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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 THE GEOMETRICAL INTERPRETATION}

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\centerline{\Largebf OF SOME RESULTS OBTAINED BY CALCULATION}

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\centerline{\Largebf WITH BIQUATERNIONS}

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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,
   5 (1853), pp.\ 388--390.)}

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

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\centerline{\largerm 2000}

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\centerline{\largeit On the Geometrical Interpretation of some
Results obtained by calculation}

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\centerline{\largeit with Biquaternions.}

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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}

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\centerline{Communicated February~28, 1853.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~5 (1853), pp.\ 388--390.]}

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Sir William R. Hamilton, LL.D., gave an account of the
geometrical interpretation of some results obtained by
calculation with biquaternions.

In this communication {\it bivectors\/} were employed, and were
shown to conduct to interesting conclusions.  The conception of
such bivectors,
$$\rho + \sqrt{-1} \rho',$$
where $\rho$ and $\rho'$ denote two geometrically real vectors,
and $\sqrt{-1}$ is the {\it old\/} and ordinary (or commutative)
imaginary of common algebra, and generally of
{\it biquaternions\/} such as
$$q + \sqrt{-1} q',$$
where $q$ and $q'$ are real quaternions, interpretable,
geometrically on the author's principles, had occurred to him
many years ago; and the remark which he made to the Academy
in November, 1844 (see the Proceedings of that date), respecting
the representations, in his Calculus, of the geometrically
{\it unreal\/} tangents to a sphere from an internal point, as
having {\it positive squares}, belonged essentially to this
theory of bivectors.  In the same year, the more general theory
of biquaternions had occurred to him, in connexion with what in
his theory presented themselves as the {\it imaginary roots}, or
purely symbolical solutions, of a certain quadratic equation in
quaternions.  Notices on the subject have since appeared in his
subsequent papers, in the Proceedings of the Academy, and in the
Philosophical Magazine: and a fuller statement of the theory will
be found in his (as yet unpublished) Lectures on Quaternions, of
which many sheets have long since been distributed among his
friends and others in the University.  On the present occasion he
has employed {\it bivectors with null squares}, such as
$$i + hj, \quad\hbox{or}\quad j + hk,$$
where $i$,~$j$,~$k$ are the {\it peculiar\/} symbols of the
quaternion calculus, observing the laws communicated by him to
the Academy in November, 1843, while $h$ is used as a temporary
and abridged symbol for the {\it old\/} imaginary $\surd -1$.  In
fact the rules of this calculus give
$$(j + hk)^2 = j^2 + h (jk + kj) + h^2 k^2
   =  -1 + 0 h + (-1) (-1) = 0,$$
$h$ being a {\it free\/} (or commutative) factor in any
multiplication, as in algebra, but $jk$ being $ = i = -kj$, while
$$h^2 = i^2 = j^2 = k^2 = -1.$$
Thus, at least for any numerical exponent~$x$, we have the
simplification,
$$(1 + j + hk)^x = 1 + x (j + hk),$$
which Sir W.~R.~H. states that he has found useful in a part of a
geometrical investigation, respecting the interpretation of
certain continued fractions in quaternions, of the form
$$u_x = \left( {b \over a +} \right)^x u_0,$$
already mentioned by him to the Academy on a former occasion, and
specially for the case when $\alpha^4 + 4 \beta^2 = 0$, in the
fraction
$$\rho_x = \left( {\beta \over \alpha +} \right)^x \rho_0,$$
where the vector~$\beta$ is supposed to be perpendicular to
$\alpha$ and $\rho_0$, and therefore also to $\rho_x$.

By the investigation referred to, he has found, among others, the
following results.  Let $C$ and $D$ be two given
points, and $P$ an assumed point.  Perpendicular to
$D P$ draw $C Q$, towards a given hand,
and such that the rectangle
$C Q \mathbin{.} D P$
may be equal to a given rectangle
$C C' D D'$.
From $Q$ derive $R$, as $Q$ has been derived
from $P$, and conceive the process repeated without end.
Then, I., the locus of the alternate points
$P, R, T,\ldots $
is one circle, and the locus of the other alternate points
$Q, S, U,\ldots $
is another circle.  II.  These two circular loci have the top
$C' D'$ of the given rectangle for the common radical
axis, of themselves and of the given circle described on
$C D$ as diameter.  III.  The centres of the two
alternate loci are harmonic conjugates with respect to the given
circle.  IV.  If from two fixed summits of the two loci chords be
drawn to the successive points, and prolonged (if necessary) till
they meet the radical axis in other points
$P'$,~$Q'$,~\&c.; if also a summit~$F$ of the
given circle be suitably chosen (on the line of the three
centres), then the two lines $F P'$,
$C Q'$ will cross in one point on the given circle,
the two lines $F Q'$, $C R'$ in another
point thereon, and so on for ever: and the same thing holds for
the lines
$D P'$, $F Q'$, or
$D Q'$, $F R'$, \&c.
Particular forms of these theorems have been published in the
Phil.\ Mag.\ for this month (February, 1853), but only
for the case when the top of the rectangle, or the radical axis,
meets the given circle in two {\it real points},
$A$,~$B$, in which case the derived points
$Q, R,\ldots $ {\it converge\/} towards the
point~$B$ nearer to $C$.  In the contrary case there
can be {\it no convergence}, but there may be {\it circulation in
a period}.  For if we then denote by $V$ one of the two
common points of the system of common orthogonals, and by
$W$ the point of contact of the given circle with a tangent
drawn from the middle point between them, the angle
$P' V Q'$ or $Q' V R'$
will be constant, and equal to $V F W$; so that
if this latter angle be commensurate with a right angle, the
points $P' Q' R' \, \ldots $, and therefore
also the points $P Q R \, \ldots $ will recur
in a certain periodical order.  These conclusions have been by
Sir W.~R. Hamilton obtained as results of his quaternion
analysis; but he believes that it will not be found difficult to
confirm them by a purely geometrical process, founded on the
known theory of homographic divisions.\footnote*{{\sc Note},
{\it added during printing}.  Since the foregoing communication
was made, the author has seen how to obtain such {\it
geometrical\/} proofs, or confirmations, of all the foregoing
results.}

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