the  Principle  of  the  Conservation  of  Energy.  125 
uu 
is  obtained.     Putting  this  value  of  —  into  the  equation 
r  \cc         /' 
we  get 
y=  el  (r-r0f  p       r  +  r0  _  u0*0\ _  x  \ 
r  \r—p  \  r0  r         cc  J        /' 
dV  =  ee'     r0-p  ee'      A      (3     o  P  Wo\  *W) 
rfr  "~   r     (r—p)2      (>'— p)2\        \  r/  tt  /    cc 
12. 
According  to  the  last  section,  there  exists  an  equation  between 
the  relative  velocity  u  and  the  relative  distance  r  of  two  particles 
moving  anyhow  in  space  under  the  action  of  their  reciprocal 
forces,  namely  the  equation 
=  r~-?'o(P  +  r  +  ro  «o«<A 
r— p  \r0         r        cc  )' 
uu      r  — 
cc 
in  which  p  denotes  a  constant  that  is  positive  for  two  similar  par- 
ticles, and  negative  for  two  dissimilar  particles. 
Now  from  this  there  follow  results  relative  to  the  free  motions 
of  two  particles  in  space,  which  move,  under  the  influence  of  their 
own  reciprocal  action,  with  unequal  velocities  in  a  direction  per- 
pendicular to  the  straight  line  joining  them,  quite  similar  to 
those  arrived  at  in  relation  to  the  motions  considered  in  section  10 
in  the  direction  of  the  straight  line  r.  There  results,  in  fact,  in 
this  case  also,  a  distinction  between  two  states  of  aggregation 
for  two  similar  particles — namely,  a  state  of  aggregation  in  which 
the  two  particles  move  in  such  a  way  as  to  return  periodically 
into  the  same  position  relatively  to  each  other,  and  a  state  of 
aggregation  in  which  the  two  particles  move  so  as  to  become 
always  more  and  more  distant  from  each  other  and  never  return 
to  the  same  position.  No  transition  from  one  state  of  aggrega- 
tion to  the  other  takes  place  so  long  as  the  two  particles  move 
only  under  the  influence  of  their  own  reciprocal  forces. 
13. 
A  rotation  of  the  two  particles  about  each  other  implies  the 
existence  of  a  certain  attracting  force  if  the  two  particles  are  to 
remain  at  a  constant  distance  from  each  other  during  this  rota- 
tion ;  and  this  attracting  force  required  for  the  rotation  increases, 
for  the  same  distance,  according  to  the  square  of  velocity  of  ro- 
tation. According  to  this,  one  would  expect  that,  for  two  similar 
electrical  particles  at  a  distance  r0<p  (at  which  they  attract  each 
