1)(\alpha - 2)
        \over 1 \mathbin{.} 2 \mathbin{.} 3} \tau^3
   + \cdots \over \displaystyle
      1 - {\alpha(\alpha - 1) \over 1 \mathbin{.} 2} \tau^2
   + \cdots};$$
and observe that these two real and rational functions $\phi_1$
and $\phi_2$ of single real quantities have always real
inverses, $\phi_1^{-1}$ and $\phi_2^{-1}$, at least if the
operation $\phi_1^{-1}$ be performed on a positive quantity,
while the function~$\phi_1^{-1}(r^2)$ has but one real and
positive value, and the function~$\phi_2^{-1}(t)$ has $\alpha$
real values; we see that the determination of $x$ and $y$ in the
equation
$$x + \sqrt{-1} y = \root \alpha \of {a + \sqrt{-1} b},$$
comes ultimately to the calculation of the following real
functions of single real variables, of which the inverse
functions are rational:
$$x^2 + y^2 = \phi_1^{-1}(a^2 + b^2);\quad
  {y \over x} = \phi_2^{-1} \left( {b \over a} \right);$$
and to the extraction of a single real square-root, which gives
$$\eqalign{
x &= \pm \sqrt{\vphantom{\biggl(}}
      \left\{ { \phi_1^{-1} (a^2 + b^2) \over
         \displaystyle
         1 + \left( \phi_2^{-1} {b \over a} \right)^2 }
         \right\},\cr
y &= \pm \left( \phi_2^{-1} {b \over a} \right) \mathbin{.}
         \sqrt{\vphantom{\biggl(}}
      \left\{ { \phi_1^{-1} (a^2 + b^2) \over
         \displaystyle
         1 + \left( \phi_2^{-1} {b \over a} \right)^2 }
         \right\}.\cr}$$
Now, notwithstanding the importance of those two particular forms
of rational functions $\phi_1$ and $\phi_2$ which present
themselves in separating the real and imaginary part of the
radical $\root \alpha \of {a + \sqrt{-1} b}$, and of which the
former is a power of a single real variable, while the latter is
the tangent of a multiple and real arc expressed in terms of the
single and real arc corresponding; it may appear with reason
that these functions do not possess such an eminent prerogative
of simplicity as to entitle the {\it inverses of them alone\/} to
be admitted into elementary algebra, to the exclusion of the
inverses of all other real and rational functions of single real
variables.  And since the general equation of the fifth degree,
with real or imaginary coefficients, has been reduced, by
Mr.~{\sc Jerrard}'s\footnote*{{\it Mathematical Researches}, by
George B. Jerrard, Esq., A.B.; printed by William Strong,
Clare-street, Bristol.}
method, to the system of the two real equations
$$x^5 - 10 x^3 y^2 + 5 x y^4 + x = a,\quad
  5 x^4 y - 10 x^2 y^3 + y^5 + y = b,$$
it ought, perhaps, to be now the object of those who interest
themselves in the improvement of this part of algebra, to inquire
whether the dependence of the two real numbers $x$ and $y$, in
these two last equations, on the two real numbers $a$ and $b$,
cannot be expressed by the help of the real inverses of some new
real and rational, or even transcendental functions of single
real variables; or, (to express the same thing in a practical, or
in a geometrical form,) to inquire whether the two sought real
numbers cannot be calculated by a finite number of tables of
single entry, or constructed by the help of a finite number of
curves: although the argument of {\sc Abel} excludes all hope
that this can be accomplished, if we confine ourselves to those
particular forms of rational functions which are connected with
the extraction of radicals.

It may be proper to state, that in adopting, for the
convenience of others, throughout this paper, the usual language
of algebraists, especially respecting real and imaginary
quantities, the writer is not to be considered as abandoning the
views which he put forward in his Essay on Conjugate Functions,
and on Algebra as the Science of Pure Time, published in the
second Part of the seventeenth volume of the Transactions of
the Academy: which views he still hopes to develop and
illustrate hereafter.

He desires also to acknowledge, that for the opportunity of
reading the original argument of {\sc Abel}, in the first volume
of {\sc Crelle}'s Journal, he is indebted to the kindness
of his friend Mr.~{\sc Lubbock}; and that his own remarks were
written first in private letters to that gentleman, before they
were thrown into the form of a communication to the Royal Irish
Academy.

