Dielectric  Strength  of  Air.  251 
potentials  V1  and  V2  is  given  by 
v^i  +  g)^  i-?4'-3   q2„-2 
V,(l+y)'*  1-g4'-1      ,,_!  ri3x 
The  most  important  case  is  when 
Vx=  —  V2  =  V/2,  and  in  this  case 
.     ^  ^t  i-^-        •  •    (i4) 
\Y 
her 
V 
"  2a' 
(i+ 
■  i- 
■qJ 
7(1  +  9*- 
-1 
-i)2 
f- 
-1 
X 
f, 
/: 
2a    1-y    f  (1  +  s2"-1)25      '   •     •     •     ^ 
and  a;  is  the  minimum  distance  between  the  two  spheres.  It 
is  convenient  to  tabulate  /  for  various  values  o£  x/a.  We 
see  that  /  is  the  factor  which  converts  the  average  electric 
intensity  in  the  line  joining  the  centres  of  the  two  spheres 
into  the  maximum  electric  intensity.  In  measuring  dielectric 
strengths  electricians  as  a  rule  merely  give  the  average  electric 
intensity,  assuming  that  /  is  unity  whatever  may  be  the  shape 
of  the  electrodes. 
Another  practical  case  is  when  one  of  the  spheres  is  main- 
tained at  zero  potential.  In  this  case  let  us  suppose  that 
V1  =  V,  and  V2  =  0.     Hence  by  (13) 
here 
Rm= 
X   7" 
A- 
.  x(i+(Jy 
a    l  —  q 
'  oc 
s 
1 
1- 
-  qAn~ 
g±n- 
-3 
-sy. 
:'T 
(16) 
In  practice  it  is  very  difficult  to  make  certain  that  one  sphere 
is  at  zero  potential,  and  so  this  method  of  testing  dielectric 
strengths  is  not  advisable. 
We  may  write  (13)  in  the  form 
^=~A-^(2f-A) 
=  ^/1  +  2I;(/1-/).    •    •    (N) 
lb  <6 
Thus  if  we  can  calculate  the  values  of  /and  /3  for  any  given 
value  of  x/a  we  have  completely  solved  the  problem. 
