changes in radium A, the values of λ_{1}, λ_{2}, λ_{3} were taken as 3·85 × 10^{-3}, 5·38 × 10^{-4}, 4·13 × 10^{-4} respectively, i.e., the times required for each successive type of matter to be half transformed are about 3, 21, and 28 minutes respectively. The ordinates of the curves represent the relative number of atoms of the matter A, B, and C existing at any time, and the value of n, the original number of atoms of the matter A deposited, is taken as 100. The amount of matter B is initially zero, and in this particular case, passes through a maximum about 10 minutes later, and then diminishes with the time. In a similar way, the amount of C passes through a maximum about 37 minutes after removal. After an interval of several hours the amount of both B and C diminishes very approximately according to an exponential law with the time, falling to half value after intervals of 21 and 28 minutes respectively. 198. Case 2. A primary source supplies the matter A at a constant rate and the process has continued so long that the amount of the products A, B, C, . . . has reached a steady limiting value. The primary source is then suddenly removed. It is required to find the amounts of A, B, C, . . . remaining at any subsequent time t.
In this case, the number n_{0} of particles of A, deposited per second from the source, is equal to the number of particles of A which change into B per second, and of B into C, and so on. This requires the relation
n_{0} = λ_{1}P_{0} = λ_{2}Q_{0} = λ_{3}R_{0} (6),
where P_{0}, Q_{0}, R_{0} are the maximum numbers of particles of the matter A, B, and C when a steady state is reached.
The values of P, Q, R at any time t after removal of the source are given by equations of the same form as (3) and (5) for a short exposure. Remembering the condition that initially
P = P_{0} = n_{0}/λ_{1},
Q = Q_{0} = n_{0}/λ_{2},
R = R_{0} = n_{0}/λ_{3},
