GENERAL  ELECTROSTATIC  THEORY.         47

By  solving  the  n  equations  (34),  each  of  the  charges  may  be
expressed  as  a  linear  function  of  all  the  voltages.     Thus


(0

(2)


(35)


the  s's  being  functions  of  the  />'s.
Each  s  has  two  subscripts  and  denotes  the  ratio  of  the  charge
upon  the  conductor  with  the  first  subscript  to  the  voltage  to  Ao
from  the  conductor  with  the  second  subscript  when  the  voltage
to  A0  from  each  of  all  the  conductors  except  that  with  the  second
subscript  is  zero  (all  the  conductors  except  that  with  the  second
subscript  connected  metallically  with  Ao).
Those  coefficients  with  the  two  subscripts  equal  are  called
coefficients  of  capacity,  while  those  with  the  subscripts  unequal
are  called  coefficients  of  induction.
The  /'s  and  s's  are  functions  only  of  the  configuration  of  the
conductors  and  of  the  dielectric  constant  of  the  medium  in  which
the  system  is  placed.
51.  The  Coefficients.  Any  two  coefficients  of  potential  with
the  same  subscripts  in  different  order  are  identical.  That  is

Pu=Piu  (36)

To  prove  this  relation,  let  all  the  conductors  of  the  system
Al  •  •  •  An  except  Ah  and  Al  remain  insulated  and  uncharged  (alge-
braically). The  energy  of  the  dielectric  within  Ao  when  Ah  and
Al  have  the  charges  qh  and  ql  will  be  independent  of  the  manner
in  which  the  field  is  established.  The  energy  obtained  by
charging  Ah  first  and  then  Al  may  therefore  be  equated  to  that
obtained  by  charging  Al  first  and  then  Ah.  Denoting  the  energy
in  the  first  case  by  Wl  and  that  in  the  second  case  by  W2,  we
have


48  ELEMENTS   OF    ELECTROMAGNETIC    THEORY.
