This must obviously be the case, for otherwise there would be a destruction or creation of matter by the mere process of separation of the source from its products; but, by hypothesis, neither the rate of supply from the source, nor the law of change of the products, has been in any way altered by removal. Substituting the values of _P_, _Q_, _R_ from equations (7), (8), and (9), we obtain $$ \frac {P_1} {P₀} = 1 − e^{–λ_1 t} $$, $$ \frac {Q_1} {Q₀} = 1 − \frac {(λ_1 e^{–λ_2 t} - λ_2e^{–λ_1 t})} {λ_1 − λ_2} $$, $$ \frac {R_1} {R₀} = 1 − λ_3 (ae^{–λ_1 t} + be^{–λ_2 t} + ce^{–λ_3 t}) $$, where _a_, _b_, and _c_ have the values given after equation (9). The curves representing the increase of _P_, _Q_, _R_, are thus, in all cases, complementary to the curves shown in Fig. 73. The sum of the ordinates of the two curves of rise and decay at any time is equal to 100. We have already seen examples of this in the case of the decay and recovery curves of Ur X and Th X. =201. Activity of a mixture of products.= In the previous calculations we have seen how the number of particles of each of the successive products varies with the time under different conditions. It is now necessary to consider how this number is connected with the activity of the mixture of products. If _N_ is the number of particles of a product, the number of particles breaking up per second is λ_N_, where λ is the constant of change. If each particle of each product, in breaking up, emits one α particle, we see that the number of α particles expelled per second from the mixture of products at any time is equal to λ₁_P_ + λ₂_Q_ + λ₃_R_ + ..., where _P_, _Q_, _R_, ... are the numbers of particles of the successive products _A_, _B_, _C_, .... Substituting the values of _P_, _Q_, _R_ already found from any one of the four cases previously considered, the variation of the number of α particles expelled per second with the time can be determined. The ideal method of measuring the activity of any mixture of radio-active products would be to determine the number of α or β particles expelled from it per second. In practice, however, this is inconvenient and also very difficult experimentally. Certain practical difficulties arise in endeavouring to compare the activity of one product with another. We shall see later that, in many cases, all of the successive products do not emit α rays. Some give out β and γ rays alone, while there are several “rayless” products, that is, products which do not emit either α, β, or γ rays. In the case of radium, for example, radium _A_ gives out only α rays, radium _B_ no rays at all, while radium _C_ gives out α, β, and γ rays. In practice, the relative activity of any individual product at any time is usually determined by relative measurements of the saturation ionization current produced between the electrodes of a suitable testing vessel.