% 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}

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\centerline{\Largebf AND BIQUADRATIC EQUATION, WHICH}

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\centerline{\Largebf IS SATISFIED BY THE SYMBOL OF LINEAR}

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\centerline{\Largebf OPERATION IN 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,
   8 (1864), pp.\ 190--191)}

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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 the Existence of a Symbolic and Biquadratic
Equation, which is}
\centerline{\sc satisfied by the Symbol of Linear Operation in
Quaternions.}

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\centerline{Sir William Rowan Hamilton.}

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\centerline{Read June 23rd, 1862.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~viii (1864), pp.\ 190--191.]}

\bigskip

1.
In a recent communication (of June~9, 1862), I showed how the
general Linear and Quaternion Function of a Quaternion could be
expressed, under a standard quadrinomial form; and how that
function, when so expressed, could be inverted.

\bigbreak

2.
I have since perceived, that whatever {\it form\/} be adopted, to
represent the {\it Linear Symbol of Quaternion Operation\/} thus
referred to, that {\it symbol\/} always satisfies a certain
{\it Biquadratic Equation}, with {\it Scalar Coefficients}, of
which the {\it values\/} depend upon the particular {\it
constants\/} of the {\it Function\/} above referred to.

\bigbreak

3.
This result, with the properties of the {\it Auxiliary Linear\/}
and {\it Quaternion Functions\/} which which it is connected,
appears to me to constitute the most remarkable accession to the
{\it Theory of Quaternions proper}, as distinguished from their
{\it separation\/} into {\it scalar\/} and {\it vector parts},
and from their {\it application to Geometry\/} and {\it Physics},
which has been made since I had first the honour of addressing
the Royal Irish Academy on the subject, in the year 1843.

\bigbreak

4.
The following is an outline of one of the proofs of the existence
of the biquadratic equation, above referred to.  Let
$$fq = r
   \eqno (1)$$
be a given linear equation in quaternions; $r$ being a given
quaternion, $q$ a sought one, and $f$ the symbol of a linear or
distributive operation: so that
$$f(q + q') = fq + fq',
   \eqno (2)$$
whatever two quaternions may be denoted by $q$ and $q'$.

\bigbreak

5.
I have found that the {\it formula of solution\/} of this
equation (1), or the formula of {\it inversion\/} of the
{\it function}, $f$, may be thus stated:
$$nq = n f^{-1} r = F r;
   \eqno (3)$$
where $n$ is a {\it scalar constant\/} depending for its
{\it value}, and $F$ is an auxiliary and {\it linear symbol\/} of
operation depending for its {\it form\/} (or rather for the
{\it constants\/} which it involves), on the {\it particular
form\/} of $f$; or on the special values of the {\it constants},
which enter into the composition of the {\it particular
function}, $fq$.

\bigbreak

6.
We have thus, independently of the particular quaternions, $q$ and
$r$, the equations,
$$F f q = n q,\quad F f r = n r;
   \eqno (4)$$
or, briefly and symbolically,
$$F f = f F = n.
   \eqno (5)$$

\bigbreak

7.
Changing next $f$ to $f_c  = f + c$, that is to say, proposing
next to resolve the {\it new linear equation},
$$f_c q = fq + cq = r,
   \eqno (6)$$
where $c$ is an {\it arbitrary scalar}, I find that the
{\it new\/} formula of solution, or of inversion, may be thus
written:
$$f_c F_c = n_c;
   \eqno (7)$$
where
$$F_c = F + c G + c^2 H + c^3,
   \eqno (8)$$
and
$$n_c = n + n' c + n'' c^2 + n''' c^3 + c^4;
   \eqno (9)$$
$G$ and $H$ being the symbols (or characteristics) of {\it two
new linear operations}, and $n'$, $n''$, $n'''$ denoting
{\it three new scalar constants}.

\bigbreak

8.
Expanding then the symbolical product $f_c F_c$, and comparing
powers of $c$, we arrive at {\it three new symbolical equations},
namely, the following:
$$f G + F = n';\quad fH + G = n'';\quad f + H = n''';
   \eqno (10)$$
by elimination of the symbols, $F$, $G$, $H$, between which and
the equation (5), the {\it symbolical biquadratic},
$$0 = n - n'f + n'' f^2 - n''' f^3 + f^4,
   \eqno {\sc (A)}$$
is obtained.

\bye
