﻿Theory 
  of 
  Gravity. 
  651 
  

  

  four-dimensional 
  equations 
  of 
  fluid 
  motion. 
  This 
  is 
  done 
  by 
  

   writing 
  on 
  the 
  left 
  of 
  (30) 
  the 
  expression 
  

  

  d 
  

  

  ( 
  Div 
  Sr 
  + 
  -z- 
  Ss 
  Jdv 
  ds 
  dr. 
  

  

  We 
  have 
  then 
  the 
  following 
  four-dimensional 
  equations 
  of 
  

   motion 
  : 
  

  

  ¥m 
  2 
  , 
  

  

  dr 
  

  

  d 
  ( 
  - 
  ds 
  \_ 
  -1 
  d£ 
  -i 
  (dxd$_jcls_d<j> 
  dJ\ 
  dp 
  m 
  _ 
  Q 
  ] 
  | 
  (34) 
  

  

  ' 
  \ 
  dr/ 
  <^t' 
  \cZt 
  ds 
  dr 
  ds 
  ds 
  ) 
  ds 
  

  

  d 
  _dv 
  t-j 
  ds 
  d 
  I 
  

  

  dr' 
  ~~ 
  dr 
  dr 
  ds' 
  ' 
  

  

  Now, 
  just 
  as 
  in 
  the 
  three 
  dimensions 
  of 
  § 
  2, 
  so 
  it 
  may 
  be 
  

   shown 
  here 
  that 
  for 
  any 
  complete 
  circuit 
  

  

  ^Si 
  (kV^+a-hiiF 
  yr-fkhn 
  2 
  ^ 
  + 
  a~hn$\ 
  ds} 
  =0. 
  

  

  So 
  that 
  the 
  circulation 
  round 
  any 
  four-dimensional 
  circuit 
  

   remains 
  constant. 
  We 
  are 
  concerned, 
  however, 
  with 
  a 
  

   motion 
  steady 
  in 
  four 
  dimensions 
  and 
  wish 
  to 
  determine 
  

   under 
  what 
  conditions 
  it 
  may 
  be 
  irrotational. 
  From 
  (34) 
  it 
  

   can 
  be 
  deduced 
  that 
  in 
  any 
  steady 
  motion 
  whatever 
  the 
  

   quantity 
  

  

  

  is 
  a 
  constant 
  along 
  any 
  one 
  stream-line. 
  This 
  is 
  quite 
  

   analogous 
  to 
  corresponding 
  results 
  in 
  ordinary 
  hydro- 
  

   dynamics. 
  If 
  the 
  motion 
  is 
  irrotational 
  the 
  above 
  quantity 
  

   is 
  an 
  absolute 
  constant 
  everywhere, 
  

  

  Pm 
  + 
  ikW^J-QJ'j~mJ=-k 
  M 
  . 
  . 
  (35) 
  

  

  If 
  in 
  (34) 
  the 
  value 
  of 
  p 
  m 
  is 
  substituted 
  and 
  if, 
  to 
  corre- 
  

   spond 
  with 
  § 
  2, 
  we 
  write 
  

  

  k 
  2 
  a 
  2 
  = 
  ec 
  2 
  , 
  

  

  then 
  it 
  can 
  easily 
  be 
  shown 
  that 
  the 
  first 
  three 
  equations 
  (34) 
  

   become 
  identical 
  with 
  (20). 
  The 
  last 
  equation 
  (34) 
  is 
  now 
  

  

  ecru( 
  T 
  <ni 
  + 
  V^c 
  ) 
  = 
  cruE. 
  

  

  {|« 
  + 
  v«) 
  

  

  