different velocities, which have ranges in air lying between 0 and the maximum range. The ionization of the particles per unit path is greatest near the end of its range, and decreases somewhat as we approach the radiant source. A complex source of rays like radium gives out four types of rays, each of which has a different but distinct range.
From this theory it is possible to calculate approximately the decrease of current to be observed when sheets of metal foil are placed over a large area of radio-active substance. This is the method that has been employed to obtain the curves of Figs. 35 and 38.
Suppose a very thin layer of simple radio-active matter is employed (for example a bismuth plate covered with radio-tellurium or a metal plate made active by exposure to the presence of the thorium or radium emanations) and that the ionization vessel is of sufficient depth to absorb the [Greek: alpha] rays completely.
Let d be the thickness of the metal plate, ρ its density compared with air. Consider a point P close to the upper side of the plate. The range of the particles moving from a point, when the path makes an angle θ with the normal at P, is a - ρd sec θ, where a is the range in air. The rays coming from points such that the paths make an angle with the normal greater than cos^{-1} (ρd/a) will thus be absorbed in the plate. By integrating over the circular area under the point P, it is easy to show that the total ionization in the vessel is proportional to
[integral]_{0}^{cos^{-1} (ρd/a)} 2π sin θ cos θ(a - ρd sec θ) dθ = π(a - ρd)^2/a.
The curves showing the relation between current and distance of metal traversed should thus be parabolic with respect to d. This is approximately the case for a simple substance like radio-tellurium. The curve for a thick layer of radium would be difficult to calculate on account of the complexity of the rays, but we know from experiment that it is approximately exponential. An account of some recent investigations made to determine the range of velocity over which the [Greek: alpha] particle is able to ionize the gas is given in Appendix A. The results there given strongly support the theory of absorption of the [Greek: alpha] rays discussed above.
