﻿Wave-Resistance 
  of 
  a 
  Ship. 
  117 
  

  

  are 
  now 
  available, 
  and 
  tables* 
  of 
  Y 
  and 
  Y 
  L 
  have 
  been 
  calcu- 
  

   lated 
  by 
  Mr. 
  B. 
  A. 
  Smith, 
  who 
  has 
  kindly 
  prepared 
  tables 
  of 
  

   kJ 
  — 
  Y 
  and 
  /cJ 
  1 
  — 
  Y 
  x 
  , 
  appended 
  to 
  the 
  present 
  paper. 
  

   We 
  now 
  have 
  

  

  - 
  5? 
  = 
  T-k 
  f 
  V** 
  {F 
  (• 
  +#.) 
  + 
  F(«-*)}dk 
  

  

  where 
  

  

  F(s 
  + 
  yu,) 
  = 
  /eJ 
  (s+fl)- 
  Y 
  (s 
  + 
  /a) 
  

  

  6 
  ~t~ 
  fit 
  

  

  and 
  the 
  expression 
  for 
  the 
  mutual 
  resistance 
  is 
  

  

  ^f 
  a 
  a' 
  0' 
  re-^{F(s 
  + 
  l 
  u)+F(s-ri}dn, 
  

   w^V 
  s/r 
  Jo 
  

  

  where 
  

  

  r=g(z 
  + 
  z 
  f 
  )/v 
  2 
  

  

  For 
  elements 
  at 
  opposite 
  ends 
  of 
  the 
  ship 
  s 
  will 
  in 
  general 
  

   be 
  large 
  compared 
  with 
  unity 
  and 
  with 
  yjkr, 
  and 
  in 
  this 
  

   case 
  we 
  can 
  put 
  

  

  KJ 
  (s 
  + 
  fi) 
  ~ 
  Y 
  o(* 
  + 
  aO='y/ 
  2 
  ^ 
  ) 
  Sin 
  {i 
  ~" 
  (s 
  + 
  ^} 
  q 
  * 
  P 
  " 
  

   and 
  so 
  for 
  (s—/ju), 
  and 
  then 
  approximately 
  

  

  H 
  = 
  — 
  - 
  ■ 
  sin 
  (-. 
  5)1 
  e-» 
  2 
  > 
  lr 
  cos 
  fi 
  d/j, 
  

  

  v 
  2rs 
  \4 
  / 
  J 
  

  

  and 
  the 
  resistance 
  is 
  

  

  = 
  - 
  p 
  ^' 
  2 
  <t 
  a* 
  0' 
  sin 
  ( 
  J 
  -g(x—J)/# 
  l*-«7(*+*W 
  . 
  

  

  Now 
  if 
  Z 
  is 
  the 
  length 
  of 
  the 
  free 
  wave 
  which 
  travels 
  with 
  

   the 
  velocity 
  of 
  the 
  ship 
  

  

  v*=gl/27r, 
  

  

  * 
  ' 
  Messenger 
  of 
  Mathematics,' 
  1896. 
  

  

  