HALL. — ELECTRIC CONDUCTION AND THERMOELECTRIC ACTION. 85 



probably so. Accordingly we get, as the total energy of our - electrons 

 at temperature T of the metal, 



B ~ lj r(^ T+r+ *) + i(^ T+r ) (17) 



= I(| + X ^ RT+ l+\ (!_«)#. 



In looking for the amount of heat energy which is absorbed from the 



metal by our - electrons in their passage from T to T + dT through 

 e 



the metal, we must remember that x is a variable, increasing, accord- 

 ing to my assumption, with rise of temperature. We have 



(IE 

 adT = jfdT, 



whence 



a = e [_2 If {RT) + df (XRI) + dT + dT ~ ^rr} (18) 



It may be well just here to inquire whether the (B) electrons, 

 which we have found to be inadequate when taken alone, are needed 

 at all, — whether the (A) electrons acting alone would serve our 

 purpose. Let us accordingly assume, for the moment at least, that 

 v and x are each equal to zero. That part of F which is dependent 

 on the free electrons will also be zero in this case, while the other part 

 of F, the part depending on the attraction of the atoms as a function 

 of temperature, will be included in <J>, provided (— <£) is now defined 

 as the gain of potential energy of an electron in being taken from an 

 atom in the metal to a point outside the metal. Accordingly we get 

 from (18) 



a = I [jT (1 + p) T " + '^ Tiw ~ n ] ( 19 > 



The first term within the brackets is + , according to our assumption 

 regarding R as a function of T. In the second term the factor h^ is — , 

 as we have seen before, but the factor w is also — , if, as seems probable, 

 $ diminishes numerically with rise of temperature. We seem, then, 

 with (19) to have no provision for negative values of a, such as, we 

 know, occur in metals. 



