﻿Airi/^s 
  Ttitegral 
  to 
  the 
  Bessel 
  Functions. 
  

   expanding 
  the 
  Bessel 
  functions, 
  we 
  obtain 
  

  

  or 
  

  

  leading 
  to 
  

  

  by 
  a 
  property 
  of 
  the 
  Gamma 
  functions. 
  

   Again, 
  

  

  = 
  ,- 
  I 
  die 
  cos 
  (iv^ 
  —pic) 
  with 
  p 
  = 
  ; 
  

  

  = 
  I 
  IV 
  sin 
  lo^dw 
  = 
  — 
  V 
  ( 
  .=, 
  ), 
  

  

  Jo 
  2\/3 
  \3/ 
  

  

  and 
  finally, 
  when 
  a 
  is 
  positive, 
  

   j 
  die 
  . 
  cos 
  (?6'^ 
  — 
  (Tic) 
  

  

  Moreover, 
  when 
  p 
  is 
  positive, 
  

  

  /I 
  00 
  

  

  I 
  (it(;cos 
  (?t'^ 
  + 
  pic) 
  

  

  By 
  the 
  property 
  of/a+Zs 
  indicated 
  above, 
  there 
  exists 
  also 
  

   a 
  relation 
  of 
  the 
  form 
  

  

  dwe-''^-''''' 
  + 
  1 
  dwi\\\{w^ 
  — 
  Gw) 
  

  

  