﻿the 
  Ions 
  of 
  Gases. 
  359 
  

  

  by 
  which 
  it 
  moves 
  away 
  will 
  determine 
  the 
  degree 
  of 
  

   irregularity 
  which 
  causes 
  the 
  recombination 
  which 
  we 
  

   suppose 
  to 
  go 
  on 
  as 
  a 
  sort 
  of 
  leak. 
  To 
  determine 
  A 
  as 
  a 
  

   function 
  of 
  the 
  density 
  of 
  the 
  gas 
  containing 
  the 
  ions 
  is 
  a 
  

   problem 
  in 
  the 
  calculus 
  of 
  probabilities 
  known 
  as 
  that 
  of 
  the 
  

   random 
  walk. 
  A 
  man 
  walks 
  a 
  distance 
  I 
  in 
  any 
  direction 
  

   from 
  an 
  origin 
  0, 
  then 
  he 
  walks 
  the 
  same 
  distance 
  in 
  any 
  

   other 
  direction, 
  and 
  so 
  on 
  till 
  he 
  has 
  taken 
  n 
  walks. 
  It 
  is 
  

   required 
  to 
  find 
  the 
  probability 
  that 
  his 
  final 
  distance 
  from 
  

   lies 
  between 
  r 
  and 
  r 
  + 
  dr. 
  Evidently 
  this 
  is 
  the 
  same 
  as 
  

   our 
  ionic 
  problem, 
  for 
  the 
  ion 
  after 
  n 
  free 
  paths 
  of 
  average 
  

   length 
  I 
  is 
  at 
  some 
  required 
  average 
  distance 
  from 
  0, 
  its 
  

   original 
  position 
  in 
  the 
  initial 
  imaginary 
  uniform 
  distribu- 
  

   tion. 
  The 
  solution 
  has 
  been 
  given 
  by 
  Eayleigh 
  (' 
  Nature 
  j'^ 
  

   Ixxii. 
  1905, 
  p. 
  318 
  ; 
  Phil. 
  Mag. 
  [5] 
  x. 
  1880, 
  p. 
  73, 
  xlvii. 
  

   1899, 
  p. 
  2-1^) 
  for 
  the 
  case 
  when 
  n 
  is 
  large. 
  The 
  required 
  

   probability 
  is, 
  when 
  Z 
  = 
  1, 
  

  

  ^e-'"\dr 
  (21) 
  

  

  n 
  

  

  and 
  the 
  probability 
  that 
  the 
  distance 
  from 
  is 
  greater 
  than 
  

   r 
  is 
  e~^''\ 
  If 
  we 
  introduce 
  I 
  explicitly 
  the 
  exponent 
  

   becomes 
  —r'^jV^n. 
  In 
  comparing 
  results 
  for 
  different 
  sub- 
  

   stances 
  or 
  for 
  the 
  same 
  substance 
  at 
  different 
  pressures, 
  we 
  

   ought 
  to 
  take 
  n 
  to 
  be 
  the 
  number 
  of 
  paths 
  described 
  in 
  unit 
  

   length, 
  because, 
  though 
  the 
  departures 
  from 
  uniformity 
  

   recur 
  with 
  a 
  frequency 
  proportional 
  to 
  the 
  molecular 
  velocity, 
  

   each 
  lasts 
  a 
  time 
  inversely 
  proportional 
  to 
  the 
  velocity. 
  For 
  

   this 
  reason 
  the 
  element 
  of 
  time 
  does 
  not 
  enter 
  into 
  the 
  com- 
  

   parison. 
  Thus 
  nl 
  is 
  to 
  be 
  constant 
  for 
  the 
  same 
  substance 
  at 
  

   different 
  pressures 
  and 
  also 
  for 
  different 
  substances. 
  We 
  

   have 
  now 
  to 
  find 
  how 
  r 
  is 
  to 
  be 
  chosen 
  in 
  order 
  that 
  the 
  

   comparison 
  may 
  be 
  statistically 
  correct. 
  The 
  agencies 
  

   maintaining 
  uniformity 
  are 
  proportional 
  to 
  I 
  the 
  mean 
  free 
  

   path, 
  for 
  instance, 
  D 
  the 
  coefi&cient 
  of 
  diffusion 
  is 
  propor- 
  

   tional 
  to 
  /. 
  But 
  the 
  same 
  agencies 
  produce 
  departures 
  from 
  

   uniformity 
  temporarily, 
  while 
  they 
  are 
  maintaining 
  average 
  

   uniformity, 
  just 
  as 
  the 
  thermal 
  velocities, 
  which 
  keep 
  the 
  

   average 
  pressure 
  of 
  a 
  gas 
  constant 
  and 
  also 
  its 
  average 
  

   density 
  constant, 
  at 
  the 
  same 
  time 
  cause 
  those 
  temporary 
  

   local 
  variations 
  of 
  pressure 
  and 
  density 
  which 
  occur 
  when 
  

   two 
  molecules 
  collide. 
  As 
  regards 
  departures 
  from 
  uni- 
  

   formity, 
  then 
  the 
  proper 
  unit 
  for 
  measuring 
  its 
  amount 
  

   linearly 
  is 
  the 
  mean 
  free 
  path 
  of 
  a 
  molecule. 
  Thus 
  r 
  must 
  

   be 
  proportional 
  to 
  I. 
  Accordingly 
  r^jPn 
  takes 
  the 
  form 
  cl 
  

   where 
  c 
  is 
  a 
  constant. 
  Let 
  Zq 
  be 
  the 
  value 
  of 
  I 
  when 
  the 
  

  

  2B2 
  

  

  