242  Mr.  A.  Russell  on  the 
Professor  A.  Heydweiller*  carried  out  a  valuable  set  of 
experiments  on  sparking  distances  in  1892.  He  also  uses 
KirchhofFs  formula  without,  however,  proving  it.  Tables 
of  the  numerical  values  of  the  electric  intensity  when 
the  spheres  are  various  distances  apart  and  when  they 
are  at  equal  and  opposite  potentials  are  calculated.  The 
formula?  are  applied  to  his  own  experimental  results,  but 
as  he  does  not  discriminate  between  the  cases  when  they 
are  and  when  they  are  not  applicable,  and  neglects  the  '  lost 
volts/  the  results  vary  widely.  The  experimental  results 
analysed  in  Table  VI.  below  are  taken  from  this  paper. 
In  Mascart  and  Joubert's  '  Lecons  sur  FElectricite  et  le 
Magnetisme,'  vol.  ii.  p.  610  (1897),  a  neat  proof  of  a  series 
formula  for  the  maximum  electric  intensity  between  two 
unequal  spheres  is  indicated.  KirchhorFs  results  are  also 
quoted,  and  the  formulae  are  applied  with,  however,  indifferent 
success. 
In  this  paper  the  author  gives  a  simple  proof  by  Kelvin's 
method  of  images  of  KirchhofFs  series  formula.  He  shows 
by  elementary  algebra  that  it  can  be  expressed  quite  approxi- 
mately enough  for  all  practical  purposes  by  a  simple  formula. 
He  has  also  calculated  complete  tables  which  enable  any  one 
to  write  down  at  once  the  maximum  value  of  the  electric 
intensity  between  twro  equal  spheres  whatever  may  be  their 
potentials. 
In  some  of  the  experiments  analysed  below  cylindrical 
electrodes  are  used  ;  it  is  therefore  necessary  to  get  the 
formula  for  this  case  also.  It  will,  however,  be  more  in- 
structive to  consider  the  very  simplest  mathematical  cases 
first,  and  thus  we  shall  be  able  to  form  a  clearer  picture  of 
the  phenomena  that  happen  in  the  more  difficult  practical 
cases. 
8.   The  Electric  Intensity  between  Two  Concentric  Spheres. 
In  the  case  of  a  spherical  condenser  we  have  a  metallic 
sphere  concentric  with  a  metallic  spherical  envelope.  If  the 
radius  of  the  inner  sphere  be  a  and  the  inner  radius  of  the 
outer  sphere  be  b,  we  have 
dv       g 
~dr  =  P' 
where  v  is  the  potential  at  a  distance   r  from  their  common 
centre,  and  q  is  the  charge  on  the  inner  sphere.     Hence  we 
*  Wiedemann's  Annahn,  vol.  xlviii,  p.  785  (1893). 
