286        Lord  Rayleigh  on  the  Production  of  Vibrations 
The  duration  of  impact,  therefore,  varies  inversely  as  the 
fifth  root  of  the  initial  relative  velocity  "  *. 
80  long  as  the  condition  is  satisfied  that  the  duration  of 
the  impact  is  very  long  in  comparison  with  the  free  periods, 
vibrations  will  not  be  excited  in  a  sensible  degree,  the  energy 
remains  translational,  and  Newton's  laws  find  application. 
It  would  be  of  great  interest  if  we  could  enfranchise  our- 
selves from  this  restriction.  It  is  hardly  to  be  expected  that 
a  complete  solution  of  the  problem  will  prove  feasible,  but  I 
have  thought  that  it  would  be  worth  while  to  inquire  into 
the  circumstances  of  the  first  appearance  of  sensible  vibra- 
tions. We  should  then  be  in  a  better  position  to  appreciate 
at  least  the  range  over  which  Newton's  laws  may  be  expected 
to  hold. 
In  the  case  of  spheres  the  vibrations  to  be  considered  are 
those  of  the  ."  second  class  "  investigated  by  Lamb  f .  The}' 
involve  spherical  harmonic  functions  of  the  various  orders, 
limited  in  the  present  case  to  the  zonal  kind.  But  for  each 
order  there  are  an  infinite  number  of  modes  corresponding 
to  greater  or  less  degrees  of  subdivision  along  the  radius. 
The  first  appearance  of  vibrations  will  be  confined  to  those 
of  longest  period,  of  which  the  most  important  is  of  the 
second  order.  In  this  mode  the  sphere  vibrates  symmetrically 
with  respect  both  to  a  polar  axis  and  to  the  equatorial  plane, 
the  greatest  compression  along  the  axis  synchronizing  with 
the  greatest  expansion  at  the  equator.  In  what  follows  we 
shall  denote  by  <£i,  <£2?  &c«  the  radial  displacements  at  the 
pole  (point  of  contact)  corresponding  to  the  several  modes, 
the  first  (f)^  being  appropriated  to  that  mode  in  which  the 
sphere  moves  as  a  rigid  body  (spherical  harmonic  of  order  1), 
the  next  cf>2  to  the  mode  of  the  second  order  above  described 
which  gives  the  principal  vibration. 
Since  there  is  no  force  of  restitution  corresponding  to  <£1? 
the  equation  for  it  takes  the  simple  form 
«i  #1  =  Po? ($) 
P0  being  as  before  the  total  pressure  between  the  spheres  at 
any  time,  and  ax  a  coefficient  of  inertia — in  this  case  the 
simple  mass  of  a  sphere.  On  the  other  hand,  the  equations 
for  <p2  &c.  are  of  the  form 
a„fa  +  c2<t>2=F0 (9) 
*  Love,  loc.  cit.  p.  154. 
f  Proc.  Lond.  Math.  Soc.  vol.  xiii.  p.  189  (1882). 
