12  Prof.  W.  Weber  on  Electricity  in  relation  to 
when  m=0,  the  following  value  is  easily  got,  namely 
and  consequently  the  sum 
r0—rr2(m  +  m!)       r0  +  r      "1* 
W_V=-=  +  r, 
3ni  +  ro'""°  0# 
This  sum  always  retains  the  same  value  as  long  as  the  values  of 
r0  and  a0  remain  unchanged — that  is,  so  long  as  the  system  of 
the  two  particles  undergoes  neither  loss  nor  gain  of  energy  from 
without.  The  external  kinetic  energy  of  such  a  detached  system 
amounts  separately  to  a  constant  sum. 
Now  the  same  thing  holds  good  also  for  two  electrical  particles 
e,  ef ;  for  their  potential,  taken  with  the  negative  sign  and  added 
to  their  kinetic  energy,  gives  in  like  manner  always  the  same 
sum  so  long  as  the  particles  constitute  a  detached  system. 
*  The  force  with  which  the  two  particles  mutually  act  on  each  other, 
dV 
namely  -r- ,  divided  by  m,  gives  the  acceleration  of  the  particle  m — that  is, 
]     dV 
—  •  -T-  ;  divided  by  m'  it  gives  the  acceleration  of  the  particle  m',  namely 
_L  ._.  Consequently  that  part  of  the  relative  acceleration  of  the  two 
m'     dr 
particles  which  arises  from  their  mutual  action  is  I  —  +  ~,J  -j-,  while 
that  part  of  the  relative  acceleration  of  the  two  particles  which  arises  from 
xec 
their  rotation  about  one  another  is  represented  by  — .     If  now  this  last 
portion  be  subtracted  from  the  total  acceleration  ^,  the  following  equa- 
tion results : 
du_uci__/\_.    1  \  dV 
dt       r  ""  \  m      m'J  dr  ' 
Putting  r=rQ  and  «  =  «0  for  the  instant  at  which  «  =  0,  we  obtain  the  ex- 
pression 
ecr=ec0rQ 
as  applicable  for  the  case  in  which  the  only  forces  acting  on  the  two  par- 
ticles are  those  due  to  their  mutual  action.  Accordingly  we  get,  by  inte- 
grating the  above  differential  equation  after  it  has  been  multiplied  by 
2dr=2udt, 
.  \  1  \     o'/l     i    l\fmm'      mm'\ 
+  «Woro    —  -■ —  =  2  — +  — )> 
\rr      rQrJ        \m.       m/\  r  r0  I 
_rjiZH2(m^m^      r°+rnc  tt  )-  ^o-r /2(m+m')_rn-)-r      \ 
"     r     \      r0  r       "  V        r0     \       r  r0        / 
uu 
and  hence 
uu 
