2*±       Lord  Rayleigh  on  the  -Production  of  \  'ibrations 
But  it  is  with  Hertz's*  solution,  under  certain  conditions, 
of  the  problem  of  impinging  curved  bodies  with  which  I  am 
now  more  concerned.  He  commences  with  the  purely  statical 
problem  of  contact  under  pressure.  Thus  if  two  equal  spheres 
of  similar  material  be  pressed  together  with  a  given  force  P0, 
the  surfaces  of  contact  are  moulded  to  a  plane  ;  and  it  is 
required  to  find  the  radius  of  the  circle  of  contact,  and  more 
especially  the  distance  («)  through  which  the  centres  (or 
other  points  remote  from  the  place  of  contact)  approach  one 
another.  It  appears  that  the  relation  between  P0  and  a,  is 
simply 
Po  =  *i«d (1) 
where  /r2  depends  only  on  the  forms  and  materials  of  the  two 
bodies.     In  the  particular  case  above  mentioned. 
k   =  y/2r-  E (2) 
"      3(1-^)  w 
where   r  is  the  radius   of   the   spheres,  E  Young's  modulus, 
and  a  Poisson's  ratio. 
In  applying  this  result  to  impacts  Hertz  proceeds  : — "  It 
follows  both  from  existing  observations  and  from  the  results 
of  the  following  considerations,  that  the  time  of  impact,  i.  e. 
the  time  during  which  the  impinging  bodies  remain  in  con- 
tact, is  very  small  in  absolute  value ;  yet  it  is  very  large 
compared  with  the  time  taken  by  waves  of  elastic  deforma- 
tion in  the  bodies  in  question  to  traverse  distances  of  the 
order  of  magnitude  of  that  part  of  their  surfaces  which  is 
common  to  the  two  bodies  when  in  closest  contact,  and  which 
we  shall  call  the  surface  of  impact.  It  follows  that  the 
elastic  state  of  the  two  bodies  near  the  point  of  impact 
during  the  whole  duration  of  impact  is  very  nearly  the  same 
as  the  state  of  equilibrium  which  would  be  produced  by 
the  total  pressure  subsisting  at  any  instant  between  the 
two  bodies,  supposing  it  to  act  for  a  long  time.  If,  then,  we 
determine  the  pressure  between  the  two  bodies  by  means  of 
the  relation  which  we  previously  found  to  hold  between  this 
pressure  and  the  distance  of  approach  along  the  common 
normal  of  two  bodies  at  rest,  and  also  throughout  the  volume 
of  each  body  make  use  of  the  equations  of  motion  of  elastic 
solids,  we  can  trace  the  progress  of  the  phenomenon  very 
exactly.  We  cannot  in  this  way  expect  to  obtain  general 
laws;   but  we  may  obtain  a  number  of  such  if  we  make  the 
*  Journal  fur  reine  und  cmgewandte  MathematiJc,  xcii.  p.  156  (1881) ; 
Hertz's  Miscellaneous  Papers.  English  edition,  p.  146.  A  good  account 
is  given  by  Love,  he.  cit. 
