by  Forces  of  Relatively  Long  Duration.  2S?> 
further  assumption  that  the  time  of  impact  is  also  large 
compared  with  the  time  taken  by  elastic  waves  to  traverse 
the  impinging  bodies  from  end  to  end.  When  this  condition 
is  fulfilled,  all  parts  of  the  impinging  bodies,  except  those 
infinitely  close  to  the  point  of  impact,  will  move  as  parts  of 
rigid  bodies  ;  we  shall  show  from  our  results  that  the  con- 
dition in  question  may  be  realised  in  the  case  of  actual 
bodies."  The  above-mentioned  condition  may  in  fact  always 
be  satisfied  by  taking  the  relative  velocity  of  impact  to  be 
sufficiently  small. 
The  solution  of  the  problem,  thus  limited,  is  now  easily 
found.  For  the  case  of  two  spheres  the  relative  acceleration 
a  is  connected  with  P0  by  the  equation 
Po  =-«/*! •        (3). 
where  L\  —  (?/?j  +  m2)/m1m2, 
and  nix,  m2  are  the  masses  of  the  spheres.  Eliminating  P0 
between  (1)  and  (3),  we  get 
«  +  kx  k2a%  =  0, (4) 
and  on  integration  as  the  equation  of  energy 
'a?  -  «02  +  z  /•,  h  »-S  =  0 (5) 
«0  being  the  relative  velocity  before  impact. 
"  The  greatest  compression  takes  place  when  a.  vanishes, 
and  if  ax  be  the  value  of  a  at  this  instant 
4  /,,  L, 
(6> 
Before  the  instant  of  greatest  compression  the  quantity  a 
increases  from  zero  to  a  maximum  al9  and  a.  diminishes  from 
a  maximum  a0  to  zero.  After  the  instant  of  greatest  com- 
pression a.  diminishes  from  olx  to  zero  and  a  increases  to  a0. 
The  bodies  then  separate,  and  the  velocity  with  which  thev 
rebound  is  equal  to  that  with  which  they  approach.  This 
result  is  in  accord  with  Newton's  Theory.  It  might  have 
been  predicted  from  the  character  of  the  fundamental 
assumptions." 
"  The  duration  of  the  impact  is 
r«i du 2*1  ri     dx 
J  o  v7^2  --ihh2cd)  ~>o  Jo  n/(1-^) 
s"1  (2-9432),    .     .     (7 
,y        \  \ 
:,  X/TT  .  I   (g)        a, 
5  «0    rcA) 
where  olx  -is  given  by  ((>).- 
