﻿642 
  Prof. 
  S. 
  B. 
  McLaren 
  on 
  a 
  

  

  requires 
  L 
  to 
  be 
  an 
  invariant 
  under 
  the 
  Lorentz-Einstein 
  

   substitution 
  

  

  a) 
  a 
  /3 
  2 
  (1-* 
  2 
  ) 
  = 
  1 
  .... 
  (A) 
  

  

  c 
  being 
  the 
  velocity 
  of 
  light. 
  

   The 
  simplest 
  solution 
  is 
  to 
  write 
  

  

  Z 
  = 
  i 
  / 
  d 
  2 
  (u 
  2 
  -c 
  2 
  ) 
  (2) 
  

  

  In 
  varying 
  the 
  distribution 
  of 
  matter 
  remember 
  that 
  

  

  8(pdv) 
  = 
  (3) 
  

  

  The 
  condition 
  (3) 
  is 
  associated 
  with 
  the 
  equation 
  of 
  con- 
  

   tinuity 
  

  

  )|+Div 
  P 
  u=0 
  (4) 
  

  

  The 
  equations 
  of 
  motion 
  are 
  now 
  deduced 
  by 
  the 
  ordinary 
  

   methods 
  of 
  the 
  calculus 
  of 
  variations. 
  They 
  are 
  

  

  ~( 
  P 
  2 
  udo)+V&yc 
  2 
  -ip 
  2 
  tf)dv 
  = 
  0. 
  . 
  . 
  (5) 
  

   Here 
  d 
  d 
  , 
  _ 
  

  

  so 
  that 
  ti 
  denotes 
  here 
  and 
  always 
  the 
  time 
  rate 
  of 
  change 
  

  

  at 
  a 
  moving 
  point. 
  

   (4) 
  is 
  equivalent 
  to 
  

  

  ?W=0 
  00 
  

  

  Thus 
  (5) 
  becomes 
  

  

  p|>(^) 
  + 
  V(i|oV-i 
  P 
  2 
  u 
  2 
  ) 
  = 
  0. 
  ... 
  (7) 
  

  

  According 
  to 
  (5) 
  the 
  momentum 
  per 
  unit 
  volume 
  varies 
  as 
  

   the 
  pressure 
  varies 
  as 
  

  

  p 
  2 
  u, 
  

  

  i^-u 
  2 
  ). 
  

   Further, 
  the 
  energy 
  is 
  per 
  unit 
  volume 
  

  

  For 
  J, 
  QfWdv 
  + 
  ipPMv) 
  

  

  =pu^( 
  P 
  VL) 
  dv 
  + 
  p<?%pdv+ 
  (J/) 
  2 
  u 
  2 
  -f 
  ipVJDivu 
  

  

  = 
  _uV(|pV- 
  ip 
  2 
  u 
  2 
  ) 
  dv 
  - 
  p 
  2 
  c 
  2 
  DiYudv 
  + 
  \ 
  (p 
  2 
  u 
  2 
  + 
  \p 
  V) 
  Div 
  ndv 
  

   = 
  -ViY{u(ip 
  2 
  c 
  2 
  -ip 
  2 
  u 
  2 
  )}dv. 
  

  

  