﻿Ions 
  in 
  Gases 
  under 
  Different 
  Pressures, 
  291 
  

  

  Therefore 
  the 
  total 
  decrease 
  in 
  the 
  number 
  of 
  ions 
  in 
  the 
  

   volume 
  bbiCyC 
  in 
  the 
  time 
  t 
  

  

  -Fm-^> 
  

  

  -A„,.«{,.^^ 
  kg 
  ,Ci±i±^)}. 
  

  

  If 
  we 
  let 
  T 
  be 
  the 
  time 
  that 
  it 
  takes 
  the 
  number 
  of 
  ions 
  to 
  

   fall 
  to 
  half 
  their 
  value, 
  then 
  the 
  total 
  decrease 
  in 
  the 
  time 
  T 
  

  

  But 
  the 
  maximum 
  number 
  of 
  ions 
  is 
  equal 
  to 
  

   ZA 
  ly 
  l+ 
  ^Ja/^ 
  by 
  equation 
  (4) 
  ; 
  

  

  r 
  .__ 
  /d 
  + 
  l 
  + 
  Td^w^ 
  

  

  ••• 
  A 
  1 
  ftT{l-Td*Vlog,( 
  d 
  + 
  TdV 
  J> 
  )} 
  

  

  This 
  equation 
  for 
  obtaining 
  the 
  value 
  of 
  T 
  is 
  incapable 
  of 
  

   an 
  exact 
  solution. 
  An 
  approximate 
  solution 
  may, 
  however, 
  

   be 
  obtained 
  as 
  follows 
  : 
  — 
  

  

  where 
  B 
  = 
  Td\/q 
  * 
  n 
  

   ,.A 
  l9 
  „T{z-Blo 
  & 
  (l 
  + 
  ^)}=i/A 
  1 
  (l 
  + 
  ^) 
  x 
  /^; 
  

  

  .-. 
  by 
  expanding 
  log^l+ 
  j^b) 
  

  

  as 
  a 
  first 
  approximation, 
  provided 
  that 
  I 
  is 
  small 
  compared 
  

   with 
  d 
  + 
  B. 
  

  

  It 
  is 
  found 
  by 
  supplying 
  the 
  experimental 
  values 
  of 
  the 
  

  

  quantities 
  involved 
  in 
  the 
  expression 
  -7-775;, 
  that 
  the 
  second 
  

  

  term 
  in 
  the 
  expansion 
  of 
  the 
  logarithm 
  amounts 
  to 
  approxi- 
  

  

  U2 
  

  

  