PHYSICS: E. H. HALL 
167 
r, the mean life of an electron in the free state, being terminated only 
by collision with an ion, is proportional to {T c) -7- n; that is, 
T = q T -T- n c, (1) 
where g is a constant. 
Let P = the number of free electrons produced per unit volume per 
second. They are supposed to be produced by collisions, — usually 
by collisions of atoms with atoms, though sometimes by collisions of * 
atoms with ions. 
A little consideration shows that 
Pt== n, or P = n-T-T = n'^c-T-qT. (2) 
Let B = the total number of ion heat-paths begun per unit volume 
per second. 
Let F = the total number of ion heat-paths finished, to collision with 
atom or ion, per unit volume per second. 
F < B, because of collisions like the one here indicated (fig. 3), in 
which the ion captures a free electron and ceases to be an ion before it 
reaches the atom it is approaching. 
FIG. 3 
Let d = the mean duration of one heat-path of an atom or an ion. 
It is inversely proportional to the 'proper frequency' of atomic vibra- 
tion and may therefore be taken as independent of the temperature. 
If an ion could cease to be such only by collision with a free electron, 
the mean length of life of the ions would be the same as r, the mean 
length of life of the free electrons. But an ion may cease to be such by 
collision with an atom, and the time required for its approach to such a 
collision is 0. Hence the probability^ that an ion will survive as such 
till it has collision with an atom is e~^^^. Accordingly we have 
F = Be-^^' ' (3) 
We have now to consider various types of collisions, between atoms, 
between atoms and ions, or between ions. Some of these may yield 
free electrons. 
Since all the types of colKsion in which ions figure are here shown, we 
have 
B = P + F (4) 
