18 Dr. N. Bohr : Theory of Decrease of Velocity of 

 where 



+ 00 +00 



r N 1 C z sin xz 7 ^ f cos #£ , M , . N 



CO —00 



in which /(a?) is the same function as above. 



While for the motion perpendicular to the direction of the 

 particle the energy transferred is always smaller than the 

 energy calculated by considering the electrons as free, this 

 is not the case for the motion parallel to the path of the 

 particle. 



For the total energy transferred to the electron by the 

 collision we have now 



Q=Q 1 + Q 2 =^|>(f > ), ... (2) 



where P(a') ==/ 2 (#) + g 2 (x) is equal to 1 for x = 0, and decreases 

 very rapidly for increasing values of x, when x is great ; for 

 x = we notice that P'(#) = 0. 



Let us now consider a particle passing through matter. 

 Let us assume that the numbers of atoms per unit volume is 

 N, and that each atom contains r electrons of frequency n. 

 Let further a be a constant, great in comparison with X, but 

 small in comparison with Y/n (see p. 15), we then get for the 

 total energy dT transferred to the electrons when the particle 

 travels through a distance dx 



dT = Nr f" r Qo^Trpdp + f Q27rpdpl dx ; 

 by help of (1) and (2) we get 



47H? 2 E 



dT = 



-TfjM>(T>>- 



mY 

 Neglecting (X/a) 2 (see above), we get 



CO 



a I = r^ — loo- { — I + I - r(z)dz dx. 



an 

 V 



00 



7rn 477^ 2 E 2 Nr r, /a\ , /an\ n /an\ (\ _,,. . ~i 



fl= ^-K-)- Io «(v ) • p (v )-J ^--•n--)^*. 



V 



an . 



According to our assumption, v is very small, and we can 



