Velocity of Swiftly Moving Electrified Particles. 607 



Denoting by Q the value for Q obtained by putting p — 

 in (1), we get, integrating (30) from Q = W to Q = Qo, 



, 2Tre 2 E 2 ^?iA.v 



Aw = 



m 



^^.(L_L\ m) 



Qo=;-^r 2 . (32) 



where ^ 2mM 2 V 



(m+M) 



If we consider a substance in which the different electrons 

 correspond to different values for W, we get instead of (31) 

 simply 



. 27re 2 E 2 NA.i' » / 1 1\ ,_n 



Aw= mV , -S(w-Q j' * ' ^ 33) 



Sir J. J. Thomson showed that the formula (31) with 

 close approximation can explain the relative number of ions 

 produced by a and (3 rays. If, however, in (31) we in- 

 troduce the values for W calculated from the observed 

 ionization potentials, and the values for the number of elec- 

 trons in the atoms which were found to agree with the 

 calculations in section 4-, we obtain absolute values for A^ 

 which are several times smaller than the ionization observed. 

 It appears, however, that this disagreement may be explained 

 by considering the secondary ionization produced by the 

 electrons expelled from the atoms in the direct collisions 

 with the a aud /3 particles. In Sir J. J. Thomson's paper it 

 is argued that this secondary ionization seems to be very 

 small compared with the direct ionization, since the tracks 

 of a and /3 particles on C. T. R. Wilson's photographs show 

 only very few branches. A calculation, however, indicates 

 that the ranges of the great number of the secondary rays 

 able to ionize are so short that they probably would escape 

 observation. The rays in question wiil be the electrons ex- 

 pelled with an energy greater than \V, and will be due to 

 collisions in which the a or /3 particle loses an amount of 

 energy greater than 2W. The number of such collisions 

 is given by (31) if W is replaced by 2W. Let this number 

 be A 2 w The total energy lost by the particle in the col- 

 lisions in question is equal to 



fQo 2 7 r^E 2 NnA^ 1 Q _ Q 



approximately. The mean value of the energy of the elec- 

 trons expelled is therefore P = W(2 log (Q /2W)-1). For 

 hydrogen and a rays from radium C this gives approximately 



