Radium _D_ Thorium Actinium (Active) _C_ _C_ (final) (final) &c. Each product on this scheme is the parent of the product below it. Since only two products have been observed in the active deposit of thorium and actinium, thorium _C_ and actinium _C_ respectively refer to their final inactive products. It will be shown in the next chapter that, as in the case of thorium, an intermediate product exists between actinium and its emanation. From analogy to the products Th X and Ur X, this substance is termed “actinium X.” =196. Theory of Successive Changes.= Before considering the evidence from which these changes are deduced, the general theory of successive changes of radio-active matter will be considered. It is supposed that the matter _A_ changes into _B_, _B_ into _C_, _C_ into _D_, and so on. Each of these changes is supposed to take place according to the same law as a monomolecular change in chemistry, _i.e._, the number _N_ of particles unchanged after a time _t_ is given by $$ M = N₀ e^{–λt} $$, where _N₀_ is the initial number and λ the constant of the change. Since _dN_/_dt_ = -λ_N_, the rate of change at any time is always proportional to the amount of matter unchanged. It has previously been pointed out that this law of decay of the activity of the radio-active products is an expression of the fact that the change is of the same type as a monomolecular chemical change. Suppose that _P_, _Q_, _R_ represent the number of particles of the matter _A_, _B_, and _C_ respectively at any time _t_. Let λ₁, λ₂, λ₃ be the constants of change of the matter _A_, _B_, and _C_ respectively. Each atom of the matter _A_ is supposed to give rise to one atom of the matter _B_, one atom of _B_ to one of _C_, and so on. The expelled “rays” or particles are non-radio-active, and so do not enter into the theory. It is not difficult to deduce mathematically the number of atoms of _P_, _Q_, _R_, ... of the matter _A_, _B_, _C_, ... existing at any time _t_ after this matter is set aside, if the initial values of _P_, _Q_, _R_, ...are given. In practice, however, it is generally only necessary to employ three special cases of the theory which correspond, for example, to the changes in the active deposit, produced on a wire exposed to a constant amount of radium emanation and then removed, (1) when the time of exposure is extremely short compared with the period of the changes, (2) when the time of exposure is so long that the amount of each of the products has reached a steady limiting value, and (3) for any time of exposure.