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  V. 
  Light 
  Distribution 
  round 
  the 
  Focus 
  of 
  a 
  Lens, 
  at 
  

   various 
  Apertures. 
  By 
  L. 
  Silberstein, 
  Ph.D., 
  Lecturer 
  

   in 
  Natural 
  Philosophy 
  at 
  the 
  University 
  of 
  Rome 
  *. 
  

  

  Bibliographic 
  and 
  'Introductory. 
  

  

  fl^HE 
  distribution 
  of 
  the 
  intensity 
  of 
  light 
  in 
  the 
  

   -L 
  neighbourhood 
  of 
  a 
  caustic 
  has 
  been 
  studied 
  by 
  Sir 
  G. 
  

   Airy 
  as 
  early 
  as 
  in 
  1838 
  (Camb. 
  Phil. 
  Trans, 
  vol. 
  vi.), 
  for 
  

   the 
  case, 
  however, 
  of 
  an 
  unlimited 
  beam 
  only. 
  Some 
  of 
  

   the 
  effects 
  of 
  spherical 
  aberration 
  of 
  limited 
  beams 
  upon 
  the 
  

   central 
  intensity 
  and 
  the 
  definition 
  of 
  the 
  image 
  have 
  been 
  

   investigated 
  by 
  Lord 
  Rayleigh 
  in 
  1879 
  (Phil. 
  Mag. 
  vol. 
  viii. 
  

   pp. 
  403-411). 
  The 
  chief 
  problem 
  considered 
  by 
  him 
  relates 
  

   to 
  a 
  beam 
  of 
  cylindrical 
  waves 
  of 
  rectangular 
  section, 
  their 
  

   aberration 
  being 
  assumed 
  proportional 
  to 
  the 
  cube 
  of 
  the 
  

   lateral 
  coordinate 
  x. 
  The 
  solution 
  is 
  reduced 
  to 
  the 
  evalua- 
  

   tion 
  of 
  an 
  integral 
  of 
  the 
  form 
  \ 
  cos 
  (ax 
  + 
  bx 
  2 
  )dx. 
  Availing 
  

   himself 
  of 
  the 
  numerical 
  results 
  of 
  the 
  mechanical 
  quadratures 
  

   recorded 
  in 
  Airy's 
  paper, 
  Lord 
  Rayleigh 
  calculates 
  and 
  draws 
  

   three 
  intensity 
  curves 
  for 
  the 
  focal 
  plane 
  (loc. 
  cit. 
  p. 
  406), 
  

   corresponding 
  to 
  the 
  case 
  of 
  no 
  aberration 
  (6 
  = 
  0), 
  and 
  to 
  

   those 
  in 
  which 
  the 
  marginal 
  aberrations 
  amount 
  to 
  £ 
  and 
  

   ^ 
  period. 
  The 
  practically 
  important 
  consequence 
  drawn 
  

   from 
  the 
  aspect 
  of 
  these 
  curves 
  is 
  that 
  " 
  aberration 
  begins 
  

   to 
  be 
  distinctly 
  mischievous 
  when 
  it 
  amounts 
  to 
  about 
  a 
  

   quarter-period. 
  - 
  " 
  In 
  the 
  next 
  case 
  studied, 
  that 
  of 
  sym- 
  

   metrical 
  aberration 
  proportional 
  to 
  a- 
  4 
  , 
  Lord 
  Rayleigh 
  

   calculates, 
  by 
  the 
  aid 
  of 
  a 
  series, 
  the 
  intensity 
  at 
  the 
  central 
  

   point 
  only. 
  Passing, 
  finally, 
  to 
  beams 
  of 
  circular 
  section, 
  

   he 
  limits 
  himself 
  again 
  to 
  the 
  calculation 
  of 
  the 
  central 
  

   intensity, 
  viz. 
  in 
  the 
  case 
  of 
  axially 
  symmetric 
  aberration 
  

   proportional 
  to 
  the 
  fourth 
  power 
  of 
  the 
  distance, 
  and 
  finds 
  

   that, 
  as 
  in 
  the 
  preceding 
  cases, 
  aberration 
  begins 
  to 
  be 
  

   prejudicial 
  when 
  it 
  mounts 
  up 
  to 
  a 
  quarter 
  of 
  a 
  period. 
  

   This 
  result 
  has 
  since 
  become 
  widely 
  known, 
  having 
  been 
  

   incorporated 
  into 
  the 
  Enc. 
  Brit.-f 
  and 
  several 
  English 
  

   text-books. 
  In 
  1884 
  Lommel 
  investigated 
  the 
  distribution 
  

   of 
  illumination 
  in 
  the 
  diffraction 
  image 
  of 
  a 
  point 
  given 
  by 
  a 
  

   circular 
  aperture 
  J. 
  The 
  problem 
  in 
  this 
  case 
  reduces 
  to 
  the 
  

   evaluation 
  of 
  integrals 
  of 
  the 
  form 
  

  

  f 
  J 
  (aa) 
  cos 
  (bx 
  2 
  )xdx, 
  jj 
  (a#) 
  sin 
  (ba 
  2 
  )xdx 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  + 
  Cf. 
  Lord 
  Rayleigh's 
  article 
  on 
  "Diffraction 
  of 
  Light" 
  in 
  the 
  

   11th 
  ed. 
  of 
  Enc. 
  Brit. 
  vol. 
  viii. 
  pp. 
  238-255. 
  

   X 
  E. 
  Lommel, 
  Bayer. 
  Akad. 
  d. 
  Wiss. 
  vol. 
  xv. 
  

  

  