﻿276 
  Dr. 
  J. 
  W. 
  Nicholson 
  on 
  Bessel 
  Functions 
  

  

  error 
  not 
  greater 
  than 
  — 
  , 
  I\ 
  is 
  the 
  real 
  part 
  of 
  I 
  3 
  , 
  where 
  

  

  z 
  

  

  J 
  s 
  = 
  ( 
  r 
  j 
  dw 
  - 
  ex 
  P- 
  ~ 
  * 
  i 
  w 
  * 
  + 
  P 
  w 
  }> 
  

   in 
  which 
  p 
  = 
  n 
  — 
  z 
  . 
  ( 
  - 
  1 
  (17) 
  

  

  Now 
  when 
  n 
  and 
  z 
  are 
  great, 
  

  

  /-»00 
  

  

  - 
  I 
  ,,— 
  h# 
  — 
  ~sinhl 
  

  

  Jo 
  

  

  dO 
  

  

  is 
  o£ 
  order 
  - 
  , 
  and 
  may 
  also 
  therefore 
  be 
  ignored. 
  Thus 
  

   z 
  

  

  to 
  order 
  — 
  , 
  when 
  n— 
  z. 
  (— 
  J 
  is 
  not 
  large 
  compared 
  with 
  

   unity, 
  

  

  J 
  »^)=^-(J)*[ 
  cos|^ 
  + 
  «^(|)%.|^, 
  . 
  (18) 
  

  

  which 
  is 
  a 
  multiple 
  o£ 
  Airy's 
  Integral 
  *. 
  

  

  In 
  another 
  connexion, 
  Stokes 
  has 
  considered 
  the 
  properties 
  

   of 
  this 
  integral 
  in 
  great 
  detail. 
  The 
  tables 
  given 
  by 
  Stokes 
  

   and 
  Airy 
  may 
  be 
  at 
  once 
  transformed 
  into 
  tables 
  of 
  the 
  

   Bessel 
  functions 
  whose 
  order 
  and 
  argument 
  are 
  nearly 
  equal. 
  

   If 
  tables 
  be 
  required 
  to 
  a 
  higher 
  order 
  than 
  this 
  integral 
  will 
  

   furnish, 
  the 
  necessary 
  correction 
  may 
  be 
  readily 
  made 
  by 
  

   keeping 
  certain 
  terms 
  ignored 
  in 
  the 
  above 
  discussion, 
  but 
  

   this 
  will 
  not 
  usually 
  be 
  necessarv. 
  We 
  note 
  that 
  to 
  

  

  order 
  — 
  , 
  

   n 
  

  

  J«(ft) 
  = 
  — 
  • 
  ( 
  — 
  I 
  \ 
  cos 
  ID 
  Z 
  . 
  dw 
  

  

  \ 
  c 
  o 
  

  

  = 
  T(^\.2-H~*7r- 
  1 
  n-K 
  . 
  . 
  . 
  (19) 
  

  

  But 
  this 
  formula, 
  although 
  only 
  formally 
  proved 
  true 
  to 
  

   this 
  -order, 
  has 
  in 
  reality 
  a 
  much 
  wider 
  application 
  in 
  practice, 
  

   a 
  property 
  shared 
  by 
  the 
  companion 
  formulae 
  proved 
  in 
  this 
  

   paper. 
  Thus 
  on 
  comparison 
  with 
  tables 
  f, 
  the 
  expression 
  

   last 
  written 
  gives 
  J 
  7 
  (7) 
  correctly 
  to 
  three 
  places 
  of 
  decimals, 
  

  

  * 
  Airy, 
  Cambr. 
  Phil. 
  Trans, 
  vol. 
  vi. 
  p. 
  379, 
  vol. 
  viii, 
  p. 
  595. 
  Stokes, 
  

   Camb. 
  Phil. 
  Trans, 
  vol. 
  ix 
  ; 
  Math. 
  & 
  Phys. 
  Papers, 
  ii, 
  p. 
  329 
  et 
  seg. 
  

   f 
  E. 
  g. 
  Gray 
  and 
  Matthews' 
  Treatise 
  on 
  Bessel 
  Functions. 
  

  

  