﻿A 
  Special 
  Case 
  of 
  Gaseous 
  Conduction. 
  913 
  

  

  --— 
  = 
  A7r(n 
  1 
  — 
  n 
  2 
  )e 
  (3) 
  

  

  i/e 
  = 
  X(n 
  1 
  k 
  1 
  + 
  n 
  2 
  Jc 
  2 
  )— 
  l 
  —\I) 
  1 
  7i 
  1 
  -~D 
  2 
  n 
  2 
  J 
  ... 
  (4) 
  

  

  with 
  the 
  boundary 
  conditions 
  

  

  rix 
  = 
  when 
  x 
  = 
  / 
  ; 
  n 
  2 
  = 
  when 
  x=—l. 
  . 
  . 
  (5) 
  

  

  I 
  am 
  unable 
  to 
  make 
  any 
  material 
  progress 
  towards 
  a 
  

   solution 
  unless 
  many 
  simplifying 
  assumptions 
  are 
  made. 
  

   Let 
  us 
  assume 
  in 
  the 
  first 
  place 
  that 
  k 
  1 
  ~k 
  2 
  and 
  D!=D 
  2 
  . 
  

   X 
  must 
  then 
  be 
  a 
  function 
  of 
  even 
  powers 
  of 
  x. 
  Let 
  us 
  now 
  

   confine 
  our 
  attention 
  to 
  those 
  cases 
  when 
  the 
  current 
  is 
  

   nearly 
  saturated 
  and 
  assume 
  that 
  in 
  these 
  cases 
  X 
  has 
  the 
  

   form 
  

  

  X 
  = 
  X 
  + 
  2<7r/^ 
  2 
  , 
  (6) 
  

  

  the 
  second 
  term 
  being 
  small 
  compared 
  with 
  the 
  first. 
  We 
  

   now 
  obtain 
  from 
  (3) 
  and 
  (4) 
  

  

  ox 
  f 
  N 
  

  

  n 
  x 
  — 
  n 
  2 
  = 
  — 
  •> 
  (7) 
  

  

  n 
  ' 
  + 
  n 
  > 
  = 
  ^Ki 
  < 
  8 
  > 
  

  

  Hence 
  from 
  (5) 
  

  

  ^t'tt-^ 
  ..... 
  O) 
  

  

  Adding 
  (1) 
  and 
  (2) 
  and 
  putting 
  x=0, 
  we 
  get 
  

  

  ,,^-d™ 
  (10) 
  

  

  In 
  virtue 
  of 
  the 
  assumption 
  that 
  the 
  second 
  term 
  of 
  (6) 
  is 
  

   always 
  small 
  compared 
  with 
  the 
  first, 
  it 
  appears 
  that 
  we 
  may 
  

   write 
  approximately 
  

  

  9 
  *& 
  i 
  + 
  l>V 
  

  

  2q 
  =^'-M 
  ^ 
  

  

  and 
  

  

  Y=2X 
  l 
  • 
  (12) 
  

  

  But 
  

  

  I 
  = 
  2qel 
  (13) 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  26. 
  No. 
  155. 
  Nov. 
  1913. 
  3 
  Q 
  

  

  