﻿Postulate 
  of 
  the 
  Theory 
  of 
  Relativity. 
  165 
  

  

  above-mentioned 
  hypothesis 
  or 
  mechanical 
  theory 
  and 
  the 
  

   relativistic 
  one. 
  It 
  is 
  indeed 
  possible 
  to 
  see 
  that 
  some 
  

   method, 
  one 
  moreover 
  already 
  in 
  use, 
  adopted 
  for 
  the 
  verifi- 
  

   cation 
  of 
  Doppler's 
  principle 
  may 
  serve 
  for 
  the 
  solution 
  of 
  

   the 
  above-quoted 
  problem. 
  

  

  In 
  order 
  to 
  see 
  this, 
  let 
  us 
  consider 
  a 
  luminous 
  source 
  S 
  

   which 
  emits 
  waves 
  of 
  length 
  X 
  and 
  of 
  frequency 
  n 
  moving 
  

   towards 
  the 
  observer 
  fixed 
  at 
  (fig. 
  1). 
  If 
  we 
  suppose 
  

  

  Fig. 
  1. 
  

   f 
  S* 
  A 
  A 
  Q 
  . 
  R 
  B 
  f 
  

  

  «. 
  C 
  ^.^.y^ 
  4 
  c 
  *~V*-* 
  

  

  that 
  the 
  waves 
  are 
  transmitted 
  through 
  a 
  stationary 
  sether, 
  

   the 
  n 
  waves 
  emitted 
  in 
  a 
  second 
  by 
  S 
  will 
  be 
  distributed 
  

   over 
  tbe 
  segment 
  f$ 
  / 
  A=c—v. 
  In 
  the 
  same 
  time 
  all 
  the 
  n 
  

   waves 
  distributed 
  in 
  the 
  segment 
  OB 
  = 
  c 
  will 
  have 
  passed 
  

   through 
  : 
  we 
  have 
  therefore 
  

  

  c— 
  v 
  c 
  , 
  c 
  

  

  = 
  — 
  , 
  or 
  n 
  =n 
  . 
  

  

  n 
  n 
  c 
  — 
  v 
  

  

  If 
  we 
  put 
  v\c 
  = 
  (B 
  and 
  neglect 
  terms 
  of 
  higher 
  order 
  than 
  the 
  

   first 
  in 
  /3 
  we 
  have 
  

  

  n'=h(l+/3). 
  

  

  The 
  new 
  wave-length 
  is 
  obtained 
  by 
  the 
  relations 
  

   c 
  = 
  n\ 
  = 
  n'\' 
  : 
  

  

  A'=X(l-0). 
  

  

  If 
  now 
  instead 
  of 
  the 
  hypothesis 
  of 
  a 
  stationary 
  medium 
  we 
  

   adopt 
  the 
  ballistic 
  or 
  emissive 
  hypothesis 
  of 
  which 
  we 
  have 
  

   spoken 
  above, 
  we 
  shall 
  find 
  that 
  in 
  one 
  second 
  the 
  n 
  waves 
  

   emitted 
  by 
  S 
  will 
  be 
  distributed 
  over 
  the 
  segment 
  S'A' 
  = 
  c. 
  

   In 
  the 
  same 
  time 
  there 
  will 
  pass 
  through 
  0, 
  n' 
  waves 
  which 
  

   will 
  be 
  distributed 
  over 
  the 
  segment 
  0B' 
  = 
  c 
  + 
  r. 
  We 
  have, 
  

   therefore, 
  

  

  C 
  =^ 
  r 
  l 
  \orn' 
  = 
  n(l+/3). 
  

   n 
  n 
  

  

  And 
  since 
  c 
  = 
  n\ 
  and 
  c 
  + 
  r 
  = 
  n'\' 
  we 
  see 
  that, 
  in 
  this 
  case, 
  

  

  As 
  regards 
  the 
  frequency 
  we 
  arrive, 
  therefore, 
  at 
  the 
  same 
  

   conclusions 
  (with 
  the 
  exception 
  of 
  the 
  terms 
  in 
  /3 
  2 
  ) 
  whether 
  

   we 
  adopt 
  the 
  sethereal 
  or 
  the 
  ballistic 
  hypothesis 
  ; 
  but 
  for 
  

   the 
  wave-length 
  we 
  obtain 
  different 
  values 
  from 
  the 
  two 
  

  

  