31 



III. On the Laws of Crystalline Reflexion and Refraction. By James 

 Mac Cullagh, Fellow of Trinity College, Dublin. 



Read 9th January, 1837. 



W HEN a ray of light, which has been polarised in a given plane, suffers reflexion 

 and refraction at the surface of a transparent medium, the rays Into which it Is di- 

 vided are found to be polarised in certain other planes ; and it becomes a question 

 to determine the positions of these planes, as well as the relative Intensities of the 

 different rays ; or, in theoretical language, to find the direction and magnitude of 

 the reflected and refracted vibrations, supposing those of the Incident vibration 

 to be given. The transparent medium may be either a singly refracting sub- 

 stance, such as glass, or a doubly refracting crystal like Iceland spar. "When the 

 medium Is of the first kind, the problem is comparatively simple, being, in fact, 

 nothing more than a particular case of the problem which we have to consider when 

 the medium is supposed to be of the second kind. In the progress of knowledge 

 it was natural that the simpler question should be first attended to ; and accord- 

 ingly Fresnel, during his brief and brilliant career, found time to solve it. But 

 the general problem, relative to doubly refracting media, had not been attempted 

 by any one, when, in the year 1834, my thoughts were turned to the subject. I 

 then recollected a conclusion to which I had been led some years before, and which, 

 on this occasion, proved of essential service to me. Being fond of geometrical con- 

 structions, I amused myself, when T first became acquainted with Fresnel's theo- 

 ries, by throwing his algebraical expressions, whenever I could. Into a geometrical 

 form ; and treating In this way the well-known formulae In which he has embodied 

 his solution of the problem just alluded to, I obtained a remarkable result, which 

 gave me the first view of the principle that I have since employed under the name 

 of the principle of the equivalence of vibrations. In order to state this result 



