Dielectric  Strength  of  Air.  241 
for  reasons  given  above,  only  considering  experiments  with 
large  electrodes,  at  appreciable  distances  apart,  we  find  that 
tbe  maximum  values  of  the  electric  intensity  at  the  moment 
of  the  disruptive  voltage  is  practically  constant  for  distances 
varying  from  a  millimetre  up  to  15  centimetres,  and  for 
voltages  varying  between  4  and  160  kilovolts. 
2.  Historical, 
Nearly  all  experimenters  have  used  equal  spherical  elec- 
trodes. It  is  therefore  necessary  to  be  able  to  write  down  at 
once  the  value  of  the  electric  intensity  between  two  spheres 
whatever  may  be  their  potentials.  Kirchhoff*  in  a  very 
valuable  paper  has  shown  how  to  obtain  from  Poisson's  f 
equations  an  expression  for  the  maximum  value  of  the  electric 
intensity  in  the  form  of  an  infinite  series.  Unfortunately 
this  paper  can  only  be  understood  by  those  who  are 
thoroughly  familiar  with  Jacobins  theorems  in  Elliptic 
Functions,  and  so  the  important  results  contained  in  it  are 
known  to  few  physicists.  In  1890,  Professor  A.  Schuster  % 
published  a  table  giving  the  value  of  the  maximum  electric 
intensity  between  two  spheres  when  one  was  at  potential  V 
and  the  other  at  potential  zero.  He  gives,  however,  no  proof 
of  the  formula,  merely  referring  to  KirchhofFs  work.  He 
reduces  the  infinite  series  formula  for  the  case  of  two  spheres 
close  together,  given  by  Kirchhoff,  into  a  remarkably  simple 
algebraical  form,  and  shows  that,  when  the  spheres  are  at 
potentials  Y  and  0,  it  applies  with  sufficient  accuracy  for 
practical  purposes  up  to  a  distance  between  them  equal  to 
one-filth  of  their  radius.  In  what  follows  it  will  be  shown 
that  this  Kirchhoff-Schuster  formula  applies  with  very  con- 
siderable accuracy,  when  the  potentials  are  +  V/2  and  —  Y/2, 
up  to  a  distance  apart  equal  to  their  radius.  This  is  proved 
by  actually  calculating  the  values  of  the  series,  as  it  is 
difficult  to  see  from  KirchhofFs  method  of  proof  what  are  the 
limitations  of  his  formula.  By  considering  the  equipotential 
surfaces  round  two  particles  having  equal  and  opposite 
charges,  the  author  shows  how  the  first  two  terms  of  the 
Kirch h off-Schuster  formula  can  be  found  very  simply. 
*  Crelle's  Journal,  1860,  "  Ueber  die  Vertheilung  der  Elektricitat  auf 
zwei  leitenden  Kugeln,"  p.  89 ;  Gesammelte  Abhandlungen,  p.  78. 
t  Memoires  de  CInstitut  Imperial  de  France,  "  Sur  la  Distribution  de 
l'Electricite  a  la  Surface  des  Corps  Conducteurs."  Head  9th  May  and 
3rd  Aug.  1812. 
\  A.  Schuster,  "  The  Disruptive  Discharge  of  Electricity  through 
Gases,"  Phil.  Mag.  vol.  xxix.  p.  192  (Feb.  1890). 
Phil.  Mag.  S.  6.  Vol.  11.  No.  62.  Feb.  1906.  R 
