﻿Wave-Be 
  shtance 
  of 
  a 
  Ship. 
  Ill 
  

  

  we 
  get 
  

  

  /( 
  ' 
  *'-»*■ 
  J 
  )_ 
  J/«' 
  SA 
  + 
  sin 
  ink 
  

  

  X 
  cos 
  m(l- 
  — 
  3c)d% 
  tig 
  dm 
  

   4 
  /-"r 
  00 
  f* 
  w' 
  cosh 
  n'(z-h) 
  cosh 
  n'(f- 
  A) 
  

  

  0» 
  f*i 
  

   -oo 
  Jo 
  

  

  t 
  |o 
  J-ooJo 
  2n'A 
  + 
  sinh 
  2n'A 
  

  

  X 
  cos 
  m(%—cc)d^ 
  dfj 
  dm. 
  

  

  In 
  particular, 
  suppose 
  the 
  depth 
  of 
  water 
  infinite, 
  we 
  have 
  

   then 
  

  

  nh 
  = 
  r7T 
  + 
  €, 
  

  

  lidn 
  = 
  7r, 
  

   tan 
  n/i 
  = 
  tan 
  e, 
  

   cos 
  n 
  (z 
  — 
  li) 
  = 
  ( 
  — 
  ) 
  * 
  cos 
  (nz 
  — 
  e) 
  , 
  

   2n 
  + 
  (sin2rf)//i=27i, 
  

   n 
  tan 
  e 
  = 
  — 
  km 
  2 
  [k 
  = 
  v 
  2 
  jg\ 
  , 
  

   — 
  2km 
  2 
  n 
  

  

  sin 
  2e 
  = 
  

  

  cos 
  2e 
  

  

  rc 
  2 
  + 
  Fm 
  4 
  ' 
  

  

  n 
  2 
  — 
  k 
  2 
  m 
  4 
  

  

  n 
  2 
  + 
  k 
  2 
  m 
  4 
  ' 
  

   Also 
  tanh 
  n!h 
  = 
  1, 
  

  

  n 
  = 
  km 
  2 
  , 
  

  

  cosh 
  n'(z~h) 
  cosh 
  n 
  f 
  (%— 
  K) 
  1 
  _ 
  n 
  , 
  (z+ 
  ^ 
  

  

  2n'h 
  + 
  sinh2n'A 
  

  

  = 
  te 
  

  

  The 
  result 
  receives 
  some 
  confirmation 
  from 
  what 
  has 
  been 
  

   observed 
  with 
  torpedo-boats 
  at 
  high 
  speeds. 
  It 
  has 
  been 
  

   found 
  that 
  the 
  total 
  resistance 
  varies 
  as 
  a 
  power 
  of 
  the 
  velocity 
  

   which 
  at 
  first 
  is 
  nearly 
  the 
  second, 
  but 
  which, 
  increasing 
  to 
  a 
  

   maximum, 
  ultimately 
  becomes 
  less 
  than 
  the 
  second 
  *. 
  A 
  

   very 
  simple 
  investigation, 
  given 
  below, 
  shows 
  that 
  in 
  shallow 
  

   water, 
  if 
  we 
  neglect 
  all 
  but 
  the 
  long 
  waves, 
  the 
  wave-resist- 
  

   ance 
  varies 
  ultimately 
  as 
  the 
  first 
  power 
  of 
  the 
  velocity. 
  

  

  I 
  may 
  mention 
  that 
  somewhat 
  similar 
  work 
  to 
  that 
  of 
  the 
  

   present 
  paper 
  gives 
  a 
  theory 
  of 
  the 
  damping 
  of 
  the 
  oscilla- 
  

   tions 
  of 
  ships 
  due 
  to 
  wave-making. 
  This 
  I 
  hope 
  to 
  give 
  in 
  a 
  

   subsequent 
  paper. 
  

  

  * 
  White, 
  p. 
  470. 
  

  

  