﻿of 
  Equal 
  Order 
  and 
  Argument. 
  369 
  

  

  When 
  n 
  is 
  an 
  even 
  integer, 
  we 
  find 
  that 
  

  

  •/'o 
  Jo 
  

  

  The 
  integral 
  on 
  the 
  right 
  does 
  not 
  seem 
  to 
  be 
  expressible 
  

   by 
  elementary 
  functions 
  , 
  but 
  Ave 
  have 
  

  

  c 
  

  

  J 
  oG/)<fy 
  = 
  l- 
  \ 
  J 
  (y)dy 
  

  

  CO 
  

  

  sin 
  Q-7T— 
  n) 
  

  

  /(±7T*) 
  

  

  by 
  integrating 
  the 
  ordinary 
  asymptotic 
  expansion 
  of 
  J 
  (y). 
  

   The 
  following 
  table 
  indicates 
  the 
  mode 
  of 
  increase 
  of 
  

  

  n\ 
  5 
  n 
  (nx)dx 
  to 
  its 
  limit 
  (namely 
  J) 
  as 
  n 
  increases 
  through 
  

   Jo 
  

   the 
  odd 
  integral 
  values 
  1, 
  3, 
  5, 
  7, 
  ... 
  . 
  

  

  n. 
  

  

  n 
  \ 
  J 
  n 
  (nx)dx. 
  

   Jo 
  

  

  n. 
  

  

  n 
  \ 
  J 
  Jnx)dx. 
  

  

  Jo 
  

  

  1 
  

  

  0-2348023 
  

  

  13 
  

  

  0-3175245 
  

  

  3 
  

  

  0-2878694 
  

  

  15 
  

  

  0-3190390 
  

  

  5 
  

  

  0-3020013 
  

  

  17 
  

  

  0-3202397 
  

  

  7 
  

  

  0-3087657 
  

  

  19 
  

  

  0-3212187 
  

  

  9 
  

  

  0-3128134 
  

  

  21 
  

  

  0-3220347 
  

  

  11 
  

  

  0-3155424 
  

  

  23 
  

  

  0-3227271 
  

  

  4. 
  We 
  can 
  obtain 
  a 
  closer 
  approximation 
  to 
  the 
  integral 
  

   in 
  the 
  following 
  manner 
  : 
  — 
  

   When 
  6 
  is 
  small 
  we 
  have 
  

  

  0+ 
  sin 
  cos 
  0-2<9 
  2 
  cot 
  0cv> 
  8/9 
  5 
  /45, 
  

   <9 
  2 
  -sin 
  2 
  <9cv>0Y3, 
  

   2 
  - 
  sin 
  2 
  6 
  cos 
  2 
  6 
  cv>40 
  4 
  /3, 
  

   F(0)cvd4(9 
  3 
  /(V3). 
  

  

  Hence 
  from 
  (6) 
  we 
  find 
  that 
  7r 
  — 
  tan 
  -1 
  { 
  2r 
  sin 
  0/(1— 
  r 
  2 
  )} 
  

   is 
  a 
  function 
  of 
  6 
  approximately 
  equal 
  to 
  

  

  ^-07(5v/3), 
  

  

  and 
  the 
  complete 
  expansion 
  of 
  this 
  function 
  involves 
  even 
  

   powers 
  of 
  6 
  only. 
  

  

  