or the number of systems changing in unit time is proportional to the number unchanged at that time. In the case of recovery of activity, after an active product has been removed, the number of systems changing in unit time, when radio-active equilibrium is produced, is equal to λ_N₀_. This must be equal to the number _q₀_ of new systems applied in unit time, or _q₀_ = λ_N₀_, _q₀_ and λ = ------; _N₀_ λ has thus a distinct physical meaning, and may be defined as the proportion of the total number of systems present which change per second. It has different values for different types of active matter, but is invariable for any particular type of matter. For this reason, λ will be termed the “_radio-active constant_” of the product. We are now in a position to discuss with more physical definiteness the gradual growth of Th X in thorium, after the Th X has been completely removed from it. Let _q₀_ particles of Th X be produced per second by the thorium, and let _N_ be the number of particles of Th X present at any time _t_ after the original Th X was removed. The number of particles of Th X which change every second is λ_N_, where λ is the radio-active constant of Th X. Now, at any time during the process of recovery, the rate of increase of the number of particles of Th X = the rate of production − the rate of change; that is _dN_ ------ = _q₀_ − λ_N_. _dt_ The solution of this equation is of the form $$ N = ae^{–λt} + b $$, where _a_ and _b_ are constants. Now when _t_ is very great, the number of particles of Th X present reach a maximum value _N₀_. Thus, since _N_ = _N₀_ when _t_ = infinity, _b_ = _N₀_; since _N_ = 0 when _t_ = 0, _a_ + _b_ = 0; hence _b_ = -_a_ = _N₀_, and the equation becomes $$ \frac {N} {N₀} = 1 − e^{–λt} $$ . This is equivalent to the equation already obtained in section 130, since the intensity of the radiation is always proportional to the number of particles present. =134. Influence of conditions on the rate of decay.= Since the activity of any product, at any time, may be taken as a measure of the rate at which chemical change takes place, it may be used as a means of determining the effect of conditions on the changes occurring in radio-active matter. If the rate of change should be accelerated or retarded, it is to be expected that the value of the radio-active constant λ will be increased or decreased, _i.e._ that the decay curve will be different under different conditions.