﻿Theory 
  of 
  Lubrication. 
  3 
  

  

  suppose 
  that 
  when 
  v 
  = 
  0, 
  u= 
  — 
  U, 
  and 
  that 
  when 
  y 
  = 
  A, 
  

  

  u 
  = 
  0. 
  Thus 
  * 
  ' 
  

  

  ••=^|-('-f)^ 
  • 
  • 
  ■ 
  « 
  

  

  The 
  whole 
  flow 
  of 
  liquid, 
  regarded 
  as 
  incompressible, 
  between 
  

   and 
  A 
  is 
  

  

  f 
  * 
  , 
  A 
  3 
  dp 
  AU 
  

  

  where 
  Q 
  is 
  a 
  constant, 
  so 
  that 
  

  

  cLv~ 
  -ir[ 
  h 
  -lT) 
  W 
  

  

  If 
  we 
  suppose 
  the 
  passage 
  to 
  be 
  absolutely 
  blocked 
  at 
  a 
  place 
  

   where 
  x 
  is 
  negatively 
  great, 
  we 
  are 
  to 
  make 
  Q 
  = 
  and 
  (4) 
  

   gives 
  the 
  rise 
  of 
  pressure 
  as 
  x 
  decreases 
  algebraically. 
  But 
  

   for 
  the 
  present 
  purpose 
  Q 
  is 
  to 
  be 
  taken 
  finite. 
  Denoting 
  

   2Q/U 
  by 
  H, 
  we 
  write 
  (4) 
  

  

  g 
  = 
  -¥VH).. 
  . 
  . 
  . 
  (5, 
  

  

  When 
  y 
  = 
  0, 
  we 
  get 
  from 
  (3) 
  and 
  (5) 
  

  

  du 
  tt^A 
  — 
  3H 
  , 
  c 
  

  

  PTy^^—W-' 
  (6) 
  

  

  which 
  represents 
  the 
  tangential 
  traction 
  exercised 
  by 
  the 
  

   liquid 
  upon 
  the 
  moving 
  plane. 
  

  

  It 
  may 
  be 
  remarked 
  that 
  in 
  the 
  case 
  of 
  a 
  simple 
  shearing 
  

   motion 
  Q 
  = 
  ^AU, 
  making 
  H 
  = 
  A, 
  ami 
  accordingly 
  

  

  dp/dx 
  = 
  Q, 
  dujdy 
  = 
  ~Ulh. 
  

  

  Our 
  equations 
  allow 
  for 
  a 
  different 
  value 
  of 
  Q 
  and 
  a 
  pressure 
  

   variable 
  with 
  x. 
  

  

  So 
  far 
  we 
  have 
  regarded 
  A 
  as 
  absolutely 
  constant. 
  But 
  it 
  

   is 
  evident 
  that 
  Reynolds' 
  equation 
  (5) 
  remains 
  approximately 
  

   applicable 
  to 
  the 
  lubrication 
  problem 
  in 
  two 
  dimensions 
  even 
  

   when 
  A 
  is 
  variable, 
  though 
  always 
  very 
  small, 
  provided 
  that 
  

   the 
  changes 
  are 
  not 
  too 
  sudden, 
  x 
  being 
  measured 
  circum- 
  

   ferentially 
  and 
  y 
  normally 
  to 
  the 
  opposed 
  surfaces. 
  If 
  the 
  

   whole 
  changes 
  of 
  direction 
  are 
  large, 
  as 
  in 
  the 
  journal-bearing 
  

   with 
  a 
  large 
  arc 
  of 
  contact, 
  complication 
  arises 
  in 
  the 
  

   reckoning 
  of 
  the 
  resultant 
  forces 
  operative 
  upon 
  the 
  solid 
  

   parts 
  concerned 
  ; 
  but 
  this 
  does 
  not 
  interfere 
  with 
  the 
  appli- 
  

   cability 
  of 
  (5) 
  when 
  A 
  is 
  suitably 
  expressed 
  as 
  a 
  function 
  

   of 
  x. 
  In 
  the 
  present 
  paper 
  we 
  confine 
  ourselves 
  to 
  the 
  case 
  

  

  B 
  2 
  

  

  