﻿278 
  Dr. 
  J. 
  W. 
  Nicholson 
  on 
  Bessel 
  Functions 
  

  

  and 
  is 
  the 
  real 
  part 
  o£ 
  I 
  3 
  , 
  provided 
  that 
  

  

  I 
  3 
  exp. 
  — 
  L?nr 
  =\ 
  dO 
  . 
  exp. 
  i 
  . 
  {r 
  sin 
  — 
  nd}. 
  

   Jo 
  

  

  Therefore, 
  as 
  in 
  the 
  previous 
  calculation, 
  to 
  order 
  - 
  , 
  

  

  I 
  3 
  . 
  exp. 
  — 
  inir 
  = 
  I 
  - 
  \ 
  I 
  dw 
  . 
  exp. 
  —. 
  1 
  \ 
  ty 
  3 
  + 
  n 
  — 
  z 
  I 
  - 
  J 
  w 
  I 
  ; 
  

   and 
  if 
  p 
  is 
  very 
  small, 
  it 
  follows 
  that 
  

  

  ^-i-(!K 
  r 
  (i)"(--ff) 
  +, 
  MJ)~KT)---l'H 
  

  

  On 
  reduction 
  with 
  (22), 
  

  

  J/ 
  \ 
  J-l 
  , 
  • 
  -1-2 
  

  

  _„(£) 
  = 
  h 
  sin 
  U7T 
  — 
  - 
  

  

  7T 
  7T 
  

  

  2\ 
  . 
  2tt 
  / 
  2tt 
  

  

  + 
  r! 
  r 
  (37 
  sm 
  ¥ 
  cos 
  l 
  n7r 
  " 
  j) 
  

  

  + 
  |! 
  j 
  r(|) 
  S 
  i„^cos(n.-^) 
  + 
  ...}(24) 
  

  

  When 
  ?i 
  is 
  an 
  integer 
  this 
  makes 
  J- 
  n 
  (z) 
  = 
  ( 
  — 
  ) 
  n 
  J 
  n 
  (z) 
  in 
  

   accordance 
  with 
  the 
  original 
  definition 
  o£ 
  the 
  functions. 
  

  

  We 
  note 
  that 
  to 
  order 
  — 
  , 
  

  

  J_ 
  n 
  ( 
  n 
  ) 
  = 
  r(i^.2*.3-*.7r- 
  1 
  .n-3.cos/n7r--|), 
  (25) 
  

  

  to 
  which 
  the 
  remarks 
  made 
  on 
  (19) 
  apply. 
  

   It 
  follows 
  readily 
  that 
  when 
  p 
  is 
  small, 
  

  

  f 
  J. 
  n 
  — 
  cosn7rJ 
  n 
  "\ 
  

  

  Yn 
  ^-"* 
  7r 
  \_ 
  BinnTT 
  jn 
  = 
  int. 
  

  

  p 
  being 
  n 
  — 
  2 
  . 
  ( 
  — 
  j 
  , 
  to 
  order 
  — 
  , 
  

   \z 
  / 
  n 
  

  

  