Moving Electrified Particles passing through Matter. 19 



therefore put P( =~ I = 1, and further take the limit for the 



integral to be and co , (P'(0) = 0). 



Putting 



we thus get 



r 



Jo 



dl = ^ — loerl ^ ., ]d;i'. 



mV 2 °\neE{M + m)J 



I have calculated k by help of the above formulae for f(x)' 

 and found 



£=1-123. 



If we assume that the atoms contain electrons corresponding 

 to different frequencies, and if we denote the frequencies of 

 the r electrons in each atom by n v n 2 , ... n n we get 



,„ 4™ 2 E 2 N , «S' , ( V'&Mm > (v . 



Since cZT is equal to the decrease in the kinetic energy of 

 the particle, i. e., in JMV 2 , we have 



dV _ 47r fc 2 E 2 N -J* / V 8 ftMm \ 



<fo ~ wMF ,=i g U,«E(M+«i)J' * W 



In establishing the formula (4) we have only considered 

 the interaction between the particle and the electrons, and 

 not the interaction between the particle and the central 

 charge in the atoms ; as Darwin * has shown, the effect of 

 the latter interaction will, however, be negligibly small in 

 comparison with the former : this conclusion will hold un- 

 altered for the theory in the form it is given here. 



The formula (4) expresses the rate of decrease of velocity 

 of moving electrified particles as a function of the velocity 

 of the particles and the number and frequencies of the 

 electrons. 



If V is very great we can neglect the variation in the 

 logarithmic term, and get for the relation between V" and 

 the distance the particles have travelled through matter, 

 denoting the velocity for .i* = by V , 



V *-Vz 4 = £M?, (5) 



where 



1 6tt? 2 E 2 N « / V 3 *Mm \ 



.=i ° g WE(MtMi)/ 



a = 



rn 



M 



* Darwin, loc. cit. p. 905. 

 C 2 



