﻿

  of 
  Equal 
  Argument 
  and 
  Order. 
  277 
  

  

  and 
  J 
  10 
  (10) 
  to 
  four 
  places. 
  This 
  indicates 
  that 
  n 
  does 
  not 
  

   need 
  to 
  be 
  very 
  great 
  for 
  the 
  formulae 
  to 
  give 
  good 
  approxi- 
  

   mations, 
  although 
  these 
  approximations 
  rapidly 
  become 
  more 
  

   valid 
  as 
  n 
  increases. 
  

  

  This 
  comparison 
  made 
  in 
  the 
  case 
  in 
  which 
  the 
  formulae 
  

   will 
  necessarily 
  be 
  least 
  accurate, 
  is 
  of 
  more 
  practical 
  value 
  

   than 
  a 
  refined 
  analysis 
  of 
  the 
  exact 
  validity 
  of 
  the 
  investi- 
  

   gation, 
  which 
  would 
  be 
  extremely 
  cumbrous. 
  

  

  By 
  an 
  expansion 
  of 
  (18), 
  it 
  is 
  readily 
  shown 
  that 
  if 
  

  

  J 
  " 
  ^ 
  = 
  s 
  • 
  (r) 
  {r 
  (3) 
  cos 
  6 
  - 
  + 
  £ 
  r 
  (3) 
  cos 
  T 
  

  

  + 
  g 
  r 
  (l) 
  cos 
  ? 
  + 
  --} 
  ( 
  20 
  > 
  

  

  when 
  n 
  and 
  z 
  are 
  nearly 
  equal. 
  

  

  The 
  expansion 
  of 
  J- 
  n 
  (~) 
  may 
  be 
  derived 
  in 
  a 
  similar 
  

   manner 
  from 
  the 
  formula 
  (z 
  positive) 
  

  

  J_ 
  n 
  <» 
  = 
  - 
  ( 
  "cos 
  [z 
  sin 
  6 
  + 
  n0)d0 
  + 
  sin 
  ^ 
  7r 
  ^ 
  e 
  «e-zsinhe 
  cW 
  

   Jo 
  * 
  

  

  = 
  - 
  l 
  + 
  sin»:r^ 
  (say) 
  (21) 
  

  

  But, 
  to 
  the 
  same 
  order 
  as 
  in 
  ail 
  the 
  above 
  calculations, 
  

   I, 
  = 
  J" 
  d0exp.-fn-^0-^6 
  3 
  \ 
  

  

  where 
  /? 
  has 
  its 
  previous 
  value, 
  and 
  is 
  small. 
  

   Again, 
  

  

  I 
  T 
  = 
  ( 
  ' 
  cos 
  (z 
  sin 
  6 
  + 
  n0)dO 
  

   Jo 
  

  

  — 
  1 
  ' 
  cos 
  {z 
  sin 
  + 
  rnr— 
  n6) 
  dO, 
  

  

  Jo 
  

  

  