﻿Velocity of Electrons in Gases. 385 



mass m of a molecule of the gas that all directions of 

 motion of a molecule become equally probable after a 

 collision with an ion, it was shown that formula (2) reduces to 



U=Xel/m'u, (3) 



as in this case it may be seen that 



l + X m 



m 



(4) 



It will be observed that formula (2) reduces to (1) when 

 A, is zero, that is when m is small compared with m\ so that 

 either of these two formulae may be applied to the case of an 

 electron moving in a gas. Mayer, however, selects formula 

 (3) to find the velocity of ions of small mass or electrons, 

 although it is definitely stated in my paper that formula (3) 

 refers to large ions, and the relation (4) on which it depends 

 can only hold when m is greater than m! '. As the correct 

 formula (1; for electrons differs by the factor m'/m from 

 formula (3), it is unreasonable to expect the latter formula to 

 give the velocity of an electron. 



The above formulae, obtained by simple considerations 

 when the velocities of agitation of all the ions are taken as 

 being the same, are of course not absolutely exact. There 

 is a numerical factor by which the expressions should be 

 multiplied in order to allow for the variations of the velocity 

 of agitation about the mean velocity. In the most interesting 

 case, which is that of electrons moving in a uniform electric 

 field, the value of the numerical factor is about '9, but it has 

 not been determined exactly. The determination of this 

 factor is very difficult, as the distribution of the velocities of 

 agitation of the electrons depends on the energy of an 

 electron which is lost in a collision, and experiments show 

 that the proportion of the total energy of an electron which 

 is thus lost depends on the velocity. This problem has been 

 fully considered by F. B. Pidduck (Proceedings of the 

 London Mathematical Society, ser. 2, vol. xv. pt. 2, 1915), 

 who shows that under certain conditions the proportion of 

 the velocities which differ largely from the mean velocity 

 of agitation is much less than the proportion indicated by 

 Maxwell's formula for the distribution. 



It appears that the error introduced by taking the velocities 

 of agitation as being all equal to the mean velocity may be no 

 greater than when the velocity distribution is taken as being 

 the same as that given by Maxwell's formula. 



[n order to obtain an exact formula for the velocity U it 

 Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. 2 



