Suppose that _q₀_ particles of new matter are produced per second from a given mass of matter. The rate of emission of energy due to the particles produced in the time _dt_, is, at the moment of their formation, equal to _Kq₀__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 _T_. The activity _dI_, due to the matter produced during the time _dt_ at the time _t_, decays according to an exponential law during the time _T_ − _t_ that elapses before its activity is estimated, and in consequence is given by $$ dI = Kq₀e^{–λ (T-t)} dt $$, where λ is the constant of decay of activity of the active matter. The activity _I_{T}_ due to the whole matter produced in the time _T_ is thus given by $$ I_t = \int₀^T Kq₀e^{–λ (T-t)} dt $$ $$ = \frac {Kq₀} {λ} (1 − e^{–λ T}) $$ . The activity reaches a maximum value _I₀_ when _T_ is very great, and is then given by _Kq₀_ _I₀_ = ----- λ thus $$ \frac {I_T} {I₀} = 1 − e^{–λ T} $$ . This equation agrees with the experimental results for the recovery of lost activity. Another method for obtaining this equation is given later in section 133. A state of equilibrium is reached when the rate of loss of activity of the matter already produced is balanced by the activity supplied 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 separated 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 may be. If _I₀_ is the initial activity of the separated product, the activity _I_{t}_ after an interval _t_ is given by $$ \frac {I_T} {I₀} = e^{–λt} $$ . Thus, the two assumptions—of uniform production of active matter and of the decay of its activity in an exponential law from the moment of its formation—satisfactorily explain the relation between the curves of decay and recovery of activity.