\bigbreak

\centerline{\sc addition}

\nobreak\medskip

Since the foregoing paper was communicated, the writer has seen,
in the first Part of the Philosophical Transactions for
1837, an essay entitled ``Analysis of the Roots of Equations,''
by a mathematician of a very high genius, the Rev.~R.~{\sc
Murphy}, Fellow of Caius College, Cambridge; who appears to have
been led, by the analogy of the expressions for roots of
equations of the first four degrees, to conjecture that the five
roots
$x_1 \, x_2 \, x_3 \, x_4 \, x_5$ of the general equation of the
fifth degree,
$$x^5 + a x^4 + b x^3 + c x^2 + d x + e = 0,
   \eqno (1)$$
can be expressed as finite irrational functions of the five
arbitrary coefficients $a$,~$b$,~$c$,~$d$,~$e$, as follows:
$$\left. \eqalign{
x_1 &= {-a \over 5} + \root 5 \of {} \alpha
                    + \root 5 \of {} \beta
                    + \root 5 \of {} \gamma
                    + \root 5 \of {} \delta,\cr
x_2 &= {-a \over 5} + \omega   \root 5 \of {} \alpha
                    + \omega^2 \root 5 \of {} \beta
                    + \omega^3 \root 5 \of {} \gamma
                    + \omega^4 \root 5 \of {} \delta,\cr
x_3 &= {-a \over 5} + \omega^2 \root 5 \of {} \alpha
                    + \omega^4 \root 5 \of {} \beta
                    + \omega   \root 5 \of {} \gamma
                    + \omega^3 \root 5 \of {} \delta,\cr
x_4 &= {-a \over 5} + \omega^3 \root 5 \of {} \alpha
                    + \omega   \root 5 \of {} \beta
                    + \omega^4 \root 5 \of {} \gamma
                    + \omega^2 \root 5 \of {} \delta,\cr
x_5 &= {-a \over 5} + \omega^4 \root 5 \of {} \alpha
                    + \omega^3 \root 5 \of {} \beta
                    + \omega^2 \root 5 \of {} \gamma
                    + \omega   \root 5 \of {} \delta,\cr}
   \right\}
   \eqno (2)$$
$\omega$ being an imaginary fifth-root of unity, and
$\alpha$ $\beta$ $\gamma$ $\delta$ being the four roots of an
auxiliary biquadratic equation,
$$\left. \eqalign{
\alpha &= \alpha' + \surd \beta'
                  + \surd \gamma'
                  + \surd \delta',\cr
\beta  &= \alpha' + \surd \beta'
                  - \surd \gamma'
                  - \surd \delta',\cr
\gamma &= \alpha' - \surd \beta'
                  + \surd \gamma'
                  - \surd \delta',\cr
\delta &= \alpha' - \surd \beta'
                  - \surd \gamma'
                  + \surd \delta';\cr}
   \right\}
   \eqno (3)$$
in which $\beta'$ $\gamma'$ $\delta'$ are the three roots of the
auxiliary cubic equation,
$$\left. \eqalign{
\beta'  &= \alpha'' + \root 3 \of {} \beta''
                    + \root 3 \of {} \gamma'',\cr
\gamma' &= \alpha'' + \theta   \root 3 \of {} \beta''
                    + \theta^2 \root 3 \of {} \gamma'',\cr
\delta' &= \alpha'' + \theta^2 \root 3 \of {} \beta''
                    + \theta   \root 3 \of {} \gamma'';\cr}
   \right\}
   \eqno (4)$$
$\theta$ being an imaginary cube-root of unity, and $\beta''$,
$\gamma''$ being the two roots of an auxiliary quadratic,
$$\left. \eqalign{
\beta''  &= \alpha''' + \surd \alpha^{\rm IV},\cr
\gamma'' &= \alpha''' - \surd \alpha^{\rm IV}.\cr}
   \right\}
   \eqno (5)$$
And, doubtless, it is allowed to represent any five arbitrary
quantities $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ by the system of
expressions (2) (3) (4) (5), in which $a$, $\omega$ and $\theta$
are such that
$$a = - (x_1 + x_2 + x_3 + x_4 + x_5),
   \eqno (6)$$
$$\omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0,
   \eqno (7)$$
$$\theta^2 + \theta + 1 = 0,
   \eqno (8)$$
provided that the auxiliary quantities
$\alpha$ $\beta$ $\gamma$ $\delta$
$\alpha'$ $\beta'$ $\gamma'$ $\delta'$
$\alpha''$ $\beta''$ $\gamma''$
$\alpha'''$ $\alpha^{\rm IV}$
be determined so as to satisfy the conditions
$$\left. \eqalign{
5 \root 5 \of \alpha
   = x_1 + \omega^4 x_2
         + \omega^3 x_3
         + \omega^2 x_4
         + \omega   x_5,\cr
5 \root 5 \of \beta
   = x_1 + \omega^3 x_2
         + \omega   x_3
         + \omega^4 x_4
         + \omega^2 x_5,\cr
5 \root 5 \of \gamma
   = x_1 + \omega^2 x_2
         + \omega^4 x_3
         + \omega   x_4
         + \omega^3 x_5,\cr
5 \root 5 \of \delta
   = x_1 + \omega   x_2
         + \omega^2 x_3
         + \omega^3 x_4
         + \omega^4 x_5,\cr}
   \right\}
   \eqno (9)$$
$$\left. \eqalign{
4 \alpha' &= \alpha + \beta + \gamma + \delta,\cr
4 \beta'  &= \alpha + \beta - \gamma - \delta,\cr
4 \gamma' &= \alpha - \beta + \gamma - \delta,\cr
4 \delta' &= \alpha - \beta - \gamma + \delta,\cr}
   \right\}
   \eqno (10)$$
$$\left. \eqalign{
3 \alpha''
   &= \beta' + \gamma' + \delta',\cr
3 \root 3 \of \beta''
   &= \beta' + \theta^2 \gamma' + \theta   \delta',\cr
3 \root 3 \of \gamma''
   &= \beta' + \theta   \gamma' + \theta^2 \delta',\cr}
   \right\}
   \eqno (11)$$
$$\left. \eqalign{
2 \alpha'''             &= \beta'' + \gamma'',\cr
2 \surd \alpha^{\rm IV} &= \beta'' - \gamma''.\cr}
   \right\}
   \eqno (12)$$
But it is not true that the four auxiliary quantities
$\alpha'$,~$\alpha''$,~$\alpha'''$,~$\alpha^{\rm IV}$,
determined by these conditions, are symmetric functions of the
five quantities $x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$, or rational
functions of $a$,~$b$,~$c$,~$d$,~$e$, as Mr.~{\sc Murphy} appears
to have conjectured them to be.

