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  XXXII. 
  On 
  the 
  Absorption 
  of 
  Light 
  hi 
  Heterogeneous 
  Media.. 
  

   By 
  P. 
  G. 
  Cutting 
  *: 
  

  

  PHOTOGKAPHIO 
  density 
  depends 
  upon 
  the 
  size 
  and 
  

   number 
  of 
  the 
  imbedded 
  silver 
  grains 
  and 
  to 
  a 
  slight 
  

   extent 
  upon 
  their 
  form 
  and 
  distribution 
  as 
  well. 
  The 
  

   mathematical 
  problem 
  of 
  relating 
  density 
  to 
  grain 
  is 
  

   obviously 
  to 
  be 
  treated 
  by 
  probability 
  theory 
  rather 
  than 
  by 
  

   infinitesimal 
  analysis. 
  The 
  solution 
  here 
  presented 
  will 
  be 
  

   of 
  interest 
  to 
  students 
  of 
  the 
  theories 
  of 
  radiation 
  and 
  of 
  

   entropy 
  in 
  discontinuous 
  systems, 
  in 
  that 
  it 
  is 
  a 
  much 
  simpler 
  

   problem 
  treated 
  by 
  similar 
  probability 
  methods. 
  

  

  Suppose 
  snowflakes 
  of 
  a 
  given 
  size 
  to 
  be 
  falling 
  with 
  

   perfect 
  irregularity 
  upon 
  a 
  surface. 
  When 
  a 
  given 
  number 
  

   per 
  unit 
  area 
  have 
  fallen, 
  what 
  will 
  be 
  the 
  mean 
  relative 
  

   areas 
  covered 
  and 
  uncovered 
  ? 
  In 
  the 
  plate 
  grain 
  problem 
  

   as 
  in 
  the 
  snowflake 
  problem, 
  the 
  distribution 
  is 
  completely 
  

   irregular, 
  but 
  in 
  a 
  volume 
  instead 
  of 
  a 
  plane. 
  The 
  grains 
  

   are 
  contained 
  in 
  a 
  layer 
  10 
  to 
  20 
  /ju 
  thick 
  and 
  are 
  themselves 
  

   0*5 
  to 
  3 
  yu. 
  in 
  diameter, 
  irregular 
  in 
  outline 
  and 
  fairly 
  

   uniform 
  in 
  area 
  in 
  any 
  one 
  plate. 
  The 
  grains 
  are 
  not. 
  

   crystals, 
  but 
  aggregates 
  of 
  finely 
  divided 
  silver 
  resembling 
  

   platinum-black 
  or 
  soot, 
  of 
  very 
  high 
  absorbing 
  and 
  low 
  

   reflecting 
  power. 
  This 
  reflecting 
  power 
  has 
  not 
  yet 
  been 
  

   directly 
  determined, 
  but 
  estimates 
  based 
  on 
  scattering 
  make 
  

   it 
  well 
  under 
  2 
  per 
  cent. 
  In 
  the 
  following 
  discussion 
  both 
  

   reflexion 
  and 
  diffusion 
  are 
  neglected, 
  though 
  both 
  may 
  be 
  

   readily 
  entered 
  in 
  the 
  equations. 
  

  

  Consider 
  the 
  absorbing 
  body 
  divided 
  into 
  layers 
  about 
  

   one 
  grain 
  thick, 
  parallel 
  with 
  the 
  surface, 
  so 
  that 
  there 
  will 
  

   be 
  but 
  a 
  negligible 
  amount 
  of 
  overlapping 
  of 
  grains 
  in 
  any 
  

   one 
  layer. 
  That 
  certain 
  grains 
  lie 
  partly 
  in 
  two 
  successive 
  

   layers 
  is 
  of 
  no 
  consequence, 
  since 
  in 
  the 
  equations 
  they 
  are 
  

   counted 
  but 
  once 
  in 
  the 
  layer 
  in 
  which 
  their 
  greater 
  bulk 
  

   lies. 
  In 
  the 
  first 
  layer 
  let 
  there 
  be 
  n 
  x 
  grains 
  per 
  unit 
  area 
  

   and 
  let 
  a 
  x 
  be 
  their 
  average 
  projected 
  area. 
  Then 
  the 
  

   probability 
  of 
  a 
  ray 
  of 
  light 
  being 
  stopped 
  by 
  this 
  layer 
  is 
  

   the 
  ratio 
  of 
  the 
  covered 
  to 
  the 
  total 
  area, 
  or 
  as 
  n 
  l 
  a 
  1 
  to 
  1. 
  

   Similarly 
  the 
  probability 
  of 
  a 
  ray 
  passing 
  the 
  first 
  layer 
  is 
  

   1— 
  niOi. 
  In 
  other 
  words, 
  the 
  absorption 
  and 
  transmission 
  

   coefficients 
  are 
  respectively 
  the 
  'probabilities 
  of 
  being 
  stopped 
  

   and 
  of 
  being 
  passed. 
  

  

  In 
  the 
  second 
  layer 
  let 
  the 
  corresponding 
  quantities 
  be 
  

   n 
  2 
  and 
  a 
  2 
  , 
  in 
  the 
  third 
  n 
  z 
  and 
  a 
  z 
  and 
  so 
  on. 
  For 
  brevity 
  call 
  

   the 
  product 
  n 
  l 
  a 
  1 
  = 
  A 
  l5 
  &c. 
  Now 
  the 
  only 
  manner 
  in 
  which 
  a 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  