19 



Let each molecule emit N a particles per second. iV is a 

 very small fraction. Then the number emitted by each particle 

 between the inclinations and ^ + 8^ is 



27rsin^8^ N^inOSO 



i-rr 2 



Hence, finally, the number of a particles whose ranges in air 

 after emergence lie between r and r + Sr, and which have 

 inclinations to the normal varying from to + SO, is 



JV sin On^ cos OSrSO 



2^ 



The limits of are 0, and such a value of that the 

 a particles which come from the very surface of the radio-active 

 material and move at this inclination to the normal have a 

 range r in the air after penetrating the absorbing sheet. This 

 value of is given by the equation D secO + r = R. 



Integrating between these limits we find that the total 

 number of a particles whose ranges lie between r and r + Sr 



= — il \Sr 



Each of these moves over a range r in air. If, as a first approxi- 

 mation, it be supposed that in doing so it makes /r ions, then 

 we find that the whole ionisation (i) is obtained by integrat- 

 ing this expression with respect to r, having inserted the fac- 

 tor Ir, between the limits R — D and 0. The result is 



Nln, 



\{R-D){R-?.D)+ 2Z>Mog|| 



8s 

 If 7 = the ionisation when D = 0, then 



1= R'' 



8.S 



Hence 



=(-9(-?) 



«7/=|i--| |i- — I +^'i%'-^ 



From this formula a curve may be plotted showing the relation 

 between /// and DjR. 



This result is obtained on the assumption that the ionisation 

 caused by the a particle is proportional to the distance traversed, 

 in other words that the ionisation is independent of the particle's 

 velocity. This is not actually the case. I have shown (" Phil. 

 Mag.," Nov., 1905) that the ionisation is inversely proportional to 

 the square of the velocity. Assuming, therefore, that the ionisation 



