30 Mr. R. A. Houstoun on Total Reflexion at the 



vibration to the amplitude of the incident vibration at M ; 

 and ps" the similar ratio at ~N. Let S s be the ratio of the 

 amplitude o£ the refracted to the amplitude of the original 

 vibration at M. Those ratios are here complex. Write 



p s = r s e^s; B s = d s e iA s' ; p,"= r,W, 



where r s , d s , and r s " are real. Theu A s , A/, and A s " are the 

 changes of phase which the waves undergo. 



Let us represent the incident wave in the direction H M 

 by sin T. Then the reflected wave in the direction MQ is 

 r s sin(T + A s ). The wave which enters the glass at P has 

 been twice refracted and once reflected. Its phase has been 

 retarded by an amount L owing to its having to traverse the 

 distance MNP. Then to take the absorption into account, we 

 multiply by a factor k, like L dependent on the thickness of 

 the film. The wave which enters the glass at P is therefore 

 M«V,"sin(T + 2A/ + A/ / — L); and the resultant reflected 

 wave, polarized at right angles to the plane of incidence, is 



r.sin (T-f A,) +M s VVsin (T + 2A/ + A/'-L). 



Similarly the resultant reflected wave, polarized in the plane 

 of incidence, is 



r p sin (T + A p ) + HV sin (T + 2A/ + A/' - L) . 



When we sum the two simple harmonic motions, of which 

 each of these waves consists, we find that their phases are 



T_^A j_f -i kd * r »" sin (2A/-4 - A, "-A,- L) . T , Q ,_, 



T + A.+ tan ^T^T^co^^ (0 



and 



T* A p+ tan-* 5> 2 " f (i % A + £'-, A \ L \ , = T + A p + P (8) 



p r p -\-kd P 2 r p " cos (2 A p ' + A p —A p — L) w 



The relative phase-difference which we measure is approxi- 

 mately the difference of these two expressions. 



If ol x and y l are the sine and cosine of the angle of incidence 

 in the glass, and if a 2 7 2 and a 3 y 3 are defined by 



n^i = n(l — iK.)oc 2 = a 3 , 

 then (Voigt's Kompendium, ii. page 6±6) 



a l Yl — «272 _ a 2 Yj ~ a 1 72 



Os = ; Op — 



a l7l + a 2 y 2 ^7i + *i72 



g _ 2^7! g __ 2^7! 



«l7l-t"«272 P «27l+^l72 