In fact, the conditions just mentioned give, in the first place,
expressions for $\alpha$,~$\beta$, $\gamma$, $\delta$,~$\alpha'$,
as functions of the five roots $x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$,
which functions are rational and integral and homogeneous of the
fifth dimension; they give, next, expressions for
$\beta'$,~$\gamma'$,~$\delta'$,~$\alpha''$,
as functions of the tenth dimension; for
$\beta''$,~$\gamma''$,~$\alpha'''$, of the thirtieth; and for
$\alpha^{\rm IV}$, of the sixtieth dimension.  And
Mr.~{\sc Murphy} has rightly remarked that the function
$\alpha^{\rm IV}$ may be put under the form
$$\alpha^{\rm IV}
  = k {\sc a}_1^2 \mathbin{.} {\sc a}_2^2 \mathbin{.} {\sc a}_3^2
         \mathbin{.} {\sc a}_4^2 \mathbin{.} {\sc a}_5^2
         \mathbin{.} {\sc a}_6^2
         \mathbin{.} {\sc b}_1^2 \ldots {\sc b}_6^2
         \mathbin{.} {\sc c}_1^2 \ldots {\sc c}_6^2
         \mathbin{.} {\sc d}_1^2 \ldots {\sc d}_6^2
         \mathbin{.} {\sc e}_1^2 \ldots {\sc e}_6^2,
   \eqno (13)$$
in which $k$ is a numerical constant, and
$$\left. \eqalign{
{\sc a}_1
   &= x_2 - x_4 + \omega (x_3 - x_4) + \omega^2 (x_3 - x_5),\cr
{\sc a}_2
   &= x_3 - x_2 + \omega (x_5 - x_2) + \omega^2 (x_5 - x_4),\cr
{\sc a}_3
   &= x_4 - x_5 + \omega (x_2 - x_5) + \omega^2 (x_2 - x_3),\cr
{\sc a}_4
   &= x_5 - x_3 + \omega (x_4 - x_3) + \omega^2 (x_4 - x_2),\cr}
   \right\}
   \eqno (14)$$
$$\left. \eqalign{
{\sc a}_5
   &= x_2 - x_5 + (\omega^2 + \omega^3) (x_3 - x_4),\cr
{\sc a}_6
   &= x_3 - x_4 + (\omega^2 + \omega^3) (x_5 - x_2);\cr}
   \right\}
   \eqno (15)$$
these six being the only linear factors of
$\displaystyle \sqrt{\vphantom{\biggl(}}
      {\alpha^{\rm IV} \over k}$
which do not involve $x_1$.  But the expression (14) give,
by (7),
$$\eqalignno{
\left( {\omega \over 1 + \omega} \right)^2
      {\sc a}_1 {\sc a}_2 {\sc a}_3 {\sc a}_4 \hskip-72pt \cr
   &= \{ x_2^2 + x_3^2 + x_4^2 + x_5^2
         - (x_2 + x_5)(x_3 + x_4) \}^2 \cr
   &\mathrel{\phantom{=}}
      + \{ (x_2 - x_5)^2 + (x_2 - x_4)(x_5 - x_3) \}
        \{ (x_3 - x_4)^2 + (x_2 - x_3)(x_5 - x_4) \};
   &(16)\cr}$$
and the expressions (15) give
$${\omega^3 \over 1 + \omega} {\sc a}_5 {\sc a}_6
  = (x_3 - x_4)^2 + (x_2 - x_5)(x_3 - x_4)
      - (x_2 - x_5)^2;
   \eqno (17)$$
the part of $\alpha^{\rm IV}$, which is of highest dimension
relatively to $x_1$, is therefore of the form
$$\eqalignno{
   & {\sc n} x_1^{48} ( \{ x_2^2 + x_3^2 + x_4^2 + x_5^2
         - (x_2 + x_5)(x_3 + x_4) \}^2 \cr
   &\quad + \{ (x_2 - x_5)^2 + (x_2 - x_4)(x_5 - x_3) \}
            \{ (x_3 - x_4)^2 + (x_2 - x_3)(x_5 - x_4) \})^2 \cr
   &\quad \times
            \{ (x_3 - x_4)^2 + (x_2 - x_5)(x_3 - x_4)
               - (x_2 - x_5)^2 \}^2,
   &(18)\cr}$$
${\sc n}$ being a numerical coefficient; and consequently the
coefficients, in $\alpha^{\rm IV}$, of the products
$x_1^{48} x_2^{11} x_3$ and $x_1^{48} x_2 x_3^{11}$ are,
respectively $-6 {\sc n}$ and $-4 {\sc n}$; they are therefore
unequal, and $\alpha^{\rm IV}$ is not a symmetric function of
$x_1$,~$x_2$,~$x_3$,~$x_4$,~$x_5$.

