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



The solution of this equation subject to the condition that 



x = and -=— = for t= — co , is * 

 at 



If*. dx C t 



x = - I sin n(t— -z) . (f> (z)dz ; -r— = 1 cos w(£ — 2) . <f>(z)dz. 



— oo — ao 



We have in the above expression assumed that the electron 

 was at rest before the collision with the particle ; if we 

 assume that the electrons are in motion in the atoms 

 before the collision (the dimension of their orbits must, how- 

 ever, for the justification of the above calculations be small 

 in proportion to p ; for the fulfilment of this condition see 

 later on p. 20), the effect will only be an introduction of 



some terms in the expressions for x and -jr which will 



again disappear in the expression for the mean value of the 

 energy transferred. 



For the sum of the kinetic energy of the electron at the 

 time t, and its potential energy due to its displacement 

 relative to the rest of the atom, we have now 



m /dos\ 2 mn 2 9 m \~ f* , , s 1 ~1 2 



— oo 



+ j\\ sin nz.cf)(z)dzj. 

 - 00 

 For the energy transferred to the electron by the collision, 

 due to motion perpendicular to the path of the particle, we 

 now get, observing that in this case (f)(z) is an even function 



of z, 



+00 



Qi—21 I cos nz ' < t > ^ z ^ 2 \ ' 

 — 00 

 and introducing for <j>(z) 



+ 00 



Z m 



r C cos nz ~[ 2 



n - 2* 2 E* ( T \ 



* See Lord Rayleigh, ' Theory of Sound/ i. p. 75. For the following 

 analysis compare also J. H. Jeans, ' Kinetic Theory of Gases,' p. 198. 



