192 On a Dynamical Theory of 



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 '", all the roots are real, the re- 

 fraction 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 diameters 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 perpendicular to the axis of #'. 



With regard to the ellipse in which the vibrations are per- 

 formed, it may be worth while to observe, that if it be projected 

 perpendicularly 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 re- 

 spective 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 ether. 



The new laws here announced are, properly speaking, laws 

 of double refraction, and are necessary to complete our know- 

 ledge of that subject. Between them and the laws of Fresnel 

 a curious analogy 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 be dwelt upon in this abstract, as 

 nothing is now more easy than to form the equations which con- 

 tain them. In fact, the difficulties which formerly surrounded 

 the problem of reflexion, even in the simplest cases, have com- 

 pletely disappeared, since the author made known the conditions 

 which must be fulfilled at the separating surface of two media. 



