756  Prof.  W.  H.  Bragg  on  the 
The  a  particles,  when  they  emerge  into  the  air,  will  pene- 
trate to  distances  depending  on  the  quantity  o£  matter 
traversed  before  emerging.  Consider  in  the  first  place  all 
those  whose  ranges  in  air  after  emergence  lie  between  r  and 
r  +  dr.  These  move  at  various  inclinations  to  the  surface: 
the  number  emerging  at  any  inclination  depends  on  that 
inclination,  and  may  be  reckoned  as  follows. 
Consider  only  those  whose  inclinations  to  the  normal  lie 
between  0  and  0  +  d0.  All  these  come  from  a  layer  of  a 
certain  thickness  at  a  certain  depth  below  the  surface.  The 
depth  does  not  concern  us,  but  the  thickness  does,  for  we 
need  to  know  the  number  of  radioactive  atoms  in  the  layer. 
Let  n  be  the  number  of  radioactive  atoms  in  a  cub.  cent, 
of  the  material,  n0  the  number  of  molecules  in  a  cub.  cent,  of  air. 
An  a  particle  loses  the  same  range  in  traversing  a  distance 
&r  in  air,  as  in  traversing  a  distance  n0Sr/ns  in  the  radioactive 
/ 
AbsordiTiy  laz/er. 
/ 
Radio-active  material 
p                            N 
P 
0 
F' 
material.  Hence  if  PP;  is  the  layer  in  question,  0  the 
radioactive  atom,  OS  the  course  of  the  a  particle,  we  have 
OQ  =  n0hrjns  and  ON"  =  n0Br  cos  0/ns.  This  last  expression 
is  also  the  volume  of  the  layer  from  which  the  a  particles 
come,  since  we  are  considering  unit  area  of  the  material  ;  and 
therefore  the  number  of  radioactive  atoms  in  it  is  n08r  cos  0/s. 
Let  each  atom  emit  N  a  particles  per  second.  N  is  a  very 
small  fraction.  Then  the  number  emitted  by  each  particle 
between  the  inclinations  0  and  0-\-d0  is 
]ST  .  2tt  sin  0  .  S0/4tt  =  N  sin  6  .  h6j2. 
Hence  finally  the  number  of  a  particles  whose  ranges  in 
air  after  emergence  lie  between  r  and  r-\-8r,  and  which  have 
inclinations  to  the  normal  varying  from  6  to  0-\-B0,  is 
N  sin  6  .  n0  cos  08rS6/'2s. 
The  limits  of  0  are  zero  and  such  a  value  of  6  that  the  a 
particles  which  come  from  the  very  surface  of  the  radioactive 
