﻿Airy^s 
  Integral 
  to 
  the 
  Bessel 
  Functions. 
  15 
  

  

  expressions 
  o£ 
  disturbances 
  in 
  wave 
  theory 
  originating 
  from 
  

   a 
  body 
  and 
  passing 
  off 
  into 
  space. 
  

   It 
  may 
  be 
  at 
  once 
  shown 
  that 
  

  

  ^n{^)^\Q^e-"l[f,^f,-,f,] 
  . 
  . 
  ..(36) 
  

  

  where 
  the 
  functions 
  /have 
  the 
  usual 
  argument, 
  if 
  z 
  and 
  n 
  do 
  

   not 
  differ 
  widely 
  in 
  comparison 
  Avith 
  zh. 
  Therefore, 
  if 
  the 
  

   Bessel 
  functions 
  of 
  order 
  J- 
  have 
  the 
  appropriate 
  argument 
  cr, 
  

   whether 
  n 
  be 
  integral 
  or 
  not, 
  

  

  K„(--) 
  = 
  \ 
  ©*-""'^3\/3 
  { 
  V3J-i- 
  V3Ji 
  -. 
  Ji-. 
  J-i 
  } 
  

   which 
  may 
  be 
  written 
  

  

  But 
  by 
  the 
  selected 
  definition, 
  for 
  any 
  argument 
  

  

  and 
  thus 
  when 
  n 
  and 
  z 
  are 
  large, 
  and 
  n-^z 
  not 
  very 
  large 
  in 
  

   comparison 
  with 
  z^, 
  

  

  K„(-') 
  = 
  i(^. 
  -- 
  n^%Q.n-z..\/'ln-z) 
  

  

  (38) 
  

  

  if 
  z 
  be 
  greater 
  than 
  n. 
  

  

  The 
  modification 
  necessary 
  in 
  this 
  result, 
  as 
  in 
  (31-33), 
  

   when 
  z 
  is 
  less 
  than 
  w, 
  may 
  be 
  obtained 
  at 
  once. 
  The 
  accuracy 
  

   of 
  all 
  the 
  formulae 
  is 
  as 
  stated 
  above. 
  

  

  The 
  following 
  tables 
  have 
  been 
  calculated 
  from 
  the 
  results 
  

   given 
  by 
  Airy, 
  with 
  the 
  aid 
  of 
  the 
  formula 
  

  

  Airy 
  employs 
  the 
  integral 
  in 
  the 
  form 
  

  

  /•■oo 
  

  

  I 
  TT 
  

  

  I(??i)=l 
  dw 
  COS- 
  {w^—mw) 
  . 
  . 
  (39) 
  

  

  Jo 
  ^ 
  

  

  which 
  is 
  identical 
  with 
  

  

  I(„0=g)V:{— 
  gyi. 
  . 
  . 
  . 
  (40) 
  

   SO 
  that 
  

  

  *w.(^y.|-..(?yi 
  . 
  . 
  . 
  m, 
  

  

  