the  Principle  of  the  Consei'vation  of  Energy. 
stant  cc,  such  that 
x 
.2 
* 
a       cc    dt2 
*  If  e  and  e'  denote  the  masses  of  the  particles  e  and  e',  oc  and  /3  the  velo- 
cities of  f  in  the  direction  of  r  and  at  right  angles  thereto,  and  od  and  /3'  the 
same  velocities  for  c',  so  that  «-«'=  ~  is  the  relative  velocity  of  the  two 
particles,  then 
i*(«*+#)+5e'(*'*'+/3'/3') 
is  the  total  n's  viva  of  the  two  particles.    If  we  now  put  for  ec 
and  for  «' 
e-f-e  e+e' 
eot+f'at'       e'(oc—  at') 
"7+7"  e+e'    ' 
we  get  the  total  vis  viva  of  the  two  particles  represented  as  the  sum  of  two 
parts  in  the  following  manner — namely, 
1    ee'       dr2 
the  first  part  of  which,  or •  -pr>  is  the  relative  vis  viva  of  the  par- 
ticles which  was  denoted  above  by  x.  But  a  is  also  a  relative  vis  viva  of 
the  same  particles,  namely  that  which  corresponds   to  a  definite  relative 
velocity  c,  so  that  a  ■=  — .  cc.    Hence  we  get  -  =  —  —  ,as  was  given 
3  2  e+e'  &     a      cc    dt*  & 
above. 
It  may  be  further  observed  that  the  second  part  of  the  above  sum, 
namely  ^  - —  f  ;~+  e/3/3  +  f'/3'/3'  ,  may  be  again  represented,  after 
another  subdivision,  as  the  sum  of  two  parts,  thus 
1     ee'       ds2       lrKe'«')2,        ,    ,,     ~1 
where  j-  represents  the  velocity  with  which  the  two  particles  move  rela- 
tively to  each  other  in  space  perpendicularly  to  r,  while  y  represents  the 
velocity,  perpendicular  to  r,  of  the  centre  of  gravity  of  the  two  particles. 
"We  thus  get  the  total  vis  viva  of  the  two  particles  divided  into  three  parts — 
namelv, 
•      1    <*'     <*** 
..       1     ee'      ds2 
1    [-(«*+*'*')> 
p^+(.+0?y] 
the /rsf  of  which,  namely  'o~T~r'  IF'  rePrcsents  tue  relative  vis  viva  cf 
