﻿652 
  Prof. 
  S. 
  B. 
  McLaren 
  on 
  a 
  

  

  This 
  is 
  simply 
  the 
  equation 
  of 
  energy 
  in 
  three 
  dimensions, 
  

   and 
  is 
  deducible 
  from 
  the 
  others. 
  Thus 
  all 
  the 
  equations 
  of 
  

   motion 
  return 
  to 
  their 
  three 
  dimensional 
  forms. 
  The 
  electro- 
  

   magnetic 
  equations 
  are 
  unaltered. 
  Finally, 
  variation 
  of 
  J 
  

   in 
  (31) 
  gives 
  

  

  (v 
  2 
  -[ 
  2 
  Jy+47rm 
  = 
  (36) 
  

  

  /can 
  be 
  recognized 
  as 
  the 
  Newtonian 
  potential. 
  The 
  only 
  

   equation 
  in 
  which 
  the 
  fourth- 
  dimension 
  finally 
  intrudes 
  

   is 
  (35). 
  This 
  may 
  be 
  written 
  

  

  Now 
  at 
  the 
  surface 
  p 
  m 
  is 
  zero. 
  Hence 
  (35) 
  becomes 
  

  

  ±eo 
  a 
  ((?-u*) 
  = 
  k 
  m 
  -fnJ, 
  ...... 
  (38) 
  

  

  and 
  if 
  the 
  left-hand 
  side 
  of 
  (38) 
  is 
  regarded 
  as 
  the 
  fluid 
  

   pressure, 
  then 
  at 
  the 
  surface 
  there 
  seem 
  to 
  be 
  the 
  applied 
  

   pressure 
  lc 
  m 
  and 
  the 
  applied 
  tension 
  mj. 
  The 
  latter 
  is 
  

   equivalent 
  (taking 
  m 
  to 
  be 
  an 
  absolute 
  constant) 
  to 
  the 
  

   gravitational 
  volume 
  force 
  

  

  mdvS/J 
  (39) 
  

  

  and 
  the 
  absolute 
  weight 
  of 
  all 
  matter 
  is 
  the 
  same 
  per 
  unit 
  

   volume. 
  The 
  applied 
  pressure 
  postulated 
  by 
  Poincare 
  is 
  the 
  

   pressure 
  k 
  m 
  . 
  

  

  §5. 
  The 
  Newtonian 
  Potential 
  a 
  Velocity 
  Potential. 
  

  

  The 
  theory 
  of 
  gravitation 
  given 
  in 
  last 
  section 
  is 
  purely 
  

   formal. 
  By 
  bringing 
  in 
  a 
  fourth 
  dimension 
  Maxwell* 
  s 
  

   scalar 
  potential 
  <fi 
  is 
  replaced 
  by 
  a 
  scalar 
  potential 
  J. 
  Max- 
  

   well's 
  F 
  and 
  c/> 
  become 
  the 
  four 
  components 
  of 
  a 
  single 
  

   vector. 
  It 
  is 
  now 
  possible 
  to 
  introduce 
  Riemann's 
  idea 
  of 
  a 
  

   fluid 
  aether. 
  In 
  (31) 
  the 
  terms 
  containing 
  differentials 
  of 
  J 
  are 
  

   just 
  those 
  arising 
  in 
  the 
  ir 
  rotational 
  motion 
  of 
  such 
  a 
  fluid 
  

   as 
  that 
  of 
  § 
  1 
  when 
  subject 
  to 
  no 
  external 
  forces. 
  If 
  we 
  

   write 
  as 
  in 
  (13) 
  

  

  VJ=/m=(47ry 
  

  

  dr 
  

  

  Y 
  (40) 
  

  

  then 
  (36), 
  if 
  m 
  is 
  zero, 
  can 
  be 
  interpreted 
  as 
  the 
  equation 
  

   of 
  continuity 
  just 
  as 
  (14) 
  in 
  § 
  1. 
  

  

  