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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 2000.

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\centerline{\Largebf ON SOME RESULTS OBTAINED BY THE}

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\centerline{\Largebf QUATERNION ANALYSIS RESPECTING}

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\centerline{\Largebf THE INSCRIPTION OF ``GAUCHE''}

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\centerline{\Largebf POLYGONS IN 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.\ 380--387.)}

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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 some Results obtained by the Quaternion
Analysis respecting the}

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\centerline{\largeit inscription of ``gauche'' Polygons in
Surfaces of the Second Order.}

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

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\centerline{Communicated June~25, 1849.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), pp.\ 380--387.]}

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Sir William Rowan Hamilton communicated to the Academy some
results, obtained by the quaternion analysis, respecting the
{\it inscription of gauche polygons in surfaces of the second
order}.

If it be required to inscribe a rectilinear polygon
${\sc p}, {\sc p}_1, {\sc p}_2,\ldots \, {\sc p}_{n-1}$
in such a surface, under the conditions that its $n$ successive
sides
${\sc p} {\sc p}_1, {\sc p}_1 {\sc p}_2,\ldots \,
   {\sc p}_{n-1} {\sc p}$,
shall pass respectively through $n$ given points,
${\sc a}_1, {\sc a}_2,\ldots \, {\sc a}_n$, the analysis of
Sir W.~R.~H. conducts to {\it one}, or to {\it two real lines},
as containing the first corner~${\sc p}$, according as the
number~$n$ of sides is {\it odd\/} or {\it even\/}: while, in the
latter of these two cases, the two real right lines thus found
are {\it reciprocal polars\/} of each other, with reference to
the surface in which the polygon is to be inscribed.  Thus, for
the inscription of a plane triangle, or of a gauche pentagon,
heptagon, \&c., in a surface of the second order, where three,
five, seven, \&c, points are given upon its sides, a single right
line is found, which may or may not intersect the surface; and
the problem of inscription admits generally of two real
{\it or\/} of two imaginary solutions.  But for the inscription
of a gauche quadrilateral, hexagon, octagon, \&c., when four,
six, eight, \&c.\ points are given on its successive sides, two
real right lines are found, which (as above stated) are polars of
each other; and therefore, if the surface be an ellipsoid, or a
hyperboloid of {\it two\/} sheets, the problem admits generally
of two real {\it and\/} of two imaginary solutions: while if the
surface be a hyperboloid of {\it one\/} sheet, the four solutions
are then, in general, together real, or together imaginary.

When a gauche polygon, or polygon with $2m + 1$ sides, is to be
inscribed in an ellipsoid or in a double-sheeted hyperboloid, and
when the single straight line, found as above, lies wholly
outside the surface, so as to give two imaginary solutions of the
problem as at first proposed, this line is still not useless
geometrically; for its reciprocal polar intersects the surface in
two real points, of which each is the first corner of an
inscribed decagon, or polygon with $4m + 2$ sides, whose $2m + 1$
pairs of {\it opposite sides\/} intersect each other respectively
in $2m + 1$ given points,
${\sc a}_1, {\sc a}_2,\ldots \, {\sc a}_{2m+1}$.
Thus when, in the well-known problem of inscribing a triangle in
a plane conic, whose sides shall pass through three given points,
the known rectilinear locus of the first corner is found to have
no real intersection with the conic, so that the problem, as
usually viewed, admits of no real solution, and that the
inscription of the {\it triangle\/} becomes geometrically
{\it impossible\/}; we have only to conceive an ellipsoid, or a
double-sheeted hyperboloid, to be so constructed as to contain
the given conic upon its surface; and then to take, with respect
to this surface, the polar of this known right line, in order to
obtain two {\it real\/} or geometrically possible solutions of
{\it another\/} problem, not less interesting: since this
rectilinear polar will cut the surface in two real points, of
which each is the first corner of an {\it inscribed gauche
hexagon\/} whose {\it opposite sides intersect\/} each other in
the three points proposed.  (It may be noticed that the three
{\it diagonals\/} of this gauche hexagon, or the three right
lines joining each corner to the opposite one, intersect each
other in {\it one common point}, namely, in the pole of the given
plane.)

