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% School of Mathematics, Trinity College, Dublin 2, Ireland
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\centerline{\Largebf ON ``GAUCHE'' CURVES OF THE THIRD DEGREE}
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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, 8 (1864),
pp.\ 331--334.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 2000}
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\centerline{{\sc On ``Gauche'' Curves of the Third Degree.}}
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\centerline{\largerm Sir William Rowan Hamilton}
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\centerline{[Communicated April 27th, 1863.]}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~viii (1864), pp.~331--334.]}
\bigbreak
The following letter, addressed to the President, by
Sir {\sc W.~R.~Hamilton}, was read:---
\bigbreak
\line{\hfil {\it Observatory}, 27 April 1863.}
\nobreak\bigskip
{\sc My dear Mr.~President},---I have been wishing for your
permission to report, through you, to the Royal Irish Academy,
some of the results to which I have lately arrived, while
extending the applications of Quaternions, in connexion with my
forthcoming {\it Elements}.
\bigbreak
I.
One set of such results relates to those {\it gauche curves of
the third degree}, which appear to have been first discovered,
described, and to some extent applied, by Professor M\"{o}bius,
in the {\it Barycentric Calculus\/} (1827), and afterwards
independently by M.~Chasles, in a Note to his {\it Aper\c{c}u
Historique sur l'Origine et le Developpement des M\'{e}thodes en
G\'{e}om\'{e}trie\/} (1837); and for which our countryman,
Dr.~Salmon, who has done so much for the Classification of Curves
in Space, has proposed the short but expressive name of
{\it Twisted Cubics}.
\bigbreak
II.
A particular curve of that class presented itself to me in an
investigation more than ten years ago, and some account of it was
given in my {\it Lectures\/}, and (I think) to the Academy also,
in connexion with the problem of {\it Inscription of Polygons\/}
in surfaces of the second order. I gave its vector equation,
which was short, but was not sufficiently {\it general}, to
represent {\it all\/} curves in space of the third degree: nor
had I, at the time, any aim at such representation. But I have
lately perceived, and printed (in the {\it Elements\/}), the
strikingly simple, and yet complete equation,
$${\rm V} \alpha \rho + {\rm V} \rho \phi \rho = 0,$$
which represents {\it all twisted cubics}, if only a point of the
curve be taken, for convenience, as the origin: $\phi \rho$
denoting that {\it linear and vector function\/} of a vector,
which has formed the subject of many former studies of mine, and
$\alpha$ being a constant vector, while $\rho$ is a variable one.
\bigbreak
III.
It is known that a twisted cubic can in general be so chosen, as
to pass through {\it any six points of space}. It is therefore
natural to inquire, what is the {\it Osculating Twisted Cubic\/}
to a given curve of double curvature, or the one which has, at
any given place, a {\it six-point contact\/} with the curve. Yet
I have not hitherto been able to learn, from any book or friend,
that even the {\it conception\/} of the problem of the
determination of {\it such\/} an osculatrix, had occurred to any
one before me. But it presented itself naturally to me lately,
in the course of writing out a section on the application of
quaternions to curves; and I conceive that I have completely
resolved it, in {\it three\/} distinct ways, of which {\it two\/}
seem to admit of being geometrically described, so as to be
understood without diagrams or calculation.
\bigskip
IV.
It is known that the {\it cone of chords\/} of a twisted cubic,
having its vertex at any one point of that curve, is a {\it cone
of the second order}, or what Dr.~Salmon calls briefly a
{\it quadric cone}. If, then, a point~${\sc p}$ of a {\it given
curve\/} in space be made the vertex of a cone of chords of
{\it that curve}, the quadric cone which has its vertex at
${\sc p}$, and has {\it five-side contact\/} with {\it that
cone}, must {\it contain\/} the osculating cubic sought. I have
accordingly determined, by my own methods, the {\it cone\/} which
is thus {\it one locus\/} for the cubic: and may mention that I
find {\it fifth differentials\/} to enter into its equation,
{\it only\/} through the {\it second differential\/} of the
{\it second curvature}, of the given curve in space.
{\it This\/} may perhaps have not been previously perceived,
although I am aware that Mr.~Cayley and Dr.~Salmon, and probably
others, have investigated the problem of {\it five-point
contact\/} of a {\it plane conic\/} with a plane curve.
\bigbreak
V.
It is known also that {\it three quadric cylinders\/} can be
described, having their generating lines parallel to the three
(real or imaginary) {\it asymptotes\/} of a twisted cubic, and
wholly {\it containing\/} that gauche curve. My {\it first
method}, then, consisted in seeking the (necessarily real)
{\it direction\/} of {\it one\/} such {\it asymptote}, for the
purpose of determining a {\it cylinder\/} which, as a {\it second
locus}, should contain the osculating cubic sought. And I found
a {\it cubic cone}, as the locus for the generating line (or
edge) of such a cylinder, through the given point~${\sc p}$ of
osculation: and proved that of the {\it six right lines}, common
to the quadric and the cubic cones, {\it three\/} were
{\it absorbed\/} in the {\it tangent\/} to the given curve at
${\sc p}$.
