﻿Theory 
  of 
  Lubrication. 
  7 
  

  

  For 
  example, 
  if 
  ?z 
  = 
  2, 
  so 
  that 
  the 
  curve 
  of 
  the 
  second 
  sur- 
  

   face 
  is 
  part 
  of 
  a 
  common 
  parabola, 
  P 
  is 
  a 
  maximum 
  at 
  

  

  P 
  = 
  0'163^- 
  2 
  , 
  (24) 
  

  

  when 
  A 
  = 
  2*3. 
  The 
  departure 
  from 
  (18) 
  with 
  k 
  — 
  2'2 
  is 
  

   but 
  small. 
  In 
  order 
  to 
  estimate 
  the 
  curvature 
  involved 
  we 
  

   may 
  compare 
  j?(h 
  1 
  -\-h 
  2 
  ) 
  with 
  the 
  middle 
  ordinate 
  of 
  the 
  

   curve, 
  viz. 
  

  

  im(a 
  + 
  b) 
  2 
  = 
  i\ 
  *//h+ 
  */(2'3A|) 
  j 
  2 
  = 
  1-58 
  h, 
  

  

  which 
  is 
  but 
  little 
  less 
  than 
  

  

  ifa 
  + 
  hs) 
  = 
  Pi(l 
  4- 
  2-3) 
  = 
  1-65^. 
  

  

  It 
  appears 
  that 
  curvature 
  following 
  the 
  parabolic 
  law 
  is 
  of 
  

   small 
  advantage. 
  

  

  I 
  have 
  also 
  examined 
  the 
  case 
  of 
  w 
  = 
  cc. 
  It 
  is 
  perhaps 
  

   simpler 
  and 
  comes 
  to 
  the 
  same 
  to 
  assume 
  

  

  h=ef> 
  x 
  (25) 
  

  

  The 
  integrals 
  required 
  in 
  (7), 
  (8) 
  are 
  easily 
  evaluated. 
  

   Thus 
  

  

  Cdx 
  e-^-e-w 
  _ 
  k 
  2 
  -l 
  

  

  J 
  h 
  2 
  2/3 
  ~2£*V' 
  

  

  Cdx 
  e 
  -*fr—e-W> 
  P-l 
  

  

  A 
  3 
  3)8 
  ~ 
  3/31*1^ 
  

  

  making 
  R= 
  j^^ 
  (26) 
  

  

  In 
  like 
  manner 
  

  

  'xdx 
  k 
  2 
  (l 
  + 
  2/3a)-l-2/3b 
  

  

  j 
  

  

  h 
  2 
  ~ 
  4/3 
  2 
  &'V 
  

  

  x 
  dx 
  _ 
  k\l 
  + 
  3/3a) 
  -l-Z/3b 
  

  

  9£Wit 
  

  

  Jxdx 
  _ 
  

   IF- 
  

  

  Using 
  these 
  in 
  (7), 
  we 
  get 
  on 
  reduction 
  

   or, 
  since 
  /3c 
  = 
  log 
  k, 
  

  

  