﻿Principle 
  of 
  Relativity 
  and 
  2^ 
  on- 
  Newtonian 
  Mechanics. 
  521 
  

   the 
  electromagnetic 
  theory, 
  is 
  diminished 
  in 
  the 
  ratio 
  

  

  ^^ 
  — 
  - 
  — 
  — 
  if 
  thev 
  are 
  moving 
  perpendicular 
  to 
  the 
  line 
  

  

  1 
  ' 
  . 
  . 
  1-/3^ 
  

  

  joining 
  them 
  : 
  and 
  in 
  the 
  ratio 
  — 
  - 
  — 
  if 
  moving 
  parallel 
  to 
  

  

  the 
  line 
  joining 
  them. 
  

  

  From 
  the 
  standpoint 
  of 
  the 
  principle 
  o£ 
  relativity, 
  one 
  of 
  

   the 
  most 
  interesting 
  quantities 
  in 
  mechanics 
  is 
  the 
  so-called 
  

   kinetic 
  energy, 
  which 
  is 
  the 
  increase 
  in 
  energy 
  attributed 
  to 
  

   a 
  body 
  when 
  it 
  is 
  set 
  in 
  motion 
  with 
  respect 
  to 
  an 
  arbitrarily 
  

   chosen 
  point 
  of 
  rest. 
  Knowing 
  the 
  change 
  of 
  the 
  mass 
  with 
  

   velocity 
  as 
  given 
  by 
  equation 
  (1), 
  the 
  general 
  equation 
  for 
  

   kinetic 
  energy 
  *, 
  E', 
  may 
  readily 
  be 
  shown 
  to 
  be, 
  

  

  E' 
  = 
  '"o<vi^-0 
  ^^) 
  

  

  From 
  equations 
  1 
  and 
  2 
  we 
  may 
  derive 
  one 
  of 
  the 
  most 
  

   interesting 
  consequences 
  of 
  the 
  principle 
  of 
  relativity. 
  If 
  E 
  

   is 
  the 
  total 
  energy 
  (including 
  internal 
  energy) 
  of 
  a 
  body 
  in 
  

   motion, 
  and 
  Eg 
  is 
  its 
  energy 
  at 
  rest, 
  the 
  kinetic 
  energy 
  E' 
  is 
  

   equal 
  to 
  E— 
  E^ 
  and 
  equation 
  (2) 
  may 
  be 
  written. 
  

  

  Moreover, 
  we 
  may 
  write 
  equation 
  (1) 
  in 
  the 
  form 
  

  

  and 
  dividing 
  (3) 
  by 
  (4), 
  

  

  (5) 
  

  

  In 
  other 
  words 
  when 
  a 
  body 
  is 
  in 
  motion 
  its 
  energy 
  and 
  

   mass 
  are 
  both 
  increased, 
  and 
  the 
  increase 
  in 
  energy 
  is 
  equal 
  

   to 
  the 
  increase 
  in 
  mass 
  multiplied 
  by 
  the 
  square 
  of 
  the 
  

   velocity 
  of 
  light. 
  From 
  the 
  conservation 
  laws 
  we 
  know 
  that 
  

   when 
  a 
  body 
  is 
  set 
  in 
  motion 
  and 
  thus 
  acquires 
  mass 
  and 
  

   energy, 
  these 
  must 
  come 
  from 
  the 
  environment. 
  So 
  also 
  

   when 
  a 
  moving 
  body 
  is 
  brought 
  to 
  rest 
  it 
  must 
  give 
  up 
  mass 
  

   as 
  well 
  as 
  energy 
  to 
  the 
  environment. 
  The 
  mass 
  thus 
  acquired 
  

   by 
  the 
  environment 
  is 
  independent 
  of 
  the 
  particular 
  form 
  

  

  * 
  Consider 
  a 
  body 
  moving 
  with 
  the 
  velocity 
  v 
  subjected 
  to 
  a 
  force 
  f 
  

   in 
  the 
  line 
  of 
  its 
  motion. 
  Its 
  momentum 
  M 
  and 
  its 
  kinetic 
  energy 
  E' 
  

   will 
  be 
  changed 
  by 
  the 
  amounts 
  <ZM 
  =fdt, 
  c^E' 
  =fdl=fvdt. 
  Hence 
  

   .d¥J 
  = 
  vdM. 
  or 
  substituting 
  mv 
  for 
  M. 
  dE' 
  = 
  vivdv+v^dm. 
  Eliminating 
  

   m 
  between 
  this 
  equation 
  and 
  equation 
  (1), 
  and 
  integrating, 
  gives 
  at 
  once 
  

   the 
  above 
  equation 
  (2). 
  

  

  Phil, 
  Mag, 
  !S. 
  6. 
  Vol. 
  18. 
  No. 
  106. 
  Oct, 
  1909. 
  2 
  N 
  

  

  