since the number of constraint equations (conservation of momentum and energy for each collision) equals the number of variables. However, depending upon the mass ratios, not all of these processes may be allowed. That is, conservation of energy and momentum may forbid one or more of these scenarios — or, in the cases studied below, all three may be forbidden. In 1987 the senior author of this paper devised a simple diagram to show how the available processes depend on the mass ratios [7]. This diagram, which has come to be known as the “Lieber diagram,” is shown in Fig. 2. The symmetry of the diagram about the 45 ◦ line may be attributed to time-reversal symmetry. The curved boundaries are simple rectangular hyperbolas. Related diagrams have been given in [3, 8]. The quantum mechanical picture is more complicated [1–4]. Because of the uncertainty principle and the necessity of using wave packets, the momenta of the particles and their positions are spread out. At the high energies considered here, the Born approximation might be considered a good approximation. With Coulomb potentials it leads to a differential cross section with a large peak in the forward direction. The total cross section falls off as E − $^{6}$. However the second Born term dominates the first at sufficiently high energies because it falls off only as E − 5 . $^{5}$, as shown in the dissertation of Drisko [9]. This unique phenomenon occurs because of the Thomas double-collision process, which gives rise to a pole in the integrand of the second Born term when the propagator is on shell. The singularity gives rise to a second peak in the differential cross section at the Thomas angle which emerges from the background when the projectile energy is sufficiently large. The Thomas peak has been observed experimentally [10] in the pure three-body case of electron capture in proton-hydrogen scattering. An earlier atomic physics observation [11] utilized capture of one electron from helium in proton helium scattering - the scenario of Rutherford, Henderson and Thomas - and so is somewhat less clear-cut because of the presence of two electrons. Even earlier there was an observation of the Thomas peak in a molecular collision, as proposed by Bates [12]: a proton is captured from a methane molecule in the process p + $^{CH$_{4}$}$→ CH$_{3}$ + H + 2 . The Thomas peak occurs at 46 degrees. The question to be explored in this paper is the behavior of collisions for which all three Thomas double-collision processes are kinematically forbidden. Is capture possible by more than two collisions? We answer this question in the affirmative. Atomic processes which are Thomas-forbidden are rare because of the particular masses of the electron and proton. 3