at Low Pressures. 
259 
The theory of the Hall effect in salt solutions has been worked 
out by Donnan (Phil. Mag., Nov. 1898), and Larmor ( Aether and 
Matter, p. 301) gives the theory for a completely dissociated salt 
solution. 
In air at low pressures the amount of dissociation is relatively 
very small and the ionisation does not depend merely on the 
concentration as in salt solutions. 
Nevertheless it is easy to show that, provided the influence of 
the walls of the discharge tube is neglected, Larmor’s theory for 
completely dissociated salt solutions applies without modification 
to the uniform positive column. 
Let v x (= k x X) be the velocity along the tube of the positive 
ions, and v 2 (= k 2 X) that of the negative ions. 
Then if i is the current density at any point 
i — Xe (k-giry + k 2 n 2 ), 
where n x and n 2 are the numbers of positive and negative ions 
in unit volume respectively, and e is the charge carried by 
an ion. 
Let the axis of the discharge tube AB be denoted by x , and 
the perpendicular horizontal direction of the Hall effect by z. 
dX 
Then — 0, and since Z is small and probably uniform, we 
may put ^ = 0, whence n 2 , and i = Xne (k x + k 2 ). 
CLZ 
In the uniform positive column the ionisation is everywhere 
equal to the recombination or q= an 2 where q is the rate of 
ionisation and a the constant of recombination. Now the most 
important peculiarity of the positive column is that X is inde- 
pendent or nearly so of i, so that 
q = an 2 = a 
X 2 e 2 (k x + k 2 ) 2 
is simply proportional to the square of the current density. Hence 
if owing to the Hall effect the transverse distribution of the 
current is changed the ionisation will adjust itself automatically 
to the new conditions and still be everywhere equal to the recom- 
bination. 
The ionisation and recombination may therefore be neglected 
and the calculation becomes identical with that for a completely 
dissociated salt solution. The result for this case as Larmor 
shows ( loc . cit.) is 
zAHX(h-h). 
If z is reckoned positive in the direction in which a current going 
