% 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 THE EXISTENCE OF A SYMBOLIC AND}

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\centerline{\Largebf BIQUADRATIC EQUATION WHICH IS SATISFIED}

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\centerline{\Largebf BY THE SYMBOL OF LINEAR OR DISTRIBUTIVE}

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\centerline{\Largebf OPERATION ON A QUATERNION}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Philosophical Magazine, 24 (1862), pp.\ 127--128.)}

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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 the Existence of a Symbolic and Biquadratic Equation
which is satisfied by the Symbol of Linear or Distributive
Operation on a Quaternion.  By\/} {\largerm Sir}
{\largesc William Rowan Hamilton}, {\largeit LL.D.\
\&c.}\footnote*{Communicated by the Author.}

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\centerline{[{\it The London, Edinburgh and Dublin Philosophical
Magazine and Journal of Science,}}
\centerline{4th series, vol.~xxiv (1862), pp. 127--128.]}

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1.
As early as the year 1846, I was led to perceive the existence of
a certain {\it symbolic\/} and {\it cubic equation}, of the form
$$0 = m - m' \phi + m'' \phi^2 - \phi^3,
   \eqno (1)$$
in which $\phi$ is used as a symbol of {\it linear\/} and
{\it vector operation\/} on a {\it vector}, so that
$\phi \rho$ denotes a vector depending on $\rho$, such that
$$\phi(\rho + \rho') = \phi \rho + \phi \rho',
   \eqno (2)$$
if $\rho$ and $\rho'$ be any two vectors; while $m$, $m'$ and
$m''$ are {\it three scalar constants}, depending on the
{\it particular\/} form of the linear and vector function
$\phi \rho$, or on the (scalar or vector) constants which enter
into the composition of that function.  And I saw, of course,
that the problem of {\it inversion\/} of such a {\it function\/}
was at once given by the formula
$$m \phi^{-1} = m' - m'' \phi + \phi^2,
   \eqno (3)$$
---the required assignment of the inverse function,
$\phi^{-1} \rho$, being thus reduced to the performance of a
limited number of {\it direct operations}.

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2.
Quite recently I have discovered that the far more general
{\it linear\/} (or distributive) and {\it quaternion function of
a quaternion\/} can be {\it inverted}, by an analogous process,
or that there always exists, for any {\it such\/} function $fq$,
satisfying the condition
$$f(q + q') = fq + fq',
   \eqno (4)$$
where $q$ and $q'$ are any two quaternions, a {\it symbolic\/}
and {\it biquadratic\/} equation, of the form
$$0 = n - n' f + n'' f^2 - n''' f^3 + f^4,
   \eqno (5)$$
in which $n$, $n'$, $n''$, and $n'''$ are {\it four scalar
constants}, depending on the particular composition of the linear
function~$fq$; and that therefore we may write generally this
{\it Formula of Quaternion Inversion},
$$n f^{-1} = n' - n'' f + n'' f^2 - f^3.
   \eqno (6)$$

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3.
As it was in the Number of the Philosophical Magazine for
July~1844 that the first {\it printed\/} publication of the
Quaternions occurred (though a paper on them had been previously
read to the Royal Irish Academy in November 1843), I have thought
that the Editors of the Magazine might perhaps allow me thus to
put on record what seems to myself an important addition to the
theory, and that they may even allow me to add, in a Postscript
to this communication, so much as may convey a distinct conception
of the {\it Method\/} which I have pursued.

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Observatory of Trinity College, Dublin,

June~27, 1862.

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