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 (see equation 8, section 198). The value of Q, in this case, does not increase after removal, but at once commences to diminish. The activity, in consequence, decreases from the moment of removal, but more slowly than would be given by an exponential law. The activity finally decays exponentially, as in the previous case, falling to half value in 11 hours.

In the previous case we have discussed the activity curve obtained when both the active and inactive product have comparatively rapid rates of transformation. In certain cases which arise in the analysis of the changes in actinium and radium, the rayless product has a rate of change extremely slow compared with that of the active product. This corresponds to the case where the active matter B is supplied from A at a constant rate. The activity curve will thus be identical in form with the recovery curves of Th X and Ur X, that is, the activity I at any time t will be represented by the equation I_{t}/I_{0} = 1 - e^{-λ_{2}t}, where I_{0} is the maximum value of the activity and λ_{2} the constant of change of B.

204. In this chapter we have considered the variation with time, under different conditions, of the number of atoms of the successive products, when the period and number of the changes are given. It has been seen that the activity curves to be expected under various conditions can be readily deduced from the simple theory. In practice, however, the investigator has been faced with the much more difficult inverse problem of deducing the period, number, and character of the products, by analysis of the activity curves obtained under various conditions.

In the case of radium, where at least seven distinct changes occur, the problem has been one of considerable difficulty, and a solution has only been possible by devising special physical and chemical methods of isolation of some of the products.

We shall see later that two rayless changes occur in radium and actinium and one in thorium. It is at first sight a very striking fact that the presence of a substance which does not emit rays can be detected, and its properties investigated. This is only possible when the rayless product is transformed into another