【正文】 1. Introduction The simple ionization collision of a hydrogenic atom by the impact of a structureless particle, the “threebody problem”, is one of the oldest unsolved problems in physics. The twobody problem was analyzed by Johannes Kepler in 1609 and solved by Isaac Newton in 1687. The threebody problem, on the other hand, is much more plicated and cannot be solved analytically, except in some particular cases. In 1765, for instance, Leonhard Euler discovered a “collinear” solution in which three masses start in a line and remain linedup. Some years later, Lagrange discovered the existence of five equilibrium points, known as the Lagrange points. Even the most recent quests for solutions of the threebody scattering problem use similar mathematical tools and follow similar paths than those travelled by astronomers and mathematicians in the past three centuries. For instance, in the centerofmass reference system, we describe the threebody problem by any of the three possible sets of the spatial coordinates already introduced by Jacobi in 1836. All these pairs are related by lineal point canonical transformations, as described in [1]. In momentum space, the system is described by the associated pairs (kT,KT), (kP,KP) and (kN,KN). Switching to the Laboratory reference frame, the final momenta of the electron of mass m, the (recoil) target fragment of mass MT and the projectile of mass MP can be written in terms of the Jacobi impulses Kj by means of Galilean transformations [1] For decades, the theoretical description of ionization processes has assumed simplifications of the threebody kinematics in the final state, based on the fact that ? in an ion–atom collision, one particle (the electron) is much lighter than the other two, ? in an electron–atom or positron–atom collision, one particle (the target nucleus) is much heavier than the other two. For instance, based on what is known as Wick’s argument, the overwhelming majority of the theoretical descriptions of ion–atom ionization collisions uses an impactparameter approximation, where the projectile follows an undisturbed straight line trajectory throughout the collision process, and the target nucleus remains at rest [2]. It is clear that to assume that the projectile follows a straight line trajectory makes no sense in the theoretical description of electron or positron–atom collisions. However, it is usually assumed that the target nucleus remains motionless. These simplifications of the problem were introduced in the eighteenth century. The unsolvable threebody problem was simplified, to the socalled restricted threebody problem, where one particle is assumed to have a mass small enough not to influence the motion of the other two particles. Though introduced as a means to provide approximate solutions to systems such as Sun–pla–et within a Classical Mechanics framework, it has been widely used in atomic physics in the socalled impactparameter approximation to ion–atom ionization collisions. Another simplification of the threebody problem widely employed in the nieenth century assumes that one of the particles is much more massive than the other two and remains in the center of mass unperturbed by the other two. This approximation has been widely used in electron–atom or positron–atom ionization collisions. 2. The multiple differential cross section A kinematically plete description of a threebody continuum finalstate in any atomic collision would require, in principle, the knowledge of nine variables, such as the ponents of the momenta associated to each of the three particles in the final state. However, the condition of momentum and energy conservation reduces this number to five. Furthermore, whenever the initial targets are not prepared in any preferential direction, the multiple differential cross section has to be symmetric by a rotation of the threebody system around the initial direction of motion of the projectile. Thus, leaving aside the internal structure of the three fragments in the final state, only four out of nine variables are necessary to pletely describe the scattering process. Therefore, a plete characterization of the ionization process may be obtained with a quadruple differential cross section: There are many possible sets of four variables to use. For, instance, we can chose azimuthal angles of the electron and of one of the other two particles, the relative angle between the planes of motion, and the energy of one particle. Such a choice is arbitrary, but plete in the sense that any other set of variables can be related to this one. A similar choice of independent variables has been standard for the description of atomic ionization by electron impact, both theoretically and experimentally [3] and [4]. A picture of the very general quadruple differential cross section is not feasible. Thus, it is usually necessary to reduce the number of variables in the cross section. This can be achieved by fixing one or two of them at certain particular values or conditions. For instance, we might arbitrarily restrict ourselves to describe a coplanar (. = 0) or a collinear motion (. = 0 and θ1 = θ2), so as to reduce the dependence of the problem to three or two independent variables, respectively. The other option is to integrate the quadruple differential cross section over one or more variables. The former has been widely used to study electron–atom collisions, while the latter has been the main tool to characterize ion–atom and positron–atom ionization collisions. Particularly important has been the use of single particle spectroscopy, where the momentum of one of the particles is measured. 3. Single particle momentum distributions In ionization by positron impact it is feasible to study the momentum distri