258  Mr.  A.  Eussell  on  tlie 
two  concentric  spheres,  it  is  of  importance  to  know  in  what 
cases  coronaB  can  be  formed.  The  problem  is  now  much 
more  difficult  as  the  coronse  are  only  approximately  spherical, 
the  maximum  thickness  o£  the  stratum  of  conducting  air 
round  each  electrode  being  on  the  line  joining  the  centre  of 
the  two  spheres. 
If  we  make  the  assumption  that  the  surrounding  air  is 
broken  down  to  the  same  depth  at  every  point  on  the  surface 
of  either  electrode,  we  can  find  whether  the  value  of  Rwl  in- 
creases or  diminishes  with  this  depth.  In  the  former  case  a 
disruptive  discharge  will  certainly  ensue,  and  a  fortiori  it 
will  ensue  in  the  actual  case  of  two  spherical  electrodes,  as 
the  actual  breakdown  begins  at  the  centre  of  the  spherical 
face,  raising,  as  it  were,  a  small  blister  at  that  point,  and  so 
Rm  must  be  greater  owing  to  the  greater  curvature. 
When  the  distance  between  the  spheres  is  greater  than  the 
radius  a,  we  have,  to  an  accuracy  of  1  in  a  1000, 
Bm=J{i(l  +  */«0W(2 +*/«)}, 
where  a  is  1*077,  provided  that  xja  is  less  than  7. 
Hence 
T      _  V  r      1  1  2acc       | 
m ~  2    {  d-2a  +  a  +  d(d-2a)  J 
V  r  1+* 
=  i  J 
a       d  J 
2    \d-2a 
and  therefore  dRm      V  r  2(1  + a)        1] 
da    ~  2    \  (d-2af      a2  j 
when  V  and  d  are  constants.  Hence,  for  values  of  cl  less 
than  a(2  +  A/2(l  +  «),  that  is,  for  values  of  d  less  than  4'04«, 
R  increases  as  a  increases,  and  thus,  on  our  assumption,  a 
disruptive  discharge  will  ensue. 
If  the  spheres  be  not  further  apart  than  twice  their  diameter 
we  should  therefore  expect  a  disruptive  discharge  to  ensue 
the  moment  Rm  became  Rmax.  For  large  spheres,  experiment 
shows  that  this  is  the  case  up  to  a  distance  apart  equal  to 
about  three  times  their  diameter.  For  greater  distances  apart, 
the  moment  RBl  attains  the  value  Rmax<  the  air  in  the  neigh- 
bourhood of  that  point  is  broken  down  and  a  partial  corona 
is  formed,  the  value  of  RWi  at  the  surface  of  the  corona  being- 
less  than  1^^.  Iu  these  cases,  as  the  equipotential  surfaces 
are  no  longer  spheres,  we  cannot  apply  our  formulae. 
