Dielectric  Strength  of  Air.  255 
Knowing  the  values  of  /  we  can  find  the  values  of  /, 
easily  by  means  of  an  elliptic  integral  series  which  is  quoted 
in  KirchhofPs  paper.     It  can  be  shown  that  * 
oo  an— 1     1__/»2n— i  l>k'~K* 
f  ^  }  9  (l+^-1)2~       7T2       ' 
Hence  it  follows  from  (15)  and  (16)  that 
a?     (1  +  ?)2      /**'K2 
-         /x     x  +  ka     kk'K2  00 
=^  V  a  •  ^T  •-?*■•       •     •     •     (23) 
The  values  of  k,  Jcr,  and  K  can  easily  be  found  by  well-known 
formulae.  Let  us  suppose,  for  instance,  that  we  wish  to  find 
the  value  of  fx  when  x\a  is  0*5.  We  have  already  found 
that /is  1-1726  and  q  is  05. 
Nowf 
4/2K/tt  =  1  +  2  2  2? 
i 
=  1  +  1  +  0-125  +  0-003906 
+  0-000031+  .  .  . 
=  2-1289. 
We  also  have 
%  v^2KA//w  =  l  +  2S  {-)nqn2 
i 
=  1-1  +  0-12503-0-00391+  .. 
=  0-12112. 
Thus  F=(0-12112/2-1289)2=:0-0032369 
and  k=  Vl-^/2  =  0-99999. 
Hence  M'K2/tt2  =  0*01662. 
Thus  finally  by  (16) 
/!  =  M726+{9/(4  \/2)  [(0-01662) 
=  1-1990. 
This  agrees  with  the  value  of  f\  found  by  direct  calculation 
from  (16). 
*  A.  Enneper,  EUiptische  Functionen,  p.  180. 
f  A.  G.  Greenhill,  'Elliptic  Functions/  p.  303. 
\  Greenhill,  p.  303. 