If we seek to inscribe a polygon of $4m$ sides in a surface of
the second order, under the condition that its opposite sides
shall intersect respectively in $2m$ given points, the quaternion
analysis conducts generally to two polar right lines, as loci of
the first corner, which lines are the same with those that would
be otherwise found as loci of the first corner of an inscribed
polygon of $2m$ sides, passing respectively through the $2m$
given points.  Thus, {\it in general}, the polygon of $4m$ sides,
found as above, is merely the polygon of $2m$ sides, with
{\it each side twice traversed\/} by the motion of a point along
its perimeter.  But if a certain {\it condition\/} be satisfied,
by a certain {\it arrangement of the $2m$ given points\/} in
space; namely, if the last point~${\sc a}_{2m}$ be on that real
right line which is the locus of the first corner of a real or
imaginary inscribed polygon of $2m - 1$ sides, which pass
respectively through the first $2m - 1$ given points
${\sc a}_1,\ldots \, {\sc a}_{2m-1}$;
then the inscribed polygon of $4m$ distinct sides becomes not
only possible but {\it indeterminate}, its first corner being in
this case allowed to take {\it any position on the surface}.  For
example, if the two triangles
${\sc p}' {\sc p}_1' {\sc p_2}'$,
${\sc p}'' {\sc p}_1'' {\sc p_2}''$
be inscribed in a conic, so that the corresponding sides
${\sc p}' {\sc p}_1'$ and ${\sc p}'' {\sc p}_1''$
intersect each other in ${\sc a}_1$;
${\sc p}_1' {\sc p}_2'$ and ${\sc p}_1'' {\sc p}_2''$
in ${\sc a}_2$; and
${\sc p}_2' {\sc p}'$ and ${\sc p}_2'' {\sc p}''$
in ${\sc a}_3$; and if we take a fourth point~${\sc a}_4$ on the
right line ${\sc p}' {\sc p}''$, and conceive any surface of the
second order constructed so as to contain the given conic; then
{\it any point\/}~${\sc p}$, on this surface, is fit to be the
first corner of a plane or gauche {\it octagon},
${\sc p} \, {\sc p}_1 \, \ldots \, {\sc p}_7$,
inscribed in the surface, so that the first and fifth sides
${\sc p} {\sc p}_1$, ${\sc p}_4 {\sc p}_5$ shall intersect in
${\sc a}_1$; the second and sixth sides in ${\sc a}_2$; the third
and seventh sides in ${\sc a}_3$; and the fourth and eighth in
${\sc a}_4$.  And generally if $2m$ given points be points of
intersection of opposite sides of {\it any one\/} inscribed
polygon of $4m$ sides, the {\it same $2m$ points\/} are then fit
to be intersections of opposite sides of {\it infinitely many
other\/} inscribed polygons, plane or gauche, of $4m$ sides.  A
very elementary example is furnished by an inscribed plane
quadrilateral, of which the two points of meeting of opposite
sides are well known to be {\it conjugate}, relatively to the
conic or to the surface, and are adapted to be the points of
meeting of opposite sides of infinitely many other inscribed
quadrilaterals.

When {\it all the sides but one}, of an inscribed gauche polygon,
pass through given points, the {\it remaining side\/} may be said
{\it generally\/} to be {\it doubly tangent\/} to a real or
imaginary {\it surface of the fourth order}, which separates
itself into {\it two\/} real or imaginary {\it surfaces of the
second order}, having real or imaginary {\it double contact\/}
with the original surface of the second order, and with each
other.  If the original surface be an ellipsoid~$({\sc e})$, and
if the number of sides of the inscribed polygon,
${\sc p} {\sc p}_1,\ldots \, {\sc p}_{2m}$, be odd, $= 2m + 1$,
so that the number of fixed points
${\sc a}_1,\ldots \, {\sc a}_{2m}$
is even, $= 2m$, then the two surfaces enveloped by the last side
${\sc p}_{2m} {\sc p}$ are a {\it real inscribed
ellipsoid\/}~$({\sc e}')$, and a {\it real exscribed hyperboloid
of two sheets\/}~$({\sc e}'')$; and these three surfaces
$({\sc e})$~$({\sc e}')$~$({\sc e}'')$ touch each other at the
{\it two real points\/} ${\sc b}$,~${\sc b}'$, which are the
first corners of two inscribed polygons
${\sc b} {\sc b}_1,\ldots \, {\sc b}_{2m-1}$ and
${\sc b}' {\sc b}_1',\ldots \, {\sc b}_{2m-1}'$,
whose $2m$ sides pass respectively through the $2m$ given points
$({\sc a})$.  If these three surfaces of the second order be cut
by any three planes parallel to either of the two common tangent
planes at ${\sc b}$ and ${\sc b}'$, the sections are three
{\it similar and similarly placed ellipses\/}; thus ${\sc b}$ and
${\sc b}'$ are two of the four {\it umbilics\/} of the
ellipsoid~$({\sc e}')$, and also of the
hyperboloid~$({\sc e}'')$, when the original surface~${\sc e}$ is
a {\it sphere}.  The {\it closing chords\/}
${\sc p}_{2m} {\sc p}$ touch a series of real {\it curves\/}
$({\sc c}')$ on $({\sc e}')$, and {\it also\/} another series of
real curves $({\sc c}'')$ on $({\sc e}'')$, which curves are the
{\it ar\^{e}tes de rebroussement\/} of two series of
{\it developable surfaces}, $({\sc d}')$ and $({\sc d}'')$, into
which latter surfaces the closing chords arrange themselves; but
these two sets of developable surfaces are {\it not\/} generally
{\it rectangular\/} to each other, and consequently the closing
chords themselves are {\it not generally perpendicular to any one
common surface}.  However, when $({\sc e})$ is a sphere, the
developable surfaces cut it in two series of curves,
$({\sc f}')$,~$({\sc f}'')$, which everywhere cross each other at
right angles; and generally at any point~${\sc p}$ on
$({\sc e})$, the tangents to the two curves $({\sc f}')$ and
$({\sc f}'')$ are parallel to two conjugate semidiameters.

