﻿Experiment 
  relating 
  to 
  the 
  Drift 
  of 
  the 
  Miher. 
  23 
  

  

  The 
  denominator 
  may 
  be 
  simplified 
  so 
  that 
  in 
  this 
  case 
  

  

  _ 
  Acqs 
  d 
  (V 
  2 
  - 
  v 
  2 
  ) 
  ( 
  Y 
  2 
  - 
  w 
  2 
  ) 
  (V 
  2 
  - 
  w' 
  2 
  ) 
  

  

  P= 
  V 
  2 
  Q 
  ^[{V 
  2 
  -v 
  2 
  yGo^ 
  x 
  + 
  \{ 
  V2 
  - 
  u2 
  )^X 
  + 
  '^vcos 
  x 
  } 
  2 
  ] 
  

  

  10. 
  This 
  seems 
  the 
  proper 
  place 
  to 
  give 
  a 
  formal 
  proof 
  of 
  

   the 
  statement 
  made 
  above 
  from 
  general 
  considerations 
  that 
  

   the 
  apparent 
  frequencies 
  of 
  each 
  wave-train 
  with 
  respect 
  to 
  a 
  

   point 
  moving 
  with 
  the 
  apparatus 
  are 
  the 
  same. 
  

  

  The 
  velocity 
  of 
  such 
  a 
  point 
  relative 
  to 
  the 
  first 
  train 
  is 
  

  

  V 
  + 
  Usin(a-A). 
  

   Its 
  apparent 
  frequency 
  is 
  therefore 
  

  

  V 
  i-Usin(a-A) 
  

  

  \ 
  

  

  Now 
  A 
  = 
  <f>i 
  l 
  , 
  Hence 
  

  

  V 
  -j- 
  U 
  sin 
  (a 
  — 
  A) 
  = 
  V 
  + 
  v 
  cos 
  0/ 
  — 
  u 
  sin 
  </>/. 
  

  

  Substituting 
  for 
  <£/ 
  in 
  terms 
  of 
  $ 
  1? 
  it 
  will 
  be 
  seen 
  that 
  

  

  V 
  2 
  — 
  v 
  2 
  

   Y 
  + 
  Usin 
  (a 
  — 
  A) 
  = 
  — 
  jy~ 
  {V 
  — 
  r 
  cos^x— 
  wsin^i}-. 
  

  

  Again, 
  (j> 
  x 
  = 
  <j> 
  + 
  <j>' 
  — 
  ^tt. 
  Substituting 
  this 
  and 
  expressing 
  

   (/>' 
  in 
  terms 
  of 
  its 
  angle 
  of 
  incidence 
  <£ 
  + 
  6, 
  

  

  Y 
  — 
  v 
  cos 
  (f>i 
  — 
  u 
  sin 
  <£ 
  2 
  

  

  — 
  -p 
  — 
  \ 
  V 
  + 
  w 
  cos 
  (<£ 
  + 
  #) 
  — 
  (v 
  cos 
  4- 
  u 
  sin 
  <£ 
  ) 
  sin 
  (<f> 
  + 
  0)\; 
  

   whence 
  

   V+ 
  U 
  sin 
  («-A) 
  = 
  ~^ 
  2 
  ' 
  ""/ 
  (V^ysinfl-Mooa 
  6») 
  

  

  = 
  ^Y--Ucos(0--«)^ 
  

   Similarly, 
  it 
  can 
  be 
  shown 
  that 
  

  

  Y 
  + 
  Usin(a-B)=^{Y-Ucos(<9-«)}; 
  

  

  A 
  

  

  whence 
  

  

  V 
  + 
  Usin 
  («-A) 
  _ 
  V 
  + 
  U 
  sinQ-B) 
  _ 
  Y-Ucos 
  [6 
  -a) 
  

   A,] 
  X 
  2 
  A, 
  

  

  which 
  shows 
  that 
  the 
  frequencies 
  are 
  equal. 
  

   But 
  further, 
  

  

  Y— 
  U 
  cos 
  (6-a) 
  =Y-w 
  cos 
  0-r 
  sin 
  0. 
  

  

  