Second Surface of a Thin Plane Parallel Plate. 25 



are all different, and that total reflexion takes place at the 

 second face. The mathematics is rather complicated ; the 

 rigorous formula in the case of silver iodide was found 

 capable of taking a relatively simple form and agreed perfectly 

 with the experimental result. An approximate formula was 

 used for the silver film. The necessary formulae are given 

 in Voigt's Kompendium, ii. p. 643 and the following pages, 

 but must be considerably transformed. The formula for the 

 silver iodide will be derived from first principles, as that can 

 be done more shortly than is generally supposed by means of 

 the reciprocal property of Maxwell's equations, and will be 

 compared with experiment. Then the case of the silver films 

 will be treated from the formulae in the Kompendium. 



When the wave in the glass falls on the film on the hypo- 

 tenuse, it gives rise to a reflected wave in the glass, to a 

 refracted wave in the film that is repeatedly reflected back 

 and forwards in the film, and to a wave in the air beyond. 

 We write down exponentials to represent those waves and 

 substitute them in the boundary conditions. Consider the 

 magnetic vector as the light vector. We can consider 

 separately the components polarized in and perpendicular to 

 the plane of incidence. Let us consider the component per- 

 pendicular to the plane of incidence. We can write the 

 incident and reflected waves in the glass 



Then summing all the waves in the film incident on the 

 second surface, and all those reflected by it, we may write 

 them 



E 2 e T <* : R 2 ^ 2 . 



Let the wave in air be represented by 

 where T _ '2ir / ,v sin fa 4- z cos fa 



_ Z7T / x sm fa -f z cos fa\ 



etc. 



t is the period of the vibration, v 1 the velocity, and fa the 

 angle of incidence in the medium. The z axis is perpen- 

 dicular to the surfaces and is measured positive from glass to 

 air. Let I be the thickness of the film, /j, the magnetic 

 inductivity, and K the electric inductivity of the medium in 

 question. 



If H is the component of the magnetic vector perpendicular 

 to the plane of incidence, the boundary conditions are that 



H and -rr—r- go through the boundaries z = and z = I 

 K dz tt & 



