﻿120 
  Mr. 
  J. 
  H. 
  Michell 
  on 
  the 
  

  

  then 
  the 
  integral 
  is 
  found 
  to 
  be 
  *620, 
  and 
  the 
  resistance 
  is 
  

   R 
  = 
  940 
  lbs. 
  wt. 
  about. 
  

  

  This 
  seems 
  to 
  be 
  about 
  what 
  one 
  would 
  expect 
  from 
  the 
  

   experimental 
  results 
  available; 
  but 
  I 
  know 
  of 
  no 
  formula 
  with 
  

   which 
  to 
  compare 
  it, 
  and 
  experiment 
  alone 
  can 
  decide 
  whether 
  

   the 
  theory 
  has 
  numerical 
  value. 
  Of 
  course 
  the 
  method 
  of 
  

   successive 
  approximations 
  can 
  be 
  applied 
  if 
  necessary. 
  

  

  To 
  examine 
  the 
  case 
  of 
  the 
  ship 
  in 
  shallow 
  water 
  in 
  which 
  

   all 
  but 
  the 
  long 
  waves 
  are 
  neglected, 
  we 
  may 
  proceed 
  as 
  

   follows 
  : 
  — 
  We 
  make 
  the 
  motion 
  steady 
  as 
  before 
  and 
  use 
  the 
  

   same 
  set 
  of 
  axes. 
  The 
  pressure-equation 
  gives 
  at 
  the 
  surface 
  

  

  g% 
  -f 
  v 
  -p 
  =0, 
  as 
  before 
  ; 
  

  

  dx 
  

   while 
  the 
  equation 
  of 
  continuity 
  for 
  long 
  waves 
  gives 
  

  

  £{M-"tS}+|{c*-o^}-o. 
  

  

  where 
  h 
  is 
  the 
  depth 
  of 
  the 
  water, 
  and 
  this 
  is 
  

  

  Differentiating 
  with 
  respect 
  to 
  x 
  and 
  substituting 
  for 
  <j) 
  

   from 
  the 
  pressure-equation, 
  we 
  obtain 
  

  

  or 
  v_,n 
  ** 
  + 
  <**? 
  =0 
  

  

  - 
  G 
  V) 
  dx 
  2 
  ^° 
  dtf 
  U 
  ' 
  

  

  where 
  c 
  is 
  the 
  velocity 
  of 
  free 
  long 
  waves. 
  

  

  The 
  ship 
  being 
  wall-sided, 
  and 
  extending 
  to 
  the 
  bottom 
  of 
  

   the 
  water, 
  the 
  kinematic 
  equation 
  over 
  the 
  ship 
  is 
  

  

  d$ 
  _ 
  dr) 
  

  

  dy 
  dx 
  1 
  

  

  which 
  on 
  differentiation 
  with 
  respect 
  to 
  x 
  and 
  use 
  of 
  the 
  

   pressure-equation 
  gives 
  

  

  d 
  A 
  -. 
  2^ 
  

   y 
  dy 
  dx 
  2 
  * 
  

  

  and, 
  of 
  course, 
  

  

  § 
  =0 
  

  

  dy 
  

  

  over 
  the 
  rest 
  of 
  the 
  plane 
  y 
  = 
  0. 
  

  

  