116 Mr. G. H. Livens on the Electron 



If, then, we write 



/(?, V, t, *, y, *, t)-Ae-**=4>(& v, ?, *, y, z, Q, 



then the function <j> is the mathematical expression for the 

 change which an external force or difference of temperature 

 produces in the state of motion of the system of electrons. 

 We have now to calculate the number of collisions which 

 occur during the small interval of time between t and t + dt 

 in the group of electrons in which there are in all 



m=6dVdv 



electrons with their velocity components in the specified 

 range. 



Let us first of all find the part of this number SN of 

 electrons, which, after travelling for a time t after the 

 instant considered have not yet struck against an atom, 

 a number that is evidently some function of r. If we denote 

 this part by SN ; , then during the next succeeding interval dr 

 a certain part d hW of this number will be disturbed in their 

 motion by collision, and since this part will be proportional 

 to SN' and to dr we may write it 



PSWdr, 



where /3 is a constant, which may, however, be a function of 

 the velocity components (f, rj, f) and the position in the 

 metal (w, y, s). Hence, during the small interval dr the 

 number 8N' changes by 



d8N'=-j38Wdr, 

 so that we have 



because SN' = SN for t = 0. We thus find that the number 

 of the group 8N of electrons which collide between the 

 instants t and t + dr is 



so that the average time that elapses before collision for all 

 the electrons in the group is 



/"»0O -J 



Tm = \ jSe-^Tdr =-. 



Hence we may say that the number of the group of 

 electrons which collide in the interval between r and 

 T-\-dr is 



5N -I , 

 — e T m ai. 



