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polarization, as Fresnel had supposed, were found, on the contrary, 

 to be parallel to that plane, as Mac Cullagh himself had sup- 

 posed. 



" He was enabled also from the same theory to deduce again, in a 

 far easier manner, all the beautiful geometrical laws of crystalline 

 reflection and refraction, which he had formerly laid before the 

 Royal Irish Academy in 1837, and for which that body awarded him 

 the honorary distinction of the Conyngham Medal, which I have 

 before alluded to. And they fully confirmed the acute prophecy 

 then made by his sagacious mind, on finding to his astonishment, 

 that a law of reflection depended for its existence on the existence 

 of a law of propagation ; when he said that the law of vis viva 

 which he had assumed at the outset could not be a fundamental, 

 but rather a secondary law, and remarked that perhaps the next step 

 in physical optics would be the deduction, as parts of one system, 

 of all the laws, both of propagation and reflection, from some higher 

 and more general law, containing them both as particular cases : 

 anticipations which were singled out for special attention in the 

 Address delivered by Sir W. Rowan Hamilton, on the occasion 

 already referred to. How little, perhaps, did Professor Mac Cullagh 

 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 crea- 

 tive mind ! 



" In the general case of total reflection at the surface of a crys- 

 tal, he afterwards showed, by a most ingenious employment of ima- 

 ginary quantities, that the refraction was still double, and never 

 more than double ; and be showed that the directions of the re- 

 fracted rays remained always the same, whatever were the incidence, 

 provided it gave total reflection. Again, as he had done for the case 

 of ordinary reflection by means of his beautiful theorem of the 

 polar plane, so in the case of total reflection he determined the two 

 directions of polarization, in a given incident plane polarized vva\-e, 

 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 difficult case. And finally, by 

 means of the circular sections of the ellipsoid apsidal to the surface 

 of indices, he showed how to determine completely, in plane, posi- 



