Prof. M'CulIagh on Crystalline Reflexion anil Refraction. 229 



Phil. Mag. S. 3, vol. xvi. p. 229) the author had given the general prin- 

 ciples for solving all questions relative to the propagation of light in 

 a given medium, or its reflexion and refraction at the separating sur- 

 face of two media ; but he had applied them only to the common case 

 of waves, which suffer no diminution of intensity in their progress, 

 and in which the vibration may be represented by the sine or cosine 

 of an arc multiplied by a constant quantity. Some months after that 

 paper was read, it occurred to him that he might obtain new and 

 important results by substituting in his differential equations of mo- 

 tion a more general expression for the integral, that is (as usual in 

 such problems), by making the displacements proportional to the 

 sine or cosine of an arc, multiplied by a negative exponential, of 

 which the exponent should be a linear function of the coordinates. 

 Such vibrations would become very rapidly insensible, and would 

 therefore be fitted to represent the disturbance which, in the case 

 of total reflexion, takes place immediately behind the reflecting sur- 

 face ; and the law3 of this disturbance being thus discovered, the 

 laws of polarization in the totally reflected light would also become 

 known, by means of the general formulae which the author had esta- 

 blished 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 formulas of Fresnel for total reflexion at the surface 

 of an ordinary medium. 



The existence of vibrations represented by an expression contain- 

 ing a negative exponential as a factor, had been recognized by other 

 writers, and was indeed sufficiently indicated by the phaenomenon of 

 total reflexion ; but it was impossible to obtain the laws of such vi- 

 brations, so long as the general equations for the propagation of light 

 were unknown. 



The method of deducing these equations was given in the abs- 

 tract of the author's former paper (see Proceedings, as above) ; but 

 as they were not there stated, it may be well to transcribe them 

 here. If then we put 



X = —i — ^l, Y = ^ — ^1, Z=^ — tm . ' . (\) 



dz dy dx dz' dy dx 



and suppose the axes of coordinates to be the principal axes of the 

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



(2) 



