﻿518 
  Messrs. 
  G. 
  N. 
  Lewis 
  and 
  R. 
  C. 
  Tolman 
  on 
  the 
  

  

  ball, 
  as 
  measured 
  by 
  A, 
  is 
  the 
  same 
  as 
  the 
  chancre 
  in 
  velocity 
  

   of 
  the 
  other 
  ball, 
  as 
  measured 
  by 
  B. 
  This 
  being 
  the 
  case, 
  

   the 
  observer 
  A, 
  considering 
  himself 
  at 
  rest, 
  concludes 
  that 
  the 
  

   real 
  change 
  in 
  velocity 
  of 
  the 
  ball 
  h 
  is 
  different 
  from 
  that 
  of 
  

   his 
  own, 
  for 
  he 
  remembers 
  that 
  while 
  the 
  unit 
  of 
  length 
  is 
  tlie 
  

   same 
  in 
  this 
  transverse 
  direction 
  in 
  both 
  systems, 
  the 
  unit 
  of 
  

   time 
  is 
  longer 
  in 
  the 
  moving 
  system. 
  

  

  Velocity 
  is 
  measured 
  in 
  centimetres 
  per 
  second, 
  and 
  since 
  

   the 
  second 
  is 
  longer 
  in 
  the 
  moving 
  system, 
  while 
  the 
  centi- 
  

   metre 
  in 
  the 
  direction 
  which 
  we 
  are 
  considering 
  is 
  the 
  same 
  

   in 
  both 
  systems, 
  the 
  observer 
  A, 
  always 
  using 
  the 
  units 
  of 
  his 
  

   own 
  system, 
  concludes 
  that 
  the 
  change 
  in 
  velocity 
  of 
  the 
  ball 
  

  

  h 
  is 
  smaller 
  in 
  the 
  ratio 
  :j 
  than 
  the 
  change 
  in 
  velocity 
  

  

  of 
  the 
  ball 
  a. 
  The 
  change 
  in 
  velocity 
  of 
  each 
  ball 
  multiplied 
  

   by 
  its 
  mass 
  gives 
  its 
  change 
  in 
  momentum. 
  Now, 
  from 
  

   the 
  law 
  of 
  conservation 
  of 
  momentum, 
  A 
  assumes 
  that 
  

   each 
  ball 
  experiences 
  the 
  same 
  change 
  in 
  momentum, 
  and 
  

   therefore 
  since 
  he 
  has 
  already 
  decided 
  that 
  the 
  ball 
  h 
  has 
  expe- 
  

   rienced 
  a 
  smaller 
  change 
  of 
  velocity 
  in 
  the 
  ratio 
  _^l 
  l/i, 
  he 
  

  

  must 
  conclude 
  that 
  the 
  mass 
  of 
  the 
  ball 
  in 
  system 
  h 
  is 
  greater 
  

   than 
  that 
  of 
  his 
  own 
  in 
  the 
  ratio 
  ■ 
  — 
  , 
  In 
  eeneraL 
  

  

  therefore, 
  we 
  must 
  assume 
  that 
  the 
  mass 
  of 
  a 
  body 
  increases 
  

   with 
  its 
  velocity. 
  We 
  must 
  bear 
  in 
  mind, 
  however, 
  as 
  in 
  all 
  

   other 
  cases, 
  that 
  the 
  motion 
  is 
  determined 
  with 
  respect 
  to 
  

   some 
  point 
  arhitrarily 
  chosen 
  as 
  a 
  point 
  of 
  rest. 
  

  

  If 
  m 
  is 
  the 
  mass 
  of 
  a 
  body 
  in 
  motion 
  and 
  m^ 
  its 
  mass 
  at 
  

   rest, 
  we 
  have 
  * 
  

  

  m 
  

  

  mo 
  Vl— 
  /3' 
  

  

  (1) 
  

  

  The 
  only 
  opportunity 
  of 
  testing 
  experimentally 
  (.he 
  change 
  

   of 
  a 
  body's 
  mass 
  with 
  its 
  velocity 
  has 
  been 
  afforded 
  by 
  the 
  

   experiments 
  on 
  the 
  mass 
  of 
  a 
  moving 
  electron 
  or 
  j3 
  particle. 
  

   The 
  actual 
  measurements 
  were 
  indeed 
  not 
  of 
  the 
  mass 
  

  

  of 
  the 
  electron, 
  but 
  of 
  the 
  ratio 
  of 
  charge 
  to 
  mass 
  I— 
  V 
  

  

  « 
  \m} 
  

  

  * 
  This 
  equation 
  and 
  others 
  developed 
  in 
  this 
  section 
  are 
  identical 
  with 
  

   those 
  obtained 
  through 
  an 
  entirely 
  different 
  course 
  of 
  reasoning 
  by 
  Lewis 
  

   (Phil. 
  Mag. 
  xvi. 
  p. 
  705, 
  1908). 
  The 
  equations 
  were 
  thei'e 
  obtained 
  for 
  

   systenas 
  in 
  motion 
  with 
  respect 
  to 
  a 
  point 
  at 
  absolute 
  rest. 
  We 
  shall 
  

   show 
  here, 
  however, 
  that 
  they 
  are 
  true, 
  whatever 
  arbitrary 
  point 
  is 
  

   selected 
  as 
  a 
  point 
  of 
  rest. 
  

  

  