﻿426 
  Absorption 
  of 
  Light 
  in 
  Heterogeneous 
  Media. 
  

  

  T 
  is 
  the 
  fraction 
  of 
  the 
  light 
  transmitted 
  by 
  the 
  entire 
  thick- 
  

   ness 
  of 
  film, 
  1 
  — 
  A 
  is 
  the 
  corresponding 
  transmission 
  of 
  any 
  

   single 
  layer 
  one 
  grain 
  deep, 
  while 
  A 
  is 
  the 
  fraction 
  absorbed, 
  

   •on 
  an 
  average, 
  in 
  any 
  single 
  layer. 
  The 
  final 
  column, 
  10 
  A, 
  

   is 
  what 
  the 
  absorption 
  of 
  a 
  film 
  ten 
  grains 
  deep 
  (an 
  ordinary 
  

   film) 
  would 
  be 
  if 
  each 
  grain 
  were 
  fully 
  effective. 
  Thus, 
  for 
  

   unit 
  density, 
  the 
  transparency 
  is 
  10 
  per 
  cent., 
  the 
  absorption 
  

   of 
  a 
  single 
  layer 
  20 
  per 
  cent, 
  and 
  there 
  are 
  present 
  2*1 
  times 
  

   ;as 
  many 
  grains 
  as 
  would 
  completely 
  cover 
  a 
  surface. 
  

  

  Taking 
  the 
  particular 
  case 
  of 
  unit 
  density 
  and 
  A 
  x 
  = 
  20 
  

   per 
  cent. 
  (0*20) 
  and 
  computing 
  the 
  absorption 
  of 
  the 
  first 
  n 
  

   flayers 
  by 
  the 
  formula 
  

  

  B 
  1 
  ...,= 
  l~T 
  n 
  =l-(l^A)», 
  

  

  B 
  1 
  = 
  A=0-20, 
  B 
  12 
  = 
  2A-A 
  2 
  = 
  0-40-0-04 
  = 
  0-36; 
  that 
  is, 
  in 
  

   i;he 
  first 
  two 
  layers 
  36 
  per 
  cent, 
  is 
  covered 
  and 
  4 
  per 
  cent, 
  of 
  

   this 
  covered 
  twice. 
  In 
  the 
  first 
  three 
  layers, 
  

  

  B^.^aA-SA^A^O-GO-O-^ 
  + 
  0-008 
  = 
  0-488, 
  

  

  the 
  probability 
  of 
  overlapping 
  is 
  0*12, 
  of 
  triple 
  overlapping 
  

   •of 
  grains 
  is 
  0*008. 
  Tabulating 
  for 
  ten 
  layers 
  : 
  — 
  

  

  n. 
  

  

  Tn. 
  

  

  Bn. 
  

  

  AB* 
  (%). 
  

  

  A' 
  1 
  . 
  

  

  wP 
  10 
  . 
  

  

  1 
  

  

  0794 
  

  

  0-206 
  

  

  — 
  

  

  0-214 
  

  

  2-06 
  

  

  2 
  

  

  0633 
  

  

  0-367 
  

  

  161 
  

  

  0-04 
  

  

  1-80 
  

  

  3 
  

  

  0501 
  

  

  0-499 
  

  

  13-2 
  

  

  0008 
  

  

  0-96 
  

  

  4 
  

  

  0-398 
  

  

  0-602 
  

  

  10-3 
  

  

  00016 
  

  

  0-336 
  

  

  5 
  

  

  316 
  

  

  0-684 
  

  

  8-2 
  

  

  0-0 
  3 
  32 
  

  

  0-080 
  

  

  6 
  

  

  0-251 
  

  

  0749 
  

  

  6-5 
  

  

  00 
  4 
  64 
  

  

  0013 
  

  

  7 
  

  

  0-200 
  

  

  0-800 
  

  

  5-1 
  

  

  00 
  4 
  13 
  

  

  00017 
  

  

  8 
  

  

  0-159 
  

  

  0-841 
  

  

  41 
  

  

  0-0 
  5 
  26 
  

  

  00 
  3 
  12 
  

  

  9 
  

  

  0126 
  

  

  0-874 
  

  

  33 
  

  

  00 
  6 
  51 
  

  

  0-0 
  5 
  52 
  

  

  10 
  

  

  0100 
  

  

  0-900 
  

  

  26 
  

  

  o-o 
  c 
  io 
  

  

  o-o 
  c 
  io 
  

  

  The 
  second 
  and 
  third 
  columns 
  give 
  respectively 
  the 
  

   (transparency 
  and 
  absorption 
  of 
  the 
  first 
  n 
  layers. 
  The 
  fourth 
  

   ■column 
  gives 
  the 
  increment 
  of 
  absorption 
  due 
  to 
  the 
  

   nth 
  layer. 
  The 
  fifth 
  column 
  gives 
  the 
  probability 
  of 
  n 
  

   grains 
  overlapping 
  in 
  n 
  layers. 
  The 
  last 
  column 
  gives 
  the 
  

   probability 
  of 
  n 
  grains 
  being 
  in 
  line 
  in 
  all 
  ten 
  layers, 
  it 
  is 
  

   A"10!/n!(10->i)!. 
  

  

  Rochester, 
  N.Y., 
  

  

  February, 
  1913. 
  

  

  