Theory of Metallic Conduction. 555 



3. The bases of the Drude-Thomson theory. 



The form of the general differential equation satisfied by 

 the fundamental function and the form of the function itself 

 deduced from this equation are susceptible of a very simple 

 interpretation in a form which will be of immediate use to 

 us. It has already been remarked by Jeans* that the effect 

 of the persistence of velocities may simply be interpreted as 

 implying that on the average the free path motion of the 

 electrons is longer in the ratio 1:1 — e. We might, therefore, 

 obtain a sufficiently effective picture of the electronic motions, 

 applicable in all cases where the actual dynamical charac- 

 teristics of the collisions are of small importance (that is, 

 in all so-called free path phenomena) by imagining that the 

 free paths of all the electrons are lengthened in the corre- 

 sponding ratio ^ (which may of course be a function of 



the velocity), and that the collisions take place as with elastic 

 spheres at the ends of these extended paths, so that there is 

 no further persistence of the velocity. It follows, therefore, 

 that any discussion given in terms of the simpler theory, 

 which assumes the electrons and atoms to be elastic spheres, 

 can be immediately generalized to the previous extent by 

 the proper choice of t,„, the mean time of description of a 

 free path for an electron with the typical velocity. We shall, 

 therefore, confine our remarks to this simpler case. 



We now turn to a consideration of the alternative but 

 necessarily equivalent method of attack in these problems 

 which is suggested by Thomson and Drude. This method is 

 equally effective in giving in a simple manner the most 

 general form for the velocity distribution, although, so far 

 as I am aware, it has never previously been used for that 

 purpose. 



In terms of the theory which regards the electrons and 

 atoms as hard elastic spheres, the fundamental assumption 

 underlying the theories of Thomson and Drude is that the 

 collisions of the electrons with the atoms completely oblite- 

 rate any regularity existing in the statistical motion of the 

 electrons, a steady state being attained when the organizing 

 effect of the external circumstances is just balanced by the 

 disorganizing collisions. The calculations on the basis of 

 this theory, therefore, depend essentially on the fact that 

 the distribution of velocities among any number of electrons, 

 when taken each immediately after its last collision just 

 preceding a certain instant, is precisely that specified by 

 * ' Dynamical Theory of Gases.' (Cambridge, 1904.) 



