the  Principle  of  the  Conservation  of  Energy.  127 
distance,  but  approach  each  other  from  r=r0  to  r=0  with  a  velo- 
i  /         r  r  01  01      1  \ 
city  which  increases  from  w  =  0  to  u=+/  lcc+  °  °  °  °  •  -  J— that 
is  to  say,  becomes  infinite,  if  the  velocity  of  rotation  a0  differed 
from  nothing  for  the  instant  at  which  r=r0.  From  this  it  fol- 
lows that  the  interval  of  time  0  in  which  the  two  particles  ap- 
proach each  other  from  the  distance  r=r0  to  r  =  0  has  a  finite 
value.  The  fact  that  for  the  instant  at  which  r  becomes  equal 
to  0  the  value  of  the  relative  velocity  of  the  two  particles  becomes 
*/<-+*?*■!)-*•• 
signifies  here  only  that  this  relative  velocity  is  to  be  henceforward 
taken  as  a  velocity  of  separation  =  +  go  ,  whereas  it  was,  up  to 
this  point,  a  veloctiy  of  approach  =  —  co  .  This  being  premised, 
it  easily  follows  that,  in  a  second  equal  interval  of  time  0,  the 
two  particles  will  separate  from  each  other  again  from  the  dis- 
tance r  =  0  to  the  distance  r  =  r0.  The  interval  of  time  20,  in 
which  the  two  particles  approach  each  other  with  increasing  ve- 
locity from  the  distance  r  =  r0  to  r=0  and  then  separate  again 
from  the  distance  r  =  0  to  r  =  ?*0,  may  be  called  the  time  of  oscil- 
lation of  the  atomic  pair  formed  of  the  two  electrical  particles. 
There  still  remains  the  problem  of  determining  the  time  of  os- 
cillation 20  of  such  an  atomic  pair. 
This  time  of  oscillation  can  be  readily  deduced  from  the  equa- 
tion 
™  =  r~~ro(P  +  ro  +  r  .  *o"(A 
cc       r — p  \rn         r         cc  J' 
if  it  be  assumed  that  therein  r0  is  not  greater  than  p. 
For  if  we  first  consider  the  limiting  case  in  which  r0=pj  it 
follows  from  the  above  equation  that 
uu  =  cc  +  «0a0  +  pct0ct0  .  - ; 
and  hence,  putting  u=  -j-> 
■=—dr\/ 7- 
V    pa0«0+(cc  +  «oa< 
dt: 
From  this  we  obtain,  by  integration, 
0=  _  Cdr  a  / - 
Je       V   pa0a0+(cc-fa0a°) 
+jq  •      r—v~v  •    \ —   ■  —«-  r 
Accordingly  we  get : — 
