By William R. Hamilton

Hamilton's first years of mathematical research were dedicated to
the creation of a theory of geometric optics, whose fundamental
principles were based on the use of a *characteristic
function*. This theory was expounded and developed in four
substantial papers:---

- Theory of Systems of Rays
- Supplement to an Essay on the Theory of Systems of Rays
- Second Supplement to an Essay on the Theory of Systems of Rays
- Third Supplement to an Essay on the Theory of Systems of Rays

Hamilton's first published paper was *Part First* of

which appeared in the Transactions of the Royal Irish Academy, volume 15 (1828), pp. 69-174.Theory of Systems of Rays.

The *Theory of Systems of Rays* (*Part First*) is
available here in the following formats:

- Plain TeX (347 kB)
- DVI (528 kB)
- PostScript (898 kB)
- PDF (1598 kB)

This paper was written whilst Hamilton was an undergraduate at
Trinity College, Dublin. (He had earlier submitted for
publication a paper entitled *On Caustics*, but
this was not accepted by the Royal
Irish Academy in the form in which it was presented: the referees
felt that the methods by which some of the conclusions had been
arrived at needed to be explained in more detail.)

The *Theory of Systems of Rays* was originally intended to
consist of three parts, and the analytical table of contents
describes all three parts. However only *Part First* was
published in Hamilton's lifetime. (Part Second appeared
posthumously in *The Mathematical Papers of Sir William Rowan
Hamilton, Volume I: Geometrical Optics*, edited by A. W.
Conway and J. L. Synge, and published by Cambridge University
Press in 1931.)

An Account of a Theory of Systems of Rays, written by Hamilton himself, is to be found in chapter VI of the Life of Sir William Rowan Hamilton by R. P. Graves.

Hamilton subsequently published three supplements to this essay.

Hamilton developed his approach to geometrical optics in the

which appeared in the Transactions of the Royal Irish Academy, volume 16, part 1 (1830), pp. 1-61.Supplement to an Essay on the Theory of Systems of Rays.

The *Supplement to an Essay on the Theory of Systems of
Rays* is available here in the following formats:

- Plain TeX (167 kB)
- DVI (240 kB)
- PostScript (464 kB)
- PDF (762 kB)

Hamilton subsequently published two further supplements to this essay.

The methods of the *Supplement to an Essay on the Theory of
Systems of Rays* were further developed in the

which appeared in the Transactions of the Royal Irish Academy, volume 16, part 2 (1831), pp. 93-125.Second Supplement to an Essay on the Theory of Systems of Rays.

The *Second Supplement to an Essay on the Theory of Systems of
Rays* is available here in the following formats:

- Plain TeX (79 kB)
- DVI (122 kB)
- PostScript (288 kB)
- PDF (488 kB)

The *Second Supplement* was followed by a further
supplement.

The most developed presentation of Hamilton's approach to geometrical optics is to be found in the

which appeared in the Transactions of the Royal Irish Academy, volume 17, (1837), pp. 1-144.Third Supplement to an Essay on the Theory of Systems of Rays.

The *Third Supplement to an Essay on the Theory of Systems of
Rays* is available here in the following formats:

- Plain TeX (606 kB)
- DVI (861 kB)
- PostScript (1437 kB)
- PDF (2463 kB)

The *Third Supplement* was completed in 1832. It
commences with an exposition of the theory of Hamilton's
characteristic function *V* (considered as a function of the
coordinates both the initial and final point of the light ray)
and the auxiliary functions *W* (first introduced in the
*Supplement to an Essay on the
Theory of Systems of Rays*) and *T*. This is
followed by a detailed discussion of aberration. The paper
concludes with a discussion of the relationship between
Hamilton's theory of the characteristic function and the wave
theory of light. The theory is applied to the refraction of
light in biaxal crystals (such as arragonite), further developing
the theory of refraction in such crystals formulated by Fresnel,
and Hamilton predicts the occurrence of the phenomenon of
*conical refraction*,
a prediction that was subsequently verified experimentally by
Humphrey Lloyd.

Hamilton published in 1833 an expository article, entitled
On an General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function,
in the Dublin University Review. This article
provides an introduction to the theory of the characteristic
function with which the *Third Supplement* commences.

Links:

D.R. Wilkins(

School of Mathematics

Trinity College, Dublin