The {\it centres\/} of the three surfaces of the second order are
placed on {\it one straight line\/}; and every closing chord
${\sc p}_{2m} {\sc p}$ is {\it cut harmonically\/} at the points
where it touches the two surfaces $({\sc e}')$,~$({\sc e}'')$, or
the two curves $({\sc c}')$,~$({\sc c}'')$, which are the
{\it ar\^{e}tes\/} of the two developable surfaces
$({\sc d}')$,~$({\sc d}'')$, passing through that chord
${\sc p}_{2m} {\sc p}$.  In a certain class of {\it cases\/} the
three surfaces $({\sc e})$,~$({\sc e}')$,~$({\sc e}'')$ are all
of {\it revolution}, round one common axis; and when this
happens, the curves $({\sc c}')$,~$({\sc c}'')$,
$({\sc f}')$,~$({\sc f}'')$ are certain {\it spires\/} upon these
surfaces, having this {\it common character}, that for any one
such spire {\it equal rotations\/} round the axis give
{\it equal anharmonic ratios\/}: or that, more fully, if on a
spire~$({\sc c}')$, for example, there be taken two pairs of
points ${\sc c}_1'$,~${\sc c}_2'$ and
${\sc c}_3'$,~${\sc c}_4'$, and if these be projected on the
axis ${\sc b} {\sc b}'$ in points ${\sc g}_1'$,~${\sc g}_2'$ and
${\sc g}_3'$,~${\sc g}_4'$, then the rectangle
${\sc b} {\sc g}_1' \mathbin{.} {\sc g}_2' {\sc b}'$
will be to the rectangle
${\sc b} {\sc g}_2' \mathbin{.} {\sc g}_1' {\sc b}'$, as
${\sc b} {\sc g}_3' \mathbin{.} {\sc g}_4' {\sc b}'$, to
${\sc b} {\sc g}_4' \mathbin{.} {\sc g}_3' {\sc b}'$,
if the dihedral angle ${\sc c}_1' {\sc b} {\sc b}' {\sc c}_2'$
be equal to the dihedral angle
${\sc c}_3' {\sc b} {\sc b}' {\sc c}_4'$.
In another extensive class of cases the hyperboloid or two sheets
$({\sc e}'')$ reduces itself to a pair of planes, touching the
given ellipsoid~$({\sc e})$ in the points ${\sc b}$ and
${\sc b}'$; and then the prolongations of the closing chords,
${\sc p}_{2m} {\sc p}$, all meet the right line of intersection
of these two tangent planes: or the inscribed
ellipsoid~$({\sc e}')$ may reduce itself to the right line
${\sc b} {\sc b}'$, which is, in that case, crossed by all the
closing chords.  For example, if the first four sides of an
inscribed gauche pentagon pass respectively through four given
points, which are all in one common plane, then the fifth side of
the pentagon intersects a fixed right line in that plane.

An example of {\it imaginary envelopes\/} is suggested by the
problem of inscribing a gauche quadrilateral, hexagon, or polygon
of $2m$ sides in an ellipsoid, all the sides but the last being
obliged to pass through fixed points.  In this problem the
{\it last side\/} may be said to touch two imaginary surfaces of
the second order, which intersect each other in two real or
imaginary conics, situated in two real planes; and when these two
conics are real, they touch the original ellipsoid in two real
and common points, which are the two positions of the first
corner of an inscribed polygon, whose sides pass through the
$2m - 1$ fixed points.  Every rectilinear tangent to
{\it either\/} conic is a closing chord ${\sc p}_{2m-1} {\sc p}$;
but no position of that closing chord, which is not thus a
tangent to one or other of these conics, is intersected
{\it anywhere\/} by {\it any\/} infinitely near chord of the
system.  These results were illustrated by an example, in which
there were three given points; one conic was the known envelope
of the fourth side of a plane inscribed quadrilateral; and this
was found to be the {\it ellipse de gorge\/} of a certain
single-sheeted hyperboloid, a certain section of which
hyperboloid, by a plane perpendicular to the plane of the
ellipse, gave the {\it hyperbola\/} which was, in this example,
the {\it other\/} real conic, and was thus situated in a plane
{\it perpendicular\/} to the plane of the ellipse.  And to
illustrate the {\it imaginary\/} character of the
{\it enveloped surfaces}, or the general non-intersection (in
this example) of infinitely near positions of the closing chords
in space, {\it one\/} such chord was selected, and it was shown
that all the infinitely near chords, which made with {\it this\/}
chord equal and infinitesimal angles, were generatrices (of one
common system) of an infinitely thin and single-sheeted
hyperboloid.

