250  Mr.  A.  Russell  on  the 
and  thus,  since  Qi  =  Vi«i,  we  have 
•  q,^^^.    .  .  .  (9) 
Hence  also  ,-,  ,         N   9„ 
Q/=-av/ifar.     .     .     .     (10) 
The  electric  intensity  between  the  two  spheres  will 
obviously  have  its  maximum  values  Rm  at  L  and  M, 
and  thus, 
a2  "*"  (a-w2)2       *  *  *    '    (a— u»+i)j 
Q/  Q» 
(d  — a  —  u-l)  (d  —  a-unf 
Now  by  (8) 
Qn  _  Qn+l(Mn+l/a) 
(d—a—Un)2  (a—Un+i)2   ' 
and  hence 
Q-i   ,   <>  Q»+i       a  +  un+ 
IU=^-+2 
a2  a     *  (a  —  ^+i)2' 
Substituting  for  un+i  and  Qn+i  their  values  from  (5)  and 
(9)  we  get 
_v,  o+?)'g  i-?*-3    2„_2         (11) 
The  value  of  the  electric  intensity  R„'  at  M  is  given  by 
{d  —  a)2  (d  —a  —  iin)2 
Q/  Q,/ 
[a— ui)a       '  '         (a  —  un!)2 
Noticing  that  d—un=a2jun  and  that  Qn/Q/  =  —  a/unf, 
we  find  that 
*■    ~       a      1-q    T(l-t'/"-1)22        '       *     {     ' 
We  can  write  down  the  values  o£  R,n  and  RTO'  when  the 
spheres  X  and  Y  are  at  potentials  0  and  Y2  in  a  similar 
manner.  Hence  by  the  principle  of  superposition  we  find 
that   the  electric  intensity  at   L    when    the   spheres  are    at 
