These effects receive a general explanation on the views already put forward. When the radium is placed in the closed vessel, the emanation is given off at a constant rate and gradually diffuses throughout the enclosure. Since the time taken for diffusion of the emanation through tubes of ordinary size is small compared with the time required for the activity to be appreciably reduced, the emanation, and also the excited activity due to it, will be nearly equally distributed throughout the vessel.
The luminosity due to it should thus be equal at each end of the tube. Even with a capillary tube connecting the two bulbs, the gas continuously given off by the radium will always carry the emanation with it and cause a practically uniform distribution.
The gradual increase of the amount of emanation throughout the tube will be given by the equation
N_{t}/N_{0} = 1 - e^{-λt},
where N_{t} is the number of emanation particles present at the time t, N_{0} the number present when radio-active equilibrium is reached, and λ is the radio-active constant of the emanation. The phosphorescent action, which is due partly to the radiations from the emanation and partly to the excited activity on the walls, should thus reach half the maximum value in four days and should practically reach its limit after three weeks' interval.
The variation of luminosity with different distances between the screens is to be expected. The amount of excited activity deposited on the boundaries is proportional to the amount of emanation present. Since the emanation is equally distributed, the amount of excited activity deposited on the screens, due to the emanation between them, varies directly as the distance, provided the distance between the screens is small compared with their dimensions. Such a result would also follow if the phosphorescence were due to the radiation from the emanation itself, provided that the pressure of the gas was low enough to prevent absorption of the radiation from the emanation in the gas itself between the screens.
