408 MR. J. C. MAXWELL GARNETT ON 



On taking the modulus and substituting for e from (17) we obtain 



|C| 2 = 1 - 47T c//X sec . w 2 /c ...... . (22). 



Secondly, suppose that the incident light is polarised perpendicular to the plane of 

 incidence, a, /8, y being the magnetic force. 

 The incident wave is 



n = 0, /3 = exp [ij? \t (x sin + z cos 0)/c}], y = 0. 



The transmitted wave is 



a = 0, /8 = C exp \ip {t (x sin + z cos 0)/c}], y = 0. 



The velocity V inside the film is connected with the angle of refraction by the 

 equations 



V' - .2 / cos2< /" _j_ 5in 2< \ c3fc/ 



e e' = ee' + sin 2 (e' - e) - 



The final result after using the two sets of boundary conditions is 



/-( wr (^ sec 6 \ .->/>/ , \ an e' 1 1 



C = 1 -^ cos 2 (t 1) + snrt' , J-, 



whence, using (17), we obtain 



r 2/c + tan ^6'-n - . ; (23)> 



X 



When the light is normally incident, the crystalline character of the film does not 

 manifest itself, and we have from (22) or (23) 



|C| 2 = 1 - 4ird/X. n*K ........ (24). 



The absorption of directly incident light by a thin film is therefore governed 



by n z K. 



Owing to the difficulty of knowing whether any particular metal film whose changes 

 of colour have been observed, but whose thickness has not been recorded, for 

 example, the films observed by Professor R. W. WOOD or by Mr. G. T. BEILBY 

 (loc. cit.), is to be regarded as very thin for the purpose of this section, formulae for 

 thick films will now be found. 



We consider here only the case of directly incident plane -polarised light, and 

 proceed to obtain an equation corresponding to (24), reserving the full discussion of 

 the behaviour of thick films under oblicpuely incident light till later. Using the axes 

 shown in fig. 10, suppose that 



Incident wave is 



E = 0, exp {ip (t z/c)}, ; H = exp (ip (t z/c}, 0, 0. 

 Reflected wave 



E = 0, B exp {ip (t + z/c)}, ; H = B exp {ip (t + z/c)}, 0, 0. 



