﻿4 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  where 
  one 
  surface 
  (at 
  ,y 
  = 
  0) 
  may 
  be 
  treated 
  as 
  absolutely 
  

   plane. 
  The 
  second 
  surface 
  is 
  supposed 
  to 
  be 
  limited 
  at 
  x=a 
  

   and 
  at 
  x 
  = 
  b, 
  where 
  h 
  is 
  equal 
  to 
  A 
  x 
  and 
  h 
  2 
  respectively, 
  and 
  

   the 
  pressure 
  at 
  both 
  these 
  places 
  is 
  taken 
  to 
  be 
  zero. 
  

   For 
  the 
  total 
  pressure, 
  or 
  load, 
  (P) 
  we 
  have 
  

  

  = 
  1 
  pdx=—\ 
  x^~-dx. 
  

  

  Ja 
  Ja 
  dx 
  

  

  dp 
  

  

  on 
  integration 
  by 
  parts 
  with 
  regard 
  to 
  the 
  evanescence 
  of 
  

   p 
  at 
  both 
  limits. 
  Hence 
  by 
  (5) 
  

  

  P 
  C 
  h 
  xdx 
  jjCKxdx 
  

   Again, 
  by 
  direct 
  integration 
  of 
  (5), 
  

  

  by 
  which 
  H 
  is 
  determined. 
  It 
  is 
  the 
  thickness 
  of 
  the 
  layer 
  

   at 
  the 
  place, 
  or 
  places, 
  where 
  p 
  is 
  a 
  maximum 
  or 
  a 
  minimum. 
  

   A 
  change 
  in 
  the 
  sign 
  of 
  U 
  reverses 
  also 
  that 
  of 
  P. 
  

  

  Again, 
  if 
  x 
  be 
  the 
  value 
  of 
  x 
  which 
  gives 
  the 
  point 
  of 
  

   application 
  of 
  the 
  resultant 
  force, 
  

  

  so 
  that 
  

  

  #.P=| 
  p^dt' 
  = 
  ij 
  x 
  2 
  -J?-dx, 
  

  

  x.V 
  ^C 
  b 
  x 
  2 
  dx 
  C 
  b 
  x 
  2 
  dx 
  

  

  3^U 
  J 
  a 
  V 
  *J 
  B 
  A3 
  • 
  ' 
  • 
  • 
  • 
  (9) 
  

  

  By 
  (7), 
  (8), 
  (9) 
  x 
  is 
  determined. 
  

  

  As 
  regards 
  the 
  total 
  friction 
  (F), 
  we 
  have 
  by 
  (6) 
  

  

  ,W?- 
  3H 
  J> 
  • 
  • 
  ■ 
  <-> 
  

  

  Comparing 
  (7) 
  and 
  (10), 
  we 
  see 
  that 
  the 
  ratio 
  of 
  the 
  tota 
  

   friction 
  to 
  the 
  total 
  load 
  is 
  independent 
  of 
  p 
  and 
  o/U. 
  And, 
  

   since 
  the 
  right-hand 
  members 
  of 
  (7) 
  and 
  (10) 
  are 
  dimen- 
  

   sionless, 
  the 
  ratio 
  is 
  also 
  independent 
  of 
  the 
  linear 
  scale. 
  

   But 
  if 
  the 
  scale 
  of 
  h 
  only 
  be 
  altered, 
  F/P 
  varies 
  as 
  li. 
  

  

  We 
  may 
  now 
  consider 
  particular 
  cases, 
  of 
  which 
  the 
  

   simplest 
  and 
  the 
  most 
  important 
  is 
  when 
  the 
  second 
  surface 
  

  

  