﻿2 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  In 
  all 
  these 
  investigations 
  the 
  question 
  is 
  treated 
  as 
  two- 
  

   dimensional. 
  For 
  instance, 
  in 
  the 
  case 
  of 
  the 
  journal 
  the 
  

   width 
  — 
  axial 
  dimension 
  — 
  of 
  the 
  bearing 
  must 
  be 
  large 
  in 
  

   comparison 
  with 
  the 
  arc 
  of 
  contact, 
  a 
  condition 
  not 
  usually 
  

   fulfilled 
  in 
  practice. 
  But 
  Michell* 
  has 
  succeeded 
  in 
  solving 
  

   the 
  problem 
  for 
  a 
  plane 
  rectangular 
  block, 
  moving 
  at 
  a 
  

   slight 
  inclination 
  over 
  another 
  plane 
  surface, 
  free 
  from 
  this 
  

   limitation, 
  and 
  he 
  has 
  developed 
  a 
  system 
  of 
  pivoted 
  bearings 
  

   with 
  valuable 
  practical 
  results. 
  

  

  It 
  is 
  of 
  interest 
  to 
  consider 
  more 
  generally 
  than 
  hitherto 
  

   the 
  case 
  of 
  two 
  dimensions. 
  In 
  the 
  present 
  paper 
  attention 
  

   is 
  given 
  more 
  especially 
  to 
  the 
  case 
  where 
  one 
  of 
  the 
  opposed 
  

   surfaces 
  is 
  plane, 
  but 
  the 
  second 
  not 
  necessarily 
  so. 
  /As 
  an 
  

   alternative 
  to 
  an 
  inclined 
  plane 
  surface, 
  consideration 
  is 
  given 
  

   to 
  a 
  broken 
  surface 
  consisting 
  of 
  two 
  parts, 
  each 
  of 
  which 
  is 
  

   parallel 
  to 
  the 
  first 
  plane 
  surface 
  but 
  at 
  a 
  different 
  distance 
  

   from 
  it. 
  .-fit 
  appears 
  that 
  this 
  is 
  the 
  form 
  which 
  must 
  be 
  

   approached 
  if 
  we 
  wish 
  the 
  total 
  pressure 
  supported 
  to 
  be 
  a 
  

   maximum, 
  when 
  the 
  length 
  of 
  the 
  bearing 
  and 
  the 
  closest 
  

   approach 
  are 
  prescribed. 
  In 
  these 
  questions 
  we 
  may 
  anti- 
  

   cipate 
  that 
  our 
  calculations 
  correspond 
  pretty 
  closely 
  with 
  

   what 
  actually 
  happens, 
  — 
  more 
  than 
  can 
  be 
  said 
  of 
  some 
  

   branches 
  of 
  hydrodynamics. 
  

  

  In 
  forming 
  the 
  necessary 
  equation 
  it 
  is 
  best, 
  following 
  

   Sommerfeld, 
  to 
  begin 
  with 
  the 
  simplest 
  possible 
  case. 
  The 
  

   layer 
  of 
  fluid 
  is 
  contained 
  between 
  two 
  parallel 
  planes 
  at 
  

   y 
  — 
  and 
  at 
  y 
  — 
  h. 
  The 
  motion 
  is 
  everywhere 
  parallel 
  to 
  x, 
  

   so 
  that 
  the 
  velocity-component 
  u 
  alone 
  occurs, 
  v 
  and 
  w 
  being 
  

   everywhere 
  zero. 
  Moreover 
  u 
  is 
  a 
  function 
  of 
  y 
  only. 
  The 
  

   tangential 
  traction 
  acting 
  across 
  an 
  element 
  of 
  area 
  repre- 
  

   sented 
  by 
  dx 
  is 
  /jb(du/dy)dx, 
  where 
  fx 
  is 
  the 
  viscosity, 
  so 
  

   that 
  the 
  element 
  of 
  volume 
  {dx 
  dy) 
  is 
  subject 
  to 
  the 
  force 
  

   fjb{d 
  2 
  u[dy 
  2 
  ) 
  dx 
  dy. 
  Since 
  there 
  is 
  no 
  acceleration, 
  this 
  force 
  is 
  

   balanced 
  by 
  that 
  due 
  to 
  the 
  pressure, 
  viz. 
  — 
  {dp/dx)dx 
  dy, 
  and 
  

   thus 
  

  

  dp 
  _ 
  

  

  dx~ 
  r 
  dy" 
  

  

  In 
  this 
  equation 
  p 
  is 
  independent 
  of 
  y, 
  since 
  there 
  is 
  in 
  this 
  

   direction 
  neither 
  motion 
  nor 
  components 
  of 
  traction, 
  and 
  (1), 
  

   which 
  may 
  also 
  be 
  derived 
  directly 
  from 
  the 
  general 
  hydro- 
  

   dynamical 
  equations, 
  is 
  immediately 
  integrable. 
  We 
  have 
  

  

  u 
  =k 
  d 
  £y 
  +A+B 
  ^- 
  • 
  • 
  • 
  • 
  ( 
  2 
  ) 
  

  

  where 
  A 
  and 
  B 
  are 
  constants 
  of 
  integration. 
  We 
  now 
  

   * 
  Zeitschr.f. 
  Math.t. 
  52. 
  p. 
  123 
  (1905). 
  

  

  7* 
  XX 
  C 
  1 
  ) 
  

  

  