The possibility of increasing the current density of a beam of fast electrons passes through glass cone-shaped channels was demonstrated in . But the fraction of the electrons that passed through the conical channels without loss of the initial energy was not cleared up. Measurements of X-ray spectra generated by transmitted electrons in copper target mounted in vicinity of capillary output were performed for a detailed study of the contactless passage of 10 keV electrons through conical capillaries. All the measurements were made at different tilting angles with respect to the incident beam axis. It is shown that a significant part of electrons retains its initial energy even at the angles exceeding geometric opening of the capillary.