500     Prof.  Wilson  and  Mr.  Gold :  Electrical  Conductivity  of 
But  -1  — .     Hence 
at  p 
r-yopCMpt=       _  +  _  5  +  4«-&.      .    .    (4) 
The  solution  of  this  equation  is 
-^  0sm(^-«) 
{(l""$S)^2+^}2 
(5) 
where  /„      Airdt 
pm 
tan  a  =  ^ 
A 
If  a  P.D.  V=V0sin_p£  is  applied  to  a  condenser  of  capa- 
city C,  the  current  is  given  by  the  equation  z=.CV0^?  cos  pt. 
For  the  flame,  if  A  and  m  are  both  negligible  (5)  becomes 
V 0p  cos  pt 
±ird      ' 
so  that  the  apparent  capacity  is  —  -z  per  unit  area,     Now  - 
is  the  amplitude  of  vibration  of  the  negative  ions  so  that  dp 
must  be  the  amount  of  electricity  flowing  during  a  half- 
vibration. 
Let  i  =  -  ^  so  that     Q  =  ~ ° ,  sin  pt. 
dt  4zird 
Then  we  have,  integrating  from  0  to  ir, 
so  that  the  apparent  capacity  per  unit  area  is 
4irrf       V 
If  p  =  0  this  makes  the  capacity  zero,  whereas  it  should  be 
—  -p..     This  is  due  to  the  omission  of-r =-,  which  would  do 
47tD  47r  dt 
be  negligible  if  p  were  very  small.     If,  however,  we  take 
^P/SttYq   to    be    not    the    apparent    capacity  but   the   in- 
crease   in   the    apparent    capacity    due    to    the    presence    of 
the  ions,  then  no   error  will  be  made  even  if  p  be   small. 
The   quantity    -=-  —  —    which  has    been    determined  is  pro- 
portional,  as  we  have  seen,  to  the  increase  in  the  apparent 
