by  Forces  of  Relatively  Long  Duration.  287 
c2  &c.  being  coefficients  of  stability  to  be  treated  as  large. 
This  form  applies  to  all  the  lower  modes,  for  which  the  force 
of  collision  operating  at  any  moment  may  be  treated  as  a 
whole.  By  equation  (1)  of  Hertz's  theory  P0  =  k2  at,  but 
now  that  we  are  admitting  the  possibility  of  vibrations  a 
must  be  reckoned  no  longer  from  the  centre,  but  from  a 
point  which  is  at  once  near  the  surface  and  yet  distant  from 
it  by  an  amount  large  in  comparison  with  the  diameter  of 
the  circle  of  contact.     We  may  write 
i*=^i  +<£2  + (10) 
inclusion  being  made  of  the  coordinates  of  the  lower  modes 
only.  The  sum  of  all  the  coordinates  would  be  zero,  since 
(in  the  case  of  equal  spheres)  the  pole  does  not  move.     Thus 
Po  =  M2fr +  2*, +..,)* (11) 
hi  the  first  approximation  c2  &c.  are  regarded  as  infinite, 
so  that  cj)2  &c  vanish.  P0  reduces  to  k2  (2(^)1,  and  so  from 
(8) 
oi&=M2tWt, (12/ 
the  solution  of  which  gives  fa  as  in  Hertz's  theory.  If  P0 
be  regarded  as  a  known  function  of  the  time,  fa  is  deter- 
mined by  (9);  but  it  may  be  well  at  this  stage  to  ascertain 
how  far  P0  is  modified  in  a  second  approximation.  Retain- 
ing for  brevity  (j>2  only,  we  have  approximately  $3  =  P0/<"2. 
Hence 
«,&  =  i,(2fc)*-[  l+|f(24>i)*}>    .     .     (13) 
and  we  infer  that  P0  is  changed  by  a  term  of  the  order  c2~1. 
We  will  now  pass  on  to  consider  the  general  problem 
of  a  vibrator  whose  natural  vibrations  are  very  rapid  in 
comparison  with  the  force  which  operates.  We  write  (0)  in 
the  form 
4>  +  n2  c/>  =  P0/%  =  <S>,    .     .      .     .      (14) 
where  n2  =  c2/a2,  and  is  to   be   treated  as  very  great.     If  d> 
and  0  vanish  when  t  =  0,  the  solution  of  (14)  is  * 
</>  =  *  P  sin  n  (t-tf)  <S>4=4.  df.     .     .     .      (15) 
nJ  o 
*  <  Theory  of  Sound,'  §  66. 
