﻿514: 
  Messrs. 
  G. 
  N. 
  Lewis 
  and 
  E,. 
  C. 
  Tolman 
  on 
  the 
  

  

  Thus 
  to 
  each 
  observer 
  it 
  seems 
  that 
  the 
  other's 
  clock 
  is 
  running 
  

   too 
  slowly. 
  

  

  This 
  divergence 
  o£ 
  opinion 
  evidently 
  depends 
  not 
  so 
  much 
  

   on 
  the 
  fact 
  that 
  the 
  tv^o 
  systems 
  are 
  in 
  relative 
  motion, 
  but 
  

   on 
  the 
  fact 
  that 
  each 
  observer 
  arbitrarily 
  assumes 
  that 
  his 
  

   own 
  system 
  is 
  at 
  rest. 
  If, 
  however, 
  they 
  both 
  decide 
  to 
  call 
  

   A^s 
  system 
  at 
  rest, 
  then 
  both 
  will 
  agree 
  that 
  in 
  the 
  two 
  

   experiments 
  the 
  light 
  passes 
  over 
  the 
  paths 
  opo 
  and 
  mnm^ 
  

   respectively, 
  and 
  that 
  B's 
  clock 
  runs 
  more 
  slowly 
  that 
  A's. 
  

   In 
  general, 
  whatever 
  point 
  may 
  be 
  arbitrarily 
  chosen 
  as 
  a 
  

   point 
  of 
  rest, 
  it 
  will 
  be 
  concluded 
  that 
  any 
  clock 
  in 
  motion 
  

   relative 
  to 
  this 
  point 
  runs 
  too 
  slowly. 
  

  

  Consider 
  fig. 
  1 
  again, 
  assuming 
  system 
  a 
  at 
  rest. 
  We 
  

   have 
  shown 
  that 
  it 
  is 
  necessary 
  to 
  assume 
  that 
  B's 
  clock 
  runs 
  

   more 
  slowly 
  than 
  A's 
  in 
  the 
  ratio 
  of 
  the 
  lengths 
  of 
  the 
  path 
  

   opo 
  to 
  the 
  pathmn??!^; 
  in 
  other 
  words, 
  the 
  second 
  of 
  B's 
  clock 
  

   is 
  longer 
  than 
  the 
  second 
  of 
  A's, 
  in 
  the 
  ratio 
  7nnm' 
  to 
  opo. 
  This 
  

   ratio 
  between 
  the 
  two 
  paths 
  will 
  evidently 
  depend 
  on 
  the 
  

   relative 
  velocity 
  of 
  the 
  two 
  systems, 
  v, 
  and 
  on 
  the 
  velocity 
  of 
  

   light, 
  c. 
  

  

  Obviously 
  from 
  the 
  figure, 
  

  

  (opY 
  = 
  (hiy 
  — 
  {mny—{miy. 
  

   Dividing 
  by 
  {mny, 
  

  

  [mny 
  (inny 
  ' 
  

  

  But 
  the 
  distance 
  ml 
  is 
  to 
  the 
  distance 
  mn 
  as 
  v 
  is 
  to 
  c. 
  

   Hence 
  

  

  mn 
  1 
  

  

  Denoting 
  the 
  important 
  ratio 
  - 
  by 
  the 
  letter 
  y8, 
  we 
  see 
  

  

  that 
  in 
  general 
  a 
  second 
  measured 
  by 
  a 
  moving 
  clock 
  

   bears 
  to 
  a 
  second 
  measured 
  by 
  a 
  stationary 
  clock 
  the 
  ratio 
  

   _ 
  1 
  

  

  Whatever 
  assumption 
  the 
  observers 
  A 
  and 
  B 
  may 
  make 
  as 
  

   to 
  their 
  motion, 
  it 
  is 
  obvious 
  that 
  their 
  measurements 
  of 
  

   length, 
  at 
  least 
  in 
  a 
  direction 
  perpendicular 
  to 
  their 
  line 
  of 
  

   relative 
  motion, 
  will 
  lead 
  to 
  no 
  disagreement. 
  For 
  evidently, 
  

   if 
  each 
  observer 
  with 
  a 
  measuring-rod 
  determines 
  the 
  distance 
  

   from 
  his 
  system 
  to 
  the 
  other, 
  the 
  two 
  determinations 
  must 
  

  

  