Hamilton's Papers on Geometrical Optics

William Rowan Hamilton's first years of mathematical research were devoted to the study of geometrical optics. One of the notable results of that research was Hamilton's prediction, in 1832, of the phenomenon of conical refraction at the surfaces of biaxal crystals.

Listed below are William Rowan Hamilton's papers on Optics: the Theory of Systems of Rays and its three supplements, his British Association lectures and an expository article on optics, and other short papers on Optics.

These papers, with the exception of On some Quaternion Equations connected with Fresnel's wave surface, are included in The Mathematical Papers of Sir William Rowan Hamilton, Volume I: Geometrical Optics, edited for the Royal Irish Academy by A. W. Conway and J. L. Synge, and published by Cambridge University Press in 1931. That volume also contains some previously unpublished manuscripts, and extracts from Hamilton's notebooks; these include the first part of On Caustics, submitted by Hamilton to the Royal Irish Academy at the age of nineteen, but not accepted for publication by the Academy, and Part Second of the Theory of Systems of Rays.

• 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

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

The Theory of Systems of Rays was Hamilton's first published mathematical paper. A summary of the objectives of this paper, entitled 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. The Theory of Systems of Rays is concerned with geometrical optics. Although the paper had three parts, only the first part of the paper was printed in Hamilton's lifetime; this dealt with the properties of systems of rays under reflection. The paper includes a proof of the theorem that states that if light rays are emitted from a point, or perpendicular to some surface, and are reflected one or more times, then the final rays of the system are perpendicular to a series of surfaces (the `Theorem of Malus'). It also discusses in detail the caustic curves and surfaces when light rays are reflected from flat or curved mirrors, and concludes with a detailed discussion of the density of the light near the caustic surfaces. A key feature of Hamilton's approach is the use of a characteristic function, V, with the property that, in an isotropic medium, the light rays are perpendicular to the level surfaces of V. This paper had been written whilst Hamilton was an undergraduate at Trinity College, Dublin, and was presented to the Royal Irish Academy on April 23, 1827, and ordered to be printed. (Hamilton was still an undergraduate at the time.) He had earlier submitted a paper, On Caustics, which was read on December 13, 1824, but that paper had not been accepted for publication in the form in which it was presented.

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

In this Supplement, Hamilton extends his theory to systems in which the light is refracted. He begins by deriving the `Euler-Lagrange' equations for the light rays from the `Principle of Least Action' (which corresponds to or generalizes Fermat's `Principle of Least Time'). He then introduces an auxiliary function, W, for those homogeneous systems in which the refractive index does not depend locally on position. This auxiliary function W is a function of the direction cosines of the final ray. Hamilton discusses the relationships between the characteristic function V, the auxiliary function W, and their partial derivatives of the first and second orders, and generalizes an number of the results of the Theory of Systems of Rays to refracted systems.

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

In the Second Supplement, Hamilton integrates the partial differential equations satisfied by his characteristic function, using expansions in power series and a theorem of Laplace. He applies his results to the study of plane systems and systems of revolution, and derives a formula due to John Herschel for the aberration of a thin lens.

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

The Third Supplement begins with the presentation of Hamilton's theory of the characteristic and auxiliary functions, in its most general and refined form. (An account of the basic theory of the characteristic function is also to be found in the expository paper On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function.) The characteristic function of a ray is considered as a function of the coordinates of the initial and final coordinates of the ray, and satisfies a partial differential equation of the first order with respect to the final coordinates, and a corresponding partial differential equation of the first order with respect to the initial coordinates. (These partial differential equations are analogous to the `Hamilton-Jacobi equation' of dynamics.) Hamilton investigates the relations amongst the partial differential coefficients, of the first and second orders, of the characteristic and auxiliary functions. He investigates the behaviour of these functions under change of coordinates. He investigates the effect of aberration, characterising its effects by means of certain `elements of arrangement'. He discusses the relationship between his theory of the characteristic function and the undulatory theory of light, and in particular applies his theory to the study of the propagation of light in biaxal crystals such as arragonite. He predicts that such crystals will exhibit the phenomenon of conical refraction at the surfaces of biaxal crystals.

