354  Mr'  milner  on  the  Communication  of 
computation.  The  other  principle,  that  the  relative  ve- 
locity of  a and  B is  not  altered  by  the  ftroke,  is  neither 
to  be  demonftrated  nor  confirmed  by  experience;  it  is  a 
diredt  confequence  of  the  definition  of  elafticity.  Again, 
fuppcfe  a and  /3  to  reprefent  the  refpedtive  velocities  of 
A and  B after  the  ftroke,  and  from  thefe  data  it  is  eafily 
inferred,  that  a«2  + b/32=a^'  +b^:  for  a-b  is  equal  to 
,/3— a,  becaufe  a-b  is  the  relative  velocity  before,  and 
■jS— a the  relative  velocity  after,  the  ftroke.  And  Aa+nb 
is  equal  to  hot  + b/3,  becaufe  thefe  quantities  reprefent  the 
fum  of  the  motions  before  and  after  the  ftroke  refpec- 
tively ; and  from  thefe  equations  the  above  equation  is 
deduced,  (hewing,  that  in  elaftic.  bodies  the  fum  of  the 
two  bodies  multiplied  by  the  fquares  of  their  abfolute 
velocities,  is  not  altered  by  the  ftroke. 
The  fame  theorem  A__ ^ ■ 0 
V p 
may  be  demonftrated  s 
geometrically  in  the  following  manner.  Let  the  veloci- 
ties of  A and  B be  reprefented  by  ad,  ab,  refpedtively; 
and  let  g be  their  center  of  gravity,  when  placed  at  b 
and  d ; the  velocity  of  A after  the  ftroke  will  be  repre- 
sented by  B£-,  if'&g  betaken  equal  toon,  and  the  velo- 
city of  b by  ab  + 2BG.  From  the  nature  of  the  center  of 
'gravity  axgd=bxbg,  and  ax  gd  x 4 ag=b  x bgx  4AG- 
