In 1833 Hamilton published his Third Supplement to an Essay on the Theory of Systems of Rays in the Transactions of the Royal Irish Academy. This paper is included in vol. 17 of the Transactions, part 1 (1837), pp. 1--144. This paper concludes with several articles describing the mathematical theory underlying Hamilton's prediction of the phenomenon of conical refraction.
An extract has been made which includes the following sections of the paper: the introduction, the table of contents, articles 1 to 3 (properties of the characteristic function and its derivatives), and articles 26 to 31 (connection with the undulatory theory of light, refraction in biaxal crystals, conical refraction).
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Some aspects of the theory of mathematical optics developed by Hamilton are also set out in the papers On some Results of the View of a Characteristic Function in Optics and On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function.