Royal Society. 227 



both of propagation and reflexion from some higher and more gene- 

 ral law, containing them both as particular cases : anticipations 

 which were singled out for special attention, in the Address deli- 

 vered by Sir W. Rowan Hamilton, on the occasion already referred 

 to. How little perhaps did Professor MacCullagh then know that 

 both of his own prophecies were destined to be so soon fulfilled, 

 and both by the powers of his own mighty and creative mind ! 



In the general case of total reflexion at the surface of a crystal, 

 he afterwards showed, by a most ingenious employment of imagi- 

 nary quantities, that the refraction was still double, and never more 

 than double ; and he showed that the directions of the refracted rays 

 remained always the same, whatever were the incidence, provided it 

 gave total reflexion. Again, as he had done for the case of ordinary 

 reflexion by means of his beautiful theorem of the polar plane, so 

 in the case of total reflexion he determined the two directions of 

 polarization, in a given incident jilane polarized wave, which would 

 give uniradial refracted rays, by means not of a polar plane, but of 

 a polar cylinder, which he succeeded in showing was the analogous 

 surface in the more difiicult cases. 



In the particular case of total reflexion at the surface of an ordi- 

 nary medium, the whole theory of total reflexion became exceedingly 

 simple, and that case is left by him completed. He showed that 

 whatever were the incidence, the refracted wave was always perpen- 

 dicular to the intersection of the plane of incidence, and of the sur- 

 face of the crystal ; he showed that the axes of the elhpse of vibra- 

 tion, projected on the plane of incidence, were parallel and perpen- 

 dicular to that line; 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 at once the limiting angle of total reflexion : and, 

 finally, he got out the two empirical formulse of Fresnel, for the ac- 

 celeration of the refracted i)hase over the incident, and the subse- 

 quent equal acceleration of the reflected phase over the refracted ; 

 the one for the case of the incident light polarized in the plane of 

 incidence, and the other for the same polarized in the perpendicular 

 plane. For all cases, whether of propagation or of reflexion, ordi- 

 nary or total, the whole theory, as he has left it, is analytically 

 complete : but the geometrical interpretations in the general case of 

 total reflexion at the surface of a crystal present very great difficult 

 ties. Many of these his acute intellect had with great labour sur- 

 mounted ; he had been working hard at the subject for the last four 

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

 commenced a new paper for the Irish Transactions, embodying the 

 results of his latest investigations. 'I'hc heading of this paper, which 

 I have already mentioned, remains in his own handwriting. It is 

 Ijelieved that several of his manuscripts on other subjects are in the 

 possession of his family, although it was not his custom to preserve 

 many written jjupers. 



Q2 



