Suppose that P, Q, R represent the number of particles of the matter A, B, and C respectively at any time t. Let λ_{1}, λ_{2}, λ_{3} be the constants of change of the matter A, B, and C respectively. Each atom of the matter A is supposed to give rise to one atom of the matter B, one atom of B to one of C, and so on. The expelled "rays" or particles are non-radio-active, and so do not enter into the theory. It is not difficult to deduce mathematically the number of atoms of P, Q, R, . . . of the matter A, B, C, . . . existing at any time t after this matter is set aside, if the initial values of P, Q, R, . . . are given. In practice, however, it is generally only necessary to employ three special cases of the theory which correspond, for example, to the changes in the active deposit, produced on a wire exposed to a constant amount of radium emanation and then removed, (1) when the time of exposure is extremely short compared with the period of the changes, (2) when the time of exposure is so long that the amount of each of the products has reached a steady limiting value, and (3) for any time of exposure. 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 = λ_{1}P - λ_{2}Q (1).
Similarly dR/dt = λ_{2}Q - λ_{3}R (2).
Substituting in (1) the value of P in terms of n,
dQ/dt = λ_{1}ne^{-λ_{1}t} - λ_{2}Q.
