the  Principle  of  the  Conservation  of  Energy.  7 
and  from  this,  by  differentiation,  the  expression  for  the  force    . 
<*V  £?7I     r**9_,\  ,  9eJ   f    dfr 
IF' •      7?\cc'{th       l)  +  ~rcc'J,-dF> 
or,  if  we  again  put  -j-  for  fr, 
^  =  ^.V 1 -—•—  +  —  •  —  •  —^ 
dr  ~rr\        cc     dt2      cc     dt      dr )' 
for  which  we  may  write 
£Y  -  eA(\  _  L  *?  4.  *L    d^\ 
dr  ~  rr  \        cc  dt2       cc     dt2  J 
From  this  it  appears  that 
r  \cc    dt~        ) 
is  a  function  whose  differential  coefficient  with  respect  to  r  re- 
presents the  repulsive  force  between  the  two  particles  e  and  e\ 
dr 
where  r  and -7  denote  respectively  their  distance  and  relative  velo- 
ee1  (  1     dr2        \ 
city  regarded  as  functions  of  the  time.  But  since  —  ( —  •  --rk  ~1  J 
becomes  equal  to  nothing  when  e  and  e7  are  separated  infinitely 
ee'  I  1     dr2        \ 
far  from  each  other,  —I -73  —  1 )  is  the  potential  of  the  elec- 
r  \cc     dt  j 
trical  particles  e  and  e1 — that  is  to  say,  the  work  which  is  ex- 
pended in  causing  the  particles  to  approach  each  other  from  an 
infinite  distance  while  under  the  action  of  their  mutual  repulsion, 
dr 
and  to  arrive  at  the  distance  r  with  the  relative  velocity  —  *. 
It  likewise  results  from  the  foregoing  that  the  work,  which  is 
expended  when  a  given  relative  arrangement  and  state  of  motion 
of  a  system  of  particles  e,  e'  are  changed  to  another  arrangement 
and  another  state  of  motion,  depends  only  on  the  initial  and 
*  This  law  of  electrical  potential  has  also  been  taken  as  his  starting-point 
by  Beer  in  his  '  Introduction  to  Electrodynamics '  (see  Eirdeitung  in  die 
Elektrostatik,  die  Lehre  vom  Mognetismus  und  die  Elektrodynamik,  von 
August  Beer.  Nach  dem  Tode  des  Verfassers  herausgegeben  von  Julius 
Plucker  :  Braunschweig,  1865.  S.  250).  The  placing  of  the  law  of  poten- 
tial in  the  foreground  as  the  fundamental  law,  and  deriving  the  law  of  force 
from  it,  ought  not  to  give  rise  to  any  misgiving.  We  have  in  many  respects 
a  better  justification  for  speaking  of  the  phjsical  existence  of  the  work  ex- 
pressed by  the  potential  than  for  speaking  of  the  physical  existence  of  a 
force,  as  to  which  all  we  can  say  is  that  it  tends  to  change  the  physical  rela- 
tions of  bodies. 
