the  Principle  of  the  Conservation  of  Energy.  131 
tides,  when  r=r0, 
dr       roiro-p)\  ccj' 
du=l     cc    fp      Q«o«0y 
#       2ro-plr0  "^      cc  /' 
an 
since 
d  that,  when  also  a0=  a  /  _  ££?  (which  has  now  a  real  value, 
V         2r0 
—pz=—2{--h—j —  is  positive  for  dissimilar  particles), 
=  0;  according  to  which,  when  r=r0  and  «0=a /  —  (^ 
V        2r0 
the  two  particles  in  their  rotation  about  each  other  remain  always 
at  the  same  distance  (  =  r0)   apart,  a  case  which  with  two  similar 
particles  cannot  occur  at  all. 
It  follows,  however,  further  from  the  equation 
du 
dt 
uu 
cc 
r— p  \r0         r       cc  /' 
rtTrfLrtLt 
or,  when  we  put  w  for  the  constant  value Q_o_o_p  from  ^ 
pec 
following  equation, 
r  — 
•S-CHH-^IH. 
P 
that  besides  the  value  r  =  r0,  for  which  w  =  0  is  given,  there  is 
nr 
in  general  also  another  value  of  r,  namely  — — — ,  for  which  like- 
rQ     n 
wise  w  =  0. 
These  two  values  of  r,  however,  for  which  w=0,  differ  from 
each  other  sometimes  to  a  greater  and  sometimes  to  a  smaller 
r 
extent,  according  to  the  value  of  n\  and  when  n=  £  (that  is  to 
say,  when  a0=  k/  —  ^ — ),  they  coincide  completely;  and  it  is 
only  when  the  two  values  of  r  for  which  u  =  0  coincide  thus  that 
the  previously  mentioned  case  occurs,  for  which  we  have  at  the 
du 
same  time  w=0  and  —  =  0;  and  consequently  the  two  particles, 
while  revolving  round  each  other,  remain  at  the  same  distance. 
In  all  other  cases  in  which  the  velocity  w  =  0  (as,  for  example, 
when  r  =  2n—x,  where  x<n)  there  is  also  a  second  value  of  r 
nx  du 
— inthiscase2/H , — for  which  also  the  velocity  u  =  0.     -j- 
n  ~ ■—  x  az 
K2 
