730  Prof.  J.  S.  Townsend  on  the  Field  of  Force 
between  the  electrode  and  the  same  point  in  the  gas  when  the 
wire  is  removed. 
In  the  following  investigation  it  will  be  assumed  that  the 
conduction  in  the  gas  is  produced  in  the  manner  which  has 
been  explained  in  the  previous  paper.  According  to  the 
theory,  practically  all  the  ionization  is  produced  by  the  colli- 
sions of  positive  and  negative  ions  with  the  molecules  of '  the 
gas.  On  this  supposition  the  following  result  has  been 
obtained  : 
exn1xu  =  ix  Z(V)  x  |  [Z(V)]      xfixd 
where  Z(V)  =  eJo  ,  i  being  the  current  per  unit  area  of 
the  electrodes,  x  the  distance  of  any  point  in  the  gas  from 
the  negative  electrode,  e  the  charge  on  an  ion,  u  and  v  the 
velocities  of  the  negative  and  positive  ions  respectively, 
%  the  number  of  negative  ions  per  c.c.  at  any  time  at  the 
point  x,  n2  the  number  of  positive  ions,  a  the  number  of 
molecules  of  the  gas  ionized  by  a  negative  ion  in  moving 
through  a  distance  of  one  centimetre  under  the  force  X,  /3  the 
number  ionized  by  a  positive  ion  under  similar  conditions. 
It  will  suffice  for  present  purposes  to  consider  very  small 
currents,  so  that  the  potential  will  not  be  appreciably  affected 
by  the  charges  on  the  ions,  and  the  electric  force  will  be 
practically  uniform  between  the  plates.  As  the  quantities 
a,  /3,  u,  and  v  depend  only  on  the  electric  force  and  the 
pressure  of  the  gas,  these  quantities  will  not  vary  with  the 
distance  x,  so  that  the  above  equation  reduces  to  the  form 
exn1xu=~^(e(a-p)x-l). 
a— #v  ' 
The  equation 
e  X  n2  x  v=  ^-  (etf—X-*)-!) 
may  be  similarly  found. 
The  ratio 
»i«_£v    6(a^)x-l 
x 
n9v       a.       1  —  e<0-*)(«-*) 
represents  the  proportion  of  the  current  carried  by  the  nega- 
tive ions  to  that  carried  by  the  positive  ions.  The  values 
of  a  and  /3  have  been  found  over  considerable  ranges  of 
forces  and  pressures,  so  that  with  the  aid  of  the  above 
formula  it  is  possible  to  find  the  ratio  of  the  currents  carried 
by  the  ions  of  opposite  sign  at  various  points  in  the  gas. 