On a View of Mathematical Optics (British Association Report, Oxford 1832, 2nd edition published 1835, pp. 545-547.)

Hamilton gave a very brief outline of his approach to the study of optics at the meeting of the British Association for the Advancement of Science at Oxford in 1832.

On some Results of the View of a Characteristic Function in Optics (British Association Report, Cambridge 1833, published 1834, pp. 360-370.)

At the meeting of the British Association for the Advancement of Science at Cambridge in 1833, Hamilton described the fundamental principles of his approach to the study of optics via the use of the characteristic function V. He introduces also the auxiliary function T, and outlines the method by which this function can be employed in the study of optical instruments that are symmetric under rotations about some axis of revolution. He states some theorems concerning the nature of aberrations for such an optical instrument. Hamilton's account concludes with a discussion of conical refraction at the surfaces of biaxal crystals.

On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function (Dublin University Review, 1833, pp. 795-826.)

This expository article falls into two quite distinct halves. The first half is an account of the history of the study of optics, and is non-mathematical in character. The second half is an exposition of the basic principles of Hamilton's approach to the study of optics through the use of a characteristic function. There is no discussion of conical refraction. The article concludes with a brief indication of how this approach to optics can be adapted to the study of dynamical problems.

On the Effect of Aberration in prismatic Interference (Philosophical Magazine, 3rd series, volume 2 (1833), pp. 191-194.)

This paper concerns an experiment on the refraction of light by a prism, conducted by Mr. Potter, and described by him in the Philosophical Magazine. Potter had claimed that the results of his experiment were inconsistent with the wave theory of light. Hamilton here disputes a part of Mr. Potter's analysis of the problem.

On the undulatory Time of Passage of Light through a Prism (Philosophical Magazine, 3rd series, volume 2 (1833), pp. 284-287.)

Hamilton presents in more detail the mathematical analysis he had employed in relation to Mr. Potter's experiment.

Note on Mr. Potter's Reply (Philosophical Magazine, 3rd series, volume 2 (1833), p. 371.)

This is a very brief response to Mr. Potter's reply regarding Hamilton's analysis of his experiment.

Remarques de M. Hamilton, Directeur de l'Observatoire de Dublin, sur un Mémoire de M. Plana inséré dans le Tome VII de la Correspondance Math. (Extrait d'une Lettre) (Correspondance Mathématique et Physique, tome 8 (1834), pp. 37-40.)

This extract from a letter, published in Quetelet's Correspondance Mathématique et Physique concerns a claim by Plana, of Turin, that Malus had been right in supposing that light rays emitted from a point were in general no longer orthogonal to a series of surfaces after two reflections. Hamilton refutes Plana's claim, by showing how this orthogonality can in fact deduced, both from Plana's approach to the problem, and by his own methods.

On the Focal Lengths and Aberrations of a thin Lens of Uniaxal Crystal, bounded by Surfaces which are of Revolution about its Axis (Philosophical Magazine, 3rd series, volume 19 (1841), pp. 289-294.)

Hamilton describes some applications of his optical method, based on research which he had undertaken a decade earlier.

On a Mode of deducing the Equation of Fresnel's Wave (Philosophical Magazine, 3rd series, volume 19 (1841), pp. 381-383.)

Hamilton describes a method for obtaining the equation of the wave surface of light in biaxal crystals proposed by Fresnel, based on a report he had prepared a decade earlier when refereeing a paper by Mac Cullagh on the subject. In a footnote, Hamilton notes that Fresnel himself had published a much simpler approach, of which he had not previously been aware.

On some Quaternion Equations connected with Fresnel's wave surface (Proceedings of the Royal Irish Academy, volume 7 (1862) pp. 122-124, 163.)

Hamilton applies quaternion methods to derive the equation for Fresnel's wave surface.

Links:

• Sir William Rowan Hamilton (1805-1865)
• History of Mathematics