﻿516 
  Messrs. 
  G. 
  N. 
  Lewis 
  and 
  E. 
  C. 
  Tolman 
  on 
  the 
  

  

  the 
  point 
  o£ 
  view 
  o£ 
  a 
  person 
  considered 
  at 
  rest, 
  how^ever, 
  we 
  

   have 
  just 
  seen 
  that 
  the 
  path 
  is 
  increased 
  by 
  the 
  larger 
  ratio 
  

  

  ^. 
  In 
  order 
  to 
  account 
  for 
  this 
  larger 
  difference, 
  we 
  

  

  must 
  assume 
  that 
  the 
  unit 
  of 
  length 
  in 
  the 
  moving 
  system 
  

  

  has 
  been 
  shortened 
  in 
  the 
  ratio 
  ^ 
  . 
  

  

  We 
  thus 
  see 
  that 
  a 
  metre-stick, 
  which, 
  when 
  held 
  perpen- 
  

   dicular 
  to 
  its 
  line 
  o£ 
  motion, 
  has 
  the 
  same 
  length 
  as 
  a 
  

   metre-stick 
  at 
  rest, 
  will 
  be 
  shortened 
  when 
  turned 
  parallel 
  

  

  to 
  the 
  line 
  o£ 
  motion 
  in 
  the 
  ratio 
  ^ 
  ~^ 
  , 
  and 
  indeed 
  any 
  

  

  moving 
  body 
  must 
  be 
  shortened 
  in 
  the 
  direction 
  o£ 
  its 
  motion 
  

   in 
  the 
  same 
  ratio 
  *. 
  

  

  Let 
  us 
  emphasize 
  once 
  more, 
  that 
  these 
  changes 
  in 
  the 
  

   units 
  o£ 
  time 
  and 
  length, 
  as 
  well 
  as 
  the 
  changes 
  in 
  the 
  units 
  

   of 
  mass, 
  force, 
  and 
  energy 
  which 
  we 
  are 
  about 
  to 
  discuss, 
  

   possess 
  in 
  a 
  certain 
  sense 
  a 
  purely 
  factitious 
  significance 
  ; 
  

  

  * 
  Certain 
  of 
  Einstein's 
  other 
  deductions 
  from 
  the 
  principle 
  of 
  relativity 
  

   will 
  not 
  be 
  needed 
  in 
  the 
  development 
  of 
  this 
  paper, 
  but 
  may 
  be 
  directly 
  

   obtained 
  by 
  the 
  methods 
  here 
  employed. 
  For 
  example, 
  the 
  principle 
  of 
  

   relativity 
  leads 
  to 
  certain 
  curious 
  conclusions 
  as 
  to 
  the 
  comparative 
  

   readings 
  of 
  clocks 
  in 
  a 
  system 
  assumed 
  to 
  be 
  in 
  motion. 
  Consider 
  two 
  

   systems 
  in 
  relative 
  motion. 
  An 
  observer 
  on 
  system 
  a 
  places 
  two 
  care- 
  

   fully 
  compared 
  clocks, 
  unit 
  distance 
  apart, 
  in 
  the 
  line 
  of 
  motion, 
  and 
  has 
  

   the 
  time 
  on 
  each 
  clock 
  read 
  when 
  a 
  given 
  point 
  on 
  the 
  other 
  system 
  

   passes 
  it. 
  An 
  observer 
  on 
  system 
  h 
  performs 
  a 
  similar 
  experiment. 
  The 
  

   (.lifierence 
  between 
  the 
  readings 
  of 
  the 
  two 
  clocks 
  in 
  one 
  system 
  must 
  be 
  

   the 
  same 
  as 
  the 
  difference 
  in 
  the 
  other 
  system, 
  for 
  by 
  the 
  principle 
  of 
  

   relativity, 
  the 
  relative 
  velocity 
  v 
  of 
  the 
  systems 
  must 
  appear 
  the 
  same 
  to 
  

   an 
  observer 
  in 
  either. 
  However, 
  the 
  observer 
  A, 
  considering 
  himself 
  at 
  

   rest, 
  and 
  familiar 
  with 
  the 
  change 
  in 
  the 
  units 
  of 
  length 
  and 
  time 
  in 
  the 
  

   moving 
  system 
  which 
  we 
  have 
  already 
  deduced, 
  expects 
  that 
  the 
  velocity 
  

   determined 
  by 
  B 
  will 
  be 
  greater 
  than 
  that 
  which 
  he 
  himself 
  observes 
  

  

  in 
  the 
  ratio 
  ^ 
  _ 
  , 
  since 
  he 
  has 
  concluded 
  that 
  B's 
  unit 
  of 
  time 
  is 
  

  

  longer, 
  and 
  his 
  unit 
  of 
  length 
  in 
  this 
  direction 
  is 
  shorter, 
  each 
  by 
  a 
  factor 
  

   involving 
  V'l 
  — 
  /3^. 
  The 
  only 
  possible 
  way 
  in 
  which 
  A 
  can 
  explain 
  this 
  

   discrepancy 
  is 
  to 
  assume 
  that 
  the 
  clocks 
  which 
  B 
  claims 
  to 
  have 
  set 
  

   together 
  are 
  not 
  so 
  in 
  reality. 
  In 
  other 
  words 
  he 
  has 
  to 
  conclude 
  that 
  

   clocks 
  which 
  in 
  a 
  moving 
  system 
  appear 
  to 
  be 
  set 
  together 
  really 
  read 
  

   differently 
  at 
  any 
  instant 
  (in 
  stationary 
  time), 
  and 
  that 
  a 
  given 
  clock 
  is 
  

   " 
  slower" 
  than 
  one 
  immediately 
  to 
  the 
  rear 
  of 
  it 
  by 
  an 
  amount 
  propor- 
  

   tional 
  to 
  the 
  distance. 
  From 
  what 
  has 
  preceded 
  it 
  can 
  be 
  readily 
  shown 
  

   that 
  if 
  in 
  a 
  moving 
  system 
  two 
  clocks 
  are 
  situated, 
  one 
  in 
  front 
  of 
  the 
  

   other 
  by 
  a 
  distance 
  /, 
  in 
  units 
  of 
  this 
  svstem, 
  the 
  difference 
  in 
  setting 
  will 
  

  

  Iv 
  

   be 
  ^ 
  . 
  From 
  this 
  point 
  Einstein's 
  equations 
  concerning 
  the 
  addition 
  of 
  

  

  velocities 
  also 
  follow 
  directly. 
  

  

  