Hamilton, in his first years of mathematical research, had constructed an extensive theory of mathematical optics. He published a series of long papers in the Transactions of the Royal Irish Academy on this Theory of Systems of Rays, which associated to any optical system a certain characteristic function whose properties determined the behaviour of the system. Hamilton gave an account of the basic principles of his theory in an expository article On a General method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function, published in the Dublin University Review and Quarterly Magazine in 1833. (This article does not however contain an account of the theory of conical refraction.)
Hamilton could apply this theory to determine the relationship between the angles of incidence and refraction for rays of light entering or leaving a medium such as a biaxal crystal, in which the speed of light in the medium is dependent on the direction of the ray. Fresnel had developed a theory which described the shape of a wave propagating from a point source in a biaxal crystal, and thus determined the dependence of the speed of light in a biaxal crystal on the direction of the ray. On applying his theory with the equations for the wave surface in a biaxal crystal proposed by Fresnel, Hamilton discovered that, according to his theory, if a single ray of light entered or emerged from a biaxal crystal in a certain direction, then that ray of light would be refracted into a hollow cone of rays. This is the phenomenon known as conical refraction.
In the case of external conical refraction a single ray of light leaving a biaxal crystal would be refracted into an emergent cone of rays. In the case of internal conical refraction a single ray of light entering a biaxal crystal would be refracted into a cone of rays within the crystal; if the faces of the crystal are parallel then that cone would emerge as a hollow cylinder of rays.
Hamilton described his prediction when he presented the concluding part of his Third Supplement to an Essay on the Theory of Systems of Rays to the Royal Irish Academy on the 22nd of October 1832. He asked Humphrey Lloyd, the Professor of Natural and Experimental Philosophy at Trinity College, Dublin, to try to verify the prediction experimentally. Lloyd had difficulty in obtaining a suitable crystal, but, after obtaining a good specimen of arragonite, succeeded in observing external conical refraction on the 14th of December. He was subsequently able to observe internal conical refraction. Humphrey Lloyd gave an account of his experiments in two papers published in the London and Edinburgh Philosophical Magazine in February and March, 1833.
At the meeting of the British Association for the Advancement of Science at Cambridge in 1833, both Hamilton and Lloyd gave accounts of the discovery of conical refraction. Hamilton's account is contained in his report On some Results of the View of a Characteristic Function in Optics. Lloyd's report is simply entitled On Conical Refraction. The mathematical theory underlying Hamilton's prediction is to be found in the Third Supplement to an Essay on the Theory of Systems of Rays.
Hamilton's biographer, Robert Perceval Graves, gave an account of the discovery of conical refraction in Chapter XIII in the first volume of his Life of Sir William Rowan Hamilton.