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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, 1999.

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\centerline{\Largebf ON A VIEW OF MATHEMATICAL OPTICS}

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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, Oxford 1832,
   pp.\ 545--547.)}

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

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\centerline{\largerm 1999}

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{\largeit\noindent
On a View of Mathematical Optics.  By\/}
{\largesc William R. Hamilton,}
{\largeit Royal Astronomer of Ireland, \&c.}

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\centerline{[{\it Report of the First and Second Meetings of the
British Association for the Advancement of}}
\centerline{{\it Science; at York in 1831, and at Oxford in 1832.}
(John Murray, London, 1835), 545--547.]}

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The Memoirs on Systems of Rays, which have been presented by me
to the Royal Irish Academy, and of which some have been published
in the XVth and XVIth volumes of the {\it Transactions} of that
Academy, contain a view of mathematical optics, which appears to
me to be analogous to the view taken by Descartes of algebraical
geometry, and likely to lead those who shall adopt it to
analogous changes of method.  It has been thought desirable, by
the Mathematical Committee of the British Association for the
Advancement of Science, that a short statement of this view of
optics should be given in the forthcoming publication of that
body.  Such a statement, therefore, I shall now offer, as briefly
as I can; endeavouring only to communicate the view itself, and
abstaining from giving any account of the results to which it has
conducted me.

The {\it general problem} that I have proposed to myself in
optics, is to {\it investigate the mathematical consequences of
the law of least action\/}: a general law of vision, in which are
included, as it is well known, all the particular conditions of
reflexion and refraction, gradual and sudden, ordinary and
extraordinary.  And the {\it central idea} from which my whole
{\it method} flows, is the idea of {\it one radical or
characteristic relation for each optical system of rays}, that
is, for each combination of straight or bent, or curved paths,
along which light is supposed to be propagated according to the
law of least action.  This characteristic relation, being
different for different systems, and being such that the
mathematical properties of the system can all be deduced from it,
in the same manner as the method invented by Descartes for the
algebraical solution of geometrical problems, flows all from the
central idea of one radical relation, for each plane curve, or
curved surface, in the form of which relation are included all
the properties of the curve or the surface.  In the radical
relation thus contemplated by Descartes, in his view of
algebraical geometry, the related things are elements of position
of a variable point which has for locus a curve or a surface; and
the number of these related elements is either two or three.  In
the relation contemplated by me, in my view of algebraical
optics, the related things are, in general, in number, eight: of
which, six are elements of position of two variable points of
space, considered as visually connected; the seventh is an index
of colour; and the eighth, which I call the
{\sc characteristic function},---because I find that in the
manner of its dependence on the seven foregoing are involved all
the properties of the system,---is the {\it action} between the
two variable points; the word {\it action} being used here, in
the same sense as in that known law of vision which has been
already mentioned.  I have assigned, for the variation of this
characteristic function, corresponding to any infinitesimal
variations in the positions on which it depends, a fundamental
formula; and I consider as {\it reducible to the study of this
one characteristic function, by the means of this one fundamental
formula, all the problems of mathematical optics}, respecting all
imaginable combinations of mirrors, lenses, crystals and
atmospheres.  And though, among these problems of mathematical
optics, it is not here intended to include investigations
respecting the {\it ph{\ae}nomena of interference}, yet it is
easy to perceive, from the nature of the quantity which I have
called the characteristic function, and which in the hypothesis
of undulations is {\it the time of propagation of light from one
variable point to another}, that the study of this function must
be useful in such investigations also.  My own researches,
however, have been hitherto chiefly directed to the consequences
of the law of least action, and to the properties of optical
systems, and systems of rays in general.  And having stated, in
the foregoing remarks, the {\it view} that has guided these
researches, I must refer, for the {\it results}, to the volumes
already mentioned, of the Royal Irish Academy, and to the XVIIth
volume, not yet published, in which a third supplement to my
Essay on the Theory of Systems of Rays has been ordered by the
Academy to be printed.

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