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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf ON ADDITIONAL APPLICATIONS OF}

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\centerline{\Largebf QUATERNIONS TO SURFACES OF THE}

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\centerline{\Largebf SECOND ORDER}

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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), pp.\ 14--19.)}

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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 additional Applications of Quaternions to
Surfaces of the Second Order}

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

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\centerline{Communicated November~30, 1847.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), pp.\ 14--19.]}

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Sir W. Rowan Hamilton gave an account of some additional
applications of Quaternions to Surfaces of the Second Order.

In the Abstract printed as part of the Proceedings of the Academy
for July~20, 1846, the following equation of the ellipsoid (there
numbered 44),
$${\rm T} ( \iota \rho + \rho \kappa) = \kappa^2 - \iota^2,
   \eqno (1)$$
was given, as a transformation of this other equation of the same
surface (there marked 35):
$${\rm T} (\alpha \rho + \rho \alpha + \beta \rho - \rho \beta) = 1;
   \eqno (2)$$
which was itself deduced by transforming, according to the rules
of quaternions, the formula
$$(\alpha \rho + \rho \alpha)^2 - (\beta \rho - \rho \beta)^2 = 1;
   \eqno (3)$$
this last quaternion form of the equation of the ellipsoid having
been previously exhibited to the Academy, at its meeting of
December~8, 1845.  (See the equation numbered 21, in the
Proceedings of that date.)  The symbols $\alpha$, $\beta$, denote
two constant vectors; the symbols $\iota$, $\kappa$, denote two
other constant vectors, connected with them by the relations
$$\alpha + \beta = {\iota \over \iota^2 - \kappa^2},\quad
  \alpha - \beta = {\kappa \over \iota^2 - \kappa^2},
   \eqno (4)$$
where $\iota^2 - \kappa^2$ is a negative scalar; and $\rho$
denotes a variable vector, drawn from the centre to the surface
of the ellipsoid: while ${\rm T}$ is the characteristic of the
operation of taking the {\it tensor\/} of a quaternion.

If a new variable vector~$\nu$ be defined, as a function of the
three vectors $\iota$, $\kappa$, $\rho$, by the equation
$$(\kappa^2 - \iota^2)^2 \nu
   =  (\kappa^2 + \iota^2) \rho + \iota \rho \kappa + \kappa \rho \iota,
   \eqno (5)$$
it results from the general rules of this calculus that this new
vector~$\nu$ will satisfy each of the two following equations:
$${\rm S} \mathbin{.} \nu \rho = 1;\quad
  {\rm S} \mathbin{.} \nu \, {\rm d} \rho = 0;
   \eqno (6)$$
which give also these two other equations, of the same kind with
them, and differing only by the interchange of the two symbols
$\rho$ and $\nu$:
$${\rm S} \mathbin{.} \rho \nu = 1;\quad
  {\rm S} \mathbin{.} \rho \, {\rm d} \nu = 0;
   \eqno (7)$$
where ${\rm d}$ is the characteristic of differentiation, and
${\rm S}$ is that of the operation of taking the scalar part of a
quaternion.  The equations (6) shew that $\nu$ is the vector, of
which the reciprocal~$\nu^{-1}$ represents in length and in
direction the perpendicular let fall from the common origin of
the variable vectors here considered on the plane which touches
at the extremity of the vector~$\rho$ the locus of that variable
extremity; so that $\nu^{-1}$ is here a symbol for the
perpendicular let fall from the centre of the ellipsoid on the
tangent plane to that surface: and $\nu$ itself denotes, in
length and in direction, the reciprocal of that perpendicular, so
that it may be called the {\it vector of proximity\/} of the
tangent plane, or of the element of the surface of the ellipsoid,
to the centre regarded as an origin.  Accordingly, the equation
here marked (5) was given in the Abstract of July, 1846 (where it
was numbered 45), as a formula for determining what was there
also called the vector of proximity of the tangent plane of the
ellipsoid.  It may now be seen that the symbolical connexion
between the two equations above marked (6), and the two other
equations lately numbered (7), corresponds to, and expresses, in
this Calculus, under what may be regarded as a strikingly simple
form, the known connexion of {\it reciprocity\/} between any two
surfaces, of which one is the locus of the extremities of
straight lines drawn from any fixed point, so as to be in their
directions perpendicular to the tangent planes of the other
surface, and in their lengths inversely proportional to those
perpendiculars: from the perception of which {\it general\/}
relation of reciprocity between surfaces, exemplified previously
for the case of two reciprocal ellipsoids by that great
geometrical genius (Professor Mac Cullagh), whose recent and
untimely loss we all so deeply deplore, the author of the present
communication was led to announce to the Academy, in October,
1832, the existence of certain {\it circles of contact\/} on
Fresnel's wave, which he saw to be a necessary consequence of the
existence of certain {\it conical cusps\/} on another and
reciprocal surface.  A very elegant {\it geometrical\/} proof of
the same general theorem of reciprocity was given afterwards, in
the Transactions\footnote*{See the beautiful paper entitled,
``Geometrical Propositions applied to the Wave Theory of Light.
By James Mac Cullagh, F.~T.~C.~D.''  Read June~24, 1833.
Transactions of the Royal Irish Academy, vol.~xvii.}
of this Academy by Professor Mac Cullagh himself.

