﻿114 
  Mr. 
  J. 
  H. 
  Michell 
  on 
  the 
  

  

  where 
  X 
  = 
  mv 
  2 
  jg, 
  

  

  and 
  

  

  1=1 
  \ 
  ffa 
  z 
  ) 
  e 
  -^i» 
  2 
  cos 
  Xgxjv 
  2 
  dx 
  dz, 
  

  

  JO 
  J 
  — 
  00 
  

  

  r* 
  oo 
  /» 
  oo 
  

   J 
  = 
  | 
  I 
  /(^ 
  ~) 
  e 
  -***/•• 
  s 
  i 
  n 
  ty#/„* 
  <fo 
  <fe. 
  

  

  ^0 
  . 
  —oo 
  

  

  If 
  the 
  ship 
  is 
  similarly 
  formed 
  at 
  bow 
  and 
  stern 
  1 
  = 
  0, 
  the 
  

   origin! 
  being 
  at 
  midship. 
  

  

  We 
  can 
  now 
  prove 
  that 
  the 
  resistance 
  vanishes 
  when 
  the 
  

   velocity 
  is 
  infinite. 
  

  

  Observe 
  that 
  

  

  f 
  /(*, 
  z) 
  e~W*dz 
  = 
  F 
  (*) 
  ( 
  6-**i+ 
  dz 
  

   Jo 
  Jo 
  

  

  9 
  

  

  - 
  2 
  -F(^), 
  .... 
  (7) 
  

  

  where 
  F(#) 
  is 
  less 
  than 
  the 
  greatest 
  value 
  of 
  f(x, 
  z) 
  

   for 
  a 
  given 
  value 
  of 
  x 
  ; 
  and, 
  therefore, 
  if 
  we 
  substitute 
  a 
  

   large 
  number 
  t 
  instead 
  of 
  go 
  as 
  the 
  upper 
  limit 
  of 
  X, 
  the 
  

   part 
  of 
  R 
  neglected 
  is 
  of 
  order 
  not 
  greater 
  than 
  

  

  gf 
  00 
  dX 
  

  

  " 
  1 
  v 
  

  

  or 
  v 
  t 
  

  

  2/— 
  2 
  

  

  and 
  this 
  vanishes 
  when 
  v 
  = 
  c© 
  if 
  we 
  take 
  

  

  2/3 
  

  

  -(,)' 
  

  

  In 
  the 
  part 
  of 
  R 
  retained 
  ty/v* 
  is 
  small 
  throughout, 
  so 
  that 
  

   we 
  may 
  expand 
  the 
  circular 
  functions 
  and 
  write 
  

  

  /*oo 
  /^oo 
  

  

  1=1 
  j 
  /(a?, 
  z) 
  e- 
  k 
  ^h 
  2 
  dx 
  d 
  z 
  

  

  >0 
  J-oo 
  

  

  -j 
  2 
  n 
  oo 
  /^oo 
  

  

  - 
  ^ 
  X 
  2 
  M 
  j 
  j\x, 
  z) 
  a* 
  e 
  -W 
  dx 
  dz 
  + 
  . 
  . 
  . 
  

  

  foo 
  /*00 
  

  

  L- 
  (J 
  «^ 
  00 
  

  

  /(^ 
  2> 
  3 
  e- 
  A2 
  ^ 
  /v2 
  d<r<fe-f- 
  . 
  . 
  . 
  

  

  and 
  

  

  Mil 
  

  

  