﻿G. 
  F. 
  Becker 
  — 
  Computing 
  Diffusion. 
  

  

  281 
  

  

  space 
  into 
  which 
  it 
  diffuses 
  may 
  be 
  regarded 
  as 
  infinite. 
  The 
  

   quality 
  at 
  any 
  distance 
  measured 
  perpendicularly 
  to 
  the 
  initial 
  

   plane 
  is 
  then 
  proportional 
  to 
  the 
  area 
  of 
  the 
  " 
  probability 
  

   curve" 
  taken 
  between 
  certain 
  limits. 
  Now 
  this 
  area 
  between 
  

   any 
  limits 
  has 
  repeatedly 
  been 
  computed 
  and 
  tabulated, 
  because 
  

   it 
  is 
  of 
  importance 
  in 
  the 
  astronomical 
  discussion 
  of 
  refraction, 
  

   in 
  the 
  theory 
  of 
  probabilities, 
  etc. 
  Thus 
  it 
  is 
  only 
  needful 
  to 
  

   apply 
  these 
  tables 
  to 
  find 
  the 
  distribution 
  of 
  quality 
  at 
  any 
  

   time 
  or 
  for 
  any 
  distance. 
  

  

  1. 
  

  

  Probability 
  Curve. 
  

  

  For 
  this 
  simplest 
  case 
  of 
  diffusion 
  let 
  c 
  be 
  the 
  initial 
  concen- 
  

   tration 
  and 
  let 
  also 
  

  

  q 
  = 
  

  

  then 
  the 
  

   simply* 
  

  

  quality 
  

  

  v 
  in 
  terms 
  of 
  the 
  

  

  initial 
  concentration 
  is 
  

  

  

  tfdq 
  

  

  Here 
  q 
  appears 
  as^the 
  abscissa 
  of 
  the 
  probability 
  curve. 
  The 
  

   value 
  of 
  the 
  integral 
  in 
  this 
  expression 
  has 
  been 
  tabulated, 
  but 
  

   more 
  usually 
  it 
  is 
  the 
  area 
  from 
  zero 
  to 
  q 
  instead 
  of 
  that 
  from 
  

   q 
  to 
  infinity 
  which 
  is 
  computed. 
  That 
  is 
  a 
  mere 
  matter 
  of 
  

   detail 
  ; 
  for 
  the 
  whole 
  area 
  from 
  zero 
  to 
  infinity 
  is 
  simply 
  Vtt/2, 
  

   so 
  that 
  a 
  knowledge 
  of 
  either 
  part 
  of 
  the 
  area 
  leads 
  immedi- 
  

   ately 
  to 
  the 
  determination 
  of 
  the 
  other. 
  If 
  the 
  accessible 
  

   tables 
  have 
  the 
  usual 
  form, 
  it 
  is 
  only 
  necessary 
  to 
  write 
  the 
  

   equation 
  as 
  follows 
  : 
  — 
  

  

  Q 
  

  

  v 
  2 
  i 
  

  

  = 
  1 
  - 
  / 
  e-<?dq. 
  

  

  M* 
  

  

  There 
  is 
  thus 
  no 
  mystery 
  or 
  difficulty 
  about 
  computing 
  dif- 
  

   fusion 
  when 
  k 
  is 
  known. 
  Of 
  the 
  three 
  quantities 
  x, 
  t, 
  and 
  v, 
  

  

  * 
  The 
  truth 
  of 
  this 
  theorem 
  is 
  easily 
  tested 
  ; 
  but 
  its 
  proof 
  does 
  not 
  come 
  within 
  

   the 
  scope 
  of 
  this 
  note. 
  

  

  