290        Lord  Rayleigh  on  the  Production  of  Vibrations 
which  is  of  order  n~l7  or  less.     In  the  first  part  the  relation 
between  dt'  and  da  is,  as  in  (7), 
(iff ^ (27) 
v("o  —  %  ki  k?  av 
If  we  exclude  the  terminal  parts  of  the  range,  the  integral 
would  be  of  order  w_1,  or  less,  so  that  it  is  only  the  terminal 
parts  that  contribute  to  the  leading  term.  For  the  beginning 
we  see  from  (27),  or  independently,  that 
u  =  u0t, (2S) 
nearly,  so  that  for  this  terminal  region 
Umn(t  —  t')u-idt' 
a0-i  f  .         Ccosnt'd(nt')  Tsinnt'd  (nt')\ 
=  -^V  4  sin  nt    y-TjST — ■  ~  cos  nt  I j—^i — L  \  . 
When  we  suppose  n  very  great,  the  limits  of  integration 
maybe  identified  with  zero  and  infinity;  and  further  by  a 
known  theorem 
J,=°  sin  x  dx       P*  cos  x  dx  _      n    \ 
o       \/x  J  o       \/x 
Thus,  so  far  as  it  depends  upon  the  early  part  of  the 
collision, 
M^J^fooB  (««+.*»).    ■     •     •     (29) 
There  will  be  a  similar  term  due  to  the  end  of  the  collision, 
derivable  from  (29)  by  replacing  nt  with  n  (t  —  r). 
If,  as  I  think  must  be  the  case,  (29)  gives  the  leading- 
term  in  the  expression  for  a  vibration,  the  next  question  is  as 
to  the  order  of  magnitude  of  the  corresponding  energy  in 
comparison  with  the  energy  before  collision,  viz.  \  Mass  x  a02, 
or  §  irpr^  a02. 
The  maximum  kinetic  energy  of  the  vibration  is  given  by 
3        -2       _  9tt  k£al 
■2  «2<P  max.  —  o9  5    J 
