﻿670 
  Prof. 
  S. 
  B. 
  McLaren 
  on 
  a 
  

  

  Under 
  the 
  Lorentz-Einstein 
  substitution 
  

  

  is 
  an 
  invariant 
  ; 
  and 
  

  

  dv 
  a 
  ds 
  a 
  

  

  is 
  the 
  "volume" 
  of 
  aether 
  which 
  has 
  flowed 
  across 
  the 
  section 
  

   dv 
  a 
  of 
  a 
  tube, 
  ds 
  a 
  being 
  the 
  distance 
  the 
  sether 
  has 
  moved 
  

   along 
  it. 
  Also 
  (24) 
  may 
  be 
  written 
  

  

  So 
  that 
  dv 
  a 
  ds 
  a 
  varies 
  as 
  p 
  a 
  dv 
  a 
  . 
  Hence 
  the 
  equation 
  of 
  con- 
  

   tinuity 
  is 
  expressed 
  by 
  making 
  

  

  p 
  a 
  dv 
  a 
  

  

  a 
  constant 
  along 
  the 
  tube, 
  that 
  is 
  by 
  (88). 
  

  

  I 
  now 
  write 
  the 
  electromagnetic 
  equations 
  in 
  a 
  form 
  due 
  

   to 
  Hargreaves 
  (Camb. 
  Phil. 
  Trans, 
  vol. 
  xxi. 
  p. 
  107)* 
  

  

  [J 
  (E 
  ± 
  dy 
  dz 
  + 
  E 
  2 
  dz 
  dx 
  + 
  E 
  3 
  dxdy 
  ""j 
  

  

  — 
  H 
  x 
  dx 
  cdt 
  — 
  H 
  9 
  dy 
  cdt 
  —H*dz 
  cdt) 
  — 
  , 
  

  

  Y- 
  (9°) 
  

  

  J 
  J 
  ( 
  H 
  x 
  dy 
  dz 
  + 
  H 
  2 
  dz 
  dx 
  + 
  B 
  3 
  dx 
  dy 
  

  

  + 
  E 
  1 
  dx 
  cdt 
  + 
  E 
  2 
  dy 
  cdt 
  + 
  E 
  3 
  dz 
  cdt) 
  = 
  0, 
  J 
  

  

  The 
  expressions 
  on 
  the 
  left 
  of 
  (90) 
  are 
  differential 
  in- 
  

   variants. 
  Apply 
  (90) 
  to 
  a 
  region 
  bounded 
  by 
  the 
  tubes 
  of 
  

   flow 
  or 
  by 
  equipotentials 
  where 
  J 
  or 
  t 
  a 
  have 
  constant 
  values. 
  

  

  Then 
  by 
  (87) 
  write 
  in 
  (90) 
  

  

  dt 
  l 
  — 
  Pa 
  ldt 
  a> 
  

  

  The 
  equations 
  (90) 
  yield 
  the 
  results 
  

  

  dS 
  a 
  is 
  any 
  element 
  of 
  area 
  lying 
  in 
  the 
  space 
  defined 
  by 
  a 
  

   constant 
  value 
  of 
  t 
  a 
  ; 
  dx 
  a 
  is 
  an 
  element 
  of 
  length 
  of 
  the 
  curve 
  

   bounding 
  this 
  area. 
  Write 
  

  

  E 
  = 
  p 
  a 
  E 
  a 
  , 
  H 
  = 
  /?«H„, 
  

   * 
  See 
  H. 
  Bateman, 
  Proc. 
  Lond. 
  Math. 
  Soc. 
  ser. 
  2, 
  vol.viii. 
  p. 
  227. 
  

  

  