1 88 On a Dynamical Theory of 



reflected light would also become known, by means of the 

 general formulae which the author had established for all cases 

 of reflexion at the common surface of two media. 



The present supplement is the fruit of these considerations. 

 It contains the complete theory of the new kind of vibrations, 

 not only in ordinary media, but in doubly-refracting crystals ; 

 and also the complete discussion of the laws of total reflexion at 

 the first or second surface of a crystal, including, as a particular 

 case, the well-known empirical formulee of Fresnel for total 

 reflexion at the surface of an ordinary medium. 



The existence of vibrations represented by an expression con- 

 taining a negative exponential as a factor had been recognised 

 by other writers, and was indeed sufficiently indicated by the 

 phenomenon of total reflexion ; but it was impossible to obtain 

 the laws of such vibrations, so long as the general equations for 

 the propagation of light were unknown. 



The method of deducing these equations was given in the 

 abstract of the author's former Paper ;* but as they were not there 

 stated, it may be well to transcribe them here. If then we put 



= ( h_ ( % Y=^- Z=- (1) 



dz dy' dx dz' dy dx' 



and suppose the axes of co-ordinates to be the principal axes of 

 the crystal, the equations in question may be thus written : 



= z _ z 

 l dz 3 dx' 



dt 2 dx dy ' 

 and if we further put 



dz dy ' dx dz' dy dx ' 



* See Proceedings, 9th December, 1839 (supra, p. 157). 



