﻿424 
  Mr. 
  P. 
  G. 
  Nutting 
  on 
  the 
  

  

  ray 
  may 
  pass 
  through 
  all 
  layers 
  is 
  to 
  pass 
  each 
  layer 
  

   separately, 
  hence 
  the 
  probability 
  of 
  passing 
  all 
  layers 
  is 
  the 
  

   continued 
  product 
  

  

  (l-A 
  1 
  )(l-A 
  2 
  )....(l-A„)==T 
  M 
  # 
  # 
  (1) 
  

  

  o£ 
  the 
  probabilities 
  of 
  passing 
  each 
  separate 
  layer. 
  This 
  is 
  

   the 
  transparency 
  T 
  m 
  of 
  the 
  whole 
  sheet. 
  The 
  corresponding 
  

   absorption 
  B 
  m 
  is 
  the 
  complementary 
  quantity 
  

  

  B 
  m 
  =l-T 
  m 
  . 
  (2) 
  

  

  It 
  may 
  be 
  noted 
  that 
  the 
  absorption 
  of 
  the 
  whole 
  is 
  not 
  the 
  

   product 
  (A!A 
  2 
  ...A 
  TO 
  ) 
  of 
  the 
  probabilities 
  of 
  absorption 
  in 
  

   the 
  various 
  layers 
  since 
  the 
  action 
  is 
  not 
  alike 
  in 
  all 
  layers, 
  

   a 
  ray 
  may 
  be 
  passed 
  by 
  several 
  layers 
  to 
  be 
  stopped 
  in 
  

   another. 
  The 
  above 
  product 
  (AjA 
  2 
  ... 
  A 
  m 
  ) 
  is 
  the 
  probability 
  

   of 
  possible 
  stoppage 
  in 
  all 
  layers, 
  i. 
  e. 
  the 
  probability 
  per 
  

   unit 
  area 
  of 
  a 
  continuous 
  train 
  of 
  grains 
  lying 
  one 
  behind 
  

   the 
  other, 
  through 
  all 
  the 
  successive 
  m 
  layers. 
  In 
  fact, 
  

   if 
  the 
  value 
  of 
  T 
  m 
  in 
  (1) 
  be 
  written 
  in 
  (2), 
  multiplied 
  

   out 
  and 
  grouped 
  according 
  to 
  the 
  number 
  of 
  A's 
  multiplied 
  

   together, 
  then 
  each 
  group 
  gives 
  the 
  probability 
  of 
  2, 
  3 
  ... 
  m 
  

   grains 
  overlapping. 
  

  

  In 
  the 
  special 
  case 
  of 
  all 
  layers 
  alike 
  in 
  number 
  and 
  size 
  

   of 
  grain, 
  the 
  transparency 
  of 
  all 
  m 
  layers 
  will 
  be 
  

  

  T 
  Hl 
  = 
  (l-A)-, 
  (3) 
  

  

  since 
  in 
  (1) 
  A 
  1 
  =-A 
  2 
  = 
  ... 
  =A 
  m 
  . 
  This 
  corresponds 
  to 
  Beers 
  

   Law 
  in 
  ordinary 
  Optics. 
  

  

  Photographic 
  density 
  D 
  has 
  of 
  late 
  years 
  been 
  precisely 
  

   defined 
  by 
  the 
  relation 
  

  

  D=-log 
  10 
  T 
  (4) 
  

  

  T 
  being 
  the 
  transparency 
  in 
  the 
  sense 
  used 
  above. 
  The 
  

   value 
  of 
  T 
  m 
  in 
  either 
  (1) 
  or 
  (3) 
  may 
  be 
  substituted 
  in 
  (4) 
  

   according 
  to 
  conditions. 
  Equations 
  (1) 
  and 
  (4) 
  give 
  the 
  

   general 
  relation 
  sought 
  between 
  density 
  and 
  the 
  number, 
  

   size 
  and 
  distribution 
  of 
  grain. 
  

  

  In 
  all 
  ordinary 
  practice 
  the 
  size 
  and 
  distribution 
  of 
  grain 
  

   throughout 
  the 
  film 
  is 
  so 
  uniform 
  that 
  (3) 
  gives 
  a 
  very 
  close 
  

   approximation 
  indeed. 
  In 
  this 
  case 
  

  

  D=-mlog(l-A) 
  (5) 
  

  

  If 
  further 
  A 
  is 
  so 
  small 
  that 
  the 
  overlapping 
  of 
  grains 
  is 
  

   negligible, 
  as 
  is 
  the 
  case 
  with 
  low 
  and 
  medium 
  densities, 
  

  

  D=-mA 
  = 
  ?7ina 
  (6) 
  

  

  