2(30 
MR. J. S. TOWNSEND ON THE DIFFUSION OF IONS PRODUCED IN AIR 
number of ions which come through the tube without touching the sides to the 
© © 
number which enter is—- 
It = -193E 
7-31KZ -44-5KZ 
+ -0243E + &c. 
( 1 ). 
where K is the coefficient of diffusion of the ions into the gas. B, Z the length of the 
tube, a its radius, and V the mean velocity of the gas, B, along the tube. This 
value of B is obtained on the supposition that the velocity, W, of the gas, B, at a 
distance r from the axis of the tube is given by the formula W = 2Y(ar — r~)/a~. 
Let Pij and IL be the values of ft equation (1) corresponding to lengths Z : and 
Z 2 , then y — Bj/IL is the ratio of the number of ions which come through tubes of 
lengths Z| and Z : , the same uniform distribution entering each. 
The ratio y can be easily determined experimentally, and the value of Iv can be 
found from the formula— 
y = .(2)> 
if the loss of conductivity arises principally from diffusion. 
Before we proceed to estimate the loss due to recombination or mutual repulsion, 
we may here point out the advantages of having a small density of ionisation, and 
of using tubing of small bore. 
The ratio y, equation (2), is independent of the number of ions used. It shows 
that the proportion of ions which are lost in passing through the tube is the same 
for large and small conductivities. When reduction in conductivity is caused by 
recombination or mutual repulsion, this law no longer holds. In both these cases 
the absolute loss is proportional to the square of the number present, so that the 
proportion of ions lost is proportional to the density of ionisation. 
The ratio, y , is a function of KZj/2crY, when Z 1 /Z. : is constant. Let c be a value of 
KZ,/2rUV which gives the value of y between '3 and ’6 that can be found accurately 
by experiment. If the mean time, Z t /Y, during which the gas is in the longer tube 
be given, the radius can be selected so as to make KZ 1( /2a 2 Y = C. In dealing witli 
the corrections to be applied to y, we need only to consider the effect of recombination 
or mutual repulsion in the longer tube. In the apparatus which was used, Z L = 4 
centims., Z 2 = *5 centim. The processes which give rise to errors take place during 
the time Z^V, and the rate at which they destroy the conductivity is independent of 
the radius of the tubing. The errors therefore arising from either of the processes 
which we are considering can be reduced to any desired extent by reducing the 
density of ionisation, or reducing the bore of the tubing. 
Recombination. 
Among the methods of producing conductivity with which we are dealing, it is 
only in the case of ionisation produced by the radio-active substance that both 
