---- = -λNt,
dt
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 λN₀. This must be equal to the number q₀ of new systems applied in unit time, or
q₀ = λN₀,
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 λN, 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