﻿Wave-Resistance 
  of 
  a 
  Ship. 
  115 
  

  

  Now 
  

  

  j 
  f(x,z)dx=\ 
  ^da=0, 
  

  

  the 
  ship 
  being 
  of 
  finite 
  length 
  and 
  rj 
  = 
  at 
  both 
  ends. 
  

   Hence, 
  using 
  the 
  formula 
  (7) 
  

  

  I=A$ 
  + 
  B^(J)V... 
  

  

  j=a 
  '^ 
  +b 
  ' 
  x 
  C) 
  ?+ 
  • 
  • 
  • 
  • 
  

  

  and 
  

  

  The 
  successive 
  terms 
  are 
  of 
  orders 
  

  

  -7 
  2 
  l0gV 
  2 
  , 
  V~ 
  10 
  / 
  3 
  , 
  .. 
  

  

  and 
  all 
  vanish 
  when 
  v 
  = 
  go. 
  The 
  resistance 
  therefore 
  ulti^ 
  

   mately 
  vanishes. 
  Of 
  course 
  this 
  result 
  is 
  only 
  proved 
  for 
  a 
  

   ship 
  which 
  is 
  very 
  short 
  in 
  comparison 
  with 
  the 
  depth 
  of 
  the 
  

   water. 
  

  

  We 
  now 
  proceed 
  to 
  the 
  reduction 
  of 
  the 
  integral 
  which 
  

   gives 
  the 
  resistance 
  due 
  to 
  two 
  elements 
  of 
  the 
  surface. 
  

  

  Consider 
  two 
  elementary 
  areas 
  cr, 
  a' 
  at 
  (x, 
  z) 
  f 
  (sc' 
  ', 
  z') 
  on 
  the 
  

   side 
  of 
  the 
  ship, 
  and 
  let 
  0, 
  6 
  f 
  be 
  the 
  inclinations 
  of 
  the 
  hori- 
  

   zontal 
  lines 
  in 
  these 
  areas 
  to 
  the 
  axis 
  of 
  x. 
  The 
  resistance 
  

   due 
  to 
  these 
  two 
  elements 
  is 
  

  

  C*"° 
  X 
  2 
  

   ■' 
  6 
  f 
  \ 
  e 
  -«*+*> 
  2 
  cos 
  \g(x-a/)/v* 
  .- 
  d\, 
  

  

  or, 
  say, 
  

  

  -&*■ 
  aa'Oe'Ce 
  2 
  cos 
  e\ 
  , 
  X 
  * 
  d\. 
  

  

  TTV 
  2 
  U 
  ^A 
  2 
  -! 
  

  

  Now, 
  writing 
  for 
  the 
  moment 
  z 
  = 
  x 
  -f 
  iy 
  } 
  and 
  taking 
  the 
  

   integral 
  

  

  J 
  

  

  18Z 
  

  

  dz 
  

  

  \/l-z< 
  

  

  around 
  the 
  circuit 
  enclosing 
  the 
  region 
  x 
  and 
  y 
  positive, 
  we 
  

   get 
  

  

  f 
  00 
  e 
  f°° 
  e** 
  f* 
  e~* 
  x 
  

  

  \ 
  dx 
  + 
  i\ 
  dx—i 
  \ 
  , 
  dx=0, 
  

  

  J 
  Vl-^ 
  2 
  ) 
  l 
  Vx*-± 
  J 
  Vl+x 
  2 
  

  

  