% 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 A THEOREM CONCERNING}

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\centerline{\Largebf POLYGONIC SYNGRAPHY}

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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), p.\ 474--475.)}

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

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\centerline{\largerm 2000}

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\centerline{\largeit A Theorem concerning Polygonic Syngraphy.}

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

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\centerline{Communicated June~13, 1853.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~5 (1853), p.\ 474--475.]}

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Professor Sir William Rowan Hamilton exhibited the following
Theorem, to which he had been conducted by that theory of
geometrical {\it syngraphy} of which he had lately submitted to
the Academy a verbal and hitherto unreported sketch, and on which
he hopes to return in a future communication.

\bigbreak

{\it Theorem}.
Let $A_1, A_2,\ldots \, A_n$ be any $n$ points
(in number odd or even) assumed at pleasure on the $n$ successive
sides of a closed polygon
$B B_1 B_2 \, \ldots \, B_{n-1}$
(plane or gauche), inscribed in any given surface of the second
order.  Take any three points, $P$,~$Q$,~$R$,
on that surface, as initial points, and draw from each a system
of $n$ successive chords, passing in order through the $n$
assumed points $(A)$, and terminating in three other
superficial and final points, $P'$,~$Q'$,~$R'$.
Then there will be (in general) {\it another\/} inscribed and
closed polygon,
$C C_1 C_2 \, \ldots \, C_{n-1}$,
of which the $n$ sides shall pass successively, in the same
order, through the same $n$ points $(A)$; and of which the
initial point~$C$ shall also be connected with the
point~$B$ of the former polygon, by the relations
$$    {a e l \over b c}
         {\beta \gamma \over \alpha \epsilon \lambda}
   =  {a' e' l' \over b' c'}
         {\beta' \gamma' \over \alpha' \epsilon' \lambda'},\quad
      {b f m \over c a}
         {\gamma \alpha \over \beta \zeta \mu}
   =  {b' f' m' \over c' a'}
         {\gamma' \alpha' \over \beta' \zeta' \mu'},\quad
      {c g n \over a b}
         {\alpha \beta \over \gamma \eta \nu}
   =  {c' g' n' \over a' b'}
         {\alpha' \beta' \over \gamma' \eta' \nu'};$$
where
$$\multieqalign{
a  &= Q R, &
b  &= R P, &
c  &= P Q, \cr
e  &= B P, &
f  &= B Q, &
g  &= B R, \cr
l  &= C P, &
m  &= C Q, &
n  &= C R, \cr
a' &= Q' R', &
b' &= R' P', &
c' &= P' Q', \cr
e' &= B P', &
f' &= B Q', &
g' &= B R', \cr
l' &= C P', &
m' &= C Q', &
n' &= C R'; \cr}$$
while
$\alpha \, \beta \, \gamma \, \epsilon \, \zeta \, \eta \,
   \lambda \, \mu \, \nu$, and
$\alpha' \, \beta' \, \gamma' \, \epsilon' \, \zeta' \, \eta' \,
   \lambda' \, \mu' \, \nu'$,
denote the semidiameters of the surface, respectively parallel to
the chords
$a \, b \, c \, e \, f \, g \, l \, m \, n$,
$a' \, b' \, c' \, e' \, f' \, g' \, l' \, m' \, n'$.

As a very particular {\it case\/} of this theorem, we may suppose
that $P Q' R P' Q R'$
is a plane hexagon in a conic, and $B C$ its Pascal's
line.

\bye

