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

 By WILLIAM R. HAMILTON, Royal Astronomer of Ireland, 

 Sfc. ^c. 



Read October 25, 1830. 



INTRODUCTION. 



T.HE present Supplement contains the integration of some partial 

 differential equations, to which I have been conducted by the view 

 of mathematical optics, proposed in my former memoirs. Accord- 

 ing to that view, the geometrical properties of an optical system of 

 rays, may be deduced by analytic methods, from the form of one 

 characteristic function ; of which the partial differential coeffi- 

 cients of the first order, taken with respect to the three rectangular 

 coordinates of any proposed point of the system, are, in the case of 

 ordinary light, equal to the index of refraction of the medium, mul- 

 tiplied by the cosines of the angles which the ray passing through 

 the point makes with the axes of coordinates: and as these cosines 

 are connected by the known relation that the sum of their squares is 



VOL. XVI. P 



