﻿of 
  Equal 
  Argument 
  and 
  Order. 
  215 
  

  

  that 
  depending 
  on 
  10 
  3 
  , 
  and 
  the 
  remaining 
  portions 
  are, 
  in 
  all 
  

   cases, 
  o£ 
  even 
  less 
  moment. 
  Neglecting 
  the 
  latter 
  for 
  the 
  

   present, 
  

  

  i 
  t 
  =(*yy(*'dw.&$' 
  -*k+/wo}. 
  . 
  . 
  . 
  (13) 
  

  

  ThllS 
  ni 
  

  

  i 
  3 
  = 
  g;. 
  ( 
  i 
  4 
  -i 
  5 
  ), 
  

  

  where 
  

  

  /"»00 
  

  

  I 
  4 
  = 
  j 
  dw 
  . 
  exp. 
  — 
  L-{w 
  z 
  -\-pw}, 
  . 
  . 
  . 
  (14) 
  

  

  Jo 
  

  

  I 
  5 
  = 
  1 
  dw 
  . 
  exp. 
  — 
  1{ 
  w 
  3 
  -f 
  /nc}, 
  

  

  x 
  = 
  -.(|" 
  

  

  But 
  I 
  5 
  has 
  an 
  expansion 
  in 
  powers 
  of 
  the 
  small 
  quantity 
  p 
  

   whose 
  leading 
  term 
  is 
  

  

  P 
  

  

  to 
  . 
  e 
  , 
  

  

  or, 
  by 
  the 
  usual 
  asymptotic 
  evaluation 
  of 
  an 
  integral 
  of 
  

   this 
  type, 
  

  

  1 
  

  

  e~ 
  lk3 
  + 
  

   3*V 
  

  

  of 
  order 
  -^ 
  at 
  most. 
  If 
  p 
  were 
  not 
  small, 
  the 
  order 
  of 
  

  

  I 
  5 
  would 
  be 
  higher 
  in 
  l/~. 
  

  

  Thus, 
  subject 
  to 
  an 
  error 
  not 
  greater 
  than 
  z~ 
  f 
  , 
  I 
  5 
  may 
  be 
  

   neglected 
  in 
  comparison 
  with 
  I 
  4 
  , 
  and 
  therefore 
  

  

  I 
  3 
  = 
  (Jjl 
  4 
  = 
  (Q*f 
  " 
  fe. 
  exp. 
  -« 
  { 
  W 
  '+p«.}. 
  (15) 
  

  

  The 
  leading 
  term 
  of 
  the 
  error 
  involved 
  in 
  the 
  expression 
  of 
  

   I 
  3 
  in 
  the 
  form 
  (13) 
  is 
  I 
  6 
  , 
  where 
  

  

  I^ffl 
  .f^jC 
  vtdw.ex$. 
  -t{t*>+pw}. 
  . 
  (16) 
  

  

  The 
  integral 
  in 
  this 
  expression 
  has 
  order 
  zero 
  in 
  s, 
  and 
  

  

  1 
  . 
  _z 
  

  

  the 
  error 
  has 
  therefore 
  an 
  order 
  - 
  , 
  which 
  is 
  again 
  z 
  * 
  

  

  relatively 
  to 
  that 
  of 
  the 
  portion 
  retained. 
  Finally, 
  with 
  an 
  

  

  