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 $$ \frac {I_t} {I₀} = 1 − e^{–λ_2 t} $$, where _I₀_ is the maximum value of the activity and λ₂ 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 substance which emits rays; for the variation of the activity of the latter may be such as to determine not only the period but also the physical and chemical properties of the parent product. In the two following chapters the application of the theory of successive changes will be shown to account satisfactorily for the complicated processes occurring in the radio-elements. Footnote 294: Ramsay, _Proc. Roy. Soc._ p. 470, June, 1904; _C. R._ 138, June 6, 1904. Footnote 295: _Phil. Mag._ February, 1904. CHAPTER X. TRANSFORMATION PRODUCTS OF URANIUM, THORIUM, AND ACTINIUM. =205.= In the last chapter the mathematical theory of successive changes has been considered. The results there obtained will now be applied to explain the radio-active phenomena observed with uranium, thorium, actinium, radium, and their products. Transformation products of Uranium.