﻿116 
  Mr. 
  J. 
  H. 
  Michell 
  on 
  tlie 
  

  

  and, 
  therefore 
  *, 
  realizing, 
  

  

  Ji 
  \/x*-i 
  Jo 
  Vl+X* 
  Jo 
  Vi-x 
  2 
  

  

  = 
  I 
  g-« 
  sinhe 
  ^q_ 
  I 
  2 
  sin 
  (5 
  sin 
  0) 
  d0 
  

  

  Jo 
  Jo 
  

  

  • 
  b«J,(«)-Y,(«) 
  > 
  (8) 
  

  

  where 
  *:= 
  log 
  2— 
  y 
  = 
  -11593 
  

   From 
  which 
  

  

  i 
  2 
  

  

  since 
  

  

  = 
  [«J 
  (,)-Y 
  ( 
  S 
  )} 
  

   -i{«Ji(*)-Yi(0), 
  

  

  Jo"W+Jo'(«)A 
  + 
  Jo(s)=0 
  

  

  J,'(*)*»-Ji(0. 
  

  

  and 
  so 
  for 
  Y 
  (s). 
  

   Now 
  taking- 
  

   ly 
  1 
  e~ 
  Tk2 
  cos 
  s\ 
  , 
  - 
  . 
  

   and 
  putting 
  

  

  1 
  r°° 
  

  

  ^-rA2__ 
  __^ 
  I 
  g-M 
  2 
  Mr 
  cog 
  X^ 
  dfl, 
  

  

  1 
  f 
  •/»*._ 
  ^ 
  

  

  H 
  = 
  I7Tr) 
  J. 
  ^ 
  / 
  neosX(, 
  +/ 
  .) 
  +oo.X(.- 
  / 
  J 
  }^n 
  

  

  = 
  r-7=- 
  f 
  ^ 
  2/4r 
  {* 
  J 
  (* 
  + 
  /*) 
  - 
  Y 
  (* 
  + 
  ^) 
  

   2 
  V^ 
  Jo 
  

  

  . 
  +Kj 
  (s—fj,)—Y 
  (s—fi)\dfi, 
  . 
  (9) 
  

  

  from 
  equation 
  (8). 
  

  

  From 
  which 
  H 
  can 
  be 
  readily 
  calculated 
  by 
  mechanical 
  

   quadrature 
  in 
  the 
  case 
  in 
  which 
  4r 
  is 
  not 
  large, 
  and 
  this 
  is 
  

   the 
  case 
  for 
  ordinary 
  ships. 
  Elaborate 
  tables 
  of 
  J 
  and 
  J\ 
  

  

  * 
  See 
  Gray 
  and 
  Mathews, 
  ' 
  Bessel 
  Functions/ 
  p. 
  65, 
  and 
  Ex. 
  18, 
  

   p. 
  230. 
  The 
  formula 
  (8) 
  was 
  given 
  by 
  Weber. 
  

  

  