﻿8 
  Dr. 
  J. 
  W. 
  Nicholson 
  071 
  the 
  Relation 
  of 
  

  

  Writing 
  ^ 
  -x^ 
  

  

  ^ 
  f=P 
  ^F, 
  

  

  then 
  on 
  reduction 
  

  

  Taking 
  a 
  new 
  independent 
  variable 
  y 
  defined 
  by 
  p^ 
  = 
  ?/^ 
  

   after 
  further 
  reduction, 
  

  

  This 
  is 
  identical 
  with 
  the 
  normal 
  form 
  of 
  the 
  equation 
  for 
  

   the 
  Bessel 
  function 
  I„(^), 
  provided 
  that 
  

  

  5 
  _1_ 
  2 
  _1 
  

  

  36" 
  4 
  '' 
  ^'' 
  ''~3* 
  

  

  A 
  solution 
  of 
  the 
  original 
  equation 
  is 
  therefore 
  

  

  ■ 
  /=^%(&) 
  (^) 
  

  

  In 
  tliis 
  formula, 
  I„ 
  denotes 
  the 
  function 
  

  

  I"0^)=2nr(n4-1) 
  i^ 
  + 
  SniTi 
  + 
  l"^ 
  2^2! 
  (n 
  + 
  l)(n 
  + 
  2) 
  + 
  ---|'^^^ 
  

   where, 
  if 
  ^h[x) 
  be 
  the 
  usual 
  Bessel 
  function 
  of 
  order 
  n, 
  

  

  i,.(,.) 
  = 
  r"J,.(...) 
  (9) 
  

  

  If 
  \^n{^) 
  denote 
  the 
  corresponding 
  function 
  with 
  the 
  sign 
  

   of 
  n 
  changed 
  throughout, 
  fi 
  must 
  be 
  expressible 
  in 
  the 
  form 
  

  

  /.= 
  .^{AI,(V0+BI<f^|)}. 
  

   Similarly, 
  writing 
  p=z—(T^ 
  

   \ 
  die 
  . 
  cos 
  (w^ 
  — 
  aw) 
  

  

  A 
  and 
  B 
  may 
  be 
  at 
  once 
  determined. 
  When 
  <t 
  = 
  0, 
  

  

  