HALL. — ELECTRIC CONDUCTION AND THERMOELECTRIC ACTION. 75 



Now in place of the equation — = — -jjrf dl we have 



p R'T R'$ T' 



But p = R'Tinn, and so 



dp _ dR' dn dT _ f dT 

 ~p : : ~R ~w + ~f ~~ ~ W$ ' Y' 



dR' , dn ( f , AdT 



whence W + ^ = ~ (lTp + 1 JT> 



or, if we multiply both R' and /' by m, and so get R and / for a single 

 electron, 



dR dn- ff \dT _ 



When we observe that /, /?, and i? are all variables, the integra- 

 tion of this equation appears at first to present difficulties. But two 

 of our fundamental assumptions, expressed in equation (1), give 

 Rn = k T q , where A- and q are constants, and the only wav to make 



7/ 



this equation agree with equation (5) is to have the factor ( v*--. + 1 

 a constant. Thus we get by the integration of (5) 



Rn = JcT~(rp + 1 ) =1cT«, (6) 



and so 



n 2 /r-M 



(7) 



In dealing with our detached piece of wire I shall use F to indicate 

 virtual potential for a single electron, — that is, the total potential due 

 to attractions and repulsions of electric charges together with the 

 attractions of the metal atoms, 1 * as exerted on a single electron. If the 

 distance I is measured from the cold end of the wire, we have 



* dF , a dT f dF 



J = Yj, ancl P = -JT> so tnat q = -Jf- ( 8 ) 



We have seen that equation (6) is a necessary consequence of 

 equations (1) and (5). As (1) expresses merely certain fundamental 



14 If the assumption of an attraction of the metal for the electrons, as a 

 function of temperature, were omitted, the word virtual as applied to the 

 potential would be omitted. 



