190 On a Dynamical Theory of 



where <j> has the same signification as before, and 



e = e 



the vibrations being now elliptical, whereas in the former case 

 they were rectilinear. In these elliptic vibrations the motion 

 depends not only on the distance of the vibrating particle from 

 the plane whose equation is 



Ix + my + nz = 0, (7) 



but also on its distance from the plane expressed by the equation 



and if the constants in the equation of each plane denote the 

 cosines of the angles which it makes with the co-ordinate planes, 

 we shall have A for the length of the wave, and s for the velocity 

 of propagation ; while the rapidity with which the motion is ex- 

 tinguished, in receding from the second plane, will depend upon 

 the constant r. The constants p and q may be any two conju- 

 gate semi- diameters of the ellipse in which the vibration is per- 

 formed ; the former making, with the axes of co-ordinates, the 

 angles a, /3, 7, the latter the angles a', j3', 7'. 



As vibrations of this kind cannot exist in any medium, unless 

 they are maintained by total reflexion at its surface, we shall 

 suppose, in order to contemplate their laws in their, utmost 

 generality, that a crystal is in contact with a fluid of greater re- 

 fractive power than itself, and that a ray is incident at their 

 common surface, at such an angle as to produce total reflexion. 

 The question then is, the angle of incidence being given, to de- 

 termine 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 transmitted 

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

 (the origin of co-ordinates being upon it at the point of inci- 

 dence) 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 re- 



