﻿14 
  Prof. 
  H. 
  Nagaoka 
  on 
  the 
  Diffraction 
  Phenomena 
  in 
  the 
  

  

  and 
  when 
  lies 
  external 
  to 
  the 
  circle 
  

  

  ■ 
  i 
  7 
  9 
  4a(a 
  + 
  S) 
  

  

  p 
  e 
  =(2a 
  + 
  8)^l-k 
  e 
  2 
  sm 
  2 
  cj>, 
  where 
  k 
  2 
  = 
  ^ 
  a 
  ^ 
  

  

  Thus 
  k 
  2 
  is 
  always 
  less 
  than 
  1, 
  whether 
  the 
  point 
  be 
  inside 
  or 
  

   outside 
  the 
  circle. 
  

  

  We 
  can 
  easily 
  prove 
  that 
  

  

  (2a-8)(l-^ 
  2 
  sin 
  2 
  </)) 
  

  

  d 
  ° 
  = 
  (/ 
  + 
  (2 
  fl 
  -S)(l-tfrin«A))fo* 
  

  

  (hi 
  \ 
  

   ^ 
  "^ 
  i 
  — 
  z. 
  2 
  * 
  • 
  2 
  a 
  ) 
  ^ 
  ^ 
  or 
  an 
  internal 
  point, 
  

   J. 
  t* 
  /Cj- 
  sin 
  Q)/ 
  

  

  and 
  

  

  dd 
  

  

  = 
  J 
  1 
  — 
  , 
  2 
  e 
  . 
  2 
  J 
  d(/> 
  for 
  an 
  external 
  point, 
  

   hi 
  and 
  */ 
  being 
  defined 
  by 
  

  

  hi 
  = 
  VI 
  - 
  *> 
  2 
  = 
  

  

  hi 
  = 
  VI 
  - 
  * 
  * 
  = 
  

  

  2a 
  -5' 
  

   8 
  

  

  2a 
  + 
  8' 
  

   The 
  intensity 
  at 
  an 
  internal 
  point 
  becomes 
  

  

  ^Jo 
  Jo 
  r 
  

  

  ir 
  

  

  For 
  an 
  external 
  point 
  we 
  get 
  

  

  where 
  p'=OP', 
  p=OP, 
  and 
  «=ZAOT. 
  

   Putting 
  the 
  values 
  of 
  /o 
  e 
  and 
  d#, 
  we 
  obtain 
  

  

  i>-^(W+VW)(i 
  -i-jJ^H 
  ■ 
  

  

  If 
  the 
  point 
  be 
  not 
  very 
  near 
  the 
  rim 
  of 
  the 
  circle, 
  we 
  

   can 
  put 
  

  

  2 
  

  

  J 
  2 
  0>) 
  + 
  J 
  i 
  2 
  (p) 
  = 
  — 
  nearly, 
  

  

  r 
  

  

  * 
  See 
  Halphen, 
  Traite 
  des 
  Fonctions 
  Elliptiques, 
  torn. 
  i. 
  

  

  