﻿Theory 
  of 
  Lubrication, 
  5 
  

  

  also 
  is 
  flat, 
  but 
  inclined 
  at 
  a 
  very 
  small 
  angle 
  to 
  the 
  first 
  

   surface. 
  We 
  take 
  

  

  h=mx, 
  (11) 
  

  

  and 
  we 
  write 
  for 
  convenience 
  

  

  b—a 
  = 
  c, 
  h 
  2 
  /Jii 
  = 
  b/a 
  = 
  k, 
  . 
  . 
  . 
  (12) 
  

  

  so 
  that 
  

  

  m 
  = 
  (k-l)h,/c 
  (13) 
  

  

  We 
  find 
  in 
  terms 
  of 
  c, 
  &, 
  and 
  h 
  t 
  

  

  H 
  =fi 
  a*) 
  

  

  ^_ 
  P-l-2Hog& 
  - 
  

  

  io 
  - 
  ( 
  p_l 
  } 
  l 
  og 
  Z;_2(A-l) 
  2 
  ' 
  • 
  • 
  " 
  ^ 
  

  

  P~~ 
  c 
  3(*+l)log£— 
  6(A-1)' 
  ' 
  ' 
  V 
  4) 
  

  

  U 
  being 
  positive, 
  the 
  sign 
  of 
  P 
  is 
  that 
  of 
  

  

  2(&-l) 
  

  

  If 
  A 
  :> 
  1, 
  that 
  is 
  when 
  h 
  2 
  >- 
  h 
  l9 
  this 
  quantity 
  is 
  positive. 
  

   For 
  its 
  derivative 
  is 
  positive, 
  as 
  is 
  also 
  the 
  initial 
  value 
  when 
  

   k 
  exceeds 
  unity 
  but 
  slightly. 
  In 
  order 
  that 
  a 
  load 
  may 
  be 
  

   sustained, 
  the 
  layer 
  must 
  be 
  thicker 
  where 
  the 
  liquid 
  enters. 
  

  

  In 
  the 
  above 
  formulae 
  we 
  have 
  taken 
  as 
  data 
  the 
  length 
  

   of 
  the 
  bearing 
  c 
  and 
  the 
  minimum 
  distance 
  hi 
  between 
  the 
  

   surfaces. 
  So 
  far 
  k, 
  giving 
  the 
  maximum 
  distance, 
  is 
  open. 
  

   It 
  may 
  be 
  determined 
  by 
  various 
  considerations. 
  Reynolds 
  

   examines 
  for 
  what 
  value 
  P, 
  as 
  expressed 
  in 
  (15), 
  is 
  a 
  maxi- 
  

   mum, 
  and 
  he 
  gives 
  (in 
  a 
  different 
  notation) 
  k 
  = 
  2'2. 
  For 
  

   values 
  of 
  k 
  equal 
  to 
  2*0, 
  2* 
  J, 
  2*2, 
  2*31 
  find 
  for 
  the 
  coefficient 
  

   of 
  c 
  2 
  jhi 
  2 
  on 
  the 
  right 
  of 
  (14) 
  respectively 
  

  

  •02648, 
  -02665, 
  -02670, 
  -02663. 
  

  

  In 
  agreement 
  with 
  Reynolds 
  the 
  maximum 
  occurs 
  when 
  

   k 
  = 
  2'2 
  nearly, 
  and 
  the 
  maximum 
  value 
  is 
  

  

  P 
  = 
  0-1602^f 
  2 
  (18) 
  

  

  hi 
  

  

  It 
  should 
  be 
  observed 
  — 
  and 
  it 
  is 
  true 
  whatever 
  value 
  be 
  

   taken 
  for 
  k 
  — 
  that 
  P 
  varies 
  as 
  the 
  square 
  of 
  c\h\. 
  

  

  