97 



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

 ami important results by substituting in his difllerential 

 equations of motion a more general expression for the inte- 

 gral, 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 vibra- 

 tions 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 surface ; and the laws of this distui'bance being 

 thus discovered, the laws of polarization in the totally re- 

 flected 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 considera- 

 tions. It contains the complete theory of the new kind of 

 vibrations, not only in ordinary media, but in doubly refract- 

 ing crystals; and also the complete discussion of the law^s of 

 total reflexion at the first or second surface of a crystal, in- 

 cluding, as a particular case, the well known empirical for- 

 mulas of Fresnel for total reflexion at the surface of an ordi- 

 nary medium. 



The existence of vibrations represented by an expression 

 containing a negative exponential as a factor, had been re- 

 cognized by other writers, and was indeed suflSciently indi- 

 cated 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 un- 

 known. 



The method of deducing these equations was given in 

 the abstract 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 



_ rfn <1l _dZ, _dl d^ _dji (y 



""- d'z djf ^ ~ dx dz' ' dtj dx' ^ 



