184 CONTINUOUS PRODUCTION OF RADIO-ACTIVE MATTER [CH. 
to the particles produced in the time df, is, at the moment of their 
formation, equal to Kq dt where K is a constant. 
It is required to find the activity due to the whole matter 
produced after the process has continued for a time 7. 
The activity dZ, due to the matter produced during the time dé 
at the time t, decays according to an exponential law during the 
time 7’—t that elapses before its activity is estimated, and in 
consequence is given by 
dI = Kqe-* dt, 
where ) is the constant of decay of activity of the active matter. 
The activity 7; due to the whole matter produced in the time 7’ is 
thus given by 
The activity reaches a maximum value J, when T is very great, 
and is then given by 
pet 
nr 
Thus i = mae 
This equation agrees with the experimental results for the 
recovery of lost activity. 
A state of equilibrium is reached when the rate of loss of 
activity of the matter already produced is balanced by the activity 
supphed by the production of new active matter. According to 
this view the radio-active bodies are undergoing change, but the 
activity remains constant owing to the action of two opposing 
processes. Now if this active matter can at any time be sepa- 
rated from the substance in which it is produced, the decay of 
its activity, as a whole, should follow an exponential law with 
the time, since each portion of the matter decreases in activity 
according to an exponential law with the time, whatever its age 
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