% 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 THE INSCRIPTION OF CERTAIN}

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\centerline{\Largebf ``GAUCHE'' POLYGONS IN SURFACES}

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\centerline{\Largebf OF THE SECOND DEGREE}

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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.\ 325--326.)}

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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 Inscription of certain ``Gauche''
Polygons in Surfaces}

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\centerline{\largeit of the Second Degree.}

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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.\ 325--326.]}

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In the same note to the Secretary, it was requested by Sir
William R. Hamilton that the Academy might be informed of a
theorem respecting the inscription of certain gauche polygons, in
surfaces of the second degree, which he had lately communicated
to the Council.  This theorem was obtained by the method of
quaternions, and included, as a particular case, the
following:---``If the first, second, third, and fourth sides of a
gauche nonagon, inscribed in a surface of the second order, be
respectively parallel to the fifth, sixth, seventh, and eighth
sides of that nonagon, and also to the first, second, third, and
fourth sides of a gauche quadrilateral, inscribed in the same
surface; then the plane containing the first, fifth and ninth
corners of the nonagon will be parallel to the plane which
touches the surface at the first corner of the quadrilateral.''

More generally the theorem here referred to shews that for the
inscribed quadrilateral we may substitute a gauche polygon with
any even number, $2n$, of sides; and for the nonagon, another
gauche polygon, with $4n + 1$ sides, connected with that polygon
of $2n$ sides, by the same law of construction as that which had
connected the nonagon with the quadrilateral; and that then the
tangent plane to the surface at the first corner of the polygon
of $2n$ sides, will be parallel to the plane through the first,
middle, and last corners $(1, 2n + 1, 4n + 1)$ of the polygon of
$4n + 1$ sides.

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