﻿Postulate 
  of 
  the 
  Theory 
  of 
  Relativity. 
  169 
  

  

  therefore 
  to 
  examine 
  the 
  raj 
  reflected 
  from 
  a 
  moving 
  mirror 
  

   by 
  an 
  interference 
  method 
  simpler 
  than 
  that 
  on 
  which 
  the 
  

   action 
  of 
  the 
  diffraction 
  depends, 
  as 
  has 
  been 
  said 
  above. 
  

  

  Before 
  expounding 
  this 
  method 
  it 
  is 
  well 
  to 
  recall 
  that 
  

   many 
  theoretical 
  researches 
  have 
  been 
  made 
  on 
  the 
  influence 
  

   of 
  the 
  motion 
  of 
  the 
  mirror 
  upon 
  the 
  reflected 
  luminous 
  

   wave, 
  amongst 
  them 
  those 
  of 
  Abraham, 
  Brown, 
  Edser, 
  

   Harnack, 
  Larmor, 
  Planck. 
  These 
  researches 
  make 
  of 
  the 
  

   problem 
  either 
  a 
  simply 
  geometrical 
  investigation, 
  or 
  an 
  

   application 
  of 
  the 
  electromagnetic 
  theory 
  of 
  light. 
  But 
  

   without 
  discussing 
  the 
  result 
  of 
  these 
  researches 
  we 
  may 
  

   accept 
  the 
  conclusion 
  of 
  Harnack 
  * 
  respecting 
  the 
  frequency 
  

   of 
  the 
  vibrations 
  reflected 
  by 
  a 
  mirror 
  in 
  uniform 
  motion. 
  

   Let 
  v 
  be 
  the 
  velocity 
  of 
  the 
  latter, 
  normal 
  to 
  its 
  plane, 
  

   reckoned 
  as 
  positive 
  towards 
  the 
  source 
  ; 
  c 
  the 
  velocity 
  of 
  

   the 
  luminous 
  ray 
  in 
  vacuo 
  w 
  r 
  hich 
  makes 
  the 
  angle 
  of 
  inci- 
  

   dence 
  6 
  with 
  the 
  mirror 
  ; 
  ??, 
  n' 
  the 
  frequencies 
  of 
  the 
  ray 
  

   before 
  and 
  after 
  reflexion, 
  the 
  source 
  and 
  observer 
  being 
  at 
  

   rest. 
  If 
  we 
  put 
  /3 
  = 
  v/c 
  we 
  shall 
  have 
  

  

  1 
  + 
  2/3 
  cos 
  0+/3 
  2 
  

   n 
  =7i- 
  

  

  1-/3 
  2 
  

  

  which, 
  neglecting 
  the 
  terms 
  in 
  fi 
  2 
  , 
  reduces 
  to 
  

  

  n 
  / 
  = 
  n(l 
  + 
  2/3cos<9). 
  

  

  This 
  relation 
  is 
  the 
  same 
  as 
  that 
  of 
  Ketteler 
  f, 
  which 
  was 
  

   employed 
  by 
  Belopolski 
  % 
  in 
  his 
  investigation 
  of 
  the 
  Doppler 
  

   effect, 
  and 
  follows 
  simply 
  from 
  the 
  consideration 
  that 
  the 
  

   image 
  of 
  the 
  source 
  moves 
  with 
  the 
  velocity 
  2v 
  in 
  the 
  direc- 
  

   tion 
  of 
  the 
  normal 
  to 
  the 
  mirror 
  and, 
  consequently, 
  the 
  

   component 
  of 
  this 
  velocity 
  in 
  the 
  direction 
  of 
  the 
  reflected 
  

   ray 
  is 
  2v 
  cos 
  6. 
  

  

  If 
  now 
  we 
  suppose 
  that 
  the 
  ray 
  is, 
  by 
  suitable 
  arrange- 
  

   ments, 
  reflected 
  with 
  the 
  incidence 
  0, 
  k 
  times 
  from 
  several 
  

   mirrors 
  in 
  motion 
  with 
  the 
  velocity 
  v, 
  we 
  shall 
  have 
  

  

  n 
  =n(l 
  + 
  2£/3cos<9). 
  

  

  Therefore, 
  according 
  to 
  the 
  hypothesis 
  of 
  constant 
  velocity 
  

   of 
  light, 
  neglecting 
  the 
  terms 
  in 
  /S 
  2 
  we 
  shall 
  have 
  

  

  V 
  = 
  \(l-2££cos0). 
  

   If, 
  on 
  the 
  contrary, 
  we 
  suppose 
  that 
  the 
  velocity 
  of 
  the 
  

  

  * 
  Ann. 
  d. 
  Phys. 
  xxxix. 
  p. 
  1053 
  (1912) 
  and 
  xlv. 
  p. 
  547 
  (1915). 
  

  

  t 
  Astronomische 
  Undulationstheorie. 
  

  

  % 
  Communications 
  Ace. 
  Russe, 
  xiii. 
  p. 
  451 
  (1900). 
  

  

  