586 Dr. N. Bohr on the Decrease of 



mechanism o£ transfer of energy from the a. or ft particle 

 to the electrons are correct, we should expect that the formula 

 (5) will hold for absorption of a. rays in the lightest elements,, 

 and for ft rays also for the absorption in the heavier elements- 

 In case of ft rays it must, however, be remembered that the 

 formula (1) is deduced under the assumption that V is small 

 compared with the velocity of light. We shall return to 

 this question in Section 3, when we have considered the 

 probability variation in the loss of energy suffered by the 

 single particles. 



§ 2. The probability distribution of the losses of energy 

 suffered by the single ol or ft particles. 



The questions to be discussed in this section are intimately 

 connected with the probability of the presence of a given 

 number of particles at a given moment in a small limited 

 part of a large space, in which a large number of the 

 particles are distributed at random. This problem has been 

 investigated by M. v. Smoluchowski*, who has shown that 

 the probability for the presence of n particles is given by 



W(«)=J.-, (6) 



where e is the basis for the natural logarithm and a> is the 

 mean value of the number of particles to be expected in the 

 part of the space under consideration. If co is very large 

 this probability distribution is to a high degree of approxi- 

 mation represented by the formula 



W(s)ds= x /™6-*<* s2 ds, .... (7) 



where s is defined bj r w = a>(l 4 s), and W(s)ds denotes the 

 probability that s has a value between s and s + ds. 



In the paper cited K, Herzfeld uses the formula (7) in 

 calculating the probability distribution of the distance R 

 which an a particle of a given initial velocity will penetrate 

 through a gas before it is stopped. Herzfeld makes the simple 

 assumption that a certain number of collisions with the gas 

 molecules is necessary to stop the particle, and he takes this 

 number A to be equal to the total number of ions formed 

 by the particle in the gas. Now the number of collisions 

 suffered by an a particle in penetrating a given distance 

 through the gas is the same as the number of molecules 



* Boltzmann-Festschrift, 1904, p. 626 ; see also H. Bateman, Phil. 

 Mag. xxi. p. 746(1911). 



