30 
PHYSICS: E. H. HALL 
I shall in the present paper discuss the matter on the assumption that the 
'free' electrons within the metals are the only ones moving progressively in 
the maintenance of a current and the only ones taking part in thermo-electric 
action. For the sake of generality I shall not assume that these electrons 
have the same kinetic energy as gas molecules or even have the same 
kinetic energy in different metals at the same temperature. In fact, I shall 
in using the equation 
pv = CRT, (1) 
where v is the volume of a gram of free electrons within a metal and G is 
the number of electrons in this gram, assume that R may be a function of T 
and may vary from metal to metal. I shall follow common practice in sup- 
posing n, the number of free electrons per unit volume, to be greater in 
some metals than in others and to increase in all metals with rise of temper- 
ature. I shall omit, as being unnecessary for a general explanation of 
thermo-electric action, the assumption of a specific attraction of metals for 
electrons. 
In a detached bar of metal a, kept hot at one end and cold at the other, 
there is a higher free-electron pressure at the hot end than at the cold end, 
partly because of the greater value of n, partly because of the higher average 
heat energy of the electrons, at the hot end. Hence there is a mechanical 
tendency of the free electrons to move toward the cold end of the bar. But 
it cannot be said with confidence that this tendency, if unrestricted, would 
result in a condition of equal pressure from end to end of the bar; for it is 
quite possible, as I have pointed out in a previous paper, 1 that it would pro- 
duce the condition of equilibrium which holds in 'thermal effusion. ' 
I shall make use in turn of two alternative hypothesis; 
(A) That the mechanical tendency is toward equality of pressure, repre- . 
sented by the equation 
11R T = a constant, 
from end to end of the bar. 
(B) That the mechanical tendency is toward thermal effusion equilibrium, 
represented by the equation 
n (RT)* = a constant, 
from end to end of the bar. 
Under neither hypothesis will there be any sufficient movement of electrons 
to produce an appreciable alteration of n from its natural value at any point, — 
that is, the value it would have at this point if the whole bar were at the tem- 
perature of this point. For the free electrons are mingled, naturally, with 
an equal number of positive metal ions which are not free to move through 
the bar, and therefore these electrons are powerfully restrained from any 
large movement down the pressure gradient. The proper distribution of an 
