radio-active equilibrium is produced, is equal to λN_{0}. This must be equal to the number q_{0} of new systems applied in unit time, or
q_{0} = λN_{0},
and λ = q_{0}/N_{0};
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 q_{0} 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 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/dt = q_{0} - λN.
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 N_{0}.
Thus, since N = N_{0} when t = [infinity],
b = N_{0};
since N = 0 when t = 0,
a + b = 0;
hence b = -a = N_{0},
and the equation becomes
N/N_{0} = 1 - e^{-λt}.
This is equivalent to the equation already obtained in section 130,
