RECENT STATISTICAL THEORIES 739 



members of this new equation over the entire velocity-space. The 

 integral on the right then vanishes by reason of (118), and for the 

 integrals on the left we have: 



a I dT(dfo/di) cos e + jdT^(dfo/dx) cos 6 = 0. (123) 



By obvious transformations we get: 



a j dr-^cos^ ^ ^T r^^/o^ cos2 ^ = 0. (124) 



Integrating over the angles: 



Leaving the second term as it is, but integrating the first by parts, 

 we find that as /o vanishes (whichever statistics we use) at one limit 

 and V at the other limit of integration, we get: 



-|a f4TrfoV-'-v'dv+^^ f iwfoV-v'dv = 0. (126) 



The integrals are written in this curious fashion, to bring out the 

 feature that they are proportional to the mean values of functions — 

 the functions v~'^ and v, respectively — averaged over the electrons in 

 question; which is to say, all the electrons contained in the compart- 

 ment dxdydz of coordinate-space, to which equation (117) has reference. 

 They are in fact equal to the products of these mean values, to wit 

 the mean reciprocal speed and the mean speed, by the number ndxdydz 

 of the electrons in the compartment dxdydz. Rewriting (125) accord- 

 ingly, with overlinings to signify averages, and dividing out the 

 factors dxdydz and 1/3, we get: 



— 2anv-^ -f d{nv)ldx = 0, (127) 



and this is the equation for the acceleration a or the accelerating 

 field E = male, required to counteract the electric current which 

 otherwise would be produced in the presence of the gradient d{nv)ldx 

 of the quantity nv. This is the gradient which evokes the hypothetical 

 electric field; gradients of temperature or of concentration act indi- 

 rectly, by making nv vary. 



With the classical statistics the development is extremely simple, 

 for V depends on temperature only, while n may be varied at will. 



