Refraction  of  Light  by  intensely  Opaque  Matter.  331 
in  which  the  coefficient  of  x  varies  as  «.  For  instance,  let 
a?  —  —  The  effect  on  reflection  wonld  be  insensible  to  ordi- 
nary observation,  though  the  opacity  is  so  great  as  to  halve  the 
light  within  a  distance  equal  to  the  wave-length  in  air.  Thus 
it  is  evident  that,  in  order  to  aid  in  reflection,  opacity  must  be 
very  extreme. 
"We  have  hitherto  supposed  that  the  reflection  takes  place  at 
the  bounding  surface  of  the  opaque  medium  and  air ;  but  it  is 
easy  to  adapt  our  formulae  so  as  to  express  the  result  when  the 
first  medium,  still  supposed  transparent,  or  at  least  not  very 
opaque,  has  a  refractive  index  pJ  different  from  unity.  The  only 
change  required  is  to  write  R-^/z/  for  R.  Thus  at  perpendicular 
incidence, 
tan/=JL(*  +  £). 
■*       COS  a  \/jl'       a/ 
If  the  reflection  be  still  so  good  as  to  allow  of  the  neglect  of  the 
second  term,  we  have 
tan  f  =  —. • 
J       \tJ  cos  a 
The  reflection  when  light  strikes  from  glass  on  silver  would  be 
considerably  less  perfect  than  when  the  first  medium  is  air;  in 
fact  the  percentage  not  reflected 
2  u}  cos  a 
=  lTTarrr-R-aPPr0X- 
So  much  for  vibrations  perpendicular  to  the  plane  of  inci- 
dence. When  we  pass  to  the  consideration  of  vibrations  in  that 
plane,  we  are  embarrassed  by  difficulties,  of  the  kind  met  with  in 
the  theory  of  ordinary  reflection,  here  presenting  themselves  in 
an  aggravated  form.  If,  following  Green,  we  assumed  the  equa- 
tions of  motion  applicable  to  elastic  solids  with  the  addition  of 
terms  proportional  to  the  velocity  to  represent  the  frictional  loss, 
and  further  supposed  that  the  rigidity  is  the  same  in  the  two 
media,  while  the  compressibility  is  indefinitely  small,  we  should 
arrive  at  results  differing  only  from  his  by  the  substitution  of  an 
imaginary  for  a  real  refractive  index.  But  we  know  from  expe- 
riment that  Green's  results  are  not  verified  for  transparent  media 
without  a  modification  of  doubtful  significance,  and  of  magnitude 
increasing  rapidly  with  \i.  It  is  therefore  useless  to  attempt  to 
apply  Green's  results.  The  only  other  course  appears  to  be  to 
start  from  FresnePs  tangent-formula,  and  transform  that,  as  we 
have  done  the  one  involving  sines,  by  the  introduction  of  a  com- 
plex refractive  index  (and  angle  of  refraction) .  This  is  what  has 
been  done  by  Cauchy  and  Eisenlohr.      Following  a  process 
