% 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.
\magnification=\magstep1
\vsize=227 true mm \hsize=170 true mm
\voffset=-0.4 true mm \hoffset=-5.4 true mm
\def\folio{\ifnum\pageno>0 \number\pageno \else\fi}
\font\Largebf=cmbx10 scaled \magstep2
\font\largerm=cmr12
\font\largeit=cmti12
\font\largesc=cmcsc10 scaled \magstep1
\def\multieqalign#1{\null\,\vcenter{\openup1\jot \mathsurround=0pt
\ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
&&\quad \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
\crcr#1\crcr}}\,}
\pageno=0
\null\vskip72pt
\centerline{\Largebf A THEOREM CONCERNING}
\vskip12pt
\centerline{\Largebf POLYGONIC SYNGRAPHY}
\vskip24pt
\centerline{\Largebf By}
\vskip24pt
\centerline{\Largebf William Rowan Hamilton}
\vskip24pt
\centerline{\largerm (Proceedings of the Royal Irish Academy,
5 (1853), p.\ 474--475.)}
\vskip36pt
\vfill
\centerline{\largerm Edited by David R. Wilkins}
\vskip 12pt
\centerline{\largerm 2000}
\vskip36pt\eject
\null\vskip36pt
\centerline{\largeit A Theorem concerning Polygonic Syngraphy.}
\vskip 6pt
\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
\bigskip
\centerline{Communicated June~13, 1853.}
\bigskip
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~5 (1853), p.\ 474--475.]}
\bigskip
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