SO PROCEEDINGS OF THE AMERICAN ACADEMY. 



We have to consider the passage of electrons through the region 

 lying between two parallel plane surfaces, each of which is isothermal 

 and also equipotential, the temperatures being T and T -\- dT, re- 

 spectively. We have to take account of three changes in the energy 

 condition of the electrons during that passage, change of kinetic 

 energy due to the change of temperature, change of potential energy of 

 surrounding electron gas pressure, change of potential energy F due 

 to attractions and repulsions. If R were constant, the first two 

 changes together would equal the mass of the electrons multiplied by 

 the rise of temperature and by the specific heat, C p , of the electron gas, 

 at constant pressure. We may indeed, if we please, use the terms 

 C v and C p ; but, as neither of the specific heats is a constant when R 

 is variable, we should find them of no great service. It is to be noted, 

 however, that, even when R is variable, we have, c being velocity of 

 mean square of the electrons and energy being reckoned in ergs, 



kinetic energy per gram — \ vmnc z , 



R7 1 



and pv " " " = = \mnch, 



m 



whence Ice. " " " = f — , (11) 



m 



Ice. per electron = f RT, (12) 



pv energy per electron = RT. (13) 



When, therefore, we have the electromagnetic unit quantity of 



electricity in the form of - electrons entering the region in question at 



the T surface and issuing at the T -\- dT surface, we have 



ti 3 



gain of Ic c. energy = — 



(RT) - (RT) 



T+dT T 



3 f R+Tf\lT, 



2e 



gain of pv energy = - ( R + T -j^jdT, 



(IF 1 

 gain of F energy (see eq. (S)) = - = -(/ -4- /3) dT. 



The total gain of energy by the electrons, which must equal the 

 heat energy absorbed from the metal between the two planes T and 



T + dT, is 



5 f„ JtR\ ;„ . 1 



cdT = fill + T— dT + - (/ - 0)dT, (14) 



