﻿Diffusion 
  to 
  Conducting 
  Gases. 
  471 
  

  

  The 
  equation 
  ( 
  1) 
  thus 
  becomes 
  

  

  4/9 
  ^ 
  e 
  « 
  2 
  . 
  (2n 
  — 
  l)7rx 
  

  

  = 
  - 
  J 
  - 
  }L 
  2, 
  - 
  — 
  — 
  sin- 
  

  

  7T 
  n 
  -i 
  

  

  2n 
  — 
  1 
  a 
  

  

  This 
  value 
  of 
  p 
  is 
  unaltered 
  by 
  changing 
  x 
  into 
  a—x, 
  

   showing, 
  as 
  it 
  should, 
  that 
  at 
  any 
  time 
  the 
  distribution 
  is 
  

   symmetrical 
  with 
  respect 
  to 
  a 
  plane 
  midway 
  between 
  the 
  two 
  

   plates. 
  

  

  Let 
  q 
  t 
  denote 
  the 
  mass 
  of 
  the 
  gas 
  A 
  which 
  remains 
  mixed 
  

   with 
  B 
  after 
  the 
  gases 
  have 
  been 
  allowed 
  to 
  remain 
  between 
  

   the 
  two 
  plates 
  for 
  a 
  time 
  t, 
  and 
  ^o 
  the 
  initial 
  mass 
  of 
  A 
  be- 
  

   tween 
  the 
  two 
  plates. 
  We 
  have 
  

  

  ft 
  = 
  Jo 
  ^ 
  _ 
  3 
  % 
  e 
  

  

  q 
  p 
  a 
  ~tt\ 
  =0 
  (2n-lf 
  

  

  Hence 
  

  

  — 
  (2)l-l)2,r2 
  K 
  « 
  

  

  8n=oo 
  5 
  

   ^ 
  € 
  « 
  2 
  

  

  2. 
  Let 
  the 
  gases 
  be 
  contained 
  inside 
  a 
  cylinder 
  of 
  radius 
  a. 
  

   The 
  differential 
  equation 
  then 
  becomes 
  

  

  k 
  d 
  dp 
  dp 
  

  

  r 
  dr 
  dr~dt 
  7 
  

  

  where 
  r 
  is 
  the 
  cylindrical 
  coordinate 
  which 
  denotes 
  the 
  

   distance 
  of 
  any 
  point 
  from 
  the 
  axis. 
  

  

  Let 
  p=f(r)e~ 
  a2Kt 
  , 
  and 
  we 
  obtain 
  the 
  equation 
  

  

  1 
  d 
  df 
  9J 
  , 
  

  

  - 
  ~r 
  r-y- 
  = 
  —a'f 
  

   r 
  dr 
  dr 
  

  

  to 
  determine/. 
  

  

  The 
  solution 
  of 
  this 
  equation 
  is 
  

  

  /=AJoM+BV 
  (W), 
  

  

  and 
  since 
  the 
  gas 
  has 
  a 
  finite 
  pressure 
  at 
  the 
  centre 
  we 
  must 
  

   reject 
  the 
  second 
  term, 
  so 
  that 
  we 
  get 
  

  

  p 
  = 
  2AJ 
  (W)e- 
  a 
  H 
  

  

  The 
  condition 
  that 
  p 
  = 
  Q 
  at 
  the 
  surface 
  is 
  satisfied 
  if 
  a 
  be 
  so 
  

   determined 
  as 
  to 
  satisfy 
  the 
  equation 
  J 
  (a? 
  i 
  ) 
  = 
  0. 
  

   Hence 
  

  

  p 
  = 
  Ax 
  J 
  (oL 
  1 
  r)e-^ 
  2Kt 
  + 
  A 
  3 
  J 
  (a 
  2 
  r)e- 
  a 
  * 
  2l(t 
  4- 
  &c, 
  

  

  where 
  a« 
  i? 
  aa 
  2 
  , 
  &c. 
  are 
  the 
  roots 
  of 
  J 
  (x) 
  — 
  Q. 
  

  

  2 
  K 
  2 
  

  

  