178 The Aether as an Elastic Solid. 



To explain their method, we shall suppose the incident 

 light to be polarized in the plane of incidence. According to 

 Fresnel's sine-law, the amplitude of the light (polarized in this 

 way) reflected from a transparent body is to the amplitude of 

 the incident light in the ratio 



_ sin (i - r) 

 sin (i + r)' 



where i denotes the angle of incidence and r is determined from 

 the equation 



sin i = ft sin r. 



MacCullagh and Cauchy assumed that these equations hold good 

 also for reflexion at a metallic surface, provided the refractive 

 index /* is replaced by a complex quantity 



IJL = v(l *v/ 1) say, 



where v and K are to be regarded as two constants characteristic 

 of the metal. We have therefore 



tan i - tan r (ju 2 - sin 2 i)% - cos i 



jj ,.-,.. - - - -- " 



tan i + tan r (fj 2 - sin 2 i)k + cos i 

 If then we write 



so that equations defining U and v are obtained by equating 

 separately the real and the imaginary parts of this equation, we 

 have 



Ue^ ~ l - cos i 

 J 



TT v \/ 1 



Ue v + cos 

 and this may be written in the form 



where 



-=. 2 U* + cos 2 ^ - 2 U cos v cos i 

 U" + cos 2 * + 2 U cos v cos i 

 2 U cos i sin v 



tang = 



U* - cos 2 * 



