232 Royal Irish Academy. 



each pair will correspond to a refracted system, and we shall have, 



for the first, 



sin i sin w , n x 



tanwxs , s = — — r, r = st>; . . . . (9.) 



u sine 



and for the second, 



. sini . sin w' , , , /in \ 



tanw' = — r , s' = , r" = s'v'. . . . (10.) 



u' sin i 



When i lies between i' and a certain smaller angle i", two of the 

 roots will be real, and two imaginary. The real roots correspond 

 to waves which follow the law of Fresnel ; the imaginary roots give 

 a single wave, following the other laws just mentioned. 



Lastly, when i is less than i", all the roots are real, the refraction 

 is entirely regulated by Fresnel's law, and the reflexion by the laws 

 already discovered and published by the author. 



If the crystal be uniaxal, and all the values of z' imaginary, the 

 ordinary wave normal will coincide with the axis of x' ; whilst the 

 extraordinary wave normal and the axis of z' will be conjugate dia- 

 meters of the ellipse in which the index-surface is cut by the plane 

 of incidence. 



When a = b = c, the crystal becomes an ordinary medium ; there 

 is then only single refraction, and the refracted wave is always per- 

 pendicular to the axis of x' . 



With regard to the ellipse in which the vibrations are performed, 

 it may be worth while to observe, that if it be projected perpendi- 

 cularly on the plane of incidence, the projected diameters which are 

 parallel to the surface of the crystal and to the wave plane will, in 

 all cases, be conjugate to each other, and their respective lengths 

 will be in the proportion of r to unity. The vibrations, it is obvious, 

 are not performed in the plane of the wave, though they take place 

 without changing the density of the aether. 



The new laws here announced are, properly speaking, laws of 

 double refraction, and are necessary to complete our knowledge of 

 that subject. Between them and the laws of Fresnel a curious ana- 

 logy exists, founded on the change of real into imaginary constants. 



The laws of the total reflexion, which accompanies the new kind 

 of refraction, need not to be dwelt upon in this abstract, as nothing 

 is now more easy than to form the equations which contain them. 

 In fact, the difficulties which formerly surrounded the problem of re- 

 flexion, even in the simplest cases, have completely disappeared, 

 since the author made known the conditions which must be fulfilled 

 at the separating surface of two media. 



In what precedes, it has been supposed that the reflexion and re- 

 fraction take place at the first surface of the crystal, because this is 

 the more difficult and complicated of the two cases into which the 

 question resolves itself. But it will usually happen in practice that 

 a ray which has entered the crystal will suffer total reflexion at the 

 second surface, while the new kind of vibration is propagated into 

 the air without. The refracted wave will then be always perpendi- 

 cular to the axis of x { ; the fcwo reflected rays, within the crystal, 



