100 
PHYSICS: E. H. HALL 
obliged to suppose a dead-lock between the potential gradient and this differ- 
ential specific attraction. The conception of electron circulation, with thermal 
conduction (or convection), in a detached unequally heated bar survives the 
admission of specific attraction; but the whole matter now becomes more 
involved. 
In addition to the potential, P, due to electric charge, we must now think 
of a potential, P a , due to the differential attraction of the unequally heated, 
unhomogeneous, metal for the associated electrons, and also of a potential, 
Pf, due to the differential attraction of the metal for the free electrons. 
Both classes of molecules are subject to the charge-potential P, but elec- 
trons (A) only are subject to the potential P a , and electrons (F) only are 
subject to the potential Pf. 
Under hypothesis (A): If we assume, as hypothesis (A), that the mechanical 
tendency of the free electrons, if acting without electric forces, would produce 
equality of pressure from end to end of the bar, the condition of equilibrium 
(see fig. 1) in a detached bar hot at one and cold at the other is 
g.fl + d{P + Fb) .nedl) + nmdAne = - (1) 
dl dl J J dl 
where (dp/dl) is the gradient of mechanical pressure of the free electrons 
along the bar of length /, n is the number-density of the free electrons in the 
metal, m is the mass and e the charge of an electron, \x is the coefficient of 
mobility of the free electrons through the metal, and k a is the electric con- 
ductivity of the metal, so far as conductivity is due to the electrons (A). 
A simple formula, 
\_ 
where G is (1/m), connects /j, with the free-electron specific conductivity, 
kf. Evidently 
G — nv, (3) 
where v is the volume of one gram of free electrons in the metal. 
Very simple operations, using equations (2) and (3), derive from (1) the 
form 
r*h 
- -^-.dP a = ~- T~.vdp, (4) 
kf Ge J k a + kf 
*. + *, J k K + 
in which the integration extends from the hot end (h) to the cold end (c) 
of the bar. 
The first member of this equation is the amount of reversible work that 
would be done by or on the unit quantity of electricity in passing through the 
bar, if it were made part of a closed thermo-electric circuit. This is something 
different from, probably smaller than, (P c — P h ), which is the charge poten- 
