the  Principle  of  the  Conservation  of  Energy. 
15 
AVe  understand,  in  fact,  by  the  potential  of  two  particles,  the 
amount  of  work  which,  in  consequence  of  the  mutual  action  of 
the  two  particles,  is  done  when  they  are  transferred  in  any  way 
whatever  from  an  infinite  distance  to  the  actually  existing  dis- 
dr 
tance  r  with  the  actually  existing  relative  velocity  -=-• 
It  is,  however,  evident  that  wo?'k  is  done,  in  consequence  of 
the  mutual  action  of  the  two  particles,  not  only  during  their 
transference  from  a  greater  distance  to  the  distance  r,  but  also 
during  their  transference  from  a  smaller  distance  to  the  distance 
r.  And  there  is  no  obvious  reason  why  the  energy  ascribed  to 
the  system  should  be  made  to  depend  on  the  work  done  in  the 
former  case,  and  not  on  that  done  in  the  latter  case  also. 
For  example,  if  the  first  quantity  of  work  were  denoted,  ac- 
cording  to  Section  4,  by  V,  and  the  second  by  - V,  the  poten- 
r 
tial  energy  ascribed  to  the  system  might  be  taken  as  the  differ - 
p — r                   r 
ence  of  these  two  amounts  of  work,  namely  =  - V— V= V. 
This  difference  of  the  two  amounts  of  work  is  evidently  the 
quantity  of  work  which  is  done,  in  consequence  of  the  mutual 
action  of  the  two  particles,  during  their  transference  from  the 
limiting  value  of  small  distances  to  the  limiting  value  of  great 
distances — that  is  to  say,  the  value  which   — V=  —  (l ) 
assumes  when  r  is  taken  therein  as  equal  to  the  limiting  value 
of  small  distances,  or  when  we  put  r=p,  where  p  denotes  the 
limiting  value  of  small  distances.     According  to  this,  therefore, 
ee  r        uu\         r 
this  difference  of  the  two  quantities  of  work-=.  —  M J  =  — V. 
In  order  to  determine  in  this  way  the  potential  energy  of  a 
system  of  two  electrical  particles  when  the  first  quantity  of  work 
above  referred  to  is 
^■7(5-4 
it  is  only  necessary  further,  for  the  determination  of  the  second 
quantity  of  work,  to  determine  the  value  of  p — that  is,  of  the 
smaller  distance  which  is  to  be  taken  account  of  in  that  portion 
of  the  work: 
Now  this  smaller  distance,  equally  with  the  greater  distance, 
must  be  determined  on  its  own  account,  independently  of  the 
actually  existing  conditions  of  the  two  particles.  This  was  done 
in  the  case  of  the  greater  distance  by  assigning  to  it  an  infinitely 
great  value ;  in  the  case  of  the  smaller  distance  the  same  thins: 
