It is important, for many applications, to have reliable data on the magnitudes of the cross-sections for excitation and ionization of atoms and ions by electrons. In part I the usual approximations (those of Born and of Oppenheimer) which are made to obtain theoretical values are critically examined. It is pointed out that the assumption of separable bound wave functions may often lead to considerable errors. In the case of the Oppenheimer approximation the errors may even be such as to give results violating the principle of detailed balancing. Circumstances in which these errors are likely to be serious are analyzed, and precautions which may be taken to reduce them are proposed. The conditions under which the approximations are likely to fail, even when exact bound wave func- tions are used, appear to be related to the magnitude of certain coupling terms which are ignored in obtaining the approximations. The usefulness of certain conservation theorems which limit the possible size of collision cross-sections is also pointed out. A summary of those general properties of inelastic cross-sections which are reliably given by the theory is included. In part II the available experimental data are compared with the predictions of the Born and Oppenheimer approximations. The collision processes studied include the following: excitation of H, He, Na, Ne and Hg; ionization of H
2
, He, Ne, Hg, Ni (
K
) and Ag (K and LUI). The investigation shows that the Born approximation is the one that should generally be used in the treatment of transitions which can take place without electron exchange having to be invoked. For these the approximation achieves a considerable degree of success. As far as can be judged from the comparison data available, the main defects are that the maxima of the predicted cross-section energy curves tend to be too pronounced, and to be located too close to the critical potentials. In the case of transitions involving a reversal of electron spin the Oppenheimer approximation must be used. Unfortunately, it proves to be very unsatisfactory. Thus for non-hydrogenic systems it may give very different results according to whether a prior or a post interaction is adopted. It leads to frequent violations of the conservation theorem and cannot be relied upon even to give the detailed shape of cross-section against energy curves. By generalizing from the evidence collected, an attempt is made to specify the conditions under which the Born and Oppenheimer approximations are most reliable; on this basis, proposals for systemization are made. Attention is drawn to the fact that some (but by no means all) of the observed excitation functions possess an extremely sharp peak just beyond the critical potential. The theory seems unable to reproduce this peculiar feature. It does not appear in the observed ionization functions.
SUPERCRITICAL POMERON AND EIKONAL REPRESENTATION OF THE DIFFRACTION SCATTERING AMPLITUDE