﻿Fundamental 
  Laws 
  of 
  Matter 
  and 
  Energy. 
  715 
  

  

  testing 
  this 
  point 
  would 
  be 
  very 
  great, 
  of 
  course, 
  but 
  perhaps 
  

   not 
  insurmountable. 
  

  

  The 
  plausibility 
  of 
  our 
  fundamental 
  assumption 
  which 
  led 
  

   directly 
  to 
  equation 
  (15) 
  has 
  been 
  greatly 
  increased 
  by 
  the 
  

   agreement 
  between 
  this 
  equation 
  and 
  Kaufinann's 
  results, 
  

   and 
  also 
  perhaps 
  by 
  the 
  similarity 
  between 
  this 
  equation 
  and 
  

   those 
  deduced 
  from 
  electromagnetic 
  theory. 
  But 
  the 
  simplest 
  

   as 
  well 
  as 
  the 
  strongest 
  evidence 
  of 
  the 
  correctness 
  of 
  our 
  

   point 
  of 
  view 
  comes 
  from 
  a 
  consideration 
  of 
  the 
  non-New- 
  

   tonian 
  equation 
  for 
  kinetic 
  energy. 
  

  

  The 
  integration 
  of 
  equation 
  (14) 
  obviously 
  does 
  not 
  yield 
  

   the 
  simple 
  Newtonian 
  equation, 
  

  

  E'==l/2mv 
  2 
  . 
  

  

  This 
  equation 
  must 
  be 
  replaced 
  by 
  one 
  that 
  is 
  obtained 
  as 
  

   follows 
  : 
  — 
  

  

  Let 
  a 
  body, 
  which 
  at 
  rest 
  has 
  the 
  mass 
  m 
  , 
  be 
  given 
  the 
  

   velocity 
  v. 
  As 
  its 
  internal 
  energy 
  changes, 
  its 
  mass 
  will 
  

   change 
  according 
  to 
  equation 
  (7), 
  and 
  

  

  E' 
  

  

  where 
  E' 
  is 
  the 
  acquired 
  kinetic 
  energy 
  and 
  m 
  — 
  m 
  h 
  the 
  

   increase 
  in 
  m3ss. 
  

  

  Eliminating 
  m 
  between 
  this 
  equation 
  and 
  (15) 
  gives 
  

  

  E'=7?iY 
  2 
  [l-(l-/3 
  2 
  ) 
  12 
  ] 
  (16) 
  

  

  This 
  is 
  the 
  general 
  formula 
  for 
  the 
  kinetic 
  energy 
  of 
  a 
  moving- 
  

   body. 
  As 
  usual 
  ft 
  represents 
  v/Y, 
  the 
  ratio 
  of 
  this 
  velocity 
  

   to 
  the 
  velocity 
  of 
  light. 
  

  

  From 
  equations 
  (10), 
  (15), 
  and 
  (16) 
  may 
  be 
  constructed 
  

   the 
  whole 
  science 
  of 
  non-Newtonian 
  dynamics, 
  of 
  which 
  

   Newtonian 
  dynamics 
  furnishes 
  a 
  limiting 
  case, 
  namely, 
  for 
  

   velocities 
  which 
  are 
  negligible 
  in 
  comparison 
  with 
  the 
  velocity 
  

   of 
  light. 
  

  

  For 
  example, 
  expanding 
  (16) 
  by 
  the 
  binomial 
  theorem 
  gives 
  

  

  E'=mVW+&8*.....) 
  (17) 
  

  

  For 
  low 
  values 
  of 
  ft 
  the 
  higher 
  terms 
  may 
  be 
  neglected 
  and 
  

  

  E'=imv 
  2 
  . 
  

  

  That 
  is, 
  the 
  limit 
  approached 
  by 
  the 
  kinetic 
  energy 
  of 
  a 
  

   body 
  as 
  the 
  velocity 
  approaches 
  zero 
  is, 
  as 
  in 
  ordinary 
  

   mechanics, 
  one 
  half 
  the 
  product 
  of 
  the 
  mass 
  and 
  the 
  square 
  

   of 
  the 
  velocity. 
  At 
  the 
  other 
  limit 
  of 
  velocity 
  when 
  ft 
  = 
  1, 
  

   it 
  follows 
  from 
  (16) 
  that 
  

  

  E'=mV 
  2 
  (18) 
  

  

  