100 



the motion is extinguished, in receding from the second plane, 

 will depend upon the constant r. The constants/; and q may 

 be any two conjugate semidiameters of the ellipse in which 

 the vibration is performed ; the former making, with the axes 

 of coordinates, the angles a, /3, 7, the latter the angles a!, 



/3'.7'- 



As vibrations of this kind cannot exist in any medium, 



unless they are maintained by total reflexion at its surface, 

 we shall sujjpose, in order to contemplate their laws in their 

 utmost generality, that a crystal is in contact with a fluid 

 of greater refractive power than itself, and that a ray is 

 incident at their common surface, at such an angle as to pro- 

 duce total reflexion. The question then is, the angle of in- 

 cidence being given, to determine the laws of the disturbance 

 within the crystal. 



The author finds that the refraction is still double, and 

 that two distinct and separable systems of vibration are trans- 

 mitted into the crystal. He shows that the surface of the 

 crystal itself (the origin of coordinates being upon it at the 

 point of incidence) must coincide with the plane expressed 

 by equation (8), a circumstance which determines the three 

 constants/, g, h. The plane expressed by (7) is parallel to 

 the plane of the refracted wave ; and a normal, drawn to it 

 through the origin, lies in the plane of incidence, making 

 with a perpendicular to the face of the crystal an angle w 

 which may be called the angle of refraction, so that, if i be 

 the angle of incidence, we have 



sin w = 5 sin i, 



the velocity of propagation in the fluid being regarded as 

 unity. 



To each refracted wave, or system of vibration, corres- 

 ponds a particular system of values for r, s, w. These the 

 author shows how to determine by means of the index-sur- 

 face (the reciprocal of Fresnel's wave surface) which he has 

 employed on other occasions, (Transactions of the Academy, 



