﻿Theory 
  of 
  Gravity. 
  649 
  

  

  at 
  Q 
  if 
  the 
  distance 
  in 
  three-dimensional 
  space 
  between 
  

   P 
  and 
  Q 
  is 
  greater 
  than 
  light 
  can 
  travel 
  in 
  the 
  interval 
  o£ 
  

   time 
  between 
  the 
  two 
  instants. 
  This 
  assumes 
  the 
  velocity 
  o£ 
  

   light 
  to 
  be 
  the 
  maximum 
  physical 
  velocity. 
  

  

  Let 
  £p 
  and 
  t§ 
  denote 
  the 
  times 
  and 
  rj? 
  and 
  Tq 
  the 
  vector 
  

   coordinates 
  of 
  P 
  and 
  Q. 
  If 
  

  

  (r 
  P 
  -v 
  q 
  y-c%t 
  r 
  -t 
  q 
  y>o, 
  . 
  . 
  . 
  (i.) 
  

  

  then 
  P 
  is 
  neither 
  " 
  before 
  " 
  nor 
  " 
  after 
  " 
  Q. 
  

   If 
  (rp 
  _ 
  rQ 
  y_ 
  c2 
  ( 
  tp 
  _^)2 
  <0 
  ,. 
  . 
  . 
  (II.) 
  

  

  then 
  P 
  is 
  before 
  or 
  after 
  Q 
  according 
  as 
  £q 
  is 
  greater 
  or 
  less 
  

   than 
  £ 
  P 
  . 
  The 
  expression 
  on 
  the 
  left 
  of 
  (I.) 
  and 
  (II.) 
  is 
  an 
  

   invariant 
  under 
  the 
  Lorentz-Einstein 
  substitution. 
  

  

  Further, 
  provided 
  (II.) 
  holds, 
  the 
  sign 
  of 
  t? 
  —t^ 
  is 
  unaltered 
  

   by 
  that 
  substitution 
  if 
  we 
  assume 
  again 
  that 
  all 
  velocities 
  

   are 
  less 
  than 
  that 
  of 
  light. 
  When 
  the 
  relations 
  " 
  before 
  " 
  or 
  

   " 
  after 
  " 
  hold 
  at 
  all 
  between 
  two 
  instants 
  P 
  and 
  Q 
  they 
  are 
  

   invariant 
  relations. 
  

  

  The 
  theory 
  of 
  relativity, 
  so 
  unjustly 
  accused 
  of 
  abstract- 
  

   ness 
  and 
  paradox, 
  keeps 
  with 
  unswerving 
  fidelity 
  to 
  the 
  safe 
  

   path 
  of 
  experience 
  and 
  common-sense. 
  

  

  § 
  4, 
  Electrodynamics 
  in 
  Four 
  Dimensions. 
  

  

  I 
  propose 
  now 
  to 
  extend 
  to 
  four 
  dimensions 
  the 
  formulae 
  

   (16) 
  and 
  (1). 
  

  

  SffijfiLtdvtdsidT^O 
  (27) 
  

  

  i 
  „ 
  (8 
  „,-,(* 
  + 
  v„,)- 
  + 
  , 
  8 
  .,-.(*- 
  i+ 
  gf 
  

  

  - 
  (&•■) 
  ->(~1 
  - 
  V*,)' 
  -(SV) 
  - 
  1 
  (Curl 
  FO 
  2 
  

  

  +4 
  *w{©- 
  + 
  (D'-4 
  

  

  Here 
  the 
  spatial 
  coordinates 
  are 
  x 
  u 
  y 
  l5 
  ,?-,, 
  s 
  l9 
  and 
  the 
  

   time 
  is 
  t 
  x 
  . 
  I 
  have 
  reserved 
  the 
  vector 
  notation 
  for 
  three 
  

   dimensions. 
  

  

  There 
  is 
  in 
  (28) 
  a 
  vector 
  whose 
  components 
  are 
  F, 
  fa 
  and 
  

   a 
  scalar 
  J 
  x 
  . 
  It 
  will 
  be 
  noticed 
  that 
  all 
  four 
  coordinates 
  enter 
  

   into 
  (28) 
  on 
  exactly 
  the 
  same 
  terms 
  ; 
  the 
  space 
  is 
  in 
  fact 
  

  

  (28) 
  

  

  