Velocity of Saiftly Moving Electrified Particles. 587 



present in a tubular space round the path of the particle. 

 The probability distribution of: the number of collisions can 

 therefore be obtained from the above formulae, if for a> we 

 introduce the mean value of the number of collisions. Since 

 A is supposed to be very great the variation in the ranges R 

 of the single particle will be very small. The probability 

 that R has a value between R (l + s) and R (l + s + ds), 

 where R is the mean value of the ranges, will therefore, on 

 Herzfeld's assumption, be simply given by (7) if we put 

 co = A. On the present theory the calculations cannot be 

 performed quite so simply. The total number of collisions 

 is not supposed to be sharply limited, but it is supposed that 

 the amount of energy lost by the a or /3 particle in collisions 

 with the electrons will depend on the distance of the electron 

 from the path of the particle, and will decrease continuously 

 for an increase of this distance. In order to apply con- 

 siderations similar to Herzfeld's, it is therefore necessary to 

 divide the collisions up into groups in such a way that the 

 amount of energy lost by the particles will be very nearly 

 equal for all the collisions inside each group. 



Consider an a or /3 particle penetrating through a thin 

 sheet of some substance of thickness A.r, and let us divide 

 the number of collisions of the particle with the electrons 

 into a number of groups in such a way that the distance p 

 has a value between p r and p r+ i for the collision in the rth 

 group. 



Let us now for the present assume that it is possible in 

 this way to divide the collisions into groups so that the 

 number in each group is large at the same time as the dif- 

 ference between any two values for the energy Q lost by a 

 collision in the same group is small. Let the value for Q 

 corresponding to the rth group be Q,. and let the mean value 

 of the number of collisions in this group be A,., and the 

 actual number of: collisions in this group suffered by the 

 given « or (3 particle be A,.(l-f-s r ). The total energy lost 

 by the particle in passing through the sheet in question is 

 then given by 



AT=2Q r A r (l + 5r ). 



From this we get, denoting the mean value of AT by A T, 

 AT-A T=2Q,A^. 

 Since the A's are large numbers, we get from (7) for the 

 probability that s r has a value between s r and s r -¥ds r , 



WW4=\/^- w i* 



