﻿J" 
  

  

  Jo 
  

  

  of 
  Equal 
  Order 
  and 
  Argument. 
  365 
  

  

  The 
  associated 
  function 
  

  

  'i 
  

   J 
  n 
  (nx)da?, 
  

  

  o 
  

  

  which 
  does 
  not 
  occur 
  in 
  many 
  of 
  the 
  physical 
  problems 
  in 
  

   which 
  the 
  other 
  Bessel 
  functions 
  present 
  themselves, 
  appears 
  

   to 
  play 
  a 
  prominent 
  part 
  in 
  connexion 
  with 
  various 
  series 
  

   arising 
  in 
  the 
  theory 
  of 
  Electromagnetic 
  Radiation, 
  and 
  

   consequently 
  Professor 
  Schott 
  has 
  asked 
  me 
  to 
  determine 
  

   whether 
  there 
  is 
  any 
  approximate 
  formula 
  analogous 
  to 
  the 
  

   results 
  

  

  t 
  / 
  n 
  r 
  tt) 
  t 
  r/ 
  n 
  3*r(#) 
  

  

  tt2 
  3 
  3 
  6 
  ti 
  3 
  7r2% 
  f 
  

  

  This 
  note, 
  in 
  which 
  I 
  prove 
  the 
  remarkably 
  simple 
  result 
  

   that 
  

  

  ""l 
  1 
  

  

  J 
  n 
  (n#)^oo— 
  , 
  (1) 
  

  

  is 
  the 
  outcome 
  of 
  his 
  inquiry. 
  A 
  closer 
  approximation 
  is 
  

   given 
  in 
  § 
  4, 
  but 
  it 
  involves 
  the 
  gamma 
  function 
  of 
  1/3 
  ; 
  

   this 
  more 
  precise 
  result 
  is 
  

  

  Jn{nx)d 
  ^^-¥5Jray 
  

  

  In 
  order 
  to 
  obtain 
  this 
  approximate 
  formula 
  I 
  propose 
  to 
  

   employ 
  not 
  the 
  elementary 
  methods 
  which 
  I 
  have 
  used 
  

   elsewhere* 
  in 
  connexion 
  with 
  J„(n) 
  and 
  J„'(n), 
  but 
  the 
  

   methods 
  which 
  depend 
  on 
  the 
  contour 
  integrals 
  of 
  Debye 
  ; 
  

   the 
  latter 
  methods 
  yield 
  the 
  desired 
  result 
  with 
  a 
  much 
  

   smaller 
  expenditure 
  of 
  labour. 
  

  

  2. 
  We 
  take 
  the 
  well-known 
  contour 
  integral 
  

  

  J„(n*)= 
  5^.f 
  (0+) 
  /' 
  I 
  «- 
  I/ 
  V"- 
  1 
  A, 
  

  

  (in 
  which 
  the 
  contour 
  starts 
  from 
  -co, 
  encircles 
  the 
  origin 
  

   once 
  counter-clockwise, 
  and 
  then 
  returns 
  to 
  -co), 
  and 
  on 
  

   integrating 
  under 
  the 
  integral 
  sign 
  we 
  get 
  

  

  r 
  

  

  r 
  

  

  i 
  

  

  

  JLf 
  (0+) 
  I 
  

  

  dt. 
  

  

  ft7nj_ 
  fl0 
  t 
  2 
  — 
  1 
  

   * 
  See 
  the 
  last 
  of 
  the 
  papers 
  cited. 
  

  

  