102 



If the crystal be uuiaxal, and all the values of 2' imaginary, 

 the ordinary wave normal will coincide with the axis of a:' ; 

 whilst the extraordinary wave normal and the axis of s' will 

 be conjugate diameters of the ellipse in which the index 

 surface is cut by the plane of incidence. 



When a = 6 = c, the crystal becomes an ordinary me- 

 dium ; there is then only single refraction, and the refracted 

 wave is always perpendicular 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 perpendicularly on the plane of incidence, the pro- 

 jected 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 ether. 



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 thejlaws of 

 Fresnel a curious analogy exists, founded on the change of 

 real into imaginary constants. 



The laws ofthe 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 

 contain them. In fact, the difficulties which formerly sur- 

 rounded the problem of reflexion, even in the simplest cases, 

 have completely disappeared, since the author made known 

 the conditions which must be fulfilled at the separating sur- 

 face of two media. 



In what precedes, it has been supposed that the reflexion 

 and refraction take place at the^r*^ 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 



