228 BELL SYSTEM TECHNICAL JOURNAL 



of motion of the electron. Consider in the hrsi place a mixture of 

 electrons and massive neutral molecules, assumed perfectly elastic, 

 in which the persistence of velocities of the electrons after collision 

 is negligible. If an electric field A'*'"' operates in the x direction and 

 if the state of motion is a steady one, we can compute the energy w 

 taken from the wa\e by a single electron at any time after a collision 

 at the time t\ and before the next collision. Let this time after /i be 

 T. If the mean frequenc\' of collisions is/, the time r between colli- 

 sions will be distributed according to the law 



/* 



-fr 



and we shall obtain the mean energy taken from the wave per collision 

 by multiplying w by the above expression, integrating from zero to 

 infinity with respect to r and then performing an average o\er all the 

 times /,. The result of this is that the mean energy loss per col- 

 lision is sini[)ly 



'^~ 2mn- P + n^ 



and consequently the loss per second is / times this. If we equate 

 this to rv^, which is also the rate at which energy is being dissipated, 

 we find that r = nif, which is therefore the resistance term to be inserted 

 in the equation of motion of the electron. 



If the convection current is carried partly by heavier ions, it will 

 not be annulled at each collision and all the energy derived from the 

 field will not be lost on impact. 



The foregoing computation assumes as ob\ious that energy is lost 

 from the wave at a rate equal to the number of collisions times the 

 average energy which the electron takes from the wave between 

 collisions. The second method is somewhat more general. The 

 mean velocity at a time / is found for electrons which collided last 

 in an interval at /i. This is evidently- a function of the velocity 

 persisting through the last collision and hence of the average velocity 

 before the impact; so that if the average velocity before collision 

 was V, that after impact would be 5 v, in which 5 is a number less 

 than unity, depending on the relative masses and the nature of the 

 collision. Averaging for all values of /, before / and using the same 

 law of distribution assuiTied above, the mean velocity of the ions 

 since the last collision is obtained. \i\ comjjarison with the solu- 

 tion obtained for the velocity of forced oscillation in which the re- 

 sistive force is rv, we find that r = ;w/(l— 6). For the special case of 

 electrons, B may be taken equal to zero, hence' r = w;/. For the case 



