5=Ao^a»+c*>. 
Refraction  of  Light  by  intensely  Opaque  Matter.         335 
we  must  be  prepared  to  distinguish  between  the  effects  of  den- 
sity and  opacity,  though  these  are  in  the  same  direction  so  far 
as  regards  the  magnitude  of  the  reflection  &c. 
Let  us  then  consider  analytically  the  behaviour  of  a  thin  me- 
tallic plate  when  light  is  incident  normally  upon  it.  Above  let 
the  disturbance  be 
£_  ei(aj-+cO  _|_  ]}  6i(-ax+ct)  . 
below  the  plate, 
-v 
In  the  interior  we  must  introduce  both  kinds  of  exponentials,  in 
order  to  represent  the  reflection  from  the  second  surface.     Thus 
where  at=^ia  as  before. 
The  conditions  to  be  satisfied  are  the  continuity  of  f  and  — 
at  the  two  surfaces  of  separation,  viz.  when  <2?=0  and  when 
x=  — 8,  which  give  four  simple  equations  for  the  determination 
of  By,  A',  B'3  x\2.     On  elimination  of  A',  B',  we  obtain 
j.  _   __  (fju2— l) i  sm /xaS 
1  2fi  cos /xaB  +  (/j? -\- 1)  i  sin  /xaS 
A2- 
cos 
fiaS  +  I  — —  j  i  sin  fxaS 
These  expressions  contain  the  ordinary  results  for  transparent 
plates.     Considering  /jl  real,  the  reflected  wave  is 
((*?  —  1)  sin  fiaS 
where 
ei(-ax+ct+e) 
\f4p*  cos2  nah  +  {ij?  + l)2  sin2  /juiS 
u2  +  l 
tan  e= - —  tan  uao. 
2/t 
Similarly  the  transmitted  wave  is 
1 
ei(ax+ct+aS+e>) 
a/ cos2 fiaB -f ( ^— — J  sin2 fj,a8 
where 
jx2-fl 
tan  e'=  —  '—^ tan  fiaS. 
If  fMah  be  very  small,  the  expression  for  the  wave  becomes  ap- 
proximately 
