About 5 hours after removal the second term in the brackets becomes very small, and the activity after that time will decay nearly according to an exponential law with the time, falling to half value in 11 hours. For any time of exposure _T_, the activity at time _t_ after the removal (see equation 11, section 199) is given by $$ \frac {I_t} {I₀} = \frac {Q} {Q_T} = \frac {ae^{–λ_2 t} - be^{–λ_1 t}} {a − b} $$, where _I₀_ is the initial value of the activity, immediately after removal, and $$ a = \frac {1 − e^{–λ_2 T}} {λ_2} $$, $$ b = \frac {1 − e^{–λ_1 T}} {λ_1} $$, By variation of _T_ the curves of variation of activity for any time of exposure can be accurately deduced from the equation, when the values of the two constants λ₁, λ₂ are substituted. Miss Brooks[306] has examined the decay curves of excited activity for thorium for different times of exposure and has observed a substantial agreement between experiment and theory. [Illustration: Fig. 78.] The results are shown graphically in Fig. 78. The maximum value of the activity is, for each time of exposure, taken as 100. The theoretical and observed values are shown in the Figure. =208. Analysis of the decay and recovery curves of Th X.= The peculiarities of the initial portions of the decay and recovery curves of Th X and thorium respectively (Curves _A_ and _B_, Fig. 47, p. 221), will now be considered. It was shown that when the Th X was removed from the thorium by precipitation with ammonia, the radiation increased about 15 per cent. during the first day, passed through a maximum, and then fell off according to an exponential law, decreasing to half value in four days. At the same time the activity of the separated hydroxide decreased for the first day, passed through a minimum, and then slowly increased again, rising to its original value after the lapse of about one month.