﻿Theory 
  of 
  Lubrication, 
  11 
  

  

  the 
  ratio 
  of 
  the 
  two 
  thicknesses 
  of 
  the 
  layer 
  (7i 
  2 
  /^i), 
  and 
  that 
  

   by 
  (34) 
  

  

  c 
  2 
  / 
  Cl 
  = 
  2-588 
  (41) 
  

  

  This 
  defines 
  the 
  form 
  of 
  the 
  upper 
  surface 
  which 
  gives 
  the 
  

   maximum 
  total 
  pressure 
  when 
  the 
  minimum 
  thickness 
  and 
  

   the 
  total 
  length 
  are 
  given, 
  and 
  it 
  is 
  the 
  solution 
  of 
  the 
  problem 
  

   as 
  proposed. 
  But 
  it 
  must 
  not 
  be 
  overlooked 
  that 
  it 
  violates 
  

   the 
  supposition 
  upon 
  which 
  the 
  original 
  equation 
  (5) 
  was 
  

   founded. 
  The 
  solution 
  of 
  an 
  accurate 
  equation 
  would 
  pro- 
  

   bably 
  involve 
  some 
  rounding 
  off 
  of 
  the 
  sharp 
  corners, 
  not 
  

   greatly 
  affecting 
  the 
  numerical 
  results. 
  

  

  The 
  distance 
  x 
  of 
  the 
  centre 
  of 
  pressure 
  from 
  the 
  narrow 
  

   end 
  is 
  given 
  by 
  

  

  #=0-4262c, 
  (42) 
  

  

  differing 
  very 
  little 
  from 
  the 
  value 
  found 
  in 
  (20). 
  From 
  

   (10) 
  with 
  use 
  of 
  (38) 
  we 
  get 
  

  

  F 
  _ 
  4c(&-l)» 
  _ 
  4c 
  

   ^U 
  h 
  1 
  (l 
  + 
  2¥-M*) 
  (2A 
  + 
  1JV 
  * 
  * 
  ^ 
  6) 
  

   and 
  

  

  P 
  c(2k-3) 
  [ 
  } 
  

  

  If 
  £ 
  = 
  1-87, 
  

  

  F/P= 
  4-091 
  hjc, 
  ..... 
  (45) 
  

  

  a 
  little 
  less 
  than 
  was 
  found 
  in 
  (21). 
  The 
  maximum 
  total 
  

   pressure 
  and 
  the 
  corresponding 
  ratio 
  F/P 
  are 
  both 
  rather 
  

   more 
  advantageous 
  in 
  the 
  arrangement 
  now 
  under 
  discussion 
  

   than 
  for 
  the 
  simply 
  inclined 
  line. 
  But 
  the 
  choice 
  would 
  

   doubtless 
  depend 
  upon 
  other 
  considerations. 
  

  

  The 
  particular 
  case 
  treated 
  above 
  is 
  that 
  which 
  makes 
  P 
  a 
  

   maximum. 
  We 
  might 
  inquire 
  as 
  to 
  the 
  form 
  of 
  the 
  curve 
  

   for 
  which 
  F/P 
  is 
  a 
  minimum, 
  for 
  a 
  given 
  length 
  and 
  closest 
  

   approach 
  to 
  the 
  axis 
  of 
  x. 
  In 
  the 
  expression 
  corresponding 
  

   with 
  (32), 
  instead 
  of 
  a 
  product 
  of 
  two 
  linear 
  factors, 
  the 
  

   coefficient 
  of 
  Bh 
  will 
  involve 
  a 
  quadratic 
  factor 
  of 
  the 
  form 
  

  

  Bzh+CW+Dx+m+JF, 
  .... 
  (46) 
  

  

  so 
  that 
  the 
  curve 
  is 
  again 
  hyperbolic 
  in 
  the 
  general 
  sense. 
  

   But 
  its 
  precise 
  determination 
  would 
  be 
  troublesome 
  and 
  

   probably 
  only 
  to 
  be 
  effected 
  by 
  trial 
  and 
  error. 
  It 
  is 
  unlikely 
  

   that 
  any 
  great 
  reduction 
  in 
  the 
  value 
  of 
  F/P 
  would 
  ensue. 
  

  

  