in Singly and Doubly Refracting Media, 515 



the right-hand side. Accordingly we obtain forthwith 

 A E sin E + A R sin R = 2A D sin B , 

 (A E sin E — A R sin R ) cos e = SA D sin D n cos r y 

 (A E cos E + A R cos # R ) cos e = SA D cos r (cos # D r" (36) 



+ tan A tan r), j 

 A E cos 0% — A R cos # R = SAp cos D ?z. 



For isotropic media, therefore for A = 0, the first two of 

 these equations refer to the first principle case (0 E = 90°), the 

 last two to the second (0 E = O°). 



The coefficient of weakening -r- , to be derived from equa- 



tions (36), has been sufficiently tested by experiment. And 

 if, on the one hand, the amplitude (AdA d ) of the light issuing 

 from the hinder surface of a plane-parallel plate after double 

 refraction is deduced immediately by means of the general 

 relation 



Ad Ad = A E — A R , 



on the other hand Neumann has likewise amply discussed the 

 interesting subject of the internal reflexion of crystals. Our 

 fundamental theorems would of course lead to identical equa- 

 tions, whereas Cauchy's very ambiguous theory of continuity 

 may, it is true, be extended to the occurrences at the fore sur- 

 face, by suitable auxiliary assumptions, but not to those at 

 the hinder surface of a crystal. But since with this the advan- 

 tage claimed for it, that it establishes the elliptical polarization 

 of transparent media on a mechanical basis, becomes quite 

 illusory, that polarization being far more perfectly derived di- 

 rectly from the foregoing formulae, its principles also may now 

 prove to be outwardly specious, but intrinsically untenable 

 and arbitrary. 



I may, then, well conclude this investigation with the propo- 

 sition that the hypothesis of the convibration of the ponderable 

 particles harmonizes the verified formulae of Neumann with 

 Fresnel's view concerning the position of the polarization- 

 plane. 



Generalization of the Results obtained. The Fundamental 

 Law of Dispersion. 



15. The two equations into which the difFerential equation 

 (11a) divides can without difficulty be extended so that they may 

 be applied immediately to absorbent as to transparent media. 

 If, namely, instead of the abscissa r (above designated by oc) 

 referred to the direction of the propagation, we introduce the 



2 L 2 



