336  The  Hon.  J.  W.  Strutt  on  the  Reflection  and 
this  case  there  is  no  loss  of  intensity  in  the  transmitted  light, 
a2— 1 
and  the  retardation  is  — ^ — 8. 
But  if  (Jk  be  complex  (equal  to  Reia), 
_(RV<«-l)ism(Reia08)  
B,= 
2Re'«  cos  (Rela«8)  +  i(R2e2ta  + 1)  sin  (Reia«8) 
The  intensity  of  the  reflected  light  is  to  be  found  by  multiplying 
By  by  the  quantity  derived  from  it  by  changing  the  sign  of  i. 
The  numerator  of  the  resulting  fraction  is 
(R4-2R2 cos  2«+ 1)  sin  (Re^aS)  sin  (Re-fa«8) . 
The  product  of  sines  is  the  half  of 
cos{2Rsina  .iaS}  —  cos{2Rcosa .  aS} 
__  1  (e2RSina.a5  +  6-RSina.«5j.  _  cog  |2R  COS  U  .  tf8j-  . 
We  may  infer  that  the  intensity  of  the  reflected  light  is  nearly 
proportional  to 
2cosi2Rcosa.«Sj- 
c2Rsina.  aS    i    g— 2Rsina.a5 
For  transparent  media  the  sum  of  exponentials  reduces  to  the 
constant  2,  but  for  opaque  media  it  increases  rapidly  with  8. 
After  the  first,  corresponding  to  8  =  0,  the  minima  are  no  longer 
zero,  and  soon  all  fluctuation  becomes  insensible. 
Another  effect  of  the  exponential  terms  is  to  displace  the  po- 
sition of  the  maxima  and  minima  with  respect  to  8.  They  tend 
to  occur  earlier  than  they  otherwise  would  do.  Tn  our  ignorance 
of  the  values  of  the  constants  it  seems  hardly  worth  while  to  fol- 
low the  result  more  minutely. 
The  transformation  of  A2  when  fi  is  complex  leads  to  a  long 
expression  ;  and  I  will  therefore  confine  myself  to  the  particular 
case  of  a  very  thin  layer,  whose  thickness  does  not  amount  to 
more  than  a  small  fraction  of  the  wave-length. 
Let  the  reduced  expression  for  the  transmitted  wave  be 
A^ax+ct)-.    (amplitude)   ei(ax+aS  +  ct+e')a 
Then  e'  is  given  by  the  equation 
D'-D 
tan  e' 
i(D'  +  D); 
if  we  denote  the  denominator  of  the  expression  for  A2  by  D,  that 
derived  from  it  by  changing  the  sign  of  i  by  D'.     Now 
