﻿Interpretation 
  oj 
  the 
  Bessel 
  Function 
  of 
  Zero 
  Order. 
  435 
  

  

  The 
  complete 
  wave 
  at 
  any 
  point 
  is, 
  therefore, 
  the 
  real 
  

   part 
  of 
  e 
  { 
  Pt 
  {A 
  x 
  e- 
  iax 
  + 
  B 
  x 
  e 
  iax 
  }. 
  

  

  Now 
  to 
  solve 
  equations 
  (14) 
  and 
  (15). 
  

  

  Differentiating 
  both 
  with 
  regard 
  to 
  x, 
  (14) 
  gives 
  (neglecting 
  

   subscripts 
  to 
  A 
  and 
  B) 
  

  

  — 
  ■- 
  A. 
  B 
  e2iax 
  

  

  dx 
  - 
  2x 
  + 
  ~Jx~ 
  ■ 
  • 
  • 
  • 
  C 
  16 
  ) 
  

   dB 
  B 
  A 
  e~ 
  2iax 
  

  

  (17) 
  

  

  dx 
  2x 
  2x 
  

  

  Differentiate 
  (16) 
  and 
  (17) 
  with 
  regard 
  to 
  x, 
  

  

  dx 
  2 
  + 
  2x 
  dx 
  2x 
  2 
  - 
  2x 
  'dx 
  +±J 
  ( 
  2x 
  2x 
  2 
  ( 
  * 
  U 
  ° 
  j 
  

  

  d^B^^dB 
  JB_ 
  e~ 
  2iax 
  dA 
  _ 
  2iax 
  f 
  —2ia 
  _1_") 
  . 
  

  

  dx 
  2 
  ~ 
  2x 
  dx 
  + 
  2x 
  2 
  ~ 
  ~2x~ 
  dx 
  ■ 
  I 
  2a* 
  ~~ 
  2a 
  2 
  J 
  '* 
  

  

  (19) 
  

  

  eliminating 
  B 
  and 
  — 
  from 
  (18) 
  by 
  means 
  of 
  (16) 
  and 
  (17) 
  

  

  d 
  2 
  A 
  l^dA 
  A_,^|B 
  ^ 
  e 
  -2iax 
  x 
  

  

  ^ 
  2+ 
  2^^~2^ 
  = 
  ~2^7 
  12#~ 
  2# 
  J 
  

   (. 
  2# 
  2x 
  2 
  J 
  (. 
  «# 
  -^ 
  J 
  

  

  A 
  /V 
  _i\/^A 
  A\ 
  1 
  ( 
  dA 
  L 
  A\ 
  

  

  4.x 
  2 
  + 
  V 
  m 
  x)[dx 
  + 
  2x) 
  + 
  2x 
  \dx 
  + 
  2x 
  ) 
  

   i. 
  e. 
  d 
  2 
  A 
  dA/ 
  a 
  . 
  1\ 
  iaA 
  /OA 
  . 
  

  

  d^ 
  2 
  --dx-{ 
  2ia 
  -x)--x~=°> 
  ' 
  ' 
  (20) 
  

  

  similarly 
  eliminating 
  A 
  from 
  (16), 
  (17). 
  (19), 
  

  

  d?B 
  _ 
  J. 
  dB 
  B_ 
  e~ 
  2iax 
  r 
  A 
  Be 
  2iax 
  -) 
  

   dx 
  2 
  ~~2xdx 
  + 
  2^? 
  = 
  ~Jx~~ 
  1 
  " 
  2x 
  + 
  ~1^T 
  j 
  

  

  2# 
  \ 
  x 
  ) 
  X 
  dx 
  2x 
  ) 
  

  

  ~ 
  4a; 
  2 
  + 
  V 
  m+ 
  x)[~ 
  dx 
  ~ 
  2x) 
  + 
  2x\ 
  dx 
  " 
  2a; 
  J 
  

  

  ( 
  a 
  . 
  , 
  1WB 
  iaB 
  A 
  , 
  M 
  ^ 
  

  

  + 
  \ 
  2ia 
  +x)dx- 
  + 
  -x-=°- 
  • 
  ■ 
  ^ 
  

  

  i. 
  *. 
  _^B 
  /«. 
  . 
  l\dB 
  iaB 
  

  

  dx 
  2 
  

  

  (20) 
  and 
  (21) 
  are 
  the 
  two 
  differential 
  equations 
  to 
  determine 
  

   A 
  and 
  B. 
  

  

  