﻿108 
  

  

  Mr. 
  J. 
  H. 
  Michell 
  on 
  the 
  

  

  ultimate 
  vanishing 
  of 
  the 
  resistance 
  has 
  not, 
  so 
  far 
  as 
  I 
  know, 
  

   been 
  anticipated. 
  From 
  general 
  considerations 
  it 
  is 
  clear 
  

   that, 
  so 
  far 
  as 
  the 
  wave-form 
  is 
  concerned, 
  the 
  effect 
  of 
  

   increasing 
  the 
  velocity 
  is 
  the 
  same 
  as 
  that 
  of 
  decreasing 
  the 
  

   acceleration 
  of 
  gravity, 
  and, 
  if 
  gravity 
  vanishes, 
  there 
  is 
  no 
  

   propagation 
  of 
  waves 
  ; 
  but 
  this 
  is 
  not 
  quite 
  the 
  theorem 
  to 
  

   be 
  obtained. 
  

  

  Fiff. 
  1. 
  

  

  x 
  

  

  y/A 
  

  

  Take 
  the 
  vertical 
  median 
  plane 
  of 
  the 
  ship 
  as 
  y 
  = 
  } 
  and 
  

   the 
  surface 
  of 
  the 
  undisturbed 
  water 
  as 
  s 
  = 
  0, 
  the 
  axis 
  Ox 
  

   being 
  in 
  the 
  direction 
  of 
  motion 
  of 
  the 
  ship 
  and 
  Oz 
  vertically 
  

   downwards. 
  We 
  may 
  suppose 
  the 
  ship 
  at 
  rest 
  and 
  the 
  water 
  

   moving 
  backwards 
  with 
  uniform 
  velocity 
  v 
  apart 
  from 
  the 
  

   wave-disturbance. 
  The 
  motion 
  is 
  assumed 
  steady 
  and 
  the 
  

   velocity 
  potential 
  written 
  — 
  vx-\-(f>. 
  Since 
  the 
  inclination 
  of 
  

   the 
  ship's 
  surface 
  to 
  the 
  plane 
  y 
  = 
  is 
  everywhere 
  small, 
  <f> 
  

   will 
  be 
  small, 
  and 
  we 
  shall 
  neglect 
  the 
  squares 
  of 
  the 
  

   velocities 
  due 
  to 
  </> 
  in 
  comparison 
  with 
  their 
  first 
  powers. 
  At 
  

   the 
  surface 
  of 
  the 
  water 
  let 
  f 
  be 
  the 
  depression 
  at 
  (#, 
  y) 
  

   below 
  the 
  mean 
  level. 
  Then 
  

  

  d(j> 
  

  

  dK 
  

  

  dz 
  dx 
  

  

  (i) 
  

  

  is 
  the 
  kinematic 
  surface 
  condition, 
  and 
  

  

  p/p 
  + 
  \(f" 
  —gK— 
  const, 
  

   the 
  equation 
  of 
  pressure, 
  which, 
  since 
  

  

  = 
  ^_2t;-£, 
  (q. 
  p.) 
  

  

  gives 
  

  

  dx^ 
  

  

  dx 
  * 
  

  

  