HALL. — ELECTRIC CONDUCTION AND THERMOELECTRIC ACTION. 77 



The differential expression of this law is 



pdR pdT 

 dp = | -g + g ~f (1 °) 



That is, the difference of pressure which the free electrons in a stratum 

 of thickness dl can bear without drift, and without the help of electrical 

 force, is 



pdR pdR 



2 R + 2 T' 



If the difference of pressure on the two faces of the stratum is greater 

 than this quantity, we must, in order to have equilibrium, balance the 

 excess of dp by the electrical force —fndl acting up the temperature 

 incline. Thus we get 



pdR pdT 



whence, as 

 we have 



n = p + RT, and = dT -f- eH, 



dp /_ f/7 7 1 dR dT 

 p R(3' T ' - R + 2 7' * 



As ;j — »i?r, we get from this equation, by substituting for p and dp, 



1 dRcIn__ff \dT ■ 



Now the assumptions which give equation (1) give also the equation 



f#tt = ki T l x k n T"= h'TV, (1') 



and in order to make (5') and (1') agree we must, as in the case of 

 equation (5), treat (J -f- R /?) as a constant. Accordingly we have 



Ri n =k r ' r~fe+*) =JfeT«', (6') 



where fc' and q' are constants. 



Not knowing which of the two conditions, (9) or (9'), represents the 

 gas-pressure tendency of the free electrons the more nearly, I shall 

 try each of the two corresponding equations, (6) and (6'), in turn. 

 But before doing this, I must call attention to the fact that neither 

 form of this tendency can make any great change in what we may call 

 the natural values of n\ and »2, — that is, the values which would be 

 found in two detached pieces of the same metal, one at 7\, the other 

 at T-2. For, if the number of the free electrons is not very small 