\bigbreak
VI.
In fact, I found that this tangent, say ${\sc p} {\sc t}$, was a
{\it nodal side\/} (or ray) of the cubic cone; and that
{\it one\/} of the {\it two tangent planes\/} to that cone, along
that side, was the {\it osculating plane\/} to the curve, which
plane also touched the quadric cone along that {\it common
side\/}: while the same side was to be {\it counted\/} a
{\it third time}, as being a line of {\it intersection}, namely,
of the quadric cone with the {\it second branch\/} of the cubic
cone, the tangent plane to which branch was found to cut the
first branch, or the quadric cone, or the osculating plane to the
curve, at an angle of which the trigonometric {\it cotangent\/}
was equal to {\it half the differential of the radius of second
curvature, divided by the differential of the arc\/} of the same
given curve.
\bigbreak
VII.
It might then have been thus expected that a {\it cubic
equation\/} could be assigned, of an algebraical {\it form}, but
involving fifth differentials in its {\it coefficients}, which
should determine the {\it three planes}, tangential to the curve,
which are parallel to the three asymptotes of the sought twisted
cubic: and then, with the help of what had been previously done,
should assign the {\it three quadric cylinders\/} which wholly
{\it contain\/} that cubic.
\bigbreak
VIII.
Accordingly, I succeeded, by quaternions, in forming such a cubic
equation, for {\it curves in space\/} generally: and its
correctness was tested, by application to the case of the
{\it helix}, the fact of the {\it six-point contact\/} of my
osculating cubic with which well-known curve admitted of a very
easy and elementary verification. I had the honour of
communicating an outline of my results, so far, to Dr.~Hart, a
few weeks ago, with a permission, or rather a request, which was
acted on, that he should submit them to the inspection of
Dr.~Salmon.
\bigbreak
IX.
Such, then, may be said briefly to have been my {\it first
general method\/} of resolving this new problem, of the
determination of the twisted cubic which {\it osculates}, at a
given point, to a given curve of double curvature. Of my
{\it second method\/} it may be sufficient here to say, that it
was suggested by a recollection of the expressions given by
Professor M\"{o}bius, and led again to a {\it cubic equation},
but this time for the determination of a {\it coefficient}, in
a development of a comparatively {\it algebraical kind}. For the
moment I only add, that the {\it second method\/} of solution,
above indicated, bore also the test of verification by the
{\it helix\/}; and gave me generally {\it fractional
expressions\/} for the co-ordinates of the osculating twisted
cubic, which admitted, in the case of the helix, of elementary
verifications.
\bigbreak
X.
Of my {\it third general method}, it may be sufficient at this
stage of my letter to you to say, that it consists in assigning
the {\it locus of the vertices\/} of all the {\it quadric cones},
which have {\it six-point contact\/} with a given curve in space,
at a given point thereof. I find this locus to be a {\it ruled
cubic surface}, on which the tangent~${\sc p} {\sc t}$ to the
curve is a {\it singular line}, counting as a {\it double line\/}
in the intersection of the surface with any plane drawn
{\it through it\/}; and such that if the same surface by cut by a
plane drawn {\it across it}, the {\it plane cubic\/} which is the
section has generally a {\it node}, at the point where the plane
crosses that line: although this node degenerates into a
{\it cusp}, when the cutting plane passes through the
point~${\sc p}$ itself.
\bigbreak
XI.
And I find, what perhaps is a new {\it sort\/} of result in these
questions, that the {\it intersection\/} of this {\it new cubic
surface\/} with the former {\it quadric cone}, consists only of
the {\it right line\/}~${\sc p} {\sc t}$ itself, and of the
{\it osculating twisted cubic\/} to the proposed curve in space.
\bigbreak
XII.
These are only {\it specimens\/} of {\it one set\/} (as above
hinted) of recent results obtained through quaternions; but at
least they may serve to mark, in some small degree, the respect
and affection, to the Academy, and to yourself, with which I
remain,
\nobreak\bigskip
\line{\hfil\vbox{\halign{#\hfil\cr
My dear Mr.~President,\cr
\quad Faithfully yours,\cr
\quad\quad {\sc William Rowan Hamilton}\cr}}}
\nobreak\bigskip
{\it The Very Rev.~Charles Graves, D.~D., P.R.I.A.,\par
\quad Dean of the Chapel Royal, \&c.}
\bye