728 BELL SYSTEM TECHNICAL JOURNAL 



or a temperature-gradient, or which varies in its chemical nature 

 from place to place, as might an alloy. If in such a case we orient 

 the axis of x parallel to the gradient of the variable quantity — be it 

 electric potential, temperature, or whatever else — we must expect ^ 

 to enter differently into the distribution-function from -q and $. 



Various arguments show that as a rule the departures from the 

 standard function must be rather small. Lorentz therefore postulated 

 that in the presence of a gradient directed parallel to the :r-axis, 

 the actual distribution should differ from the standard function fo {v) 

 — this he of course assumed to be Maxwell's — by virtue of a small 

 additive term, a new function of v multiplied by the velocity-compo- 

 nent ^: 



f=U{v) + ^g{v), (92) 



and he proceeded to determine the new function g by the condition 

 that / should remain constant in time despite the collisions of the 

 electrons with the atoms. More precisely, he found for each of the 

 three cases with which we shall be concerned a function g, such that 

 the distribution-function obtained by adding ^g to /o conforms to 

 that condition. This justifies the procedure. 



Much the simplest case of the three is that of a uniform metal at a 

 uniform temperature, subject to an electric field; for here the distri- 

 bution-function need not vary from place to place. It will be well 

 to go through the reasoning in this instance, though the formula for 

 conductivity in which it leads differs but little from (91). 



It is required, to find a function g of the combination (^^ -j_ ^2 _|_ ^2)1/2 

 such that if at any moment the distribution (/o + H) prevails, it 

 continues unchanged throughout time — the number of particles in 

 any compartment or cell of the velocity-space (the momentum-space 

 of the earlier pages, with its unit of length altered in the ratio m : \) 

 stays constant. Choose a compartment enclosed between planes ^ 

 and ^ -\- d^, r] and r? + rfr?, f and f + d^. Call it the compartment C. 

 Its volume is d^drjd^, which to save a profusion of Greek letters I 

 will usually denote by dr. The number of particles in it is f-dr. 

 This number must remain unchanged, though individual particles 

 are constantly moving into and out of C in either of two ways — by 

 "drift" and by "collision." 



Owing to the field E, all of the particles have a steady acceleration 

 a = eE/m, because of which they are continually and continuously 

 drifting from cell to cell. One easily sees that the number which 

 thus drift out of C per unit time (it is best to think of "unit time" 

 not as one second, but as a period small compared with the mean 



