250 Lord Rayleigh on the Reflexion of Light 



These two equations determine p and /z, when r, s are given. 

 Since the elimination of /jl leads to a quadratic in p 2 , it is 

 evident that there are four admissible values +Pi, ±^2j corre- 

 sponding to waves of given periodicity, whose trace on the 

 plane of separation moves with a given velocity. Of these 

 two (say with the + sign) are waves approaching the surface, 

 and two are waves receding from it. If we limit ourselves to 

 a single incident wave ( + ^1), we shall have still to take into 

 account two reflected waves corresponding to — pi,— p 2 . The 

 equations show that the value of \x is the same whether p be 

 positive or negative ; we shall suppose that /a x corresponds to 

 ± p 1} ^ to ±p 2 . 



In applying the equations to the second medium we have 

 to change the sign of B ; and it is evident that they are 

 satisfied by the same values of p as before, and that the pre- 

 ceding values of //. are to be taken negatively. Hence in the 

 second medium — fx x corresponds to +pi 9 — fi* to +p 2 . For 

 the purposes of our present problem, where there is no 

 incident wave in the second medium, we are concerned only 

 with -j- p x and + p 2 . 



The complete specification of the system of waves corre- 

 sponding to a single incident wave (^) in the first medium is 

 thus : — 

 Incident wave : 



/, ff,h=(r, A * 1 ,- Pl )e 1 «' l >.*+«»+-> ; • • • (39) 



Tivo reflected waves : 



fg,h=(r, ft, Pl ) ffji-Pe+rz+H) 



+ (r, fi 2 , p 2 ) 0'V<^+**+rf> ; . . (40) 

 Two refracted * waves : 



/, g, h= (r,-to,- Fl ) <V^+-+*0 



+ (r,-/* 2 ,- p 2 ) 2e i(P*z+r Z+st ) t m m ( 41 j 



The next step is the introduction of the boundary con- 

 ditions (A). The continuity off requires that 



e i + / + 0" = 1 + 2 (42) 



The continuity of R, or D/i, or h, gives with equal facility 



PiQi-piP-p*0"=pA+P20» • • • (43) 

 Again, the continuity of Q, equal to Bf+Cg in the first 



* It should be noticed that one of the refracted waves is not refracted 

 in the literal sense, being parallel to the incident wave. 



