170 
PHYSICS: E. H. HALL 
From (5) and (6) we get F = an? P (e^^^ ^ - 1), (7) 
where a = (y q) and b = (yd q), both constant. 
We next assume that the relation of to T is of the form 
n =^ kV (8), where k and i are constants. 
From (7) and (8) comes F = ak'P''-'-'^ - (e^^n^'-o-^) _ i). (9) 
where b k T (i — 0.5), the exponent of e, is merely another expression 
for the original -^ r). 
The lowest temperature to be considered will be called To, and it will 
be defined as the temperatures at which F = ^ B, which means that 
only one-half of the ion heat-paths begun are completed by the ions as 
such.^ 
d_ 
This makes = 2, and so F = a¥ T^^* ^'^^ . Higher temperatures 
will be expressed as multiples of Tq. 
Giving to i the values 0, 0.125, 0.25, 0.50, 1.00 and 1.50, in turn, 
we get from (9) values of F which are indicated by the curves in figure 5. 
If, instead of assuming c X T*, as in (6) we take c ^ T, as in (6'), 
we get, by making i = 0, 0.125, 0.25, 0.50, and 1.00, in turn, values 
of F which are indicated by the curves, in figure 6. 
I have sought to account for the temperature relations of conductiv- 
ity by combining with the considerations just presented the hypothesis 
of an atomic vibration consisting of a simple harmonic motion prema- 
turely ended by collision of atom with atom or with an ion. This at- 
tempt, though not entirely successful, seemed not altogether hopeless 
until the experiments of Bridgman showed the atoms to be far less rigid 
than the kind of vibration in question required them to be. 
1 It should be, according to Conlomb's law, about 5 X 10"^ dyne, corresponding to a 
potential gradient of about 3 X 10^ volts per centimeter. 
2 This is a doubtful proposition, and possibly some ingenious variation of it would make 
the ion theory more successful in dealing with the temperature relation. 
3 The question here is analogous to the inquiry what is the probability that a gas mole- 
cule will go a distance x without collision, the mean free path being l; d corresponds to 
X and T to /. 
^ Jeans, J. H., Dynamical Theory of Gases, 1916, §559. 
^ This does not imply a high degree of ionization. If we assume the electrons to have a 
mean kinetic energy equal to that of gas molecules at the same temperature, the mean 
velocity of the electrons will be about 350 times as great as that of the atoms. Hence, if 
the mean free path of an electron is equal to the mean heat path of an atom or ion, a single 
electron will have about 350 collisions during the time d. If g be the degree of ionization,— 
that is, the ratio of n to the number of atoms per unit volume, — the total number of col- 
lisions per second of electrons with ions will be (350 -f- d) n g. This is the number of ion 
heat-paths interrupted per second per unit volume by the free electrons. The total number 
