Velocity of Swiftly Moving Electrified Particles. 589 



will express also the probability that a particle in order to 

 lose the energy AoT will penetrate through a layer o£ thick- 

 ness between Aa?=A A'(l + s) and A.i'-f- dx = A #(l + s + ds),. 

 where Ao«£ = A T/<£. In order to find the probability 

 W(R)eZR, that a particle in order to lose all its energy will 

 penetrate through a layer of thickness between R and 

 R + <:ZR, let us now divide the interval from to T in a great 

 number of small steps A]T, A2 '1' ••• and let us for the rth step 

 denote the quantities corresponding to A.r, ?/, <£, and s by 

 A,-' 1 , u r , <f>.,., and s r . The distance through which a given 

 particle will penetrate is equal to 



From this we get, denoting the mean value oE the ranges of 

 the particles by R , 



A T 



Iii exactly the same manner as that used in obtaining (8) 



(R-RJ- 



we now get 



where 



or simply 



W(R>/R = (2ttU)--'6 2U dR, . . (12) 



n (g) ...... (13) 



where the differential coefficient stands for the mean value 



A t f 



The equations (b'j and (9) and consequently also (12) and 

 (13) are deduced under the assumption that the collisions 

 suffered by the swiftly moving particle in penetrating a thin 

 sheet can be divided into groups in such a way that the 

 variation of Q for each group is small, while at the same 

 time the number of collisions in the group is large. The 



condition for this is that the quantity \ = dA I— - is large 



/ ^ 

 compared with unity. Substituting from (1) and (3) we 

 get 



\=7rNnA#(jP 2 + a 2 ) (14) 



