% 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 QUATERNION PROOF OF A THEOREM OF}

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\centerline{\Largebf RECIPROCITY OF CURVES IN SPACE}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (British Association Report, 1862,
   Part~II, p.~4.)}

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

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\centerline{\largerm 2000}

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{\largeit\noindent
Quaternion Proof of a Theorem of Reciprocity of Curves in Space.  By\/}
{\largerm Sir} {\largesc William Rowan Hamilton},
{\largeit LL.D., \&c.}

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\centerline{[{\it Report of the Thirty-Second Meeting of the British
Association for the Advancement of}}
\centerline{{\it Science; held at Cambridge in October 1862}.}
\centerline{(John Murray, London, 1863), Part~II, p.~4.]}

\bigbreak

Let $\phi$ and $\psi$ be any two vector functions of a scalar
variable, and $\phi'$, $\psi'$, $\phi''$, $\psi''$ their derived
functions, of the first and second orders.  Then each of the two
systems of equations, in which $c$ is a scalar constant,
$$(1) \, \ldots \quad
  {\rm S} \phi   \psi = c,\quad
  {\rm S} \phi'  \psi = 0,\quad
  {\rm S} \phi'' \psi = 0,$$
$$(2) \, \ldots \quad
  {\rm S} \psi   \phi = c,\quad
  {\rm S} \psi'  \phi = 0,\quad
  {\rm S} \psi'' \phi = 0,$$
or each of the two vector expressions,
$$(3) \, \ldots \quad
  \psi = {c V \phi' \phi'' \over S \phi \phi' \phi''},\quad
  (4) \, \ldots \quad
  \phi = {c V \psi' \psi'' \over S \psi \psi' \psi''},$$
includes the other.

If then, from any assumed origin, there be drawn lines to
represent the reciprocals of the perpendiculars from that point
on the osculating planes to a first curve of double curvature,
those lines will terminate on a second curve, from which we can
{\it return\/} to the first by a precisely similar process of
construction.

And instead of thus taking the {\it reciprocal\/} of a {\it
curve\/} with respect to a {\it sphere}, we may take it with
respect to {\it any surface\/} of the {\it second order}, as is
probably well known to geometers, although the author was lately
led to perceive it for himself by the very simple
{\it analysis\/} given above.

\bye