and the recovery curve by
where λ is the radio-active constant of the emanation.
This relation is to be expected, since the decay and recovery curves of the emanation are determined by exactly the same conditions as the decay and recovery curves of Ur X and Th X. In both cases there is:
(1) A supply of fresh radio-active particles produced at a constant rate.
(2) A loss of activity of the particles following an exponential law with the time.
In the case of Ur X and Th X, the active matter produced manifests its activity in the position in which it is formed; in this new phenomenon, a proportion of the active matter in the form of the emanation escapes into the surrounding gas. The activity of the emanation, due to a thorium compound kept in a closed vessel, thus reaches a maximum when the rate of supply of fresh emanation particles from the compound is balanced by the rate of change of those already present. The time for recovery of half the final activity is about 1 minute, the same as the time taken for the emanation, when left to itself, to lose half its activity.
If q₀ is the number of emanation particles escaping into the gas per second, and N₀ the final number when radio-active equilibrium is reached, then ([section 133]),
q₀ = λN₀.