and the activity _I_ at any time of the two together is proportional to λ₁_P_ + _K_λ₂_Q_, where _K_ is the ratio of the ionization of B compared with that of A. Then $$ \frac {I_t} {I₀} = \frac {λ_1 P + Kλ_2 Q} {λ_1 n₀} $$ $$ = e^{–λ_1 t} (1 + \frac {Kλ_2} {λ_2 − λ_1} (1 − e^{-(λ_2 − λ_1) t})) $$, where _I₀_ is the initial activity due to _n₀_ particles of Th X. By comparison of this equation with the curve of variation of the activity of Th X with time, shown in Fig. 47, it is found that _K_ is almost ·44. It must be remembered that the activity of the emanation and Th X are included together, so that the activity of thorium B is about half of the activity of the two preceding products. The calculated values of _I_{t}_/_I₀_ for different values of _t_ are shown in the second column of the following table, and the observed values in the third column. Time Theoretical Observed value value 0 1·00 1·00 ·25 days 1·09 — ·5 „ 1·16 — 1 „ 1·15 1·17 1·5 „ 1·11 — 2 „ 1·04 — 3 „ ·875 ·88 4 „ ·75 ·72 6 „ ·53 ·53 9 „ ·315 ·295 13 „ ·157 ·152 [Illustration: Fig. 80.] The theoretical and observed values thus agree within the limit of error in the measurements. The theoretical curve is shown in Curve _A_, Fig. 80 (with the observed points marked, for comparison). The curve _B_ shows the theoretical curve of the decay of the activity of Th X and the emanation, supposing there is no further change into the active deposit. Curve _C_ shows the difference curve between the curves _A_ and _B_, _i.e._ the proportion of the activity at different times due to the active deposit. The activity due to the latter thus rises to a maximum about two days after removal of the Th X, and then decays with the time at the same rate as the Th X itself, _i.e._ the activity falls to half value every four days. When _t_ exceeds four days, the term $$ e^{-(λ_2 − λ_1) t} $$ in the theoretical equation is very small. The equation of decay after this time is therefore expressed by $$ \frac {I_t} {I₀} = (1 + \frac {Kλ_2} {λ_2 − λ_1}) e^{–λ_1 t} $$, _i.e._ the activity decays according to an exponential law with the time.