On the double Refraction of Light in a crystallized Medium, according 

 to the Principles of Fresnel. By JAMES Mac CULLAGH, Esq. 

 Communicated by the Rev. Dr. Sadleir, S. F. T. C, D,, Vice 

 President. 



Read June 21st, 1830. 



The mathematical difficulties under which the beautiful and inter- 

 esting theory of Fresnel has hitherto laboured, are well known, and 

 have been regarded as almost insuperable. He tells us, in his 

 Memoir, (see the Memoirs of the Royal Academy of Sciences of 

 Paris, torn-, vii. p. 136), that the calculations, by which he assured 

 himself of the truth of his construction for finding the surface of the 

 wave, were so tedious and embarrassing, that he was obliged to omit 

 them altogether. A direct demonstration has since been supplied by 

 M. Ampere, (Annales de Chimie et de Physique, tom. xxxix. 

 p. 1 13) ; but his solution is excessively complicated and difficult. 



Judging from the simplicity and elegance of the results, that there 

 must be some simple method of arriving at them, I have been led to 

 consider the subject with the attention which it merits, and have 

 succeeded in discovering a method by which the whole may be ex- 

 plained with that simplicity which is characteristic of every theory 

 that is founded in nature. 



In the following paper I shall give a brief view of this method, 

 sufficient to enable those who are acquainted with the mechanical 

 principles laid down in the original memoir of Fresnel, to trace, at a 

 glance, the connexion between the several parts of his theory. For 

 this purpose it will be convenient to premise the following geometri- 

 cal Lemmas. Lemma 1 . If a, h, c, be the Semiaxes of an Ellip- 



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