﻿Theory 
  of 
  Gravity. 
  657 
  

  

  surface 
  in 
  a 
  region 
  where 
  s 
  has 
  any 
  given 
  value. 
  Thus 
  for 
  

   any 
  such 
  surface 
  the 
  value 
  of 
  the 
  induction 
  

  

  is 
  a 
  constant. 
  I 
  assume 
  that 
  this 
  constant 
  is 
  not 
  to 
  be 
  varied. 
  

  

  Hence 
  pEtfS 
  = 
  0. 
  

  

  This 
  condition 
  and 
  the 
  equations 
  (06) 
  are 
  satisfied 
  by 
  

  

  §E 
  = 
  CurlgA, 
  $H=^SA 
  + 
  VS«. 
  

  

  Here 
  SA 
  and 
  Sco 
  are 
  any 
  continuous 
  functions. 
  

  

  I 
  shall 
  confine 
  myself 
  to 
  steady 
  four-dimensional 
  states. 
  

   (55) 
  may 
  now 
  be 
  written 
  

  

  (Tl 
  i 
  i 
  H 
  (^ 
  s 
  A 
  + 
  vSc 
  °) 
  ~ 
  e 
  Curi 
  ba 
  } 
  dv 
  ds 
  dr 
  = 
  °* 
  

  

  Omitting 
  the 
  time 
  element 
  dr, 
  this 
  last 
  is 
  equivalent 
  to 
  

  

  fff{£( 
  HSA 
  ) 
  +v 
  (^ 
  H 
  "[H)}^^ 
  

  

  Hence 
  ^? 
  = 
  - 
  Curl 
  E, 
  DivH 
  = 
  0. 
  . 
  . 
  (58) 
  

  

  as 
  

  

  And 
  (57) 
  reduces 
  to 
  

  

  {^■(HSA|+V(«»H-[SAE])Trff7d*=0. 
  • 
  (59) 
  

  

  Remembering 
  (26) 
  the 
  equations 
  (52) 
  and 
  (58) 
  are 
  seen 
  to 
  

   be 
  the 
  electromagnetic 
  equations 
  (59) 
  will 
  give 
  the 
  sur- 
  

   face 
  conditions. 
  It 
  will 
  be 
  convenient 
  to 
  give 
  it 
  a 
  three- 
  

   dimensional 
  form. 
  That 
  is 
  

  

  (yr^|-^(H8A) 
  + 
  V(^H-[SAE])|^^ 
  = 
  0. 
  

  

  The 
  in 
  te 
  oration 
  with 
  respect 
  to 
  the 
  time 
  has 
  as 
  its 
  upper 
  

   limit 
  the 
  moment 
  when 
  the 
  surface 
  of 
  matter 
  crosses 
  the 
  

   volume 
  dv. 
  If 
  Un 
  is 
  the 
  normal 
  velocity 
  of 
  the 
  surface 
  

   element 
  dS, 
  then 
  (59) 
  becomes 
  

  

  ^{c- 
  1 
  J7 
  n 
  H8A-8amH 
  + 
  n[8AE]}dS<fc 
  = 
  0. 
  . 
  (60) 
  

  

  n 
  is 
  a 
  unit 
  vector 
  drawn 
  outwards 
  normally 
  from 
  the 
  surface 
  

   of 
  matter. 
  

  

  