If the recovery curve is produced backwards to meet the vertical axis, it does so at a minimum of 25 per cent., and the above conclusions hold more accurately, if the recovery is assumed to start from this minimum. This is clearly shown by Fig. 48, where the percentages of activity recovered, reckoned from the 25 per cent. minimum, are plotted as ordinates. In the same figure the decay curve, after the second day, is shown on the same scale. The activity of the Th X decays with the time according to an exponential law, falling to half value in about four days. If _I₀_ is the initial activity and _I{t}_ is the activity after a time _t_, then $$ \frac {I₀} {I_t} = e^{–λt} $$, where λ is a constant and _e_ the natural base of logarithms. The experimental curve of the rise of activity from a minimum to a maximum value is therefore expressed by the equation $$ \frac {I_t} {I₀} = 1 − e^{–λt} $$, where _I₀_ is the amount of activity recovered when the state of constant activity is reached, _I_{t}_ the activity recovered after a time _t_, and λ is the _same constant_ as before. =129. Uranium X.= Similar results were obtained when uranium was examined. The Ur X was separated by Becquerel’s method of successive precipitations with barium. The decay of the separated activity and the recovery of the lost activity are shown graphically in Fig. 49. A more detailed discussion of this experiment is given in section 205. [Illustration: Fig. 49.] The curves of decay and recovery exhibit the same peculiarities and can be expressed by the same equations as in the case of thorium. The time-rate of decay and recovery is, however, much slower than for thorium, the activity of the Ur X falling to half its value in about 22 days. A large number of results of a similar character have been obtained from other radio-active products, separated from the radio-elements, but the cases of thorium and uranium will suffice for the present to form a basis for the discussion of the processes that are taking place in radio-active bodies. =130. Theory of the phenomena.= These processes of decay and recovery go on at exactly the same rate if the substances are removed from the neighbourhood of one another, or enclosed in lead, or placed in a vacuum tube. It is at first sight a remarkable phenomenon that the processes of decay and recovery should be so intimately connected, although there is no possibility of mutual interaction between them. These results, however, receive a complete explanation on the following hypotheses: (1) That there is a constant rate of production of fresh radio-active matter by the radio-active body; (2) That the activity of the matter so formed decreases according to an exponential law with the time from the moment of its formation.