780  Prof.  J.  J.  Thomson  on  the 
velocities  of  B  parallel  to  x  before  and  after  the  collision, 
M  a  1VF  
where  cj>  is  the  angle  between  the  plane  containing  b  and  V 
and  that  containing  V  and  x.  Averaging,  the  term  containing 
cos  cf)  will  disappear,  and  we  have 
2M,u  1 
1+    e" 
V   MJi,   t 
AMx  +  M,/ 
If  there  are  N  of  the  interatomic  corpuscles  per  unit 
volume,  the  number  of  collisions  in  which  b  is  between  b  and 
b  +  db  made  by  a  corpuscle  B  when  it  travels  over  a  distance 
Ax  is  NA#  .  2irb  .  db.  Hence,  if  U  is  the  sum  of  the  values 
of  u  for  the  B  corpuscles  per  unit  volume,  and  A(U)  the 
change  in  U  in  the  distance  Ax, 
A/TTN  OTTAT       A  Ml  P*'  2irb  Jb  ,-v 
LVlx  __  p  Virbctb 
J01+    *2    VMi  +  M,/ 
the  upper  limit  being  determined  by  the  condition  that  B 
comes  into  collision  with  the  A  corpuscles  one  at  a  time,  so 
that  the  shortest  distance  between  B  and  the  corpuscle  with 
which  it  comes  into  collision  must  be  small  compared  with  a, 
the  distance  between  two  corpuscles.  If  r  is  the  shortest 
distance  between  the  A  and  B  corpuscles,  we  can  easily  show 
that 
1- 
>2       2e2M1+M2 
r2~Y2r    MxMg    ' 
Putting  r  =  a,  we  see  that  b'  is  of  order 
/  2e2  Mx  +  MA* 
°V     V2a    M!M2  J  ' 
Integrating  the  expression  on  the  right-hand  side  of 
equation  (6),  we  get 
dV_     on   N.M,  ^(M1  +  M2)2        /       Wy  MLM2  \»\ 
dx~  Mi  +  M2  V^MxMj) 2      g  V   +    *2    \Mi  +  M2  /  /• 
Since  the  logarithmic  term  only  varies  slowly,  we  may  put 
for  bx  any  quantity  of  the  same  order  without  greatly  affecting 
the  result,  putting 
/       2V2M1+M?\J 
b-a[}~  ATP7J 
^  ,TTs__TT4!rNf_4(M1+ML)        /aV»_M^_       \ 
