Supplement to an Essay on the Theory of Systems of Rays. By 

 WILLIAM R. HAMILTON, A. B., M. R. I. A., M. Ast. 

 Soc. Lond., Hon. M. Soc. Arts for Scotland, Andrews Pro- 

 fessor of Astronomy in the University of Dublin, and Royal 

 Astronomer of Ireland. 



Read April 26, 1830. 



INTRODUCTION. 



The present supplement contains some developments of a view of 

 Mathematical Optics, which was proposed by me in the foregoing 

 volume of the Transactions of this Academy. According to that 

 view, the geometrical properties of an optical system of rays, whether 

 straight or curved, whether ordinary or extraordinary, may be 

 deduced by analytic methods, from one fundamental formula, and one 

 characteristic function : the formula being an expression for the vari- 

 ation which the definite integral, called action, receives, when the 

 coordinates of its limits vary ; and the characteristic function being 

 this integral itself, considered as depending on those coordinates. 

 Although this view was stated, and the formula announced, in the 

 Jable of Contents prefixed to my preceding Memoir, yet the demon- 

 stration was not given in the part already published, except for the 



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