It has been shown that in uranium and thorium compounds there is a continuous production of active matter which keeps the compound in radio-active equilibrium. The changes by which the active matter is produced must be chemical in nature, since the products of the action are different in chemical properties from the matter in which the changes take place. The activity of the products has afforded the means of following the changes occurring in them. It now remains to consider the connection between the activity at any time, and the amount of chemical change taking place at that time. In the first place, it is found experimentally that the saturation ionization current _i_{t}_, after the active product has been allowed to decay for a time _t_, is given by $$ \frac {i_t} {i₀} = e^{–λt} $$, where _i₀_ is the initial saturation current and λ the constant of decay. Now the saturation current is a measure of the total number of ions produced per second in the testing vessel. It has already been shown that the α rays, which produce the greater proportion of ionization in the gas, consist of positively charged particles projected with great velocity. Suppose for simplicity that each atom of active matter, in the course of its change, gives rise to one projected α particle. Each α particle will produce a certain average number of ions in its path before it strikes the boundaries or is absorbed in the gas. Since the number of projected particles per second is equal to the number of atoms changing per second, the number of atoms _n_{t}_ which change per second at the time _t_ is given by $$ \frac {n_t} {n₀} = e^{–λt} $$, where _n₀_ is the initial number which change per second. On this view, then, the law of decay expresses the result that the number of atoms changing in unit time, diminishes according to an exponential law with the time. The number of atoms _N_{t}_ which remain _unchanged_ after an interval _t_ is given by $$ N_t = \int_t^{\infty} n_t dt $$ $$ = \frac {n₀} {λ} e^{–λt} $$ . If _N₀_ is the number of atoms at the beginning, $$ N₀ = \frac {n₀} {λ} $$, Thus $$ \frac {N_t} {N_₀} = e^{–λt} $$ (1). or the law of decay expresses the fact that the _activity of a product at any time is proportional to the number of atoms which remain unchanged at that time_. This is the same as the law of monomolecular change in chemistry, and expresses the fact that there is only one changing system. If the change depended on the mutual action of two systems, the law of decay would be different, since the rate of decay in that case would depend on the relative concentration of the two reacting substances. This is not so, for not a single case has yet been observed in which the law of decay was affected by the amount of active matter present. From the above equation (1) _dN_{t}_ ------- = –λ_N_{t}_, _dt_