% 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 QUATERNIONS AND THE DETERMINATION}

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\centerline{\Largebf OF THE DISTANCES OF ANY RECENTLY}

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\centerline{\Largebf DISCOVERED COMET OR PLANET FROM}

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\centerline{\Largebf THE EARTH}

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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,
   4 (1850), p.~75.)}

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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 Application of Quaternions to the
Determination of the Distance}
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\centerline{\largeit of any recently discovered Comet or
Planet from the Earth.}

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

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\centerline{Communicated February~28th, 1848.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), p.~75.]}

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The following notice was communicated by Sir William Rowan
Hamilton, of a Paper ``on the Application of Quaternions to the
Determination of the Distance of any recently discovered Comet or
Planet from the Earth.''

This celebrated problem is treated in this paper by means of the
formul{\ae} which were communicated to the Academy by the author,
in July, 1845.  The chief step consists in a very easy deduction,
from those formul{\ae}, of the equation:
$${a \over c} \left( {M \over a^3} - {M \over b^3} \right)
   = {{\rm S} \mathbin{.} \gamma \gamma' \gamma''
         \over {\rm S} \mathbin{.} \gamma \gamma' \alpha};$$
where $c$ is the sought distance of the comet (or planet) from
the earth; $M$ is the mass of the sun, and $a$ and $b$ are the
distances of earth and comet from that body; $\alpha$ is the
heliocentric vector-unit of the earth, and $\gamma$ is the
geocentric vector-unit of the comet; while $\gamma'$, $\gamma''$
are the first and second differential coefficients of $\gamma$,
taken with respect to the time, and determined, along with
$\gamma$ itself, from three successive observations: and
${\rm S}$ is the characteristic of the operation of taking the
scalar part of a quaternion.  The second member of the equation
admits of being geometrically interpreted as a ratio of two
pyramids, and can in various ways be transformed by the rules of
the calculus of quaternions.

\bye