248 Lord Bayleigh on the Reflexion of Light 



Incident wave : 



satisfying 



Reflected wave : 



f—q0i e Kp'x+qy+st) } g = —p f & e Kp'x+qy+st) g # ^9) 



Refracted wave : 



f—qQ^x+qy+st)^ g = _p 1 Q ie Kp } x+qy+8t) # _ (3Q) 



The coefficient of the time (5) is necessarily the same 

 throughout on account of the periodicity ; and the coefficient 

 of y is the same, since the traces of all three waves upon the 

 plane of separation # = must move together. The relations 

 between p, q, s ; p' ', q, s ; p v q, s are to be obtained by sub- 

 stitution in the differential equations. Of these the equation 

 in h is satisfied identically, since R=0. The other equations 

 for the upper medium are by (7), (8), (25), 



These must be satisfied by the incident and reflected waves. 

 On substitution we find that both equations lead to the same 

 conditions, viz. : — 



s*=Aq' 2 -2Bpq + Cp 2 , .... (31) 



a quadratic equation of which the two roots give p and p' in 

 terms of q and s. 



In the second medium we get in like manner for the re- 

 fracted wave 



* 2 =A£ 2 + 2Bp 1 q + Gp l % .... (32) 



the sign of B being changed. Equating the two values of S* 2 , 

 we find 



G(p^p^ = 2Bq(p+p l ), 



or C(p- Pl )=2Bq (33) 



We have now to consider the boundary conditions (A). The 

 functions E, and b vanish throughout ; but it remains to pro- 

 vide for the continuity of/, Q, and c, when #=0. The first 

 of these conditions gives at once 



l + 6 , = l (34) 



Again, the continuity of Q ; equal to B/+C# in the first 