Conceive that any rectilinear polygon of $n$ sides,
${\sc b} {\sc b}_1, \ldots \, {\sc b}_{n-1}$, has been
inscribed in any surface of the second order, and that $n$ points
${\sc a}_1,\ldots \, {\sc a}_n$ have been assumed on its $n$
sides,
${\sc b} {\sc b}_1,\ldots \, {\sc b}_{n-1} {\sc b}$.
Take then at pleasure any point~${\sc p}$ upon the same surface,
and draw the chords
${\sc p} {\sc a}_1 {\sc p}_1,\ldots \,
   {\sc p}_{n-1} {\sc a}_n {\sc p}_n$,
passing respectively through the $n$ points~(${\sc a}$).  Again
begin with ${\sc p}_n$, and draw, through the same $n$
points~$({\sc a})$, $n$ other successive chords,
${\sc p}_n {\sc a}_1 {\sc p}_{n+1},\ldots \,
    {\sc p}_{2n-1} {\sc a}_n {\sc p}_{2n}$.
Again, draw the $n$ chords,
${\sc p}_{2n} {\sc a}_1 {\sc p}_{2n+1},\ldots \,
   {\sc p}_{3n-1} {\sc a}_n {\sc p}_{3n}$.
Draw tangent planes at ${\sc p}_n$ and ${\sc p}_{2n}$, meeting
the two new chords ${\sc p} {\sc p}_{2n}$ and
${\sc p}_n {\sc p}_{3n}$ in points ${\sc r}$,~${\sc r}'$; and
draw any rectilinear tangent ${\sc b} {\sc c}$ at ${\sc b}$.
Then one or other of the two following theorems will hold good,
according as $n$ is an {\it odd\/} or {\it even\/} number.  When
$n$ is {\it odd}, the three points ${\sc b} {\sc r} {\sc r}'$
will be situated in one straight line.  When $n$ is {\it even},
the three pyramids which have ${\sc b} {\sc c}$ for a common
edge, and have for their edges respectively opposite thereto the
three chords
${\sc p} {\sc p}_{2n}$,~${\sc p}_{2n} {\sc p}_n$,~${\sc p}_n {\sc p}_{3n}$,
being divided respectively by the squares of those three chords,
and multiplied by the squares of the three respectively parallel
semidiameters of the surface, and being also taken with algebraic
signs which it is easy to determine, have their sum equal to
zero.  Both theorems conduct to a form of Poncelet's construction
(the present writer's knowledge of which is derived chiefly from
the valuable work on Conic Sections, by the Rev. George Salmon,
F.~T.~C.~D.), when applied to the problem of inscribing a polygon
in a plane conic: and the second theorem may easily be stated
generally under a {\it graphic\/} intesd of a {\it metric\/}
form.

The analysis by which these results, and others connected with
them, have been obtained, appears to the author to be
sufficiently simple, as least if regard be had to the novelty and
difficulty of some of the questions to which it has been thus
applied; but he conceives that it would occupy too large a space
in the Proceedings, if he were to give any account of it in
{\it them\/}: and he proposes, with the permission of the
Council, to publish his calculations as an appendage to his
Second Series of Researches respecting Quaternions, in the
Transactions of the Academy.  He would only further observe, on
the present occasion, that he has made, in these investigations,
a frequent use of expressions of the form
${\sc q} + \surd (-1) {\sc q}'$, where $\surd (-1)$ is the
{\it ordinary imaginary\/} of the older algebra, while ${\sc q}$
and ${\sc q}'$ are {\it two different quaternions}, of the kind
introduced by him into analysis in 1843, involving the {\it three
new imaginaries\/} $i$,~$j$,~$k$, for which the fundamental
formula,
$$i^2 = j^2 = k^2 = ijk = -1,$$
holds good.  (See the Proceedings of November 13th, 1843.)

And Sir W.~R. Hamilton thinks that the name
``{\sc biquaternion},'' which he has been for a considerable time
accustomed to apply, in his own researches, to an expression of
this form ${\sc q} + \surd (-1) {\sc q}'$, is a designation more
appropriate to such expressions than to the entirely different
(but very interesting) octonomials of Messrs.\ J.~T. Graves and
Arthur Cayley, to which {\it Octaves\/} the Rev.\ Mr.~Kirkman, in
his paper on {\it Pluquaternions}, has suggested (though with all
courtesy towards the present author), that the name of
{\it biquaternion\/} might be applied.

\bye