﻿8 
  Prof. 
  H. 
  Nagaoka 
  on 
  the 
  Diffraction 
  Phenomena 
  in 
  the 
  

  

  Of 
  these 
  three 
  expressions 
  (A), 
  (B), 
  (C), 
  we 
  shall 
  have 
  

   occasion 
  to 
  use 
  the 
  second 
  form 
  (B) 
  most 
  frequently, 
  as 
  it 
  

   expresses 
  the 
  nature 
  of 
  the 
  curve 
  y 
  = 
  J 
  2 
  (x) 
  +J 
  x 
  2 
  (x) 
  in 
  the 
  

   simplest 
  manner. 
  It 
  shows 
  that 
  the 
  curve 
  is 
  to 
  the 
  first 
  

   approximation 
  a 
  rectangular 
  hyperbola, 
  

  

  «*=§, 
  (D) 
  

  

  if 
  x 
  be 
  not 
  small. 
  To 
  the 
  second 
  approximation, 
  we 
  have 
  to 
  

  

  cos 
  2i 
  v 
  

   introduce 
  the 
  term 
  z— 
  . 
  which 
  gives 
  the 
  curve 
  an 
  undu- 
  

  

  7TX 
  2 
  ' 
  & 
  

  

  lating 
  appearance 
  ; 
  the 
  effect 
  of 
  the 
  third 
  and 
  fourth 
  terms 
  is 
  

   still 
  smaller, 
  so 
  that 
  for 
  practical 
  purposes 
  it 
  is 
  sufficient 
  to 
  

   assume 
  the 
  mean 
  curve 
  to 
  be 
  an 
  hyperbola, 
  as 
  it 
  is 
  very 
  

   tedious 
  to 
  push 
  the 
  calculation 
  to 
  the 
  fourth 
  decimal 
  place. 
  

   To 
  show 
  the 
  difference 
  in 
  y 
  n 
  calculated 
  from 
  (B) 
  and 
  (D), 
  I 
  

   give 
  the 
  following 
  table 
  : 
  — 
  

  

  #!= 
  3-832, 
  ^=0-1622 
  by 
  (B), 
  yi 
  = 
  0-1661 
  bv 
  (D). 
  

  

  a? 
  2 
  = 
  7-016, 
  # 
  2 
  =0'0921 
  * 
  „ 
  ?/ 
  2 
  =0-0907 
  " 
  „ 
  

  

  .273 
  = 
  10-173, 
  y 
  3 
  =0-0624 
  „ 
  #3 
  = 
  0-0626 
  „ 
  

  

  ^ 
  = 
  13-324, 
  # 
  4 
  =0-0477 
  „ 
  # 
  4 
  =0-0478 
  „ 
  

  

  Thus 
  the 
  coincidence 
  becomes 
  closer 
  with 
  increasing 
  values 
  

   of 
  X. 
  

  

  4. 
  Intensity 
  at 
  the 
  Centre 
  of 
  a 
  Circular 
  Image. 
  

  

  Equation 
  (II. 
  a) 
  shows 
  that 
  the 
  intensity 
  of 
  light 
  at 
  the 
  centre 
  

   of 
  a 
  luminous 
  disk 
  as 
  seen 
  through 
  a 
  telescope 
  is 
  given 
  by 
  

  

  IW 
  = 
  l-J„ 
  2 
  W-J 
  I 
  s 
  (r) 
  

   = 
  l-y. 
  

  

  If 
  the 
  disk 
  be 
  divided 
  by 
  a 
  series 
  of 
  concentric 
  circles 
  of 
  

   radius 
  x 
  n 
  into 
  a 
  number 
  of 
  zones, 
  whose 
  breadth 
  is 
  equal 
  to 
  

   the 
  difference 
  between 
  the 
  successive 
  roots 
  of 
  J 
  1 
  fa?)=0, 
  we 
  

   find 
  that 
  the 
  illumination 
  at 
  the 
  centre 
  due 
  to 
  each 
  of 
  these 
  

   zones 
  is 
  given 
  by 
  the 
  height 
  of 
  the 
  corresponding 
  step 
  in 
  

   the 
  curve 
  # 
  = 
  Jq 
  2 
  (x) 
  + 
  J 
  1 
  2 
  (x). 
  The 
  diminution 
  in 
  the 
  height 
  

   of 
  these 
  steps, 
  with 
  increasing 
  x, 
  shows 
  that 
  the 
  effect 
  of 
  the 
  

   zone 
  of 
  nearly 
  the 
  same 
  breadth 
  varies 
  almost 
  inversely 
  pro- 
  

   portional 
  to 
  its 
  distance 
  from 
  the 
  centre. 
  The 
  same 
  reasoning 
  

   will 
  apply 
  to 
  a 
  circular 
  sector. 
  

  

  The 
  following 
  table 
  gives 
  the 
  intensity 
  at 
  the 
  centre 
  of 
  the 
  

   luminous 
  disk 
  whose 
  radius 
  is 
  equal 
  to 
  the 
  root 
  of 
  J 
  x 
  (r*) 
  = 
  : 
  — 
  

  

  