The same defect of symmetry may be more easily proved for the
case of the function~$\alpha'$, by observing that when $x_1$
and $x_5$ are made $= 0$, the expression
$$\eqalignno{
4. 5^5 \mathbin{.} \alpha'
   &= (x_1 + \omega   x_2
           + \omega^2 x_3
           + \omega^3 x_4
           + \omega^4 x_5)^5 \cr
   &\mathrel{\phantom{=}}
    + (x_1 + \omega^2 x_2
           + \omega^4 x_3
           + \omega   x_4
           + \omega^3 x_5)^5 \cr
   &\mathrel{\phantom{=}}
    + (x_1 + \omega^3 x_2
           + \omega   x_3
           + \omega^4 x_4
           + \omega^2 x_5)^5 \cr
   &\mathrel{\phantom{=}}
    + (x_1 + \omega^4 x_2
           + \omega^3 x_3
           + \omega^2 x_4
           + \omega   x_5)^5
   &(19)\cr}$$
becomes
$$\eqalignno{
     &   (x_2 + \omega x_3 + \omega^2 x_4)^5
       + (x_2 + \omega^2 x_3 + \omega^4 x_4)^5
       + (x_2 + \omega^3 x_3 + \omega   x_4)^5
       + (x_2 + \omega^4 x_3 + \omega^3 x_4)^5 \cr
     &\quad =
      4 x_2^5 - 5 x_2^4 (x_3 + x_4)
    - 10 x_2^3 (x_3^2 + 2 x_3 x_4 + x_4^2)
    - 10 x_2^2 (x_3^3 + 3 x_3^2 x_4 - 12 x_3 x_4^2 + x_4^3) \cr
     &\quad \mathrel{\phantom{=}}
       - 5 x_2 (x_3^4 - 16 x_3^3 x_4 + 6 x_3^2 x_4^2
                  + 4 x_3 x_4^3 + x_4^4) \cr
     &\quad \mathrel{\phantom{=}}
       + 4 x_3^5 - 5 x_3^4 x_4 - 10 x_3^3 x_4^2
       - 10 x_3^2 x_4^3 - 5 x_3 x_4^4 + 4 x_4^5,
   &(20)\cr}$$
which is evidently unsymmetric.

The elegant analysis of Mr.~{\sc Murphy} fails therefore to
establish any conclusion opposed to the argument of {\sc Abel}.

\bye

