HALL. — ELECTRIC CONDUCTION AND THERMOELECTRIC ACTION. 69 



molecules. 4 This innovation seems to be justified by its success 

 in meeting certain requirements of the situation. 



The conclusion to which my reflections, carried on with the help of 

 some mathematical machinery, have led me is, that the free electrons, 

 though present and essential for thermoelectric action, are of relatively 

 small importance in mere electric conduction. 



Fundamental Assumptions. 



I shall assume that n, the number of free electrons per cu. cm. of 

 the metal at any temperature, T absolute, can be expressed as n = 

 k n T", and that R, of the free-electron gas-equation pv = R T, " reck- 

 oned for a single electron," 5 can be expressed as R= Jc r T p ,k n , u, k„ 

 and p being constants. These two assumptions give 



Rn = k T T" X k n T> = lcT«, (1) 



where k and q are constants. 



For a single molecule of an ordinary gas the value of R is about 

 137 X 10 -18 . I assume that R for an electron has a very much smaller 

 value than this at low temperatures. 



Whether any metal really satisfies the conditions indicated by (1) 

 through any great range of temperature may well be doubted; but, 



4 The "law of equipartion of energy," a law more familiar, perhaps, in the 

 breach than in the observance, doubtless requires that electrons acting as gas 

 particles among other gas particles of a different class shall attain the same 

 mean translators energy as the latter, provided the two classes of particles 

 remain distinct from each other in their encounters. But if electrons collide 

 with metal atoms containing or made up of electrons, and if during a collision 

 it frequently or usually happens that the electron enters an atom and stays 

 there, displacing another electron, there seems to be no reason for supposing 

 that the mean translatory energy of the free electrons will equal that of the 

 atoms. 



5 Let us, for one gram of electrons, write pv = R'T. Then, taking m as the 

 mass per electron and c as the "velocity of mean square," we have 



p = $ mm 2 = R'T + v = R'T mn. 

 From these relations we get 



p = n{R'm)T = nRT, 

 an expression which will be used frequently hereafter, and 



cy.(RT)l. 



This proportionality must replace in this paper the simpler relation, cxT 1 ', 

 which holds for an ordinary gas. This substitution is highly important. 

 For example, if p of equation (1) is 1, so that Rs.T, we have cccT, and so 

 §mc 2 <x T 2 ; that is, the thermal capacity of the electron isoc T, and its heat- 

 energy content ce T 2 , if p = 1. 



