P. Curie and Danne made the important observation that the curve of decay _C_, corresponding to that shown in Fig. 88, for a long exposure, could be accurately expressed by an empirical equation of the form $$ \frac {I_t} {I₀} = ae^{–λ_3 t} − (a − 1) e^{–λ_2 t} $$, where λ₂ = 5·38 × 10⁻⁴ (sec)⁻¹ and λ₃ = 4·13 × 10⁻⁴ (sec)⁻¹, and α = 4·20 is a numerical constant. I have found that within the limit of experimental error this equation represents the decay of excited activity of radium for a long exposure, measured by the β rays. The equation expressing the decay of activity, measured by the α rays, differs considerably from this, especially in the early part of the curve. Several hours after removal the activity decays according to an exponential law with the time, decreasing to half value in 28 minutes. This fixes the value of λ₃. The constant α and the value of λ₂ are deduced from the experimental curve by trial. Now we have already shown (section 207) that in the case of the active deposit from thorium, where there are two changes of constants λ₂ and λ₃, in which only the second change gives rise to a radiation, the intensity of the radiation is given by $$ \frac {I_t} {I₀} = \frac {λ_2} {λ_2 − λ_3} e^{–λ_3 t} − \frac {λ_3} {λ_2 − λ_3} e^{–λ_2 t} $$ for a long time of exposure (see equation 8, section 198). This is an equation of the same form as that found experimentally by Curie and Danne. On substituting the values λ₂, λ₃ found by them, $$ \frac {λ_2} {λ_2 − λ_3} = 4\cdot3 $$, and $$ \frac {λ_1} {λ_1 − λ_3} = 3\cdot3 $$ . Thus the theoretical equation agrees in form with that deduced from observation, and the values of the numerical constants are also closely concordant. If the first as well as the second change gave rise to a radiation, the equation would be of the same general form, but the value of the numerical constants would be different, the values depending upon the ratio of the ionization in the first and second changes. If, for example, it is supposed that both changes give out β rays in equal amounts, it can readily be calculated that the equation of decay would be $$ \frac {I_t} {I₀} = \frac {\cdot5λ_2} {λ_2 − λ_3} e^{–λ_3 t} − \cdot5 (\frac {λ_3} {λ_2 − λ_3} - 1) e^{–λ_2 t} $$ . Taking the values of λ₂ and λ₃ found by Curie, the numerical factor $$ e^{–λ_2 t} $$ becomes 2·15 instead of 4·3 and 1·15 instead of 3·3. The theoretical curve of decay in this case would be readily distinguishable from the observed curve of decay. The fact that the equation of decay found by Curie and Danne involves the necessity of an initial rayless change can be shown as follows:—