﻿Theory 
  of 
  Gravity. 
  669 
  

  

  Compare 
  this 
  with 
  the 
  Local-Time 
  of 
  Lorentz. 
  If 
  a 
  rigid 
  

   material 
  system 
  moved 
  with 
  the 
  velocity 
  u 
  o£ 
  the 
  aether 
  at 
  

   any 
  one 
  point, 
  the 
  time 
  t 
  1 
  local 
  to 
  this 
  system 
  would 
  satisfy 
  

   the 
  equation 
  

  

  #! 
  = 
  (l-u 
  2 
  /c 
  2 
  )-* 
  (dt 
  -udr/c 
  2 
  ). 
  

  

  u 
  is 
  here 
  a 
  constant, 
  the 
  velocity 
  at 
  the 
  point 
  chosen. 
  

  

  Also 
  the 
  density 
  of 
  the 
  aether 
  measured 
  relative 
  to 
  the 
  

   system 
  would 
  be 
  given 
  at 
  this 
  point 
  by 
  

  

  p 
  8 
  =(l-u 
  2 
  /c 
  2 
  )-i{ 
  /0 
  -puxi/ 
  C 
  2 
  }=(l-Ti 
  2 
  /c 
  2 
  )*p. 
  • 
  (86) 
  

   Hence 
  

  

  p 
  a 
  dt± 
  = 
  p(dt—udvlc 
  2 
  ), 
  

  

  and 
  within 
  an 
  infinitesimal 
  range 
  of 
  any 
  point 
  of 
  the 
  aether 
  

   the 
  local 
  time 
  of 
  Lorentz 
  and 
  the 
  aether 
  time 
  t 
  a 
  defined 
  by 
  

   (85) 
  are 
  connected 
  by 
  

  

  dt 
  a 
  = 
  p 
  a 
  dt 
  1 
  (87) 
  

  

  In 
  the 
  four-dimensional 
  space 
  of 
  Minkowski 
  any 
  given 
  value 
  

   of 
  f 
  x 
  defines 
  a 
  Euclidean 
  space. 
  This 
  space 
  is 
  a 
  tangent 
  to 
  

   the 
  region 
  defined 
  by 
  any 
  value 
  of 
  t 
  a 
  . 
  The 
  two 
  regions 
  

   touch 
  at 
  the 
  point 
  from 
  which 
  the 
  local 
  time 
  ^ 
  is 
  taken. 
  

  

  The 
  position 
  of 
  a 
  material 
  particle 
  in 
  three 
  dimensions 
  has 
  

   so 
  far 
  been 
  defined 
  simply 
  by 
  the 
  coordinates 
  of 
  the 
  point 
  

   in 
  which 
  a 
  four-dimensional 
  stream-line 
  is 
  cut 
  by 
  the 
  region 
  

   for 
  which 
  t 
  x 
  has 
  any 
  given 
  value. 
  In 
  the 
  same 
  way 
  the 
  point 
  

   in 
  which 
  any 
  stream-line 
  cuts 
  the 
  three-dimensional 
  space 
  

   given 
  by 
  any 
  value 
  of 
  J 
  or 
  ta 
  is 
  now 
  to 
  define 
  the 
  position 
  of 
  

   a 
  material 
  particle. 
  

  

  Let 
  dv 
  a 
  be 
  the 
  volume 
  of 
  any 
  element 
  of 
  aether. 
  The 
  

   equation 
  of 
  continuity 
  is 
  

  

  2jtG»>J 
  = 
  (88) 
  

  

  For 
  take 
  any 
  four-dimensional 
  tube 
  of 
  flow 
  of 
  the 
  aether. 
  

   The 
  section 
  made 
  by 
  giving 
  any 
  constant 
  value 
  to 
  J 
  is 
  a 
  

   three-dimensional 
  volume 
  dv 
  a 
  . 
  ds 
  a 
  , 
  the 
  element 
  of 
  length 
  

   of 
  a 
  tangent 
  to 
  the 
  tube, 
  is 
  given 
  by 
  

  

  K 
  = 
  cdt 
  x 
  , 
  (89) 
  

  

  where, 
  as 
  before, 
  t 
  x 
  is 
  the 
  local 
  time 
  o£ 
  Lorentz. 
  

  

  