﻿118 
  Mr. 
  J. 
  H. 
  Michell 
  on 
  the 
  

  

  and 
  the 
  formula 
  can 
  be 
  written 
  

  

  Vl(x 
  — 
  x') 
  L* 
  J 
  

  

  This 
  gives 
  a 
  maximum 
  resistance, 
  approximately, 
  when 
  

  

  x 
  — 
  x 
  = 
  (n 
  + 
  7/8)/ 
  \n 
  an 
  integer] 
  

  

  and. 
  a 
  maximum 
  assistance 
  when 
  

  

  a;— 
  a/=(n 
  + 
  3/8)l. 
  

  

  These 
  formulas 
  correspond 
  to 
  the 
  interference 
  of 
  the 
  bow 
  

   and 
  stern 
  waves, 
  which 
  has 
  been 
  so 
  skilfully 
  discussed 
  by 
  

   Mr. 
  R. 
  E. 
  Froude. 
  When 
  the 
  two 
  elements 
  are 
  on 
  the 
  same 
  

   vertical 
  cross-section 
  of 
  the 
  ship, 
  another 
  form 
  of 
  reduction 
  

   may 
  be 
  given. 
  Putting 
  x 
  — 
  .v/ 
  = 
  0, 
  the 
  integral 
  to 
  be 
  con- 
  

   sidered 
  is 
  

  

  Ji 
  VX 
  2 
  -1 
  

  

  Put 
  X» 
  = 
  j(l+/i), 
  

  

  so 
  that 
  7 
  ^ 
  1 
  da 
  

  

  dX 
  = 
  

  

  2^2 
  Vl+/*' 
  

   and 
  A 
  2 
  — 
  1 
  = 
  J(/i— 
  1) 
  ; 
  

  

  whence 
  

  

  du 
  

  

  ^i 
  

  

  or, 
  if 
  a 
  = 
  cosh 
  <£, 
  

  

  G 
  = 
  i 
  e- 
  r 
  ' 
  % 
  \ 
  e~l' 
  cosh 
  »V» 
  d(f> 
  

  

  where 
  K 
  is 
  the 
  Bessel 
  function, 
  so 
  indicated 
  by 
  Gray 
  and 
  

   Mathews 
  (pp. 
  67, 
  90). 
  

   Hence 
  

  

  [ 
  

  

  since 
  

  

  

  -rtf 
  

  

  K 
  2 
  dX 
  

  

  dGc 
  

  

  

  

  e 
  

  

  

  Vx 
  2 
  -i 
  

  

  dr 
  

  

  

  

  

  

  

  = 
  |,-" 
  2 
  {K, 
  

  

  )0V2) 
  - 
  

  

  -Ko'^/2)} 
  

  

  

  

  

  = 
  ie-lz{K, 
  

  

  Mi) 
  - 
  

  

  -K^/2)}. 
  

  

  

  

  

  K 
  '=K 
  U 
  

  

  

  

  