Dielectric  Strength  of  Air.  273 
Denoting  Lambert's  series  S  j-      -    by  F(</)  we  get 
K ,,  =  a  1= ''*  {  P  (?)  -  2  F  (<f  )  +  F  fa*)  } 
and 
-K1,)=((--^/F(,/)-F(^)|, 
By  a  known  transformation  due  to  Clausen  *  we  have 
and  this  series  can  be  very  readily  computed. 
For  instance,  when  x  is  0*5  we  find  by  (1)  that  q  is  also 
0*5,  and 
F (0-5)  =1-6067,  F(0-25)  =  O4210,  and  F(0-0625)  =0*0709. 
Hence  we  find  that 
Ki-!=  l"2dMa  and  K12  =  —05252a. 
The  capacity  between  the  two  spheres  f  is 
(K1.1-K1.2)/2  =  0-8893a 
and  the  capacity  J  for  equal  potentials  is 
2(Ki-i  +  Ki.2)  =  l"4565a. 
To  reduce  these  values  to  microfarads  we  divide  by  900.000, 
a  being  measured  in  centimetres. 
When  the  potentials  foUow  the  harmonic  law  we  have 
A1=(K1.1V1  +  K1.2Y2)ftJ 
A^tKnVj  +  KwVJo), 
where  A:  and  A2  are  the  effective  values  of  the  capacity 
currents  flowing  to  the  two  spheres  respectively ;  V1?  V2  *ne 
effective  values  of  their  potentials,  and  (o/2ir  the  frequency  of 
the  alternating  pressures. 
For  instance,  suppose  that  we  have  two  20  cm.  spherical 
electrodes  5  cms.  apart,  and  suppose  that  the  effective  value 
of  the  P.D.  between  them  is  90  kilovolts.    Then,  if  w  be  1000, 
*  Crelle's  Journal,  vol.  iii.  p.  9o,  quoted  in  Jaeobi's  Fundamenta 
JSova,  pp.  187-188. 
f  Russell,  '  Alternating-  Currents,'  vol.  i.  p.  92. 
\  "Russell,  'Alternating  Currents,'  vol.  i.  p.  393. 
Phil.  Mag.  8.  6.  Vol.  11.  No.  62.  Feb.  1906.  T 
