% 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 THEOREM RESPECTING ELLIPSOIDS,}

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\centerline{\Largebf OBTAINED BY THE METHOD OF 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,
   4 (1850), pp.\ 349--350.)}

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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 a Theorem respecting Ellipsoids,
obtained by the Method}

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\centerline{{\largeit of Quaternions.}}

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

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\centerline{Communicated May~28, 1849.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), pp.\ 349--350.]}

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The following theorem respecting ellipsoids, obtained by the
method of quaternions, was communicated by Sir William Rowan
Hamilton, in a note to the Secretary of Council:

``On the mean axis of a given ellipsoid, as the major axis,
describe an ellipsoid of revolution, of which the equatorial
circle shall be touched by those tangents to the principal
section of the given ellipsoid (in the plane of the focal
hyperbola), which are parallel to the umbilicar diameters.  In
this equatorial circle, and in every smaller and parallel circle
of the new ellipsoid thus constructed, conceive that indefinitely
many quadrilaterals are inscribed, for each of which one pair of
opposite sides shall be parallel to the given umbilicar
diameters, while the other pair of opposite sides shall be
parallel to the asymptotes of the focal hyperbola.  Then {\it the
intersection of the first pair of opposite sides of the inscribed
quadrilateral will be a point on the surface of the given
ellipsoid}.

``I may remark that the distance of either focus of the new
ellipsoid from the common centre of the new and old ellipsoids,
will be equal to the perpendicular let fall from either of the
two points, which were called ${\sc t}$ and ${\sc v}$ in a recent
note and diagram, on the umbilicar semidiameter ${\sc a} {\sc
u}$, or on that semidiameter prolonged; while the distance of the
umbilic~${\sc u}$ from the foot of either of these two
perpendiculars, that is, the projection of either of the two
equal tangents to the focal hyperbola, ${\sc t} {\sc u}$,
${\sc u} {\sc v}$, on the umbilicar
semidiameter~${\sc a} {\sc u}$, or on that semidiameter
prolonged, will be the minor semiaxis, or the radius of the
equator, of the new ellipsoid (of revolution).

``This new ellipsoid {\it touches\/} the old one at the ends of
the given mean axis; but it also {\it cuts\/} the same old or
given ellipsoid, in a system of two ellipses, contained in planes
perpendicular to the asymptotes of the focal hyperbola.

``If the semiaxes of the given ellipsoid be $a$,~$b$,~$c$, the
common distance of the two {\it foci\/} of the new or derived
ellipsoid (of revolution) from the common centre of the two
ellipsoids, is expressed by the formula
$$e = {\surd (a^2 - b^2) \surd (b^2 - c^2)
         \over \surd (a^2 - b^2 + c^2)}.
   \eqno (1)$$

``And I venture, although with diffidence, to propose the name of
the {\sc two medial foci}, for the two points thus determined on
the mean axis~$2b$ of the ellipsoid $a$,~$b$,~$c$.  If their
vectors be denoted by $\pm \epsilon$, the equation of that
original ellipsoid may be thus written:
$${\rm T} (\lambda_1 + \epsilon) + {\rm T} (\lambda_1 - \epsilon)
   =  2b;
   \eqno (2)$$
or thus,
$${\rm T} (\lambda_1 - \epsilon)
   =  b + b^{-1} \, {\rm S} \mathbin{.} \epsilon \lambda_1;
   \eqno (3)$$
where
$$\lambda_1
   = {\rho \eta + \theta \rho \over \eta + \theta};\quad
  \epsilon
   = {2 {\rm V} \mathbin{.} \eta \theta
         \over {\rm T} (\eta + \theta)};
   \eqno (4)$$
$\eta$, $\theta$, $\rho$, having the same signification as in
notes recently read; while $e$ may perhaps be called the
{\sc medial eccentricity} of the ellipsoid $a \, b \, c$.

``In a future communication I may be induced to return on the
quaternion analysis employed, and to submit to the Academy some
account of it.''

\bye


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