the  Principle  of  the  Conservation  of  Energy.  1 1 
as  an  internal  motion  of  the  particles  of  bodies.  But  if  we  are 
dealing  with  a  system  of  two  elementary  particles  (that  is  to  say, 
particles  such  that  there  can  be  no  motion  within  them),  it  is  ob- 
vious that  in  the  case  of  such  a  system  thermal  energy  has  no 
existence,  and  energy  of  motion  and  potential  energy  alone  remain. 
Lastly,  the  potential  energy  is  that  part  of  the  energy  which 
depends  on  the  existing  potential;  and  a  special  determination  is 
needed  of  the  way  in  which  potential  energy  depends  upon  the 
potential,  exactly  as,  in  the  case  of  the  energy  of  motion,  it  is 
needful  to  determine  the  special  way  in  which  it  depends  on 
movement. 
Now  this  special  determination  has  been  made  by  equating 
potential  energy  (without  regard  to  the  sign)  and  potential*. 
The  justification  for  this  proceeding  has  been  found  in  the  fact 
that  the  potential  is  a  magnitude  which  is  homogeneous  with 
kinetic  energy,  which,  when  taken  with  the  negative  sign  and 
added  to  the  kinetic  energy,  gives  always  the  same  sum,  so  long 
as  the  two  particles  constitute  a  detached  system  which  does 
not  undergo  either  gain  or  loss  of  energy  from  without. 
For  instance,  if  we  have  a  system  of  two  ponderable  particles 
m,  m',  its  potential  is 
and  the  internal  vis  viva,  or  the  internal  kinetic  energy  of  the 
system,  is 
w      1    mm'    .  . 
where  w  =  -/-is  the  relative  velocity  of  the  two  particles,  and  a 
dt 
the  difference  of  the  velocities  in  space  perpendicularly  to  ?*. 
But,  for  such  a  detached  system,  if  we  put  r  =  r0  and  <*  =  a0 
*  The  sign  of  the  potential,  V,  is  so  determined  that  positive  values  of 
rl  V 
_ indicate  repelling  forces ;  the  sign  of  the  potential  energy  is  fixed  by 
dr 
the  sign  of  the  work  which  is  clone,  in  consequence  of  the  mutual  action  of 
the  particles,  when  the  two  particles  are  separated  from  the  distance  r  to 
an  infinite  distance.     Consequently,  for  two  ponderable  particles  m,  rri ,  the 
potential  is  V= ,  and  the  potential  energv  =— For  two  elec- 
r  r 
...         ee'   /I     dr2      \ 
trical  particles  e,  e    the  potential  is   =  —  (-—  '^5  —  1 )»  and  the  poten- 
ee'  f         1     df*\ 
tial  energy  »7V~5"iF/' 
