=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 _n₀_ 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 _n₀_ = λ₁_P₀_ = λ₂_Q₀_ = λ₃_R₀_ (6), where _P₀_, _Q₀_, _R₀_ 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_ = _P₀_ = _n₀_/λ₁, _Q_ = _Q₀_ = _n₀_/λ₂, _R_ = _R₀_ = _n₀_/λ₃, it can readily be shown that $$ P = \frac {n₀} {λ_1} e^{–λ_1 t} $$ .... (7), $$ Q = \frac {n₀} {λ_1 − λ_2} (\frac {λ_1} {λ_2} e^{–λ_2 t} − e^{–λ_1 t}) $$ .... (8), $$ R = n₀ (ae^{–λ_1 t} + be^{–λ_2 t} + ce^{–λ_3 t}) $$ .... (9), where $$ a = \frac {λ_2} {(λ_1 − λ_2) (λ_1 - λ_3)} $$, $$ b = \frac {–λ_1} {(λ_1 − λ_2) (λ_2 - λ_3)} $$, $$ c = \frac {λ_1 λ_2} {λ_3 (λ_1 − λ_3) (λ_2 − λ_3)} $$ . [Illustration: Fig. 73.] The relative numbers of atoms of _P_, _Q_, _R_ existing at any time are shown graphically in Fig. 73, curves _A_, _B_, _C_ respectively. The number of atoms _R₀_ is taken as 100 for comparison, and the values of λ₁, λ₂, λ₃ are taken corresponding to the 3, 21, and 28-minute changes in the active deposit of radium. A comparison with Fig. 72 for a short exposure brings out very clearly the variation in the relative amounts of _P_, _Q_, _R_ in the two cases. Initially the amount of _R_ decreases very slowly. This is a result of the fact that the supply of _C_ due to the breaking up of _B_ at first, nearly compensates for the breaking up of _C_. The values of _Q_ and _R_ after several hours decrease exponentially, falling to half value in 28 minutes. =199.= CASE 3. _Suppose that a primary source has supplied the matter A at a constant rate for any time T and is then suddenly removed. Required the amounts of A, B, C at any subsequent time._ Suppose that _n₀_ particles of the matter _A_ are deposited each second. After a time of exposure _T_, the number of particles _P_{T}_ of the matter _A_ present is given by $$ P_T = n₀ \int₀^T e^{–λ_1 t} dt = \frac {n₀} {λ_1} (1 - e^{–λ_1 T}) $$ . At any time _t_, after removal of the source, the number of particles _P_ of the matter _A_ is given by $$ P = P_T e^{–λ_1 t} = \frac {n₀} {λ_1} (1 − e^{–λ_1 T}) e^{–λ_1 t} $$ .