﻿I 
  

  

  4 
  Prof. 
  H. 
  Nagaoka 
  on 
  the 
  Diffraction 
  Phenomena 
  in 
  the 
  

  

  Denoting 
  the 
  limits 
  of 
  integration 
  with 
  respect 
  to 
  r 
  by 
  r 
  Q 
  

   and 
  r 
  l9 
  which 
  are 
  generally 
  functions 
  of 
  6, 
  we 
  obtain 
  

  

  ^.f(Jo 
  2 
  (n)+Ji 
  2 
  (»'o)-Jo 
  s 
  ('- 
  1 
  )-Ji 
  9 
  (n))^ 
  (HO 
  

  

  which 
  at 
  the 
  centre 
  of 
  the 
  image 
  of 
  a 
  circular 
  source 
  of 
  

   radius 
  r 
  reduces 
  to 
  

  

  I=l-J 
  «(r)-JiV)» 
  • 
  • 
  • 
  • 
  (!!•«) 
  

   and 
  at 
  the 
  vertex 
  of 
  a 
  circular 
  sector 
  including 
  angle 
  a, 
  

  

  I=Jl(l-J,«(r)-J,V))- 
  • 
  • 
  • 
  (II-*) 
  

  

  ZlT 
  

  

  2. 
  On 
  the 
  Curve 
  y 
  = 
  J 
  2 
  (x) 
  + 
  Jx 
  2 
  (x). 
  

  

  Before 
  entering 
  into 
  further 
  discussion, 
  it 
  will 
  be 
  worth 
  

   while 
  to 
  examine 
  the 
  term 
  J 
  2 
  (r) 
  + 
  J 
  1 
  2 
  (r), 
  which 
  enters 
  into 
  

   the 
  expressions 
  for 
  the 
  intensity 
  of 
  the 
  diffracted 
  light. 
  

   Although 
  the 
  values 
  of 
  J 
  *(r) 
  and 
  Ji\r) 
  are 
  in 
  themselves 
  

   oscillating, 
  the 
  sum 
  of 
  these 
  two 
  functions 
  presents 
  a 
  remark- 
  

   able 
  aspect, 
  as 
  will 
  be 
  easily 
  seen 
  by 
  representing 
  it 
  as 
  a 
  

   curve, 
  

  

  y=J,*(*)+'J 
  1 
  '(«). 
  

  

  In 
  the 
  first 
  place, 
  the 
  curve 
  is 
  confined 
  to 
  the 
  positive 
  part 
  

   of 
  the 
  ordinate 
  ; 
  further, 
  the 
  relation 
  

  

  J 
  *(«) 
  + 
  2^0) 
  + 
  2J 
  2 
  2 
  (^) 
  + 
  . 
  . 
  . 
  = 
  1 
  

  

  shows 
  that 
  y 
  cannot 
  be 
  greater 
  than 
  1, 
  which 
  it 
  will 
  attain 
  

   only 
  for 
  x 
  = 
  Q. 
  

  

  Since 
  dy 
  _ 
  _ 
  2 
  Ji 
  2 
  (aQ 
  

  

  dx 
  x 
  

  

  and 
  d?y 
  __ 
  _o 
  J 
  i 
  2 
  M 
  /oj 
  _ 
  ^M 
  \ 
  

  

  dx 
  2 
  x 
  \ 
  ° 
  x 
  y 
  

  

  we 
  see 
  that 
  points 
  corresponding 
  to 
  the 
  roots 
  x 
  n 
  of 
  J^(x)= 
  

   are 
  the 
  points 
  of 
  inflexion, 
  and 
  have 
  tangents 
  parallel 
  to 
  

   the 
  axis 
  of 
  x. 
  The 
  coordinates 
  of 
  these 
  points 
  are 
  # 
  n 
  , 
  J 
  \xn). 
  

  

  From 
  the 
  nature 
  of 
  the 
  roots 
  of 
  Ji(#) 
  =0, 
  we 
  see 
  that 
  these 
  

   points 
  occur 
  at 
  nearly 
  equal 
  intervals 
  of 
  the 
  abscissas 
  little 
  

   greater 
  than 
  it. 
  

  

  In 
  addition 
  to 
  these, 
  we 
  find 
  another 
  series 
  of 
  inflexion- 
  

   points 
  given 
  by 
  the 
  equation 
  

  

  3JlW 
  =2J 
  (*); 
  

  

  x 
  

  

  