736 BELL SYSTEM TECHNICAL JOURNAL 



range of position, coming into dxdydz from the region adjacent to it 

 on the side towards smaller or greater values of x, according as ^ is 

 positive or negative. By the same sort of reasoning as led to the 

 term cxdf/d^ in (93), one sees that the net rate at which ds loses electrons 

 in this way is ^df/dx. Equating the two, we have: 



^df/dx ^ b -a = av/l)g(v), (116) 



and when we put/o for /as before, on the ground that the term (^g) in 

 the full expression for /makes but a small contribution to the left-hand 

 member, we have all that is required for computing g from (116) 

 and PF from (115) for whatever standard function we elect. 



At this point, however, there arises a difficulty. If having adopted 

 this way of determining g we proceed to compute the electric current 

 / in the metal by formula (110) we find that it is not zero. The 

 reasoning has led to the conclusion that wherever there is a net current 

 of heat in a metal, there is also a net current of electricity. This 

 conclusion is not in accord with experiment. Yet there is apparently 

 no other way to circumvent it, than to suppose that when a gradient 

 of temperature is maintained in a metal there arises a spontaneous 

 internal electric field, of just such a magnitude as to counteract the 

 electric current which would otherwise persist. The gradient of 

 temperature calls forth a gradient of potential; the actual distribution- 

 function is the one which is stable under both these gradients com- 

 bined. In the bookkeeping of the compartment ds, the net gain 

 from impacts (b — a) is balanced against the sum of the net loss 

 through drift in the coordinate-space (^df/dx) and the net loss through 

 drift in the velocity-space {adf/d^). Putting these statements into 

 the form of equations, and denoting by E the hypothetical electric 

 field and by a(= eE/m) the acceleration which it imparts to each 

 electron, we have: 



aidf/dO + m/dx) =b -a = Uvll)g (117) 



J/e = j JJd^dvdteg = 0, (118) 



a pair of equations for determining E and the function g. 



Lorentz, adopting the Maxwell-Boltzmann function for /o, solved 

 the equations, and obtained: 



w 8 nlk'T (dT\ \ k dT 



