a.  Particles  of  Uranium  and  Thorium.  Ihl 
material  and  move  at  this  inclination  to  the  normal  have  a 
range  r  in  the  air  after  penetrating  the  absorbing  sheet. 
This  value  of  6  is  given  by  D  sec  6  +  r  =  R. 
Integrating  between  these  limits  we  find  that  the  total 
number  of  a.  particles  whose  ranges  lie  between  r  and  r  +  Br 
4*     I  [R-r)2  I 
Each  of  these  moves  a  distance  r  through  the  air  of  the 
ionization  chamber  before  it  ceases  to  ionize.  Now  I  have 
shown  (Phil.  Mag.  Nov.  1905)  that  the  a  particle  spends 
energy  on  ionization  at  a  rate  which  is  inversely  proportional 
to  the  energy  which  it  possesses ;  so  that  we  may  say  that 
Be  =  kSr/e,  where  e  is  the  energy  of  the  particle  and  Be 
the  energy  spent  in  traversing  a  distance  Br.  Hence  we  find 
that  e  <x  \/(r+c),  where  c  is  a  constant.  The  value  of  the 
latter  I  have  shown  (loc.  cit.)  to  be  1*33.  Hence  the  ioni- 
zation produced  by  the  a  particle  in  the  last  r  cm.  before  it 
ceases  to  ionize  may  be  written 
Z(\/r+l-33-v/i-33). 
Even  if  this  expression  should  prove  to  be  based  on  im- 
perfect theory,  it  nevertheless  expresses  the  actual  fact  very 
nearly. 
Finally,  therefore,  the  whole  ionization  (  =  i) 
the  proper  limits  being  given  to  r  in  the  integral. 
After  some  reduction  the  value  of  this  integral  can  be  found 
to  be 
(f(R-D  +  rf)?-fd-|-Ilv/rf—Dv/R+5=D"+2D\/a 
+     ^_     loo-  v^x/W  +  \/R  +  d=D)  1 
V&  +  d     °         \/D(\/R  +  o]  +  \/d)~     i' 
If  we  put  D  =  0,  we  obtain  the  value  of  the  current  (I) 
when  the  radioactive  material  is  uncovered,  viz.: 
Simpler  formulae  may  be  found  by  neglecting  the  variation 
of  ionization  with  velocity.     If  we  put  the  ionization  due  to 
TSlnp 
±s 
