142 
PHYSICS: E. H. HALL 
Proc. N. a. S. 
energy, which affects free electrons only; (5) the Pa potential energy, 
which affects the associated electrons only. We must take account also 
of the gain of energy involved in the ionization which may occur in the 
current from T to T -\- dT, due to the increase in the ratio {kf -^ k) with 
rise of temperature. 
Under hypothesis {A): All changes due to change of {kf -^ k) being 
considered last, we have as the change of pv energy, (see equation (3)) 
{kf k){m ^ e)d{pv) = {kf k){R ^ e)dT. (a) 
The gain of thermal kinetic energy by the free electrons is 
{kf -^ k){l -^ e).^-RdT. {h) 
The gain of P potential energy is dP. {c) 
The gain of Pf potential energy is {kf k)dPf. {d) 
The gain of Pa potential energy is {ka k)dPa {e) 
The gain of energy through ionization is 
\d{kf ^ k) = {I e)yd{kf - k) . (/) 
The sum of all these quantities is the Thomson-effect heat absorbed 
between T and T -\- dT; that is 
adT = (a) + (6) + {c) + {d) + {e) + (/) (5) 
From this we get 
^k_f 5R Vkf dPf k_a dPa dPl y d{kf -f- k) 
k ' 2 e Ik ' dT ^ k ' dT ~^ drj ^ e ' dT ' ^ ^ 
From the conditions of equilibrium, under hypothesis {A), in a detached 
bar like CH we have^ 
Subtracting (7) from (6) we get 
_ kf 3_R _kj RT dn y d{kf -^ k) 
~ k '2 e k ' ne 'dT^ e ' dT ' 
Under hypothesis {B) we get 0.5R instead of the first R in (7), and this 
gives 2R instead of 1.5R in the first term of (8). Hypothesis {B) makes 
no other change in (7) or (8). 
I shall now try to put equation (8) into a form suitable for dealing with 
the values of a found by Bridgman^ in his experiments on a large number 
of metals. In this undertaking I assume that for present purposes the 
following equations hold above 0° centigrade: 
n = zT^, (9) 
and 
{kf-r-k)=C + Cit + C2t^ (10) 
where q, C, Ci and C2 are constants, and / is temperature on the ordinary 
centigrade scale. 
