% 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 AN EQUATION OF THE ELLIPSOID}

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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.\ 324--325.)}

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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 an Equation of the Ellipsoid.}

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

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\centerline{Communicated April~9, 1849.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), p.\ 324--325.]}

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The Secretary of Council read the following communication from
Sir William Rowan Hamilton, on an equation of the ellipsoid.

``A remark of your's, recently made, respecting the form in which
I first gave to the Academy, in December, 1845, an equation of the
ellipsoid by quaternions,---namely, that this form involved only
{\it one\/} asymptote of the focal hyperbola,---has induced me to
examine, simplify, and extend, since I last saw you, some
manuscript results of mine on that subject; and the following new
form of the equation, which seems to meet your requisitions, may,
perhaps be shewn to the Academy tonight.  This new form is the
following:
$${\rm T} {\rm V} {\eta \rho - \rho \theta
      \over {\rm U} (\eta - \theta)}
   =  \theta^2 - \eta^2.
   \eqno (1)$$

``The constant vectors $\eta$ and $\theta$ are in the directions
of the two asymptotes required; their symbolic sum
$\eta + \theta$, is the vector of an umbilic; their difference,
$\eta - \theta$, has the direction of a cyclic normal; another
umbilicar vector being in the direction of the sum of their
reciprocals, $\eta^{-1} + \theta^{-1}$, and another cyclic normal
in the direction of the difference of those reciprocals,
$\eta^{-1} - \theta^{-1}$.  The lengths of the semiaxes of the
ellipsoid are expressed as follows:
$$a = {\rm T} \eta + {\rm T} \theta;\quad
  b = {\rm T} (\eta - \theta);\quad
  c = {\rm T} \eta - {\rm T} \theta.
   \eqno (2)$$

``The focal ellipse is given by the system of the two equations
$${\rm S} \mathbin{.} \rho \, {\rm U} \eta
   =  {\rm S} \mathbin{.} \rho \, {\rm U} \theta;
   \eqno (3)$$
and
$${\rm T} {\rm V} \mathbin{.} \rho \, {\rm U} \eta
   =  2 {\rm S} \surd ( \eta \theta );
   \eqno (4)$$
where ${\rm T} {\rm V} \mathbin{.} \rho \, {\rm U} \eta$ may be
changed to ${\rm T} {\rm V} \mathbin{.} \rho \, {\rm U} \theta$;
and which represent respectively a plane, and a cylinder of
revolution.  Finally, I shall just add what seems to me
remarkable,---though I have met with several similar results in my
unpublished researches,---that the focal hyperbola is adequately
represented by the {\it single\/} equation following:
$${\rm V} \mathbin{.} \eta \rho \mathbin{.}
      {\rm V} \mathbin{.} \rho \theta
   =  ( {\rm V} \mathbin{.} \eta \theta )^2.\hbox{''}
   \eqno (5)$$

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