﻿Theory 
  of 
  Lubrication. 
  9 
  

  

  liquid. 
  If 
  we 
  suppose 
  that 
  h 
  becomes 
  h 
  + 
  Sh, 
  where 
  S 
  is 
  the 
  

   symbol 
  of 
  the 
  calculus 
  of 
  variations, 
  (8) 
  gives, 
  

  

  *p£d-^wffidm+mfg=0, 
  . 
  . 
  (30) 
  

   and 
  from 
  (7) 
  

  

  6>TJ=J 
  3E 
  aH 
  JlT'- 
  ' 
  (31) 
  

  

  the 
  integrations 
  being 
  always 
  over 
  the 
  length. 
  Elimi- 
  

   nating 
  SH, 
  we 
  get 
  

  

  The 
  evanescence 
  of 
  BY 
  for 
  all 
  possible 
  variations 
  Sh 
  would 
  

   demand 
  that 
  over 
  the 
  whole 
  range 
  either 
  

  

  \h~3xdx 
  _„ 
  

  

  But 
  this 
  is 
  not 
  the 
  requirement 
  postulated. 
  It 
  suffices 
  

   that 
  the 
  coefficient 
  of 
  Sh 
  on 
  the 
  right 
  of 
  (32) 
  vanish 
  over 
  

   that 
  part 
  of 
  the 
  range 
  where 
  h>h 
  i 
  , 
  and 
  that 
  it 
  be 
  negative 
  

   when 
  h=h 
  l9 
  _ 
  so 
  that 
  a 
  positive 
  Sh 
  in 
  this 
  region 
  involves 
  a 
  

   decrease 
  in 
  P, 
  a 
  negative 
  Sh 
  here 
  being 
  excluded 
  a 
  priori. 
  

   These 
  conditions 
  may 
  be 
  satisfied 
  if 
  we 
  make 
  h 
  = 
  h 
  1 
  from 
  

   # 
  = 
  at 
  the 
  edge 
  where 
  the 
  layer 
  is 
  thin 
  to 
  a?=c 
  1} 
  where 
  c 
  x 
  is 
  

   finite, 
  and 
  7i=|H 
  over 
  the 
  remainder 
  of 
  the 
  range 
  from 
  c 
  x 
  to 
  

   Ci-\-c 
  2y 
  where 
  e 
  1 
  + 
  c 
  2 
  =c, 
  the 
  whole 
  length 
  concerned 
  (fig. 
  2). 
  

   For 
  the 
  moment 
  we 
  regard 
  c 
  x 
  and 
  c 
  2 
  as 
  prescribed. 
  

  

  Fig. 
  2. 
  

  

  ■$«<- 
  

  

  h 
  2 
  

  

  U 
  

  

  For 
  the 
  first 
  condition 
  we 
  have 
  by 
  (8) 
  

  

  2 
  7j 
  _tt 
  c 
  T 
  Mi 
  2 
  + 
  c 
  2/V 
  

  

  so 
  that 
  

  

  c 
  2 
  / 
  Cl 
  = 
  P(2&-3), 
  (34) 
  

  

  determining 
  £, 
  where 
  as 
  before 
  k 
  = 
  h 
  2 
  /h 
  1 
  . 
  The 
  fulfilment 
  of 
  (34) 
  

  

  