﻿of 
  Equal 
  Argument 
  and 
  Order. 
  279 
  

  

  and 
  in 
  particular, 
  

  

  Y,(n)=-r(i).2r*3*#T*. 
  • 
  . 
  (27) 
  

  

  with 
  the 
  same 
  criticisms 
  as 
  (25, 
  19). 
  

  

  When 
  p 
  is 
  not 
  small 
  in 
  comparison 
  with 
  unity, 
  the 
  results 
  

   are 
  best 
  left 
  as 
  integrals 
  to 
  be 
  calculated 
  by 
  Airy's 
  method 
  *. 
  

  

  Tf 
  

  

  ■*- 
  L 
  /too 
  

  

  F(» 
  = 
  | 
  cos 
  (w* 
  + 
  pw)dw 
  . 
  . 
  . 
  . 
  (2S) 
  

  

  *, 
  

  

  /0) 
  = 
  l 
  sm(w*+pw)dw. 
  . 
  . 
  . 
  (29) 
  

  

  Jo 
  

  

  p=^.(J)* 
  (30) 
  

  

  Then 
  

  

  J»(--) 
  = 
  ^-g) 
  i 
  F(p)- 
  ..... 
  (31) 
  

  

  J. 
  llW 
  =g) 
  i 
  { 
  C0 
  ^F( 
  P 
  ) 
  + 
  ^/(p)} 
  

  

  + 
  g-) 
  1 
  . 
  s 
  ^ 
  . 
  £ 
  { 
  F 
  („ 
  -'-?) 
  + 
  ./(„ 
  -v)} 
  (32) 
  

  

  T.(*) 
  = 
  -(j)*{«tp( 
  /m 
  -t)4»-t/(p«-t)]. 
  . 
  (33) 
  

  

  These 
  may 
  be 
  proved 
  by 
  the 
  application 
  of 
  contour 
  inte- 
  

   gration 
  to 
  the 
  integral 
  previously 
  called 
  I 
  2 
  . 
  

  

  The 
  reduction 
  of 
  the 
  Bessel 
  functions 
  to 
  a 
  dependence 
  on 
  

   Airy's 
  integral 
  and 
  its 
  associate 
  is 
  important 
  in 
  that 
  it 
  

   furnishes 
  a 
  means 
  of 
  comparing 
  the 
  results 
  of 
  the 
  ordinary 
  

   theory 
  of 
  diffraction 
  problems, 
  as 
  developed 
  by 
  Fresnel 
  and 
  

   others, 
  with 
  those 
  of 
  the 
  electromagnetic 
  theory. 
  The 
  latter 
  

   has 
  not 
  hitherto 
  led 
  to 
  these 
  integrals, 
  but 
  their 
  connexion, 
  

   through 
  the 
  Bessel 
  functions 
  to 
  which 
  dynamical 
  theory 
  

   naturally 
  leads, 
  may 
  now 
  be 
  seen. 
  

  

  * 
  Loc. 
  cit. 
  

  

  