% 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 A NEW AND GENERAL METHOD OF}

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\centerline{\Largebf INVERTING A LINEAR AND QUATERNION}

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\centerline{\Largebf FUNCTION OF A QUATERNION}

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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.\ 182--183)}

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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 a New and General Method of Inverting a Linear
and Quaternion}
\centerline{\sc Function of a Quaternion.}

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\centerline{Sir William Rowan Hamilton.}

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\centerline{Read June 9th, 1862.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~viii (1864), pp.\ 182--183.]}

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Let $a$, $b$, $c$, $d$, $e$ represent any five quaternions, and
let the following notations be admitted, at least as temporary
ones:---
$$ab - ba = [ab],\quad S [ab] c = (abc);$$
$$(abc) + [cb] \, Sa + [ac] \, Sb + [ba] \, Sc = [abc];$$
$$Sa [bcd] = (abcd);$$
then it is easily seen that
$$[ab] = - [ba];\quad
  (abc) = - (bac) = (bca) = \hbox{\&c.};$$
$$[abc] = - [bac] = [bca] + \hbox{\&c.};$$
$$(abcd) = - (bacd) = (bcad) = \hbox{\&c.};$$
$$0 = [aa] = (aac) = [aac] = (aacd),\enspace \hbox{\&c.}$$

We have then these two Lemmas respecting Quaternions, which
answer to two of the most continually occurring transformations of
vector expressions:---
$$\quad\vcenter{\halign{\hfil #$\ldots$\quad
      &$\displaystyle #$\hfil \cr
I&
0 = a(bcde) + b(cdea) + c(deab) + d(eabc) + e(abcd),\cr
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or I${}'$&
e(abcd) = a(ebcd) + b(aecd) + c(abed) + d(abce);\cr
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and II&
e(abcd) = [bcd] \, Sae - [cda] \, Sbe + [dab] \, Sce
      - [abc] \, S de;\cr}}$$
as may be proved in various ways.

Assuming therefore {\it any four\/} quaternions $a$, $b$, $c$,
$d$, which are {\it not\/} connected by the relation,
$$(abcd) = 0,$$
we can {\it deduce\/} from them four others,
$a'$,~$b'$,~$c'$,~$d'$, by the expressions,
$$a' (abcd) = f [bcd],\quad b' [abcd] = -f[cda],\enspace \hbox{\&c.},$$
where $f$ is used as the characteristic of a linear or
{\it distributive quaternion function\/} of a quaternion, of
which the form is supposed to be given; and thus the {\it general
form\/} of {\it such\/} a function comes to be represented by the
expression,
$$\hbox{V}\ldots\quad
  r = fq = a' \, Saq + b' \, Sbq + c' \, Scq + d' \, Sdq;$$
involving {\it sixteen scalar constants}, namely those contained
in $a' \, b' \, c' \, d'$.

The {\it Problem\/} is to {\it invert\/} this {\it
function\/}~$f$; and the {\it solution\/} of that problem is
easily found, with the help of the new Lemmas I.\ and II., to be
the following:---
$$\eqalign{
  \hbox{VI}\ldots\quad
q(abcd) (a'b'c'd')
   &= (abcd) (a' b' c' d') f^{-1} r \cr
   &= [bcd] (r b' c' d') + [cda] (r c' d' a')
      + [dab] (r d' a' b') + [abc] (r a' b' c');\cr}$$
of which solution the correctness can be verified, {\it \`{a}
posteriori}, with the help of the same Lemmas.

Although the foregoing problem of {\it Inversion\/} had been
{\it virtually\/} resolved by Sir W.~R.~H.\ many years ago,
through a reduction of it to the corresponding problem respecting
{\it vectors}, yet he hopes that, as regards the Calculus of
{\it Quaternions\/}, the new solution will be considered to be an
important step.  He is, however, in possession of a general
{\it method\/} for treating questions of this class, on which he
may perhaps offer some remarks at the next meeting of the
Academy.

\bye