hjj  Forces  of  Relaticely  Long  Duration.  289 
I£  <3>  and  its  differential  coefficients  up  to  a  high  order  are 
continuous  within  the  range  o£  integration  and  vanish  at  the 
limits,  the  leading  term  in  the  development  of  (16)  is  of 
high  inverse  power  in  ??.  An  extreme  case  of  this  kind  is 
considered  by  Mr.  Jeans  *,  who  takes 
In  this  case  the  solution  involves  the  factor  e~nG,  smaller 
when  n  is  great  than  any  inverse  power  of  n.  But  the  force 
is  not  here  limited  to  a  finite  range  o£  time. 
The  application  of  these  results  to  the  problem  of  the  col- 
lision of  equal  elastic  spheres  is  not  quite  so  straightforward  as 
had  been  expected.     In  (9), 
®=?0/a2  =  a2-lk2*i,     .     .  (23) 
a.  denoting,  as  in  (I),  (5),  (6),  (7),  the  approach  of  the  spheres. 
The  terminal  values  of  a  and  of  at  are  zero.     Again, 
so  that  d<§  J  dt  vanishes  at  the  limits  of  the  range.     But 
aW^=^"  \dt)  +*«aW 
=  4  ;  02*'1  —  f()  h  k2  *%     .     .     .     (25) 
use  being  made  of  (4),  (5)  ;  and  the  first  part  of  this  becomes 
infinite  at  the  limits  where  a.  =  0. 
Equations  (18),  (19)  give 
and  in  this  we  have  now  to  consider  the  two  parts 
(  sin  n  (t  —  tr)  a.-i  dt'    and    \  sin  n  (t — t')  a2  dt' . 
For  the  second  we  get  on  integration  by  parts,  since  a2 
vanishes  at  both  limits, 
1  f  /.       ,r\  da2    ,  , 
-;-jooB-(t-0Sr*'> 
*  '  Dynamical  Theoiy  of  Gases,'  Cambridge,  1904,  §  241.  I  should 
perhaps  mention  that  most  of  the  results  of  the  present  paper  were 
obtained  before  I  was  acquainted  with  Mr.  Jeans'  work. 
Phil.  Mag.  S.  6.  Vol.  11.  No.  62.  Feb.  1906.  U 
(20) 
