1881.] the Electromagnetic Theory of Light. 1G5 



and (19) and (21) become 



fxS sin 2 $ cos + /u, 1 ^ 1 sin 2 fa cos t = p,'S' sin 2 fa cos #'. 



Also jr cos CP= p.V 2 sin cos <£ 



= yu./) 2 sin 2 cp sin cos cp. 



Thus (20) gives 



/z$ sin 2 cp sin cos cp — /Lt^sin 2 <£ sin X cos (^ 



= /u,'/S" sin 2 fa (sin 0' cos fa + tan ^ sin fa). 



Again Fcos CQ = Fcos 6 cos (£ = p cos cos <£ sin cp. 



Thus (23) becomes 



S cos cos cp sin cp — S t cos t cos <^» 1 sin fa = $' cos 0' cos </>' sin <£'. 



We will re-write these four equations in two pairs, remembering 

 that since the first medium is isotropic, cp = <p x . 



(Scos + S t cos t ) p, sin 2 <f> = 8' cos & pi sin 2 fa (24)) 



(#cos — S t cos 0j) cos cp sin cp = 8' cos & cos fa sin fa (25) J 



{S sin + S, sin 0J sin <f> = S' sin & sin f (26)] 



($ sin — 8 X sin 0J /a sin 2 <£ cos cp > . 



= /n'$' sin 2 $' (sin 0' cos </>' + tan ^ sin fa (27) j 



If both media be isotropic we have here four unknowns, viz. 

 S v S', lt #'and these equations are sufficient to determine them. 



If the second medium be a crystal, 0' is not unknown, it has 

 one of two definite values depending only on the position of the 

 refracted wave with reference to the optic axes. Thus we have 

 only three variables in our four equations and they require modi- 

 fication. Now we assumed that there was ouly one refracted wave. 

 In general when light falls on a crystal there are two, we must 

 therefore introduce into each of our four equations terms involving 

 quantities S", 0", fa', p," in the same way as 6', fa, p! are involved, 

 6" being a known function of fa'. Again if the first medium also 

 be crystalline, X is not variable, and we are again reduced to three 

 variables and four equations. Moreover terms involving the angle 

 between the ray and the wave normal in the incident and reflected 

 rays would be introduced into our fourth equation (27). 



But we know that we have two reflected waves, and we must 

 add to our equations terms involving S 2 , 2 , </> 2 , p,^ in the same 

 manner as S t , lf cp l , p. x are involved. 



VOL. IV. pt. in. 12 



