COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 407 



First let the incident light be polarised in the plane of incidence, so that the incident 

 wave is 



X = 0, Y = exp [ip {t (x sin 6 + 2 coa 0)/c}], Z = 0, 



= - cos 6 exp [tp {/. - (a: sin 6 + 2 cos 0)/c}], = 0, y = sin exp [ t p {...]]. 

 The reflected wave is 



X = 0, Y = B exp [ t p {/ - (x sin - 2 cos 0)/c}], Z = 0, 



a = B cos exp [ip {t - (x sin - 2 cos 0)/c}], /3 = 0, y = B sin exp[ t p {...}]. 

 Inside the film, i.e., between 2 = and 2 = d, 



X = 0, Y = A' exp [ip {t. (x sin $ -f 2 cos <)/Vj] 



+ B' exp [ip {f (x sin <f> z cos <)/V}], Z = 0, 



= - -y * {A'exp[ t p {< - (xsin^, + 2C os <)/}] - B'exp[ip {...}]}, ft = 0, 



y = sm0{A'exp[...] + B'exp[...J{. 

 Transmitted wave 



X = 0, Y = C exp [ip {^ - (.T sin + z cos 0)/c}], Z = 0, 



= C cos exp [ip {t (x sin + 2 cos 0)/c}], /S = 0, y = ( ' sin exp [. . . ]. 

 In these expressions we have 



V' 2 = c-/t and sin <ffV = sin 0/c ....... (a). 



Since Y and a are continuous at z = 0, 



1+B = A' + B' and (1 - B) cos = (A' - B') c/V cos ^ . . (1)). 



Since Y and a are continuous at 2 = d, if we replace 

 exp [i ip rf/V cos ^] by 1 ip dfV cos (/>, and exp [ ip d/c cos 0] by 1 ip djc. cos 0, 



we obtain, when the square of 2ir d/\ is neglected, the equations 



{ A' + B' - (A' - B') ip d/V cos <j>] = C (I - ip d/c cos 0)1 

 c cos 4>jV {(A' - B') - (A' + B') tp d/V cos <^} = cos 0C (1 - ^ rf/c cos 0)J 



From the last pair of equations we find, neglecting squares of pd/c, that 

 A'+B' =C 



on using (a) ; then eliminating A', B', B from the equations (b) and (d), we finally 



have 



C = 1 - ITT d/\ . sec . (e 1). 



