﻿44 
  Dr. 
  L. 
  Silberstein 
  on 
  Light 
  Distribution 
  

  

  the 
  curves 
  mark, 
  by 
  their 
  centres, 
  the 
  apertures 
  corre- 
  

   sponding 
  to 
  v 
  = 
  0'l, 
  0*2, 
  and 
  so 
  on, 
  up 
  to 
  v 
  = 
  2>'0. 
  The 
  

   solitary 
  circlets 
  near 
  v 
  = 
  l*0, 
  2*0, 
  3*0 
  belong 
  to 
  p 
  = 
  0, 
  i.e. 
  to 
  

   the 
  central 
  or 
  focal 
  curve 
  which 
  is 
  the 
  usual 
  Cornu 
  spiral. 
  

   It 
  will 
  be 
  seen 
  that 
  the 
  curve 
  [1] 
  deviates 
  but 
  little 
  from 
  the 
  

   central 
  one. 
  The 
  intensity, 
  proportional 
  to 
  the 
  square 
  of 
  

   distance 
  from 
  the 
  origin, 
  increases 
  with 
  v, 
  or 
  with 
  the 
  square 
  

   of 
  the 
  aperture, 
  first 
  rapidly 
  and 
  then 
  more 
  and 
  more 
  slowly, 
  

   and 
  reaches 
  its 
  maximum 
  a 
  little 
  beyond 
  v=l'2. 
  Then 
  it 
  

   falls 
  to 
  a 
  minimum 
  which 
  is 
  considerably 
  smaller 
  than 
  that 
  

   maximum, 
  and 
  so 
  on. 
  The 
  spiral 
  character 
  of 
  the 
  curve 
  [1] 
  

   would 
  go 
  on 
  for 
  over 
  3000 
  windings 
  (v 
  over 
  140) 
  which, 
  

   however, 
  for 
  any 
  reasonable 
  lens, 
  would 
  lie 
  much 
  beyond 
  

   the 
  limit-aperture 
  (h 
  = 
  r/n) 
  ; 
  the 
  spiral 
  would 
  reach 
  its 
  cusp, 
  

   or 
  first 
  zero 
  of 
  J 
  , 
  much 
  beyond 
  that 
  limit. 
  The 
  next 
  curve, 
  

   [10], 
  is 
  very 
  soon 
  deprived 
  of 
  its 
  simple 
  spiral 
  character 
  ; 
  the 
  

   Bessel 
  function 
  vanishes 
  already 
  at 
  v 
  = 
  1*465, 
  where 
  & 
  = 
  x>, 
  

   and 
  the 
  curve 
  has 
  a 
  cusp 
  in 
  the 
  neighbourhood 
  of 
  which 
  the 
  

   stations 
  i* 
  = 
  l'l, 
  1*2, 
  etc. 
  are 
  more 
  and 
  more 
  crowded. 
  Then 
  

   the 
  curve 
  emerges 
  from 
  the 
  singular 
  point 
  in 
  an 
  elegant 
  

   fashion 
  and 
  follows 
  on 
  a 
  nearly 
  circular 
  path 
  ; 
  it 
  is 
  drawn 
  

   up 
  to 
  v 
  = 
  3'0 
  only, 
  but 
  even 
  beyond 
  that 
  it 
  would 
  never 
  go 
  

   far 
  away 
  from 
  the 
  spot. 
  The 
  intensity, 
  for 
  [10], 
  reaches 
  its 
  

   maximum 
  a 
  little 
  beyond 
  v 
  = 
  27 
  ; 
  the 
  value 
  of 
  this 
  maximum 
  

   is 
  only 
  about 
  J 
  of 
  that 
  of 
  the 
  curve 
  [1] 
  . 
  The 
  curve 
  [20] 
  has 
  

   one 
  cusp 
  at 
  v 
  = 
  0*3662, 
  and 
  another 
  at 
  v 
  = 
  1*929. 
  It 
  also 
  is 
  

   drawn 
  up 
  to 
  v 
  = 
  3'0. 
  Finally, 
  the 
  curve 
  [30] 
  has 
  in 
  the 
  

   interval 
  studied 
  as 
  many 
  as 
  three 
  cusps, 
  one 
  between 
  v 
  = 
  0*l 
  

   and 
  0*2, 
  another 
  between 
  0*8 
  and 
  09, 
  and 
  yet 
  another 
  between 
  

   2*1 
  and 
  2*2. 
  Neither 
  the 
  [30]-, 
  nor 
  the 
  "[20] 
  -curve 
  goes 
  far 
  

   away 
  from 
  the 
  origin. 
  The 
  illumination 
  is 
  here 
  very 
  scanty, 
  

   especially 
  at 
  v==l"2 
  which 
  is 
  the 
  most 
  favourable 
  aperture 
  for 
  

   the 
  centre, 
  and 
  also 
  for 
  [1] 
  . 
  Other 
  details 
  can 
  be 
  seen 
  from 
  

   the 
  drawing 
  itself. 
  Here 
  but 
  two 
  more 
  remarks 
  : 
  First, 
  that 
  

   it 
  seems 
  very 
  doubtful 
  whether 
  there 
  is 
  at 
  all 
  a 
  curve 
  re- 
  

   passing 
  exactly 
  through 
  the 
  origin, 
  i. 
  e. 
  whether 
  there 
  is 
  

   at 
  all 
  a 
  rigorously 
  dark 
  ring 
  round 
  the 
  focus 
  of 
  the 
  lens 
  

   (physically 
  it 
  is 
  enough 
  that 
  the 
  light 
  becomes 
  very 
  weak 
  

   already 
  for 
  [20] 
  ) 
  . 
  And, 
  second, 
  that 
  the 
  best 
  illumination 
  at 
  

   the 
  focus 
  and, 
  at 
  the 
  same 
  time, 
  the 
  best 
  definition 
  is 
  reached 
  

   a 
  little 
  beyond 
  v= 
  1*2. 
  Opening 
  the 
  plano-convex 
  lens 
  so 
  far 
  

   as 
  t* 
  = 
  2 
  , 
  0, 
  for 
  instance, 
  would 
  not 
  only 
  darken 
  the 
  focus, 
  but 
  

   make 
  the 
  intensity 
  at 
  [10] 
  nearly 
  as 
  great 
  as 
  at 
  the 
  focus. 
  

  

  In 
  the 
  process 
  of 
  constructing 
  the 
  curves 
  of 
  ng. 
  3 
  many 
  

   of 
  the 
  values 
  of 
  A, 
  B 
  needed 
  have 
  been 
  calculated 
  by 
  

   quadratures 
  ; 
  to 
  make 
  the 
  set 
  complete 
  all 
  others 
  have 
  also 
  

  

  