552 Mr. G. H. Livens on the Electron 



motions oE the electrons of the specified group. In this 

 case, therefore, the differential equation above is to be inter- 

 preted with 



a-b=- r {f-f ). 



If, however, we wish to include the full effect of the 

 persistence of the velocities, we must modify this equation in 

 some such manner as the following argument indicates. It 

 is to be noticed that if we neglect the persistence of velocities, 

 the average or expected value of any electron moving at any 

 time t with velocity (£, 77, f) when taken immediately after 

 its next collision will be zero; if we do not neglect the per- 

 sistence, however, the expected value of the velocity will be 

 f(f, rj, f), e being a factor (a function in general of the 

 resultant velocity) which measures the persistence. In other 

 words, if there were SN electrons moving with this velocity 

 before collision there will still be eSN moving with the same 

 velocity after the collisions, one for each electron, have 

 taken place. The collisions have thus reduced the number 

 of electrons in the group by the factor (1 — e) only. It 

 follows, then, by the help of the argument used above that 



«-6=~(/-/«)- 



The argument for this can be put in a slightly more direct 

 and perhaps more rigorous form*. If the typical collision 

 is one in which p is the perpendicular on the initial asymptote 

 of the relative path and yfr is the azimuth angle of this path, 

 then it is easily shown that 



b — a— I I [f —f)updpd^, 



<Jq Jo 

 where 



/=/(?, v, o, 



and (f ', 77', f) denotes the velocity of the electron after the 

 typical collision. Now if in the most general case we can 

 write 



/=/„(l + £), 



and if also the general dynamical nature of the collision is 

 such as to leave the resultant velocity of an electron unaltered, 

 then we shall have 



/o =/o> 

 so that then 



C2 ir r « 



^'o 

 * See, for example, 0. W. Richardson, Phil. Mag. July 1912. 



