% 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\tensc=cmcsc10
\font\sevensc=cmcsc10 scaled 700
\newfam\scfam \def\sc{\fam\scfam\tensc}
\textfont\scfam=\tensc \scriptfont\scfam=\sevensc
\font\largesc=cmcsc10 scaled \magstep1

\pageno=0

\null\vskip72pt

\centerline{\Largebf ADDITIONAL THEOREMS RESPECTING}

\vskip12pt

\centerline{\Largebf CERTAIN RECIPROCAL SURFACES}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (Proceedings of the Royal Irish Academy,
   4 (1850), p.\ 192--193.)}

\vskip36pt

\vfill

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

\vskip 12pt

\centerline{\largerm 2000}

\vskip36pt\eject

\null\vskip36pt

\centerline{\largeit Additional Theorems respecting certain
Reciprocal Surfaces.}

\vskip 6pt

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

\bigskip

\centerline{Communicated June~26, 1848.}

\bigskip

\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), p.\ 192--193.]}

\bigskip

Sir W.~R. Hamilton stated the following additional theorems
respecting certain reciprocal surfaces, to which his own methods
have conducted him.

If a plane quadrilateral $A B C D$ be
inscribed in a given sphere, so that its four sides may be
constantly parallel to four given straight lines; and if
$E$,~$F$ be the two points of meeting of the two
pairs of opposite sides, namely, $E$ the meeting of the
sides $A B$, $C D$, and $F$ the
meeting of $B C$, $D A$ (prolonged if
necessary); then the locus of the point~$E$ will be one
ellipsoid, and the locus of the point~$F$ will be another
ellipsoid reciprocal thereto.

And other pairs of reciprocal surfaces of the second degree may
be generated in like manner, by changing the sphere to other
surfaces of revolution of the second degree.

For instance, a pair of reciprocal cones of the second degree may
be generated as the loci of two points $E$,~$F$,
which are, in like manner, the points of meeting of the opposite
sides of a plane quadrilateral $A B C D$,
inscribed in a circular section of a right-angled cone of
revolution, with their directions in like manner constant.  And a
pair of reciprocal hyperboloids (whether of one or of two sheets)
may, in like manner, be generated from an equilateral hyperboloid
of revolution (of one or of two sheets).

The writer may take this opportunity of mentioning a result which
lately occured to him, respecting two {\it arbitrary}, but
{\it reciprocal\/} conical surfaces, of which each is the locus
of all the normals to the others, erected at their common vertex;
namely, that two such cones have always one common conical
surface of centres of curvature.

\bye