As respects the reciprocal ellipsoid, of which the vector~$\nu$,
in the equation lately marked (5), denotes a semidiameter, it may
be mentioned here that, with the same significations of the
symbols, the following equation holds good:
$$(2 \beta {\rm S} \mathbin{.} \alpha \beta)^2
   =  (\beta {\rm S} \mathbin{.} \beta \nu)^2
       + ( {\rm V} \mathbin{.} \beta {\rm V} \mathbin{.} \alpha \nu)^2;
   \eqno (8)$$
with equations for other central surfaces of the second order,
regarded as reciprocals of central surfaces, which differ only in
the signs of their terms from this equation (8).  The author
proposes, in a future continuation of the present communication,
to illustrate this new form, as regards the processes of
obtaining and of interpreting it.  Meanwhile he desires to submit
to the notice of the Academy the following construction, for
generating a system of two reciprocal ellipsoids, by means of a
moving sphere, to which his own methods have conducted him,
although it may turn out to have been already otherwise
discovered.  Let then a sphere of constant magnitude, with
centre~$E$, move so that it always intersects two fixed and
mutually intersecting straight lines, $AB$, $AB'$, in four
points, $L$,~$M$,~$L'$,~$M'$, of which $L$ and $M$ are on $AB$
and $L'$ and $M'$ are on $AB'$; and let one diagonal $LM'$, of
the inscribed quadrilateral $LMM'L'$, be constantly parallel to a
third fixed line $AC$, which will oblige the other diagonal $ML'$
of the same quadrilateral to move parallel to a fourth fixed line
$AC'$.  Let $N$ be the point in which the diagonals intersect,
and draw $AF$ equal and parallel to $EN$; so that $AENF$ is a
parallelogram: then {\it the locus of the centre~$E$ of the
moving sphere is one ellipsoid, and the locus of the opposite
corner~$F$ of the parallelogram is another ellipsoid reciprocal
thereto}.  These two ellipsoids have a common centre, namely, the
point~$A$; and a common mean axis, which is equal to the diameter
of the moving sphere.  Two sides, $AE$, $AF$, of the
parallelogram $AENF$, are thus two semidiameters, which may be
regarded as reciprocal to each other, one of the one ellipsoid,
and the other of the other.  It is, however to be observed, that
they fall at opposite sides of the principal plane, containing
the four fixed lines, and that, therefore, it may be proper to
call them more fully {\it opposite reciprocal semidiameters\/};
and to call the points $E$ and $F$, in which they terminate, {\it
opposite reciprocal points}.  The two other sides $EN$, $FN$, of
the same varying parallelogram, are the normals to the two
ellipsoids, meeting each other in the point~$N$, upon the same
principal plane.  In that plane, the two former fixed lines,
$AB$, $AB'$, are the axes of the two cylinders of revolution
which are circumscribed about the first ellipsoid; and the two
latter fixed lines, $AC$, $AC'$, are the two cyclic normals of
the same first ellipsoid: while the diagonals $LM'$, $ML'$, of
the inscribed quadrilateral in the construction, are the axes of
the two circles on the surface of that ellipsoid, which circles
pass through the point~$E$, that is through the centre of the
moving sphere, and which are also contained upon the surface of
another sphere, having its centre at the point~$N$: all which is
easily adapted, by suitable interchanges, to the other or
reciprocal ellipsoid, and flows with great facility from the
quaternion equations above given.

It may not be out of place to mention, on this occasion, although
for the present without its demonstration, another simple
geometrical construction connected with a surface of the second
order, and derived from the same calculus of quaternions.  This
construction is adapted to determine the cone of revolution which
osculates, along a given side, to a cone of the second degree;
but it will perhaps be most easily understood by considering it
as serving to assign the interior pole of the small circle on a
sphere, which osculates at a given point~$T$, to a given spherical
conic.  Let the given cyclic arcs be $AC$, $AC'$, extending from
one of the two points~$A$ of their own mutual intersection to the
tangent arc $CTC'$, which is well known to be bisected at the
point of contact~$T$.  On the normal arc $NTP$, drawn through
that given point~$T$, let fall a perpendicular arc $AN$; draw
$NC$, or $NC'$, and erect $CP$ or $C'P$, perpendicular thereto,
and meeting the normal arc in $P$: the point~$P$, thus
determined, will be the pole, or spherical centre of curvature,
which was required.

\bye