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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 A METHOD PROPOSED BY}

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\centerline{\Largebf PROFESSOR BADANO FOR THE SOLUTION}

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\centerline{\Largebf OF ALGEBRAIC EQUATIONS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   2 (1844), pp.~275--276.)}

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

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\centerline{\largerm 2000}

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\noindent
{\largeit On a Method proposed by Professor Badano for the
Solution of Algebraic Equations.\par}

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\centerline{Sir William R. Hamilton}

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\centerline{Communicated 4 August 1842.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~II (1844), pp.~275--276.]}

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The President made a communication respecting a method which had
been lately proposed by Professor Badano of Genoa, for the
solution of algebraical equations of the fifth and higher
degrees.\footnote*{{\it Nuove Ricerche sulla Risoluzione
Generale delle Equazioni Algebriche} del P. Gerolamo Badano,
Carmelitano scalzo, Professore di Matematica nella R. Universita
di Genova.  Genova, Tipografia Ponthenier, 1840.  See also an
`Appendice' to the same work.}

Lagrange had shown that the function
$$t^5 = (x' + \omega x'' + \omega^2 x'''
      + \omega^3 x^{\rm IV} + \omega^4 x^{\rm IV})^5$$
receives only twenty-four different values, for all possible
changes of arrangement of the five quantities,
$x',\ldots \, x^{\rm V}$, if $\omega$ be an imaginary root of
unity, so that
$$\omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0.$$

Professor Badano has proposed to express these twenty-four values
by certain combinations of quadratic and cubic radicals,
suggested by the theory of biquadratic equations, and having the
following for their type:
$$\eqalign{
t^5 &= H_1 + \surd H_2
         + \root 3 \of { H_3 + \surd H_4 }
         + \root 3 \of { H_5 - \surd H_6 } \cr
   &\mathrel{\phantom{=}}
         + \surd \{ H_7 + \surd H_8
            + \root 3 \of { H_9 + \surd H_{10} }
            + \root 3 \of { H_{11} - \surd H_{12} } \} \cr
   &\mathrel{\phantom{=}}
         + \surd \{ H_{13} + \surd H_{14}
            + \theta   \root 3 \of { H_{15} + \surd H_{16} }
            + \theta^2 \root 3 \of { H_{17} - \surd H_{18} } \} \cr
   &\mathrel{\phantom{=}}
         + \surd \{ H_{19} + \surd H_{20}
            + \theta^2 \root 3 \of { H_{21} + \surd H_{22} }
            + \theta   \root 3 \of { H_{23} - \surd H_{24} } \};\cr}$$
$\theta$ being here an imaginary cube-root of unity.  He contends
that the twenty-four quantities, $H_1,\ldots \, H_{24}$, are all
symmetric functions of the five quantities
$x',\ldots \, x^{\rm V}$; and that they are connected among
themselves by the sixteen relations
$$H_3    = H_5,\quad
  H_4    = H_6,\quad
  H_7    = H_{13} = H_{19},\quad
  H_8    = H_{14} = H_{20},\quad
  H_9    = H_{15} = H_{21},$$
$$H_{10} = H_{16} = H_{22},\quad
  H_{11} = H_{17} = H_{23},\quad
  H_{12} = H_{18} = H_{24},\quad
  H_9    = H_{11},\quad
  H_{10} = H_{12}.$$
Sir W. Hamilton examines, in great detail, the composition of the
two conjugate quantities $H_4$, $H_6$, which are each of the
thirtieth dimension relatively to the five original quantities
$x',\ldots \, x^{\rm V}$; though each is symmetric relatively to
four of them.  He finds also that these two quantities $H_4$ and
$H_6$ are not generally equal to each other, but differ by the
sign of an imaginary radical, namely,
$$(\theta - \theta^2) (\omega - \omega^2 - \omega^3 + \omega^4)
   = \sqrt{-15},$$
when they are fully developed, in consistency with Professor
Badano's definitions.  Analogous results are obtained for the two
quantities $H_3$, $H_5$; and these general results are verified
by applying them to a particular system of numerical values of
the five quantities $x',\ldots \, x^{\rm V}$.  It is shown also
that the three quantities $H_7$, $H_{13}$, $H_{19}$, are neither
independent of the arrangement of those five quantities $x$, nor
(generally) equal to each other.  And thus, although $H_1$ is
symmetric, and $H_2$ vanishes, Sir W.~H.\ conceives it to be
proved that Professor Badano's expressions, for the twenty-four
values of Lagrange's function $t^5$, give no assistance towards
the solution of the general equation of the fifth degree, and
therefore that the same method could not be expected to resolve
equations still more elevated, even if we were not in possession
of an {\it \`{a} priori\/} proof that no root of any general
equation above the fourth degree can be expressed as a function
of its coefficients, by any finite combination of radicals and
rational functions.

\bye
