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tion, and figure, the two ellipses of vibration corresponding to any 

 given incident wave polarized in any azimuth, and incident at any 

 angle greater than the angle of total reflection. 



"In the particular case of total reflection at the surface of an 

 ordinary medium, the whole theory of total reflection became ex- 

 ceedingly simple, and that case was left by him completed. He 

 showed that whatever were the incidence, the refracted wave was 

 always perpendicular to the intersection of the planes of incidence, 

 and of the surface of the crystal. He showed that the axes of the 

 ellipse of vibration projected on the plane of incidence, were pa- 

 rallel and perpendicular to that line, and the duration of vibration 

 the same as in the general case. He gave a beautiful construction, 

 by means of an equilateral hyperbola, touching with its vertex the 

 section of the index sphere at the point where it intersects the same 

 right line, for determining the velocity of the refracted wave, and 

 the ratio of the axes of its elliptic vibrations corresponding to any 

 given incidence. He determined the limiting angle of total reflec- 

 tion. And finally, he demonstrated the two empirical formul6e of 

 Fresnel, for the acceleration of the refracted phase over the inci- 

 dent, and the subsequent equal acceleration of the reflected phase 

 over the refracted ; the one for the case of the incident light polar- 

 ized in the plane of incidence, and the other for the same polarized 

 in the perpendicular plane. 



" For all cases, whether of propagation or of reflection, ordi- 

 nary or total, the whole theory, as he has left it to us, is analyti- 

 cally complete; but \he geometrical interpretations, in the general 

 case of total reflection at the surface of a crystal, present very great 

 difl[iculties. Many of these his acute intellect had, with great labour, 

 surmounted ; he had been working hard at the subject for the last 

 six weeks of his life, and with so much success, that he had actually 

 commenced a new paper for the Transactions of the Academy, era- 

 bodying the results of his latest investigations." 



But the services for which this Academy owes to Professor 

 Mao Cullagh a debt of lasting gratitude, are not confined to his 

 scientific labours and discoveries. His enlightened patriotism led 

 him, even at the risk of diminishing his own fame, or at least of 

 retarding his progress to celebrity, to publish his researches in Ire- 



