﻿108 
  Messrs. 
  R. 
  M. 
  Deeley 
  and 
  P. 
  H. 
  Parr 
  on 
  

  

  Appendix. 
  

  

  On 
  the 
  Steady 
  Flow 
  of 
  a 
  Viscous 
  Fluid 
  along 
  a 
  Uniform 
  

   Channel 
  under 
  the 
  action 
  of 
  Gravity. 
  

  

  Taking 
  the 
  origin 
  at 
  the 
  centre 
  of 
  the 
  surface, 
  and 
  

   using 
  rectangular 
  axes, 
  let 
  x 
  be 
  the 
  horizontal 
  distance 
  

   from 
  the 
  central 
  vertical 
  plane, 
  y 
  the 
  distance 
  below 
  the 
  

   surface, 
  at 
  right 
  angles 
  to 
  that 
  surface, 
  and 
  z 
  the 
  distance 
  

   from 
  the 
  my 
  plane 
  through 
  the 
  origin, 
  along 
  the 
  stream- 
  

   lines; 
  also 
  let 
  u. 
  u, 
  w 
  be 
  the 
  velocities 
  parallel 
  to 
  the 
  three 
  

   axes. 
  Then 
  the 
  mathematical 
  conditions 
  are 
  

  

  u 
  = 
  v 
  = 
  0, 
  1^=0 
  (24) 
  

  

  V 
  2 
  ™=-~, 
  (25) 
  

  

  *■-»+& 
  < 
  26 
  > 
  

  

  P 
  = 
  the 
  bodily 
  force 
  producing 
  motion 
  or 
  gp 
  sin 
  <j>, 
  p 
  being 
  

   the 
  density 
  and 
  (p 
  the 
  angle 
  made 
  by 
  the 
  axis 
  of 
  z 
  with 
  the 
  

   horizontal, 
  and 
  tj 
  = 
  the 
  coefficient 
  of 
  viscosity, 
  taken 
  all 
  over 
  

   the 
  section, 
  and 
  subject 
  to 
  the 
  conditions 
  that 
  

  

  w 
  = 
  at 
  all 
  points 
  on 
  the 
  boundary 
  of 
  the 
  section, 
  and 
  

  

  g?=0, 
  *heny 
  = 
  (27) 
  

  

  It 
  is 
  advantageous, 
  from 
  the 
  analytical 
  point 
  of 
  view, 
  to 
  

   have 
  uniform 
  conditions 
  all 
  round 
  the 
  boundary, 
  and 
  this 
  

   may 
  be 
  secured 
  by 
  considering 
  the 
  channel 
  to 
  be 
  the 
  lower 
  

   half 
  of 
  a 
  symmetrical 
  tunnel, 
  the 
  upper 
  half 
  being 
  the 
  

   reflexion, 
  about 
  the 
  surface, 
  of 
  the 
  lower 
  half. 
  

  

  If 
  the 
  section 
  of 
  the 
  tunnel 
  is 
  taken 
  arbitrarily, 
  the 
  dif- 
  

   ferential 
  equation 
  does 
  not 
  usually 
  admit 
  of 
  a 
  solution 
  in 
  

   finite 
  terms; 
  the 
  best 
  procedure, 
  therefore, 
  is 
  to 
  study 
  a 
  

   velocity 
  distribution 
  which 
  is 
  known 
  to 
  satisfy 
  the 
  differential 
  

   equation, 
  and 
  to 
  determine 
  the 
  particular 
  forms 
  of 
  tunnel 
  

   which 
  it 
  includes. 
  

  

  The 
  first 
  expression 
  which 
  naturally 
  presents 
  itself 
  is 
  

  

  ■w=K{l-^—/3f), 
  .... 
  (28) 
  

  

  

  