﻿Reflexion 
  of 
  Light 
  at 
  an 
  Ideal 
  Plane 
  Mirror. 
  891 
  

  

  the 
  normal 
  to 
  the 
  mirror, 
  the 
  positive 
  direction 
  of 
  the 
  normal 
  

   being 
  on 
  the 
  reflecting 
  side 
  of 
  the 
  mirror. 
  

  

  Take 
  the 
  axis 
  of 
  x 
  along 
  the 
  normal 
  and 
  let 
  the 
  electric 
  

   vector 
  in 
  the 
  incident 
  wave 
  be 
  given 
  by 
  

  

  ^ 
  . 
  r 
  X 
  cos 
  (b 
  + 
  y 
  sin 
  <f)~i 
  

  

  where 
  e 
  is 
  independent 
  of 
  (.r, 
  y, 
  z. 
  t). 
  

  

  This 
  represents 
  a 
  plane 
  wave 
  of 
  light 
  of 
  frequency 
  v 
  

   travelling 
  in 
  a 
  direction 
  opposite 
  to 
  that 
  in 
  which 
  the 
  mirror 
  

   is 
  moving. 
  The 
  normal 
  to 
  the 
  wave 
  front 
  makes 
  an 
  angle 
  

   TT 
  + 
  ^ 
  with 
  the 
  normal 
  to 
  the 
  mirror, 
  so 
  that 
  the 
  angle 
  of 
  

   incidence 
  is 
  (f>. 
  

  

  The 
  reflected 
  wave 
  is 
  given 
  by 
  

  

  where 
  * 
  

  

  T^/ 
  r 
  . 
  ,T 
  J. 
  .I'cosd)'— 
  wsind)'"l 
  

  

  veos6 
  = 
  v^\—- 
  5 
  + 
  ^5 
  -.,cos0' 
  • 
  

  

  If 
  sin 
  </) 
  = 
  v^ 
  sin 
  c^^ 
  )> 
  • 
  (I-} 
  

  

  / 
  r 
  ^^^' 
  JL' 
  . 
  c^+^'^i 
  

  

  so 
  that 
  

  

  , 
  (c^ 
  + 
  v^) 
  cos 
  <b' 
  — 
  2cv 
  

   T 
  c- 
  + 
  1"^ 
  — 
  2cv 
  COS 
  (p 
  

  

  , 
  <h 
  c 
  — 
  v, 
  <b 
  

  

  tan-^ 
  = 
  tan^, 
  

  

  2 
  c+v 
  2 
  

  

  v(c 
  + 
  V 
  COS 
  </)) 
  = 
  v'(c' 
  — 
  1? 
  COS 
  </>'), 
  

  

  ,, 
  _ 
  v(c 
  -f 
  r 
  cos 
  (/) 
  + 
  ?i 
  sin 
  <^) 
  v'{c 
  — 
  v 
  cos 
  cj) 
  ' 
  -]ru 
  sin 
  <^^) 
  _ 
  „, 
  

  

  w^here 
  u 
  is 
  the 
  component 
  of 
  w 
  along 
  the 
  axis 
  of 
  ?/. 
  

  

  77ig 
  Za5i 
  equation 
  indicates 
  that 
  the 
  apparent 
  frequencies 
  

   v" 
  ^ 
  v'" 
  of 
  the 
  incident 
  and 
  reflected 
  rays^ 
  relatively 
  to 
  an 
  observer 
  

   moving 
  with 
  the 
  same 
  velocitf/ 
  as 
  the 
  mirror, 
  are 
  the 
  same^i. 
  

  

  * 
  See 
  the 
  papers 
  by 
  Einstein 
  and 
  Hicks. 
  

  

  t 
  Cf. 
  Hicks 
  and 
  Einstein, 
  /. 
  c. 
  We 
  may 
  show 
  that 
  the 
  expression 
  on 
  

   the 
  left-hand 
  side 
  represents 
  the 
  apparent 
  frequency 
  of 
  the 
  incident 
  

   light 
  by 
  applying 
  Lorentz's 
  transformation 
  so 
  as 
  to 
  pass 
  from 
  a 
  system 
  

   moving 
  with 
  velocity 
  2c 
  to 
  a 
  system 
  at 
  rest. 
  

  

  