There is also another case of importance which is practically a converse of Case 3, viz. when the matter _A_ is supplied at a constant rate from a primary source and the amounts of _A_, _B_, _C_ are required at any subsequent time. The solution of this can, however, be deduced immediately from Case 3 without analysis. =197.= CASE 1. _Suppose that the matter initially considered is all of one kind A. It is required to find the number of particles P, Q, R of the matter A, B, C respectively present after any time t._ Then $$ P = ne^{–λ_1 t} $$, if _n_ is the number of particles of _A_ initially present. Now _dQ_, the increase of the number of particles of the matter _B_ per unit time, is the number supplied by the change in the matter _A_, less the number due to the change of _B_ into _C_, thus _dQ_/_dt_ = λ₁_P_ − λ₂_Q_ (1). Similarly _dR_/_dt_ = λ₂_Q_ − λ₃_R_ (2). Substituting in (1) the value of _P_ in terms of _n_, $$ \frac {dQ} {dt} = λ_1 ne^{–λ_1 t} − λ_2 Q $$ . The solution of this equation is of the form $$ Q = n (ae^{–λ_1 t} + be^{–λ_2 t}) $$ ......(3). By substitution it is found that _a_ = λ₁/(λ₂ − λ₁). Since _Q_ = 0 when _t_ = 0, _b_ = –λ₁(λ₂ − λ₁). Thus $$ Q = \frac {nλ_1} {λ_1 − λ_2} (e^{–λ_2 t} - e^{–λ_1 t}) $$ .... (4). Substituting this value of _Q_ in (2), it can readily be shown that $$ R = n (ae^{–λ_1 t} + be^{–λ_2 t} + ce^{–λ_3 t}) $$ ..... (5). where $$ a = \frac {λ_1 λ_2} {(λ_1 − λ_2) (λ_1 - λ_3)} $$, $$ b = \frac {- λ_1 λ_2} {(λ_1 − λ_2) (λ_2 − λ_3)} $$, $$ c = \frac {λ_1 λ_2} {(λ_1 − λ_3) (λ_2 - λ_3)} $$, [Illustration: Fig. 72.] The variation of the values of _P_, _Q_, _R_ with the time _t_, after removal of the source, is shown graphically in Fig. 72, curves _A_, _B_, and _C_ respectively. In order to draw the curves for the practical case which will be considered later corresponding to the first three changes in radium _A_, the values of λ₁, λ₂, λ₃ 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.