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

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\centerline{\Largebf ON SOME EXTENSIONS OF QUATERNIONS}

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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,
   6 (1858), pp.\ 114--115)}

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

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\centerline{\largerm 2000}

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\centerline{\sc On some Extensions of Quaternions.}

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\centerline{Sir William Rowan Hamilton.}

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\centerline{Read June 26th, 1854.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~vi (1858), pp.\ 114--115.]}

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Sir W.~R. Hamilton read a paper on some extensions of quaternions:

Besides some general remarks on associative polynomes, and on
some extensions of the modular property, Sir W.~R. Hamilton
remarked that if, in the quadrinomial expression
$${\rm Q} = w + \iota x + \kappa y + \lambda z,$$
the laws of the symbols $\iota \, \kappa \, \lambda$ be
determined by the following formula of vector-multiplication,
$$\eqalignno{
\hbox{(A)}\ldots\qquad
(\iota x + \kappa y + \lambda z) (\iota x' + \kappa y' + \lambda z')
   \hskip -12em \cr
   &=    (m_1^2 - l_2 l_3) x x'
       + (l_1 m_1 - m_2 m_3) (y z' + z y') \cr
   &   + (m_2^2 - l_3 l_1) y y'
       + (l_2 m_2 - m_3 m_1) (z x' + x z') \cr
   &   + (m_3^2 - l_1 l_2) z z'
       + (l_3 m_3 - m_1 m_2) (x y' + y x') \cr
   &   + (\iota   l_1 + \kappa  m_3 + \lambda m_2) (y z' - z y') \cr
   &   + (\kappa  l_2 + \lambda m_1 + \iota   m_3) (z x' - x z') \cr
   &   + (\lambda l_3 + \iota   m_2 + \kappa  m_1) (x y' - y x'),\cr}$$
then this expression, which he proposes to call a
{\sc quadrinome}, has many properties (associative, modular, and
others), analogous to the quaternions; which latter are indeed
only that {\it case\/} of such quadrinomes, for which,
$$l_1 = l_2 = l_3 = 1,\quad
  m_1 = m_2 = m_3 = 0,\quad
  \iota = i,\quad
  \kappa = j,\quad
  \lambda = k.$$

He has, however, found another distinct sort of associative
quadrinomial expression, which has also several analogous
properties, and for which he suggests the name of {\sc tetrads};
the product of two vectors being in it,
$$\eqalign{
\hbox{(B)}\ldots\qquad
   &  (l x + m y + n z) (l x' + m y' + n z') \cr
   &   + (\kappa  n - \lambda m) (y z' - z y')
       + (\lambda l - \iota   n) (z x' - x z')
       + (\iota   m - \kappa  l) (x y' - y x').\cr}$$

\bye