during the time that radium D is being formed. Now D does not itself give out rays, but the succeeding product E does. The products D and E are practically in radio-active equilibrium one month after D is set aside, and the variation of the β ray activity of E then serves as a measure of the variation of the parent product D. Suppose that a vessel is filled with a large quantity of radium emanation. After several hours, the product radium C, which emits β rays, reaches a maximum value, and then decreases at the same rate as the emanation loses its activity, i.e. it falls to half value in 3·8 days. If N_{1} is the number of β particles expelled from radium C at its maximum value, the total number Q_{1} of β particles expelled during the life of the emanation is given approximately by
Q_{1} = [integral]_{0}^[infinity] N_{1}e^{-λ_{1}t}dt = N_{1}/λ_{1},
where λ_{1} is the constant of change of the emanation.
After the emanation has disappeared, and the final products D + E are in radio-active equilibrium, suppose that the number of β particles N_{2} expelled per second by radium E is determined. If Q_{2} is the total number of particles expelled during the life of D + E, then Q_{2} as before is approximately given by Q_{2} = N_{2}/λ_{2} where λ_{2} is the constant of change of radium D. Now we have seen that if each particle of C and of E gives rise to one β particle, it is to be expected that
Q_{1} = Q_{2},
or λ_{2}/λ_{1} = N_{2}/N_{1}.
The ratio N_{2}/N_{1} was determined by measuring the activity due to the β rays from C and E in the same testing-vessel. Then, since N_{2}/N_{1} is known, and also the value of λ_{1}, the value of the constant of change, λ_{2}, of radium D is obtained. In this way it was calculated that D is half transformed in about 40 years.
In the above calculations it is assumed, as a first approximation, that the β rays from C and E have the same average velocity. This is probably not accurately the case, but the above number certainly serves to fix the order of magnitude of the period of the
